Existence Of Global Attractors For Abstract Evolution Equations With Fading Memory

By use of some new methods and estimate techniques, the long-time dynamical behaviors of the solution for abstract evolution equation with fading memory are discussed, and the existence of global attractors is proved in the weak topological space V_?×H×L_?² ?R⁺,V_?? and strong topological space V_2?×V_??L_?²(R⁺,V_2?) when the nonlinearity is critical and the forcing term satisfies g?H^-1??? and g? L²??? respectively. The paper includes three chapters:In chapter 1, we first introduce the developing process and backgrounds of dynamical systems, and the basic theory development and the research process of global attractors for autonomous dynamical systems. Then the main problem and research approach applied in this paper.In chapter 2, we study the long-time dynamical behavior of the solution for abstract evolution equation with fading memory when the nonlinearity is critical and the forcing term satisfies g ? H^-1???. The existence of global attractor is proved by applying the semigroup theory and the decomposition technique in the weak topological space V_?×H × L_?²?R⁺,V_??.In chapter 3, we study the long-time dynamical behavior of the solution for abstract evolution equation with fading memory when the nonlinearity is critical and the forcing term satisfies g ? L²???. The existence of global attractor is proved in the strong topological space V_2? × V_?× L_?²(R⁺, V_2?).