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

Abstract

We consider the blowup problem for $u_t={\Delta }_{p}u+\mid u{\mid }^{p–2}u$ ($x\in \Omega$, $t>0$) 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 $\Omega$ 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.