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

Abstract

Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, 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 $\sigma $ 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 $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.