Solutions to the stochastic heat equation with polynomially growing multiplicative noise do not explode in the critical regime

TL;DR

This paper proves that solutions to the stochastic heat equation with polynomially growing multiplicative noise do not explode in finite time at the critical growth rate, completing the understanding of explosion behavior at this threshold.

Contribution

It establishes that in the critical case where the noise growth rate is /2, solutions do not explode, resolving a key open problem in stochastic PDE theory.

Findings

Solutions do not explode in finite time at the critical growth rate /2.

Confirms the boundary case behavior for explosion in stochastic heat equations.

Completes the classification of explosion phenomena based on noise growth rate.

Abstract

We investigate the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t} = \Delta u(t,x) + \sigma(u(t,x))\dot{W}(t,x)$ in the critical setting where $\sigma$ grows like $\sigma(u) \approx C(1 + |u|^\gamma)$ and $\gamma = \frac{3}{2}$. Mueller previously identified $\gamma=\frac{3}{2}$ as the critical growth rate for explosion and proved that solutions cannot explode in finite time if $\gamma< \frac{3}{2}$ and solutions will explode with positive probability if $\gamma>\frac{3}{2}$. This paper proves that explosion does not occur in the critical $\gamma=\frac{3}{2}$ setting.