A Free Boundary Problem of a Semilinear Combustible System with Higher Dimension and Heterogeneous Environment

Abstract

In this paper, we investigate a free boundary problem of a semilinear combustible system with higher dimension and heterogeneous environment. Such a problem is usually used as a model to describe heat propagation in a two-component combustible mixture in which the free boundary is described by Stefan-like condition. For simplicity, we assume that the environment and solutions are radially symmetric. We use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup property and the long time behavior of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.