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

We consider stochastic equations of the prototype d u ( t , x ) = ( Δ u ( t , x ) + u ( t , x ) 1 + β ) d t + κ u ( t , x ) d W t on a smooth domain D ⊂ R d , with Dirichlet boundary condition, where β , κ are positive constants and { W t , 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.