Extinction in a finite time for solutions of a class of quasilinear parabolic equations

Abstract

We study the property of extinction in a finite time for nonnegative solutions of $\frac{1}{\mathit{q}}\frac{\partial }{\partial \mathit{t}}\mathrm{(}{\mathit{u}}^{\mathit{q}}\mathrm{)}-\nabla \mathrm{(}|\nabla \mathit{u}{|}^{\mathit{p}-2}\nabla \mathit{u}\mathrm{)}+\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathit{\lambda }}=0$ for the Dirichlet Boundary Conditions when $\mathit{q}\mathrm{>}\mathit{\lambda }\mathrm{>}0$ , $\mathit{p}⩾1+\mathit{q}$ , $\mathit{p}⩾2$ , $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}⩾0$ and Ω a bounded domain of ${\mathbb{R}}^{\mathit{N}}$ ( $\mathit{N}⩾1$ ). We prove some necessary and sufficient conditions. The threshold is for power functions when $\mathit{p}\mathrm{>}1+\mathit{q}$ while finite time extinction occurs for very flat potentials $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}$ when $\mathit{p}=1+\mathit{q}$ .