Global solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

A condition is identified that implies that solutions to the stochastic reaction-diffusion equation ∂u ∂t = Au+f(u)+σ(u)Ẇ on a bounded spatial domain never explode. We consider the case where σ 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.

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