Extinction for a Singular Diffusion Equation with Strong Gradient Absorption Revisited

Abstract

Abstract When 2 ⁢ N / ( N + 1 ) < p < 2 {2N/(N+1)<p<2} and 0 < q < p / 2 {0<q<p/2} , non-negative solutions to the singular diffusion equation with gradient absorption ∂ t ⁡ u - Δ p ⁢ u + | ∇ ⁡ u | q = 0   in  ⁢ ( 0 , ∞ ) × ℝ N \partial_{t}u-\Delta_{p}u+|\nabla u|^{q}=0\quad\text{in }(0,\infty)\times% \mathbb{R}^{N} vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition u 0 {u_{0}} decays sufficiently rapidly as | x | → ∞ {|x|\to\infty} . On the one hand, the optimal decay of u 0 {u_{0}} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that p - 1 < q < p / 2 {p-1<q<p/2} , optimal extinction rates near the extinction time are derived.