Asymptotic Behaviour Of Nonautonomous Infinite-dimension Dynamical System And Random Dynamical System

In Chapter 1, we have discussed the three concepts: global attractor for a semigroup, kernel sections for the process of a non-autonomous system and random attractors for a random dynamical system. We remark the difference and relationship among them and introduce the history of research from global attractor to random attractor.Global attractor is a crucial concept in the theory of infinite dimensional dynamical system and one of the most important discoveries in mathematics and mathematical physics during past 30 years. Kernel sections are generalization of global attractor in non-autonomous dynamical system. In Chapter 2, we have discussed the concepts of global attractor and kernel sections. We have remarked the difference and the relationship between them. At the same time, we gave the general way on estimating Hausdorff dimension of global attractor and kernel sections.In Chapter 3, using the theory and the way on estimating Hausdorff dimension of kernel sections, the author have studied a series of non-autonomous strongly damped nonlinear wave with the damping coefficient dependent on the state, with non-degenerate Kirchhoff type, with a viscoelastic term and with linear memory, respectively. The existence of kernel sections for these equations are proved. To be important, precise estimates of upper bound of Hausdorff dimension of these kernel sections are obtained. Particularly, The non-autonomous strongly damped nonlinear wave equation with linear memory determines a system with delays. It is difficult. But by introducing the new concept of "history space", we prove the existence of kernel sections and obtain a upper bound of Hausdorff dimension of the kernel sections on the new space. Here, the inflation theory on the stability of kernel sections is also discussed. We prove the existence and continuity of inflated kernel sections for the perturbed non-autonomous strongly damped wave equations. The result implies that the unperturbed non-autonomous strongly damped wave equations possesses the stable kernel sections.In Chapter 4, we introduce the basic theory of Brown motion which has to be known in the research of stochastic dynamical system. The general concepts on stochastical differential equations are given. We have studied Ornstein-Uhlenbeck process which is very important in transformating stochastical differential equations to random differential equations. We provide the theory on the existence of solutions of ordinary differential equations and partial differential equations. Finally, we have studied the notion of random attractors and obtained the theorem on estimating their Hausdorff dimension.In Chapter 5, by a series of transformation, the author obtain the existence of random attractors for random dynamical systems determined by three damped Sine-Gordon equations with different white noises, respectively. At the same time, we obtain upper bounds of Hausdorff dimension of these random attractors. All these results are very new in stochastic wave equation. They make a progress in the theoryand application of random dynamical system.