Asymptotical Behavior For Several Classes Of Stochastic Reaction-Diffusion Equations

The deterministic reaction-diffusion equations have been successfully applied in pat-tern theory and the evolution of population dynamics. For example, the study of complex systems and networks such as cells and nerves led to the birth of mathematical biology. However, all kinds of dynamic systems can be influenced by natural or man-made random factors, such as random external force, random medium, random boundary conditions, ran-dom environments, etc. In recent decades, stochastic reaction-diffusion equations attract more and more attentions of researchers. In this doctoral dissertation, we study the asymp-totical behavior for several classes of stochastic reaction-diffusion equations(we also consid-er the asymptotical behavior for deterministic equations in the last chapter). These asymp-totical behavior includes:The (L2,H1)-random attractors for stochastic reaction-diffusion equations with additive noise on unbounded domains; The (L2,H1/0)-random attractors for stochastic reaction-diffusion equations with multiplicative noise on bounded domains; The fractal dimension of random invariant sets; The exponential attractors in L2p-2(D) and H2(D) for a class of deterministic reaction-diffusion equations, where D is a bounded set in R3; The robustness of exponential attractors for a class of nonclassical diffusion equa-tions with parameter.This thesis consists of three parts:In the first part, we introduce the evolution of infinite-dimensional dynamical systems and random dynamical systems. We recall the basic notations and the known methods and results related to random attractors, and briefly describe our research results of the present paper. Then, we present some preliminary definitions and results that will be used in this thesis.The second part is the core content of this thesis. Firstly, we study the existence of (L2,H1)-random attractors for a class of stochastic reaction-diffusion equations with addi-tive noise defined on the whole space Rn, where the nonlinearity is supposed to satisfy the polynomial growth of arbitrary order p-1(p>2). To prove the (L2, H1)-asymptotically compactness, we use the cut-off technique and the tail estimate method to overcome the difficulty of the lack of compactness of Sobolev embeddings on unbounded domains. We establish a new estimate when we use the cut-off technique, and the estimate is accurate enough so that we can obtain the asymptotically compactness in any bounded ball without differentiating the equation.Secondly, we prove the existence of (L2, H1/0)-random attractors for a class of stochastic reaction-diffusion equations with multiplicative noise defined on a bounded domain D C Rn. Compare with the case for additive noise on bounded domains, it is more complicated for multiplicative noise, and the estimates on random coefficients and the Gronwall type inequality should be more accurate.Thirdly, we give a new upper bound estimate on the finite fractal dimension of random invariant sets. This is the extension of the results of the deterministic case to stochastic case. The extension is non-trivial, since, different from deterministic case, the random invariant sets and their coverings are varied with time, and we use Poincare recurrence theorem to overcome this difficulty. This new abstract result do not require differentiable property of the system and can be applied to Banach spaces, hence it has widely applications. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the fractal dimension of the random attractors.Finally, we do some research on exponential attractors for deterministic reaction-diffusion equations. These include:(1) We consider a class of reaction-diffusion equations defined on a bounded domain in R3with arbitrary polynomial growth nonlinearity/and nonhomogeneous term g. We first construct exponential attractors in H2(D) for the under-lying semigroup. Then, we obtain the exponential attractors in L2p-2(D) for any g e L2(D) by using a new approaching technique.(2) We improve the abstract results on the existence of robust exponential attractors in previous papers, and use a weaker smoothing condition in our abstract result. The modified result has more widely applications. As an application, we prove the robustness of exponential attractors in H1/0for a class of nonclassical diffusion equations with parameter, and the initial datum belongs to H1/0.In the third part, we summarize the main results and propose some problems for future research.