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 \[ \dot{u}(t\,,x)=\tfrac12 u"(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(t\,,x), \] on ${\mathbb{R}}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\epsilon >0$ such that $|b(z)|\ge |z|(\log |z|{)}^{1+\epsilon }$ 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 $|b(z)|=O(|z|{\log }_{+}|z|)$ as $|z|\to \infty$. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.

This is the submitted version of our paper

References in corpus (1)

• Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise