Asymptotic Behavior of Blowup Solutions of a Parabolic Equation with the $p$-Laplacian

Abstract

We consider the blowup problem for ut— Apu+\u p~2u (x^ O , t> 0) under the Dirichlet boundary condition and p> 2. We derive sufficient conditions on blowing up of solutions. In particular, it is shown that every non-negative and non-zero solution blows up in a finite time if the domain O is large enough. Moreover, we show that every blowup solution behaves asymptotically like a self-similar solution near the blowup time. The Rayleigh type quotient introduced in Lemma A plays an important role throughout this paper.