Title: Global solutions to 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 \[
\dot{u}(t\,,x)=\tfrac12 u"(t\,,x) + b(u(t\,,x)) + \sigma(u(t\,,x))
\xi(t\,,x), \] 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 $\vert b(z)\vert \ge|z|(\log|z|)^{1+\varepsilon}$ for
all sufficiently-large values of $|z|$. When $\sigma\equiv 0$, it is well known
that such PDEs frequently have non-trivial stationary solutions. By contrast,
Bonder and Groisman (2009) have recently shown that there is finite-time blowup
when $\sigma$ is a non-zero constant. In this paper, we prove that the
Bonder--Groisman condition is unimproveable by showing that the
reaction-diffusion equation with noise is "typically" well posed when $\vert
b(z) \vert =O(|z|\log_+|z|)$ as $|z|\to\infty$. We interpret the word
"typically" in two essentially-different ways without altering the conclusions
of our assertions.