| In this thesis,we study the time-dependent long-time dynamics of the non-damping evolution equation with fading memory under Dirichlet boundary condition,where the nonlinearity satisfies critical growth.By applying the modified pullback attrac-tors theory,asymptotic a priori estimate method and operator decomposition technique,the existence and regularity of time-dependent attractors for the non-damping abstract evolution equations with fading memory are proved.This thesis includes four chapters:In chapter one,we first introduce the developing process and backgrounds of infinite-dimensional dynamical systems,and the basic theory development and the research pro-cess of time-dependent attractors.Then,we put forward the main problem.In chapter two,some preparations are presented,including spaces and some notations for the non-damping abstract evolution equation with fading memory,preparation for the article and some abstract results.In chapter three,the existence and regularity of time-dependent attractor in weak topological spaces H lσ=Vθ+σ×Vσ×Lμ2(R+;Vθ+σ)for the non-damping abstract evolution equation with fading memory are proved,where the nonlinearity satisfies critical growth and the external forcing term only belongs to V-θ。In chapter four,the existence and regularity of time-dependent attractor in strong topological spaces H lθ+σ=V2θ+σ×Vθ+σ×Lμ2(R+;V2θ+σ)(strong time-dependent attractor)for the non-damping abstract evolution equation with fading memory are proved,where the nonlinearity satisfies critical growth and the external forcing term belongs to H. |
