Decay of mass for the equation \(u_t=\Delta u-a(x)u^p|\nabla u|^q\)
type: Item
instance of: scholarly article
title: Decay of mass for the equation \(u_t=\Delta u-a(x)u^p|\nabla u|^q\)
zbMATH Open document ID: 0963.35021
author: Ross G. Pinsky
published in: Journal of Differential Equations
publication date: 2001-06-18T00:00:00Z
review text: The paper deals with the quasilinear Cauchy problem \[ u_t =\Delta u-a(x)u^p |\nabla u|^q, x\in {\mathbb R}^d,\;t>0, \quad u(x,0)=\phi(x)\geq 0,\;x\in {\mathbb R}^d, \] with \(a>0,\) \(0\not\equiv \phi\in L^1({\mathbb R}^d)\cap C_b^{3,\alpha}({\mathbb R}^d)\) and \(p\geq 0\) and \(q\geq 1\) or \(p> 1\) and \(q=0\). It is proved a unique positive classical solvability of the above problem. Moreover, the author shows that \[ \lim_{t\to\infty} \int_{{\mathbb R}^d} u(x,t) dx=0 \] if \(a(x)\) is of the order \(|x|^m\) as \(|x|\to\infty\) and \(dp+(d+1)q\leq d+2+m.\)
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1538227
zbMATH Keywords: quasilinear Cauchy problem
zbMATH Keywords: unique positive classical solvability
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3771
OpenAlex ID: W1963794767
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DOI: 10.1006/JDEQ.2000.3771
description: scientific article; zbMATH DE number 1538227