Uniform rates of decay of solutions for a nonlinear lattice with memory

Article type: Research Article

Authors: Perla Menzala, G.^{; ;} | Muñoz Rivera, Jaime E.^; | Konotop, V.V.

Affiliations: National Laboratory of Scientific Computation, LNCC/MCT Rua Getulio Vargas 333, Petropolis, RJ, CEP 25651 070, Brazil E‐mail: perla@lncc.br | Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, Rio de Janeiro, Brazil E‐mail: rivera@lncc.br | Centro de Física Teórica e Computacional, Universidade de Lisboa, Complexo Interdisciplinar, Av. Professor Gama Pinto 2, 1649‐003 Lisbon, Portugal E‐mail: konotop@cii.fc.ul.pt

Note: [] Corresponding author.

Abstract: In this paper we study a nonlinear lattice with memory and show that the problem is globally well posed. Furthermore we find uniform rates of decay of the total energy. Our main result shows that the memory effect is strong enough to produce a uniform rate of decay. That is, if the relaxation function decays exponentially then, the corresponding solution also decays exponentially. When the relaxation kernel decays polynomially then, the solution also decays polynomially as time goes to infinity.

Journal: Asymptotic Analysis, vol. 38, no. 2, pp. 167-185, 2004