Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption

Abstract

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$ \partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times \mathbb R ^N, $$ with $m\geq1$, $q\in(0,1)$ and $\sigma > 0$, is addressed. Introducing the critical exponent $\sigma^* := 2(1-q)/(m-1)$ for $m > 1$ and $\sigma^*=\infty$ for $m=1$, extinction in finite time is known to take place for $\sigma\in [0,\sigma^*)$ and an alternative proof is provided therein. When $m > 1$ and $\sigma\ge \sigma^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of $\sigma$ and showing their sharpness.