The extinction behavior of the solutions for a class of reaction-diffusion equations

The methods of Lp estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boundnary values\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{gathered} \partial u/\partial t = \Delta u - \lambda \left| { u } \right|^{\gamma - 1} u - \beta u ((x,t) \in \Omega \times (0, + \infty )), \hfill \\ u(x,t)\left| {_{\partial \Omega \times (0, + \infty )} = 0,} \right. \hfill \\ u(x,0) = u_0 (x) \in H_0^1 (\Omega ) \cap L^{1 + \gamma } (\Omega ) (x \in \Omega ) \hfill \\ \end{gathered}$$ \end{document}. Sufficient and necessary conditions about the extinction of the solutions is given. Here γ>0, γ>0, β>0 are constants, Ω∈RN is bounded with smooth boundary ∂Ω. At last, it is simulated with a higher order equation by using the present methods.