Complete blow up for a parabolic system arising in a theory of thermal explosion in porous media

Thomas I. Hill (Department of Mathematics, University of Akron, Ohio, U.S.A.; and Department of Mathematical Sciences, University of Cincinnati, Ohio, U.S.A.)

Gregory I. Sivashinsky (School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel)

Abstract

In this paper we consider a model of thermal explosion in porous media. The model consists of two reaction-diffusion equations in a bounded domain with Dirichlet boundary conditions and describes the initial stage of evolution of pressure and temperature fields. Under certain conditions, the classical solution of these equations exists only on finite time interval after which it forms a singularity and becomes unbounded (blows up). This behavior raises a natural question whether this solution can be extended, in a weak sense, after blow up time. We prove that the answer to this question is no, that is, the solution becomes unbounded in entire domain immediately after the singularity is formed. From a physical perspective our results imply that autoignition in porous materials occurs simultaneously in entire domain.

Keywords

combustion in porous media, thermal explosion, blow up for parabolic systems

2010 Mathematics Subject Classification

35B44, 35D30, 35K51, 35K57, 80A25

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Published 21 February 2017