An \(L^p\) theory of invariant manifolds for parabolic partial differential equations on \(\mathbb{R}^d\)
type: Item
instance of: scholarly article
title: An \(L^p\) theory of invariant manifolds for parabolic partial differential equations on \(\mathbb{R}^d\)
zbMATH Open document ID: 0996.35030
author: Kazuo Kobayasi
published in: Journal of Differential Equations
publication date: 2002-11-02T00:00:00Z
review text: It is studied the existence of finite-dimensional invariant manifolds for nonlinear heat equations of the form \[ \partial u/\partial\tau=\Delta u+F(u,\nabla u)\quad\text{on }\quad {\mathbb R}^d\times [1,\infty). \] It is shown that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero, there exist finite-dimensional invariant manifolds which control the long time asymptotics of solutions. It is considered the problem for these equations in the framework of weighted Sobolev spaces of \(L^p\) type. It is obtained an \(L^\infty\) estimate of the long-time asymptotics of solutions under natural assumptions on the nonlinear term \(F\) and their initial data.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 37L25
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1744189
zbMATH Keywords: nonliner heat equations
zbMATH Keywords: long-time asymptotics
zbMATH Keywords: weighted Sobolev spaces
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4026
OpenAlex ID: W1997194551
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DOI: 10.1006/JDEQ.2001.4026
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description: scientific article; zbMATH DE number 1744189