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Multi-dimensional travelling-wave solutions of a flame propagation model. (English) Zbl 0711.35066
The existence of a travelling wave solution $$u(x_ 1+ct,y)$$ of the equation $\partial u/\partial t+\alpha (y)\partial u/\partial x_ 1=\Delta u+g(u)\text{ in } {\mathbb{R}}\times \omega$ is shown, where $$\omega$$ is a bounded and smooth open domain. The function u satisfies an elliptic equation with parameter c where both u and c are unknown. In contrast to previous papers $$c+\alpha (y)$$ may in general change the sign in the domain $$\omega$$. Conditions for the occurrence of this phenomenon which may be interpreted as an inversion of the velocity field are given. The proofs are based on a priori estimates which are derived from methods typical for the elliptic case as the maximum principle, energy estimates, and elliptic regularity.
Reviewer: P.Kröger

MSC:
 35K57 Reaction-diffusion equations 80A25 Combustion 35J60 Nonlinear elliptic equations
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References:
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