Advances in Differential Equations

Existence of multidimensional travelling fronts with a multistable nonlinearity

F. Hamel and S. Omrani

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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.

Article information

Adv. Differential Equations, Volume 5, Number 4-6 (2000), 557-582.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35A18: Wave front sets 35B35: Stability 35B40: Asymptotic behavior of solutions


Hamel, F.; Omrani, S. Existence of multidimensional travelling fronts with a multistable nonlinearity. Adv. Differential Equations 5 (2000), no. 4-6, 557--582.

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