{"title":"具有 Balakrishnan-Taylor、强和局部非线性阻尼的可扩展梁方程的指数稳定性","authors":"Zayd Hajjej","doi":"10.1007/s00233-024-10419-9","DOIUrl":null,"url":null,"abstract":"<p>We study a nonlinear Cauchy problem modeling the motion of an extensible beam </p><span>$$\\begin{aligned} \\vert y_t\\vert ^{r}y_{tt}{} & {} +\\gamma \\Delta ^2 y_{tt}+\\Delta ^2y-\\left( a+b\\vert \\vert \\nabla y\\vert \\vert ^2+c (\\nabla y, \\nabla y_t)\\right) \\Delta y\\\\{} & {} \\quad +\\Delta ^2 y_t+ d(x)h(y_t)+f(y)=0, \\end{aligned}$$</span><p>in a bounded domain of <span>\\(\\mathbb {R}^N\\)</span>, with clamped boundary conditions in either cases: when <span>\\(r=\\gamma =0\\)</span> or else when <i>r</i> and <span>\\(\\gamma \\)</span> are positive. We prove, in both cases, the existence of solutions and the exponential decay of energy.</p>","PeriodicalId":49549,"journal":{"name":"Semigroup Forum","volume":"12 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exponential stability of extensible beams equation with Balakrishnan–Taylor, strong and localized nonlinear damping\",\"authors\":\"Zayd Hajjej\",\"doi\":\"10.1007/s00233-024-10419-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study a nonlinear Cauchy problem modeling the motion of an extensible beam </p><span>$$\\\\begin{aligned} \\\\vert y_t\\\\vert ^{r}y_{tt}{} & {} +\\\\gamma \\\\Delta ^2 y_{tt}+\\\\Delta ^2y-\\\\left( a+b\\\\vert \\\\vert \\\\nabla y\\\\vert \\\\vert ^2+c (\\\\nabla y, \\\\nabla y_t)\\\\right) \\\\Delta y\\\\\\\\{} & {} \\\\quad +\\\\Delta ^2 y_t+ d(x)h(y_t)+f(y)=0, \\\\end{aligned}$$</span><p>in a bounded domain of <span>\\\\(\\\\mathbb {R}^N\\\\)</span>, with clamped boundary conditions in either cases: when <span>\\\\(r=\\\\gamma =0\\\\)</span> or else when <i>r</i> and <span>\\\\(\\\\gamma \\\\)</span> are positive. We prove, in both cases, the existence of solutions and the exponential decay of energy.</p>\",\"PeriodicalId\":49549,\"journal\":{\"name\":\"Semigroup Forum\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-03-22\",\"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/s00233-024-10419-9\",\"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/s00233-024-10419-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们研究了一个模拟可伸展梁运动的非线性考奇问题 $$\begin{aligned}\vert y_t\vert ^{r}y_{tt}{} & {}+\gamma \Delta ^2 y_{tt}+\Delta ^2y-\left( a+b\vert \vert \vert \vert ^2+c (\nabla y, \nabla y_t)\right) \Delta y\{} & {}\quad +\Delta ^2 y_t+ d(x)h(y_t)+f(y)=0, \end{aligned}$$ in a bounded domain of \(\mathbb {R}^N\), with clamped boundary conditions in either cases: when \(r=\gamma =0\) or else when r and\(\gamma \) are positive.在这两种情况下,我们都证明了解的存在和能量的指数衰减。
in a bounded domain of \(\mathbb {R}^N\), with clamped boundary conditions in either cases: when \(r=\gamma =0\) or else when r and \(\gamma \) are positive. We prove, in both cases, the existence of solutions and the exponential decay of energy.
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