Extinction and Decay Estimates of Solutions for a $p$-Laplacian Parabolic Equation with Nonlinear Source

Abstract

The extinction phenomenon of solutions for the the initial-boundary value problem of the $p$-Laplacian parabolic equation $$u_t=\mbox{div} ( |\nabla u|^{p-2}\nabla u)+\lambda|u|^{m-1}u-\beta u$$ is studied. Sufficient conditions about the extinction and decay estimates of solutions are obtained by using $L^p$-integral model estimate methods and two crucial lemmas on differential inequality. Non-extinction results are obtained by super and sub-solution method.