Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction

Handle URI:
http://hdl.handle.net/10754/562313
Title:
Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction
Authors:
Said-Houari, Belkacem; Nascimento, Flávio A Falcão
Abstract:
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
American Institute of Mathematical Sciences (AIMS)
Journal:
Communications on Pure and Applied Analysis
Issue Date:
Sep-2012
DOI:
10.3934/cpaa.2013.12.375
Type:
Article
ISSN:
15340392
Sponsors:
Doctorate student by State University of Maringa, partially supported by a grant of CNPq, BrazilThe authors thanks Prof. Marcelo Moreira Cavalcanti for many helpful comments, which improve the first version of this paper. Moreover, the first author thanks KAUST for its support.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorSaid-Houari, Belkacemen
dc.contributor.authorNascimento, Flávio A Falcãoen
dc.date.accessioned2015-08-03T10:00:28Zen
dc.date.available2015-08-03T10:00:28Zen
dc.date.issued2012-09en
dc.identifier.issn15340392en
dc.identifier.doi10.3934/cpaa.2013.12.375en
dc.identifier.urihttp://hdl.handle.net/10754/562313en
dc.description.abstractThe goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.en
dc.description.sponsorshipDoctorate student by State University of Maringa, partially supported by a grant of CNPq, BrazilThe authors thanks Prof. Marcelo Moreira Cavalcanti for many helpful comments, which improve the first version of this paper. Moreover, the first author thanks KAUST for its support.en
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)en
dc.subjectBoundary feedbacken
dc.subjectRelaxation functionen
dc.subjectSource termen
dc.subjectViscoelastic wave equationen
dc.titleGlobal existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interactionen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalCommunications on Pure and Applied Analysisen
dc.contributor.institutionDepartment of Mathematics, State University of Ceará, FAFIDAM, 62930-000 Limoeiro do Norte - CE, Brazilen
kaust.authorSaid-Houari, Belkacemen
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