Global solutions to stochastic reaction.diffusion equations with super-linear drift and multiplicative noise.

Let .(t,x) denote space.time white noise and consider a reaction.diffusion equation of the form u.(t,x)=12u"(t,x)+b(u(t,x))+.(u(t,x)).(t,x), on R+.[0,1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists .>0 such that |b(z)|.|z|(log|z|)1+. for all sufficiently-large values of |z|. When ..0, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209.215] have recently shown that there is finite-time blowup when . is a nonzero constant. In this paper, we prove that the Bonder.Groisman condition is unimprovable by showing that the reaction.diffusion equation with noise is .typically. well posed when |b(z)|=O(|z|log+|z|) as |z|... We interpret the word .typically. in two essentially-different ways without altering the conclusions of our assertions.