Intrinsic  decay rate estimates for  the wave equationwith  competing   viscoelastic and frictional  dissipative effects

Marcelo M. Cavalcanti; Valéria N. Domingos Cavalcanti; Irena Lasiecka; Flávio A. Falcão Nascimento

doi:10.3934/dcdsb.2014.19.1987

Article Contents

2014, Volume 19, Issue 7: 1987-2011. Doi: 10.3934/dcdsb.2014.19.1987

This issue Previous Article Uniform weighted estimates on pre-fractal domains Next Article On a generalized Cahn-Hilliard equation with biological applications

Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects

1.
Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR
2.
University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152
3.
Department of Mathematics, State University of Ceará- FAFIDAM, 62930-000 Limoeiro do Norte - CE

Received: April 2013

Revised: September 2013

Published: September 2014

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Abstract

Wave equation defined on a compact Riemannian manifold $(M, \mathfrak{g})$ subject to a combination of locally distributed viscoelastic and frictional dissipations is discussed. The viscoelastic dissipation is active on the support of $a(x)$ while the frictional damping affects the portion of the manifold quantified by the support of $b(x)$ where both $a(x)$ and $b(x)$ are smooth functions. Assuming that $a(x) + b(x) \geq \delta >0 $ for all $x\in M$ and that the relaxation function satisfies certain nonlinear differential inequality, it is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping. In the special case when the viscoelastic effect is active on the entire domain and the frictional dissipation is differentiable at the origin, then the overall decay rates are dictated by the viscoelasticity. The obtained decay estimates are intrinsic without any prior quantification of decay rates of both viscoelastic and frictional dissipative effects. This particular topic has been motivated by influential paper of Fabrizio-Polidoro [15] where it was shown that viscoelasticity with poorly behaving relaxation kernel destroys exponential decay rates generated by linear frictional dissipation. In this paper we extend these considerations to: (i) nonlinear dissipation with unquantified growth at the origin (frictional) and infinity (viscoelastic) , (ii) more general geometric settings that accommodate competing nature of frictional and viscoelastic damping.

Keywords:

Wave equation,
compact Riemanian manifold,
viscoelastic and frictional distributed damping,
decay rates.

Mathematics Subject Classification: Primary: 35L05, 74F05; Secondary: 35A27, 35Q93.

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