Continuous dependence on modeling for related Cauchy problems of a class of evolution equations
type: Item
instance of: scholarly article
title: Continuous dependence on modeling for related Cauchy problems of a class of evolution equations
zbMATH Open document ID: 0905.35010
author: Shannon S. Cobb
author: Karen A. Ames
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: The authors investigate the Cauchy problem \(Pu_t+ Mu= F(t,u)\) on some interval \([0,T)\), under the initial datum \(u(0)= f\in H\), where \(P\) and \(M\) are linear operators with dense domain in the Hilbert space \(H\). \(F\) stands for a (nonlinear) map from \([0,T) \times H\) into \(H\). \(P\) and \(M\) are assumed to be symmetric and commuting, while \(P\) is strictly positive. \(F\) is supposed to satisfy a Lipschitz condition in the second argument. The problem is to justify the validity of the approximation scheme based on the singular perturbation described by the equation \[ Pw_t+ Mw- \varepsilon Pw_{tt} =F(t,w), \] under initial conditions \(w(0)=f\), \(w_t (0) =g\). Under further technical assumptions one obtains in \(L^2\) estimate for the difference \(u-w\). A nonlinear differential inequality is the key for obtaining the estimate which guarantees the validity of the procedure.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 1115436
zbMATH Keywords: abstract evolution equations
zbMATH Keywords: nonlinear differential inequality
reviewed by: Constantin Corduneanu
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5597
OpenAlex ID: W2039569545
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DOI: 10.1006/JMAA.1997.5597
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description: scientific article; zbMATH DE number 1115436