June  2016, 21(4): 1203-1223. doi: 10.3934/dcdsb.2016.21.1203

A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China

Received  June 2015 Published  March 2016

A bi-spatial pullback attractor is obtained for non-autonomous and stochastic FitzHugh-Nagumo equations when the initial space is $L^2(\mathbb{R}^n)^2$ and the terminate space is $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$. Some new techniques of positive and negative truncations are used to investigate the regularity of attractors for coupling equations and to correct the essential mistake in [T. Q. Bao, Discrete Cont. Dyn. Syst. 35(2015), 441-466]. A counterexample is given for an important lemma for $H^1$-attractor in several literatures included above.
Citation: Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203
References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 643-666. doi: 10.3934/dcdsb.2013.18.643.  Google Scholar

[2]

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[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractor for Infinite-dimensional Nonautonomous Dynamical Systems, Appl. Math. Sciences, Springer, 182, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[10]

B. Gess, Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise, Annals Probability, 42 (2014), 818-864. doi: 10.1214/13-AOP869.  Google Scholar

[11]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Differ. Equ., 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5.  Google Scholar

[12]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D, 233 (2007), 83-94. doi: 10.1016/j.physd.2007.06.008.  Google Scholar

[13]

A. K. Khanmamedov, Global attractors for one dimensional p-Laplacian equation, Nonlinear Anal., 71 (2009), 155-171. doi: 10.1016/j.na.2008.10.037.  Google Scholar

[14]

A. Krause and B. X. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037.  Google Scholar

[15]

J. Li, Y.R. Li, B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in $L^p$, App. Math. Comp., 215 (2010), 3399-3407. doi: 10.1016/j.amc.2009.10.033.  Google Scholar

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Y. R. Li, H. Y. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44. doi: 10.1016/j.na.2014.06.013.  Google Scholar

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Y. R. Li, A. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[18]

Y. R. Li and B. L. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.  Google Scholar

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G. Lukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357. doi: 10.1016/j.na.2010.03.023.  Google Scholar

[20]

G. Lukaszewicz, On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, International J. Bifurcation and Chaos, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258.  Google Scholar

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H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H^1_0$, J. Differ. Equ., 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034.  Google Scholar

[22]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[23]

B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar

[24]

B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$, Tran. Am. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar

[25]

B. X. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on $\mathbbR^n$, Front. Math. China, 4 (2009), 563-583. doi: 10.1007/s11464-009-0033-5.  Google Scholar

[26]

E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Physica D, 212 (2005), 317-336. doi: 10.1016/j.physd.2005.10.006.  Google Scholar

[27]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 71 (2009), 2811-2828. doi: 10.1016/j.na.2009.01.131.  Google Scholar

[28]

B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 70 (2009), 3799-3815. doi: 10.1016/j.na.2008.07.011.  Google Scholar

[29]

G. Wang and Y. B. Tang, $(L^2,H^1)$-Random attractors for stochastic reaction-diffusion equation on unbounded domains, Abstr. App. Anal., 2013, Art. ID 279509, 23 pp. doi: 10.1155/279509.  Google Scholar

[30]

Y. H. Wang and C. K. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations, Dyn. Syst., 23 (2008), 1-16. doi: 10.1080/14689360701611821.  Google Scholar

[31]

J. Y. Yin, Y. R. Li and H. J. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in $L^q$, Appl. Math. Comput., 225 (2013), 526-540. doi: 10.1016/j.amc.2013.09.051.  Google Scholar

[32]

W. Q. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72. doi: 10.1016/j.na.2013.01.014.  Google Scholar

[33]

W. Q. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun Nonlinear Sci Numer Simulat, 18 (2013), 2707-2721. doi: 10.1016/j.cnsns.2013.03.012.  Google Scholar

[34]

W. Q. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noise, Appl. Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106.  Google Scholar

[35]

W. Q. Zhao and Y. R. Li, ($L^2$, $L^p$)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.  Google Scholar

[36]

W. Q. Zhao and Y. R. Li, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Diff. Equ., 11 (2014), 269-298. doi: 10.4310/DPDE.2014.v11.n3.a4.  Google Scholar

[37]

C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399. 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 Contin. Dyn. Syst. Series B, 18 (2013), 643-666. doi: 10.3934/dcdsb.2013.18.643.  Google Scholar

[2]

A. Adili and B. Wang, Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise, Discrete Contin. Dyn. Syst., (2013), 1-10. doi: 10.3934/proc.2013.2013.1.  Google Scholar

[3]

C. T. Anh, T. Q. Bao and N. V. Thanh, Regularity of random attractors for stochastic semilinear degenerate parabolic equations, Electric J. Differ. Equ., 207 (2012), 1-22.  Google Scholar

[4]

L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[5]

T. Q. Bao, Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains, Disrete Contin. Dyn. Syst., 35 (2015), 441-466. doi: 10.3934/dcds.2015.35.441.  Google Scholar

[6]

P. W. Bates, K. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[7]

Z. Brzezniak, T. Caraballo, J. A. Langa, Y. Li, G. Lukaszewiczd and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differ. Equ., 255 (2013), 3897-3919. doi: 10.1016/j.jde.2013.07.043.  Google Scholar

[8]

T. Caraballo and J. A. Langa, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Disrete Contin. Dyn. Syst., 6 (2000), 875-892. doi: 10.3934/dcds.2000.6.875.  Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractor for Infinite-dimensional Nonautonomous Dynamical Systems, Appl. Math. Sciences, Springer, 182, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[10]

B. Gess, Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise, Annals Probability, 42 (2014), 818-864. doi: 10.1214/13-AOP869.  Google Scholar

[11]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Differ. Equ., 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5.  Google Scholar

[12]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D, 233 (2007), 83-94. doi: 10.1016/j.physd.2007.06.008.  Google Scholar

[13]

A. K. Khanmamedov, Global attractors for one dimensional p-Laplacian equation, Nonlinear Anal., 71 (2009), 155-171. doi: 10.1016/j.na.2008.10.037.  Google Scholar

[14]

A. Krause and B. X. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037.  Google Scholar

[15]

J. Li, Y.R. Li, B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in $L^p$, App. Math. Comp., 215 (2010), 3399-3407. doi: 10.1016/j.amc.2009.10.033.  Google Scholar

[16]

Y. R. Li, H. Y. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44. doi: 10.1016/j.na.2014.06.013.  Google Scholar

[17]

Y. R. Li, A. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[18]

Y. R. Li and B. L. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.  Google Scholar

[19]

G. Lukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357. doi: 10.1016/j.na.2010.03.023.  Google Scholar

[20]

G. Lukaszewicz, On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, International J. Bifurcation and Chaos, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258.  Google Scholar

[21]

H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H^1_0$, J. Differ. Equ., 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034.  Google Scholar

[22]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[23]

B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar

[24]

B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$, Tran. Am. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar

[25]

B. X. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on $\mathbbR^n$, Front. Math. China, 4 (2009), 563-583. doi: 10.1007/s11464-009-0033-5.  Google Scholar

[26]

E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Physica D, 212 (2005), 317-336. doi: 10.1016/j.physd.2005.10.006.  Google Scholar

[27]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 71 (2009), 2811-2828. doi: 10.1016/j.na.2009.01.131.  Google Scholar

[28]

B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 70 (2009), 3799-3815. doi: 10.1016/j.na.2008.07.011.  Google Scholar

[29]

G. Wang and Y. B. Tang, $(L^2,H^1)$-Random attractors for stochastic reaction-diffusion equation on unbounded domains, Abstr. App. Anal., 2013, Art. ID 279509, 23 pp. doi: 10.1155/279509.  Google Scholar

[30]

Y. H. Wang and C. K. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations, Dyn. Syst., 23 (2008), 1-16. doi: 10.1080/14689360701611821.  Google Scholar

[31]

J. Y. Yin, Y. R. Li and H. J. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in $L^q$, Appl. Math. Comput., 225 (2013), 526-540. doi: 10.1016/j.amc.2013.09.051.  Google Scholar

[32]

W. Q. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72. doi: 10.1016/j.na.2013.01.014.  Google Scholar

[33]

W. Q. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun Nonlinear Sci Numer Simulat, 18 (2013), 2707-2721. doi: 10.1016/j.cnsns.2013.03.012.  Google Scholar

[34]

W. Q. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noise, Appl. Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106.  Google Scholar

[35]

W. Q. Zhao and Y. R. Li, ($L^2$, $L^p$)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.  Google Scholar

[36]

W. Q. Zhao and Y. R. Li, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Diff. Equ., 11 (2014), 269-298. doi: 10.4310/DPDE.2014.v11.n3.a4.  Google Scholar

[37]

C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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