A class of second order quasilinear evolution equations
type: Item
instance of: scholarly article
title: A class of second order quasilinear evolution equations
zbMATH Open document ID: 0810.34057
author: Ryo Ikehata
author: Noboru Okazawa
published in: Journal of Differential Equations
publication date: 1995-04-18T00:00:00Z
review text: Given a nonnegative selfadjoint operator \(A\) on a Hilbert space \(H\), a nonlinear operator \(f\) from \(D(A^{{1\over 2}})\) to \(H\), and a \(C^ 1\)- function \(M(\cdot)\) satisfying \(M(s)\geq m_ 0> 0\) for all \(s\geq 0\). Using a local Lipschitz continuity assumption on \(f(u)\), the author proves the existence and the uniqueness of a local strong solution of the problem \[ u''(t)+ M(| A^{{1\over 2}} u(t)|^ 2) Au(t)= f(u(t)),\quad t\geq 0,\quad u(0)= u_ 0,\quad u'(0)= u_ 1, \tag{1} \] by applying the theory of nonlinear evolution equations developed by Kato. In the case, where \(A\) is the minus Laplacian with Dirichlet conditions on \(L^ 2(\Omega)\), \(f(u)=\mu u^ 3\) and \(M(s)= \alpha+ 2\beta s\) for \(s\geq 0\) and some constants \(\alpha> 0\), \(\beta\geq 0\) and \(\mu\in \mathbb{R}\), the blowing-up property of solutions to (1) is discussed.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 696883
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: local strong solution
zbMATH Keywords: nonlinear evolution equations
zbMATH Keywords: blowing-up property
reviewed by: A. Rhandi
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1143
OpenAlex ID: W2024438491
DOI: 10.1006/JDEQ.1994.1143
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description: scientific article; zbMATH DE number 696883