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:
[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

show all references

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

[1]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[2]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[3]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[4]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[5]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[6]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[7]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[8]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[9]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[10]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[11]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[12]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[13]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[14]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[15]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[16]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[17]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[18]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[19]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[20]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]