Time-Dependent Asymptotic Behavior Of The Solutions For The Nonclassical Reaction-Diffusion Equations With And Without Linear Memory

In recent years, M.Conti, F.Di Plinio et.al. introduced the theory of time-dependent global attractors and applied it into the wave equations and oscillation equations, respectively. Based on these results we investigate the longtime behavior, regularity and asymptotic structure of solutions for nonclassical reaction-diffusion equations with and without linear memory on time-dependent spaces according to asymptotic priori estimate and operator decomposition technique.For nonclassical reaction-diffusion equations, we investigate the dynamical be-havior of the solutions when the force term g(x)?H-1(?) and the nonlinear func-tion f(u) satisfies the critical exponent growth. Firstly, we obtain the existence of time-dependent bounded absorbing sets of the process U(t,?) associated with the problem by the priori estimate; secondly, according to the operator decomposition technique, we get the existence and regularity of time-dependent global attractors; finally, combining with the theories of complete bounded trajectories, we study the asymptotic structure of time-dependent global attractors. Further, we show that the higher regularity of this attractors.For nonclassical reaction-diffusion equations with linear memory, Follow the same methods as before, we establish the existence as well as the regularity and asymptotic structure of time-dependent global attractors for a class of nonclassical diffusion equations with fading memory when the force term g(x)?L2(?). Since the time-dependent coefficient ?(t) and the memory kernel function ?(s) simultaneously contained in the problem, this makes it more difficult for us to verify the compactness of the process U(t,?).