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

Abstract

Let ξ(t,x) denote space-time white noise and consider a reaction- diffusion equation of the form u˙(t, x) = 1/2 u'' (t, x) +b (u(t, x)) +σ (u(t, x))ξ(t,x), on ℝ+ × [0, 1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists ε > 0 such that |b(z)| ≥ |z|(log |z|) 1+ε for all sufficiently-large values of |z|. When σ ≡ 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 σ 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.