Longtime Dynamics Of Dissipative Evolution Equations On Unbounded Domain

It’s known to us that many phenomena in physics can be described by dissipative non-linear evolution equation. The study of long time behavior of solutions of these equations will enhance our understanding of the complexity in nonlinear science. The central topic of infinite dimensional dynamical system is the global attractor. Topics like existence of the global attractor, further regularity and fractal dimension are mainly concerned. Especially, if one works on global attractor in regular spaces, weak dissipative or nonlocal dissipative equations, new difficulties will arise. In the dissertation, we are devoted to deal with prob-lems of this kind. The outline is as follows.In Chapter1, we introduce the main concepts and theorems in the field of infinite dimensional dynamical system briefly, including the sufficient and necessary conditions of the existence of global attractor, the description of compactness and related topics.In Chapter2, we consider the long-time behavior of solution of reaction diffusion equa-tion with arbitrary growth nonlinear term. Based on several a priori estimates, it can be shown that the unitary operator is Holder continuous from L2to H2∩L2p-2. Since the so-lution semigroup is asymptotically compact in L2, we obtain the L2－H2∩L2p-2asymp-totical compactness of semigroup, and then the L2-H2∩L2p-2global attractor. This result is sharp in the sense that the stationary point is at most in H2∩L2p-2.In Chapter3, we investigate the long-time behavior of solution of generalized KdV equation on the real line. We obtain the existence of an absorbing set in H2by establishing three energy equations. Using the Ball’s idea, we prove the asymptotical compactness of solution semigroup in H2. Moreover, we analyze the structure of attractor in some special cases. Precisely, it can be shown that the global attractor will reduce to a single stationary point if the force is relatively small compared to the dissipative coefficient.In Chapter4, we consider the surface Quasi-Geostrohic equation on the whole space. At first, applying positive lemma of fractional Laplacian, commutator estimates and Besov spaces technique, we prove the existence of global attractor in LP.In Chapter5, we obtain the precise estimate of fractal dimension of global attractor on torus by quasi-stable method of Chueshov&Lasiecka. The upper bound is decreasing function of dissipative coefficient, which conforms to physical intuition.In Chapter6, a fractional reaction diffusion equation with white noise is studied. The well-posedness of solution is proved by∈-regular approach. Applying the idea of tailed estimates, we show the existence of global random attractor in L2.