Blowup for the multiplicative stochastic heat equation with superlinear drift

TL;DR

This paper investigates conditions under which solutions to the multiplicative stochastic heat equation blow up in finite time, establishing a necessary and sufficient criterion on the drift function and extending blowup results to the whole real line.

Contribution

It proves that the Osgood criterion on the drift is essential for finite-time blowup on [0,1], and extends blowup results to the real line for unbounded initial data.

Findings

Finite Osgood criterion on b is necessary and sufficient for blowup on [0,1]

Solutions blow up instantaneously on R with initial profile u_0 ≡ 1

Comparison principle links blowup behavior on R and [0,1]

Abstract

We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +\sigma(u) \dot W$, both on $[0,1]$ and $\mathbf{R}$. In the case of $[0,1]$ we show that the finite Osgood criterion on $b$ is a necessary and sufficient condition for finite-time blowup, under fairly general conditions on $\sigma$. In the case of $\mathbf{R}$ we show instantaneous explosion when we start with initial profile $u_0\equiv 1$, extending the work of [10] which dealt with bounded $\sigma$. The second result follows from the first by a comparison result which shows that the solution on $\mathbf{R}$ stays above the corresponding solution on $[0,1]$ with Dirichlet boundary conditions.