Extinction and non-extinction for a \(p\)-Laplacian equation with nonlinear source
type: Item
instance of: scholarly article
title: Extinction and non-extinction for a \(p\)-Laplacian equation with nonlinear source
zbMATH Open document ID: 1152.35046
author: Ya Tian
author: Chunlai Mu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-10-21T00:00:00Z
review text: The phenomenon of extinction in finite time is known for the \(p\)-Laplace equation \[ {\partial u\over\partial t}= \text{div}(|\nabla u|^{p-2}\nabla u) \] in the singular case \(1< p< 2\). The authors add the source term \(\lambda u^q\) to the right-hand member; \(\lambda> 0\), \(q> 0\) and \(u\) is nonnegative. The critical exponent for extinction is J\(q= p-1\). The authors show that in the critical case the situation is determined by a nonlinear eigenvalue problem, apparently first studied by \textit{E. H. Lieb} [Invent. Math. 74, 441--448 (1983; Zbl 0538.35058)]. Lemma 2.4 has previously been proved by \textit{P. Lindqvist} [Proc. Am. Math. Soc. 109, No. 1, 157--164 (1990; Zbl 0714.35029)].
reviewed by: Peter Lindqvist
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 5354992
zbMATH Keywords: extinction
zbMATH Keywords: evolutionary \(p\)-Laplace equation' nonlinear source term
MaRDI profile type: Publication
full work available at URL: j.na.2007.08.021
OpenAlex ID: W2017885798
cites work: Energy methods for free boundary problems. Applications to nonlinear PDEs and fluid mechanics
cites work: Degenerate parabolic equations
cites work: Q3726641
cites work: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results
cites work: Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities
cites work: Extinction properties of semilinear heat equations with strong absorption
cites work: Principal eigenvalue of the p-laplacian in RN
cites work: Q4288993
cites work: Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption
cites work: The propagation of disturbances in problems of non-linear heat conduction with absorption
cites work: Q5562267
cites work: Finite extinction time for solutions of nonlinear parabolic equations
cites work: Q3025545
cites work: Q4804250
cites work: Blow-up in quasilinear parabolic equations. Transl. from the Russian by Michael Grinfeld
cites work: Nonlinear diffusion equations
cites work: Extinction and positivity for the evolution \(p\)-Laplacian equation in \(\mathbb{R}^n\)
cites work: Existence and nonexistence of solutions for \(u_ t=\text{div}(|\nabla u|^{p-2}\nabla u)+f(\nabla u,u,x,t)\)
cites work: The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation
cites work: On the Cauchy problem and initial traces for the evolution \(p\)-Laplacian equations with strongly nonlinear sources
DOI: 10.1016/J.NA.2007.08.021
description: scientific article; zbMATH DE number 5354992