Long-time Behaviors For Some Nonlinear Evolution Equations With Fading Memory

This dissertation mainly discusses the long time behaviors for some nonlinear evolution equations with fading memory.Chapter 1 mainly introduces the research methods of global attractors, re-search status for some nonlinear partial differential equations with fading memory,and gives the main research content and purpose of this dissertation.Chapter 2 is devoted to study the long-time dynamical behavior of fourth order pseudo-parabolic equation with memory when nonlinearity is critical. By using the decompose techniques and compactness transitivity theorem, we show the existence of global attractors within the past history framework.Chapter 3 concerns with the existence of pullback attractors to the non-autonomous nonclassical diffusion equations with nonlocal diffusion and nonlinear terms with subcritical growth. Under some suitable assumptions, using the energy method, we prove the existence of minimal pullback attractors for the associated process in two different frameworks. In addition, we establish some relationships between the attractors for the universe of fixed bounded sets and those associated to a universe given by another tempered condition.Chapter 4 deals with the long-time dynamics a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, we prove the existence of a global attractor. Furthermore, in the subcritical case, we apply the quasi-stability property to show that this global attractor has finite Hausdorff and fractal dimensions.Chapter 5 is considered with the long time behavior of a quasilinear viscoelas-tic equation with nonlinear damping. Firstly, we apply Galerkin method to prove the global existence and uniqueness of weak solutions. Then, we obtain the decay estimate for the energy of solutions by using energy perturbation. Finally, we apply the stability inequality to show the existence of a global attractor.