Preventing finite-time blowup in a constrained potential for reaction–diffusion equations

We examine stochastic reaction–diffusion equations of the form $\frac{\partial u}{\partial t}=Au(t,x)+f\left(u(t,x)\right)+\sigma \left(u(t,x)\right)\dot{W}(t,x)$ on a bounded spatial domain $D\subset R^d$, where $f$ models a constrained, dissipative force that keeps solutions between $-1$ and 1. To model this, we assume that $f\left(u\right),\sigma \left(u\right)$ are unbounded as $u$ approaches $\pm 1$. We identify sufficient conditions on the growth rates of $f$ and $\sigma$ that guarantee solutions to not escape this bounded set.