{"title":"On a beam model with degenerate nonlocal nonlinear damping","authors":"V. Narciso, F. Ekinci, E. Pişkin","doi":"10.3934/eect.2022048","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>This paper contains results about the existence, uniqueness and stability of solutions for the damped nonlinear extensible beam equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> \\begin{document}$ u_{tt}+\\Delta ^2u-M(\\|\\nabla u(t)\\|^2)\\Delta u+\\|\\Delta u(t)\\|^{2\\alpha}\\,|u_t|^{\\gamma}u_t = 0\\ \\mbox{ in } \\ \\Omega \\times \\mathbb{R}^+, $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\alpha>0 $\\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\gamma\\ge 0 $\\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\Omega\\subset \\mathbb{R}^n $\\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\Gamma = \\partial \\Omega $\\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id=\"M5\">\\begin{document}$ M $\\end{document}</tex-math></inline-formula> is a nonlocal function that represents beam's extensibility term. The novelty of the work is to consider the damping as a product of a degenerate and nonlocal term with a nonlinear function. This work complements the recent article by Cavalcanti et al. [<xref ref-type=\"bibr\" rid=\"b8\">8</xref>] who treated this model with degenerate nonlocal weak (and strong) damping. The main result of the work is to show that for regular initial data the energy associated with the problem proposed goes to zero when <inline-formula><tex-math id=\"M6\">\\begin{document}$ t $\\end{document}</tex-math></inline-formula> goes to infinity.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"94 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Evolution Equations and Control Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/eect.2022048","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
This paper contains results about the existence, uniqueness and stability of solutions for the damped nonlinear extensible beam equation
where \begin{document}$ \alpha>0 $\end{document}, \begin{document}$ \gamma\ge 0 $\end{document}, \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} is a bounded domain with smooth boundary \begin{document}$ \Gamma = \partial \Omega $\end{document}, and \begin{document}$ M $\end{document} is a nonlocal function that represents beam's extensibility term. The novelty of the work is to consider the damping as a product of a degenerate and nonlocal term with a nonlinear function. This work complements the recent article by Cavalcanti et al. [8] who treated this model with degenerate nonlocal weak (and strong) damping. The main result of the work is to show that for regular initial data the energy associated with the problem proposed goes to zero when \begin{document}$ t $\end{document} goes to infinity.
This paper contains results about the existence, uniqueness and stability of solutions for the damped nonlinear extensible beam equation \begin{document}$ u_{tt}+\Delta ^2u-M(\|\nabla u(t)\|^2)\Delta u+\|\Delta u(t)\|^{2\alpha}\,|u_t|^{\gamma}u_t = 0\ \mbox{ in } \ \Omega \times \mathbb{R}^+, $\end{document} where \begin{document}$ \alpha>0 $\end{document}, \begin{document}$ \gamma\ge 0 $\end{document}, \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} is a bounded domain with smooth boundary \begin{document}$ \Gamma = \partial \Omega $\end{document}, and \begin{document}$ M $\end{document} is a nonlocal function that represents beam's extensibility term. The novelty of the work is to consider the damping as a product of a degenerate and nonlocal term with a nonlinear function. This work complements the recent article by Cavalcanti et al. [8] who treated this model with degenerate nonlocal weak (and strong) damping. The main result of the work is to show that for regular initial data the energy associated with the problem proposed goes to zero when \begin{document}$ t $\end{document} goes to infinity.
期刊介绍:
EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include:
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