On the wave equation with semilinear porous acoustic boundary conditions

Handle URI:
http://hdl.handle.net/10754/562164
Title:
On the wave equation with semilinear porous acoustic boundary conditions
Authors:
Graber, Philip Jameson; Said-Houari, Belkacem
Abstract:
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Elsevier
Journal:
Journal of Differential Equations
Issue Date:
May-2012
DOI:
10.1016/j.jde.2012.01.042
Type:
Article
ISSN:
00220396
Sponsors:
The first author wishes to thank the Virginia Space Grant Consortium and the Jefferson Scholars Foundation for their support. The second author wants to thank KAUST for its support. Both authors are very grateful to Prof. Irena Lasiecka for many fruitful discussions.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGraber, Philip Jamesonen
dc.contributor.authorSaid-Houari, Belkacemen
dc.date.accessioned2015-08-03T09:46:16Zen
dc.date.available2015-08-03T09:46:16Zen
dc.date.issued2012-05en
dc.identifier.issn00220396en
dc.identifier.doi10.1016/j.jde.2012.01.042en
dc.identifier.urihttp://hdl.handle.net/10754/562164en
dc.description.abstractThe goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.en
dc.description.sponsorshipThe first author wishes to thank the Virginia Space Grant Consortium and the Jefferson Scholars Foundation for their support. The second author wants to thank KAUST for its support. Both authors are very grateful to Prof. Irena Lasiecka for many fruitful discussions.en
dc.publisherElsevieren
dc.subjectAcoustic waveen
dc.subjectBlow upen
dc.subjectEnergy decayen
dc.subjectExponential growthen
dc.subjectFinite timeen
dc.subjectRateen
dc.titleOn the wave equation with semilinear porous acoustic boundary conditionsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalJournal of Differential Equationsen
dc.contributor.institutionUniversity of Virginia, Department of Mathematics, 22904 Charlottesville, VA, United Statesen
kaust.authorSaid-Houari, Belkacemen
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