On the decay of the energy for systems with memory and indefinite dissipation

Article type: Research Article

Authors: Muñoz Rivera, Jaime E. | Naso, Maria Grazia

Affiliations: National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitadinha-Petrópolis 25651-070, Rio de Janeiro, RJ, Brazil E-mail: rivera@lncc.br | Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti 9, 25133 Brescia, Italia E-mail: naso@ing.unibs.it

Abstract: In this paper we study the asymptotic behavior of the viscoelastic system with nondissipative kernels. We show that the uniform decay of the energy depends on the decay of the kernel, the positivity of the kernel in t=0 and some smallness condition. That is, if the kernel g∈C2(${\mathbb{R}}$+) with g(0)>0, decays exponentially to zero then the solution decays exponentially to zero. On the other hand, if the kernel decays polynomially as t−p then the corresponding solutions also decays polynomially to zero with the same rate of decay.

Keywords: materials with memory, asymptotic stability, indefinite dissipation

Journal: Asymptotic Analysis, vol. 49, no. 3-4, pp. 189-204, 2006