Plane structures in thermal runaway

Abstract

We consider the problem (1) u(t) = u(xx) + e(u) when x is-an-element-of R, t > 0, (2) u (x, 0) = u0(x) when x is-an-element-of R, where u0(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combustion of a one-dimensional solid fuel. It is known that solutions of (1), (2) blow-up in a finite time T, a phenomenon often referred to as thermal runaway. In this paper we prove the existence of blow-up profiles which are flatter than those previously observed. We also derive the asymptotic profile of u(x, T) near its blow-up points, which are shown to be isolated.