The University of Montana
Department of Mathematical Sciences

Technical report #1/2005


Collapsing Heat Waves

William R. Derrick and Leonid V. Kalachev
Department of Mathematics, University of Montana
Missoula, MT 59812, USA

and

Joseph A. Cima
Department of Mathematics, University of North Carolina
Chapel Hill, NC 27599, USA

Abstract

In certain combustion models, an initial temperature profile will develop into a combustion wave that will travel at a specific wave speed. Other initial profiles do not develop into such waves, but die out to the ambient temperature. There exists a clear demarcation between those initial conditions that evolve into combustion waves and those that do not. This is sometimes called a watershed initial condition. In this paper we will show there may be numerous exact watershed conditions to the initial-Neumann-boundary value problem , with , on I = [0,1]. They are composed from the positive non-constant solutions of , with , for small values of D. We will give easily verifiable conditions for when combustion waves arise and when they do not.

Keywords: reaction-diffusion equation, combustion, domain of attraction

AMS Subject Classification: 35K57, 35K55

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