The extinction behavior of the solutions for a class of reaction-diffusion equations. (Q1596436)

This paper is devoted to the following general problem with nonnegative initial and homogeneous Dirichlet boundary values \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta u-\lambda| u|^{\gamma-1}u-\beta u,\;(x,t)\in\Omega\times(0,+\infty)\\ u(x,t)|_{\partial\Omega\times(0,+\infty)}=0\\ u(x,0)=u_0(x)\in H^1_0(\Omega)\cap L^{1+\gamma}(\Omega),\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N(N\geq 2)\) with smooth boundary, \(\lambda,\beta,\gamma\) are positive parameters. The author presents sufficient and necessary conditions about extinction of solutions of (1). Note that this problem can be viewed as mathematical model of chemotaxis system. Moreover the author indicates and discusses the possibility of an approach developed in this paper to higher-order equation.