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Invariance for stochastic reaction-diffusion equations

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  • Given a stochastic reaction-diffusion equation on a bounded open subset $\mathcal O$ of $\mathbb{R}^n$, we discuss conditions for the invariance of a nonempty closed convex subset $K$ of $L^2(\mathcal O)$ under the corresponding flow. We obtain two general results under the assumption that the fourth power of the distance from $K$ is of class $C^2$, providing, respectively, a necessary and a sufficient condition for invariance. We also study the example where $K$ is the cone of all nonnegative functions in $L^2(\mathcal O)$.
    Mathematics Subject Classification: Primary: 60H15, 60J70; Secondary: 35K57.


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