Exponential stability for the wave model with localized memory in a past history framework |
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Authors: | MM Cavalcanti VN Domingos Cavalcanti MA Jorge Silva AY de Souza Franco |
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Institution: | 1. Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil;2. Department of Mathematics, State University of Londrina, 86057-970, Londrina, PR, Brazil |
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Abstract: | In this paper we discuss the asymptotic stability as well as the well-posedness of the damped wave equation posed on a bounded domain Ω of , subject to a locally distributed viscoelastic effect driven by a nonnegative function and supplemented with a frictional damping acting on a region A of Ω, where in A. Assuming that is constant, considering that the well-known geometric control condition holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero, we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero, even in the absence of the frictional dissipative effect. In addition, in some suitable cases where the material density is not constant, it is also possible to remove the frictional damping term , that is, the localized viscoelastic damping is strong enough to assure that the system is exponentially stable. The semi-linear case is also considered. |
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Keywords: | 35A27 35B35 35L05 74Dxx Wave equation Exponential decay Locally distributed viscoelastic damping Past history |
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