Finite-time blowup and existence of global positive solutions of a semi-linear SPDE

Abstract

We consider stochastic equations of the prototype $du(t,x)=(\Delta u(t,x)+u(t,x{)}^{1+\beta })dt+\kappa u(t,x)d{W}_t$ on a smooth domain $D\subset {\mathrm{I}\mathrm{R}}^d$, with Dirichlet boundary condition, where $\beta$, $\kappa$ are positive constants and $\{W_t$, $t\ge 0\}$ is a one-dimensional standard Wiener process. We estimate the probability of finite time blowup of positive solutions, as well as the probability of existence of non-trivial positive global solutions.