Abstract:
 We consider the initial (boundary) value problems for a class of semilinear stochastic parabolic equations of Ito type in a bounded or unbounded domain. Suppose that the nonlinear term and the multiplier of the noise term are locally Lipschitz continuous. Then there exists a unique local solution in a Sobolev space. In this talk we will discuss the existence question of some positive local solutions. Under suitable conditions, such as stochastic coercivity and positive data, we first show that the solution will remain positive almost surely at each time. In addition, if the nonlinear term is positive, convex and its reciprocal being integrable, and the noise multiplier is of linear growth, it will be shown that the Lpmoment of the solution will blow up in a finite time, for any integer p greater or equal to 1. The theorems will be proved by means of some basic tools in the stochastic analysis and the theory of differential equations.
