| In this thesis,we mainly consider the asymptotic behavior of the solutions for the following non-classical diffusion equation under different assumptions:ut+(-Δ)θut+ν(-Δ)θu+∫0∞ k(s)(-Δ)θu(t-s)ds+f(u)=g,where ν is a non-negative perturbation parameter.(1)When θ=1,ν∈[0,1),and the external force term g is related to time(denoted as g(t))and satisfies the appropriate assumptions,we consider the existence and regularity of its uniform attractor.During the research process,we encountered the following difficulties:(i)Since there is a dissipative term-νΔu with perturbation parameter,it is essentially the same as non-classical diffusion equations lacking instantaneous damping.Thus,the usual methods for judging the dissipation of the equation no longer applicable.(ii)We require the nonlinear term f to satisfy a polynomial growth of arbitraryorder,which is different from the critical(or subcritical)condition,that required in many literatures.Correspondingly,we must overcome the difficulties posed by this growth condition.(iii)Furthermore,because of the existence of the memory term,the embedding Lμ2(R+,D(A))→Lμ2(R+,H01(Ω))is non-compact.We can’t directly prove the asymptotic compactness of the family of processes ｛μσ(t,τ)}t≥τ,σ∈∑ by general method,this brings the essential difficulty in proving the existence of attractors.With the help of the methods proposed by Monica C and Nguyen D,we obtain the existence of bounded absorbing set for the system in this thesis.By using a new analytical technique(lemma 2.2.1)and a non-classical operator decomposition method,it is proved that the global weak solution of the system can be decomposed into two parts,one of which corresponds to the decay of the energy functional(non-exponential decay)to 0,while the other has higher regularity.Since Lμ2(R+,D(A))(?)Lμ2(R+,H01(Ω))is not a compact embedding,and the compactness and regularity problems of the processes family can only be solved by combining with the contractive function method,thus obtaining the existence of the uniform attractor of the system on the product space H01(Ω)× Lμ2(R+,H01(Ω)).Combing with some technique,we also prove the upper semi-continuity of the uniform attractors family(with respect to ν)and kernel sections respectively.In additionally,the well-possed of this equation is obtained in Rn.(2)When k(s)=0,θ ∈(0,1],ν is related to time(i.e.ν(t))and satisfies appropriate assumptions,this means that the degree of dissipation of the system will vary with the change of time.The system essentially becomes one without instantaneous dissipation.It is more complicated to study.Hence we only consider the well-possed of the solution for the system over a certain period of time.Using the nonclassical Faedo-Galerkin method,interpolation inequality and control convergence principle,the existence and uniqueness of the global weak solution for this equations in Hθ{0<θ ≤1)are obtained. |
