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

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.