Asymptotic Behavior Of Solutions For The Nonclassical Diffusion Equation With Time-dependent Memory Kernel

Based on the theory of time-dependent global attractors introduced by M.Conti et.al.,we study the long-time dynamical behavior of solutions for the nonclassical diffusion equation with time-dependent memory kernel and critical nonlinearity in the space H01(Ω)×Lμt2(R+;H01(Ω)).This thesis mainly study the two class of equations:i)the nonclassical diffusion equation with time-dependent memory kernel:where Ω is a bounded domain with a smooth boundary (?)Ωof R3,while the external forcing g(x)∈ L2(Ω)and ht(s)is the time-dependent memory kernel function.We first obtain the well-posedness and regularity of the solution,and then we prove the existence and the regularity of the time-dependent global attractors by using the delicate integral estimation method and decomposition technique.ii)the nonclassical diffusion equation with time-dependent memory kernel and time-dependent speed of propagation:where Ω is a bounded domain with a smooth boundary (?)Ωof R3,ε(t)∈C1(R)is a decreasing bounded function,while the external forcing ∈ L2(Ω)and ht(s)is the time-dependent memory kernel function.Follow the same methods as before,we establish the existence and the regularity of the time-dependent global attractors.