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

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $\partial_t u-Δu^m+|x|^σu^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$ with $m\geq1$, $q\in(0,1)$ and $σ>0$, is addressed. Introducing the critical exponent $σ^* := 2(1-q)/(m-1)$ for $m>1$ and $σ_*=\infty$ for $m=1$, extinction in finite time is known to take place for $σ\in [0,σ^*)$ and an alternative proof is provided therein. When $m>1$ and $σ\ge σ^*$, 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 $σ$ and showing their sharpness.

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