Time-Dependent Attractors For Abstract Evolution Equations With Fading Memory

In this paper,we study the time-dependent long-time dynamics of solution for the evolution equation with fading memory by using the theory of general infinite-dimensional dynamical system.First we discuss the asymptotic behaviors of solution by means of some new results and estimation techniques,and then obtain time-dependent attractors for dynamical systems in a weak and strong topological separately.Consider the following abstract evolution equations with fading memory:In the course of the study we found that,the key of the research lies in the compactness verification of process family.However,noncompact memory space and abstract opera-tor A? all brings essential difficulties to the verification of compactness.In addition,it is difficult to apply the general dynamical system theory and the existing research frame-work directly to study the time-dependent dynamic behavior of the memory model.For the above difficulties,by applying the techniques of asymptotic a priori estimates and the methods of operator decomposition,combining with modified pullback attractors theory,we succeed in overcoming these difficulties,the existence and regularity of time-dependent attractors are obtained.