Extinction of solutions to a class of fast diffusion systems with nonlinear sources

Abstract

In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)+v^m$, $v_t=\mathrm{div}(|\nabla v{|}^{q-2}\nabla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $\Omega \subset {\mathbb{R}}^N(N\ge 1)$ is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time provided that the initial data are ``comparable"; if $mn=(p-1)(q-1)$ and $\Omega$ is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for $1<p=q<2$ and $mn<(p-1{)}^2$, the existence of at least one non-extinction solution for any positive smooth initial data is proved.