Stabilization  of two coupled wave equations with a localized singular Kelvin-Voigt damping

Stabilization of two coupled wave equations with a localized singular Kelvin-Voigt damping | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Stabilization of two coupled wave equations with a localized singular Kelvin-Voigt damping Kaïs Ammari, Fathi Hassine, Souleymane Kadri Harouna, Louis Tebou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4277529/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We consider two wave equations coupled through a singular Kelvin-Voigt damping mechanism in a bounded domain. We are interested in investigating stability issues for this system.We prove the polynomial stability of the semigroup if the damping region is big enough, and logarithmic stability of the semigroup if the damping region is an arbitrarily small nonempty open subset of the domain under consideration. The main features of our proofs: i) frequency domain approach and, ii) flow multipliers combined with extra auxiliary elliptic systems in the case of polynomial stability, or iii) Carleman estimate in the case of logarithmic stability. A numerical analysis of the spectrum of the one dimensional space semi-discretized system using mixed finite element method indicates that uniform (with respect to the mesh size) exponential decay is not to be expected. This latter result leads us to conjecture that our first polynomial stability result cannot be improved to an exponential stability one. Carleman estimate simultaneous stabilization coupled wave equations Singular Kelvin-Voigt damping Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4277529","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":294252365,"identity":"e62a14e6-f22a-4c9a-bdae-20ae2f4d7cb8","order_by":0,"name":"Kaïs Ammari","email":"","orcid":"","institution":"university of Monastir","correspondingAuthor":false,"prefix":"","firstName":"Kaïs","middleName":"","lastName":"Ammari","suffix":""},{"id":294252366,"identity":"28e8f033-a892-4d31-9290-3236adacdaa7","order_by":1,"name":"Fathi Hassine","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9klEQVRIiWNgGAWjYBAC9gYGNoYEhgNAJmMDA0OFHGEtPAeYkbWcMYZJGODXwgDWAtLVRowW9vPHHjxguCMv33+47cHHeQb2ug28zx7zMPzJw6mFJ5ndIIHhmeGGG4nthjO3GSRuO8BubszDYFCMS4s9QzKbRALDYcYNEoxt0rzb/iSYHWBjkwZqSWzAZQv/Y7AW+/n9B9uk/84xsCesRQJiS2LDgcQ2acYGA8ZthLU8NpNIMDicDPRLm2TPMaBfDrOxSc4xMMbjsMRnkj8qDtvO7z/+TOJHDdBhx9vYJN5UyOHUAgEokcCMITIKRsEoGAWjgFQAAFzCUERoexqQAAAAAElFTkSuQmCC","orcid":"","institution":"university of Monastir","correspondingAuthor":true,"prefix":"","firstName":"Fathi","middleName":"","lastName":"Hassine","suffix":""},{"id":294252367,"identity":"a13df696-3e64-4ce3-9188-95ae67a9be2b","order_by":2,"name":"Souleymane Kadri Harouna","email":"","orcid":"","institution":"University of La Rochelle","correspondingAuthor":false,"prefix":"","firstName":"Souleymane","middleName":"Kadri","lastName":"Harouna","suffix":""},{"id":294252368,"identity":"942a0c66-d762-436e-b4b8-6c35936f1924","order_by":3,"name":"Louis Tebou","email":"","orcid":"","institution":"Florida International University","correspondingAuthor":false,"prefix":"","firstName":"Louis","middleName":"","lastName":"Tebou","suffix":""}],"badges":[],"createdAt":"2024-04-16 17:12:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4277529/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4277529/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":55967860,"identity":"5282fa5a-641b-4c6b-a2a8-f3c19edc86d6","added_by":"auto","created_at":"2024-05-07 02:52:44","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":578696,"visible":true,"origin":"","legend":"","description":"","filename":"ammarihassinekadriteboucoupsingKV.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4277529/v1_covered_baac4b1f-8164-4aae-9343-da9ba3613219.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Stabilization of two coupled wave equations with a localized singular Kelvin-Voigt damping","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Carleman estimate, simultaneous stabilization, coupled wave equations, Singular Kelvin-Voigt damping","lastPublishedDoi":"10.21203/rs.3.rs-4277529/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4277529/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We consider two wave equations coupled through a singular Kelvin-Voigt damping mechanism in a bounded domain. We are interested in investigating stability issues for this system.We prove the polynomial stability of the semigroup if the damping region is big enough, and logarithmic stability of the semigroup if the damping region is an arbitrarily small nonempty open subset of the domain under consideration. The main features of our proofs: i) frequency domain approach and, ii) flow multipliers combined with extra auxiliary elliptic systems in the case of polynomial stability, or iii) Carleman estimate in the case of logarithmic stability. A numerical analysis of the spectrum of the one dimensional space semi-discretized system using mixed finite element method indicates that uniform (with respect to the mesh size) exponential decay is not to be expected. 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