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A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model. (English) Zbl 0658.35036
This paper is concerned with the existence of traveling wave solutions for reaction-diffusion equations arising in some combustion models for two-dimensional flame propagation. The equations are set in the strip \(S=\{(x,y)\in {\mathbb{R}}^ 2\), \(0<y<L\}\). They have the form \[ -\Delta u+c\alpha (y)u_ x=g(u)\quad or\quad -\Delta u+(c+\alpha (y))u_ x=g(u)\quad in\quad S, \] together with the boundary conditions \(u(- \infty,.)=0\), \(u(+\infty,.)=1\), \(u_ y(.,0)=u_ y(.,L)=0\). The functions \(g\geq 0\) and \(\alpha >0\) are given. We establish the existence of a solution, that is a pair (c,u) for each of these problems. We first prove the existence of a solution in a bounded rectangle \(R_ a=\{(x,y)\in S\), \(-a<x<ka\}\) by a topological degree method. This solution is further required to satisfy an auxiliary normalization condition. The latter is shown to allow one to pass to the limit when \(a\to +\infty\) and obtain a solution in the whole strip S.
Reviewer: H.Berestycki

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
80A32 Chemically reacting flows
35B40 Asymptotic behavior of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
47J05 Equations involving nonlinear operators (general)
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