  A condition is identified that implies that solutions to the stochastic
reaction-diffusion equation $\frac{\partial u}{\partial t} = \mathcal{A} u +
f(u) + \sigma(u) \dot{W}$ on a bounded spatial domain never explode. We
consider the case where $\sigma$ grows polynomially and $f$ is polynomially
dissipative, meaning that $f$ strongly forces solutions toward finite values.
This result demonstrates the role that the deterministic forcing term $f$ plays
in preventing explosion.
