Attractors for wave equations with degenerate memory

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• 作者：M.M. Cavalcanti^a ; ^{mmcavalcanti@uem.br" class="auth_mail" title="E-mail the corresponding author} ; L.H. Fatori^b ; ^{lucifatori@uel.br" class="auth_mail" title="E-mail the corresponding author} ; T.F. Ma^c ; ^{matofu@icmc.usp.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35L70 ; 35B41 ; 74D99
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 January 2016
• 年：2016
• 卷：260
• 期：1
• 页码：56-83
• 全文大小：373 K

文摘

This paper is concerned with the long-time dynamics of a semilinear wave equation with degenerate viscoelasticity defined in a bounded domain 惟 of R³${\mathbb{R}}^3$, with Dirichlet boundary condition and nonlinear forcing f(u)$f(u)$ with critical growth. The problem is degenerate in the sense that the function a(x)≥0$a(x)\ge 0$ in the memory term is allowed to vanish in a part of 241a3972837c10f85b51ee85658c">$\overline{\mathrm{惟}}$. When a(x)$a(x)$ does not degenerate and g   decays exponentially it is well-known that the corresponding dynamical system has a global attractor without any extra dissipation. In the present work we consider the degenerate case by adding a complementary frictional damping b(x)u_t$b(x)u_t$, which is in a certain sense arbitrarily small, such that a+b>0$a+b>0$ in 241a3972837c10f85b51ee85658c">$\overline{\mathrm{惟}}$. Despite that the dissipation is given by two partial damping terms of different nature, none of them necessarily satisfying a geometric control condition, we establish the existence of a global attractor with finite-fractal dimension.