Plane structures in thermal runaway

We consider the problem

1. u _t=u _xx+e ^u whenx ∈ ?,t > 0,
2. u(x, 0) =u ₀(x) whenx ∈ ?,

whereu ₀(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 timeT, 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 ofu(x, T) near its blow-up points, which are shown to be isolated.