{"title":"具有可变系数、局部开尔文-沃伊特阻尼和时间延迟的耦合波方程的稳定性","authors":"Houssem Herbadji, Ammar Khemmoudj","doi":"10.1007/s00233-024-10453-7","DOIUrl":null,"url":null,"abstract":"<p>We consider the stabilization of two weakly dissipative wave equations with variable coefficients, localized Kelvin–Voigt damping, mixed boundary condition and time delay. The well posedness is obtained by the semigroup method. Using a unique continuation result, we show that the system is strongly stable. Then, we show that the system is not exponentially stable. Finally, using a frequency domain approach combined with multiplier method, we establish a polynomial energy decay rate for the undelayed system. Then, we prove that the system with delay has the same decay rate.</p>","PeriodicalId":49549,"journal":{"name":"Semigroup Forum","volume":"53 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stability of coupled wave equations with variable coefficients, localised Kelvin–Voigt damping and time delay\",\"authors\":\"Houssem Herbadji, Ammar Khemmoudj\",\"doi\":\"10.1007/s00233-024-10453-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the stabilization of two weakly dissipative wave equations with variable coefficients, localized Kelvin–Voigt damping, mixed boundary condition and time delay. The well posedness is obtained by the semigroup method. Using a unique continuation result, we show that the system is strongly stable. Then, we show that the system is not exponentially stable. Finally, using a frequency domain approach combined with multiplier method, we establish a polynomial energy decay rate for the undelayed system. Then, we prove that the system with delay has the same decay rate.</p>\",\"PeriodicalId\":49549,\"journal\":{\"name\":\"Semigroup Forum\",\"volume\":\"53 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Semigroup Forum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00233-024-10453-7\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Semigroup Forum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00233-024-10453-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Stability of coupled wave equations with variable coefficients, localised Kelvin–Voigt damping and time delay
We consider the stabilization of two weakly dissipative wave equations with variable coefficients, localized Kelvin–Voigt damping, mixed boundary condition and time delay. The well posedness is obtained by the semigroup method. Using a unique continuation result, we show that the system is strongly stable. Then, we show that the system is not exponentially stable. Finally, using a frequency domain approach combined with multiplier method, we establish a polynomial energy decay rate for the undelayed system. Then, we prove that the system with delay has the same decay rate.
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