Existence Of Exponential Attractors For Nonclassical Diffusion Equations

In this paper,we consider the dynamic behavior of the global solution of the following nonclassical reaction-diffusion equations.ut-vΔut-Δu-∫0∞ k(s)Δu(t-s)ds+f(u)=g,where Ω is a bounded region with sliced smooth boundary in Ω(?)R3,the nonlinearity term f satisfies an appropriate condition,g is a given function,k(s)kernel function that satisfies the appropriate conditions and a is any positive constant.First,using Galerkin method combined with energy estimation,the existence and uniqueness of the overall strong solution and the continuous dependence on the initial value of the nonclassical nonautonomous reaction diffusion equations are studied in strong topological space D(A)×Lμ2(R;D(A)).Then,by verifying that the corresponding process family{Uσ(t,τ)},σ∈∑ of the system the overall strong solution that satisfies the condition(C)and weak continuity,thus obtaining the existence and structure of the uniform attractors.It is worth noting that the nonlinearity term f(u)satisfies any order exponential growth instead of critical exponential growth,and the initial value is uτ E D(A),it cannot be used like the weak topology space.The application of Sobolev embedding theorem or factorization is inapplicable here to obtain the high regularity or asymptotic regularity of the overall strong solution,and the asymptotic regularity of the solution cannot be obtained by the general method.Finally,we discuss the existence of exponential attractors in the H01(Ω)space for the solution semigroup corresponding to the overall weak solution in the case of autonomous and k(s)=0.The asymptotic regularity of solutions is proved in H01(Ω)by a new decomposition technique.At the same time,the existence of the global attractor of the solution semigroup corresponding to the overall weak solution of the system and the boundedness of its fractal dimension are obtained.Then we confirm the existence of the exponential attractors.M by validated the global exponentially k-dissipative and its differentiability.It is worth mentioning that the nonlinearity s satisfies the polynomial growth of arbitrary order and M is bounded in H01(Ω)∩H2(Ω).