Time-dependent Asymptotic Of Nonclassical Diffusion Equations With Delay

The theory of time-dependent global attractors is a new concept proposed by Conti,Plinio et.al.when they studied the wave equations and oscillation equation-s with time-dependent coefficients.It provides an effective method for solving the longtime behavior of solutions to such problems.Based on this theory,we investigate the longtime behavior of solutions for the nonclassical diffusion equations with delay and time-dependent coefficients by means of asymptotic a priori estimates,operator decomposition as well as the method of contractive function.We mainly consider two problems,in the first part,we discuss the longtime behavior of the solution for nonclassical diffusion equations with delay under the condition that the poly-nomial growth nonlinearity of arbitrary order.First,we obtain the well-posedness of solutions by the Faedo-Galerkin methods.Second,the presence of-?ut make the proof of the compactness of the process more difficult.We use the method of contractive function obtain the pullback asymptotic compactness,and finally prove the existence of time-dependent pullback attractor.In the second section,we inves-tigate the longtime behavior of the solution for nonclassical diffusion equations with delay and the critical exponent nonlinearity.First,we prove the well-posedness of solutions.Second,we obtain the pullback D-w limit compactness by the method of operator decomposition,and then show the existence of pullback D attractor for the family of process corresponding to the equations.