Evolution Equations and Control Theory (EECT)

Invariance for stochastic reaction-diffusion equations

Pages: 43 - 56, Volume 1, Issue 1, June 2012      doi:10.3934/eect.2012.1.43

       Abstract        References        Full Text (409.6K)       Related Articles       

Piermarco Cannarsa - Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scienti ca 1, I-00133 Roma, Italy (email)
Giuseppe Da Prato - Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I-56125 Pisa, Italy (email)

Abstract: 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 $L^2(\mathcal O)$.

Keywords:  Stochastic partial differential equations, reaction-diffusion equations, invariance of space domains.
Mathematics Subject Classification:  Primary: 60H15, 60J70; Secondary: 35K57.

Received: December 2011;      Revised: February 2012;      Available Online: March 2012.