Time-Dependent Asymptotic Behavior Of The Solutions For The Plate Equations

The theory of time-dependent global attractors were introduced by Conti,Plinio et.al.and applied into the wave equations and oscillation equations,respectively.Based on these new results,we investigate the longtime behavior of solutions for a class of Plate equations,dividing this thesis into three parts,according to the technique of asymptotic priori estimate and operator decomposition as well as the method of contractive function.In the first part,we discuss the longtime behavior of the solution for weaker damped Plate equations under the condition that the nonlinear term satisfies the critical exponent growth.First,we obtain the dissipativity of the process U(t,?)associate with the problem by a prior estimates.Second,using the technique of operator decomposition,we prove the asymptotic compactness of the process,and then the existence in the space H~2 x L~2 and regularity in the space H~3 x H~1 of time-dependent global attractors are shown.Finally,combining with the complete bounded trajectories of the process(semigroup),the asymptotic structure of the time-dependent attractor is achieved;besides,we obtain that ut is bounded in the space H2,and the bound is independent of the time-dependent parameter.In the second section,we investigate the longtime behavior of the solution for Plate equations with nonlinear damping.The presence of nonlinear damping make the proof of the coimpact ness of the process nmore difficult,and some method is restricted.In order to obtain the compactness of the process,we use the method of contractive function,and finally get the existence of time-dependent global attractorIn the third section,we study the Plate equation with linear memory and critical nonlinear function.Since the presence of the memory,we have to construct a more complex extend phase space,in which we make a priori estimates.After that,we verifying the compactness of the process by making use of the decomposition tech-nique and combining with the compactness of translation theorem.Thus we obtain the existence and regularity of the time-dependent global attractor;furthermoe,we prove the asymptotic structure of attractor,besides,we show that ut is bounded in a more high regularity space,it's bound is independent of ?(t).