Introduction
The scalar KPP equation
The non-cooperative KPP system
Non-cooperative Fisher–KPP systems: traveling
waves and long-time behavior
Léo Girardin
Laboratoire Jacques-Louis Lions, Université Paris 6
January 6th, 2017
Introduction
The scalar KPP equation
Introduction
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
A few models from mathematical biology
Introduction
The scalar KPP equation
The non-cooperative KPP system
Mutation–competition–diffusion model with local
mutations
Continuous trait model:
Cane toads equation with non-local competition
(Bénichou–Calvez–Meunier–Voituriez, 2012)
(
Rθ
∂t n − θ∂xx n − α∂θθ n = n (t, x, θ) r − θ n (t, x, θ0 ) dθ0
∂θ n (t, x, θ) = ∂θ n t, x, θ = 0 for all (t, x) ∈ R2
n function of (t, x, θ), θ ∈ θ, θ motility trait, α mutation rate, r growth
Rθ
rate and θ n (t, x, θ0 ) dθ0 total population present at (t, x).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Discrete trait model:
L–V mutation-competition-diffusion system with step-wise mutations
∂t uN − diag (θi )i∈[N ] ∂xx uN = ruN +
α
2 MLap,N uN
θN
− (θN 1N uN ) ◦ uN ,
with MLap the discrete Neumann Laplacian.
◦ Hadamard product, [N ] = {1, 2, . . . , N }.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Mutation–competition–diffusion model with long-range
mutations
Continuous trait model:
Doubly non-local cane toads equation (Arnold–Desvillettes–Prévost,
2012)
Z
∂t n − d (θ) ∂xx n = r (θ) n (t, x, θ) +
θ
n (t, x, θ0 ) K (θ, θ0 ) dθ0
θ
Z
− n (t, x, θ)
θ
θ
n (t, x, θ0 ) C (θ, θ0 ) dθ0 .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Discrete trait model:
L–V mutation-competition-diffusion system with long-range mutations
∂t u − DN ∂xx u = LN u − (CN u) ◦ u,
with:
DN = diag (d (θi ))i∈[N ] ,
LN = diag (r (θi ))i∈[N ] + θN (K (θi , θj ))(i,j)∈[N ]2 ,
CN = θN (C (θi , θj ))(i,j)∈[N ]2 .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Age-structured model with overcrowding effect
Continuous age model:
Non-linear age-structured equation (Gurtin–MacCamy, 1977)

RA
0
0
0

∂t n + ∂a n − d (a) ∂xx n = − r (a) + 0 n (t, x, a ) C (a, a ) da n
RA
n (t, x, 0) = am n (t, x, a0 ) K (a0 ) da0 for all (t, x) ∈ R2


n (t, x, A) = 0 for all (t, x) ∈ R2
n function of (t, x, a), a ∈ [0, A] age variable, am ≥ 0 maturation age,
A > am maximal age, d diffusion rate, r mortality rate, C competition
kernel and K birth rate.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Discrete age model:
Non-linear age-structured system
∂t u − DN ∂xx u = LN u − (CN u) ◦ u
with
DN = diag (d (ai ))i∈[N ] ,
LN = Lmortality,N + Lbirth,N + Laging,N ,
CN = aN +1 (C (ai , aj ))(i,j)∈[N ]2 .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Detail of LN :
jm,N = min {j ∈ [N ] | aj ≥ am } ,
Lmortality,N = −diag r (ai )i∈[N ] ,


0 . . . 0 K ajm,N
. . . K (aN )
0
...
0 


Lbirth,N = aN +1  .
..  ,
 ..
. 
0
...
0


0 0 ... ...
0

.. 
.
..
1 −1
. 


1 
.
.. 
Laging,N =
0 . . . . . . . . .
.

aN +1 
. .

.
.
..
..
..
 ..
0
0 ...
0
1 −1
Introduction
The scalar KPP equation
Structural similarities of the three examples
Common factors
Parabolic systems;
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
Structural similarities of the three examples
Common factors
Parabolic systems;
Weakly coupled (coupling only in the 0th order term);
Introduction
The scalar KPP equation
The non-cooperative KPP system
Structural similarities of the three examples
Common factors
Parabolic systems;
Weakly coupled (coupling only in the 0th order term);
Non-linear, non-cooperative;
Introduction
The scalar KPP equation
The non-cooperative KPP system
Structural similarities of the three examples
Common factors
Parabolic systems;
Weakly coupled (coupling only in the 0th order term);
Non-linear, non-cooperative;
But linearization at 0 cooperative and fully coupled (irreducible 0th
order term L);
Introduction
The scalar KPP equation
The non-cooperative KPP system
Structural similarities of the three examples
Common factors
Parabolic systems;
Weakly coupled (coupling only in the 0th order term);
Non-linear, non-cooperative;
But linearization at 0 cooperative and fully coupled (irreducible 0th
order term L);
C positive.
Introduction
The scalar KPP equation
General system
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
Non-cooperative KPP system
General system
∂t u − D∂xx u = Lu − c [u] ◦ u.
Unknown:
u:
(EKP P )
R2
→
RN
.
(t, x) 7→ (ui (t, x))i∈[N ]
Fixed parameters:
D = diag (di )i∈[N ] with di > 0 for all i ∈ [N ], L ∈ M, c ∈ C 1 RN , RN .
Stationary problem:
−Du00 = Lu − c [u] ◦ u
(SKP P ) .
[ ] composition of functions, M N × N matrices.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Assumptions
Cooperative linear part
L is such that L − min (li,i ) I is non-negative and irreducible.
i∈[N ]
Introduction
The scalar KPP equation
The non-cooperative KPP system
Assumptions
Cooperative linear part
L is such that L − min (li,i ) I is non-negative and irreducible.
i∈[N ]
Growth rate maximal at 0
c (K) ⊂ K and c (0) = 0.
K, K+ , K++ resp. non-negative, non-negative non-zero, positive cone.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Assumptions
Cooperative linear part
L is such that L − min (li,i ) I is non-negative and irreducible.
i∈[N ]
Growth rate maximal at 0
c (K) ⊂ K and c (0) = 0.
K, K+ , K++ resp. non-negative, non-negative non-zero, positive cone.
At least linear growth of the competition at infinity
2
There exists (α? , δ, c) ∈ [1, +∞) × K++ such that, for all
N
P
(α, n, i) ∈ [α? , +∞) × K+ ∩ S (0, 1) × [N ], if
li,j nj ≥ 0, then
j=1
αδ ci ≤ ci (αn).
Introduction
The scalar KPP equation
The scalar KPP equation
The non-cooperative KPP system
Introduction
The scalar KPP equation
Definition
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
The Fisher–KPP equation
Historical Fisher or Kolmogorov–Petrovsky–Piskunov equation (1937)
∂t u − d∆x u = ru(1 − u) for all (t, x) ∈ R × Rn
with d > 0, r > 0.
Introduction
The scalar KPP equation
The non-cooperative KPP system
The Fisher–KPP equation
Historical Fisher or Kolmogorov–Petrovsky–Piskunov equation (1937)
∂t u − d∆x u = ru(1 − u) for all (t, x) ∈ R × Rn
with d > 0, r > 0.
Modern F–KPP equation
∂t u − d∆x u = f (u) for all (t, x) ∈ R × Rn
with
f 0 (0) > 0,
f 0 (0)u ≥ f (u) for all u ≥ 0,
existence of M > 0 such that f (u) < 0 for all u ≥ M .
Introduction
The scalar KPP equation
Well-known results
The non-cooperative KPP system
Introduction
The scalar KPP equation
Well-known results
In order to fix the ideas, f (u) = ru(1 − u) and x ∈ R.
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
Well-known results
In order to fix the ideas, f (u) = ru(1 − u) and x ∈ R.
Upper and lower bound
Provided the initial condition u0 satisfies 0 ≤ u0 ≤ 1, then so does the
associated solution u for all t > 0.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Well-known results
In order to fix the ideas, f (u) = ru(1 − u) and x ∈ R.
Upper and lower bound
Provided the initial condition u0 satisfies 0 ≤ u0 ≤ 1, then so does the
associated solution u for all t > 0.
Stability of the stationary states
u = 0 is unstable, u = 1 is stable.
In various senses, here focusing on generalized principal eigenvalues.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Well-known results
In order to fix the ideas, f (u) = ru(1 − u) and x ∈ R.
Upper and lower bound
Provided the initial condition u0 satisfies 0 ≤ u0 ≤ 1, then so does the
associated solution u for all t > 0.
Stability of the stationary states
u = 0 is unstable, u = 1 is stable.
In various senses, here focusing on generalized principal eigenvalues.
Persistence property
Asymptotically in time and locally uniformly in space, any non-negative
non-zero solution u converges to 1.
Related result in bounded domains with Dirichlet boundary conditions
(Cantrell–Cosner, 2003).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Well-known results: traveling waves
Traveling waves
√
For any speed c ≥ c? = 2 dr, there exists a unique, up to translation,
profile φc ∈ C 2 (R) such that:
u : (t, x) 7→ φc (x − ct) is a positive solution;
φc (−∞) = 1;
φc (+∞) = 0.
φc is decreasing and satisfies:
−dφ00c − cφ0c − rφc (1 − φc ) = 0 in R.
Furthermore, for all c ∈ [0, c? ), there exists no such profile.
c? is the minimal wave speed.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Well-known results: traveling waves
Traveling waves
√
For any speed c ≥ c? = 2 dr, there exists a unique, up to translation,
profile φc ∈ C 2 (R) such that:
u : (t, x) 7→ φc (x − ct) is a positive solution;
φc (−∞) = 1;
φc (+∞) = 0.
φc is decreasing and satisfies:
−dφ00c − cφ0c − rφc (1 − φc ) = 0 in R.
Furthermore, for all c ∈ [0, c? ), there exists no such profile.
c? is the minimal wave speed.
Strong connection with the linearized equation
−dφ00c − cφ0c − rφc = 0.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Well-known results: spreading speed
Spreading speed
Let x0 ∈ R and v ∈ Cb (R, [0, 1]) be non-negative non-zero.
Then c? coincides with the spreading speed associated with the Cauchy
problem with the “front-like initial condition” u0 (x) = H(x0 − x)v(x).
In the following sense:
lim
sup
t→+∞ x∈(y,+∞)
lim
inf
u (t, x + ct) = 0 for all c ∈ (c? , +∞) and all y ∈ R,
t→+∞ x∈[−R,R]
u (t, x + ct) = 1 for all c ∈ [0, c? ) and all R > 0.
Introduction
The scalar KPP equation
The non-cooperative KPP system
The non-cooperative KPP system
Introduction
The scalar KPP equation
“Structural similarities”
The non-cooperative KPP system
Introduction
The scalar KPP equation
“Structural similarities”
Analogy 1
L − min (li,i ) I non-negative and irreducible.
i∈[N ]
0
f (0) > 0.
The non-cooperative KPP system
Introduction
The scalar KPP equation
“Structural similarities”
Analogy 1
L − min (li,i ) I non-negative and irreducible.
i∈[N ]
0
f (0) > 0.
Analogy 2
c (0) = 0 and c (K) ⊂ K.
For all u ≥ 0, f 0 (0)u ≥ f (u).
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
“Structural similarities”
Analogy 1
L − min (li,i ) I non-negative and irreducible.
i∈[N ]
0
f (0) > 0.
Analogy 2
c (0) = 0 and c (K) ⊂ K.
For all u ≥ 0, f 0 (0)u ≥ f (u).
Analogy 3
For all (v, i) ∈ K+ × [N ], if
N
P
j=1
δ
|v| ci ≤ ci (v).
For all u ≥ M , f (u) < 0.
li,j vj > 0 and if |v| ≥ α? , then:
Introduction
The scalar KPP equation
The non-cooperative KPP system
I just lied! I’m so mean!
Analogy 1 is not strong enough!
Counter-example: if
L=
−1 − ε
1
,
1
−1 − ε
for some ε > 0, then indeed L − min (li,i ) I is non-negative and
i∈[N ]
irreducible.
But all bounded non-negative non-zero solutions go extinct exponentially
fast, uniformly in space. The persistence property is lost! No hope to
have positive steady states or traveling waves!
Introduction
The scalar KPP equation
Ok, ok, here’s the answer.
Supplementary crucial assumption
λP F (L) > 0
In the counter-example, λP F (L) = −ε < 0.
The non-cooperative KPP system
Introduction
The scalar KPP equation
My results
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
Strong positivity
Theorem (G., 2016)
For all non-negative classical solutions u of (EKP P ) set in (0, +∞) × R,
if x 7→ u (0, x) is non-negative non-zero, then u is positive in
(0, +∞) × R.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Strong positivity
Theorem (G., 2016)
For all non-negative classical solutions u of (EKP P ) set in (0, +∞) × R,
if x 7→ u (0, x) is non-negative non-zero, then u is positive in
(0, +∞) × R.
Consequently, all non-negative non-zero classical solutions of (SKP P ) are
positive.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Absorbing set and upper estimates
Theorem (G., 2016)
There exists a positive function g ∈ C ([0, +∞), K++ ), component-wise
non-decreasing, such that all non-negative classical solutions u of
(EKP P ) set in (0, +∞) × R satisfy:
u (t, x) ≤ gi sup ui (0, x)
for all (t, x) ∈ [0, +∞) × R
x∈R
i∈[N ]
and furthermore, if x 7→ u (0, x) is bounded, then:
lim sup sup ui (t, x)
≤ g (0) .
t→+∞ x∈R
i∈[N ]
Introduction
The scalar KPP equation
The non-cooperative KPP system
Absorbing set and upper estimates
Theorem (G., 2016)
There exists a positive function g ∈ C ([0, +∞), K++ ), component-wise
non-decreasing, such that all non-negative classical solutions u of
(EKP P ) set in (0, +∞) × R satisfy:
u (t, x) ≤ gi sup ui (0, x)
for all (t, x) ∈ [0, +∞) × R
x∈R
i∈[N ]
and furthermore, if x 7→ u (0, x) is bounded, then:
lim sup sup ui (t, x)
≤ g (0) .
t→+∞ x∈R
i∈[N ]
Consequently, all bounded non-negative classical solutions u of (SKP P )
satisfy:
u ≤ g (0) .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Persistence or extinction dichotomy
Theorem (G., 2016)
1
Assume λP F (L) < 0. Then all bounded non-negative classical
solutions of (EKP P ) set in (0, +∞) × R converge asymptotically in
time, exponentially fast, and uniformly in space to 0.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Persistence or extinction dichotomy
Theorem (G., 2016)
1
Assume λP F (L) < 0. Then all bounded non-negative classical
solutions of (EKP P ) set in (0, +∞) × R converge asymptotically in
time, exponentially fast, and uniformly in space to 0.
2
Conversely, assume λP F (L) > 0. Then, for all bounded intervals
I ⊂ R, all bounded positive classical solutions u of (EKP P ) set in
(0, +∞) × R satisfy:
lim inf inf ui (t, x)
∈ K++ .
t→+∞ x∈I
i∈[N ]
Introduction
The scalar KPP equation
The non-cooperative KPP system
Existence of steady states
Theorem (G., 2016)
1
If λP F (L) < 0, there exists no positive classical solution of (SKP P ).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Existence of steady states
Theorem (G., 2016)
1
If λP F (L) < 0, there exists no positive classical solution of (SKP P ).
2
If λP F (L) = 0 and if, for all associated positive eigenvectors v,
c (v) > 0, there exists no bounded positive classical solution of
(SKP P ).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Existence of steady states
Theorem (G., 2016)
1
If λP F (L) < 0, there exists no positive classical solution of (SKP P ).
2
If λP F (L) = 0 and if, for all associated positive eigenvectors v,
c (v) > 0, there exists no bounded positive classical solution of
(SKP P ).
3
If λP F (L) > 0, there exists a bounded positive classical solution of
(SKP P ) provided either one of the following conditions is satisfied:
1
2
L̂ + I 0;
for all (n, α) ∈ S+ (0, 1) × [α? , +∞), αδ c ≤ c (αn).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Existence of steady states
Theorem (G., 2016)
1
If λP F (L) < 0, there exists no positive classical solution of (SKP P ).
2
If λP F (L) = 0 and if, for all associated positive eigenvectors v,
c (v) > 0, there exists no bounded positive classical solution of
(SKP P ).
3
If λP F (L) > 0, there exists a bounded positive classical solution of
(SKP P ) provided either one of the following conditions is satisfied:
1
2
4
L̂ + I 0;
for all (n, α) ∈ S+ (0, 1) × [α? , +∞), αδ c ≤ c (αn).
There exists a bounded positive classical solution of (SKP P ) if and
only if there exists a constant positive solution of (SKP P ).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Traveling waves
Definition
A traveling wave solution
of (EKP P ) is a profile–speed pair
(p, c) ∈ C 2 R, RN × [0, +∞) which satisfies:
1
2
u : (t, x) 7→ p (x − ct) is a bounded positive classical solution of
(EKP P );
lim inf pi (ξ)
∈ K+ ;
ξ→−∞
3
i∈[N ]
lim p (ξ) = 0.
ξ→+∞
Introduction
The scalar KPP equation
The non-cooperative KPP system
Traveling waves
Definition
A traveling wave solution
of (EKP P ) is a profile–speed pair
(p, c) ∈ C 2 R, RN × [0, +∞) which satisfies:
1
2
u : (t, x) 7→ p (x − ct) is a bounded positive classical solution of
(EKP P );
lim inf pi (ξ)
∈ K+ ;
ξ→−∞
3
i∈[N ]
lim p (ξ) = 0.
ξ→+∞
The traveling wave satisfies:
−Dp00 − cp0 = Lp − c [p] ◦ p in R.
(T W [c])
Introduction
The scalar KPP equation
The non-cooperative KPP system
Theorem (G., 2016)
Assume λP F (L) > 0.
1
There exists c? > 0 such that:
1
2
there exists no traveling wave solution of (EKP P ) with speed c
for all c ∈ [0, c? );
if, furthermore, Dc (v) ≥ 0 for all v ∈ K, then there exists a
traveling wave solution of (EKP P ) with speed c for all c ≥ c? .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Theorem (G., 2016)
Assume λP F (L) > 0.
1
There exists c? > 0 such that:
1
2
2
there exists no traveling wave solution of (EKP P ) with speed c
for all c ∈ [0, c? );
if, furthermore, Dc (v) ≥ 0 for all v ∈ K, then there exists a
traveling wave solution of (EKP P ) with speed c for all c ≥ c? .
All profiles p satisfy:
p ≤ g (0) and
lim inf pi (ξ)
ξ→−∞
i∈[N ]
∈ K++ .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Theorem (G., 2016)
Assume λP F (L) > 0.
1
There exists c? > 0 such that:
1
2
2
there exists no traveling wave solution of (EKP P ) with speed c
for all c ∈ [0, c? );
if, furthermore, Dc (v) ≥ 0 for all v ∈ K, then there exists a
traveling wave solution of (EKP P ) with speed c for all c ≥ c? .
All profiles p satisfy:
p ≤ g (0) and
3
lim inf pi (ξ)
ξ→−∞
∈ K++ .
i∈[N ]
All profiles are component-wise decreasing in a neighborhood of
+∞.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Corollary: estimates for c? (G., 2016)
Assume that (di )i∈[N ] is ordered. Then we have:
2
p
p
d1 λP F (L) ≤ c? ≤ 2 dN λP F (L).
If d1 < dN , both inequalities are strict. If d1 = dN , both inequalities are
equalities.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Corollary: estimates for c? (G., 2016)
Assume that (di )i∈[N ] is ordered. Then we have:
2
p
p
d1 λP F (L) ≤ c? ≤ 2 dN λP F (L).
If d1 < dN , both inequalities are strict. If d1 = dN , both inequalities are
equalities.
Furthermore, for all i ∈ [N ] such that li,i > 0, we have:
p
c? > 2 di li,i .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Corollary: estimates for c? , second part
Let q be the unique
positive eigenvector associated with
λP F µ2c? D + L satisfying the normalization 11,N q = 1.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Corollary: estimates for c? , second part
Let q be the unique
positive eigenvector associated with
λP F µ2c? D + L satisfying the normalization 11,N q = 1.
Let r ∈ RN and M ∈ M be given by the unique decomposition of L of
the form:
L = diagr + M with MT 1N,1 = 0.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Corollary: estimates for c? , second part
Let q be the unique
positive eigenvector associated with
λP F µ2c? D + L satisfying the normalization 11,N q = 1.
Let r ∈ RN and M ∈ M be given by the unique decomposition of L of
the form:
L = diagr + M with MT 1N,1 = 0.
Let (hdi , hri) ∈ (0, +∞) × R be defined as:
(hdi , hri) = dT q, rT q .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Corollary: estimates for c? , second part
Let q be the unique
positive eigenvector associated with
λP F µ2c? D + L satisfying the normalization 11,N q = 1.
Let r ∈ RN and M ∈ M be given by the unique decomposition of L of
the form:
L = diagr + M with MT 1N,1 = 0.
Let (hdi , hri) ∈ (0, +∞) × R be defined as:
(hdi , hri) = dT q, rT q .
If hri ≥ 0, then:
p
c? ≥ 2 hdi hri.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Spreading speed
Theorem (G., 2016)
Assume λP F (L) > 0.
Let x0 ∈ R and v ∈ Cb R, RN be non-negative non-zero.
Then c? coincides with the spreading speed associated with the Cauchy
problem with the “front-like initial condition” u0 (x) = H(x0 − x)v(x).
In the following sense:
!
lim
sup
t→+∞ x∈(y,+∞)
lim inf
inf
t→+∞ x∈[−R,R]
= 0 for all c ∈ (c? , +∞) and all y ∈ R,
ui (t, x + ct)
i∈[N ]
ui (t, x + ct)
∈ K++ for all c ∈ [0, c? ) and all R > 0.
i∈[N ]
Introduction
The scalar KPP equation
Key tools & technical lemmas
The non-cooperative KPP system
Introduction
The scalar KPP equation
The non-cooperative KPP system
Saturation hypersurfaces
Lemma
There exists k ∈ K++ such that, for all i ∈ [N ] and for all v ∈ K\e⊥
i :
(L (v + ki ei ) − c (v + ki ei ) ◦ (v + ki ei ))i < 0.
|αi,m mi | = αi,m mi
δ
≤ αi,m
mi
PN
j=1 li,j mj
≤
ci
(li,j )j∈[N ] ≤
ci
Introduction
The scalar KPP equation
The non-cooperative KPP system
Harnack inequalities for linear cooperative systems
Implies positivity of non-negative non-zero solutions. Special importance
of the irreducibility.
See Araposthasis–Gosh–Marcus, 1999, or Chen–Zhao, 1997, for the
elliptic case.
See Földes–Poláčik, 2009, for the parabolic case.
Introduction
The scalar KPP equation
The non-cooperative KPP system
Generalized coupled super- and sub-solutions (Pao, 1992)
Definition
(u, u) is a pair of generalized coupled super- and sub-solutions for the
elliptic system:
−Du00 = Lu − c [u] ◦ u in (−R0 , R0 ) in (−R0 , R0 )
u (±R) = u.
if:

−di u00i ≥ (L (v) − c [v] ◦ v)i
−di u00i ≤ (L (v) − c [v] ◦ v)i

ui (±R0 ) ≥ ui (±R0 )
for all v ∈ F such that vi = ui ,
for all v ∈ F such that vi = ui ,
where:
F = v ∈ C [−R0 , R0 ] , RN | u ≤ v ≤ u .
Implies existence of a solution u ∈ F .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Exponential generalized principal eigenfunctions
Lemma
Let (c, λ) ∈ R2 . We consider the equation:
−Dp00 − cp0 − (L + λI) p = 0 in R.
(T W0 [c, λ])
If there exists a classical positive solution p of (T W0 [c, λ]), then there
exists (µ, n) ∈ R × K++ such that q : ξ 7→ e−µξ n is a solution of
(T W0 [c, λ]).
Introduction
The scalar KPP equation
The non-cooperative KPP system
Definition of c?
Lemma
1
The quantity:
c? = min
µ>0
!
λP F µ2 D + L
µ
is well-defined, positive and uniquely attained.
2
For all c ∈ [0, +∞), the equation
exactly:
1
2
3
λP F (µ2 D+L)
µ
= c admits in R
no solution if c < c? ;
one solution µc? ∈ (0, +∞) if c = c? ;
two solutions (µ1,c , µ2,c ) if c > c? , which satisfy moreover
0 < µ1,c < µc? < µ2,c .
Introduction
The scalar KPP equation
The non-cooperative KPP system
Super- and sub-solution method for (T W [c])
Super-solution: as expected, p : ξ 7→ e−µ1,c ξ n [µ1,c ] where
n [µ] ∈ S++ (0, 1) is the unit eigenvector satisfying:
µ2 Dn [µ] + Ln [µ] = λP F µ2 D + L n [µ] .
Lemma
There exist ε > 0 such that, for any ε ∈ (0, ε), there exists Aε ∈ (0, +∞)
such that the function:
p : ξ 7→ max e−µ1,c ξ n [µ1,c ]i − Aε e−(µ1,c +ε)ξ n [µ1,c + ε]i , 0
,
i∈[N ]
satisfies:
−Dp00 − cp0 − Lp ≤ −c [p] ◦ p in H −1 R, RN .
Sub-solution: p.
Introduction
The scalar KPP equation
Define the convex set:
F =
v∈C
The non-cooperative KPP system
N
[−R, R] , R
|p≤v≤p
and the compact map f which associates with some v ∈ F the unique classical solution f [v] of:
n
−Dp00 − cp0 = Lp − c [v] ◦ p
p (±R) = p (±R) .
in (−R, R)
For any v ∈ F , we have:
00
−Dp
0
− cp − Lp = 0
≥ −c [v] ◦ f [v]
= −Df [v]
00
0
− cf [v] − Lf [v] ,
p (±R) ≥ p (±R) = f [v] (±R) ,
−Dp
00
0
− cp − Lp + c [p] ◦ p ≤ 0
= −Df [v]
00
− cf [v] − Lf [v] + c [v] ◦ f [v]
00
− cf [v] − Lf [v] + c [p] ◦ f [v]
≤ −Df [v]
0
0
p (±R) = f [v] (±R) .
Thus p ≤ f [v] ≤ p holds because both linear elliptic operators above supplemented with Dirichlet boundary
conditions satisfy the weak maximum principle (p being a super-solution). Hence f (F ) ⊂ F .
Schauder fixed point theorem: f admits a fixed point pR ∈ F .