Reaction-Diffusion Models in
Mathematical Biology
Kristian Kristiansen
Kongens Lyngby 2008
Technical University of Denmark
The Department of Mathematics
Building 303 S, DK-2800 Kongens Lyngby, Denmark
Phone +45 4525 3031, Fax +45 4588 1399
[email protected]
www.mat.dtu.dk
Summary
In this master thesis the well-posedness of two-component reaction-diffusion
equations with Neumann conditions is analysed together with a study of diffusion driven instabilities. Specifically, three models from mathematical biology
is considered: The Schnakenberg model, the Gierer and Meinhardt model and
the Thomas model.
The techniques of upper and lower solutions described by Pao, [Pao (1992)],
and abstract results on semigroups together with basic functional analysis are
applied to these models in order to show well-posedness. All systems are shown
to be well-posed with non-negative and bounded solutions.
Linear stability is shown to imply stability of steady-states. This is exploited
to analyse diffusion-driven instabilities. Sufficient conditions for the general,
two-component, autonomous reaction-diffusion equation to exhibit diffusiondriven instabilities at a uniform steady-state are presented.
Diffusion-driven instabilities at a uniform steady-state are analysed numerically for the Schnakenberg model. The instabilities are shown to evolve into
steady-state, heterogeneous patterns. Additionally, it is observed, and proved,
that there curves exist in parameter-space for which the corresponding systems
are equivalent in the sense that the resulting steady-states only differ by a scaling.
ii
Resumé
I dette eksamensprojekt studeres velstillethed og diffusionsdrevet instabilitet
for to-komponente reaktion-diffusions ligninger med Neumann-betingelser. Mere
specifikt betragtes tre modeller fra matematisk biology: Schnakenberg modellen,
Gierer og Meinhardt modellen og Thomas modellen.
Øvre- og nedre-løsningernes metode, beskrevet af Pao, [Pao (1992)], og abstrakte resultater fra semigruppe-teori sammen med simple funktional analyse
benyttes til at undersøge velstilletheden af disse modeller. Alle modeller vises
at være velstillede med positive og begrænsede løsninger.
Lineær stabilitet vises at medføre stabilitet af stationære løsninger. Dette udnyttes til at analysere diffusionsdrevet instabiliteter. Tilstrækkelige betingelser
præsenteres for at den generelle to-komponent, autonome reaktion-diffusions
ligning kan fremvise diffusionsdrevet instabilitet omkring en uniform stationær
løsning.
Diffusionsdrevet instabiliteter for en uniform stationær løsning analyseres
numerisk for Schnakenberg modellen. Instabiliteterne vises at udvikle sig til
stationære, heterogene løsninger. Ydermere observeres og bevises det, at der
eksisterer kurver i parameterdomænet således, at de forskellige systemer er ekvivalente i den forstand at resulterende stationære løsninger kun afviger med en
skalering.
iv
Preface
This thesis was prepared at Department of Mathematics, the Technical University of Denmark in partial fulfillment of the requirements for acquiring the
Master degree in engineering.
The thesis deals with reaction-diffusion models in mathematical biology.
These models are coupled semilinear partial differential equations and the wellposedness of such is a complicated matter. Among other things, it is the multiple steady-states that makes these models interesting mathematically, as well
as practically relevant.
The aim of the thesis, is first of all to apply two techniques to show wellposedness of the Schnakenberg model, the Thomas model and the Gierer and
Meinhardt model. Next, we wish to justify the use of linear analysis in this
infinite-dimensional case, and apply this to the Schnakenberg model and show
the existence of diffusion driven instabilities at a uniform steady-state. Finally,
we shall apply numerical studies and analyse the existence of heterogeneous
patterns, and study the bifurcation of such heterogeneous patterns.
Lyngby, January 2008
Kristian Kristiansen
vi
Acknowledgements
I would like to thank my supervisor Michael Pedersen for support and help
during the project. Thank you for taking your time with me, and thank you for
your clever guidance.
Furthermore, I would like to thank my master student colleagues at the
institute, Jacob Hersböll and Karen Markvard Martensen, for keeping my head
high and for coping with my suddenly outbursts of celebration or confusion.
Thanks to Mads Peter Sørensen, Per Grove Thomsen and Casper Skovby for
MATLAB assistance. I also owe Casper Skovby and Peder Skafte a great favour
for their LATEX-support.
Thanks to my family, especially my mother and Anna, for always respecting
my work and supporting me. Anna, also thanks for your fantastic proofreading.
Thank you all
viii
Notation
By N we shall denote the set of positive integers and by N0 the set of nonnegative integers. R denotes the real numbers, R+ and R− the positive resp.
negative real numbers. By C we denote the complex numbers.
Rn is the n-dimensional real Euclidean space. The open balls in Rn with
centre at x and radius are denoted Br (x).
d
Differentiation of functions on R is indicated by ∂x = dx
and similarly dif∂
n
ferentiation on R is indicated by ∂xj = ∂xj .
We shall also make use of the multi-index notation: When α ∈ Nn0 , α =
(α1 , · · · , αn ) then
∂ α = ∂xα11 · · · ∂xαnn ,
and by |α| we shall denote the sum α1 + · · · + αn .
We shall say that a set Ω ⊂ Rn is smooth if for each point x on the boundary
∂Ω there is a ball Br (x), r > 0, and integer i ∈ {1, . . . , n} such that
∂Ω ∩ Br (x) = {x ∈ Br (x) | xi = φ(x1 , . . . , xi−1 , xi−1 , . . . , xn )} ,
(1)
and φ is smooth, that is infinitely many times differentiable.
The set of m-times continuously differentiable functions on Ω is denoted by
C m (Ω) while those with compact support is denoted by C0m (Ω). We shall denote
the vector space of smooth functions by C ∞ (Ω) = ∩m∈N C m (Ω) and those with
compact support C0∞ (Ω) = ∩m∈N C0m (Ω), also denoted D(Ω) and referred to as
the set of test functions.
P
If we equip C m (Ω) with the supremum norm kukm,∞ = supΩ |α|≤m |∂ α u|
we obtain a Banach space which we shall denote CLm∞ (Ω).
We shall also make use of functions on product spaces Ω×(0, T ) for example.
Let p, q ≥ 0 then by C p,q (Ω × (0, T )) we denote the functions on Ω × (0, T ) that
x
are p-times continuously differentiable for all x ∈ Ω and q-times continuously
differentiable for all t ∈ (0, T ).
For 1 ≤ p < ∞ we define on a measurable set S the space Lp (S) as the
completion of C0∞ (S) in the norm
Z
1/p
p
kukp =
.
|u|
S
identifying functions that agree almost everywhere. The Lp -spaces are Banach
spaces; in particular L2 (S) is a Hilbert space with the inner product
Z
(u, v) =
uv.
S
For 1 ≤ p < ∞ and n ∈ N the Sobolev spaces W n,p (Ω) are defined as
W n,p = {u ∈ Lp (Ω)| ∂ α u ∈ Lp (Ω) for 0 ≤ |α| ≤ n where ∂ α u is the distributional
partial derivative} ,
equipped with the Banach norm
kukn,p =

 X

0≤|α|≤n
k∂ α ukpp
1/p

.
(2)

For p = 2 we write H n = W n,2 , and then the norm (2) stems from the inner
product
X
(∂ α u, ∂ α v).
(u, v)n =
0≤|α|≤n
A definition of W s,p (Ω) for s ∈ R is more complicated, and the reader is referred
to [Adams (1975)].
We define the Neumann operator ν by ψ 7→ (n · ∇)ψ, where n is the outward
unit normal of Ω given by
n=
[−∂ α1 φ(x), · · · , −∂ αi−1 φ(x), 1, −∂ αi+1 φ(x), · · · , −∂ αn−1 φ(x)]
p
1 + ∂ α1 φ(x)2 + · · · + ∂ αn−1 φ(x)2
and φ is the function from (1) and αj = (0, · · · , 1, · · · , 0) with 1 on the j’th
coordinate and zero elsewhere.
Working with systems of partial differential equation we shall often make
use of the product topology. Let B be a Banach space, then the product space
B × B is a Banach space with the product norm
k(u, v)kB 2 = kukB + kvkB ,
for all (u, v) ∈ B 2 .
Finally, tr and det denote trace resp. determinant.
xi
xii
Contents
Contents
Summary
i
Resumé
iii
Preface
v
Acknowledgements
Notation
vii
ix
1 Introduction
1
2 The approach of Pao
5
2.1
Upper and lower solutions . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Applications of the approach of upper and lower solutions . . . .
8
2.3
Summary and remarks . . . . . . . . . . . . . . . . . . . . . . . .
13
xiv
CONTENTS
3 Existence and uniqueness of two-component reaction-diffusion
system
15
3.1
Local existence and uniqueness . . . . . . . . . . . . . . . . . . .
16
3.2
Application of result on local existence to specific models . . . .
27
3.3
Global existence and boundedness . . . . . . . . . . . . . . . . .
32
3.4
Summary and remarks . . . . . . . . . . . . . . . . . . . . . . . .
46
4 Linear analysis
49
4.1
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2
Conditions for diffusion driven instability . . . . . . . . . . . . .
57
4.3
Summary and remarks . . . . . . . . . . . . . . . . . . . . . . . .
61
5 Numerical analysis
63
5.1
Results in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5.2
Summary and remarks . . . . . . . . . . . . . . . . . . . . . . . . 100
6 Conclusion
103
A Semigroups and sectorial operators
107
B Differentiation and integration in Banach Spaces
115
C Gronwall’s inequality
117
D Matlab-codes
119
D.1 main.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
D.2 cheb.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
CONTENTS
xv
D.3 operator.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
D.4 initcond.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
D.5 solver.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D.6 recmovie.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
D.7 func2d.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
xvi
CONTENTS
Chapter
1
Introduction
Reaction-diffusion equations have enjoyed a considerable amount of scientific interest. The reason for the large amount of work put into studying these
equations is not only their practical relevance, but also interesting phenomena
that can arise from such equations, such as multiple steady states and spatial
patterns and oscillating solutions, just to mention a few. The study of these
phenomena require a variety of different methods from many areas of mathematics for example bifurcation and stability theory, semigroup theory, singular
perturbations, numerical analysis and many others.
From a qualitatively point of view, a reaction-diffusion system is a mathematical model describing how the concentration of one or more substances vary
over time and space under the influence of two terms: Reaction term or source
term, in which concentration is generated or degenerated by local interaction,
diffusion term which causes the substances to spread out in space. A reactiondiffusion system is therefore an equation heuristically like
Change in concentration = Diffusion of concentration + Source term,
or algebraically
∂t u = D∆u + f(u),
in
Ω,
where Ω ⊂ Rn , together with some appropriate boundary conditions and initial
conditions. Ω may be bounded or unbounded.
2
Introduction
More specifically we shall in this work consider two-component reactiondiffusion systems on a bounded domain Ω with homogeneous Neumann conditions
∂t u − ∆u = f (t, u, v),
in Ω × (0, T ),
∂t v − d∆v = g(t, u, v),
u
ν
= 0, on ∂Ω × [0, T )
v
u = u0 ,
v = v0 , on Ω × {t = 0} .
(1.1)
(1.2)
(1.3)
Other linear boundary conditions, such as Direchlet conditions, are also of interest in some cases, but in this thesis we shall settle with the Neumann condition.
The system is called autonomous, if the reaction functions f and g do not depend upon time.
Examples of autonomous reaction functions are given by Murray, [Murray
(2002), Chapter 2],
f (r, s) = a − r + r2 s,
g(r, s) = b − r2 s,
r2
,
s(1 + kr2 )
ρrs
f (r, s) = a − r −
,
1 + r + kr2
f (r, s) = a − br +
2
g(r, s) = r2 − s,
g(r, s) = a(b − s) −
ρrs
,
1 + r + kr2
2
for (r, s) ∈ R+ , R2+ resp. R+ .
The system with the first type of reaction functions
∂t u − ∆u = a − u + u2 v
in Ω × (0, ∞),
∂t v − d∆v = b − u2 v
u
ν
= 0, on ∂Ω × [0, ∞),
v
u = u0 ,
v = v0 , on Ω × {t = 0} ,
(1.4)
(1.5)
(1.6)
is known as the Schnakenberg model and it shall be of our main interest in this
work.
The systems with the second and third type of reaction-diffusion equations
are known as the Gierer and Meinhardt model resp. Thomas model.
Although these equations are non-linear and therefore troublesome to work
with they all enjoy the feature of involving a nice parabolic differential operator.
Furthermore the nonlinearities are on the dependent variable itself and not on
its partial derivatives. A vast selection of tools are available to account for such
nonlinearities. More specifically, the equations are said to be semilinear.1
1 Formally a partial differential equation of k’th order is semilinear if the equation is linear
in the k’th order term.
3
In this thesis we shall be concerned with the techniques described by Pao,
[Pao (1992)], on upper and lower solutions and more abstract results from semigroup theory and functional analysis utilised in [Henry (1981)] and [Hollis, S. L.;
Martin, R. H.; Pierre, M (1987)]. We wish to exploit these tools to analyse the
general two-component reaction-diffusion system as well as the specific models
above, in terms of well-posedness.
A problem is well-posed in the sense of Hadamard if the following conditions
are satisfied:
(W1) Existence and uniqueness;
(W2) Existence for all times;
(W3) Continuously dependency on initial conditions.
These conditions are clearly not sufficient for a physical or biological model to
be “practically well-posed.” When working with physical parameters such as
concentrations, we cannot allow the solution to become negative. Furthermore
the solution must be bounded. We shall therefore also be interested in the
sufficient conditions for the system to satisfy:
(W4) The solution is non-negative for non-negative initial data;
(W5) The solution is bounded for all bounded initial data.
In the following chapter we shall apply the techniques of upper and lower solutions to the specific systems. We have chosen this approach, since it applies
an ad hoc approach to the subject of well-posedness of coupled, parabolic equations, and we wish to analyse to what extend this method is sufficient. The
technique is, however, not adequate to analyse (W3).
Next, we plan to apply the theory of semigroups and functional analysis to show
the well-posedness, (W1)-(W5), for the general system (1.1)-(1.3), and then in
turn apply these to the specific systems above. We shall do so by building on the
work of Hollis, S. L.; Martin, R. H.; Pierre, M., [Hollis, Martin, Pierre (1987)].
Then, we wish to justify the use of a linear analysis, and apply this to analyse
the ideas about diffusion-driven instabilities at a uniform steady-state. Specifically, we plan to apply these results to the Schnakenberg model.
Thereafter, we aim to perform a numerical analysis of the Schnakenberg
model. Our objective with the numerical analysis, is first of all to solve the system numerically and verify the well-posedness, and next to study the diffusiondriven instabilities at uniform steady-states (u∗ , v ∗ ).
Moreover, by varying the size of the domain we wish to analyse the stability
4
Introduction
and bifurcation of the steady-states.
Finally, we want to analyse how the solutions of the Schnakenberg system are
affected by the choice of (a, b, d). For example, we desire to study the possibility
of obtaining similar patterns for different (a, b, d)-values.
Chapter
2
The approach of Pao
In this chapter the techniques described by Pao, [Pao (1992)], are presented and applied to the specific autonomous reaction-diffusion models stated
in Chap. 1 in order to show existence, uniqueness, boundedness and positivity.
2.1
Upper and lower solutions
In Pao, [Pao (1992)], f and g are assumed to be continuously differentiable
and to satisfy either of the three,
(1) ∂s f (r, s) ≥ 0, ∂r g(r, s) ≥ 0; quasimonotone nondecreasing functions;
(2) ∂s f (r, s) ≤ 0, ∂r g(r, s) ≤ 0; quasimonotone nonincreasing functions;
(3) ∂s f (r, s) ≥ 0, ∂r g(r, s) ≤ 0; mixed quasimonotone functions.
For u0 , v0 ∈ CL0 ∞ (Ω) one can for the three type of functions define ordered upper
and lower solutions of (1.1)-(1.3). Here we shall only be interested in the case
of mixed quasimontone functions in which case the definition is:
6
The approach of Pao
Definition 1 U = (u, v) and U = (u, v) in
ordered upper resp. lower solutions if U ≥ U,
2
CL2,1
are called
∞ (Ω × [0, T ))
∂t u − ∆u ≥ f (u, v), ∂t u − ∆u ≤ f (u, v),
∂t v − d∆v ≥ g(u, v), ∂t v − d∆v ≤ g(u, v),
(2.1)
(2.2)
νu ≥ 0 ≥ νu,
νv ≥ 0 ≥ νv,
(2.3)
furthermore,
and finally
u(x, 0) ≥ u0 ≥ u(x, 0)
v(x, 0) ≥ v0 ≥ v(x, 0)
x ∈ Ω.
(2.4)
2
The adequateness of upper and lower solutions follows from the following
theorem.
Theorem 1 (See [Pao (1992), Theorem 8.3.3]) Let U = (u, v) and U = (u, v)
be a pair of ordered upper and lower solutions and let f and g be mixed quasimonotone. Then there exists a unique solution U = (u, v), and it is in the
sector
2
2,1
hU, Ui = Z ∈ CL∞ (Ω × [0, T )) | U ≤ Z ≤ U .
2
Proof (Sketched) The result is established without the use of distributions
and semigroups and even with a small amount of functional analysis. Instead
(n)
the result is established by considering the sequences {U }n∈N and {Un }n∈N
defined through U1 = U, U1 = U and
(∂t − ∆ + c1 ) un = f˜(un−1 , v n−1 ),
(∂t − d∆ + c2 ) v n = g̃(un−1 , v n−1 ),
(∂t − ∆ + c1 ) un = f˜(un−1 , v n−1 ), (2.5)
(∂t − d∆ + c2 ) v n = g̃(un−1 , v n−1 ),(2.6)
for n ≥ 2 where
f˜(r, s) = f (r, s) + c1 r,
g̃(r, s) = g(r, s) + c2 s,
and
c1 = inf{∂r f (r, s) | inf u ≤ r ≤ sup u},
Ω
Ω
c2 = inf{∂s g(r, s) | inf v ≤ s ≤ sup v}.
Ω
Ω
2.1 Upper and lower solutions
7
together with boundary and initial conditions as in (1.2) resp. (1.3).
We claim that the sequences {Un }n∈N and {Un }n∈N possess the monotone
property
un ≤ un+1 ≤ un+1 ≤ un
,
v n ≤ v n+1 ≤ v n+1 ≤ v n
for all x ∈ Ω
for all n ∈ N. To show this, one applies the positivity lemma of parabolic
operators, see [Pao (1992), Lemma 2.2.1]. Let w1 = u − u2 , w2 = v − v 2 . Then
by (2.5) and (2.6) and (2.1) and (2.2):
(∂t − ∆ + c1 ) w1 = ∂t u − ∆u − f (u, v) ≥ 0,
(∂t − d∆ + c2 ) w2 = ∂t v − d∆v − g(u, v) ≥ 0,
and w1 and w2 satisfy the boundary and initial conditions:
νw1 = νu ≥ 0, w1 (x, 0) = u(x, 0) − u0 ≥ 0,
νw2 = νv ≥ 0, w2 (x, 0) = v(x, 0) − u0 ≥ 0.
By the positivity lemma of parabolic operators it follows that w1 and w2 ≥ 0
and hence u2 ≤ u and v 2 ≤ v. Similarly we obtain u2 ≥ u and v 2 ≥ v.
The proof then goes by induction, again exploiting the positivity lemma.
The existence of the pointwise limits:
(u∞ , u∞ )(x) ≡ lim (un , un )(x),
n→∞
and
(v ∞ , v ∞ )(x) ≡ lim (v n , v n )(x),
n→∞
follows from the monotone properties just proved, or at least sketched. However,
it is by no means clear that these limits are solutions of the original system (1.1)(1.3) and that u∞ = u∞ and v ∞ = v ∞ . Despite this, it is shown in [Pao (1992),
sec. 8.3] that a sufficient condition is the existence of k1 and k2 such that,
f (r2 , s) − f (r1 , s) ≤ k1 (r2 − r1 ),
for
g(r, s2 ) − g(r, s1 ) ≤ k2 (s2 − s1 ),
for
inf u ≤ r1 ≤ r2 ≤ sup u,
Ω
Ω
inf u ≤ r ≤ sup u,
Ω
Ω
inf v ≤ s ≤ sup v,
Ω
Ω
inf v ≤ s1 ≤ s2 ≤ sup v.
Ω
Ω
This is indeed the case when (r, s) 7→ f (r, s) and (r, s) 7→ g(r, s) are differentiable
for
inf u ≤ r ≤ sup u, inf v ≤ s ≤ sup v,
Ω
Ω
Ω
by the mean-value theorem.
Uniqueness is proved by contradiction.
Ω
In the following section it is the aim to prove existence and uniqueness of
the reaction-diffusion systems presented in Chap. 1 by exploiting Theorem 1.
8
2.2
The approach of Pao
Applications of the approach of upper and
lower solutions
First the Schnakenberg model (1.4)-(1.6) is considered for non-negative initial
conditions.
The Schnakenberg system’s reaction functions are quasimonotone:
∂s f (r, s) = r2 ≥ 0,
∂r g(r, s) = −2rs ≤ 0,
for r, s ≥ 0. In order to utilise Theorem 1 we have to find a pair of ordered
upper and lower solutions in accordance to Definition 1. To do so we choose
v = 0. Then u has to solve the inequalities:
∂t u − ∆u ≤ f (u, 0) = a − u,
νu ≤ 0,
u ≤ u0 ,
on ∂Ω × (0, ∞).
By inspection it is seen that u = m ≡ min(a, inf Ω u0 ) is a solution.
We now use these lower solutions to obtain upper solutions. For v:
∂t v − ∆v ≥ g(u, v) = b − m2 v,
(2.7)
and boundary and initial conditions as in (2.3) resp. (2.4). We are guided to
look for solutions independent of the spatial variable; the solution to the linear
ordinary differential equation:
∂t y = b − m2 y,
y(0) = sup v0
Ω
satisfies (2.7), (2.3) and (2.4), and y is therefore
an upper solution. The solution
is y = mb2 + exp(−m2 t) supΩ v0 − mb2 . So y ≤ M ≡ max mb2 , supΩ v0 and
then we may choose v = M .
Now, u has to solve:
∂t u − ∆u ≥ f (u, v) = a − u + u2 v,
(2.8)
and again with boundary and initial conditions as in (2.3) resp. (2.4). Our only
hope is to solve the following ordinary, nonlinear differential equation:
∂t z = a − z + z 2 M,
z0 = sup u0 .
Ω
2.2 Applications of the approach of upper and lower solutions
9
Three scenarios can occur.
1. M a > 41 . The solution is:
1
1
1 + α tan
z(t) =
αt + arctan β
,
2M
2
where
1/2
α = (4M a − 1)
2M z0 − 1
β=
.
α
,
u = z satisfies (2.8) and by definition z0 ≥ u0 in Ω so it follows that u = z is an
upper solution for t ∈ [0, T ), where T is given by
π
1
αT + arctan β = ,
2
2
T > 0,
that is,
T =
2 π
− arctan β > 0.
α 2
2. M a = 41 . The solution is:
z(t) = 2a +
4a(z0 − 2a)
,
γ(2a − z0 )t + 4a
u = z is an upper solution for t ∈ [0, T ), T > 0. If z0 > 2a then T is given by
γ(2a − z0 )T + 4a = 0,
T > 0.
that is,
4a
> 0.
γ(z0 − 2a)
T =
If z0 < 2a then z is increasing monotonically and z → 2a as t → ∞ and hence
we may take T = ∞. For z0 = 2a, z(t) = 2a for all t > 0 and hence T = ∞.
3. 0 < M a < 14 . The solution is:
1
1
z(t) =
1 − α̃ tanh
α̃t + arctanh β̃
,
2M
2
1/2
α̃ = (1 − 4M a)
1 − 2M z0
.
β̃ =
α̃
,
10
The approach of Pao
Im z(t) = 0 if and only if |β̃| < 1, or:
√
√
1 + 1 − 4M a
1 − 1 − 4M a
< sup u0 <
2M
2M
Ω
(2.9)
in which case we may take T = ∞. However, if z0 is outside this bound, then
no real solution exists.
Unless M a < 14 and u0 does not satisfy (2.9), then by the analysis above,
Theorem 1 provides local existence and uniqueness. Furthermore, the solution
is in the sector h(u, v) , (u, v)i, and therefore positive.
If M a < 41 and (2.9) is satisfied, or if M = a4 and supΩ u0 ≤ 2a, then the
pair of ordered upper and lower solutions exists for all t ≥ 0 and hence, again
by Theorem 1, there exists a unique, global solution.
Similarly the stationary equations,
−∆u = a − u + u2 v,
−d∆v = b − u2 v,
(2.10)
together with the homogeneous Neumann conditions on ∂Ω, can be analysed.
Despite the fact that this problem is elliptic rather than then parabolic, the
idea of theorem Theorem 1 is still applicable: If there exists (u, v) and (u, v)
satisfying u ≤ u, v ≤ v and the inequalities,
−∆u ≤ a − u + u2 v, −∆u ≥ a − u + u2 v,
−d∆v ≤ b − u2 v, −∆v ≥ b − u2 v,
(2.11)
together with the boundary inequalities (2.3), then there exists a unique solution
in the sector h(u, v), (u, v)i, see [Pao (1992), Theorem 5.2]. There may exist a
solution outside this sector.
The usefullness of this model is partly due to the existence of multiple steadystate solutions. In turn, this is also what makes these models challenging from
a mathematical perspective. We shall in the following try to give sufficient
conditions for the model to exhibit a unique steady-state solution.
First, by solving:
0 = a − u + u2 v,
0 = b − u2 v,
we realise that the Schnakenberg model has one single positive, homogeneous
steady-state solution given by
u∗ = a + b,
v∗ =
b
2,
(a + b)
b > 0,
a + b > 0.
We shall give sufficient conditions for this solution to be the unique solution of
the steady-state problem (2.10). To do so we have to find a set of ordered upper
2.2 Applications of the approach of upper and lower solutions
11
and lower solutions satisfying the inequalities in (2.11). Guided by previous
experience, we look for solutions independent of the spatial variable and take
v = 0, u = a and v = ab2 . It is not difficult to see that these in fact solve the
inequalities. For u:
0 ≥ a − u + u2
b
.
a2
0 = a − y + y2
b
,
a2
(2.12)
The equation:
has solutions:
y± =
a2 ± a2
q
1 − 4 ab
2b
.
Thus, for
0<
1
b
≤
a
4
y+ and y− are real and positive and we may take u =
if
(2.13)
√
b
a2 +a2 1−4 a
2b
. Now, u ≥ u∗
a2
≥ a + b,
2b
or if
a≥
However, 4 >
√
1+ 12
2
1+
√
12
b.
2
and thus, as expected u > u∗ > u. Therefore:
Corollary 1 For 0 ≤ ab ≤ 14 , (u∗ , v ∗ ) is the unique steady-state solution of the
Schnakenberg system in the sector h(a, 0), (b/a2 , u)i.
2
The result cannot be improved since increasing v only decreases u.
As mentioned other appropriate reaction functions are:
r2
, g(r, s) = r2 − s, r, s > 0
s(1 + kr2 )
ρrs
ρrs
, g(r, s) = a(b − s) −
,
f (r, s) = a − r −
1 + r + kr2
1 + r + kr2
f (r, s) = a − br +
r, s ≥ 0.
12
The approach of Pao
Again all parameters are assumed positive. For the first model, the Gierer and
Meinhardt model, the partial derivatives:
∂s f (r, s) = −r2 /(s2 (1 + kr2 )) ≤ 0,
∂r g(r, s) = 2r ≥ 0,
show that the reaction functions are mixed quasimonotone on R2+ . We only
study positive initial conditions.
For u:
∂t y = a − by,
y(0) = inf u0 .
Ω
a
b , inf Ω
The solution is y(t) ≥ m1 = min
Similarly for v:
u0 > 0, so we may choose u = m1 .
∂t z = m21 − z,
z0 = inf v0 ,
Ω
to
conclude that z ≥ m2 = min(m21 , inf Ω v0 ), so we put v = m2 .
1
x2
Now, for u we realise that m2 (1+kx
2 ) → m k from below as x →
2
we consider:
∂t w = a − bw +
∞, and then
w2
1
≥ a − bw +
,
m2 k
m2 (1 + kw2 )
w(0) = sup u0 .
Ω
It follows that w ≤ M1 = max
u = M1 .
Finally, for v we consider:
a
b
+
1
bm2 k , supΩ
u0 , and hence we may take
∂t q = M12 − q,
q(0) = sup v0 ,
Ω
and take v = M2 = max(M12 , supΩ v0 ).
It is clear that m1 < M1 , and then m2 < M2 . So by Theorem 1 we have:
Corollary 2 There exists a unique solution of the Gierer-Meinhardt system
for all positive initial conditions and parameters. The solution is in the sector
hU, U)i = h(m1 , M1 ), (m2 , M2 )i, and therefore it is bounded an positive.
2
The Thomas model is an example of a system with reaction functions that
are not suitable for the approach of upper and lower solutions, since the reaction
2.3 Summary and remarks
13
functions are not monotone:
ρr
,
∂s f (r, s) = −
1 + r + ks2
∂r g(r, s) =
ρ s kr2 − 1
(1 + r + kr2 )
2,
r, s ≥ 0;
clearly ∂r g changes sign.
2.3
Summary and remarks
In this chapter we presented the techniques described by Pao, [Pao (1992)].
The techniques were applied to the reaction-diffusion models in order to prove
existence, uniqueness, positivity and boundedness. The continuously dependency on initial data was not considered as the method is not adequate for such
analysis.
For the Gierer and Meinhardt system we obtained global existence and
uniqueness of a positive solution for all parameters and positive initial conditions. Furthermore, we showed that the solutions were bounded.
For the Schnakenberg system it was only possible to obtain local and global
existence for some specific parameters and initial conditions. However, sufficient
conditions were given in order for the stationary problem to display a unique
steady state in a sector.
It was not possible to study the Thomas model by the methods described
by Pao, since the reaction-functions were not quasimonotone. For this type of
reaction functions we need to rely on a different setting. Similarly, if we want to
show global existence for the first Schnakenberg model for all positive parameters, we also need to rely on a different sitting.
In the forthcoming chapter we shall therefore apply functional analysis and
results on semigroups to the general class of two-component reaction-diffusion
equations in order to prove well-posedness under certain appropriate conditions.
14
The approach of Pao
Chapter
3
Existence and uniqueness of
two-component
reaction-diffusion system
In this chapter we shall analyse the general reaction-diffusion system:
∂t u − ∆u = f (t, u, v),
in Ω × (0, T ),
∂t v − d∆v = g(t, u, v),
u
ν
= 0, on ∂Ω × [0, T ),
v
u = u0 ,
v = v0 , on Ω × {t = 0} ,
(3.1)
(3.2)
(3.3)
on a smooth and bounded domain Ω, in terms of the well-posedness (W1)-(W5).
To do so, we apply semigroup theory and basic functional analysis. The theory
of semigroups is set forth in App. A, but for convenience we shall just recap the
most important results.
The Neumann realisation of the Laplacian L = −∆, D(L) = {u ∈ W 2,p |
νu = 0}, is sectorial, see Definition 6 in App. A, and therefore it generates an
analytical semigroup G.
For α ≥ 0 we define the fractional powers of the Helmholtz operator, H =
−∆ + I, and denote these Hα . The domain of the fractional powers of H,
D(Hα ), equipped with graph norm k · kD(Hα ) = k · kp + kHα · kp , is continuously
16
Existence and uniqueness of two-component reaction-diffusion system
embedded in CL0 ∞ (Ω) if α > n/(2p), see Lemma 9 in App. A.1
For L we obtain the following corollary to Lemma 10 in App. A.
Corollary 3 Let G be the analytic semigroup generated by −L. The following
properties hold for the semigroup G and the fractional powers of the Helmholtz
operator, H,
1◦ G(t) : Lp (Ω) → D(Hα ) for all t > 0;
2◦ kG(t)ukD(Hα ) ≤ Cα,p t−α kukp for all t > 0, u ∈ Lp (Ω);
3◦ G(t)Hα u = Hα G(t)u for all t > 0, u ∈ D(Hα ).
3.1
2
Local existence and uniqueness
The following basic hypotheses are assumed to hold:
(H1) d > 0;
(H2) u0 ≥ 0 and v0 ≥ 0 are continuous on Ω; u0 , v0 ∈ CL0 ∞ (Ω);
3
(H3) f and g are continuously differentiable functions from R+ into R with
f (t, 0, s) ≥ 0 and g(t, r, 0) ≥ 0 for all t, r, s ≥ 0;
2
(H4) There exist m > 0 and a continuous function F : R+ → R+ such that
f (t, r, s), g(t, r, s) ≤ exp(mt)F (r, s) for all t, r, s ≥ 0.
Remark 1 Compared to [Hollis, Martin, Pierre (1987)] we have added (H4) to
2
archieve local existence for (u0 , v0 ) ∈ CL0 ∞ .
2
First we prove positivity.
Proposition 1 Suppose that (H1)-(H4) are satisfied. Then the classical solution, when it exists, is non-negative for all t in its interval of existence, [0, T ].
I.e. the cone K + = {(u, v) ∈ C 2,1 (Ω × (0, T )) | u ≥ 0, v ≥ 0} is an invariant
set: (u0 , v0 ) ∈ K + ⇒ (u(t), v(t)) ∈ K + for all t > 0.
2
1 Since 0 is in the resolvent set of ρ(Hα ) the graph-norm is equivalent to kHα · k . However,
p
we shall use the graph-norm throughout.
3.1 Local existence and uniqueness
17
Proof First, we assume u0 , v0 > 0 for all x ∈ Ω. Furthermore, we let u > 0
and v > 0 for every t ∈ [0, τ ) and all x ∈ Ω and either u(x, τ ) = 0 for x in
non-empty set M ⊂ Ω while v(x, τ ) > 0 for all x ∈ Ω, or v(x, τ ) = 0 for x in
non-empty set M ⊂ Ω while u(x, τ ) > 0 for all x ∈ Ω. If such a τ < T does
not exist, then the proof is done. Now assume that it does and without loss of
generality that u(x, τ ) = 0 while v(x, τ ) > 0.
Let x0 ∈ M . It is now claimed that x0 ∈
/ ∂Ω. This follows by the maximum
principle, see [Pao (1992), Theorem 2.1.4], which states that if x0 ∈ ∂Ω is a
minimum then νu(x0 ) < 0, which contradicts u being a solution. Therefore
x0 ∈
/ Ω. x0 is therefore an interior minimum point of u, hence
∇u(x0 , τ ) = 0,
H = (∂ij u(x0 , τ )) ≥ 0,
where the second inequality states that the Hessian is semi-positive definite at
x0 . Thus,
X
−∆u(x0 , τ ) = −tr H = −
λk ≤ 0
k
and then by (3.1) it follows that,
∂t u(x0 , τ, x0 ) ≥ 0.
∂t u(x0 , τ ) > 0 would imply by continuity that u(x0 , t) < 0 for t ∈ (τ − ǫ, τ )
for some ǫ > 0, which contradicts the assumption made about u for t < τ .
Therefore ∂t u(τ, x0 ) = 0.
Similarly, if u0 (x) = 0 or v0 (x) = 0 for x ∈ M ⊂ Ω it follows ∂t u(t, x0 ) = 0,
t > 0. This finishes the proof.
Remark 2 The Schnakenberg and the Thomas systems’ reaction functions satisfy (H3) − (H4). On the other hand, the Gierer and Meinhardt system’s reaction functions are not continuous in (t, r, s) = (0, 0, 0). However, by the results
in the previous chapter or similarly by exploiting Corollary 4, see
Sec. 3.2,
we have (u, v) ≥ (m1 , m2 ) > (0, 0), where m1 = min ab , inf Ω u0 > 0 and
m2 = min(m21 , inf Ω v0 ) > 0, whenever inf Ω (u0 , v0 ) > 0. Therefore, we may
apply the reasoning below for the Gierer and Meinhardt system if we replace
(H2), (H3) and (H4) with:
(H2′ ) u0 > 0 and v0 > 0 are continuous on Ω; u0 , v0 ∈ CL0 ∞ (Ω);
(H3′ ) f and g are continuously differentiable functions from [0, ∞) × [m1 , ∞) ×
[m2 , ∞) into R with f (t, m1 , s) ≥ 0 and g(t, r, m2 ) ≥ 0 for all t ≥ 0 and
(r, s) ≥ (m1 , m2 );
18
Existence and uniqueness of two-component reaction-diffusion system
(H4′ ) There exist m > 0 and a continuous function F : [m1 , ∞) × [m2 , ∞) → R+
such that f (t, r, s), g(t, r, s) ≤ exp(mt)F (r, s) for all t ≥ 0 and (r, s) ≥
(m1 , m2 ).
2
In the setting of semigroup and the abstract initial value problem it is natural
2
to define F and G on R+ × CL0 ∞ by:
[F (t, u, v)] (x) = f (t, u(x), v(x))
,
[G(t, u, v)] (x) = g(t, u(x), v(x))
for x ∈ Ω,
t ≥ 0,
u, v ∈ CL0 ∞ .
Moreover, we let G1 and G2 be the analytical semigroup of L resp. d × L.
The following Lemma is used frequently in what follows.
Lemma 1 If u and v are continuous from [0, T ] to Lp (Ω), then the integrals:
I1 (t) =
I2 (t) =
t
Z
Z
0
t
0
G1 (t − τ )F (τ, u(τ ), v(τ ))dτ
G2 (t − τ )G(τ, u(τ ), v(τ ))dτ,
exist and I1 (t) and I2 (t) are continuous on [0, T ) with I1 (t), I2 (t) ∈ D(L) and
I1 (t), I2 (t) → 0+ in Lp for t → 0+ .
2
Proof The proof can be seen in [Henry (1981), Lemma 3.2.1].
If the reaction-diffusion system has a classical solution then u and v satisfy:
Rt
u(t) = G1 (t)u0 + 0 G1 (t − τ )F (τ, u(τ ), v(τ ))dτ,
Rt
(3.4)
v(t) = G2 (t)v0 + 0 G2 (t − τ )G(τ, u(τ ), v(τ ))dτ,
by variation of constants, which is shown by considering the L2 -valued functions
wj (τ ) = Gj (t − τ )u(τ ), j = 1, 2. wj are differentiable since Gj is analytic and u
is differentiable.2 Then by Theorem 7 in App. A:
dw1
= ∆G1 (t − τ )u(τ ) + G1 (t − τ )u′ (τ )
dτ
= ∆G1 (t − τ )u(τ ) − G1 (t − τ )∆u(τ ) + G1 (t − τ )F (τ, u(τ ), v(τ ))
= G1 (t − τ )F (τ, u(τ ), v(τ )),
and similarly for w2 . Now integration from 0 to t and the result follows.
Now, on the other hand let α > n/(2p) and u and v be continuous functions
from [0, T ] into D(Hα ) ֒→ CL0 ∞ satisfying (3.4). It is then claimed that u and v
2 We
have
1
(wj (t+h)−w(j))
h
=
1
(Gj (t+h)u(t+h)−Gj (t)u(t+h))+ h1 Gj (t)(u(t+h)−u(t)).
h
3.1 Local existence and uniqueness
19
then solves the reaction diffusion system (1.1) - (1.3). The continuity of u and
v implies continuity of t 7→ F (t, u(t), v(t)) and t 7→ G(t, u(t), v(t)). Then, the
linear problems ∂t y − ∆y = F (t, u(t), v(t)) and ∂t z − d∆z = G(t, u(t), v(t)) and
y(0) = u0 , z(0) = v0 have a unique solution, see e.g. [Henry (1981), Theorem
3.2.2], namely where y and z are given by (3.4). But then, y = u and z = v and
the assertion has been shown.
Now, all prerequisites have been establish to prove local existence of the
reaction diffusion system (1.1) - (1.3). The result is established with L2 -theory
alone. Later, when we turn to global existence we shall make use of more general
Lp -theory.
Proposition 2 (Local existence, see [Henry (1981), Theorem 3.3.1] and [Hollis, Martin, Pierre (1987)]) Given hypotheses (H1)-(H4) are satisfied then there
exist T = T (u0 , v0 ) > 0 such that the reaction diffusion system (3.1) - (3.3) has
2
a unique solution (u, v) ∈ CL0 ∞ ((0, T ]; D(Hα )) with u(0) = u0 ∈ CL0 ∞ and
v(0) = v0 ∈ CL0 ∞ .
2
Proof By the previous discussion it suffices to prove the corresponding result
for (3.4). The proof is then established by utilising Banach’s Fixed Point Theorem.
Choose α such that n/4 < α < 1, then the injection I : D(Hα ) → CL0 ∞ is
continuous by Lemma 8 in App. A. First we establish the result for initial data
in D(Hα ).
For ρ > 0 and T > 0 the following closed ball is considered:
n
o
2
Bρ (u0 , v0 ) = (u, v) ∈ CL0 ∞ ([0, T ]; D(Hα )) | ku − u0 kα , kv − v0 kα ≤ ρ .
(3.5)
By the continuity of f and g the local Lipschitz properties:
kF (t, u1 , v1 ) − F(t, u2 , v2 )k2 ≤ L11 ku1 − u2 kD(Hα ) + L12 kv1 − v2 kD(Hα )
kG(t, u1 , v1 ) − G(t, u2 , v2 )k2 ≤ L21 ku1 − u2 kD(Hα ) + L22 kv1 − v2 kD(Hα )
(3.6)
follow for 0 ≤ t ≤ T, (u1 , v1 ), (u2 , v2 ) ∈ Bρ (u0 , v0 ) and Lipschitz-constants
L11 , L12 , L21 , L22 > 0.
Furthermore, for (y, z) ∈ Bρ (u0 , v0 ) define P : [0, T ] → L2 (Ω) and Q :
[0, T ] → L2 (Ω) by
P y(t) = G1 (t)u0 +
Qz(t) = G2 (t)v0 +
Z
t
G1 (t − τ )F (τ, y(τ ), z(τ ))dτ,
0
Z
0
t
G2 (t − τ )G(τ, y(τ ), z(τ ))dτ.
20
Existence and uniqueness of two-component reaction-diffusion system
Finally, set M1 = maxt∈[0,T ] kF (t, y(0), z(0))k2 , M2 = maxt∈[0,T ] kG(t, y(0), z(0))k2
and choose T so that,
3ρ
k(G1 (h) − I)u0 kα , k(G2 (h) − I)v0 kα ≤
, 0 ≤ h ≤ T,
4
Z T
ρ
i
s−α ds ≤ , i = 1, 2,
Cα,2
(Mi + Li1 ρ + Li2 ρ)
4
0
(3.7)
(3.8)
i
where Cα,2
, i = 1, 2, are the constants in 2◦ in Corollary 3 for L resp. d × L
and p = 2.
Remark 3 Note that such T exists since G1 (h) and G2 (h) → I as h → 0+ by
2◦ in Definition 5 in App. A and
Z h
1
h1−α → 0, h → 0+ ,
s−α ds =
1−α
0
for α < 1.
2
The proof then goes as follows:
1◦ It is shown that (P, Q) maps Bρ (u0 , v0 ) into itself;
2◦ It is shown that (P, Q) is a strict contraction on Bρ (u0 , v0 ), allowing the
use of Banach’s Fixed Point Theorem to conclude the existence of a unique
fix point in Bρ (u0 , v0 );
3◦ It is shown that the result extends to initial data in CL0 ∞ .
Let (u, v) ∈ Bρ (u0 , v0 ) then,
Z t
kP u(t) − u0 kD(Hα ) = k(G1 (t) − I)u0 +
G1 (t − τ )F (τ, u(τ ), v(τ ))dτ kD(Hα )
0
Z t
kG1 (t − τ )F (τ, u(τ ), v(τ ))kD(Hα ) dτ
≤ k(G1 (t) − I)u0 kD(Hα ) +
0
≤ (using (3.7))
Z t
3ρ
≤
+
kG1 (t − τ )k (kF (τ, u(τ ), v(τ )) − F(τ, u0 , v0 )k2
4
0
+ kF (τ, u0 , v0 )k2 ) dτ
≤ (using the local Lipschitz condition and 2◦ in Corollary 3 to apply (3.8))
Z t
3ρ
1
+ Cα,2
(M1 + L11 ρ + L12 ρ)
(t − s)−α ds
≤
4
0
≤ ρ, for 0 ≤ t ≤ T,
3.1 Local existence and uniqueness
21
and similarly,
kQv(t) − v0 kD(Hα ) ≤ ρ,
showing that (P, Q) maps Bρ (u0 , v0 ) into itself. Furthermore, from 1◦ in Corollary 3 and Lemma 1 we compute:
kP u(t + h) − P u(t)kD(Hα ) = kG1 (t)(G1 (h) − I)u0
Z t
+
(G1 (h) − I) G1 (t − τ )F (τ, u(τ ), v(τ ))dτ
0
+
Z
t
t+h
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ kD(Hα )
≤ (using Lemma 1 and the strong continuity of the semigroup)
→ 0 for
h → 0+ ,
showing that P is continuous from [0, T ] into D(Hα ).
A similar computation shows that Q holds the same properties and 1◦ has
been shown.
2◦ follows from the estimate: Let (x, y), (z, w) ∈ Bρ (u0 , v0 ) then
Z t
kP x(t) − P z(t)kD(Hα ) ≤
kG1 (t − τ ) (F (t, x(t), y(t)) − F(t, z(t), w(t))) kD(Hα )
0
≤ (using 2◦ in Corollary 3 and (3.6))
Z t
1
≤ Cα,2
(t − τ )−α dτ × L11 sup kx(t) − z(t)kD(Hα )
τ ∈[0,t]
0
+
L12
sup ky(t) − w(t)kD(Hα )
τ ∈[0,t]
!
≤ (by (3.8))
≤
1
4
sup kx(t) − z(t)kD(Hα )
t∈[0,T ]
!
+ sup ky(t) − w(t)kD(Hα ) ,
τ ∈[0,t]
for every t ∈ [0, T ] and thus
sup kP x(t) − P z(t)kD(Hα ) ≤
t∈[0,T ]
1
4
sup kx(t) − z(t)kD(Hα )
t∈[0,T ]
!
+ sup ky(t) − w(t)kD(Hα ) ,
τ ∈[0,t]
22
Existence and uniqueness of two-component reaction-diffusion system
and similarly
sup kQy(t) − Qw(t)kD(Hα ) ≤
t∈[0,T ]
1
4
sup kx(t) − z(t)kD(Hα )
t∈[0,T ]
!
+ sup ky(t) − w(t)kD(Hα ) .
τ ∈[0,t]
Therefore
sup k(P, Q)(x(t), y(t)) − (P, Q)(z(t), w(t))kD(Hα )2
t∈[0,T ]
≤
1
sup k(x(t), y(t)) − (z(t), w(t))kD(Hα )2
2 t∈[0,T ]
verifying that (P, Q) is a strict contraction on Bρ (u0 , v0 ). By Banach’s Fixed
Point Theorem, (P, Q) therefore have a unique fix point in Bρ (u0 , v0 ). This is
the solution of (3.1)-(3.3) on [0, T ] with initial value (u(0), v(0)) = (u0 , v0 ) ∈
2
(D(Hα )) .
The solution can be extended to a larger interval [0, T + ǫ], ǫ > 0, by considering w1 (t) = u(t + T ) and w2 (t) = v(t + T ) where w1 and w2 solve:
∂t w1 − ∆w1 (t) = F (t + T, w1 (t), w2 (t)),
∂t w2 − d∆w2 (t) = G(t + T, w1 (t), w2 (t)).
Again the existence of w1 and w2 on an interval [0, ǫ] is assured by above. Let
[0, T ∗ ) denote the maximal interval to which the solution can be extended.
Before continuing with the proof let us assume that the proposition holds.
Then, we state and prove the following result.
Proposition 3 The system (3.1)-(3.3) has a unique solution on a maximal
interval of existence [0, T ∗ ) and there are continuous functions N1 and N2 such
that
0 ≤ u ≤ N1 (t),
0 ≤ v ≤ N2 (t),
for
in
Ω × [0, T ∗ ).
(3.9)
If T ∗ < ∞ then
lim (kuk∞ + kvk∞ ) = ∞.
t→T ∗ −
2
Proof The existence of N1 and N2 directly follows by continuity.
If
lim (kuk∞ + kvk∞ ) < ∞,
t→T ∗ −
then the solution could be extended by the procedure above beyond T ∗ ; contradiction the definition of T ∗ .
3.1 Local existence and uniqueness
23
We are now ready to extend the result to initial data in CL0 ∞ (Ω). We let
{(un0 , v0n )}n∈N be a sequence in D(Hα ) × D(Hα ) such that un0 , v0n ≥ 0 converge
to (u0 , v0 ) in L2 :
kun0 − u0 k2 , kv0n − v0 k2 → 0,
for
n → ∞.
We show that {(un , vn )}n∈N , where (un , vn ) are the solutions corresponding to
initial conditions (un0 , v0n ), is bounded in D(Hβ ), α < β < 1, and then we exploit
the compactness of the injection D(Hα ) ֒→ D(Hβ ), see Proposition 5 in App. A.
By the continuity of f and g, and by (H4), there exists a positive, continuous
function F̃ such that,
kF (t, un (t), vn (t))k∞ , kG(t, un (t), vn (t))k∞ ≤ exp(mT )F̃ (R),
(3.10)
for all un , vn ∈ D(Hβ ) whenever t ∈ [0, T ] and kun (t)k∞ , kvn (t)k∞ ≤ R. Let R
be chosen such that kun0 k∞ , kv0n k∞ ≤ R for all n ∈ N.
It is now claimed that T ∗ = inf n∈N Tn∗ > 0. To show this, we first note that
φ = G1 (t)u0 solves:
∂t φ − ∆φ = 0,
φ(0) = u0 ,
and thus by the positivity lemma if follows that kφk∞ = kG1 (t)u0 k∞ ≤ R
whenever u0 ≤ R.
Therefore by (3.4) and 0 ≤ t < Tn∗ :
Z t
n
kun (t)k∞ ≤ kG1 (t)u0 k∞ +
kG1 (t − τ )F (τ, un (τ ), vn (τ ))k∞ dτ
0
≤ (using (3.10))
≤ R + t exp(mTn∗ )B(R + 1),
whenever ku(t)k∞ , ku(t)k∞ ≤ R + 1. For t ∈ [0, exp(−mTn∗ )F̃ (R + 1)−1 ],
ku(t)k∞ ≤ R+1 and therefore by Proposition 3, Tn∗ > exp(−mTn∗ )F̃ (R+1)−1 for
all n. This inequality defines a δ(R) > 0 such that Tn∗ ≥ δ(R) > 0 for all n, which
can be seen by considering the graphs (Tn∗ , Tn∗ ) and Tn∗ , exp(−mTn∗ )F̃ (R + 1)−1 .
In the following we shall suppress the dependency of R on δ.
According to Corollary 3 in App. A we have:
Z t
kun (t)kD(Hβ ) ≤ C1,β t−β kun0 k2 +
C1,β (t − τ )−β kF (τ, u(τ ), v(τ ))k2 dτ,
0
< (using (3.10))
< ∞,
for all n ∈ N,
and t ∈ (0, δ],
and similarly for vn , showing that the sequences {un (t)}n∈N and {vn (t)}n∈N are
bounded in D(Hβ ). Then by the compactness of the injection I : D(Hβ ) →
24
Existence and uniqueness of two-component reaction-diffusion system
D(Hα ), see Proposition 5 in App. A, the existence of u(t) ∈ D(Hα ) and v(t) ∈
D(Hα ) such that
lim kunk (t) − u(t)kD(Hα ) = 0 = lim kvnk (t) − v(t)kD(Hα ) ,
k→∞
k→∞
for
t ∈ (0, δ],
follows. By the local Lipschitz property, (3.6), and the continuity of G1 (t), G2 (t)
it follows that u and v are the solutions of (1.1)-(1.3) with initial conditions
u(0) = u0 and v(0) = v0 and they are in CL0 ∞ ((0, δ]; D(Hα )).
By replacing [0, T ∗ ) with [δ, T ∗ ) and (u0 , v0 ) with
(u(δ), v(δ)) ∈ D(Hα ) × D(Hα )
and using the results already obtained when (u0 , v0 ) ∈ D(Hα ) × D(Hα ), the
proposition follows.
Remark 4 It is possible to improve the regularity of the result in Proposition 2
by showing u, v ∈ CL1 ∞ ((0, T ∗ ); D(Hα )). To do so let u(t), v(t) be given from
Proposition 2 and define uδ = u(δ) and vδ = v(δ) for some 0 < δ < T < T ∗ and
consider:
Z t
G1 (t − τ )∂t F (τ, u(τ ), v(τ ))dτ,
l1 (t) = G1 (t − δ)F (δ, uδ , vδ ) − ∆ G1 (t − δ)uδ +
0
Z t
l2 (t) = G2 (t − δ)G(δ, uδ , vδ ) − d∆ G2 (t − δ)vδ +
G2 (t − τ )∂t G(τ, u(τ ), v(τ ))dτ.
δ
for t ∈ [δ, T ]. Note that l1 , l2 ∈ D(Hα ) since by 1◦ and 2◦ in Corollary 3 every
term is so.
We may exploit the continuity of the partial derivatives of f and g to conclude
that t 7→ ∂r F (t, u(t), v(t)), ∂s F (t, u(t), v(t)) and ∂r G(t, u(t), v(t)), ∂s G(t, u(t), v(t))
are continuous from [δ, T ] into the space of bounded linear operators in L2 , defined by
L2 ∋ w 7→ ∂r F (t, u(t), v(t))w, ∂s F (t, u(t), v(t))w
and
L2 ∋ w 7→ ∂r G(t, u(t), v(t))w, ∂s G(t, u(t), v(t))w.
We define:
B11 (t) = k (∂r F ) (t, u(t), v(t))(x)k,
B21 (t) = k (∂s F ) (t, u(t), v(t))(x)k,
B12 (t) = k (∂r G) (t, u(t), v(t))(x)k,
B22 (t) = k (∂s G) (t, u(t), v(t))(x)k,
3.1 Local existence and uniqueness
25
and w1 and w2 by
w1 (t) = l1 (t) +
Z
t
G1 (t − τ ) (∂r F (τ, u(τ ), v(τ ))w1 (τ ) + ∂s G(τ, u(τ ), v(τ ))w2 (τ )) dτ,
δ
w2 (t) = l2 (t) +
Z
δ
t
G2 (t − τ ) (∂r G(τ, u(τ ), v(τ ))w1 (τ ) + ∂s G(τ, u(τ ), v(τ ))w2 (τ )) dτ.
By [Pazy (1983), Corollary 6.1.3] there exist unique w1 and w2 in CL0 ∞ ([δ, T ]; D(Hα )).
We now claim that for t ∈ [δ, T ] uh (t) = u(t+h)−u(t)
→ w1 and vh (t) =
h
v(t+h)−v(t)
→
w
as
h
→
0.
2
h
We shall prove the claim by giving all details for u and by analogy exploit
that v follows by similar arguments.
First we consider h → 0+ and rewrite uh using (3.4) with uδ and vδ as initial
conditions, that is at time t = δ:
"
Z t+h
1
G1 (t + h − δ)uδ − G1 (t − δ)uδ +
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ
uh (t) =
h
δ
#
Z t
−
G1 (t − τ )F (τ, u(τ ), v(τ ))dτ
δ
"
Z δ+h
1
G1 (t − δ)(G1 (h) − I)uδ +
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ
=
h
δ
#
Z t
Z t+h
G1 (t − τ )F (τ, u(τ ), v(τ ))dτ .
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ −
+
δ
δ+h
(3.11)
The third term in the brackets can be rewritten by a change of variables:
Z
t+h
δ+h
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ =
Z
δ
t
G1 (t − τ )F (τ + h, u(τ + h), v(τ + h))dτ.
Furthermore, by Taylor expansion of f and g we may write:
F (τ + h, u(τ + h), v(τ + h)) = F (τ, u(τ + h), v(τ + h)) + ∂t F (τ, u(τ + h), v(τ + h)) × h
+ ǫ1 (h)
= F (τ, u(τ ), v(τ )) + ∂r F (τ, u(τ ), v(τ ))(u(τ + h) − u(τ ))
+ ∂s F (τ, u(τ ), v(τ ))(v(τ + h) − v(τ )) + ǫ2 (h)
+ ∂t F (τ, u(τ + h), v(τ + h)) × h + ǫ1 (h),
where
1
h ǫi (h)
→ 0 as h → 0 uniformly in L2 -norm on [δ, T ] for i = 1, 2.
26
Existence and uniqueness of two-component reaction-diffusion system
Hence (3.11) becomes:
Z
1 δ+h
G1 (h) − I
uδ +
uh (t) = G1 (t − δ)
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ
h
h δ
Z t
Z t
ǫ1 (h) ǫ2 (h)
G1 (t − τ )
G1 (t − τ )∂t F (τ, u(τ + h), v(τ + h)) × h dτ +
+
dτ
+
h
h
δ
δ
Z t
+
G1 (t − τ ) [∂r F (τ, u(τ ), v(τ ))uh (τ ) + ∂s F (τ, u(τ ), v(τ ))vh (τ )] dτ.
δ
And then
G1 (h) − I
uδ − G1 (t − δ)∆uδ
uh − w1 = G1 (t − δ)
h
)
( Z
1 δ+h
G1 (t + h − τ )F (τ, u(τ ), v(τ ))dτ − G1 (t − δ)F (δ, uδ vδ )
+
h δ
Z t
+
G1 (t − τ ) (∂t F (τ, u(τ + h), v(τ + h)) − ∂t F (τ, u(τ ), v(τ ))) × h dτ
δ
Z t
ǫ1 (h) ǫ2 (h)
+
G1 (t − τ )
dτ
+
h
h
δ
Z t
+
G1 (t − τ )∂r F (τ, u(τ ), v(τ ))(uh − w1 )dτ
δ
Z t
+
G1 (t − τ )∂s F (τ, u(τ ), v(τ ))(vh − w1 )dτ.
δ
By (A.3) and (B.2) in App. A resp. App. B the first four terms tends to zero
in D(Hα )-norm as h → 0+ .
For h → 0− we may use:
G(−h) − I
G1 (t + h − δ) − G1 (t − δ)
uδ = G1 (t + h − δ)
uδ → G1 (t − δ)∆uδ ,
h
−h
to obtain similar results.
Therefore, we may write:
kuh − w1 kD(Hα ) ≤ θ1 (h) + K1
Z
δ
t
kuh − w1 kD(Hα ) + kvh − w2 kD(Hα ) dτ,
(3.12)
where θ1 (h) → 0 as h → 0 and
K1 = max
!
sup kG1 (t)kB11 (t), sup kG1 (t)kB21 (t) .
t∈[δ,T ]
t∈[δ,T ]
3.2 Application of result on local existence to specific models
27
In the same manner we obtain:
kvh − w2 kD(Hα ) ≤ θ2 (h) + K2
Z
t
δ
kuh − w1 kD(Hα ) + kvh − w2 kD(Hα ) dτ,
(3.13)
again θ2 (h) → 0 as h → 0 and:
K2 = max
sup
t∈[δ,T ]
kG2 (t)kB12 (t),
sup
t∈[δ,T ]
!
kG2 (t)kB22 (t)
.
Adding (3.12) and (3.13) and using Gronwall’s inequality, see App. C and
Lemma 12, we obtain:
kuh − w1 kD(Hα ) + kvh − w2 kD(Hα ) ≤ 2 (θ1 (h) + θ2 (h)) exp((t − δ) max(K1 , K2 )),
showing that uh → w1 and vh → w2 in D(Hα ) as h → 0, and hence w1 and w2
are the derivatives of u and v resp. Since w1 , w2 ∈ CL0 ∞ ([δ, T ]; D(Hα )), u and v
are continuous differentiable on [δ, T ]. Since this holds for every 0 < δ < T < T ∗
we conclude that u, v ∈ CL1 ∞ ((0, T ∗ ); D(Hα )).
2
In the following section we shall apply the results on local existence to prove
global existence of the Schnakenberg and Thomas system presented in Chap. 1.
3.2
Application of result on local existence to
specific models
By Proposition 2 and Remark 2 local existence and uniqueness follows for
the three models described in Chap. 1. To prove global existence it suffices,
according to Proposition 3, to prove that,
lim (ku(t)k∞ + kv(t)k∞ ) < ∞.
t→T ∗ −
To do so, we shall make use of the following corollary, which is an immediate
consequence of the positivity lemma for parabolic equations.
Corollary 4 Let L be an elliptic operator, for example L = ∆. If u is continuous in Ω × (0, T ) and
∂t u − Lu = f (u) ≥ g(u), in Ω × (0, T ),
νu = 0 ≥ h, on ∂Ω × (0, T ),
u = u0 ≥ l,
on
Ω × {t = 0}
28
Existence and uniqueness of two-component reaction-diffusion system
and y is a continuous solution of:
∂t y − Ly = g(y),
in
Ω × (0, T )
νy = h, on ∂Ω × (0, T ),
y = l, on Ω × {t = 0},
then
u ≤ y,
in
Ω × [0, T ).
2
Proof Define w = u − y and observe that w satisfy:
∂t w − Lw = f (u) − g(y) ≤ 0,
in Ω × (0, T )
νw = −h ≤ 0, on ∂Ω × (0, T ),
y = u0 − l ≤ 0, on Ω × {t = 0}.
Then the positivity lemma guarantees that w ≤ 0 and thus u ≤ y and the proof
is done.
Again we consider the Schnakenberg model first.
Assertion 1 For the Schnakenberg model there exist constants Γ1 and Γ2 such
that the following estimates hold:
0 ≤v ≤ Γ1 ,
0≤
kuk22
≤ Γ2 ,
(3.14)
(3.15)
for every (x, t) ∈ Ω × (0, T ∗ ).
Proof From the equation of u we readily obtain:
∂t u − ∆u = a − u + u2 v ≥ a − u
and thus by Corollary 4 the solution y of:
∂t y = a − y,
y = inf u0
x∈Ω
satisfy 0 ≤ y ≤ u. This is completely analogous to the application of Pao in
Sec. 2.2. The solution is y = a+exp(−t)(inf x∈Ω u0 −a) thus 0 ≤ min(a, inf x∈Ω u0 ) ≤
u. Let m ≡ min(a, inf x∈Ω u0 ).
For v we have
∂t v − d∆v = b − u2 v ≤ b − m2 v,
3.2 Application of result on local existence to specific models
29
and thus the solution z of
∂t z = b − m2 v,
z(0) = sup v0 ,
x∈Ω
satisfy 0 ≤ v ≤ z. The solution is z(t) = mb2 + e−m
hence:
b
0 ≤ v ≤ max
, sup v0 ,
m2 x∈Ω
2
t
supx∈Ω v0 −
b
m2
and
(3.16)
showing (3.14). For simplicity M ≡ max mb2 , supx∈Ω v0 .
Adding the equation for u and the equation for v gives:
∂t (u + v) − ∆(u + dv) = a + b − u.
Multiplying by u + v and integration over Ω give after integration by parts:
Z
Z h
i
1
2
2
∂t (u + v)2 +
(∇u) + (1 + d)∇u∇v + d (∇v)
2
Ω
Ω
Z
(3.17)
= (a + b − u)(u + v).
Ω
Now since,
1
(∇u + (1 + d)∇v)2 ≥ 0,
2
then by expanding we conclude that,
2
2
(∇u) + (1 + d)∇u∇v + d (∇v) ≥
1
1
2
2
(∇u) − (1 + d2 ) (∇v) .
2
2
(3.18)
Using (3.18) in (3.17) gives the inequality:
Z h
Z
i Z
2(a + b − u)(u + v). (3.19)
(∇u)2 − (1 + d2 ) (∇v)2 ≤
∂t (u + v)2 +
Ω
Ω
Ω
2
We estimate the (∇v) -term by use of the equation for v and (3.16): Multiply
the equation for v with v and integrate over the domain Ω to obtain:
Z
Z
Z
1
2
2
∂t
d (∇v) = (bv − u2 v)
v +
2
Ω
Ω
ZΩ
bv ≤ vol Ω bM,
≤
Ω
30
Existence and uniqueness of two-component reaction-diffusion system
and thus:
2
(1 + d )
Z
2
Ω
(∇v) ≤
Z
1
1 1
+ d vol Ω bM −
+ d ∂t
v2 .
d
2 d
Ω
Insertion into (3.19):
∂t
Z
(u + v)2 +
Ω
1
2
Z
Z
1
1
2
+ d ∂t
+ d vol Ω bM
(∇u) −
v2 +
d
d
Ω
Ω
Z
2(a + b − u)(u + v).
≤
Ω
Setting w = (u + v)2 +
∂t
Z
w+
Z
Ω
Ω
1
2
(∇u)2 ≤
1
d
Z
+ d v 2 we obtain:
2(a + b − u)(u + v) +
Ω
1
+ d vol Ω bM.
d
(3.20)
We now claim the exist E > 0 such that
1
w≤E−
2
Ω
Z
∂t
To prove the claim we add
(∂t + 1)
Z
w+
Ω
Z
Ω
R
Ω
Z
w.
(3.21)
Ω
w to both sides of (3.20):
2
(∇u) ≤
+
Z
1
+ d vol Ω bM
d
Ω
2(a + b − u)(u + v) +
Z
w
Ω
≤ (Expand the terms on the right hand side)
Z
1
1 1
≤
+ d vol Ω bM +
+d +1
v2
d
2 d
Ω
Z
Z
u2 .
2(a + b)(u + v) −
+
Ω
Ω
The second term on the right hand side can be estimated using (3.16):
Z
1 1
1 1
+d +1
+ d + 1 vol Ω M 2
v2 ≤
2 d
2 d
Ω
3.2 Application of result on local existence to specific models
31
+ d vol Ω bM + 21 d1 + d + 1 vol Ω M 2 . Then
Z
Z
Z
2
2(a + b)(u + v) − kuk22
(∇u) ≤ H +
w+
(∂t + 1)
Let H =
1
d
Ω
Ω
Ω
≤ (By Cauchy-Schwarz and Minowski inequality)
≤ H + k2(a + b)k2 (kuk2 + kvk2 ) − kuk22
1
≤ H + 8(a + b)2 vol Ω (kuk2 + kvk2 )
2
≤ (By Young’s inequality)
1
kuk22 + kvk22
≤ H + 32(a + b)4 vol Ω2 +
4Z
1
4
2
w,
≤ H + 32(a + b) vol Ω +
2 Ω
proving the claim, with E = H + 32(a + b)4 vol Ω2 .
By Gronwall’s inequality, see App. C and Lemma 11, we deduce from (3.21)
that
Z
Z
t
2
w ≤ 2E + exp −
kuk2 ≤
w0 − 2E
2
Ω
Ω
Z 1 1
2
+ d v02 .
≤ 2E +
(u0 + v0 ) +
2 d
Ω
R Putting Γ2 = 2E + Ω (u0 + v0 )2 + 21 d1 + d v02 (3.15) follows.
We now have to turn to Lp -theory. It is clear from the definition of f and
the bounds just obtained that,
kF (u(t), v(t))k1 ≤ (a + Γ1 )(vol Ω + Γ2 )
and hence for p = 1 and α < 1 it follows by Corollary 3:
Z t
Cα,1 (t − s)−α (a + Γ1 )(vol Ω + Γ2 )ds
ku(t)kD(Hα ) ≤ Cα,1 t−α ku0 k1 +
0
≤ Cα,1 t
−α
ku0 k1 + Cα,1 (a + Γ1 )(vol Ω + Γ2 )
≤ (using α < 1)
≤ Cα,1 t−α ku0 k1 + Cα,1 (a + Γ1 )(vol Ω + Γ2 )
Z
0
T∗
(t − s)−α ds
T ∗ 1−α
.
1−α
D(Hα ) ⊂ CL0 ∞ (Ω) with continuous injection for α > n/2 for p = 1, thus, for
n = 1 we have:
kuk∞ < ∞.
Therefore by contradiction it is concluded:
32
Existence and uniqueness of two-component reaction-diffusion system
Corollary 5 For n = 1 and every u0 , v0 ∈ CL0 ∞ (Ω) the Schnakenberg system,
(1.4)-(1.6), has a unique, positive, global solution, that is T ∗ = ∞.
2
In order to prove global existence for the Schnakenberg model for n = 2 and
n = 3 we need some additional results, which we shall establish in the following
section. But first let us consider the Thomas-model, that is the system with
reaction functions:
ρrs
ρrs
f (r, s) = a − r −
, g(r, s) = α(b − s) −
.
2
1 + r + kr
1 + r + kr2
The method of upper and lower solutions was insufficient for this model as the
reaction functions were not quasi-monotone. But nevertheless it is possible, for
this model, to utilise Proposition 2 and Proposition 3 to prove, by contradiction,
global existence for space variables n = 1, 2 and 3.
From:
∂t u − ∆u = f (u, v) ≤ a − u,
∂t v − d∆v = g(u, v) ≤ α(b − v),
we conclude that,
0 ≤ u ≤ max(a, supΩ u0 ),
0 ≤ v ≤ max(b, supΩ v0 ).
(3.22)
As the bound is independent of t we therefore conclude that limt→T ∗ − (ku(t)k∞ +
kv(t)k∞ ) < ∞ and therefore by contradiction:
Corollary 6 For every u0 , v0 ∈ CL0 ∞ (Ω) the Thomas system has a global solution, that is T ∗ = ∞. By (3.22) it is furthermore uniformly bounded.
2
3.3
Global existence and boundedness
In this section we shall present and extend the work of [Hollis, Martin, Pierre
(1987)] to give sufficient conditions for the system to be global and bounded.
Let us first state the theorem we aim to prove.
Theorem 2 Suppose in addition to (H1)-(H4) the following hypotheses hold:
(H5) N2 in Proposition 3 is bounded if T ∗ < ∞;
(H6) There is an η ≥ 1 and a continuous function h : [0, ∞)2 → [0, ∞) such
that |f (t, r, s)| ≤ h(t, S)(1 + r)η for all t, r, s ≥ 0 with s ≤ S;
3.3 Global existence and boundedness
33
(H7) There is an ǫ > 0 and a continuous function l : [0, ∞)2 → [0, ∞) such that
ǫr + f (t, r, s) + g(t, r, s) ≤ l(t, S) for all t, r, s ≥ 0 with s ≤ S.
Then the solution exists on Ω × (0, ∞) and (3.9), repeated here for convenience:
0 ≤ u ≤ N1 (t),
0 ≤ v ≤ N2 (t),
for
in
Ω × [0, T ∗),
holds with T ∗ = ∞.
If furthermore N2 , h and l are bounded in t,
N2 (t) ≤ N 2 ,
h(t, S) ≤ h(S),
l(t, S) ≤ l(S),
for all
t ≥ 0,
then there exist N 1 such that N1 (t) ≤ N 1 for all t ∈ [0, ∞) and so the solution
is uniformly bounded in Ω × [0, ∞).
2
Remark 5 Notice that the Schnakenberg model (1.4)-(1.6) satisfy these hypotheses:
(H5) is satisfied by (3.16) in Sec. 3.2, and the following computation,
|f (t, r, s)| = |a − r + r2 s|
≤ (a + r + r2 s)
≤ (a + 1)(1 + r + r2 S)
≤ (a + 1)(S + 1)(1 + r + r2 )
≤ (a + 1)(S + 1)(1 + r)2 ,
shows that we may take h = (a + 1)(S + 1) and thus that (H6) is satisfied.
Simply by inspection u + f (t, r, s) + g(t, r, s) ≤ l = a + b, so (H7) follows.
By Theorem 2, the solution is global and since N2 , h and l are independent
of t, u and v are uniformly bounded.
2
Remark 6 Notice that by a change of variable x̃ = d−1/2 x (1.1) is transformed
into:
˜ = f (t, u, v),
ut − d˜∆u
˜
vt − ∆v = g(t, u, v),
in Ω̃ × (0, T ),
where d˜ = 1/d and hence N2 might as well be replaced with N1 and f with g
in (H5) resp. (H6).
2
Remark 7 Compared to [Hollis, Martin, Pierre (1987)], we had to replace
• (H7′) There is a continuous function l : [0, ∞)2 → [0, ∞) such that
f (t, r, s) + g(t, r, s) ≤ l(t, S) for all t, r, s ≥ 0 with s ≤ S.
34
Existence and uniqueness of two-component reaction-diffusion system
with (H7).
2
Proof To prove the theorem a couple of lemmas are necessary. For the first
statement in the theorem we shall make use of four lemmas. The purpose of
each lemma is described below.
For 0 ≤ δ < T < ∞ we let Lp (Ω×(δ, T )) be the space of measurable functions
φ : Ω × (δ, T ) → R equipped with the Banach-norm:
"Z
#1/p
T
kφkp,δ,T =
δ
kφkpp dt
< ∞.
In the four lemmas we consider the following linear auxiliary problem:
∂t φ = Hǫ φ − θ, in Ω × (δ, T )
νφ = 0, on ∂Ω × (δ, T ),
φ = 0, on Ω × {t = T },
(3.23)
(3.24)
(3.25)
where Hǫ = −∆ + ǫI. By introducing φ(x, t) = φ(x, T − t) and θ(x, t) =
θ(x, T − t):
∂t φ = −Hǫ φ + θ, in Ω × (0, T − δ),
νφ = 0
φ=0
on ∂Ω × (0, T − δ),
on Ω × {t = 0},
it follows by the positivity lemma that φ ≥ 0, and then φ ≥ 0, whenever
θ ≥ 0. Justified by similar arguments to those in Example 1, Hǫ is sectorial
and therefore −Hǫ generates an analytic semigroup G̃ satisfying kG̃(t)φkp ≤
exp(−δt)kφkp by Theorem 7 in App. A.
The first lemma below states that the linear mappings Lp (Ω×(δ, T )) ∋ θ 7→ φ
and Lp (Ω × (δ, T )) ∋ θ 7→ ∆φ are continuous. The second and third lemma
relate φ and θ to the solution (u, v), to conclude that u ∈ Lp (Ω × (0, T ∗ )) when
T ∗ < ∞. Finally, the fourth lemma uses the result in the third lemma and (H5)
to conclude that also F ∈ Lp (Ω × (0, T ∗ )) when T ∗ < ∞.
The four lemmas are then together with the continuous embedding D(Hα ) ֒→
0
CL∞ used to conclude that
lim ku(t)k∞ < ∞,
t→T ∗ −
in turn, implying that the solution is global according to Proposition 3.
Now for the first lemma.
Lemma 2 Suppose that p ∈ (1, ∞), θ ∈ Lp (Ω × (δ, T )) and φ is the solution
to (3.23)-(3.25). Then there is a constant C(p) only depending upon the norm
such that
kφ(δ)kp , kφkp,δ,T , k∆φkp,δ,T ≤ C(p)kθkp,δ,T .
(3.26)
2
3.3 Global existence and boundedness
35
Proof By [Ladyzenskaja et al., Theorem 9.1 p. 341]:
kφkp,δ,T , k∆φkp,δ,T ≤ C(p, T − δ)kθkp,δ,T ,
(3.27)
where C may be chosen so that,
C(p, T − δ) ≤ C(p, T ′ − δ),
if 0 ≤ δ < T ≤ T ′ .
(3.28)
Applying variations of constants we get:
kφ(δ)kp = kφ(T − δ)kp = k
≤
Z
T −δ
Z
0
T −δ
G̃(T − δ − τ )θ(τ )dτ kp
exp(−δ(T − δ − τ ))kθ(τ )kp dτ
0
≤ (using Hölder’s inequality)
!1/q
Z T −δ
≤
exp(−qδ(T − δ − τ )dτ
0
−1/q
≤ (qδ)
Z
0
T −δ
kθ(τ )kpp dτ
!1/p
kθkp,δ,T ,
where the last inequality follows from the estimate:
Z
0
T −δ
exp(−qδ(T − δ − τ )dτ = (qδ)−1 (1 − exp(−qδ(T − δ))) ≤ (qδ)−1 . (3.29)
Again, by variation of constants:
kφkp,δ,T =
≤
Z
T −δ
0
Z
0
T −δ
k
Z
t
0
Z
0
G̃(t − τ )θ(τ )dτ kpp dt
t
exp(−δ(t − τ ))kθ(τ )kp dτ
p
dt
≤ (using Hölder’s inequality and p−1 + q −1 = 1)
1/p #p
1/q Z t
Z T −δ "Z t
1
1
dt
exp −p δ(t − τ ) kθ(τ )kpp
≤
exp −q δ(t − τ )
q
p
0
0
0
#
p/q Z t
Z T −δ "Z t
p
exp(−δ(t − τ ))
≤
exp(−δ(t − τ ))kθ(τ )kp dt.
0
0
0
36
Existence and uniqueness of two-component reaction-diffusion system
Then, by utilising a similar estimate as in (3.29), we obtain:
kφkp,δ,T ≤ δ −p/q
=δ
−p/q
Z
T −δ
0
Z
Z
0
T −δ
0
t
exp(−δ(t − τ ))kθ(τ )kpp dτ dt
kθ(τ )kpp
≤ δ −p/q−1 kθkpp,0,T −δ
Z
τ
T −δ
!
exp(−δ(t − τ ))dt dτ
= (using p−1 + q −1 = 1)
= δ −p kθkpp,δ,T .
(3.30)
For ∆φ, we consider ψ(t) = A(t − t0 )φ(t) for t0 ≥ 0 where A ∈ C0∞ (R) such
that
A = 0 in {0},
(3.31)
0 ≤A ≤ 1 in (0, 1),
A = 1 in [1, 2].
(3.32)
(3.33)
See Fig. 3.1.
1
0
1
2
Figure 3.1: Test function satisfying (3.31)-(3.33).
Then,
∂t ψ = A′ (t − t0 )φ + A(t − t0 )∂t φ = A′ (t − t0 )φ + A(t − t0 )∆φ + A(t − t0 )θ
= ∆ψ + A′ (t − t0 )φ + A(t − t0 )θ,
νψ = 0, on ∂Ω × {t > t0 },
ψ = 0,
in
Ω × {t = t0 }.
in Ω × {t > t0 },
3.3 Global existence and boundedness
37
We shall make use of the inequality:
(a + b)p ≤ 2p (ap + bp ),
a, b ≥ 0,
(3.34)
which follows from the estimate:
(a + b)p ≤ (2 max(a, b))p ≤ 2p max(ap , bp ) ≤ 2p (ap + bp ).
Put C2 (p) = C(p, 2), and set θ = 0 for t > T − δ. Then by (3.27) and the
triangle inequality,
k∆ψkpp,t0 ,t0 +2 ≤ C2 (p)p kθkp,t0 ,t0 +2 + kA′ k∞ kφkp,t0 ,t0 +2
≤ (using (3.34))
p
≤ 2p C2 (p)p kθkpp,t0 ,t0 +2 + kA′ kp∞ kφkpp,t0 ,t0 +2 ,
and by the choice of A:
k∆ψkpp,t0 ,t0 +2 ≥ k∆ψkpp,t0 +1,t0 +2 = k∆φkpp,t0 +1,t0 +2 .
For t0 = 0, . . . , k, . . . we have:
k∆φkpp,1,2 ≤ k∆ψkpp,0,2 ≤ 2p C2 (p)p kθkpp,0,2 + kA′ kp∞ kφkpp,0,2
..
.
k∆φkpp,k+1,k+2 ≤ k∆ψkpp,k,k+2 ≤ 2p C2 (p)p kθkpp,k,k+2 + kA′ kp∞ kφkpp,k,k+2
..
.
Since θ ∈ Lp (Ω × (0, ∞)) we have as a consequence of (3.27) that φ ∈ Lp (Ω ×
(0, ∞)) so we may sum over k to obtain:
k∆φkpp,1,∞ ≤ 21+p C2 (p)p kθkpp,0,∞ + kA′ kp∞ kφkpp,0,∞ .
And thus by (3.28),
k∆φkpp,0,∞ ≤ (21+p + 1)C2 (p)p kθkpp,0,∞ + kA′ kp∞ kφkpp,0,∞ .
Now applying (3.30) we finally obtain:
k∆φkp,δ,T ≤ C(p)kθkp,δ,T ,
1/p
where C(p) = (21+p +1)1/p C2 (p) (1 + kA′ kp∞ δ −p )
the result follows.
. Choosing C(q) = max(C(p), (qδ)−1/p , δ −p )
38
Existence and uniqueness of two-component reaction-diffusion system
Lemma 3 Suppose (H1)-(H4) are satisfied and let 0 ≤ δ < T < T ∗ . Then,
Z
Z
Z
φ(δ)u(δ),
(3.35)
φ(ǫu + f (t, u, v)) +
uθ =
Ω
Ω×(δ,T )
Ω×(δ,T )
Z
Z
Z
vθ ≤
φg(t, u, v) +
φ(δ)v(δ) + E(T ),
(3.36)
Ω×(δ,T )
Ω×(δ,T )
Ω
where
E(δ, T ) = C(|d − 1| + ǫ)N 2 (δ, T )vol Ω1/q (T − δ)1/q kθkp ,
(3.37)
and
N 2 (δ, T ) = sup N2 (t).
δ≤t<T
If T ∗ < ∞, (3.35) and (3.36) also holds for T = T ∗ .
2
Proof First we integrate φ∂t u over Ω to obtain:
Z
φ∂t u = (using equation for u)
Ω
Z
=
(φ∆u + φf (t, u, v))
Ω
= (using Green’s identity)
Z
Z
=
u∆φ +
φf (t, u, v).
Ω
(3.38)
Ω
Furthermore, by integrating φut from t = δ and t = T :
Z T
φ∂t u = (using integration by parts)
δ
= φ(T )u(T ) − φ(δ)u(δ) −
= (using (3.23))
Z
= −φ(δ)u(δ) +
δ
Z
T
u∂t φ
δ
T
u∆φ − ǫ
Z
T
δ
uφ +
Z
T
uθ,
(3.39)
δ
since φ(T ) = 0. (3.38) together with (3.39) yield:
Z
Z
Z
Z
u∆φ
φ(δ)u(δ) +
φf (t, u, v) = −
u∆φ +
Ω×(δ,T )
Ω
Ω×(δ,T )
Ω×(δ,T )
Z
Z
−ǫ
uφ +
uθ,
Ω×(δ,T )
Ω×(δ,T )
3.3 Global existence and boundedness
39
and thus
Z
uθ =
Ω×(δ,T )
Z
φ(δ)u(δ) +
Ω
Z
φ(ǫu + f (t, u, v)).
Ω×(δ,T )
Similarly for v we obtain:
Z
Z
φ(δ)v(δ) +
φg(t, u, v)
Ω
Ω×(δ,T )
Z
Z
vφ.
v∆φ + ǫ
+ (d − 1)
vθ =
Ω×(δ,T )
Z
Ω×(δ,T )
Ω×(δ,T )
Now by assumption:
(d − 1)
Z
Ω×(δ,T )
Z
vφ ≤ |d − 1|N 2 (δ, T )
Ω×(δ,T )
Z
+ ǫN 2 (δ, T )
|φ|
v∆φ + ǫ
Z
Ω×(δ,T )
|∆φ|
Ω×(δ,T )
≤ (using Hölder’s inequality)
≤ |d − 1|N 2 (δ, T )vol Ω1/q (T − δ)1/q k∆φkp,δ,T
+ ǫN 2 (δ, T )vol Ω1/q (T − δ)1/q kφkp
≤ (using (3.27))
≤ C(p)(|d − 1| + ǫ)N 2 (δ, T )vol Ω1/q (T − δ)1/q kθkp,δ,T ,
and the result follows.
In the following let C̃(p, N 2 (δ, T )) = C(p)(|d − 1| + ǫ)N 2 (δ, T )vol Ω1/q .
Lemma 4 Suppose 0 ≤ δ < T < T ∗ or ≤ T ∗ if T ∗ < ∞ and p ∈ (1, ∞). Then
u ∈ Lp (Ω × (δ, T )).
2
40
Existence and uniqueness of two-component reaction-diffusion system
Proof Adding (3.35) and (3.36) gives:
Z
Z
Z
vθ
uθ +
uθ ≤
Ω×(δ,T )
Ω×(δ,T )
Ω×(δ,T )
Z
Z
φ(δ)(u(δ) + v(δ)) +
≤
φ(ǫ + f (t, u, v) + g(t, u, v)) + E(δ, T )
Ω×(δ,T )
Ω
≤ (using (H7))
Z
Z
≤
φ(δ)(u(δ) + v(δ)) +
Ω
φl(t, N 2 (δ, T )) + E(δ, T )
Ω×(δ,T )
≤ (using Hölder’s inequality and (3.37))
≤ ku(δ)kp kφ(δ)kq + N2 (s)vol Ω1/p kφ(δ)kq
!1/p
Z
p
kφkq,δ,T + C̃(q, N 2 (δ, T ))(T − δ)1/p kθkq,δ,T
+
l(t, N 2 (δ, T ))
Ω×(δ,T )
≤ (using (3.26))
≤
ku(δ)kp C(q) + N 2 (δ, T )vol Ω1/p C(q)
+ l(t, N 2 (δ, T ))vol Ω
1/p
1/p
(T − δ)
1/p
C(q) + C̃(q, N 2 (δ, T ))(T − δ)
where l(t, N 2 (δ, T ) = supδ≤t<T l(t, N 2 (δ, T ). Therefore
Z
uθ ≤ Ĉ(p, q, N 2 (δ, T )) 1 + ku(δ)kp + (T − δ)1/p kθkq,δ,T ,
!
kθkq,δ,T ,
(3.40)
Ω×(δ,T )
where
(
Ĉ(p, q, N 2 (δ, T )) = max C(q), N 2 (δ, T )vol Ω1/p C(q),
l(t, N 2 (δ, T ))vol Ω
1/p
)
C(q) + C̃(q, N 2 (δ, T )) .
(3.41)
Since this estimate holds for every θ ∈ Lq (Ω × (δ, T )), θ ≥ 0 we conclude by
duality that u ∈ Lp (Ω × (δ, T )), u ≥ 0 and
(3.42)
Ĉ(p, q, N 2 (δ, T )) 1 + ku(δ)kp + (T − δ)1/p
is an estimate for kukp,0,T .
Remark 8 Notice that Ĉ is independent of δ and T if N 2 (δ, T ) ≤ N 2 , as will
be the case in the last part of the theorem.
2
3.3 Global existence and boundedness
41
The following lemma combines the previous lemma with the hypotheses (H6).
Lemma 5 Suppose the suppositions of Lemma 4 hold and that p > 1 and η is
as in (H6). Then there is a constant h(T, N 2 (T )) such that
Z
δ
T
kF (t, u(t), v(t))kpp dt ≤ h(T − δ, N 2 (δ, T )) 1 + kukηp
ηp,δ,T ,
whenever 0 ≤ δ < T < T ∗ or ≤ T ∗ if T ∗ < ∞.
(3.43)
2
Proof By (H6) we have:
|f (t, u, v)|p ≤ h(T, N 2 (T ))p (1 + |u(t)|)
≤ (using (3.34))
ηp
≤ 2pη h(T, N 2 (T ))p (1 + |u(t)|pη ) ,
and integration over Ω × (δ, T ):
kF (t, u, v)kpp,δ,T ≤ 2pη h(T − δ, N 2 (δ, T ))p (T − δ)vol Ω + kukηp
ηp,δ,T
pη
p
≤ 2 h(T − δ, N 2 (δ, T )) max(1, (T − δ)vol Ω) 1 + kukηp
ηp,δ,T .
Put h(T − δ, N 2 (δ, T )) = 2pη h(T − δ, N 2 (δ, T ))p max(1, (T − δ)vol Ω) and the
result follows.
Now we are ready to prove global existence. Suppose by contradiction that
T ∗ < ∞ and thus by Proposition 3:
lim ku(t)k∞ = ∞.
t→T ∗ −
(3.44)
Select α ∈ (0, 1) and p > 1 such that n/(2p) < α and αq < 1 where q is the
p
→ 1 from above as
conjugated number of p. Clearly such p exist since q = p−1
p → ∞.
Since T ∗ < ∞ we have by Lemma 4:
kukpp,0,T ∗ ≤ C1 (p)p ,
and then by Lemma 5 there is an C2 such that
kF (t, u, v)kpp,0,T ∗ ≤ C2 (p)p (1 + C1 (pη)pη ).
(3.45)
42
Existence and uniqueness of two-component reaction-diffusion system
Using the properties in Corollary 3, variation of constants, (3.4), gives:
Z t
−α
Cα,p (t − τ )−α kF (τ, u(τ, v(τ ))kp dτ
ku(t)kD(Hα ) ≤ Cα,p t ku0 kp +
0
≤ (using Hölder’s inequality)
Z t
1/q
−α
−αq
≤ Cα,p t ku0 kp + Cα,p
(t − τ )
0
×
Z
0
t
kF (τ, u(τ ), v(τ ))kpp
≤ Cα,p t
−α
ku0 kp + Cα,p
1/p
Z
T∗
∗
−αq
!1/q
∗
−αq
!1/q
(T − τ )
0
× kF(t, u, v)kp,0,T ∗
≤ (using (3.45))
≤ Cα,p t
−α
ku0 kp + Cα,p
Z
0
T∗
(T − τ )
× C2 (p)(1 + C1 (pη))1/p
Cα,p
T ∗ 1/q−α
≤ Cα,p t−α ku0 kp +
(1 − αq)1/q
× C2 (p)(1 + C1 (pη))1/p ,
(3.46)
since αq < 1, for all t ∈ (0, T ∗ ). By the continuous injection I : D(Hα ) → CL0 ∞
limt→T ∗ − ku(t)k∞ < ∞, contradicting (3.44), and T ∗ = ∞.
For the second statement in Theorem 2, regarding the boundedness of the
solutions we shall make use of two lemmas. The lemmas construct a sequence
{tn }n∈N , tn → ∞ as n → ∞, such that a bound on kF (t, u(t), v(t))kp,tn ,tn+1 can
be established independently of n. Then the proof is completed by exploiting
the continuous injection I : D(Hα ) → CL0 ∞ .
When N2 (t) ≤ N 2 for all t ≥ 0, then it follows by Remark 8, that Ĉ in
Lemma 4, (3.41), may be chosen independent of δ and T , so that it only depends on the norms. In the following we shall for convenience write Ĉ instead
of Ĉ(p, q) and set Υ = Ĉ p (Ĉ + 2)p .
~
Lemma 6 For each p ∈ (1, ∞) there are constants C(p)
and C̆(p) and a sequence {tn }n∈N such that
~
1◦ 1 < tn+1 − tn ≤ C(p),
2◦ ku(tn )kp ≤ Ĉ + 1,
3.3 Global existence and boundedness
3◦
R tn+1
tn
43
ku(t)kpp dt ≤ C̆(p).
2
Proof We may choose Ĉ so that Ĉ + 1 ≥ ku0 kp and Ĉ ≥ 1. First we claim:
If τ ≥ 0 and ku(τ )kp ≤ Ĉ + 1 then there is
τ̃ ∈ (τ, τ + Υ) such that ku(τ̃ )kp ≤ Ĉ + 1.
(3.47)
The proof is by contradiction. So we assume that ku(t)kp > Ĉ(p) + 1 for all
t ∈ (τ, τ + Υ) and so
Z τ +Υ
ku(t)kpp dt > Υ(Ĉ + 1)p .
τ
However, by (3.42) we obtain:
Z τ +Υ
ku(t)kpp dt ≤ Ĉ p (1 + ku(τ )kp + Υ1/p )p
τ
= Ĉ p (Ĉ + 2 + Ĉ(Ĉ + 2))p
= Ĉ p (Ĉ + 2)p (Ĉ + 1)p
= Υ(Ĉ + 1)p ,
and we have arrived at a contradiction.
~
We set C(p)
= 2Υ and define the sequence {tn }n∈N by t1 = 0 and
tn+1 = sup{tn ≤ t ≤ tn + 2Υ | ku(t)kp ≤ Ĉ + 1},
n ∈ N.
tn+1 is bounded from above by tn + 2Υ. Additionally, the set
Σn = {tn ≤ t ≤ tn + 2Υ | ku(t)kp ≤ Ĉ + 1}
is nonempty for every n ∈ N, since t 7→ ku(t)kp is continuous and by assumption
ku(t1 = 0)k ≤ Ĉ + 1. Therefore, the supremum exists and the sequence is welldefined.
It is left to be shown that the sequence satisfies the lower bound in 1◦ . We
claim that tn+1 − tn ≥ Υ > 1. Again by contradiction, if tn+1 − tn < Υ
then by (3.47) there is a t̃ ∈ (tn+1 , tn+1 + Υ) such that ku(t̃)kp ≤ Ĉ + 1. But
tn ≤ t̃ ≤ tn+1 + Υ ≤ tn + 2Υ and so t̃ ∈ Σn with t̃ > tn+1 contradicting that
tn+1 = sup Σn . Thus we conclude that the sequence satisfies 1◦ .
By (3.42) and
Z tn+1
ku(t)kpp dt ≤ Ĉ p (1 + ku(tn )k + (tn+1 − tn )1/p )p
tn
≤ Ĉ p (1 + Ĉ + 1 + (2Υ)1/p )p
≤ Ĉ p (Ĉ + 2 + (2Υ)1/p )p ,
we see that (3) holds with C̆ = Ĉ p (Ĉ + 2 + (2Υ)1/p )p , independent of n.
44
Existence and uniqueness of two-component reaction-diffusion system
Lemma 7 Suppose that N2 , h and l are bounded in t, p ∈ (1, ∞) and η is as
in (H6), and that {tn }n∈N is the sequence from Lemma 6 corresponding to ηp.
Then there is a constant Ċ(p, ηp), independent of n, so that,
Z tn+1
kF (t, u(t), v(t))kpp ≤ Ċ(p, ηp), for all n ∈ N.
2
tn
Proof The proof is similar to the proof of Lemma 5. The boundedness on N2 ,
h and l applied to (3.43) imply the existence of C̊(p) independent of t and n,
such that
Z tn+1
Z tn+1
ηp
p
ku(t)kηp dt ,
kF (t, u(t), v(t))kp dt ≤ C̊(p) 1 +
tn
tn
and hence by Lemma 6:
Z tn+1
kF (t, u(t), v(t))kpp dt ≤ C̊(p) 1 + C̆(ηp) .
tn
Putting Ċ(p, ηp) = C̊(p) 1 + C̆(ηp) the result follows.
We are now ready to prove the final statement in Theorem 2 about global
boundedness of the solution.
Again choose α ∈ (0, 1) and p > 1 such that n/(2p) < α and αq < 1 and let
{tn }n∈N be the sequence from Lemma 6 corresponding to ηp. Replacing u0 by
u(tn+1 ) we obtain:
ku(t)kD(Hα ) ≤ Cα,p (t − tn+1 )−α ku(tn+1 )kp
Z t
Cα,p (t − τ )−α kF (τ, u(τ ), v(τ ))kp dτ.
+
(3.48)
tn+1
for t ∈ (tn+2 , tn+3 ) and n ∈ N. Now since t − tn+1 ≥ 1 then (t − tn+1 )−α ≤ 1
and then by Lemma 6:
Cα,p (t − tn+1 )−α ku(tn+1 )kp ≤ Cα,p ku(tn+1 )kp
≤ (using Hölder’s inequality)
≤ Cα,p vol Ω ku(tn+1 )kηp
≤ Cα,p vol Ω (Ĉ(ηp) + 1).
Put R = Cα,p vol Ω (Ĉ(ηp) + 1).
For the second term in (3.48) we apply similar reasoning as in (3.46) to
obtain:
Z t
Cα,p (t − τ )−α kF (τ, u(τ ), v(τ ))kp dτ ≤ QĊ(p, ηp)1/p ,
tn+1
3.3 Global existence and boundedness
45
for some constant Q. Thus for t ∈ [tn+2 , tn+3 ], n ∈ N then
ku(t)kD(Hα ) ≤ R + QĊ(p, ηp)1/p .
Since this estimate is independent of n, and by the continuous injection I :
D(Hα ) → CL0 ∞ , we conclude, in turn, that there exist N 1 such that
u(t) ≤ N 1 ,
for all t ≥ 0, and the proof is done.
We have given sufficient conditions for (3.1)-(3.3) to satisfy (W1), (W2) and
(W4), (W5). It is left to be shown that the solution depends continuously on
the initial data. This is shown in the following theorem.
Theorem 3 The solution in Proposition 2 depends continuously on the initial
data.
2
Proof Let (u, v) and (ũ, ṽ) be the solutions corresponding to initial data (u0 , v0 )
resp. (ũ0 , ṽ0 ) in CL0 ∞ . We shall show that ũ, ṽ → u, v whenever ũ0 , ṽ0 → u0 , v0
where both limits are taken in CL0 ∞ .
As before, we choose α ∈ (0, 1) and p > 1 such that n/(2p) < α and αq < 1.
Readily from (3.4), (3.6) and 2◦ in Corollary 3:
ku(t) − ũ(t)kD(Hα ) ≤ Cα,p t−α ku0 − ũ0 kp
Z t
+
Cα,p (t − τ )−α L11 ku(τ ) − ũ(τ )kp + L12 kv(τ ) − ṽ(τ )kp dτ,
0
kv(t) − ṽ(t)k
D(Hα )
≤ Cα,p t−α kv0 − ṽ0 kp
Z t
+
Cα,p (t − τ )−α L21 ku(τ ) − ũ(τ )kp + L22 kv(τ ) − ṽ(τ )kp dτ,
0
for every t ∈ [δ, T ] where 0 < δ < T < ∞. And then by Hölder’s inequality and
(3.34):
Z t
p/q
p
−αp
p
1
1
−αq
p p
ku0 − ũ0 kp + Λ1 (p, L1 , L2 )
(t − τ )
dτ
ku(t) − ũ(t)kD(Hα ) ≤ 2 Cα,p δ
0
×
Z
0
t
ku(τ ) − ũ(τ )kpp + kv(τ ) − ṽ(τ )kpp dτ
!
p
≤ 2p Cα,p
δ −αp ku0 − ũ0 kpp + Λ̃1 (p, L11 , L12 , T )
!
Z t
p
p
ku(τ ) − ũ(τ )kD(Hα ) + kv(τ ) − ṽ(τ )kD(Hα ) dτ ,
×
0
(3.49)
46
Existence and uniqueness of two-component reaction-diffusion system
and similarly for v,
p
kv(t) − ṽ(t)kpD(Hα ) ≤ 2p Cα,p
δ −αp kv0 − ṽ0 kpp + Λ̃2 (p, L11 , L12 )
×
Z t
0
ku(τ ) −
ũ(τ )kpD(Hα )
+ kv(τ ) −
ṽ(τ )kpD(Hα )
!
dτ ,
(3.50)
for some positive constants Λ1 , Λ2 , Λ̃1 and Λ̃2 . By adding (3.49) and (3.50) we
obtain:
p
ku(t) − ũ(t)kpD(Hα ) + kv(t) − ṽ(t)kpD(Hα ) ≤ 2p Cα,p
δ −αp ku0 − ũ0 kpp + kv0 − ṽ0 kpp
p
Λ̃1 (p, L11 , L12 ) + Λ̃2 (p, L11 , L12 )
+ 2p Cα,p
Z t
×
ku(τ ) − ũ(τ )kpD(Hα ) + kv(τ ) − ṽ(τ )kpD(Hα ) dτ.
0
p
Λ1 (p, L11 , L12 ) + Λ2 (p, L11 , L12 ) , then
Now put Π = 2 Cα,p δ
and χ = 2p Cα,p
by Gronwall’s inequality, see App. C and Lemma 12:
p
−α
ku(t) − ũ(t)kpD(Hα ) + kv(t) − ṽ(t)kpD(Hα )
≤ Π ku0 − ũ0 kpp + kv0 − ṽ0 kpp exp (tχ)
→0
as ũ0 , ṽ0 → u0 , v0
in
Lp ,
for every t ∈ [δ, T ]. Since δ and T were chosen arbitrarily this shows the continuous dependency of the initial data.
By Remark 2 we therefore have:
Corollary 7 The solution of the Schnakenberg and Thomas system and the
Gierer and Meinhardt system depend continuously on the non-negative resp.
positive initial conditions.
2
3.4
Summary and remarks
In this chapter, the general two component reaction-diffusion system was considered on a smooth and bounded domain.3 First, we showed that the solution,
3 In fact, by referring to Grisvard, [Grisvard (1985)], the results in this chapter may be
extended to piecewise smooth, convex domain; for example a cube in R3 .
3.4 Summary and remarks
47
when it exists and is classical, is positive whenever f (r, s) ≥ 0 and g(r, s) ≥ 0
2
for all r, s ∈ R+ .
Next, we gave sufficient conditions for the solution to exist and to be continuous. Under these conditions, it was then shown that if the solution only
existed on a finite time-interval, it diverged in CL0 ∞ . This result was used to
prove global existence and boundedness for the Schnakenberg for n = 1 and for
the Thomas model for n = 1, 2 and 3.
Then, we gave sufficient conditions for the solution to be global and uniformly bounded. The reaction functions of the Schnakenberg model was shown
to satisfy these conditions, and therefore the solution of the Schnakenberg system was shown to exist globally and be uniformly bounded.
Finally, it was shown that under the conditions given, the solution depends
continuously on the initial data.
To sum up, sufficient conditions were given for the general reaction-diffusion
model to be well-posed in the sense of Hadamard, (W1)-(W3), and to satisfy
the “practical requirements” (W4) and (W5) in Chap. 1. In turn, this implied
that (W1)-(W5) were satisfied for all three models.
As a result of the analysis in this chapter, we can perform a linear analysis.
This is done in next chapter, where we study the stability of uniform steadystates of the autonomous reaction-diffusion model, and give sufficient conditions
for the system to exhibit diffusion driven instabilities.
48
Existence and uniqueness of two-component reaction-diffusion system
Chapter
4
Linear analysis
Alan Turing (1952) was the first to formulate an explanation of how the
patterns of animals like leopards, jaguars and zebras develop. Turing asserted
that the patterns can arise as a result of instabilities in the diffusion of morphogenetic chemicals in the animals’ skins during the embryonic stage of development.1 Mathematically he studied reaction-diffusion models and his idea
was that, if in the absence of diffusion, the solution tended to a linear stable
uniform state state then under certain conditions spatially inhomogeneous patterns could evolve by diffusion driven instability.
In this chapter we derive the conditions for the general autonomous reactiondiffusion system (1.1)-(1.3) to exhibit diffusion driven instability, also called
Turing instability. We assume that all the hypothesis of Chap. 3 are satisfied
such that the solution (u, v) exists globally and is uniformly bounded for initial
data in CL0 ∞ (Ω).
First we need some definitions and results on stability.
1 Morphogenesis
is the part of embryology that is concerned with pattern and form.
50
Linear analysis
4.1
Stability
In the following let (u∗ , v ∗ ) be a steady-state solution of the autonomous
equation (1.1).
Definition 2 (u∗ , v ∗ ) is stable if, for any ǫ > 0, there exist δ > 0 such that
any solution (u, v) with ku(0) − u∗kD(Hα ) , kv(0) − v ∗ kD(Hα ) < δ satisfies ku(t) −
u∗ kD(Hα ) , kv(t) − v ∗ kD(Hα ) < ǫ for all t > 0.
(u∗ , v ∗ ) is asymptotically stable if it is stable and
k(u(t) − u∗ , v(t) − v ∗ )kD(Hα )2 → 0.
(u∗ , v ∗ ) is unstable if it is not stable.
Now suppose:
f (u∗ + w1 , v ∗ + w2 )
w1
+ Q(w1 , w2 ),
=
A
w2
g(u∗ + w1 , v ∗ + w2 )
2
(4.1)
where A is a matrix in R2×2 , and
kQ(w1 ) − Q(w2 )kL2 ×L2 ≤ h(ρ)kw1 − w2 kD(Hα )2 ,
(4.2)
for kw1 kD(Hα )2 , kw2 kD(Hα )2 ≤ ρ,
for some function h : R+ → R+ , continuous in 0 with h(0) = 0. By subtracting
Aw on both sides we obtain an equation for the perturbed solution w,
∂t w + Lw = Q(w),
(4.3)
where L = −D∆ − A.
If the spectrum of L lies in the right half-plane, or equivalently if the linearisation,
∂t w − D∆w = Aw, ,
(4.4)
is asymptotically stable, then we say that (u∗ , v ∗ ) is linearly stable. On the
other hand if the intersection σ(L) ∩ {λ ∈ C | Re λ < 0} is non-empty, then we
say that (u∗ , v ∗ ) is linearly unstable.
The following theorem is essential in the analysis to come.
Theorem 4 We have the following implications:
(1) (u∗ , v ∗ ) is linearly stable ⇒ (u∗ , v ∗ ) is asymptotically stable;
(2) (u∗ , v ∗ ) is linearly unstable ⇒ (u∗ , v ∗ ) is unstable.
2
4.1 Stability
51
Proof (1): L is sectorial and therefore generates an analytical semigroup G.
Let η > 0 be such that Re λ > η whenever λ ∈ σ(L).
By Lemma 10 in App. A there exist Φ ≥ 1 such that
kG(t)wkD(Hα )2 ≤ Φ exp(−ηt)kwkD(Hα )2 , Φt−α exp(−ηt)kwkL2 ×L2 .
Now by (4.2) we choose ̺ > 0 and ρ > 0 so that
Z ∞
1
̺Φ
ξ −α exp(−(η − η ′ )ξ)dξ < ,
2
0
2
2
α
2
kQ(w)kL ×L ≤ ̺kwkD(H ) , for kwkD(Hα )2 ≤ ρ,
where 0 < η ′ < η.
ρ
Let kw(0)kD(Hα )2 ≤ 2Φ
. Then by continuity of the solution kw(t)kD(Hα )2 ≤
ρ on some time interval and therefore by variation of constants:
kw(t)kD(Hα )2 ≤ Φ exp(−ηt)kw(0)kD(Hα )2
Z t
(t − τ )−α exp(−η(t − τ ))kw(τ )kD(Hα )2 dτ
+ ̺Φ
0
ρ
≤ + ρ̺Φ
2
Z
t
0
(t − τ )−α exp(−η(t − τ ))dτ < ρ.
(4.5)
Again by continuity, either kw(t)kD(Hα )2 < ρ for all t > 0 or kw(t)kD(Hα )2 = ρ
for some finite t. The second case contradicts the sharp inequality in (4.5), and
thus kw(t)kD(Hα )2 < ρ for all t > 0.
Now put
Θ(t) = sup kw(ξ)kD(Hα )2 exp(η ′ ξ), t ≥ 0.
0≤ξ≤t
By (4.5):
kw(ξ)kD(Hα )2 exp(η ′ ξ) ≤ Φ exp(−(η − η ′ )t)kw(0)kD(Hα )2
Z ξ
+ Φ̺
(ξ − τ )−α exp(−(η − η ′ )(ξ − τ ))dτ × Θ(ξ)
0
≤ (using (4.2))
ρ 1
≤ + Θ(t),
2 2
for all 0 ≤ ξ ≤ t and thus
Θ(t) ≤
ρ 1
+ Θ(t),
2 2
or
Θ(t) ≤ ρ.
52
Linear analysis
Hence
kw(t)kD(Hα )2 ≤ ρ exp(−η ′ t)
showing that (u∗ , v ∗ ) is asymptotically stable.
(n) (n)
(2): We shall show that there exist ǫ > 0 and a sequence {(u0 , v0 )}n∈N ∈
(n) (n)
L2 × L2 such that k(u0 , v0 ) − (u∗ , v ∗ )kD(Hα )2 → 0 as n → ∞, but
sup k(un , vn ) − (u∗ , v ∗ )kD(Hα )2 ≥ ǫ > 0,
t≥0
∀n ∈ N,
(n)
(n)
where un and vn are the solutions corresponding to the initial data u0 , v0 .
Let us assume that 0 ∈
/ σ(L). If 0 ∈ σ(L) we may apply the reasoning
below by perturbing L. Therefore by assumption, there exist β > 0 such that
the spectrum is disjoint from the ball in C with centre 0 and radius 2β. See
Fig. 4.1.
r ≤ 2β
λ1
...
λJ
λJ+1
...
R
Figure 4.1: Eigenvalues of the operator L. The eigenvalues are assumed to be
real.
Let σ1 = σ(L) ∩ {λ ∈ C | Re λ < 0} and σ2 = σ(L)\σ1 . Notice that σ1 is a
set of finite numbers in C.
By construction L may be diagonalised such that
Lw =
∞
X
n=1
λn (w, wn )L2 ×L2 wn .
We set
X1 = span{wn | n = 1, . . . , J},
and
L2 ×L2
X2 = span{wn | n ∈ N\{1, . . . , J}}
,
such that
L2 × L2 = X1 ⊕ X2 .
The projection operators onto X1 and X2 is denoted P1 resp. P2 .
Now, let
Li restriction of L to Xi , i = 1, 2,
4.1 Stability
53
then L1 is finite dimensional, and therefore bounded and generates an analytical
semigroup G1 (t), so that there exist χ1 and χ2 , such that for t ≤ 0:
kG1 (t)wkD(Hα )2 ≤ χ1 exp(2βt)kP1 wk2 , χ2 exp(2βt)kP1 wkD(Hα )2 .
Similarly, L2 is sectorial and therefore by Lemma 10:
kG2 (t)wkD(Hα )2 ≤ χ3 exp(−βt)kP2 wk2 , χ3 t−α exp(−βt)kP2 wkD(Hα )2 ,
for t > 0.
Let χ = max{χ1 , χ2 , χ3 , 1}. Now we claim that Ξ given by the expression:
Ξ(t) = G1 (t − ξ)σ +
+
Z
t
−∞
Z
t
ξ
G2 (t − τ )P1 Q(Ξ(τ ))
G2 (t − τ )P2 Q(Ξ(τ )),
(4.6)
solves (4.3) for σ ∈ X1 and t < ξ.
To show the assertion we define the operator T by
TΞ(t) = G1 (t − ξ)σ +
+
Z
t
−∞
Z
ξ
t
G2 (t − τ )P1 Q(Ξ(τ ))dτ
G2 (t − τ )P2 Q(Ξ(τ ))dτ,
t ≤ ξ.
By similar arguments to the proof of Proposition 2, it is shown that T maps
Bρ (0) =
(
w∈
2
CL0 ∞ ((0, ∞); D(Hα ))
)
| kwkD(Hα )2 ≤ ρ ,
where ρ > 0 is such that
Z ∞
1
1 −1
−α
,
s exp(−βs)ds ≤
χk(ρ) kP1 k β + kP2 k
2
4χ
0
(4.7)
into itself. The following computation, then shows that T is a contraction on
54
Linear analysis
Bρ (0):
kTΨ(t) − TΞ(t)kD(Hα )2
Z
t
≤
χk(ρ) exp(2β(t − τ ))kP1 kkΨ(τ ) − Ξ(τ )kD(Hα )2 dτ ξ
Z t
χk(ρ) exp(−β(t − τ ))(t − τ )−α kP2 k
+
−∞
× kΨ(τ ) − Ξ(τ )kD(Hα )2 dτ
Z
t
≤ χk(ρ)kP1 k
exp(2β(t − τ ))dτ ξ
Z t
+ χk(ρ)kP2 k|
(t − τ )−α exp(−β(t − τ ))dτ |
−∞
× sup kΨ(t) − Ξ(t)kD(Hα )2
t≤ξ
Z ∞
1
ξ −α exp(−βξ)dξ
≤ χk(ρ) kP1 k β −1 + kP2 k
2
0
× sup kΨ(t) − Ξ(t)kD(Hα )2
t≤ξ
<
1
× sup kΨ(t) − Ξ(t)kD(Hα )2 ,
2 t≤ξ
by choice of ρ, and for Ψ, Ξ ∈ Bρ (0). Hence, by Banach’s Fixed Point Theorem
there is a unique fix point in Bρ (0).
To show that Ξ solves (4.3), we consider the projections of Ξ onto X1 and
X2 . First,
P1 Ξ(t) = G1 (t − ξ)σ +
Z
ξ
t
G1 (t − τ )P1 Q(Ξ(τ ))dτ
= G1 (t)G1 (−s)σ + G1 (t)
+
Z
t
Z
0
ξ
G1 (−ξ)P1 Q(Ξ(τ ))dτ
G1 (t − τ )P1 Q(Ξ(τ ))dτ,
0
for 0 ≤ t ≤ ξ,
and similarly
P2 Ξ(t) = G2 (t)
+
Z
0
t
Z
0
−∞
G2 (t − τ )P2 Q(Ξ(τ ))dτ
G2 (t − τ )P2 Q(Ξ(τ ))dτ,
for 0 ≤ t ≤ ξ.
4.1 Stability
55
Hence,
Ξ(t) = P1 Ξ(t) + P2 Ξ(t)
= G(t) G1 (−ξ)σ +
Z
ξ
+
Z
Z
G1 (−s)P1 Q(Ξ(τ ))dτ
0
−∞
+
0
G2 (t − τ )P2 Q(Ξ(τ ))dτ
!
t
G(t − τ )Q(Ξ(τ ))dτ
Z t
= G(t)Ξ(0) +
G(t − τ )Q(Ξ(τ ))dτ,
0
0
for 0 ≤ t ≤ ξ,
where Gw = G1 P1 w + G2 P2 w. By referring to (3.4) and the discussion in
Chap. 3 we conclude that Ξ solves (4.3) with
w(0) = Ξ(0).
Next, we show that kΞ(t)kD(Hα )2 ≤ 2χkσkD(Hα)2 exp(2β(t − ξ)) for t ≤ ξ.
By (4.6):
kΞ(t)kD(Hα )2 ≤ χ exp(2β(t − ξ))kσkD(Hα )2
Z ∞
1
−α
+ χk(ρ) kP1 k
ξ exp(−βξ)dξ
+ kP2 k
2β
0
× sup kΞ(τ )kD(Hα )2 ,
0≤s≤ξ
∀t ≤ ξ,
but then
sup kΞ(τ )kD(Hα )2 ≤ χ exp(2β(t − ξ))kσkD(Hα )2 +
0≤s≤ξ
1
sup kΞ(τ )kD(Hα )2 .
2 0≤s≤ξ
Therefore,
kΞ(t)kD(Hα )2 ≤ sup kΞ(τ )kD(Hα )2 ≤ 2χ exp(2β(t − ξ))kσkD(Hα )2 ,
0≤s≤ξ
∀t ≤ ξ.
(4.8)
56
Linear analysis
D(Ãα )
Θ(t)
Θ(s) − σ
0
r = 1/2kσk
D(Ãα )2
σ r = 1/2kσk
D(Ãα )
D(Ãα )2
Figure 4.2: By (4.9) we may conclude that kΞ(ξ)kD(Hα )2 ≥ 12 kσkD(Hα )2 .
Now we are ready for the final estimate. By (4.6) we obtain:
kΞ(ξ) − σkD(Hα )2 ≤
Z
ξ
−∞
χ(ξ − s)−α exp(−β(ξ − s))kP2 k
× 2χ exp(2β(ξ − s))kσkD(Hα )2 dτ
Z ξ
= 2χ2 kP2 kkσkD(Hα )2
(ξ − s)−α exp(−3β(t − τ ))dτ
≤ (using (4.7))
1
≤ kσkD(Hα )2 .
2
−∞
(4.9)
But then, kΞ(ξ)kD(Hα )2 ≥ 12 kσkD(Hα )2 , see e.g. Fig. 4.2.
The theorem now follows, since if we choose kσkD(Hα )2 ≤
by (4.8),
kw(0)kD(Hα )2 = kΞ(0)kD(Hα )2 ≤ ρ exp(−2βn) → 0,
ρ
2χ
for
and ξ = n then
n → ∞,
while
sup kw(t)kD(Hα )2 ≥ kw(n)kD(Hα )2 ≥
0≤t≤n
and therefore (u∗ , v ∗ ) is unstable.
1
kσkD(Hα )2 ,
2
∀n ∈ N
Linear stability is therefore a sufficient condition for stability of the nonlinear
system. This shall be utilised in the following section to present sufficient conditions for the general autonomous system to exhibit diffusion driven instability.
4.2 Conditions for diffusion driven instability
4.2
57
Conditions for diffusion driven instability
In the analysis to come, it turns out that the size of the domain Ω is an
important parameter. By introducing the scaling γ and x = γ 1/2 x̃ and t = γ t̃,
the system is transformed into:
˜ = γf (u, v)
∂t̃ u − ∆u
˜ = γg(u, v) in Ω̃ × (0, ∞),
∂t̃ v − d∆v
u
ν̃
= 0, on ∂ Ω̃ × [0, ∞),
v
u = u0 ,
v = v0 , on Ω̃ × {t = 0} ,
where Ω̃ = x ∈ Rn | γ −1/2 x ∈ Ω , and by varying γ we can therefore study the
influence of the domain size on the solutions. For convenience the tilde-notation
is omitted in the following.
We assume that there exists a uniform steady-state (u∗ , v ∗ ) ∈ R2+ , i.e. so
that f (u∗ , v ∗ ) = 0 = g(u∗ , v ∗ ). For notational simplicity superscript ∗ is used
when f and g or their partial derivatives are evaluated at (u∗ , v ∗ ). For example
f ∗ = f (u∗ , v ∗ ).
By Taylor expansion:
∂r f ∗ ∂s f ∗
A=
.
(4.10)
∂r g ∗ ∂s g ∗
Let us define what we precisely mean by diffusion driven instability.
Definition 3 The autonomous reaction-diffusion model has a diffusion-driven
instability at (u∗ , v ∗ ) if the following two conditions are satisfied:
(1) (u∗ , v ∗ ) is asymptotically stable in the absence of diffusion;
(2) (u∗ , v ∗ ) is unstable in the presence of diffusion.
2
First, the sufficient conditions for (1) to hold is determined.
In the absence of spatial dependency (4.4) simplifies to
∂t w = γAw,
(4.11)
where
w=
u − u∗
v − v∗
.
58
Linear analysis
The ordinary differential equation (4.11) is asymptotically stable if the eigenvalues of γA have negative real part. The eigenvalues are given by the characteristic
equation,
λ2 − γtr A λ + γ 2 det A = 0,
with solutions:
λ± =
Re λ± < 0 if and only if
1/2 i
1 h
.
γ tr A ± tr A2 − 4det A
2
tr A = ∂r f ∗ + ∂s g ∗ < 0,
det A = ∂r f ∗ ∂s g ∗ − ∂s f ∗ ∂r g ∗ > 0.
(4.12)
Therefore by Theorem 4, (1) in Definition 3 is satisfied if (4.12) holds.
For (2) in Definition 3, we consider the linear system (4.4) and determine
the conditions for linear instability and then refer to Theorem 4.
It is well-known by preliminary partial differential equation theory that the
system (4.4) can be solved by the method of product solutions. We write:
w(x, t) =
∞
X
cki exp (λki t) Xki (x)
(4.13)
i=1
where {cki }i∈N depend upon the initial conditions and {Xki }i∈N are the eigenfunctions of the Laplacian:
−∆Xki = ki2 Xki in Ω,
νXki = 0 on ∂Ω.
(4.14)
We shall call {ki }i∈N the wavenumbers. (u∗ , v ∗ ) is linearly unstable if and only
if Re λki > 0 for some i ∈ N.
It is well-known and indeed verified in Example 1 that the ki ’s are real, and
they may be arranged such that 0 ≤ k12 ≤ · · · ≤ kJ2 ≤ . . . , and ki2 → ∞ for
i → ∞.
Inserting cki exp (λki t) Xki (x) into (4.13) we obtain an equation relating λki
and ki2 :
λki I − γA + ki2 D Xki = 0, ∀i ∈ N.
To yield non-trivial solutions the coefficient matrix must be singular, that is,
det λki I − γA + ki2 D = 0.
The determinant is evaluated, and written as follows:
−γ∂r g ∗
λki − γ∂r f ∗ + ki2
det
−γ∂s f ∗
λki − γ∂s g ∗ + dki2
= λ2ki + ki2 (1 + d) − γ (∂r f ∗ + ∂s g ∗ ) λki + h(ki2 )
4.2 Conditions for diffusion driven instability
59
where
h(ki2 ) = dki4 + γ 2 det A − γ (d∂r f ∗ + ∂s g ∗ ) ki2 .
(4.15)
The solutions of (4.2) are:
where
1/2
,
2λki = −B(ki2 ) ± B(ki2 )2 − 4h(ki2 )
(4.16)
B(ki2 ) = ki2 (1 + d) − γtr A > 0,
(4.17)
by (4.12). Let us consider λ+ : R+ → C defined by (4.16):
1/2 1
.
−B(ki2 ) + B(ki2 )2 − 4h(ki2 )
λ+ (k 2 ) =
2
Now, Re λ+ (k 2 ) > 0 for some k 2 > 0 if and only if h(k 2 ) < 0. In fact, at
bifurcation the simple eigenvalue λ+ crosses the imaginary axis in 0. Hence
Hopf’s Bifurcation Theorem is not applicable, and so we do not expect there to
exist periodic solutions.2
Next, we determine the conditions for h(k 2 ) < 0 for some k 2 > 0. Since h
is continuous and h → +∞ whenever k 2 → ∞, h attains a minimum on [0, ∞).
Elementary differentiation of h(k 2 ) with respect to k 2 shows that,
"
#
2
(d∂r f ∗ + ∂s g ∗ )
2
2
2
min h(k ) = h(k̃ ) = γ det A −
,
(4.18)
R+
4d
where
k̃ 2 =
γ (d∂r f ∗ + ∂s g ∗ )
.
2d
(4.19)
Thus h(k 2 ) < 0 for some k 2 > 0 if and only if the following conditions are
satisfied:
d∂r f ∗ + ∂s g ∗ > 0
2
(d∂r f ∗ + ∂s g ∗ )
> det A.
4d
The first condition is also apparent from (4.15): Cf. the requirement that
det A > 0, h(k 2 ) can only attain negative values for some positive parameters if
d∂r f ∗ + ∂s g ∗ > 0.
From (4.12) we have tr A = ∂r f ∗ + ∂s g ∗ < 0, and therefore d∂r f ∗ + ∂s g ∗ > 0
for d > 0 implies that ∂r f ∗ > 0 while ∂s g ∗ < 0 and d > 1. Furthermore, by the
condition det A > 0 it follows that ∂s f ∗ and ∂r g ∗ must be of opposite signs.
Now, observe that by (4.19) and (4.18), we can choose γ such that λ+ (kJ2 ) >
0, some J ∈ N. Therefore by Theorem 4 we obtain:
2 This
is not precise mathematically, we shall, however, not study this further in this thesis.
60
Linear analysis
Theorem 5 If the following conditions are satisfied:
(T1) tr A < 0;
(T2) det A > 0;
(T3) d∂r f ∗ + ∂s g ∗ > 0;
(T4)
(d∂r f ∗ +∂s g∗ )2
4d
> det A;
then, there exists a positive γc , such that for γ = γc , λ+ (k12 ) = 0. For γ < γc ,
(u∗ , v ∗ ) is stable, while for γ > γc , the autonomous reaction-diffusion system
exhibits diffusion-driven instabilities at (u∗ , v ∗ ) when there exist wavenumbers
kJ2 , J ∈ N, in the range (m2 , M 2 ), where m2 and M 2 are the zeros of h(k 2 ) = 0:
h
1/2 i 
γ

m2 = 2d
(d∂r f ∗ + ∂s g ∗ ) − (d∂r f ∗ + ∂s g ∗ )2 − 4d det A


(4.20)
i
h
1/2 

γ
∗
∗
∗
∗ 2
2

2
(d∂r f + ∂s g ) + (d∂r f + ∂s g ) − 4d det A
M =
2d
In the last section of this chapter we shall give a qualitative discription of
the result of Theorem 5.
The set of parameter values satisfying (T1)-(T4), are called the Turing domain and denoted T .
In the numerical analysis we shall, for the Schnakenberg system, make explicit use of the Turing domain given by the inequalities (T1)-(T4).
As mentioned in Chap. 2, the Schnakenberg system exhibits one single positive, uniform steady-state given by
u∗ = a + b,
v∗ =
b
2,
(a + b)
b > 0,
a + b > 0.
The partial derivatives of f and g at this steady-states are
b−a
, ∂s f ∗ = (a + b)2 ,
a+b
−2b
∂r g ∗ =
, ∂s g ∗ = −(a + b)2 .
a+b
∂r f ∗ =
Inserting these into (T1)-(T4) we obtain:
0 < b − a < (a + b)3 ,
(4.21)
(a + b) > 0,
(4.22)
d(b − a) > (a + b)3 ,
(4.23)
2
3 2
4
[d(b − a) − (a + b) ] > 4d(a + b) .
(4.24)
4.3 Summary and remarks
61
Clearly (4.22) is redundant. Moreover the lower bound in (4.21) is redundant
by the presence of (4.23) and we can therefore settle with
(a + b)3 > b − a,
3
1/2
d(b − a) − (a + b) > 2d
(4.25)
2
(a + b) .
(4.26)
Or simply:
d(b − a) > (by (4.26))
> 2d1/2 (a + b)2 + (a + b)3
> (by (4.25))
> 2d1/2 (a + b)2 + b − a.
Remark 9 Obviously, (4.21) is not satisfied for ab < 41 . Therefore, for ab < 41
the unique solution (u∗ , v ∗ ) in the sector h(u, v), (u, v)i, see Chap. 2, is asymptotically stable both in absence and presence of diffusion.
2
Now, let a resp. a be the real and non-negative solutions of the cubic equations:
d(b − a) = 2d1/2 (a + b)2 + (a + b)3 ,
3
(a + b) = b − a.
(4.27)
(4.28)
It is not hard to see that given d > 1 and b > 0, then the following inclusion
holds:
Sb,d = (a, b, d) ∈ R2+ × (1, ∞) | a < a < a ⊂ T .
Of course Sb,d may be empty for some d > 1 and b > 0.
The expressions for the solutions of (4.27) and (4.28) are quite messy:
2/3
√
− d 2√
d3/2 + 27db + 3 6d5/2 b + 81d2 b2
−
a(b, d) =
d − b,
(4.29)
1/3
√
3
3/2
5/2
2
2
3 d + 27db + 3 6d b + 81d b
√
2/3
−3
27b + 3 3 + 81b2
(4.30)
a(b) =
√
1/3 − b.
3 27b + 3 3 + 81b2
4.3
Summary and remarks
In this chapter the linear stability analysis was shown to give sufficient conditions for the stability of the full nonlinear systems. This result was used
62
Linear analysis
to analyse the diffusion-driven instabilities of an autonomous reaction-diffusion
system, and the results were stated in a theorem, Theorem 5. These conditions
were used to determine relevant parameters for the Schnakenberg model.
By Theorem 5, it followed that by changing γ it is possible to change the
stability of the uniform steady-state. Therefore, γ is a bifurcation parameter.
For γ < γc , (u∗ , v ∗ ) is stable, however, for γ > γc , (u∗ , v ∗ ) may be unstable by
diffusion.
Now, we shall give a qualitative description of the result presented in Theorem 5. To do so, let us assume that for γ > γc , diffusion driven instabilities
eventually evolves into a spatially inhomogeneous steady-state solution. Then,
if the domain were to present some embryo domain and u and v represented
some morphogenetic concentrations, we would expect the kinetics and diffusion
coefficients to be fixed, meaning that γ would be the natural variable parameter reflecting the size of the embryo. Theorem 5, together with the assumption
above then asserts that for sufficient large domains, random perturbations evolve
into stable, spatial patterns. On the other hand, for sufficient small domain the
uniform steady-state is asymptotically stable for small random perturbations,
and no pattern evolves.
This is in agreement with for example mammals such as zebras, leopards or
giraffes, where the formation of coat patterns does not take place until the embryo has reached a certain size. See e.g. [Murray (2003), Chapter 3]. It is also
in accordance with the fact that small mammals are often uniform in colour;
rats, hedgehogs and so on.
The pattern of large one-coloured mammals such as elephants, can be the
result of a very fine scale pattern, so fine that essentially no patterns can be
seen. See e.g. [Murray (2003), Chapter 3, p.152].
In fact, reaction-diffusion equations have been applied from a patterning
point of view to a larger number of ecology situations, and it has been possible
to obtain similar patterns to those seen in nature, see e.g. zebra and giraffe coat
patterns in [Murray (2003), Chapter 3, Figure 3.4 and 3.5].3
3 Diffusion-driven instabilities have not only found interest in biology and ecology but also
in economics. See e.g. http://www.soc.uoc.gr/ecosud/docs/Zurich%20presentation
/Zurich%20presentation.ppt
Chapter
5
Numerical analysis
In this chapter the results from implementing the Schnakenberg system in
MATLAB in the form:
∂t u − ∆u = γ(a − u + u2 v)
in Ω × (0, ∞),
∂t v − d∆v = γ(b − u2 v)
u
ν
= 0, on ∂Ω × [0, ∞),
v
u = u0 ,
v = v0 , on Ω × {t = 0} ,
are presented.
We consider the model on the rectangle Ω = (0, 1) × 0, π2 . The rectangle
is chosen since it is a simple geometry, and it is easy to handle numerically.
Furthermore, it is advantageous that the wavenumbers are distinguishable from
eachother on this domain. On a square domain the wavenumbers are not simple.
On Ω the eigensolutions are:
Xkn,m = X cos (nπx) cos (2my) ,
with corresponding wavenumbers:
p
kn,m = (nπ)2 + (2m)2 ,
X ∈ R2
n, m = 1, . . . , N, . . . .
Chebyshev interpolation is used to approximate the Laplacian and the integration (in time) is performed by built-in MATLAB ode-solver ode15s.
64
Numerical analysis
The developed code randomly perturbs the uniform steady-state solution,
(u∗ , v ∗ ), and solve the system for the perturbed variables u − u∗ and v − v ∗ until
the solution converge, measured by a tolerance, to a steady-state.1
The implementation was verified by considering the linearised system (4.4)
for initial conditions (u0 , v0 ) = Xkn,m , and comparing the numerical solution
with the analytical solution given by:
u
= c1 V1 exp(−σ1 t) cos(nπx) + c2 V2 exp(−σ2 t) cos(2mx),
v
where (σi , Vi ), i = 1, 2, are the eigensolutions of the algebraic eigenvalue problem:
2
−kn,m
D + γA Vi = σi Vi , i = 1, 2,
D = diag (1, d) and A is as in (4.10). The constants c1 and c2 are given by the
linear system:
c1
1
V1 V2
=
.
1
c2
The numerical solution converged spectrally to the analytical solutions as the
net was refined.
Furthermore, the code evaluated the right hand side of the equation:
Z f (u, v)
.
0=
g(u, v)
Ω
which follows by integration of the stationary equations for u and v, when
a numerical steady-state solution was obtained for the full nonlinear system,
and the equation was observed to be satisfied down to a sufficient tolerance
(< 10−4 max(u∗ , v ∗ )).
A convergence analysis of the numerical solution to the full nonlinear system,
showed that a grid of 21 × 33 = 693 mesh points was adequate.
5.1
Results in Matlab
In the numerical analysis we studied the behaviour of the system for six
different (a, b, d)-values in the Turing domain. The six points in the Turing
domain are seen in Fig. 5.1.
1 In practice, a tolerance is not specified. Instead, the code integrates from 0 to a large
value T ≈ 106 . When |∂t u| ≤ ǫ, then the built-in ode15s solver takes steps with size of order
O(ǫn ), where n > 1. So in principal, we analyse the convergence by looking at the t-values: If
the solver extrapolates using small stepsizes, then the solution has not converged; if the solver
extrapolates with a large stepsize, for example from t = 1 to t = 106 , then we conclude that
the solution has converged.
5.1 Results in Matlab
65
0.7
Lower bound
Upper bound for d =10
Upper bound for d =20
Upper bound for d =40
Upper bound for d =60
Upper bound for d =80
0.6
0.5
a
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
b
2.5
3
3.5
4
Figure 5.1: The parameters where diffusion driven stability occur, is, for fixed
d > 1, the set confined by the the upper and the lower bound. Six points are
shown. The red × is for d = 10, the black is for d = 80, while the green ◦’s
are for d = 20.
66
Numerical analysis
In Sec. 5.1.1 we analyse the two points (a, b, d) = (0.05, 1, 20) and (a, b, d) =
(0.05, 1, 80). In Sec. 5.1.2 we consider the four remaining points (a, b, d) =
(0.05, 1.5, 20), (0.1, 1.5, 20), (0.2, 0.5, 20) and (0.2, 1, 20).
5.1.1
Results for d = 10 and d = 80
In Fig. 5.2, Fig. 5.3 and Fig. 5.4 three type of developments of u are seen
for a = 0.05, b = 1, d = 10 and γ = 120. Please note, that throughout it is the
perturbed u and v that is visualised, i.e. u − u∗ resp. v − v ∗ .
As expected the solution remains bounded and a spatial pattern develops.
Furthermore, the perturbed u and v are larger than −u∗ resp. −v, so u and v
are positive, as stated in Proposition 1.
The final patterns in the three figures are different, so for these parameter
values we have found multiple steady-states. What pattern, that ultimately
develops, depends on the initial conditions.
Also for v a final steady-state pattern develops. The v-pattern is similar to
the u-pattern, but where u has a local maximum and minimum, v has a local
minimum resp. maximum.
The idea of Turing described in the previous chapter was that, u > u∗ would
give rise to one colour, while u < u∗ would give rise to another colour. Therefore,
the three patterns would imply stripes, spots resp. stripes.
5.1 Results in Matlab
67
*
(u−u*)(0)
u−u for t= 0.086461
−3
x 10
8
−3
x 10
6
1.5
1.5
6
4
4
2
2
1
1
0
y
y
0
−2
−2
−4
0.5
0.5
−4
−6
−8
0
0
−6
−10
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
x
*
u−u* for t= 0.48595
(u−u )(t=∞)
0.5
0.4
1.5
1.5
0.4
0.3
0.3
0.2
0.2
1
0.1
1
0.1
0
−0.1
y
y
0
−0.1
−0.2
−0.2
0.5
0.5
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
0
0
0.2
0.4
0.6
x
0.8
1
0
0
0.2
0.4
0.6
0.8
1
−0.6
x
Figure 5.2: u at four different times for a = 0.05, b = 1, d = 10 and γ = 120. A
spatial pattern develops. The final pattern is stripes. The v-pattern is similar in
shape, but where u has a local maximum and minimum, v has a local minimum
resp. maximum.
68
Numerical analysis
*
(u−u*)(0)
u−u for t= 0.16821
−3
x 10
6
0.015
1.5
1.5
4
0.01
2
1
0
y
y
−2
0.005
1
0
−4
−0.005
0.5
0.5
−6
−0.01
−8
0
0
−10
0.2
0.4
0.6
0.8
0
0
1
−0.015
0.2
0.4
x
0.6
0.8
1
x
*
u−u* for t= 0.85823
(u−u )(t=∞)
0.8
0.8
1.5
1.5
0.6
0.6
0.4
0.4
1
1
0.2
y
y
0.2
0
0.5
−0.2
0
0.5
−0.2
−0.4
0
0
−0.6
0.2
0.4
0.6
x
0.8
1
−0.4
0
0
−0.6
0.2
0.4
0.6
0.8
1
x
Figure 5.3: u at four different times for a = 0.05, b = 1, d = 10 and γ = 120. A
spatial pattern develops. The final pattern is spots. The v-pattern is similar in
shape, but where u has a local maximum and minimum, v has a local minimum
resp. maximum.
5.1 Results in Matlab
69
*
(u−u*)(0)
u−u for t= 0.10286
−3
x 10
6
0.01
1.5
1.5
0.008
4
0.006
0.004
2
1
0.002
1
0
y
y
0
−2
−0.002
−0.004
−4
−0.006
0.5
0.5
−6
−0.008
−0.01
−8
−0.012
0
0
−10
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
x
*
u−u* for t= 0.50813
(u−u )(t=∞)
0.6
0.6
1.5
1.5
0.4
0.4
0.2
y
0
−0.2
0.5
0.2
1
0
y
1
−0.2
0.5
−0.4
−0.4
0
0
0.2
0.4
0.6
x
0.8
1
0
0
−0.6
0.2
0.4
0.6
0.8
1
x
Figure 5.4: u at four different times for a = 0.05, b = 1, d = 10 and γ = 120. A
spatial pattern develops. The final pattern is stripes. The v-pattern is similar in
shape, but where u has a local maximum and minimum, v has a local minimum
resp. maximum.
70
Numerical analysis
Fig. 5.5 shows a pattern, similar to the pattern in Fig. 5.2. The difference is
that local maximum and minimum in Fig. 5.2 are local minimum resp. maximum
in Fig. 5.5. It was also possible to obtain similar alternatives for the patterns
in Fig. 5.3 and Fig. 5.4.
(u−u*)(t=∞)
(v−v*)(t=∞)
0.6
0.2
1.5
1.5
0.15
0.4
0.1
0.2
1
0.05
1
−0.2
0.5
y
y
0
0
−0.05
−0.1
0.5
−0.15
−0.4
0
0
0.2
0.4
0.6
x
0.8
1
−0.2
0
0
−0.25
0.2
0.4
0.6
0.8
1
x
Figure 5.5: Spatial patterns for u and v, obtained for a = 0.05, b = 1, d = 10
and γ = 120. The pattern in u is similar to the pattern seen in Fig. 5.2, but local
maximum and minimum in Fig. 5.2, are now local minimum resp. maximum.
So here v is similar to u in Fig. 5.2.
5.1 Results in Matlab
71
In Fig. 5.6 the graph of Re λ+ as a function of k is seen for a = 0.05, b = 1,
d = 10 and γ = 120. The wavenumbers kn,m , are indicated with ∗’s on the
abscissa. Text-arrows describe the unstable wavenumbers.
20
k
k
10
2,0
2,1
k1,3
0
−10
k
Re λ+
−20
1,2
k
k
0,3
k
2,2
0,4
−30
−40
−50
−60
−70
−80
0
2
4
6
k
8
10
12
Figure 5.6: The graph of Re λ+ as a function of k for a = 0.05, b = 1, d = 10
and γ = 120. The wavenumbers kn,m are indicated with ∗’s on the abscissa.
72
Numerical analysis
It is not impossible to imagine, that there would exist six different steadystate patterns, each with a correspondency to the six unstable eigenfunctions.
If such steady-states would exist, then it would be likely that
(u0 , v0 ) = (ǫu , ǫv ) cos(πnx) cos(2my),
(5.1)
where 0 < ǫu ≪ u∗ and 0 < ǫv ≪ v ∗ , would be on the stable manifold of the
corresponding steady-state solution. Therefore, for the first ∗ in Fig. 5.6, we
let (u0 , v0 ) be as in (5.1). The solution corresponding to this initial condition
converged to the steady-state solution in Fig. 5.2.
Using initial conditions corresponding to the following three ∗’s in Fig. 5.6,
k0,3 , k2,0 and k2,1 , the solutions converged to the steady-states in Fig. 5.4,
Fig. 5.3 resp. Fig. 5.2.
For the fourth ∗, k1,3 , the solution converged to the solution in Fig. 5.2,
while for the fifth and sixth ∗, k2,2 resp. k0,4 , the solutions converged to the
steady-state solution shown as a surface-plot in Fig. 5.7.
*
*
(u−u )(t=∞)
(v−v )(t=∞)
1
0.4
0.2
v−v*
u−u*
0.5
0
0
−0.2
−0.4
2
−0.5
2
1.5
1
1
0.5
0.5
y
0
0
x
1.5
1
1
0.5
0.5
y
0
0
x
Figure 5.7: u and v pattern for a = 0.05, b = 1, d = 10 and γ = 120.
5.1 Results in Matlab
73
In Fig. 5.8 six different spatial patterns are seen for a = 0.05, b = 1, d = 80
and γ = 120. Compared to the previous patterns for d = 10 it is obvious, that
the increase in d is accompanied with a higher level of spatial aggregation in
the stationary patterns. The steady-state solutions are shown to involve large
peaks where the perturbed u is ≈ 5, such that the concentration is about six
times larger than u∗ = 1.05, and areas where the concentration is constant and
less than u∗ .
The corresponding v-patterns are seen in Fig. 5.9. The v-patterns are also
more spatial aggregated compared to d = 20, and again local maximum and
minimum for u is local minimum resp. maximum for v.
74
Numerical analysis
(u−u*)(t=∞)
(u−u*)(t=∞)
5
1.5
5
1.5
4.5
4.5
4
4
3.5
3.5
3
3
1
2.5
2.5
2
2
y
y
1
1.5
1.5
1
0.5
1
0.5
0.5
0.5
0
0
−0.5
−0.5
0
0
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
x
*
*
(u−u )(t=∞)
(u−u )(t=∞)
5
1.5
5
1.5
4.5
4.5
4
4
3.5
3.5
3
3
1
2.5
2.5
2
2
y
y
1
1.5
1.5
1
0.5
1
0.5
0.5
0.5
0
0
−0.5
0
0
0.2
0.4
0.6
0.8
−0.5
0
0
1
0.2
0.4
0.6
x
x
(u−u*)(t=∞)
(u−u*)(t=∞)
0.8
1
5
1.5
5
1.5
4.5
4.5
4
4
3.5
3.5
3
3
1
2.5
2.5
2
2
y
y
1
1.5
1
0.5
1.5
1
0.5
0.5
0.5
0
0
−0.5
0
0
0.2
0.4
0.6
x
0.8
1
−0.5
0
0
0.2
0.4
0.6
0.8
1
x
Figure 5.8: Six different u-patterns for a = 0.05, b = 1, d = 80 and γ = 120.
5.1 Results in Matlab
75
(v−v*)(t=∞)
(v−v*)(t=∞)
−0.25
1.5
1.5
−0.2
−0.3
−0.35
−0.3
1
−0.4
1
−0.45
y
y
−0.4
−0.5
−0.5
0.5
0.5
−0.55
−0.6
−0.6
−0.65
0
0
−0.7
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
x
*
*
(v−v )(t=∞)
(v−v )(t=∞)
−0.3
−0.3
1.5
1.5
−0.35
−0.35
−0.4
1
−0.4
1
−0.5
−0.45
y
y
−0.45
−0.5
−0.55
−0.55
0.5
0.5
−0.6
−0.6
−0.65
0
0
0.2
0.4
0.6
0.8
−0.65
0
0
1
0.2
0.4
x
0.6
0.8
1
x
(v−v*)(t=∞)
(v−v*)(t=∞)
−0.25
−0.3
1.5
1.5
−0.3
−0.35
−0.35
−0.4
−0.4
1
1
−0.45
−0.5
y
y
−0.45
−0.5
−0.55
0.5
−0.55
0.5
−0.6
−0.6
−0.65
−0.65
0
0
−0.7
0.2
0.4
0.6
x
0.8
1
0
0
0.2
0.4
0.6
0.8
1
−0.7
x
Figure 5.9: v-patterns for a = 0.05, b = 1, d = 80 and γ = 120 corresponding
to the u-patterns in Fig. 5.8.
76
Numerical analysis
In Fig. 5.10 we consider the bifurcation of the steady-states as γ is varied.
The bifurcation functional, is the Sobolev-norm H 1 of the perturbed steadystates, i.e. ku − u∗ kH 1 and kv − v ∗ kH 1 .
In the topmost figure the unstable wavenumbers are made visible. The blue
∗’s indicate the wavenumber with largest corresponding λ+ , while the ▽’s and
the ’s indicate when the corresponding wavenumber changes stability; from
stable to unstable resp. unstable to stable. In the two buttom plots the H 1 norm of the perturbed steady-states are seen, u − u∗ resp. v − v ∗ .
In the the two buttom plots several branches are seen, and in the region
where the wavenumber with largest corresponding λ+ changes, indicated by the
blue ∗’s in the topmost figure, it is seen that a single γ-value corresponds to
two or more values of the H 1 -norm. The branches correspond to different type
of solutions, and therefore in areas where we have two or more values of the
H 1 -norm there exists multiple steady-states solutions. This in accordance to
the plots above.
It is also seen, that γc = 6.5 which is in accordance to the linear analysis.
5.1 Results in Matlab
77
10
0
0
4
20
40
60
80
100
120
140
20
40
60
80
100
120
140
20
40
60
80
100
120
140
2
0
0
1.5
1
0.5
1
H −norm of v−v
*
H1−norm of u−u*
k
5
0
γ
Figure 5.10: The bifurcation of steady-states for a = 0.05, b = 1 and d = 10.
The topmost figure shows when the wavenumbers bifurcate. The topmost figure
shows when the wavenumbers bifurcate. The blue ∗’s indicate the wavenumber
with largest corresponding λ+ , while the ▽’s and the ’s indicate when the corresponding wavenumber changes stability; from stable to unstable resp. unstable
to stable. In the two buttom figures H 1 -norms of the perturbed steady-states
are shown for different γ-values. It is made clear that multiple, stable, steadystates occur near the areas where the blue ∗’s changes from one wavenumber to
another.
78
Numerical analysis
In Fig. 5.11 the similar bifurcation diagram is seen for d = 80. Again,
multiple steady-state solutions are seen, corresponding to different branches.
However, the branches are closer together, and more difficult to separate than
for d = 10. This is due to the long width of the blue ∗’s in the topmost figure,
showing the wavenumber with largest corresponding λ+ -value. At least for low
γ-values, this results in an increased propability, compared to d = 10 where
these blue ∗’s branches were shorter, of a specific type of solutions. Thus we
see long branches in both of the two lower plots. We also note that the length
between the ▽’s and the ’s in the topmost plot is larger for d = 80 than for
d = 10.
The H 1 -norms of u − u∗ for d = 80 is in general larger than those for d = 10.
As seen, this was accompanied by a higher level of spatial aggregation in the
patterns for d = 80. For v − v ∗ the norms are comparable, but the H 1 -norm of
v − v ∗ for d = 10 is bit larger than that for d = 80.
Moreover, γc = 4.5, again in accordance to the linear analysis.
5.1 Results in Matlab
79
5
0
0
15
20
40
60
80
100
120
140
20
40
60
80
100
120
140
20
40
60
80
100
120
140
10
5
0
0
1
0.5
1
H −norm of v−v
*
H1−norm of u−u*
k
10
0
γ
Figure 5.11: The bifurcation of steady-states for a = 0.05, b = 1 and d = 10. The
topmost figure shows when the wavenumbers bifurcate. The blue ∗’s indicate the
wavenumber with largest corresponding λ+ , while the ▽’s and the ’s indicate
when the corresponding wavenumber changes stability; from stable to unstable
resp. unstable to stable. In the two buttom figures H 1 -norms of the perturbed
steady-states are shown for different γ-values. It is made clear that multiple,
stable, steady-states occur near the areas where the blue ∗’s changes from one
wavenumber to another. The H 1 -norms of u − u∗ are in general larger than the
values in Fig. 5.10, while the norms for v − v ∗ are a bit smaller.
80
5.1.2
Numerical analysis
Results for d = 20
In this section we consider the four different points in the Turing domain,
(a, b, d) = (0.05, 1.5, 20), (0.1, 1.5, 20), (0.2, 0.5, 20) and (0.2, 1, 20).
In Fig. 5.12, Fig. 5.13, Fig. 5.14 and Fig. 5.15 the corresponding bifurcation
diagrams are seen. To compare the size of the H 1 -norm for the different a, b
and d-values, we have introduced the relative H 1 -norms, Hu1 and Hv1 , given by
1
ku − u∗ kH 1 ,
u∗
resp.
1
kv − v ∗ kHv1 ≡ ∗ kv − v ∗ kH 1 .
v
ku − u∗ kHu1 ≡
In each of the bifurcation diagrams multiple branches are seen. They correspond
to multiple, steady-state solutions.
In Fig. 5.12, Fig. 5.13, and Fig. 5.15 the branches are easily distinguishable,
while in Fig. 5.14 the branches are long and closer together. We also note that
the length between the ▽’s and the ’s is largest for a = 0.2 and b = 0.5.
In the bifurcation diagrams we can read off the γc -values. They all agree
with the linear analysis.
5.1 Results in Matlab
81
10
k
5
H1u−norm
0
3
40
60
80
100
120
140
160
20
40
60
80
100
120
140
160
20
40
60
80
γ
100
120
140
160
2
1
0
1
H1v −norm
20
0.5
0
Figure 5.12: The bifurcation of steady-states for a = 0.05, b = 1.5 and d = 20.
It is shown that multiple, stable, steady-states occur near the areas where the
blue ∗’s changes from one wavenumber to another.
82
Numerical analysis
10
k
5
20
40
60
80
100
120
140
160
20
40
60
80
100
120
140
160
20
40
60
80
γ
100
120
140
160
1
1
Hu−norm
0
2
0.2
v
H1−norm
0
0.4
0
Figure 5.13: The bifurcation of steady-states for a = 0.1, b = 1.5 and d = 20.
It is shown that multiple, stable, steady-states occur near the areas where the
blue ∗’s changes from one wavenumber to another. There exist γ-values greather
than γc for which the relative H 1 -norms vanish.
5.1 Results in Matlab
83
10
k
5
H1u−norm
0
6
40
60
80
100
120
140
160
20
40
60
80
100
120
140
160
20
40
60
80
γ
100
120
140
160
4
2
0
1
H1v −norm
20
0.5
0
Figure 5.14: The bifurcation of steady-states for a = 0.2, b = 0.5 and d = 20.
It is shown that multiple, stable, steady-states occur near the areas where the
blue ∗’s changes from one wavenumber to another. The Hu1 and the length
between the ▽’s and the ’s are seen to be larger than for the other parameters
considered for d = 20, see Fig. 5.12, Fig. 5.13 and Fig. 5.15.
84
Numerical analysis
k
10
5
0
1
Hu−norm
3
40
60
80
100
120
140
160
20
40
60
80
100
120
140
160
20
40
60
80
γ
100
120
140
160
2
1
0
1
0.5
v
H1−norm
20
0
Figure 5.15: The bifurcation of steady-states for a = 0.2, b = 1 and d = 20. It is
shown that multiple, stable, steady-states occur near the areas where the blue
∗’s changes from one wavenumber to another.
5.1 Results in Matlab
85
In Fig. 5.13 it is observed, that there exists γ-values greather than γc such
that no wavenumber are linearly unstable. In Fig. 5.16 the graph of λ+ is seen
as a function of γ for the first two wavenumbers, k0,1 = 2 and k1,0 = π. For
γ ≈ 15 to 20 no wavenumbers are linearly unstable. Therefore, in the bifurcation
diagram Fig. 5.13, the relative H 1 -norms for these γ-values are zero and (u∗ , v ∗ )
is stable.
2
0
Re λ+
−2
−4
k0,1 = 2
−6
k
1,0
=π
−8
−10
0
5
10
15
γ
20
25
30
Figure 5.16: The graph of Re λ+ as a function of γ for (a, b) = (0.1, 1.5) and
for the first two wavenumbers, k0,1 = 2 and k1,0 = π. For γ ≈ 15 to 20 no
wavenumbers are linearly unstable.
86
Numerical analysis
In Fig. 5.17 contour-plots of steady-state u-patterns for the four parameter
values are seen for γ = 72. The patterns for (a, b) = (0.05, 1.5) and (0.1, 1.5)
are comparable to each other and look like the eigenfunction cos(πx) cos(4y).
Equivalently, the pattern for (a, b) = (0.2, 1) looks like the eigenfunction cos(4y).
The steady-state solution for (a, b) = (0.2, 0.5), where the Hu1 -norm of u − u∗
is largest, does not compare to any eigenfunction and compared to the other
solutions it is more spatial aggregated.
(u−u*)(t=∞)
(u−u*)(t=∞)
0.8
0.4
1.5
1.5
0.6
0.3
0.4
0.2
0.1
0.2
1
1
0
y
y
0
−0.1
−0.2
−0.2
−0.4
0.5
0.5
−0.3
−0.6
−0.4
−0.8
0
0
0.2
0.4
0.6
0.8
−0.5
0
0
1
0.2
0.4
x
0.6
0.8
1
x
(a) a = 0.05, b = 1.5, d = 20, γ = 72
(b) a = 0.1, b = 1.5, d = 20, γ = 72
(u−u*)(t=∞)
(u−u*)(t=∞)
1.4
0.4
1.5
1.5
1.2
0.3
1
0.2
0.8
1
0.1
1
0
y
y
0.6
0.4
0.2
0.5
−0.1
0.5
−0.2
0
−0.3
−0.2
−0.4
0
0
0.2
0.4
0.6
0.8
1
x
(c) a = 0.2, b = 0.5, d = 20, γ = 72
0
0
0.2
0.4
0.6
0.8
1
x
(d) a = 0.2, b = 1, d = 20, γ = 72
Figure 5.17: u-pattern at four different points in the Turing domain. There is
larger level of spatial aggregation for a = 0.2, b = 0.5.
5.1 Results in Matlab
87
Fig. 5.18 shows patterns for (a, b, d) = (0.2, 0.5, 20) and γ = 120. Again,
the patterns are spatial aggregated and involve large gradients. The patterns
resemble the patterns in Fig. 5.8 for d = 80 and a = 0.05 and b = 1.
(u−u*)(t=∞)
(u−u*)(t=∞)
1.2
1.4
1.5
1.5
1.2
1
1
0.8
0.8
1
1
0.6
y
y
0.6
0.4
0.4
0.2
0.2
0.5
0.5
0
0
−0.2
−0.2
0
0
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
x
(u−u*)(t=∞)
1.4
1.5
1.2
1
0.8
1
y
0.6
0.4
0.2
0.5
0
−0.2
0
0
0.2
0.4
0.6
0.8
1
x
Figure 5.18: u-pattern for (a, b, d) = (0.2, 0.5, 20) and γ = 120. The patterns
are spatial aggregated and involve large gradients.
88
Numerical analysis
We observed above that the length between the ▽’s and the ’s in the
topmost plot of the bifurcation diagrams was larger for (a, b, d) = (0.2, 0.5, 20)
and (a, b, d) = (0.05, 1, 80) compared to the other parameters studied. This was
accompanied by a larger Hu1 -norm and a larger level of spatial aggregation in
the stationary patterns.
The length between the ▽’s and the ’s refer to the length in γ for which
the corresponding wavenumber is unstable. This length is given by
∆γk2 : R3 ⊃ T → R+
(2)
(1)
∆γk2 (a, b, d) = γk2 (a, b, d) − γk2 (a, b, d) > 0,
(2)
(1)
where γk2 (a, b, d) > γk2 (a, b, d), for fixed k 2 , are the zeros of:
λ+ (γ) =
1/2 1
−B(γ) + B(γ)2 − 4h(γ)
,
2
and B(γ) and h(γ) are given by the right hand side of (4.17) resp. (4.15). We
obtain:
(2)
(1)
γk2 (a, b, d), γk2 (a, b, d) =
1/2 k2 (d∂r f ∗ + ∂s g ∗ ) ± (d∂r f ∗ + ∂s g ∗ )2 − 4d det A
2 det A
k2
b−a
2
=
− (a + b)
d
2(a + b)2
b+a
(
)1/2 
2
b−a
,
±
d
− (a + b)2 − 4d(a + b)2
(5.2)
b+a
and then
∆γk2 (a, b, d) = ϕ(a, b, d)k 2 ,
where
1/2
1 (d∂r f ∗ + ∂s g ∗ )2 − 4d det A
det A
(
)1/2
2
b−a
1
d
− (a + b)2 − 4d(a + b)2
,
=
(a + b)2
b+a
ϕ(a, b, d) =
(5.3)
In Fig. 5.19 ϕ is shown as a contourplot for (a, b, d) ∈ T .
It is seen, that for (a, b, d) = (0.2, 0.5, 20) and (a, b, d) = (0.05, 1, 80), the
ϕ-value is large compared to the other parameter values considered. This is in
accordance to the bifurcation diagrams and the analysis above.
5.1 Results in Matlab
89
d=10
d=20
0.25
0.3
6
15
4
0.25
0.2
10
2
0.2
0
0.15
5
0.1
a
a
−2
0.15
0
−4
0.1
−6
0.05
−5
0.05
−8
0
0
0.5
1
1.5
0
0
−10
0.5
1
b
1.5
2
−10
b
d=80
0.6
70
60
0.5
50
0.4
a
40
0.3
30
20
0.2
10
0.1
0
0
0
1
2
b
3
4
−10
Figure 5.19: ϕ(a, b, d) for d = 10, 20 and 80. Outside the Turing domain, ϕ has
been set to −10. Please note the different colourbars in the three figures. The
parameter values used are indicated by ∗’s.
In Fig. 5.20 a contourplot of:
1/2
γ
(d∂r f ∗ + ∂s g ∗ )2 − 4d det A
d
)1/2
(
2
b−a
γ
2
2
d
− 4d(a + b)
− (a + b)
=
d
b+a
M 2 − m2 =
for γ = 1 is seen for d = 10, d = 20 and d = 80. M 2 − m2 is the distance in k 2
for which λ+ > 0, see Theorem 5.
It is seen that for (a, b, d) = (0.2, 0.5, 20), (0.2, 1, 20) and (0.1, 1.5, 20), M 2 −
2
m ≈ 0.2, while (a, b, d) = (0.05, 1, 10), (0.05, 1.5, 20), M 2 − m2 ≈ 0.4, and
for (a, b, d) = (0.05, 1, 80), M 2 − m2 ≈ 0.9. Therefore, this measure cannot be
used to explain the relative higher level of spatial aggregation of the patterns
obtained for (a, b, d) = (0.05, 1, 80) and (a, b, d) = (0.2, 0.5, 20). Changing γ
does not change the conclusion as M 2 − m2 ∝ γ.
90
Numerical analysis
d=10
d=20
0.25
0.6
0.3
0.5
0.25
0.4
0.2
0.8
0.7
0.2
0.6
0.5
0.15
a
a
0.3
0.15
0.4
0.1
0.3
0.2
0.1
0.1
0.05
0.2
0.05
0.1
0
0
0.5
1
1.5
0
0
0
0.5
1
b
1.5
2
0
b
d=80
0.6
0.9
0.8
0.5
0.7
0.4
0.6
a
0.5
0.3
0.4
0.2
0.3
0.2
0.1
0.1
0
0
1
2
b
3
4
Figure 5.20: M 2 − m2 for γ = 1 and d = 10, 20 and 80. Outside the Turing
domain, M 2 − m2 has been set to 0. Please note the different colourbars in the
three figures. The parameter values used are indicated by ∗’s.
5.1 Results in Matlab
91
By Fig. 5.19 we choose parameters for which ϕ is relatively large and small.
For example for (a, b, d) = (0.01, 1, 20), ϕ ≈ 16 and for (a, b, d) = (0.01, 3.5, 80),
ϕ ≈ 2. In Fig. 5.21 patterns for these two set of parameters and γ = 100 are
seen. Again we see that the relative large ϕ-value is accompanied by a relative
higher level of spatial aggregation and a relative larger Hu1 -norm; 5.96 in (a)
compared to 1.82 in (d). The Hv1 -norms in (c) and (d) are 1.04 resp. 0.99.
(u−u*)(t=∞)
(v−v*)(t=∞)
2
0
1.5
1.5
−0.1
1.5
1
0.5
0.5
−0.2
1
−0.3
y
y
1
−0.4
0.5
0
−0.5
−0.5
0
0
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
−0.6
x
(a) a = 0.01, b = 1, d = 20, γ = 100
(b) a = 0.01, b = 1, d = 20, γ = 100
*
(v−v*)(t=∞)
(u−u )(t=∞)
1
0.06
1.5
1.5
0.04
0.5
1
0.02
1
0
y
y
0
−0.5
0.5
−0.02
0.5
−1
0
0
0.2
0.4
0.6
0.8
1
x
(c) a = 0.01, b = 3.5, d = 80, γ = 100
−0.04
0
0
0.2
0.4
0.6
0.8
1
−0.06
x
(d) a = 0.01, b = 3.5, d = 80, γ = 100
Figure 5.21: stationary solutions for (a, b, d) = (0.01, 1, 20) where ϕ(a, b, d) ≈ 16
and (0.01, 3.5, 80) where ϕ ≈ 2 for γ = 100. The relative H 1 -norm of the u- and
v-pattern in (a) and (b) is 5.96 resp. 1.04, while for (c) and (d) Hu1 = 1.82 resp.
Hv1 = 0.99.
92
Numerical analysis
5.1.3
Equivalent patterns
The final aim of the numerical analysis is to look into the possibility of
obtaining patterns for different parameters that only differ by a scaling.
˜ ∈ T , and (u, v) and (ũ, ṽ) solve:
Definition 4 Let (a, b, d), (ã, b̃, d)
−∆u = a − u + u2 v,
−d∆v = b − u2 v,
resp.
−∆u = ã − u + u2 v,
˜ = b̃ − u2 v,
−d∆v
on domain Ω and with Neumann-conditions.
(u, v) and (ũ, ṽ) are equivalent if there exists real su and sv such that
u − u∗ = su (ũ − ũ∗ ),
v − v ∗ = sv (ṽ − ṽ ∗ ),
b̃
b
where (ũ∗ , ṽ ∗ ) = (ã + b̃, ã+
) and (u∗ , v ∗ ) = (a + b, a+b
).
b̃
2
It was observed that for fixed d, the patterns corresponding to (a, b)-values
along the level curves of ϕ, involved similar spatial aggregation. But, no equivalent patterns were observed.
This is due to the fact that the wavenumbers are unstable for different γvalues on the level curves of ϕ. We can ovoid this by instead fixing the position
(1)
(2)
of the zeros γk2 and γk2 , i.e. by considering the level curves of the vector func(1)
(2)
tion (γk2 , γk2 ).
˜ ∈ T . Then the curve l given by
Again let (ã, b̃, d)
s

s
˜
˜
d+t
d+t ˜ 
l(t) = 
ã,
b̃, d + t , t ∈ I ⊂ R,
(5.4)
d˜
d˜
where I is such that l(t) ∈ T ⊂ R3 for every t ∈ I, defines a continuous level
(1)
(2)
˜ This can be verified by insertion.23
curve of (γk2 , γk2 ) through (ã, b̃, d).
By comparing patterns for the different (a, b, d)-values on the curve l, we
observed the following result, which we have formulated in a theorem.
2 The
τ )2 /b̃2 )
3l
˜ + s)2 /ã2 ) or l(τ ) = (ãτ /b̃,
, b̃(ã + s)/ã, d(ã
parametrisations l(s) = ( ã+s
ã
can also be used.
˜ see (4.20)
is also a continuous level curve of (m2 , M 2 ) through (ã, b̃, d),
b̃+τ
b̃
˜ b̃ +
, d(
5.1 Results in Matlab
93
Theorem 6 For every steady-state solution (ũ, ṽ) of (1.4)-(1.6) for parame˜ there exists a steady-state solution (u, v) of (1.4)-(1.6) for every
ters (ã, b̃, d),
parameter (a, b, d) ∈ {l(t)}t∈I ⊂ T , such that (u, v) and (ũ, ṽ) are equivalent. 2
Let us first give numerical
examples
before √
proving the
√
√ theorem.
√
The (a, b, d)-values ( 2×0.2, 2×0.5, 40), ( 3×0.2, 3×0.5, 60), (0.4, 1, 80)
are all on the level curve {l}t∈I through (0.2, 0.5, 20). In Fig. 5.22, Fig. 5.23 and
Fig. 5.24, Fig. 5.25 similar u- and v-patterns are seen for the four (a, b, d)-values
for γ = 50 resp. γ = 100.
The similar patterns have the same relative H 1 -norms. For the patterns in
Fig. 5.22 the relative H 1 -norms are 2.4133 and 0.3911 for u resp. v, while for
the patterns in Fig. 5.24, they are 2.4536 and 0.6278 for u resp. v.
Now, the main observation was that each pattern in Fig. 5.22, Fig. 5.23,
Fig. 5.24 or Fig. 5.25 could be scaled to cover any of the three remaining patterns,
with an accuracy of machine precision ≈ 10−16 . This is what has been claimed
to hold in general in Theorem 6.
94
Numerical analysis
(u−u*)(t=∞)
(u−u*)(t=∞)
1.2
1.5
1.5
1.5
1
1
0.8
1
1
0.4
0.5
y
y
0.6
0.2
0.5
0.5
0
0
−0.2
0
0
0.2
0.4
0.6
0.8
−0.5
0
0
1
0.2
0.4
x
0.6
0.8
1
x
(a) a = 0.2, b = 0.5, d = 20, γ = 72
(b) a =
√
2 × 0.2, b =
*
√
2 × 0.5, d = 40, γ = 72
*
(u−u )(t=∞)
(u−u )(t=∞)
2
2.5
1.5
1.5
2
1.5
1.5
1
1
1
y
y
1
0.5
0.5
0.5
0.5
0
0
−0.5
−0.5
0
0
0.2
0.4
0.6
0.8
1
x
(c) a =
√
3 × 0.2, b =
0
0
0.2
0.4
0.6
0.8
1
x
√
3 × 0.5, d = 60, γ = 72
(d) a = 0.4, b = 1, d = 80, γ = 72
Figure 5.22: u-pattern for (a, b, d)-values on the same level curve {l}t∈I and
γ = 50. The Hu1 -norm is equal to 2.4133 for all four (a, b, d)-values.
5.1 Results in Matlab
95
*
*
(v−v )(t=∞)
(v−v )(t=∞)
1.5
1.5
−0.1
−0.1
−0.15
1
1
−0.15
y
y
−0.2
−0.25
−0.2
0.5
0.5
−0.3
−0.35
−0.25
−0.4
0
0
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
x
0.6
0.8
1
x
(a) a = 0.2, b = 0.5, d = 20, γ = 72
(b) a =
√
2 × 0.2, b =
(v−v*)(t=∞)
√
2 × 0.5, d = 40, γ = 72
(v−v*)(t=∞)
−0.02
1.5
1.5
−0.04
−0.04
−0.06
−0.06
−0.08
1
−0.08
1
−0.1
−0.1
y
y
−0.12
−0.12
−0.14
−0.16
0.5
−0.14
0.5
−0.18
−0.16
−0.2
−0.18
−0.22
0
0
0.2
0.4
0.6
0.8
1
x
(c) a =
√
3 × 0.2, b =
0
0
−0.2
0.2
0.4
0.6
0.8
1
x
√
3 × 0.5, d = 60, γ = 72
(d) a = 0.4, b = 1, d = 80, γ = 72
Figure 5.23: v-pattern corresponding to the patterns in Fig. 5.22. The Hv1 -norm
is equal to 0.3911 for all four (a, b, d)-values.
96
Numerical analysis
*
*
(u−u )(t=∞)
(u−u )(t=∞)
1.6
1.5
1.5
1.4
0.8
1.2
1
0.6
1
1
0.8
0.6
y
y
0.4
0.4
0.2
0.5
0.2
0.5
0
0
−0.2
−0.2
0
0
0.2
0.4
0.6
0.8
0
0
1
−0.4
0.2
0.4
x
0.6
0.8
1
x
(a) a = 0.2, b = 0.5, d = 20, γ = 100
√
√
(b) a = 2 × 0.2, b = 2 × 0.5, d = 40,
γ = 100
(u−u*)(t=∞)
(u−u*)(t=∞)
2
2
1.5
1.5
1.5
1.5
1
1
1
y
y
1
0.5
0.5
0.5
0.5
0
0
−0.5
0
0
0.2
0.4
0.6
0.8
1
x
(c) a =
√
3×0.2, b =
√
−0.5
0
0
0.2
0.4
0.6
0.8
1
x
3×0.5, d = 60, γ = 100
(d) a = 0.4, b = 1, d = 80, γ = 100
Figure 5.24: u-pattern for (a, b, d)-values on the same level curve {l}t∈I and
γ = 100. The Hu1 -norm is equal to 2.4536 for all four (a, b, d)-values.
5.1 Results in Matlab
97
(v−v*)(t=∞)
(v−v*)(t=∞)
0.05
0.05
1.5
1.5
0
0
−0.05
−0.05
−0.1
1
−0.15
1
−0.1
−0.15
y
y
−0.2
−0.25
−0.2
−0.3
0.5
0.5
−0.35
−0.25
−0.4
−0.3
−0.45
0
0
0.2
0.4
0.6
0.8
1
−0.5
0
0
−0.35
0.2
0.4
x
0.6
0.8
1
x
(a) a = 0.2, b = 0.5, d = 20, γ = 100
(b) a =
γ = 100
√
2 × 0.2, b =
*
√
2 × 0.5, d = 40,
*
(v−v )(t=∞)
(v−v )(t=∞)
0.05
0
1.5
1.5
0
−0.05
−0.05
1
1
−0.1
y
y
−0.1
−0.15
0.5
−0.15
0.5
−0.2
−0.2
−0.25
0
0
0.2
0.4
0.6
0.8
1
x
(c) a =
√
3×0.2, b =
√
0
0
0.2
0.4
0.6
0.8
1
−0.25
x
3×0.5, d = 60, γ = 100
(d) a = 0.4, b = 1, d = 80, γ = 100
Figure 5.25: v-pattern corresponding to the patterns in Fig. 5.24. The Hv1 -norm
is equal to 0.6278 for all four (a, b, d)-values.
98
Numerical analysis
And now for the proof of Theorem 6.
˜
Proof The equations for the perturbed variables, (ũ, ṽ), for parameters (ã, b̃, d),
are:
−∆ũ =
b
b̃ − ã
ũ + (ã + b̃)2 ṽ + 2(ã + b̃)ṽ ũ +
ũ2 + ũ2 ṽ,
ã + b̃
(ã + b̃)2
˜ =−
−d∆ṽ
2b̃
ã + b̃
ũ − (ã + b̃)2 ṽ − 2(ã + b̃)ṽ ũ −
b̃
(ã + b̃)2
ũ2 − ũ2 ṽ.
Let
q be the
q (u, v)
solution of the perturbed equations with parameters (a, b, d) =
d
d
ã, d˜b̃, d , i.e. the solution of:
d̃
q
−∆u = q
or
d
b̃
d̃
−
q
d
ã
d˜
r
d
ã +
d˜
r
d
b̃
d˜
!2
r
d
ã +
d˜
r
!
d
b̃ vu
d˜
q u+
v+2
+ dd̃ b̃
q
d
b̃
d˜
2
2
+ q
q 2 u + u v,
d
d
ã + d˜b̃
d˜
q
r !2
r !
r
r
2 dd̃ b̃
d
d
d
d
q u−
ã +
b̃ v − 2
ã +
b̃ vu
−d∆v = − q
˜
˜
˜
d
d
d
d
d
d˜
ã
+
b̃
˜
˜
d
d
q
d
b̃
d˜
2
2
− q
q 2 u − u v,
d
d
ã + d˜b̃
d˜
d
ã
d̃
r 2
b̃ − ã
d
d
−∆u =
u+
ã + b̃ v + 2
ã + b̃ vu
ã + b̃
d˜
d˜
s
d˜
b̃
2
2
+
2 u + u v,
d
ã + b̃
r 2
2
b̃
d
d
d
ã + b̃ v − 2
ã + b̃ vu
−d˜ ∆v = −
u−
d˜
d˜
d˜
ã + b̃
s
b̃
d˜
2
2
−
2 u + u v.
d
ã + b̃
(5.5)
(5.6)
5.1 Results in Matlab
99
If we multiply (5.5) and (5.6) by
q
˜ we obtain:
d/d

s 
!
! s 
rd
2 r d
˜
˜
d
d
d˜ 
b̃
−
ã

v + 2 ã + b̃
v 
u =
u + ã + b̃
u
−∆ 
d
d
d
d˜
d˜
ã + b̃
s 2 s 2 r !
d
b̃
d˜ 
d˜ 
+
v ,
u
u
+
2 
d
d
d˜
ã + b̃
s 
r ! s 
r !
r !
˜
2
d
d
d
2b̃  d 
d˜ 
˜
v =−
v − 2 ã + b̃
v 
u − ã + b̃
u
−d∆
d
d
d˜
d˜
d˜
ã + b̃
s
−
b̃
ã + b̃
s
2 
2 s 2
d˜ 
d˜ 
u
u
+
d
d
q
From this, we realise that (u, v) =
q
˜
sv = dd , and then the theorem follows.
d
ũ,
d̃
q
d̃
d ṽ
r
!
d
v .
d˜
q
d
, and hence su =
and
d˜
In the examples above we saw that the relative H 1 -norms were constant
along {l}t∈I . This is of course no coincidence.
The relative H 1 -norms of ũ and ṽ are given by
kũ − ũ∗ kHu1
1/2

Z X
1 
|∂ α (ũ − ũ∗ )|2  ,
= ∗
ũ
Ω
kṽ − ṽ ∗ kHv1 =
|α|≤1

1 
ṽ ∗
Z
X
Ω |α|≤1
1/2
|∂ α (ṽ − ṽ ∗ )|2 
.
Now, if we move along the level curve {l}t∈I , then for every (a, b, d) ∈ {l}t∈I :
ku − u∗ kHu1

1/2
Z X
1
|∂ α (u − u∗ )|2  ,
= ∗
u
Ω
|α|≤1
1/2
r Z
X
1
d
|∂ α (ũ − ũ∗ )|2  ,
=q
˜
d ∗
d
Ω
ũ
|α|≤1
d˜
= kũ − ũ∗ kHu1 ,
100
Numerical analysis
since u∗ =
∗
since v =
5.2
q
q
d ∗
ũ ,
d˜
and similarly for v,
kv − v ∗ kHv1 = kṽ − ṽ ∗ kHv1 ,
d̃ ∗
d ṽ .
Summary and remarks
Six different (a, b, d)-values were chosen, and for each of these the existence
of multiple steady-state solutions were shown. What pattern, that ultimately
developed, depended upon the initial conditions.
One might have thought that the number of different steady-state patterns,
at least modulus rotation and multiplication by −1 as illustrated in Fig. 5.5,
for fixed (a, b, d) and γ were equal to the number of unstable wavenumbers.
However, we were not able to support this claim by the numerical analysis. An
example with six unstable wavenumbers was given, however, we only found four
different patterns.
It was observed numerically, that u had a local maximum where v had a
local minimum, and u had a local minimum where v had a local maximum.
Using the H 1 -norm the bifurcation of steady-states were analysed as we
varied γ. The results were presented in bifurcation diagrams. γc from Theorem 5
could be read off from the bifurcation diagrams for the (a, b, d)-values considered.
They all agreed with Theorem 5.
Furthermore, the bifurcation diagrams showed that for some (a, b, d)-values
the patterns had relative higher Hu1 -norm. The Hu1 -norm was seen to be a
measure of the heterogeneity or the spatial aggregation of the steady-states. It
was argued, by way of examples, that the heterogeneity was related to the size
of ϕ, defined in (5.3).
It was observed numerically that all the systems, (1.4)-(1.6), with (a, b, d)(2)
(1)
values along the level curves in T of (γk2 , γk2 ) exhibit equivalent steady-state
solutions for every γ. This result was stated in a theorem, Theorem 6, and a
proof was established.
Moreover, it was observed numerically, and proved analytically, that the
relative H 1 -norms were conserved along these level curves. This also shows,
that the relative H 1 -norms are the right concepts to work with in the bifurcation
analysis.
Some patterns appeared to be similar to certain eigenfunctions. For example,
Fig. 5.26 shows a surface-plot of the pattern in Fig. 5.2. It looks like a cosine
wave in the x-direction.
k2,0 = 2π is in the range of unstable wavenumbers, see Fig. 5.6. Therefore,
5.2 Summary and remarks
101
in the hope of finding an approximate solution we defined I : R2 → R by
2
I(U, V ) = kkn,m
U cos(kn,m x) − f (U cos(kn,m x) + u∗ , V cos(kn,m x) + v ∗ )k2
2
+kdkn,m
V cos(kn,m x) − g(U cos(kn,m x) + u∗ , V cos(kn,m x) + v ∗ )k2 ,
and searched for (U, V ) 6= (0, 0) satisfying
∂U I(U, V ) = 0 = ∂V I(U, V ).
It is combersome to do any general, analytical work on these coupled equations,
as they involve many, complicated terms. Instead we used MAPLE for the specific
values a = 0.05, b = 1, d = 10 and γ = 120. No real, non-trivial (U, V ) existed.
Other parameters were also tried, but the conclusion was the same. Therefore, we do not expect this method to be sufficient to predict any approximations
for any parameters in the Turing domain. To obtain approximations a different
approach is therefore necessary.
(v−v*)(t=∞)
1
0.4
0.5
0.2
v−v*
u−u*
(u−u*)(t=∞)
0
−0.5
0
−0.2
−1
2
−0.4
2
1.5
1
1
0.5
0.5
y
0
0
x
1.5
1
1
0.5
0.5
y
0
0
x
Figure 5.26: Surf-plot of u and v. Corresponds to the contour-plots in Fig. 5.2.
102
Numerical analysis
Chapter
6
Conclusion
Two different approaches were applied to the reaction-diffusion systems to
prove existence and uniqueness.
The first approach was the approach described by Pao, [Pao (1992)], with upper and lower solutions. Hereby, local existence was obtained for the Schnakenberg system while global existence and boundedness was achieved for the Gierer
and Meinhardt system. This technique was shown to be inadequate for the
Thomas system.
The approach was also used to study the stationary equations for the Schnaken∗
berg system. It was shown that the uniform steady-state solution, (u∗ , v√
), was
a2 +a2
1−4 b
a
the unique solution in the sector of h(a, 0), (b/a2 , ui, where u =
,
2b
for b ≤ a/4. This is an interesting result, as it is often difficult to say anything
about the number of stationary solutions. Often, there exist one single, or a
“few,” stable steady-states and “many” unstable steady-states.
The approach described by Pao was shown to provide an easy, ad hoc approach to show existence and uniqueness of coupled, parabolic and elliptic partial differential equations. It also provided an approach to analyse the stationary
equations.
A major downside, however, was the fact that, the technique only applied to
quasi-monotone reaction functions.
Next the functional analysis and semigroup approach was considered. The
conditions:
104
Conclusion
(H1) d > 0;
(H2) u0 ≥ 0 and v0 ≥ 0 are continuous on Ω, that is u0 , v0 ∈ CL0 ∞ (Ω);
3
(H3) f and g are continuously differentiable functions from R+ into R with
f (t, 0, s) ≥ 0 and g(t, r, 0) ≥ 0 for all t, r, s ≥ 0;
2
(H4) There exist m > 0 and a continuous function F : R+ → R+ such that
f (t, r, s), g(t, r, s) ≤ exp(mt)F (r, s) for all t, r, s ≥ 0.
were shown to imply positivity and local existence, i.e. existence for t < T ∗ for
some 0 < T ∗ < ∞, for the general two-component reaction-diffusion equations
with Neumann-conditions.
Furthermore, it was shown that if the solution only existed locally, then it
would diverge. By contradiction, this was used to show global existence for the
Schnakenberg system, in one space dimension, and for the Thomas system.
If we furthermore imposed the conditions
(H5) The v-variable is bounded for all t ≤ T ∗ if the solution is local;
(H6) There is an η ≥ 1 and a continuous function h : [0, ∞)2 → [0, ∞) such
that |f (t, r, s)| ≤ h(t, S)(1 + r)η for all t, r, s ≥ 0 with s ≤ S;
(H7) There is an ǫ > 0 and a continuous function l : [0, ∞)2 → [0, ∞) such that
ǫr + f (t, r, s) + g(t, r, s) ≤ l(t, S) for all t, r, s ≥ 0 with s ≤ S.
then, the solution was shown to be global. If N2 , h and l were bounded in t
then the solution was additionally proved to be bounded for all t.
Next, it was shown that when (H1)-(H7) were satisfied, then the solution
depended continuously upon the initial data.
To sum it all up, it was shown that the conditions (H1)-(H7) were sufficient
conditions for the two-component reaction-diffusion system with Neumann conditions to be well-posed, in the sense of Hadamard, and to be positive and
bounded. This in turn showed that the three specific systems considered satisfied (W1)-(W7).
Compared to the approach with upper and lower solution, the use of semigroups and general functional analysis established a setting in which the entire
analysis can be made precise and complete. However, this setting involved more
work and was not in general ad hoc.
Next, linear stability and instability was proved to be sufficient conditions
for stability resp. instability of steady states of the general autonomous, twocomponent reaction-diffusion equations. This was used to study diffusion-driven
instabilities at uniform steady state.
The conditions
105
(T1) ∂r f (u∗ , v ∗ ) + ∂s g(u∗ , v ∗ ) < 0;
(T2) ∂r f (u∗ , v ∗ )∂s g(u∗ , v ∗ ) − ∂s f (u∗ , v ∗ )∂r g(u∗ , v ∗ ) > 0;
(T3) d∂r f (u∗ , v ∗ ) + ∂s g(u∗ , v ∗ ) > 0;
(T4)
(d∂r f (u∗ ,v ∗ )+∂s g(u∗ ,v ∗ ))2
4d
> ∂r f (u∗ , v ∗ )∂s g(u∗ , v ∗ ) − ∂s f (u∗ , v ∗ )∂r g(u∗ , v ∗ );
were shown, for sufficient large domains, to be sufficient for the system to exhibit
diffusion-driven instabilities at uniform steady-states (u∗ , v ∗ ) ∈ R2+ .
Finally, the Schnakenberg model was studied numerically on a rectangle.
Chebyshev polynomials were used for the Laplacian and the time integration
was performed by built in MATLAB ode-solver ode15s. In accordance to the linear
analysis diffusion-driven instabilities were seen to occur for sufficiently large
domains. These instabilities were seen to evolve into steady-state heterogeneous
patterns.
The existence of multiple steady-states were shown by the use of bifurcation
diagrams. The H 1 -norm was used as bifurcation functional.
The patterns’ dependency upon the parameters were analysed and it was
seen that in some part of the parameter domain, determined by (T1)-(T4),
the spatial aggregation of the final patterns was relatively larger. By way of
example, this was shown to be related to the size of
s
2
1
b−a
2
− 4d(a + b)2 .
ϕ(a, b, d) =
−
(a
+
b)
d
(a + b)2
b+a
The most important observation made in the numerical analysis of the Schnakenberg model was the existence of curves in the parameter domain on which the
corresponding equations exhibited equivalent steady-state solutions. This result
was stated in a theorem and a proof was established.
It was furthermore observed and proved that the relative H 1 -norms were
conserved along these lines.
This result would not have been obtained without the numerical simulations
of the models. This illustrates why a numerical analysis is indispensable in almost every analysis of non-linear problems as numerical results often give us
ideas about what to study analytically.
106
Conclusion
Appendix
A
Semigroups and sectorial
operators
The main results on existence and uniqueness of semilinear parabolic equations are based upon the fixed point theorems from functional analysis. We are
allowed to make use of these results through the setting of semigroups. In the
following we introduce the concept of semigroups and state the most relevant
results.
It is well-known that the ordinary differential equation:
∂t u = λu,
u(0) = u0 ,
has the solution:
u(t) = exp(λt)u0 ,
t ≥ 0.
(A.1)
In the theory of semigroups we shall define similar solutions for partial differential equations like
∂t u − Au = 0,
u(0) = u0 ,
(A.2)
and some initial condition u(0) = u0 , when the linear differential operator A
satisfy some suitable conditions. We shall see that the adequate conditions, for
our purpose at least, are that A is sectorial, see Definition 6.
The main idea is to rely on the abstract functional analysis and consider u as
108
Semigroups and sectorial operators
function of t with values in Banach Space, i.e. one considers u : t 7→ X where X
is some Banach space rather than u : (x, t) 7→ R or C. In this setting existence
and uniqueness can often be obtained by applying fixed point theorems, for
example the fundamental Banach’s Fixed Point Theorem.
We are now ready for a definition of a semigroup.
Definition 5 (Semigroup, see [Henry (1981), Definition 1.3.3]) An analytic
semigroup is a family of bounded, linear operators on X, {G(t)}t≥0 , satisfying:
1◦ G(0) = I, G(t)G(s) = G(t + s) for t, s ≥ 0
2◦ G(t)x → x as t → 0+ , for each x ∈
3◦ t → G(t)x is real analytic on 0 < t < ∞ for each x ∈ X
The generator of this semigroup is defined by
Bx = lim+
t→0
G(t)x − x
,
t
(A.3)
with domain
D(B) =
G(t)x − x
exist in X .
x ∈ X| lim
t
t→0+
(A.4)
If a family of bounded, linear operators on X only satisfy 1◦ and 2◦ we call the
family a C0 -semigroup. If a C0 -semigroup is compact for every t we call the
family a compact-semigroup.
2
As already mentioned the suitable condition for our operator A in this context
is that it is sectorial.
Definition 6 (Sectorial operator, see [Henry (1981), Definition 1.4.1]) Let A
be a linear operator in a Banach Space X and suppose A is closed and densely
defined. If there exist real a, ω ∈ (0, π), M ≥ 1 such that
ρ(A) ⊃ Σ = {λ ∈ C| φ ≤ arg (λ − a) ≤ π, λ 6= a}
(A.5)
and
kRλ (A)k ≤
then we say that A is sectorial.
M
,
|λ − a|
for all λ ∈ Σ
(A.6)
2
109
The adequateness of sectorial operators is implied by the following theorem.
We refer to App. B for a short introduction on integration and differentiation
in Banach spaces.
Theorem 7 (See [Henry (1981), Theorem 1.3.4]) If A is sectorial then -A is
the infinitesimal generator of an analytic semigroup, G(t).
If Re λ > a, a ∈ R, whenever λ ∈ σ(A) then for any t > 0,
kG(t)k ≤ C exp(−at),
kAG(t)k ≤
C
exp(−at).
t
Finally,
d
G(t) = −AG(t),
dt
t > 0.
(A.7)
2
Proof See e.g. [Henry (1981), Theorem 1.3.4].
When A in (A.2) is sectorial then in the abstract setting of semigroups the
solution is, according to 2. in Definition 5 and (A.7):
u(t) = G(t)u0 ,
t ≥ 0,
similar to (A.1).
In our study of reaction-diffusion equations the Laplacian is of particular
interest. The following example considers the Neumann realisation of −∆.
Example 1 (Neumann realisation of Laplacian) L = −∆ with domain
D(L) = u ∈ H 2 (Ω) | νu = 0
is a sectorial operator in L2 (Ω). Since C0∞ (Ω) ⊂ D(L) it is densely defined in
L2 (Ω).
Assertion 2 A is closed.
Proof Let {un }n∈N ⊂ D(A) such that un → u in L2 (Ω) and Aun → v in
L2 (Ω). It must be shown that u ∈ D(L) and Lu = v.
Now since un → u in L2 (Ω),
(Lun , φ) → (Lu, φ),
for all φ ∈ D(Ω)
so that v = Lu ∈ L2 (Ω) and un → u in D(L).
The trace operator ν is continuous from H 2 (Ω) into H 1/2 (Ω), see [Lions and
Magenes (1968)]. Hence
kνu − νun k1/2 → 0.
110
Semigroups and sectorial operators
But since νun = 0:
kνuk1/2 = 0 ⇔ νu = 0.
It is concluded that u ∈ D(L) with Lu ∈ L2 (Ω) and L is therefore closed.
Assertion 3 L has a countable set of real, non-negative eigenvalues {λk }k∈N
that can be arranged in a non-decreasing sequence
0 ≤ λ1 ≤ · · · ≤ λk ≤ · · ·
Proof L = L∗ which follows by integration by parts and extension by continuity:
(Lu, v) = (u, Lv),
for all u, v ∈ D(L),
(A.8)
and therefore all eigenvalues are real.
Again by integration by parts and extension by continuity:
(Lu, u) = (∇u, ∇u) = k∇uk2 ≥ 0,
for all u ∈ D(L).
from which the non-negativeness of the eigenvalues follows.
It is now claimed that K = (I − ∆)−1 exists and is compact from L2 (Ω) into
itself. To prove this consider the variational approach, where L is defined from
the sesquilinear form s(u, v) = (u, v)H 1 = (∇u, ∇v) + (u, v). By Lax-Milgram
I − ∆ is bijection of D(L) onto L2 (Ω). Furthermore for f ∈ L2 (Ω)
kuk2H 1 = s(u, u) = (f, u) ≤ kf k2 kuk2 ≤ kf k2 kukH 1
from which it follows
kKf kH 1 ≤ kf k2
and by the Rellich-Kondrachov compactness theorem K is a compact operator.
Therefore the eigenvalues of K are countable and the spectrum is a pure point
spectrum and the same therefore holds for L = −∆.
It has been shown:
Corollary 1 For any a ≤ 0 and any ω ∈ (0, π), Σ in Definition 6, is in the
resolvent set.
To prove that L is sectorial the existence of M in accordance to Definition 6
has to be proved.
111
Assertion 4 There exists M such that
kRλ (L)k ≤
M
,
|λ|
for all
λ ∈ Σ.
Proof The result follows from the following computation
k(L − λI)uk2 kuk2 ≥ (Cauchy-Schwarz inequality)
≥ |((L − λI)u, u)|
= (((L − Re λ)u, u)2 + Im λ2 kuk22 )1/2
|Im λ|kuk22 , for Re λ ≥ 0
≥
2
((Re λ) + Im λ2 )1/2 kuk22 , for Re λ ≤ 0
|Im λ|kuk22 , for Re λ ≥ 0
=
|λ|kuk22 , for Re λ ≤ 0
≥ (writing λ = |λ| exp(iθ), θ ∈ (ω, π/2) when Re λ ≥ 0)
≥ |λ| sin(ω)kuk22 ,
any ω ∈ (0, π),
and hence
kRλ (L)k ≤
sin(ω)−1
.
|λ|
The fractional powers are the key issue in the existence in Sec. 3. First a
definition.
Definition 7 (Fractional powers, see [Pazy (1983), Section 2.6]) Suppose A is a
sectorial operator with analytic semigroup G and Re λ > 0 whenever λ ∈ σ(A),
then we define the fractional powers of A by
Z ∞
1
Aα =
tα−1 G(t)dt, α > 0,
Γ(α) 0
where
sin(απ)
Γ(α) =
π
Z
∞
ξ −α exp(−ξ)dξ,
0
and
A0 = I.
2
Proposition 4 If A is sectorial in Banach space X with Re λ > 0 whenever
λ ∈ σ(A), then for any α ≥ 0, A−α is a bounded, linear operator on X which is
injective and satisfies A−α A−β = A−(α+β) .
2
112
Semigroups and sectorial operators
Proof α = 0 is trivial. Therefore let α > 0. By assumption 0 ∈ ρ(A), and
hence that A−1 exists and is injective. For every n ∈ N A−n is therefore injective.
Now for n ≥ α
A−n x = A−n+α A−α x = 0 ⇔ x = 0.
Therefore A−α is injective.
Remark 10 The name fractional powers is misleading: We shall allow α ∈ R+
and not only in Q.
2
By Example 1 H = I − ∆ is sectorial and 0 ∈ ρ(H). Therefore −H generates
an analytic semigroup and the fractional powers H−α exist and are injective.
Furthermore D(H) = D(L).
Proposition 5 α > β > 0 implies D(Hα ) ⊂ D(Hβ ) with compact injection.
Furthermore D(Hα ) is dense in D(Hβ ).
2
Proof See [Henry (1981), Theorem 1.4.8].
Lemma 8 (See [Henry (1981), Theorem 1.6.1]) D(Hα ) ⊂ CL0 ∞ (Ω) with continuous injection for α > n/4, i.e. u ∈ D(Hα ) has a version ũ ∈ CL0 ∞ (Ω) such
that
sup |ũ| ≤ kukD(Hα ) .
Ω
.
2
Proof For s > n/2 the Sobolev Theorem for smooth, bounded subsets of Rn ,
see e.g. [Grubb (2007), Corollary 6.12],
H s (Ω) ⊂ CL0 ∞ (Ω),
with
sup {|u| | x ∈ Ω} ≤ Cs kukH s ,
or short, H s (Ω) ֒→ CL0 ∞ (Ω).
Hs/2 u ∈ L2 (Ω) if and only if u ∈ H s (Ω), see e.g. [Lions and Magenes (1968)].
Therefore D(Hα ) ֒→ H s (Ω) if α = 2s > n4 .
Remark 11 In the proof of local existence it shall be used that for n = 1, 2, 3
there exist an α such that n/4 < α < 1. For n ≥ 4 and more importantly when
we later turn to global existence the L2 -theory is inadequate and one will have
to consider general Lp -spaces, 1 ≤ p < ∞. It is therefore natural to introduce
some generalisations to Lp -spaces. We do so without proof.
First of all the Neumann realisation L of the laplacian, that is the the operator acting like −∆ on smooth functions and with domain
D(L) = u ∈ W 2,p | νu = 0
113
is still sectorial in Lp . This is obtained by similar arguments to those used in the
L2 -result in Example 1, although one has to exploit Hahn-Banach’s Theorem
rather than Riesz Representation Theorem.
Lemma 8 also have an Lp -counterpart, which is shown to be at the very core
of the proof of global existence.
Lemma 9 Let p ∈ [1, ∞). Then D(Hα ) ⊂ CL0 ∞ (Ω) with continuous injection
for α > n/(2p).
2
Proof (Sketched) Again the proof goes by utilising the Sobolev Theorem, now
in Lp -setting, and the interpolation results in [Lions and Magenes (1968)]. Note that in Lp the graph norm is kukD(Hα ) = kHα ukp + kukp .
2
Reminded by the previous remark we obtain the following lemma in Lp .
Lemma 10 Let G be an analytic semigroup generated by a sectorial operator
−A, and let δ ≥ 0 be such that −A + δ is still the generator of an analytic
semigroup. The following properties then hold for the semigroup G(t) and the
fractional powers of A:
1◦ G(t) : Lp (Ω) → D(Aα ) for all t > 0;
2◦ kG(t)ukD(Aα ) ≤ Cα,p t−α exp(−δt)kukp for all t > 0, u ∈ Lp (Ω);
3◦ G(t)Aα u = Aα G(t)u for all t > 0, u ∈ D(Hα ).
2
Proof (Sketched) See e.g. [Pazy (1983), Theorem 6.13]. 1◦ is a direct consequence of G(t) being analytic. 2◦ is proved by exploiting the Closed Graph Theorem to conclude that Aα G(t) is bounded and then using Aα G(t) = Aα−n An G(t),
with n ∈ N chosen such that n−1 < α ≤ n, to estimate kAα G(t)ukp , u ∈ Lp (Ω).
Finally 3◦ can be verified using the definition of the fractional powers and of a
semigroup, see Definition 7 resp. Definition 5.
114
Semigroups and sectorial operators
Appendix
B
Differentiation and
integration in Banach Spaces
The theory of semigroups is based on the study of functions with values in
a Banach space X. Therefore for the sake of clarity it is appropriate to discuss
differentiation and integration in Banach spaces. In the following let u(t) and
v(t) be X-valued functions for all t ∈ I, i.e. u : I → X and v : I → X, where I
is open in R.
The definition of differentiation in a normed vector space is given in the
following definition. The reader is reminded that a function ǫ : N0 → X, where
N0 is an open neighbourhood of 0, is called an ǫ-function if it is continuous in
0 with ǫ(0) = 0.
Definition 8 u is (norm) differentiable at t ∈ I if there exist a linear mapping
L : N → X, N an open neighbourhood of t, and an ǫ-function such that
u(t + h) − u(t) = L(h) + ǫ(h)|h|.
for every h ∈ {d ∈ R | t + d ∈ I}.
(B.1)
2
Remark 12 Since L(h) is linear (B.1) may be rewritten as
u(t + h) − u(t)
|h|
= L(1) + ǫ(h) .
h
h
from which it is clear that L(1) can be interpreted as the differential of u at t.
It is abbreviated ∂t u(t).
2
116
Differentiation and integration in Banach Spaces
And now for the definition of integration in Banach spaces. First a theorem
is needed.
Theorem 8 Let u : I → X be continuous with values in Banach space X. Then
there exists a unique vector I(u) that satisfies:
For every ǫ > 0 there exist an δ > 0, so that
kI(u) − SD (u)kX < ǫ,
PN
for every SD (u) = i=1 u(τi )(ti − ti−1 ), where τ1 , τ2 , · · · , τN ∈ [a, b] such that
τi ∈ [ti−1 , ti ] with fineness
µ(D) = max {|ti − ti−1 | | 1 ≤ i ≤ n} < δ.
2
Proof (Sketched) The proof can be seen in [Hansen (1995), Theorem III.3.2].
Readily it comes down to showing, that for a sequence of subdivisions {Dn }n∈N
for which µ(Dn ) → 0, the corresponding sequence of sums, {SDn }n∈N , is a
Cauchy sequence in the Banach space X.
And then:
Definition 9 I(u) from Theorem 8 is called the integral of u over [a, b] ∈ I.
2
The following (usual) rules of integration do apply, see [Hansen (1995), Theorem III.3.4],
Z
Z
b
(αv(t) + βw(t))dt = α
Z
b
v(t)dt + β
v(t)dt +
Z
c
v(t)dt =
b
a
k
1
h
Z
b
a
Z
c
v(t)dtkX ≤
c+h
Z
b
w(t)dt,
a
a
a
b
Z
c
v(t)dt,
a
Z b
a
kv(t)kX ,
v(t)dt → v(c)
in X
for
h → 0.
(B.2)
Appendix
C
Gronwall’s inequality
In this appendix the Gronwall inequality in differential and integral form are
stated and proven.
Lemma 11 (Differential form) Let f , h, y and ∂t y be locally integrable functions on [0, ∞) that satisfy
∂t y ≤ f y + h,
for
t > 0,
(C.1)
y(0) = y0 ,
then
Z t
y(t) ≤ exp(F (t)) y0 +
h(τ ) exp(−F (τ ))dτ ,
0
where F (t) =
Rt
0
f (τ )dτ .
2
Proof We multiply (C.1) with the positive valued function
exp(−F (τ ))
to obtain
exp(−F (τ ))∂τ y − f (τ ) exp(−F (τ ))y = ∂τ (exp(−F (τ ))y) ≤ h(τ ) exp(−F (τ )).
118
Gronwall’s inequality
Integration with respect to τ from 0 to t yields
Z t
h(τ ) exp(−F (τ ))dτ,
exp(−F (t))y(t) ≤ y(0) +
0
and thus
Z t
y(t) ≤ exp(F (t)) y0 +
h(τ ) exp(−F (τ ))dτ .
0
Lemma 12 (Integral form) Let y be locally integrable functions on [0, ∞) satisfying
Z t
y(t) ≤ A
y(τ )dτ + B, for t > 0,
(C.2)
0
for some non-negative constants A and B. Then,
y(t) ≤ B exp(At).
Proof Set z(t) =
to Lemma 11 then
Rt
0
2
y(τ )dτ ; then ∂t z ≤ Az + B with z(0) = 0 and according
z(t) ≤
B
exp(At)(1 − exp(−At)).
A
Finally, by (C.2) we get:
y(t) ≤ B exp(At).
Appendix
D
Matlab-codes
D.1
main.m
% num solve reaction diff eqn
close all
clear all
global
global
global
global
gamma ab diag10 u0u0 v0v0 u0v0
N2 Nx Ny
d Dx Dy
a b
% u’_t = gamma*(a-u-u0+(u’+u0)^2 (v’+v0)) +u’_xx
% v’_t = gamma*(b-(u+u0)^2 (v+v0)) + d v’_xx
% zero flux
a = .05; b=1;d=80;
tspan = [0 100];
Ly = pi/2;
Lx = 1;
Nx = 16;
Ny = floor(Ly/Lx*Nx);
120
Matlab-codes
if mod(Ny,2) ~= 0; Ny = Ny+1; end
N2 = (Nx+1)*(Ny+1);
plotN = 100;
L = ((d*(b-a)-(a+b)^3)-sqrt((d*(b-a)-(a+b)^3)^2-...
4*d*(a+b)^4))/(2*d*(a+b));
M = ((d*(b-a)-(a+b)^3)+sqrt((d*(b-a)-(a+b)^3)^2-...
4*d*(a+b)^4))/(2*d*(a+b));
% Set up operator
operator
% Set up necessary parameters
para
kvec = zeros(Nx+1,Ny+1);
for n= 0:Nx;
for m = 0:Ny;
kvec(n+1,m+1) = sqrt((n*pi)^2/Lx^2+(m*pi)^2/Ly^2);
end
end
kvec = kvec(:)
kvec = sort(kvec(:));
[X,Y] = meshgrid(x,y);
Gamma = 70:1:75;
h1u = []; h1v = []; kMvec = [];
for gamma = Gamma;
gamma
tic
% Initial conditions
initcond
evalkM
[m,temp] = min(abs(kvec-kM))
out = lambdaeval(kvec(temp),a,b,d,gamma)
if real(out) < 0
kMvec = [kMvec;0];
else
kMvec = [kMvec;kvec(temp)];
end
% Solve system
D.1 main.m
%
solver
recmovie
toc
dxuend = dx*uend;
dyuend = dy*uend;
fu = (u0+uend).^2+dxuend.^2+dyuend.^2;
Fu = 1/(Lx*Ly)*trapez(fu,x,y,Nx,Ny);
dxvend = dx*vend;
dyvend = dy*vend;
fv = (v0+vend).^2+dxvend.^2+dyvend.^2;
Fv = 1/(Lx*Ly)*trapez(fv,x,y,Nx,Ny);
h1u = [h1u; Fu]; h1v = [h1v;Fv];
figure
surf(X,Y,reshape(uend,Ny+1,Nx+1))
title([num2str(gamma)])
drawnow
end
figure
axes(’position’,[.1 .1 .8 .6])
plot(Gamma,h1u,’o’,Gamma,h1v,’d’);
xlabel(’gamma’)
ylabel(’H^1 - norm’)
hold on
legend(’u_s’,’v_s’)
axes(’position’,[.1 .7 .8 .2])
plot(Gamma,kMvec,’.’)
ylabel(’Critical wave-number’)
legend(’Linear analysis prediction’)
121
122
Matlab-codes
D.2
cheb.m
% CHEB
compute D = differentiation matrix, x = Chebyshev grid
function [D,x] = cheb(N)
if N==0, D=0; x=1; return, end
x = cos(pi*(0:N)/N)’;
c = [2; ones(N-1,1); 2].*(-1).^(0:N)’;
X = repmat(x,1,N+1);
dX = X-X’;
D = (c*(1./c)’)./(dX+(eye(N+1)));
% off-diagonal entries
D = D - diag(sum(D’));
% diagonal entries
D.3 operator.m
D.3
operator.m
% Set up local discretised operators
[dx,vecx] = cheb(Nx);
dx = -(2/Lx)*dx;
[dy,vecy] = cheb(Ny);
dy = -(2/Ly)*dy;
x = -Lx*vecx’/2+Lx/2;
y = -Ly*vecy’/2+Ly/2;
I = eye(Ny+1);
dx = kron(dx,I);
I1 = [1 0;0 0]; I2 = [0 0;0 1];
Dx = sparse(kron(I1,dx)+kron(I2,dx));
I = eye(Nx+1);
dy = kron(I,dy);
Dy = sparse(kron(I1,dy)+kron(I2,dy));
123
124
Matlab-codes
D.4
initcond.m
u0 = a+b; v0 = b/(a+b)^2;
uold = zeros(Ny+1,Nx+1);
for i=1:Nx
for j = 1:Ny
m = max(i-1,j-1);
c = rand(1)*(-1)^(round(rand(1)))*m/max(m,1)*...
1/(j*i)^(2^(round(rand(1))));
uold = uold+c*cos((j-1)*pi*Y/Ly).*cos((i-1)*pi*X/Lx);
end
end
uold = reshape(uold,N2,1);
vold = zeros(size(uold));
maxu = max(abs(uold));
uold = u0/100*uold/maxu; %vold = v0/100*(-uold)/maxu/d;
figure
axes(’fontsize’,12)
contourf(X,Y,reshape(uold,Ny+1,Nx+1))
xlabel(’x’)
ylabel(’y’)
colorbar
title(’(u-u^*)(0)’)
D.5 solver.m
D.5
solver.m
% Tolerance for built-in MATLAB ODE solver
options = odeset(’RelTol’,1e-10,’AbsTol’,1e-10);
sol = [uold;vold];
[T,sol] = ode15s(@func2d,tspan,sol,options);
sol = sol’;
uend = sol(1:N2,end);
vend = sol(N2+1:2*N2,end);
clear sol T
lengthT = length(T);
tint = floor(lengthT./plotN);
t
= zeros(plotN,1);
jstart = lengthT-tint*plotN;
u = zeros(N2,plotN);
v = zeros(N2,plotN);
for j = 1:plotN
t(j) = T(jstart + j*tint);
u(:,j) = sol(1:N2,jstart+j*tint);
v(:,j) = sol(N2+1:2*N2,jstart+j*tint);
end
125
126
D.6
Matlab-codes
recmovie.m
figure
%axis tight
xlim([0 Lx])
ylim([0 Ly])
zlim([minuv maxuv])
set(gca,’nextplot’,’replacechildren’);
% Record the movie
for j = 1:plotN
uplay = u(:,j);
uplay = reshape(uplay,Ny+1,Nx+1);
vplay=v(:,j);
vplay = reshape(vplay,Ny+1,Nx+1);
surf(X,Y,uplay)
title([’t=’ num2str(t(j))])
F(j) = getframe;
end
%movie2avi(F,’animation.avi’)
D.7 func2d.m
D.7
func2d.m
function w=func(t,u)
global gamma diag10 u0u0 v0v0
global N2 Nx Ny
%global LL
global d Dx Dy
global u0v0 ab
% u u
uu = [u(1:N2);u(1:N2)];
% v -v
vv = [u(N2+1:2*N2);-u(N2+1:2*N2)];
ux = Dx*u;
ux = reshape(ux,Ny+1,2*Nx+2);
ux(:,1) = 0;
ux(:,Nx+1) = 0;
ux(:,Nx+2) = 0;
ux(:,2*Nx+2) = 0;
ux = reshape(ux,2*N2,1);
uy = Dy*u;
uy = reshape(uy,Ny+1,2*Nx+2);
uy(1,:) = 0;
uy(Ny+1,:) = 0;
uy = reshape(uy,2*N2,1);
lapl = Dx*ux+Dy*uy;
lapl(N2+1:2*N2) = d*lapl(N2+1:2*N2);
w = gamma*(ab-diag10*(u+u0v0)+(uu+u0u0).^2.*(vv+v0v0))+lapl;
127
128
Matlab-codes
Bibliography
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[4] Hansen, Vagn Lundsgaard - Moderne Analyse, 4th, 1995, World Scientific.
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edition, 1981, Springer-Verlag.
[6] Hollis, S. L.; Martin, R. H.; Pierre, M. - Global existence and boundedness in reaction-diffusion systems, Volume 18, No. 3, 1987, Siam J. Math.
Anal.
[7] Lions, J.-L.; Magenes, E. - Problèmes aux Limites non Homogènes, 2nd
edition, Vol 2, 1968, Springer-Verlag
[8] Murray, J. D. - Mathematical Biology I: An Introduction, 3rd Edition,
Volume 17, 2002, Springer-Verlag.
[9] Murray, J. D. - Mathematical Biology II: Spatial Models and Biomedical
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[10] Pao, C. V. - Nonlinear Parabolic and Elliptic Equations, 1992, Plenum
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[12] Trefethen, Lloyd N. - Spectral Methods in Matlab, Volume 1, 2002, Society for Industrial & Applied