Preventing Finite-Time Blowup in a Constrained Potential for Reaction-Diffusion Equations

We examine stochastic reaction-diffusion equations of the form $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\mathcal{A}u(t,x)+f(u(t,x))+\sigma (u(t,x))\dot{W}(t,x)$ and provide sufficient conditions on the reaction term and multiplicative noise term that guarantees solutions never explode in finite time.