Global nonexistence results for a class of hyperbolic systems

Belkacem Said-Houari; Mokhtar Kirane

doi:10.1016/j.na.2011.05.092

Global nonexistence results for a class of hyperbolic systems

Belkacem Said-Houari, Mokhtar Kirane

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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.

Original language	English (US)
Pages (from-to)	6130-6143
Number of pages	14
Journal	Nonlinear Analysis: Theory, Methods & Applications
Volume	74
Issue number	17
DOIs	https://doi.org/10.1016/j.na.2011.05.092
State	Published - Dec 2011

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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].

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2011.05.092

Cite this

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title = "Global nonexistence results for a class of hyperbolic systems",

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 {\~F}(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 {\~F}(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. {\textcopyright} 2011 Elsevier Ltd. All rights reserved.",

author = "Belkacem Said-Houari and Mokhtar Kirane",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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].",

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AU - Kirane, Mokhtar

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N2 - 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.

AB - 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.

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ER -

Global nonexistence results for a class of hyperbolic systems

Abstract

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