Study On Problems Of Global Attractors For Wave Equations With Nonlinear Damping

In this doctoral dissertation, we mainly consider the long-time behaviors for the solutions of the following nonlinear damped wave equations with critical nonlinear ity:In abstract framework, using the concept of uniformly asymptotic compactness, we establish an abstract theorem to verify the existence of uniform attractor for non-autonomous systems;furthermore, in order to describe the structure of uniform attractor, we present the concept of skew productively norm-to-weak continuous semigroup which corresponds to that of autonomous problems, and then we establish some abstract results to characterize the existence and structure of uniform attractor.To verify the necessary compactness of semigroup, in Chapter 4 we develop a new method to prove asymptotic compactness which seems tailored for hyperbolic equations.For concrete problems, in Chapter 5, we obtain the existence of the global attractor for nonlinear damped autonomous semilinear wave equations with critical nonlinearity;in Chapter 6, when the growth order of nonlinear terms is critical, we obtain the existence of the uniform attractor for non-autonomous semilinear wave equations with non-translation compact external forces. Furthermore, we obtain the structure of the uniform attractor by use of the concept of norm-to-weak continuous process.