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.