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 ∂ₜ u-Δ uᵐ+|x|σuq=0, (t,x)∈(0,∞)× R N, with m≥1, q∈(0,1) and σ > 0, is addressed. Introducing the critical exponent σ^* := 2(1-q)/(m-1) for $m > 1$ and σ^*=∞ for $m=1$, extinction in finite time is known to take place for σ∈ [0,σ^*) and an alternative proof is provided therein. When $m > 1$ and σ≥ 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.