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 $${\partial }_{t}u-\Delta u^m+|x{|}^{\sigma }{u}^q=0,(t,x)\in (0,\infty )\times {\mathbb{R}}^N,$$ with $m\ge 1$, $q\in (0,1)$ and $\sigma >0$, is addressed. Introducing the critical exponent ${\sigma }^{\ast }:=2(1-q)/(m-1)$ for $m>1$ and ${\sigma }_{\ast }=\infty$ for $m=1$, extinction in finite time is known to take place for $\sigma \in [0,{\sigma }^{\ast })$ and an alternative proof is provided therein. When $m>1$ and $\sigma \ge {\sigma }^{\ast }$, 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.