Attractors for wave equations with degenerate memory

Author(s)

Cavalcanti, M. M. ^[1] ; Fatori, L. H. ^[2] ; Ma, T. F. ^[3]

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

• 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