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2000 Non-existence of travelling front solutions of some bistable reaction-diffusion equations
H. Berestycki, F. Hamel
Adv. Differential Equations 5(4-6): 723-746 (2000). DOI: 10.57262/ade/1356651345


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.


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H. Berestycki. F. Hamel. "Non-existence of travelling front solutions of some bistable reaction-diffusion equations." Adv. Differential Equations 5 (4-6) 723 - 746, 2000.


Published: 2000
First available in Project Euclid: 27 December 2012

zbMATH: 0988.35081
MathSciNet: MR1750116
Digital Object Identifier: 10.57262/ade/1356651345

Primary: 35K57
Secondary: 35A18 , 35B05 , 35B40

Rights: Copyright © 2000 Khayyam Publishing, Inc.


Vol.5 • No. 4-6 • 2000
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