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

## TL;DR

This paper investigates the existence and behavior of solutions to a stochastic reaction-diffusion equation with super-linear drift and multiplicative noise, establishing conditions for well-posedness and blowup, and demonstrating the sharpness of previous results.

## Contribution

It proves that the known blowup condition is optimal by showing solutions are well-posed under slightly weaker growth conditions on the drift.

## Key findings

- Solutions are typically well-posed when the drift grows like |z| log|z|.
- The blowup condition for the drift is sharp and cannot be improved.
- The paper clarifies the boundary between well-posedness and blowup for these equations.

## 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.