International Scholarly Research Network ISRN Mathematical Physics Volume 2012, Article ID 659289, 14 pages doi:10.5402/2012/659289 Research Article Exponential Decay to the Degenerate Nonlinear Coupled Beams System with Weak Damping R. F. C. Lobato, 1 D. C. Pereira, 2 and M. L. Santos 1 1 Programa de P´ os-Graduac ¸˜ ao, em Matem´ atica Eestat´ ıstica, Faculdade de Matem´ atica, Universidade Federal do Par´ a, Campus Universit´ ario do Guam´ a, Rua Augusto Corrˆ ea 01, 66075-110 Bel´ em, PA, Brazil 2 Departamento de Matem´ atica, Estat´ ıstica e Inform´ atica, Universidade do Estado do Par´ a, Rua do Una 156, Tel´ egrafo, 66113-200 Bel´ em, PA, Brazil Correspondence should be addressed to M. L. Santos, ls@ufpa.br Received 10 April 2012; Accepted 3 July 2012 Academic Editors: D. D ¨ urr and W.-H. Steeb Copyright q 2012 R. F. C. Lobato et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a nonlinear degenerate coupled beams system with weak damping. We show using the Nakao method that the solution of this system decays exponentially when the time tends to infinity. 1. Introduction For the last several decades, various types of equations have been employed as some mathe- matical models describing physical, chemical, biological, and engineering systems. Among them, the mathematical models of vibrating, flexible structures have been considerably stimulated in recent years by an increasing number of questions of practical concern. Research on stabilization of distributed parameter systems has largely focused on the stabilization of dynamic models of individual structural members such as strings, membranes, and beams. This paper is devoted to the study of the existence, uniqueness, and uniform decay rates of the energy of solution for the nonlinear degenerate coupled beams system with weak damping given by K 1 x, tu tt Δ 2 u - M  ‖u‖ 2  ‖v‖ 2  Δu  u t  0 in Ω × 0,T , 1.1 K 2 x, tv tt Δ 2 v - M  ‖u‖ 2  ‖v‖ 2  Δv  v t  0 in Ω × 0,T , 1.2