Global Attractors For The Semilinear Reaction-diffusion Equations With Memory

In this paper,we disscussed the well-posedness and long-time behavior of global strong solutions on the strong topological space H01(?)×L?(R;D(A))for the following semi-linear reaction-diffusion equation with memory (?) where ?(?)Rn(n? 3)is a bounded domain with a smooth or piecewise smooth boundary (?)?,g is a given external force term and k(s)is the memory kernel function,which satisfies the appropriate conditions.First of all,by using the Faedo-Galerkin method and combining with energy estimation and analysis skills,the existence and uniqueness of global strong solutions and the continuous dependence of solutions on the initial value are proved,corresponding to the strong topological space H01(?)×L?2(R;D(A)).Next,we investigate the existence of global attractors with respect to the solution semigroup corresponding to the global strong solution of the equation.When we consider the long-time behavior of the global strong solution,there will be some difficulties.Firstly,due to the higher the regularity of the phase space,the more difficult it is to verify the compactness of the solution semigroup,so it is difficult to obtain the compactness of the system solution semigroup through general methods(such as compact embedding).Secondly,because the nonlinear term f satisfies the exponential growth of arbitrary order,we can't directly construct a contractive function to prove that the solution semigroup corresponding to the global strong solution is a contractive semigroup.The method of constructing contractive function is one of the most effective methods to prove the asymptotically compact of solution semigroup in a highly regular phase space,and the key lies in the construction of contractive function in phase space.We use effective a prior estimation and theoretical methods such as the Lebesgue control convergence theorem to verify that the solution semigroup produced by the global strong solution is a contractive semigroup in a strong topological space.This proves the existence of strong global attractors A,when the nonlinear term f satisfies the exponential growth of arbitrary order for the reaction-diffusion equation with memory in the strong topological space H01(?)×L?2(R,D(A)).In particular,in the process of proving the existence of strong global attractors,we also get the regularity of the attractor,that is,A is the bounded set of D(A)× L?2(R;D(A)).