Non-existence of travelling front solutions of some bistable reaction-diffusion equations

Abstract

This work deals with travelling fronts solutions of some reaction-diffusion equations in an infinite cylinder in dimension $\ge 2$. The problem is set in $\Sigma=\{(x_1,y)\in{\mathbb R}\times\omega\}$ where $\omega\subset{\mathbb R}^{N-1}$ is a bounded and smooth domain with outward normal $\nu$. The equations, with unknowns $c\in {\mathbb R}$ and $u\in C^2(\overline{\Sigma})$, are $$ (P) \qquad\qquad \left\{\begin{array}{rl} \Delta u-(c+\alpha(y))\ \partial_{1}u+f(u)=0 & \hbox{ in }\Sigma={\mathbb R}\times\omega\\ \displaystyle{\frac{\partial u }{\partial\nu}}=0 & \hbox{ on }\partial\Sigma= {\mathbb R}\times\partial\omega\\ u(-\infty,\cdot)=0\hbox{ and }u(+\infty,\cdot)=1 & \end{array} \right. $$ The function $\alpha \in C^0(\overline{\omega})$ is given. The nonlinearity $f$ is assumed to be of the ``bistable type": it changes sign once in $(0,1)$. Berestycki and Nirenberg [8] proved that if $\omega$ is convex then the problem has a solution. Here, by using the invariance by translation and the sliding method, we construct an example of a non-convex domain $\omega$ and of a function $\alpha$ for which we prove that $(P)$ has no solutions. This is in sharp contrast with other types of nonlinearities for which solutions exist whatever $\omega$ may be.