Extinction and Nonextinction for the Fast Diffusion Equation

Abstract

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain of $R^N$ with $N>2$. For , under appropriate hypotheses, we show that $m=p$ is the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of $-\mathrm{\Delta }$ with homogeneous Dirichlet boundary.