Traveling fronts guided by the environment for reaction-diffusion equations

Henri Berestycki; Guillemette Chapuisat; Henri Berestycki; Guillemette Chapuisat

doi:10.3934/nhm.2013.8.79

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 79-114. doi: 10.3934/nhm.2013.8.79

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Traveling fronts guided by the environment for reaction-diffusion equations

Henri Berestycki ^{1
,},
Guillemette Chapuisat ^{2
,}

1.
CAMS, UMR 8557, EHESS, 190-198 avenue de France, 75244 Paris Cedex 13
2.
LATP, UMR 7353, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13

Received: 01 May 2012 Revised: 01 March 2013
Primary: 35K57, 35C07; Secondary: 35Q92.

This paper deals with the existence of traveling fronts for the reaction-diffusion equation: $$ \frac{\partial u}{\partial t} - \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N. $$ We first consider the case $h(u,y)=f(u)-\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{|y|\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
We also study here the case where $h(y,u)=f(u)$ for $|y|\leq L_1$ and $h(y,u) \approx - \alpha u$ for $|y|>L_2\geq L_1$. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if $L_1$ is large enough and the non-existence if $L_2$ is too small.
- Traveling front,
- reaction-diffusion,
- KPP,
- population dynamics
Citation: Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations[J]. Networks and Heterogeneous Media, 2013, 8(1): 79-114. doi: 10.3934/nhm.2013.8.79

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Abstract

This paper deals with the existence of traveling fronts for the reaction-diffusion equation: $$ \frac{\partial u}{\partial t} - \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N. $$ We first consider the case $h(u,y)=f(u)-\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{|y|\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
We also study here the case where $h(y,u)=f(u)$ for $|y|\leq L_1$ and $h(y,u) \approx - \alpha u$ for $|y|>L_2\geq L_1$. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if $L_1$ is large enough and the non-existence if $L_2$ is too small.

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