An Approximation For SPDE With Non-Lipschitz Coefficients

Usually , the term"stochastic differential equation"is applied to ordinary stochas-tic differential equations. Their theory originated from the Ito? stochastic integral andIto?'s equations that were created by Japanese mathematician Ito?.K in 1940s. Gradually,stochastic differential equations developed to one of the most beautiful and fruitful branchesof stochastic analysis. In the middle of the 1970s, in various disciplines(especially, inphysics , biology , and control theory) a vast number of models were founded that couldbe described by stochastic partial differential equations. The field of stochastic analysis wasgreatly stimulated by the emergence of these new type of equations. It also attracted a greatdeal of researchers and became the most active branches of stochastic analysis.In this thesis, mild solutions of stochastic reaction-diffusion equations and semilinearwave equations with non-Lipschitz coefficients is studied. Firstly, by means of the semi-group generated by operators, the equations mentioned above can be turned into integralequations. Secondly, by use of Picard-Iteration and Gronwall-Bellman inequality, mildsolutions of above equations can be constructed and its uniqueness also can be verified. Sothe theorems of existence and uniqueness of mild solutions of these two type of SPDEsmentioned above are verified. This paper consists of three chapters. In chapter one , aspreliminary knowledge, we construct R-Wiener process in H from a nuclear operator Ron a Hilbert space H and a sequence of independent standard Brown motion in one di-mension. And then, we, by three steps, construct stochastic integral of continuous adaptedrandom fields with respect to the R-wiener process in H. Chapter 2 deals with the existenceand uniqueness theorem of mild solution of a stochastic reaction-diffusion equation withinitial-boundary value and non-Lipschitz coefficients. By taking use of Picard-Iteration, weconstruct a mild solution whose uniqueness is verified at the same time. In chapter3, by thesame ideas and methods as chapter 2, the existence and uniqueness theorem of mild solutionof a stochastic semilinear wave equation with initial-boundary value under non-Lipschitzcoefficients is researched .