Networks and Heterogeneous Media (NHM)

Traveling fronts guided by the environment for reaction-diffusion equations

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

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Henri Berestycki - CAMS, UMR 8557, EHESS, 190-198 avenue de France, 75244 Paris Cedex 13, France (email)
Guillemette Chapuisat - LATP, UMR 7353, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France (email)

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.

Keywords:  Traveling front, reaction-diffusion, KPP, population dynamics.
Mathematics Subject Classification:  Primary: 35K57, 35C07; Secondary: 35Q92.

Received: May 2012;      Revised: March 2013;      Available Online: April 2013.