Asymptotic dynamics for autonomous and nonautonomous nonlinear wave equations
Nonautonomous dynamical systems of evolution equations on domains with moving boun...
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Author(s): |
Total Authors: 3
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Affiliation: | [1] Univ Estadual Maringa, Dept Math, BR-87020900 Maringa, Parana - Brazil
[2] Univ Estadual Londrina, Dept Math, BR-86057970 Londrina, PR - Brazil
[3] Univ Sao Paulo, Inst Math & Comp Sci, BR-13566590 Sao Carlos, SP - Brazil
Total Affiliations: 3
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Document type: | Journal article |
Source: | Journal of Differential Equations; v. 260, n. 1, p. 56-83, JAN 5 2016. |
Web of Science Citations: | 6 |
Abstract | |
This paper is concerned with the long-time dynamics of a semilinear wave equation with degenerate is viscoelasticity utt - Delta u + integral(infinity)(0) g(s)div{[}a(x)del u(t - s)]ds + f(u) = h(x), defined in a bounded domain Omega of R-3, with Dirichlet boundary condition and nonlinear forcing f (u) with critical growth. The problem is degenerate in the sense that the function a(x) >= 0 in the memory term is allowed to vanish in a part of Omega. When 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), which is in a certain sense arbitrarily small, such that a + b > 0 in Omega. 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. (C) 2015 Elsevier Inc. All rights reserved. (AU) | |
FAPESP's process: | 14/08767-1 - Frictional versus viscoelastic models with history |
Grantee: | Ma To Fu |
Support Opportunities: | Research Grants - Visiting Researcher Grant - Brazil |
FAPESP's process: | 12/19274-0 - Asymptotic dynamics for autonomous and nonautonomous nonlinear wave equations |
Grantee: | Ma To Fu |
Support Opportunities: | Regular Research Grants |