Existence Of Attractors For Non-classical Diffusion Equations In Unbounded Domains

In this paper,we discussed the well-posedness and long-time behavior of the solutions in unbounded domain for the following nonclassical diffusion equation with memory:(?)Firstly,the existence and uniqueness of global weak solutions and the continuous dependence of solutions on initial value corresponding to the phase space H1(RN)×L?2(R;H1(RN))are proved in this paper.Due to the influence of the unbounded domain,we can not directly apply the classical Galerkin method to consider this problem.Therefore,we divided RN into two parts including a open ball Bn with the radius of n and the complementary set Bnc according to the method of predecessors.The wellposedeness of the equation in the open sphere Bn is equivalent to its well-posedness in the bounded domain,and the classical Galerkin method is used to obtain the result easily.So for the well-posedness of the equation of the unbounded domain RN,we only need to extend the solution in the sphere to RN,then obtain the solution sequence defined R×RN.Thus,combining theory of functional analysis,we can obtain the well-posedness of above initial value problem.Then,we discussed the existence of the global attractor corresponding to the semigroup on the basis of the well-posedness of the equation.In the process of discussion,we have the following difficulties:first,although we can use new operator decomposition techniques to get the asymptotic regularity of the solutions,however,due to the existence of memory terms,resulting in the embedding L?2(R;D(A))(?)L?2(R;H1(?))is non-compact,thus we cannot use the method of compact embedding to get compactness of the memory variable.Second,the nonlinearity satisfies the exponential growth of arbitrary order,which makes the asymptotic compactness of semigroup cannot be obtained by the classical method of solving hyperbolic equations.Third,the space variable of solutions is defined on the whole space RN,so Poincaré inequality is no longer true and the weak topology space of the solution is not of higher regularity.It has brought us great difficulty in constructing contractive function.To overcome these difficulties,we make full use of analytical techniques and constructing asymptotic compression functions on product spaces through the asymptotic regularity of solutions to obtain the existence of bounded absorbing set and the asymptotic compactness of solution semigroups,and prove the existence of global attractors in a weak topological space H1(RN)×L?2(R;H1(RN)).