Finite Dimension Global Attractor For The Coupled Unperturbed Dissipative Hamiltonian Amplitude Wave Equations

In this paper, we discuss the finite dimension global attractor for the coupled unperturbed dissipative Hamiltonian amplitude wave equations. The paper contains three parts as following:The first chapter is devoted to the summary of the dissertation. First of all, we present the background of the coupled dissipative Hamiltonian amplitude equations. Secondly, we present some basic theories of infinite dimensional dynamical systems. Thirdly, we present our goal and the main results of the paper, and methods of the dissertation.In the second chapter, we investigate the long-time behavior of the solutions to the coupled dissipative Hamiltonian amplitude equations with the periodic boundary condition. At first, by time uniform priori estimates of solutions, we obtain the existence of global solutions in the phase space E0=Hper1×Hper×Hper1×Hper(Ω) and E1=Hper2×Hper1×Hper2×Hper1(Ω) respectively. Secondly, by means of operator decomposition and energy equation, we prove the existence of global attractor A0and A1in the phase space E0and E1respectively. Finally, we prove A0=A1.In the third chapter, we estimate the upper bounds of Fractal and Hausdorff dimensions of the global attractor for the coupled unperturbed dissipative Hamilto-nian amplitude wave equations. Firstly, we consider the first variation equations of the coupled unperturbed dissipative Hamiltonian amplitude wave equations. Sec-ondly, we use Frechet differential, and then estimate uniform Lyapunov exponents of this attractor to prove the upper bounds of fractal and Hausdorff dimensions of the global attractor.