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

We consider stochastic equations of the prototype du(t,x)=(Δu(t,x)+u(t,x)1+β)dt+κu(t,x)dWt on a smooth domain D⊂RdD⊂Rd, with Dirichlet boundary condition, where ββ, κκ are positive constants and {Wt,t≥0}{Wt,t≥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.