Dynamics of nonlinear evolution
equations at resonance
Piotr Kokocki
Faculty of Mathematics and Computer Science
Nicolaus Copernicus University of Toruń
1 July 2012
STS Geometry in Dynamics
6th European Congress of Mathematicians
Introduction
Let Ω be an open subset Rn (n ≥ 1). Consider nonlinear heat equation
ut (x, t) = ∆u(x, t) + λu(x, t) + f (t, x, u(x, t)), t ≥ 0, x ∈ Ω
and nonlinear strongly damped wave equation
utt (x, t) = ∆u(x, t)+c∆ut (x, t)+λu(x, t)+f (t, x, u(x, t)), t ≥ 0, x ∈ Ω
where
I
c > 0 and λ is a real number,
I
∆ is the Laplace operator with Dirichlet boundary conditions,
I
f : [0, +∞) × Ω × R → R is a continuous map.
We will study the dynamics for these equations, and more precisely
I
the existence of T -periodic solutions (T > 0)
I
the existence of orbits connecting stationary points
in the case of rezonanse at infinity, that is, λ is an eigenvalue of −∆,
and f is bounded.
Main difficulties
In the presence of resonance, the above equations may not have periodic
solutions and orbits connecting stationary points for general nonlinearity
f . This fact has been explained in detail in my phd. To justify it, we put
f (t, x, s) := u0 (x) for x ∈ Ω, where u0 ∈ Ker(λI + ∆) \ {0}.
Therefore, if we want to look for periodic solutions and connecting orbits,
we have to find additional assumptions characterizing f , which allow us
to obtain the existence results.
I
The classical assumptions for f that guarantee the existence of
periodic solutions for the nonlinear heat and wave equations are:
• Landesman-Lazer conditions
,
• strong resonance conditions
I
.
The existence of orbits connecting stationary points for the heat and
wave equations has not been investigated so far.
Landesman-Lazer conditions
Assume that there are continuous functions f+ , f− : Ω → R such that
f+ (x) = lim f (t, x, s) and f− (x) = lim f (t, x, s).
s→+∞
s→−∞
We say that f satisfies Landesman-Lazer condition, provided
Z
Z
f+ (x)ū(x) dx +
f− (x)ū(x) dx > (<) 0
{ū>0}
{ū<0}
for ū ∈ Ker(λI + ∆), ū 6= {0}.
Strong resonance conditions
Assume that there is a continuous function f∞ : Ω → R such that
f∞ (x) =
lim
|s|→+∞
f (t, x, s) · s
for
t ∈ [0, +∞), x ∈ Ω.
We say that f satisfies strong resonance condition, provided
Z
f∞ (x) dx > (<) 0.
Ω
The following questions arise:
1. Do Landesman-Lazer and strong resonance conditions imply the
existence of orbits connecting the stationary points ?
2. Can we give new geometric conditions characterizing f , other than
Landesman-Lazer or strong resonance conditions ?
3. If this is the case, how these new conditions will affect on the
existence of periodic solutions and connecting orbits ?
4. Are these new geometric conditions more general than the classical
Landesman-Lazer or strong resonance conditions ?
Methods
Let A : X ⊃ D(A) → X be a positively defined sectorial operator on a
Banach space X and let X α for α ∈ (0, 1), be a fractional space given by
X α := D(Aα ). The nonlinear heat and wave equation may be abstractly
written in the form
(1)
u̇(t) = −Au(t) + λu(t) + F (t, u(t)),
(2)
ü(t) = −Au(t) − cAu̇(t) + λu(t) + F (t, u(t)),
t > 0,
t > 0,
where F : [0, +∞) × X α → X is a continuous map. It suffices to take
Au := −∆u and F (t, u) := f (t, ·, u(·)).
We say that these abstract equations are at resonance at infinity,
provided Ker(λI − A) 6= {0} and F is bounded.
Our goals figure on finding periodic solutions and orbits connecting
stationary points for the equations (1) and (2).
The second order equation (2) may be written as the first order equation
(3)
ẇ(t) = −Aw(t) + F(t, w(t)),
t > 0,
where A : E ⊃ D(A) → E is a linear operator on a Banach space
E := X α × X given by
D(A) := {(x, y) ∈ E | x + cy ∈ D(A)}
A(x, y) := (−y, A(x + cy) − λx)
for
(x, y) ∈ D(A),
and F : [0, +∞) × E → E is given by
F(t, (x, y)) := (0, F (t, x))
for
t ∈ [0, +∞), (x, y) ∈ E.
Let u( · ; x) : [0, +∞) → X α be a solution for (1) starting at x ∈ X α .
Similarly, let w( · ; (x, y)) : [0, +∞) → E be a solution for (3) starting at
(x, y) ∈ E.
Define the operators associated with the equations
u̇(t) = −Au(t) + λu(t) + F (t, u(t)) ∼ Φ : [0, +∞) × X α → X α ,
ü(t) = −Au(t) − cAu̇(t) + λu(t) + F (t, u(t)) ∼ Φ : [0, +∞) × E → E
which are given by the formulas
Φ(t, x) := u(t; x)
for
t ∈ [0, +∞), x ∈ X α ,
Φ(t, (x, y)) := w(t; (x, y))
for
t ∈ [0, +∞), (x, y) ∈ E.
Periodic solutions
Finding periodic solutions is reduced to finding fixed points for
translation along trajectory operators ΦT : X α → X α and
ΦT : E → E given by
ΦT := Φ(T, · ) and ΦT := Φ(T, · ).
In order to find fixed point of these operators we will use homotopy
invariants such as topological degree.
Orbits connecting stationary points
Here we assume that F : X α → X is time independent map.
By a stationary point x0 ∈ X α for equation (1) we mean a time
independent solution, that is,
Φ(t, x0 ) = x0
for
t ∈ [0, +∞).
By an orbit for equation (1) we mean a solution for semiflow Φ, that is,
a map u : R → X α satisfying
Φ(s, u(t)) = u(t + s)
for
s ≥ 0, t ∈ R.
We say that the orbit u : R → X α connects stationary points u− , u+ , if
lim u(t) = u−
t→−∞
and
lim u(t) = u+ .
t→+∞
The tool for finding the connecting orbits is the Conley index.
For the second order equation (2) the above definitions are analogous.
Plan of the studies
1. We find the geometrical conditions for nonlinear term F , which
guarantee the existence of periodic solutions and orbits connecting
stationary points for the first and second order equations.
2. We prove that conditions from point 1. generalize the classical
Landesman-Lazer and strong resonance conditions.
3. We use the conditions from point 1. to prove the index formulas and
criteria for the existence of periodic solutions for the first and second
order equations.
4. We use these conditions to prove the index formulas and criteria for
the existence of orbits connecting the stationary points for the first
and second order equations.
5. Applications to study heteroclinic orbits and periodic solutions of
partial differential equations.
Geometrical assumptions
Let A : X ⊃ D(A) → X be a positively defined sectorial operator on a
Banach space X such that the following assumptions are satisfied:
(A1) operator A has a compact resolvents,
(A2) there is a Hilbert space H with a scalar product h · , · iH and norm
k · kH and linear continuous injective map (inclusion) i : X ,→ H,
e : H ⊃ D(A)
e → H such that
(A3) there is a self-adjoint operator A
i×i
e in the sense of inclusion X × X −
Gr (A) ⊂ Gr (A)
−→ H × H.
Under the above assumptions (A1), (A2) and (A3):
1. Spectrum σ(A) of the operator A consists of the sequence of
eigenvalues
0 < λ1 < λ2 < . . . < λi < . . . ,
which is finite or λn → +∞ as n → +∞.
2. Let λ = λk . There is a decomposition of X on the direct sum of
closed spaces X := X− ⊕ X0 ⊕ X+ such that:
• X0 = Ker(λk I − A),
• X− is finite dimensional,
• if A± : X± ⊃ D(A± ) → X± is the part of A in X± , then
λk I − A+ is positive and λk I − A− is negative.
3. Since the injection X α ⊂ X is continuous, it follows that there is a
α
α
decomposition on a sum of closed spaces X α = X−
⊕ X0 ⊕ X+
,
where
α
α
X−
:= X α ∩ X− and X+
:= X α ∩ X+ .
• The part of the operator A in space Y ⊂ X is an operator AY given by
D(AY ) := {x ∈ D(A) | Ax ∈ Y }, AY x := Ax for x ∈ D(AY ).
Assumptions for the first order equations:
(
(G1)
hF (t, x + y), xiH > 0 for (t, y, x) ∈ [0, T ] × B × X0 , kxkH ≥ R,
(
(G2)
α
α
for every ball B ⊂ X+
⊕ X−
there is R > 0 such that
α
α
for every ball B ⊂ X+
⊕ X−
there is R > 0 such that
hF (t, x + y), xiH < 0 for (t, y, x) ∈ [0, T ] × B × X0 , kxkH ≥ R,
Assumptions for the second order equations:

α
α

 for every balls B1 ⊂ X+ ⊕ X− and B2 ⊂ X0 there is R > 0
such that hF (t, x + y), xiH > −hF (t, x + y), ziH
(G3)


for (t, y, z) ∈ [0, T ] × B1 × B2 , x ∈ X0 such that kxkH ≥ R.

α
α

 for every balls B1 ⊂ X+ ⊕ X− and B2 ⊂ X0 there is R > 0
such that hF (t, x + y), xiH < −hF (t, x + y), ziH
(G4)


for (t, y, z) ∈ [0, T ] × B1 × B2 , x ∈ X0 such that kxkH ≥ R.
Orbits connecting stationary points
In order to find the orbits connecting stationary points:
1. we prove index formulas expressing the Conley index for semiflows
Φ and Φ on sufficiently large ball in the terms of geometrical
assumptions (G1)–(G4),
2. we apply theorems concerning irreducible invariant sets proved by
Rybakowski (1980).
From now on we assume that λ = λk for some k ≥ 1, is an eigenvalue of
the operator A.
Furthermore, for any l ≥ 0 we define
(
0
for
dl := Pl
i=1 dim Ker(λi I − A) for
Then one can check that dk−1 = dim X− for k ≥ 1.
l = 0,
l ≥ 1.
Theorem 1 (Index formula for first order equations)
There is a closed isolated neighborhood N ⊂ X α , admissible with respect
to the semiflow Φ such that 0 ∈ int N and, for K := Inv(N, Φ), the
following statements hold:
(i) if condition (G1) is satisfied, then h(Φ, K) = Σdk ,
(ii) if condition (G2) is satisfied, then h(Φ, K) = Σdk−1 .
Theorem 2 (Index formula for second order equations)
There is a closed isolated neighborhood N ⊂ E, admissible with respect
to the semiflow Φ such that 0 ∈ int N and, for K := Inv(N, Φ), the
following statements hold:
(i) if condition (G3) is satisfied, then h(Φ, K) = Σdk ,
(ii) if condition (G4) is satisfied, then h(Φ, K) = Σdk−1 .
• If (X, x0 ) is a topological space and x0 ∈ X, then the reducible
suspension is the topological space defined by
ΣX := (X × [0, 1])/(X × {0} ∪ X × {1} ∪ {x0 } × [0, 1]).
Then
Σ0 := S 0
and Σk := Σ(Σk−1 ) for k ≥ 1.
• An invariant set in N with respect to Φ is given by
Inv(N, Φ) := {x ∈ N | there is an orbit u : R → X α for ϕ
such that u(0) = x and u(R) ⊂ N }.
• A closed invariant set K ⊂ X is called isolated, if there is a closed set
N ⊂ X such that
K = Inv(N, Φ) ⊂ int N.
Such a set N is referred to as isolated neighborhood for K.
Additionally, we assume that:
(A4) F (0) = 0 and F is differentiable at 0 and there is ν ∈ R such that
DF (0)[x] = νx for x ∈ X α .
Theorem 3 (Criterion for the first order equation)
Equation (1) admits a relatively compact nonzero orbit u : R → X α such
that
lim u(t) = 0
or
lim u(t) = 0,
t→−∞
t→+∞
provided one of the following conditions holds:
(i) condition (G1) is satisfied and λl < λ + ν < λl+1 where λl 6= λ;
(ii) condition (G1) is satisfied and λ + ν < λ1 ;
(iii) condition (G2) is satisfied, λl−1 < λ + ν < λl and λ 6= λl , where
l ≥ 2;
(iv) condition (G2) is satisfied, λ + ν < λ1 and λ 6= λ1 .
Theorem 4 (Criterion for the second order equation)
Equation (2) admits a relatively compact nonzero orbit w : R → E such
that
lim w(t) = 0
or
lim w(t) = 0,
t→−∞
t→+∞
provided one of the following conditions holds:
(i) condition (G3) is satisfied and λl < λ + ν < λl+1 where λl 6= λ;
(ii) condition (G3) is satisfied and λ + ν < λ1 ;
(iii) condition (G4) is satisfied, λl−1 < λ + ν < λl and λ 6= λl , where
l ≥ 2;
(iv) condition (G4) is satisfied, λ + ν < λ1 and λ 6= λ1 .
Periodic solutions
To find periodic solutions we will use index formulas expressing
topological degree for the operators ΦT and ΦT with respect to
sufficiently large ball, in the terms of conditions (G1)–(G4).
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ΦT is completely continuous – Leray-Schauder degree degLS ,
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ΦT is condensing – Sadovskii degree degC .
Let N be a Banach space.
• By measure of noncompactness we mean map βN : B(N ) → R given by
βN (Ω) := inf{r > 0 | Ω ⊂
kr
[
B(xi , r), where xi ∈ N for i = 1, . . . , kr }.
i=1
• A map F is completely continuous, provided F (Ω) is relatively compact for every
bounded Ω ⊂ N .
• A map F is condensing, provided there is k ∈ [0, 1) such that
βN (F (Ω)) ≤ kβN (Ω)
for every bounded Ω ⊂ N .
Theorem 5 (Index formula for the first order equations)
The following statements are true.
(i) If condition (G1) is satisfied, then there is R > 0 such that ΦT (x) 6=
x for x ∈ X α with norm kxkα ≥ R and
degLS (I − ΦT , B(0, R)) = (−1)dk .
(ii) If condition (G2) is satisfied, then there is R > 0 such that ΦT (x) 6=
x for x ∈ X α with norm kxkα ≥ R and
degLS (I − ΦT , B(0, R)) = (−1)dk−1 .
Theorem 6 (Index formula for the second order equations)
The following statements are true.
(i) If condition (G3) is satisfied, then there is R > 0 such that
ΦT (x, y) 6= (x, y) for (x, y) ∈ E with norm k(x, y)kE ≥ R and
degC (I − ΦT , B(0, R)) = (−1)dk .
(ii) If condition (G4) is satisfied, then there is R > 0 such that
ΦT (x, y) 6= (x, y) for (x, y) ∈ E with norm k(x, y)kE ≥ R and
degC (I − ΦT , B(0, R)) = (−1)dk−1 .
Application to partial differential equations
Let Ω ⊂ Rn for n ≥ 1, be an open bounded set with C ∞ boundary.
Assume that:
I
∆ is Laplace operator with Dirichlet conditions,
I
f : [0, +∞) × Ω × R × Rn → R is a continuous, bounded map,
which is lipschitz on bounded sets.
Let X := Lp (Ω) for p ≥ 1. Define Ap : X ⊃ D(Ap ) → X as linear
operator given by
D(Ap ) := W02,p (Ω)
and Ap ū := −∆ū
for
ū ∈ D(Ap ).
Then Ap is a positive definite sectorial operator.
I
Assume that α ∈ (3/4, 1) and p ≥ 2n.
Embedding theorems for fractional spaces imply that the inclusion
X α ⊂ C(Ω) is continuous.
Therefore we can define the Niemycki operator F : [0, +∞) × X α → X
given for every ū ∈ X α by the formula
F (t, ū)(x) := f (t, x, ū(x)) for t ∈ [0, +∞) and x ∈ Ω.
1. One can prove that A2 is symmetric and Ap has compact resolvents.
Hence assumption (A1) is satisfied.
2. Take H := L2 (Ω) with standard inner product and norm. Since Ω is
bounded and p ≥ 2, we derive that i : Lp (Ω) ,→ L2 (Ω) is a
continuous embedding. In consequence assumption (A2) is satisfied.
3. Using again the boundedness of Ω, one can prove that for the
e := A2 , the inclusion Ap ⊂ A
e is satisfied in the sense of
operator A
the map i × i. Therefore (A3) holds.
Theorem 7 (Properties of Niemycki operator)
Assume that f+ , f− : Ω → R are continuous functions such that
f+ (x) = lim f (t, x, s)
s→+∞
and f− (x) = lim f (t, x, s)
s→−∞
for x ∈ Ω, uniformly for t ∈ [0, +∞).
(i) If the following condition is satisfied
Z
Z
(LL1)
f+ (x)ū(x) dx +
{ū>0}
f− (x)ū(x) dx > 0
{ū<0}
for ū ∈ Ker(λI − Ap ) \ {0}, then conditions (G1) and (G3) hold.
(ii) If the following condition is satisfied
Z
Z
(LL2)
f+ (x)ū(x) dx +
{ū>0}
f− (x)ū(x) dx < 0
{ū<0}
for ū ∈ Ker(λI − Ap ) \ {0}, then conditions (G2) and (G4) hold.
Theorem 8 (Properties of Niemycki operator)
Assume that f∞ : Ω → R is a continuous function such that
f∞ (x) =
lim
|s|→+∞
f (t, x, s) · s
for x ∈ Ω, uniformly for t ∈ [0, +∞).
(i) The following inequality
Z
(SR1)
f∞ (x) dx > 0,
Ω
implies that conditions (G1) and (G3) are satisfied.
(i) The following inequality
Z
(SR2)
f∞ (x) dx < 0.
Ω
implies that conditions (G2) and (G4) are satisfied.
We consider the following equations
(4)
(5)
ut (t, x) = ∆ u(t, x) + λu(t, x) + f (t, x, u(t, x))
utt (t, x) = ∆ u(t, x) + ∆ ut (t, x) + λu(t, x) + f (t, x, u(t, x)).
Theorem 9 (Existence of periodic solutions)
1. Assume that f+ , f− : Ω → R are continuous functions such that
f+ (x) = lim f (t, x, s), f− (x) = lim f (t, x, s) for x ∈ Ω.
s→+∞
s→−∞
If one of the conditions (LL1) or (LL2)is satisfied, then equations
(4) and (5) admit T -periodic solutions.
2. Assume that there is a continuous function f∞ : Ω → R such that
f∞ (x) =
lim
|s|→+∞
f (t, x, s) · s for x ∈ Ω.
If one of the conditions (SR1) or (SR2) is satisfied, then equations
(4) and (5) admit T -periodic solution.
Assume that f : Ω × R → R is a map of class C 1 such that
I
f (x, 0) = 0 for x ∈ Ω
I
there is ν ∈ R such that ν = Ds f (x, 0) for x ∈ Ω.
Theorem 10 (Existence of heteroclinic orbits)
Assume that f+ , f− : Ω → R are continuous functions such that
f+ (x) = lim f (x, s),
s→+∞
f− (x) = lim f (x, s) for x ∈ Ω.
s→−∞
Equations (4) and (5) admit heteroclinic orbit, provided one of the
following conditions is satisfied:
(i) condition (LL1) is satisfied and λl < λ + ν < λl+1 gdzie λl 6= λ;
(ii) condition (LL1) is satisfied and λ + ν < λ1 ;
(iii) condition (LL2) is satisfied, λl−1 < λ + ν < λl and λ 6= λl , where
l ≥ 2;
(iv) condition (LL2) is satisfied, λ + ν < λ1 and λ 6= λ1 .
Theorem 11 (Existence of heteroclinic orbits)
Assume that f∞ : Ω → R is a continuous function such that
f∞ (x) =
lim
|s|→+∞
f (x, s) · s for x ∈ Ω.
Equations (4) and (5) admit heteroclinic orbit, provided one of the
following conditions holds:
(i) condition (SR1) is satisfied and λl < λ + ν < λl+1 where λl 6= λ;
(ii) condition (SR1) is satisfied and λ + ν < λ1 ;
(iii) condition (SR2) is satisfied and λl−1 < λ + ν < λl where λ 6= λl ,
l ≥ 2;
(iv) condition (SR2) is satisfied and λ + ν < λ1 .
Literatura
I
Rybakowski K. P., The homotopy index and partial differential
equations, Universitext, Springer-Verlag 1987;
I
Rybakowski K. P., Nontrivial solutions of elliptic boundary value
problems with resonance at zero, Ann. Mat. Pura Appl. 139 (1985)
237–277;
I
Rybakowski K. P., Irreducible invariant sets and asymptotically linear
functional-differential equations, Boll. Un. Mat. Ital. B (6) 3
(1984), no. 2, 245–271;
I
Landesman E. M., Lazer A. C., Nonlinear perturbations of linear
elliptic boundary value problems at resonance, J. Math. Mech. 19
(1970) 609–623.
I
Bartolo P., Benci V., oraz Fortunato D., Abstract critical point
theorems and applications to some nonlinear problems with “strong”
resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981–1012.
Lapunov functions V : X α → R and Ve : E → R for equations (4) and
(5), respectively are given by
Z
1 1/2 2
2
f (x, ū(x)) dx,
V (ū) :=
k∆ ūkL2 (Ω) − kūkL2 (Ω) −
2
Ω
Z
1 1/2 2
2
2
e
V (ū, v̄) :=
k∆ ūkL2 (Ω) + kv̄kL2 (Ω) − kūkL2 (Ω) −
f (x, ū(x)) dx.
2
Ω