On the exponential time-decay for the one-dimensional wave equation with variable coefficients
| Autor: | Anton Arnold, Sjoerd Geevers, Ilaria Perugia, Dmitry Ponomarev |
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| Přispěvatelé: | Institute of Analysis and Scientific Computing, TU Wien, University of Vienna [Vienna], Faculty of Mathematics, University of Vienna, Analyse fonctionnelle pour la conception et l'analyse de systèmes (FACTAS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria) |
| Rok vydání: | 2022 |
| Předmět: |
[PHYS.PHYS.PHYS-OPTICS]Physics [physics]/Physics [physics]/Optics [physics.optics]
Mathematics - Analysis of PDEs 74H40 [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS] Applied Mathematics FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI] General Medicine [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] Analysis [PHYS.MECA.ACOU]Physics [physics]/Mechanics [physics]/Acoustics [physics.class-ph] Analysis of PDEs (math.AP) |
| Zdroj: | Communications on Pure and Applied Analysis Communications on Pure and Applied Analysis, 2022, 21 (10), pp.3389. ⟨10.3934/cpaa.2022105⟩ |
| ISSN: | 1534-0392 1553-5258 |
| DOI: | 10.48550/arxiv.2201.04379 |
| Popis: | We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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