The Asymptotic Behaviors For Two Classes Of Evolution Equations With Fading Memory

In this paper,the long-time dynamical behaviors of two classes of equations with memory,which are weak dissipative abstract evolution equation and nonclassical diffusion equation with linear memory,are studied under Dirichlet boundary condition by using general infinite-dimensional dynamical system theory.Some known results and estimation techniques have been made for discussing the asymptotic behaviors of solution,and global attractors for corresponding dynamical systems have been obtained in weak and strong topological spaces separately.i).Consider the following dissipative abstract evolution equation without damping term aut.Since system dissipate energy via damping and linear memory term for the general ab?stract evolution equation.When the damping term is not considered,it is only fading memory term who dissipates the energy for the dynamical system above.As a result,the velocity of energy dissipation has been slowed down.Moreover,the system contains linear memory which is not compact in correlate space,finally it is hard to obtain the compactness of solution semigroup.Therefore,by using the theory of semigroup,con?tractive function and classical global attractors,the difficulty in verifying compactness has been overcome,and also the existence of global attractors in weak topological space V1×H×L?2(R+;V1)and strong topological space V2×V1×L?2(R+;V2)is obtained in this paper.ii).Consider the following nonclassical diffusion equation with fading memory When the nonlinearity satisfies arbitrary order polynomial growth,that is,the growth of the super critical exponent,the existence of global attractor in the weak topological space H01(?)×L?2(R+;H01(?))and strong topological space(H2(?)? H01(?))× L?2(R+;(H2(?)?H01((?)))is obtained by using the theory of asymptotic contraction function.