# American Institute of Mathematical Sciences

January  2015, 35(1): 441-466. doi: 10.3934/dcds.2015.35.441

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

 1 Institute of Mathematics and Scientific Computing, University of Graz, 36 Heinrichstraβe, 8010 Graz, Austria

Received  November 2013 Revised  May 2014 Published  August 2014

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.
Citation: Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
##### References:
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Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [8] J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises,, Physica D, 233 (2007), 83. doi: 10.1016/j.physd.2007.06.008. Google Scholar [9] P. E. Kloeden and J. A. Langa Flattening, Squeezing and the existence of random attractors,, Proc. R. Soc. Lond. Ser. A, 463 (2007), 163. doi: 10.1098/rspa.2006.1753. Google Scholar [10] Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,, J. Differential Equations, 245 (2008), 1775. doi: 10.1016/j.jde.2008.06.031. Google Scholar [11] Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations,, Appl. Maths. Comput., 190 (2007), 1020. doi: 10.1016/j.amc.2006.11.187. 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Chaos, 17 (2007), 1673. doi: 10.1142/S0218127407017987. Google Scholar [23] L. Xu and W. Yan, Stochastic FitzHugh-Nagumo systems with delay,, Taiwanese J. Maths., 16 (2012), 1079. Google Scholar [24] W. Zhao and Y. Li, $(L^2,L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains,, Nonlinear Anal. TMA, 75 (2012), 485. doi: 10.1016/j.na.2011.08.050. Google Scholar [25] W. Zhao, $H^1$-random attractors for stochastic reaction diffusion equations with additive noise,, Nonlinear Anal. TMA., 84 (2013), 61. doi: 10.1016/j.na.2013.01.014. Google Scholar [26] W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction diffusion equations with multiplicative noises,, Comm. Nonlinear Sci. Numer. Simulat., 18 (2013), 2707. doi: 10.1016/j.cnsns.2013.03.012. Google Scholar [27] C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,, J. Differential Equations, 223 (2006), 367. doi: 10.1016/j.jde.2005.06.008. Google Scholar

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##### References:
 [1] A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing,, Discrete Continuous Dyn. Syst., 18 (2013), 643. doi: 10.3934/dcdsb.2013.18.643. Google Scholar [2] C. T. Anh, T. Q. Bao and N. V. Thanh, Regularity of random attractors for stochastic semilinear degenerate parabolic equations,, Electronic J. Diff. Eqs., (2012), 1. Google Scholar [3] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar [4] T. Q. Bao, Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains,, submitted., (). Google Scholar [5] J. W. Cholewa and T. Dlotko, Global Attractors for Abstract Parabolic Problems,, Cambridge University Press, (2000). doi: 10.1017/CBO9780511526404. Google Scholar [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar [7] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [8] J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises,, Physica D, 233 (2007), 83. doi: 10.1016/j.physd.2007.06.008. Google Scholar [9] P. E. Kloeden and J. A. Langa Flattening, Squeezing and the existence of random attractors,, Proc. R. Soc. Lond. Ser. A, 463 (2007), 163. doi: 10.1098/rspa.2006.1753. Google Scholar [10] Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,, J. Differential Equations, 245 (2008), 1775. doi: 10.1016/j.jde.2008.06.031. Google Scholar [11] Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations,, Appl. Maths. Comput., 190 (2007), 1020. doi: 10.1016/j.amc.2006.11.187. Google Scholar [12] E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems,, Physica D, 212 (2005), 317. doi: 10.1016/j.physd.2005.10.006. Google Scholar [13] B. Wang, Pullback attractors for non-autonomous Reaction-Diffusion equations on $\mathbb R^n$,, Frontiers of Mathematics in China, 4 (2009), 563. doi: 10.1007/s11464-009-0033-5. Google Scholar [14] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains,, J. Differential Equations, 246 (2009), 2506. doi: 10.1016/j.jde.2008.10.012. Google Scholar [15] B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains,, Nonlinear Anal. TMA, 71 (2009), 2811. doi: 10.1016/j.na.2009.01.131. Google Scholar [16] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544. doi: 10.1016/j.jde.2012.05.015. Google Scholar [17] B. Wang, Attractors for reaction-diffusion equations in unbounded domains,, Physica D, 128 (1999), 41. doi: 10.1016/S0167-2789(98)00304-2. Google Scholar [18] B. Wang and X. Gao, Random attractors for wave equations on unbounded domains,, Discrete Continuous Dyn. Syst. (suppl.), (2009), 800. doi: 10.1016/j.nonrwa.2011.06.008. Google Scholar [19] G. Wang and Y. Tang, $(L^2,H^1)$-random attractors for stochastic reaction diffusion equation on unbounded domains,, Abstr. Appl. Anal., (2013). Google Scholar [20] Y. Wang and C. K. Zhong, On the existence of pullback attractors for non-autonomous reaction diffusion,, Dyn. Syst., 23 (2008), 1. doi: 10.1080/14689360701611821. Google Scholar [21] B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains,, Nonlinear Anal. TMA, 70 (2009), 3799. doi: 10.1016/j.na.2008.07.011. Google Scholar [22] B. Wang, Dynamical behavior of the almost-periodic discrete FitzHugh-Nagumo systems,, Internat. J. Bifur. Chaos, 17 (2007), 1673. doi: 10.1142/S0218127407017987. Google Scholar [23] L. Xu and W. Yan, Stochastic FitzHugh-Nagumo systems with delay,, Taiwanese J. Maths., 16 (2012), 1079. Google Scholar [24] W. Zhao and Y. Li, $(L^2,L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains,, Nonlinear Anal. TMA, 75 (2012), 485. doi: 10.1016/j.na.2011.08.050. Google Scholar [25] W. Zhao, $H^1$-random attractors for stochastic reaction diffusion equations with additive noise,, Nonlinear Anal. TMA., 84 (2013), 61. doi: 10.1016/j.na.2013.01.014. Google Scholar [26] W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction diffusion equations with multiplicative noises,, Comm. Nonlinear Sci. Numer. Simulat., 18 (2013), 2707. doi: 10.1016/j.cnsns.2013.03.012. Google Scholar [27] C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,, J. Differential Equations, 223 (2006), 367. doi: 10.1016/j.jde.2005.06.008. Google Scholar
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