Smoothness effect and decay on a class of nonlinear evolution equation
type: Item
instance of: scholarly article
title: Smoothness effect and decay on a class of nonlinear evolution equation
zbMATH Open document ID: 0783.47074
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1994-03-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73302
review text: Solutions of the nonlinear evolution equation \(u_{tt}+ M(\| A^{1/2} u(t)\|^ 2)Au+ Au_ t=0\), \(u(0)=u_ 0\), \(u_ t(0)=u_ 1\), are considered, where \(A\) is a positive selfadjoint operator on a Hilbert space. First the author proves existence and uniqueness of global solutions. The main result of the paper is concerned with the smoothing property of the solutions, i.e. under suitable conditions for \(M\) the solutions are as smooth as one can at best expect, namely \(u(t)\) is the intersection of the domains of all powers of \(A\) for all \(t>0\), no matter how smooth the initial data are. Moreover it is proven that the solutions decay uniformly as time goes to infinity.
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 202538
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: nonlinear evolution equation
zbMATH Keywords: positive selfadjoint operator on a Hilbert space
zbMATH Keywords: existence and uniqueness of global solutions
zbMATH Keywords: smoothing property
author: Jaime E. Muñoz Rivera
reviewed by: Walter Hoh
MaRDI profile type: Publication
OpenAlex ID: W2335311697
cites work: Q3732157
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DOI: 10.5802/AFST.747
description: scientific article; zbMATH DE number 202538