Slow decay of solutions in a semilinear dissipative parabolic equation
type: Item
instance of: scholarly article
title: Slow decay of solutions in a semilinear dissipative parabolic equation
zbMATH Open document ID: 0937.35082
author: Noriko Mizoguchi
published in: Journal of Differential Equations
publication date: 2000-01-04T00:00:00Z
review text: The author of this interesting paper considers a Cauchy problem \(u_t=u_{xx}-|u|^{p-1}u\) in \(\mathbb{R}\times(0,\infty)\) and \(u(x,0)= \varphi (x)\) in \(\mathbb{R}\), where \(p>1\) and \(\varphi\in L^\infty(\mathbb{R})\). The solution \(u(x,t)\) of the problem under consideration is said to decay fast as \(t\to \infty\) if \(\lim t^{1/(p-1)} u(x,t)=0\) uniformly in \(\mathbb{R}\). Otherwise the solution \(u(x,t)\) is said to decay slowly as \(t\to\infty\). It is shown with perfect precision that in case \(\varphi(x)\) does not decays faster than \(|x|^{-q} (q<2/(p-1))\) as \(x\to\infty\) or \(x\to-\infty\), then \(u\) decays slowly as \(t\to\infty\). Also an estimate of solutions near the spatial infinity for general initial data can be seen, which implies that none of zeros of \(u(x,t)\) goes to \(\pm\infty\) at each \(t>0\).
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1384460
zbMATH Keywords: fast and slow large time decay
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1384460