Well-posedness And Asymptotic Behavior For Several Stochastic Partial Differential Equations

With the progresses of the research of fluid mechanics, plasma physics, nonlinear op-tics, molecular biology and so on, the propagation of nonlinear wave in random media, external pressure, turbulence and white noise perturbation form a lot of realistic models, they exhibit many new phenomena and new features in contrast to the deterministic partial differential equations, such as the shape and propagation velocity of solitons are affected by some random factors.On the other hand, the corresponding complex systems of various natural phenomena often show some random characteristics or uncertainty characteristics, for instance, random factors maybe delay or prevent the Blow-up occur in some nonlinear systems. A lot of facts that the deterministic equations can produce some new phenomena under random perturba-tions, for example, some results on stochastic inviscid Burgers equation exhibit these new phenomena.This Ph.D. thesis considers the well-posedness and asymptotic behavior of several kinds of stochastic partial differential equations, it contains the stochastic nonlocal Swift-Hohenberg model, the stochastic nonlinear thermoelastic system coupled sine-Gordon e-quation driven by jump noise, the stochastic viscoelastic wave equations with nonlinear damping and source terms, the stochastic damped wave equation with stochastic dynami-cal boundary conditions and the nonclassical diffusion equations with fading. They have very important applications in the research of viscoelastic materials, atmosphere, the ocean, engineering and physics.This Ph.D. thesis is divided into six chapters.In Chapter1, we introduce the physical backgrounds and research progresses of several kinds of the stochastic partial differential equations, and describe some ideas about these equations we will discuss in this paper.In Chapter2, we discuss the stochastic nonlocal Swift-Hohenberg equation on bounded domains. Under appropriate initial and boundary conditions, we proved the existence and uniqueness of the strong probabilistic solution.In Chapter3, we consider the stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. This kind of system comes from engineering and physics. We get the existence and uniqueness of strong probabilistic solution to an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.In Chapter4, we study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data, the relaxation function, the indices of nonlinear damping and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions respectively.In Chapter5, we consider the stochastic damped wave equation with stochastic dynam-ical boundary conditions on a bounded domain. Since the dynamic boundary conditions to prevent the possibility of using fractional power operators to obtain estimates in Sobolev space of higher order, we can use the method of energy equation to prove the asymptotic compactness of solution, then we get the existence of a random attractor.In Chapter6, we study stochastic nonclassical diffusion equation with fading memory on a bounded domain. Sice the low regularity of the force terms, we can use the method of the decomposition of the solution operator to prove the asymptotic smoothness of the solution with the initial boundary value problem, and then we obtain the existence of a random attractor.