Abstract
We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory , J. Differential Equations (2016). Under certain extra assumptions (namely on the nonlinear term), we show the existence of a compact attracting set which provides further regularity for the global attractor and show that it consists of regular solutions.
Highlights
An elastic body perturbed from equilibrium may undergo a restoring force subject to both frictional and viscoelastic dissipation mechanisms
Unlike the methods used to prove the existence of a global attractor by virtue of the former characterization, in the latter no bounded absorbing set B nor any uniform bound on solutions is used to prove the existence of the global attractor
We have show that the global attractors associated with a wave equation with degenerate viscoelastic dissipation in the form of degenerate memory possesses more regularity than previously obtained in [4]
Summary
An elastic body perturbed from equilibrium may undergo a restoring force subject to both frictional and viscoelastic dissipation mechanisms. Unlike the methods used to prove the existence of a global attractor by virtue of the former characterization, in the latter no (explicit) bounded absorbing set B nor any (explicit) uniform bound on solutions is used to prove the existence of the global attractor It seems that an explicit bound in terms of some of the parameters of the problem (Lipschitz constant, etc.) can be given to the fractal dimension of the global attractor (see [6, Theorem 3.4.5]). These results are obtained without assuming the two damping terms satisfy a geometric control condition We aim to provide a regularity result to the global attractors found in [4] for the problem (1.1)–(1.3)
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