Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks

We consider the nonlinear wave equation in a bounded domain with a delay term in the weakly nonlinear internal feedback u ″(x, t) − Δ x u(x, t) + μ1σ(t)g 1(u ′(x, t)) + μ2σ(t)g 2(u ′(x, t − τ(t))) = 0 and prove the global existence of solutions in suitable Sobolev spaces by means of the energy metho...

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Bibliographic Details
Published in:Journal of mathematical physics 2012-12, Vol.53 (12), p.1
Main Authors: Benaissa, Abbes, Benaissa, Abdelkader, Messaoudi, Salim. A.
Format: Article
Language:English
Subjects:
Asymptotic methods
Asymptotic properties
Delay
Energy methods
Exact sciences and technology
Feedback
Inequalities
Integrals
Mathematical analysis
Mathematical methods in physics
Mathematical problems
Mathematics
Nonlinear equations
Nonlinearity
Physics
Sciences and techniques of general use
Wave equations
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Summary:We consider the nonlinear wave equation in a bounded domain with a delay term in the weakly nonlinear internal feedback u ″(x, t) − Δ x u(x, t) + μ1σ(t)g 1(u ′(x, t)) + μ2σ(t)g 2(u ′(x, t − τ(t))) = 0 and prove the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback, the weight of the term without delay and the speed of the delay. Furthermore, we study the asymptotic behavior of solutions using the multiplier method and some general weighted integral inequalities.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4765046