Long Time Behavior Of Solution For Generalized B-BBM Equation In Rn

BBM equation is a kind of important nonlinear evolution equations. It initially originated in the model of the study of nonlinear dispersion spread in water wave by Benjamin, Bona, Mahony. Moreover, the equation has a clear physical background. The study has important significance.The article presents the long time behavior of solution for the generalized BBM equation on unbounded domains: u-a△u+δ△2u-β△u+γu+▽F(u)=h(x), x∈Rn,t∈R+; u(x,0)=u0(x). x∈Rn. Furthermore, the existence of global attractors and exponential attractors for the equations are proved.The paper is arranged as follows:The first chapter presents the background and current research situation of BBM equations, and gives the main research results of this paper.The second chapter gives some definitions and lemmas relate to the paper, and brief statement related knowledge of Sobolev space.In the third chapter,it proves the existence,uniqueness and continuity of the solution for the BBM equation in unbounded area Rn(n≥1) by Galerkin and regional limit method.In the fourth chapter operator decomposed technique and the Kuratowskii a non-compacted measure are applied to study the smooth property of the sol-ution. Finally, the existence of the global attractor of the equation in H2(Rn)(n≥1)is proved.The chapter5is devoted to the proof of the existence of the exponential attractors in unbounded area H2(Rn)(n≤3).The sixth chapter presents the summary of this dissertation and prospects of the researches.