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.
Highlights
We study some properties related to the phenomenon of finite time extinction of nonnegative solutions to the initial value problem in RN for the singular diffusion equation with gradient absorption
An obvious consequence of Theorem 1.1 is the optimality of the tail behavior (1.8) for finite time extinction to occur
The technique of the proof is based on constructing suitable supersolutions with finite time extinction, on the one hand, and subsolutions which are positive everywhere, on the other hand
Summary
In the range of exponents (1.3), the phenomenon of extinction of the solution u in finite time occurs according to [8, Theorem 1.2(iii)] provided that the initial condition u0 decays sufficiently fast as |x| → ∞. An obvious consequence of Theorem 1.1 is the optimality of the tail behavior (1.8) for finite time extinction to occur. The second restriction is related to the range of the exponent q which is restricted to the smaller interval (p − 1, p/2) in Theorem 1.2 This assumption is seemingly only technical and some arguments in that direction are the following: on the one hand, for the critical case q = p − 1, the extinction rate (1.12) is already proved in [9] for radially symmetric initial data, though by a completely different technique.
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