Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains

Pages: 441 - 466, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.441

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Bao Quoc Tang - Institute of Mathematics and Scientific Computing, University of Graz, 36 Heinrichstraβe, 8010 Graz, Austria (email)

Abstract: The regularity of the pullback random attractor for a stochastic FitzHugh-Nagumo system on $\mathbb R^n$ driven by deterministic non-autonomous forcing is proved. More precisely, the pullback random attractor is shown to be compact in $H^1(\mathbb R^n)\times L^2(\mathbb R^n)$ and attract all tempered sets of $L^2(\mathbb R^n)\times L^2(\mathbb R^n)$ in the topology of $H^1(\mathbb R^n)\times L^2(\mathbb R^n)$. The proof is based on tail estimates technique, eigenvalues of the Laplace operator in bounded domains and some new estimates of solutions.

Keywords:  Random dynamical systems, random attractors, stochastic FitzHugh-Nagumo systems, unbounded domains, tail estimates.
Mathematics Subject Classification:  Primary: 35B40, 35B41, 37L30, 37L55.

Received: November 2013;      Revised: May 2014;      Available Online: August 2014.