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2000 Existence of multidimensional travelling fronts with a multistable nonlinearity
F. Hamel, S. Omrani
Adv. Differential Equations 5(4-6): 557-582 (2000). DOI: 10.57262/ade/1356651340


This article deals with the existence of solutions of $$\left\{\begin{array}{rll} \Delta u-\beta(y,c) \frac{\partial u} {\partial x_1} +f(u) & =0 & \hbox{ in }\Sigma\\ \frac{\partial u}{\partial \nu} & =0 & \hbox{ on }\partial\Sigma\\ u(-\infty,\cdot)=0,\ u(+\infty,\cdot) & =1 & \end{array}\right.$$ where $\Sigma=\{(x_1,y)\in\mathbb R \times\omega\}$ is an infinite cylinder with outward unit normal $\nu$ and whose section $\omega\subset\mathbb R^{n-1}$ is a bounded convex domain. The unknowns are the real parameter $c$ and the function $u$ (which respectively represent the speed and the profile of a travelling wave). The function $\beta$ and the nonlinear term $f:[0,1]\to \mathbb R$ are given. We investigate the case where the function $f$ changes sign several times. We prove that there exists a travelling front $(c,u)$ provided that the speeds of the travelling waves for simpler problems can be compared. The proof uses the sliding method and the theory of sub- and supersolutions. This result generalizes for higher dimensions a one-dimensional result of Fife and McLeod.


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F. Hamel. S. Omrani. "Existence of multidimensional travelling fronts with a multistable nonlinearity." Adv. Differential Equations 5 (4-6) 557 - 582, 2000.


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

zbMATH: 1217.35080
MathSciNet: MR1750111
Digital Object Identifier: 10.57262/ade/1356651340

Primary: 35J65
Secondary: 35A18 , 35B35 , 35B40

Rights: Copyright © 2000 Khayyam Publishing, Inc.


Vol.5 • No. 4-6 • 2000
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