Global nonexistence results for a class of hyperbolic systems

Handle URI:
http://hdl.handle.net/10754/561933
Title:
Global nonexistence results for a class of hyperbolic systems
Authors:
Said-Houari, Belkacem; Kirane, Mokhtar
Abstract:
Our concern in this paper is to prove blow-up results to the non-autonomous nonlinear system of wave equations utt-Δu=a(t,x)| v|p,vtt-Δv=b(t,x)|u|q,t>0, x∈RN in any space dimension. We show that a curve F̃(p,q)=0 depending on the space dimension, on the exponents p,q and on the behavior of the functions a(t,x) and b(t,x) exists, such that all nontrivial solutions to the above system blow-up in a finite time whenever F̃(p,q)>0. Our method of proof uses some estimates developed by Galaktionov and Pohozaev in [11] for a single non-autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived. Our result generalizes some important results such as the ones in Del Santo et al. (1996) [12] and Galaktionov and Pohozaev (2003) [11]. The advantage here is that our result applies to a wide variety of problems. © 2011 Elsevier Ltd. All rights reserved.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Elsevier
Journal:
Nonlinear Analysis, Theory, Methods and Applications
Issue Date:
Dec-2011
DOI:
10.1016/j.na.2011.05.092
Type:
Article
ISSN:
0362546X
Sponsors:
The first author was partially supported by the DFG project RA 504/3-3. This author wishes to thank the Department of Mathematics and Statistics, University of Konstanz for its financial support and its kind hospitality. Moreover, the two authors wish to thank the Referee and the Editor for their useful remarks and careful reading of the proofs presented in this paper. Especially, we want to thank Prof. Enzo Mitidieri for bringing our attention to Ref. [13].
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.authorKirane, Mokhtaren
dc.date.accessioned2015-08-03T09:34:26Zen
dc.date.available2015-08-03T09:34:26Zen
dc.date.issued2011-12en
dc.identifier.issn0362546Xen
dc.identifier.doi10.1016/j.na.2011.05.092en
dc.identifier.urihttp://hdl.handle.net/10754/561933en
dc.description.abstractOur concern in this paper is to prove blow-up results to the non-autonomous nonlinear system of wave equations utt-Δu=a(t,x)| v|p,vtt-Δv=b(t,x)|u|q,t>0, x∈RN in any space dimension. We show that a curve F̃(p,q)=0 depending on the space dimension, on the exponents p,q and on the behavior of the functions a(t,x) and b(t,x) exists, such that all nontrivial solutions to the above system blow-up in a finite time whenever F̃(p,q)>0. Our method of proof uses some estimates developed by Galaktionov and Pohozaev in [11] for a single non-autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived. Our result generalizes some important results such as the ones in Del Santo et al. (1996) [12] and Galaktionov and Pohozaev (2003) [11]. The advantage here is that our result applies to a wide variety of problems. © 2011 Elsevier Ltd. All rights reserved.en
dc.description.sponsorshipThe first author was partially supported by the DFG project RA 504/3-3. This author wishes to thank the Department of Mathematics and Statistics, University of Konstanz for its financial support and its kind hospitality. Moreover, the two authors wish to thank the Referee and the Editor for their useful remarks and careful reading of the proofs presented in this paper. Especially, we want to thank Prof. Enzo Mitidieri for bringing our attention to Ref. [13].en
dc.publisherElsevieren
dc.subjectBlow upen
dc.subjectCritical curveen
dc.subjectCritical exponentsen
dc.subjectSemilinear wave equationsen
dc.titleGlobal nonexistence results for a class of hyperbolic systemsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalNonlinear Analysis, Theory, Methods and Applicationsen
dc.contributor.institutionDepartment of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germanyen
dc.contributor.institutionMathématiques, Image et Applications Pole Sciences et Technologie, Université de la Rochelle, Franceen
kaust.authorSaid-Houari, Belkacemen
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