Random Attractors For Stochastic Wave Equations And Global Attractors For Two Kinds Of Lattice Systems

Infinite dimensional dynamical systems play an very important role in nonlinear science. Lattice systems and nonlinear wave equations are two kinds of very important infinite dimensional dynamical systems. Attractor (includes global attractor, random attractor) is that of central parts in studying infinite dimensional dynamical systems. The research on the attractors lies in two aspects. one aspect is the existence of the attractor, the other is the geometry properties of the attractor under the existence of the attractor, such as the Kolmogorov entropy, Hausdorff dimension and upper continuity and so on. This Ph.D. thesis focuses on research on the stochastic attractors for nonlinear wave equations and the global attractors for two kinds of infinite lattice systems such as one-dimensional Klein-Gordon-Schr(o|¨)dinger (KGS) infinite lattice system and high-dimensional dissapitive Zakharov infinite lattice system. Firstly, the author introduce the development survey and main research directions of dynamical systems and the author's research works. In Chapter 2, the author briefly introduce preliminary results and definitions, the Sobolev function spaces and frequently used inequalities such as Young's Inequality, Holder's Inequality and Gronwall's Inequality.The research works of this thesis consist of two parts.The first part of the research works is presented in Chapter 3 and 4. In Chpter 3, the author first present the existence of a random attractor of a stochastic dynamical system generated by a damped nonlinear wave equation with white noise under the Dirichlet boundary condition and estimated the explicit bound of the random attractor; and then obtain an estimate of upper bound of the Hausdorff dimension of the random attractor. The obtained upper bound of the Hausdorff dimension decreases as the damping grows and it is uniformly bounded if the derivative of nonlinearity is bounded, moreover, in this case, the upper bound of the Hausdorff dimension of the random attractor is just the upper bound of the Hausdorff dimension of the global attractor for the corresponding deterministic system without noise, i.e., in this case, the bound of the Hausdorff dimension of the random attractor is not influenced by the noise term. However, in general, the upper bound of the Hausdorff dimension of the random attractor depends on the noise term.In Chpter 4, the author consider a strongly damped sine-Gordon equations with white noise. By introducing weight norm and splitting the positivity of the linear operator in the...