Long-time Behaviors For The Solutions Of Infinite Dimensional Dissipative Dynamical Systems In Unbounded Domains

In this doctoral dissertation, we mainly consider the long-time behaviors for the solutions of infinite dimensional dissipative dynamical systems in unbounded domains. In order to include special solutions (e.g. relaxation wave solutions, constant solutions,), we choose sufficiently "large" spaces to work in, that is, locally uniform spaces. Locally uniform spaces, which can be traced back to Kato[49], have been used by many authors in studying both parabolic and hyperbolic problems (e.g., Mielke[67], Mielke & Schneider[68], Feireisl[37], Zelik[93], Chloewa &;Dlotko[16] and Arrieta, Rodriguez-Bernal, Cholewa &;Dlotko[3] etc.). It is well known that in many cases the long-time behaviors of dynamical systems, generated by evolutionary equations of mathematical physics, can be naturally described in terms of attractors of the corresponding semigroups. According to the characteristics of the locally uniform space, we firstly give a suitable definition of the global attractor in locally uniform spaces, and then develop some new a priori estimate methods for verifying the condition—asymptotic compactness or ω-limit compactness, which is the necessary condition for proving the existence of global attractors for the semigroups generated by the evolutionary equations. Applying these methods, we get a series of new and meaningful results in considering certain infinite dimensional dynamical systems.In order to obtain the necessary asymptotic compactness of the semigroup, we firstly give some characterizations for the locally uniform spaces in Chapter 2, which are also interesting for understanding locally uniform spaces, and then we develop a new a priori estimate method for verifying the necessary asymptotic compactness. In Chapter 3, according to the characteristics of locally uniform spaces we modify the definition of the global attractor for locally unform spaces and give a new scheme to prove the existence of global attractor.In Chapter 4, we consider the existence of a global attractor for strong damped semilinear wave equation with critical nonlinearity. The necessary asymptotic compactness is obtained by use of the asymptotic a priori estimate method, given in the previous Chapter 2, and some cut-off function technique.In Chapter 5, using the asymptotic a priori estimate method, we obtain the asymptotic compactness of the semigroup generated by the nonclassical diffusion equation with critical nonlinearity and then prove the existence of a global attractor. Finally, in Chapter 6, we firstly give a simple criterion(which is seems to be tailored for the weak damped semilinear wave equation) for the necessary asymptotic compactness of semigroup, and then we prove the existence of the global attractor for the weak damped semilinear wave equation with critical nonlinearity.