,
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.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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