Stabilization of a Semilinear Wave Equation with Delay

We study the wellposedness and the stabilization of solutions of a semilinear wave equation with delay and locally distributed dissipation. The novelty of this paper is that we deal with the semilinear wave equation subject to delay and locally distributed damping without smallness conditions in the...

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Bibliographic Details
Published in:Journal of dynamics and differential equations 2024-03, Vol.36 (1), p.161-208
Main Authors: Martinez, Victor Hugo Gonzalez, Marchiori, Talita Druziani, de Souza Franco, Alisson Younio
Format: Article
Language:English
Subjects:
Applications of Mathematics
Damping
Delay
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
Stabilization
Wave equations
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Summary:We study the wellposedness and the stabilization of solutions of a semilinear wave equation with delay and locally distributed dissipation. The novelty of this paper is that we deal with the semilinear wave equation subject to delay and locally distributed damping without smallness conditions in the initial data or in the delay term. In order to address this, the argumentation requires the use of Strichartz estimates and some microlocal analysis results such as propagation of microlocal defect measures and the Gárard’s linearizability property. To obtain the observability estimate in the critical case we prove a Unique Continuation Property for the semilinear wave equation and apply it to our problem. Once we establish essential observability properties for the solutions, it is not difficult to prove that the solutions decay exponentially to 0.
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-021-10120-3