doi: 10.3934/dcdsb.2018251

Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

* Corresponding author: Wenqiang Zhao

Dedicated to Peter E. Kloeden on his 70th birthday

Received  March 2018 Published  August 2018

Fund Project: This work was supported by CTBU Grant 1751041, China NSF Grant 11601046

In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space $L^2(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ is actually a compact, measurable and attracting set in $H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$. A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in $H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$, despite of the loss of the high-order integrability of the difference of solutions for this system.

Citation: Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018251
References:
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A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dynam. Syst. Ser. B, 18 (2013), 643-666. doi: 10.3934/dcdsb.2013.18.643.

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L. Arnold and B. Schmalfuss, Fixed Points and Attractors for Random Dynamical Systems, in IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, Solid Mechanics and its Applications (eds. A. Naess and S. Krenk), Springer, Dordrecht, 47 (1996), 19-28. doi: 10.1007/978-94-009-0321-0_3.

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D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, vol. 184, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

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I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

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H. Crauel and F. Flandoli, Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

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H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

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H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math. Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.

[11]

H. CunY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324. doi: 10.1016/j.na.2015.08.009.

[12]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.

[13]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.

[14]

F. FlandoliM. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53. doi: 10.1007/s00222-009-0224-4.

[15]

O. Goubet and R. M. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.

[16]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $R^n$, J. Differential Equations, 264 (2018), 7094-7137. doi: 10.1016/j.jde.2018.02.011.

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[18]

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.

[19]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[20]

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P. E. Kloeden and T. Lorenz, Pullback and forward attractors of nonautonomous difference equations, in Proceedings of ICDEAWuhan 2014 (eds. M. Bohner, Y. Ding and O. Dosly), Springer-Verlag, Heidelberg, 150 (2015), 37-48. doi: 10.1007/978-3-319-24747-2_3.

[22]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735.

[23]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference. Equ. Appl., 22 (2016), 513-525. doi: 10.1080/10236198.2015.1107550.

[24]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dynam. Syst. Ser. B, 21 (2016), 1203-1223. doi: 10.3934/dcdsb.2016.21.1203.

[25]

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), 1775-1800. doi: 10.1016/j.jde.2008.06.031.

[26]

Y. LiA. Gu and J. Li, Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.

[27]

J. NagumoS. Arimoto and S. Yosimzawa, An active pulse transimission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.

[28]

J. C. Robinson, Infinite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[29]

B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185-192.

[30]

B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in International Conference on Differential Equations (eds. B. Fiedler, K. Gröger and J. Sprekels), World Sci. Publishing, Singapore, (2000), 684-690.

[31]

B. Tang, Regularity of pullback random attractors for stochasitic FitzHugh-Nagumo system on unbounded domains, Discrete Contin. Dynam. Syst. Ser. A, 35 (2015), 441-466. doi: 10.3934/dcds.2015.35.441.

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[33]

M. J. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Boston, 1988. doi: 10.1007/978-94-009-1423-0.

[34]

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

[35]

B. Wang, Random attractors for non-autonomous stochastic wave euqations with multiplicative noises, Discrete Contin. Dynam. Syst., 34 (2014), 269-330. doi: 10.3934/dcds.2014.34.269.

[36]

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

[37]

B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[38]

Y. WangY. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems, J. Difference Equ. Appl., 14 (2008), 799-817. doi: 10.1080/10236190701859542.

[39]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwanese J. Math., 20 (2016), 589-616. doi: 10.11650/tjm.20.2016.6699.

[40]

W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_ρ^p$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045.

[41]

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

[42]

W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\mathbb{R}^N$, Comput. Math. Appl., 75 (2018), 3801-3824. doi: 10.1016/j.camwa.2018.02.031.

[43]

W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$with an unbounded additive noise, J. Math. Anal. App., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025.

[44]

W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonliner Anal., 152 (2017), 196-219. doi: 10.1016/j.na.2017.01.004.

[45]

W. Zhao, Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise, Discrete Contin. Dynam. Syst. Ser. B. doi: 10.3934/dcdsb.2018065.

[46]

S. Zhou and Z. Wang, Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, J. Math. Anal. Appl., 441 (2016), 648-667. doi: 10.1016/j.jmaa.2016.04.038.

[47]

K. Zhu and F. Zhou, Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105. doi: 10.1016/j.camwa.2016.04.004.

show all references

References:
[1]

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

[2]

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

[3]

L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[4]

L. Arnold and B. Schmalfuss, Fixed Points and Attractors for Random Dynamical Systems, in IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, Solid Mechanics and its Applications (eds. A. Naess and S. Krenk), Springer, Dordrecht, 47 (1996), 19-28. doi: 10.1007/978-94-009-0321-0_3.

[5]

D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, vol. 184, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[8]

H. Crauel and F. Flandoli, Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[9]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[10]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math. Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.

[11]

H. CunY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324. doi: 10.1016/j.na.2015.08.009.

[12]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.

[13]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.

[14]

F. FlandoliM. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53. doi: 10.1007/s00222-009-0224-4.

[15]

O. Goubet and R. M. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.

[16]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $R^n$, J. Differential Equations, 264 (2018), 7094-7137. doi: 10.1016/j.jde.2018.02.011.

[17]

A. Haraux, Two remarks on hyperbolic dissipative problems, in Nonlinear Partial Differential Equations and their Applications (eds. H. Brezis and J. L. Lions), College de France Seminar, Vol. Ⅶ, Pitman, London, 122 (1985), 161-179.

[18]

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.

[19]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[20]

P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Difference. Equ. Appl., 6 (2000), 33-52. doi: 10.1080/10236190008808212.

[21]

P. E. Kloeden and T. Lorenz, Pullback and forward attractors of nonautonomous difference equations, in Proceedings of ICDEAWuhan 2014 (eds. M. Bohner, Y. Ding and O. Dosly), Springer-Verlag, Heidelberg, 150 (2015), 37-48. doi: 10.1007/978-3-319-24747-2_3.

[22]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735.

[23]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference. Equ. Appl., 22 (2016), 513-525. doi: 10.1080/10236198.2015.1107550.

[24]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dynam. Syst. Ser. B, 21 (2016), 1203-1223. doi: 10.3934/dcdsb.2016.21.1203.

[25]

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), 1775-1800. doi: 10.1016/j.jde.2008.06.031.

[26]

Y. LiA. Gu and J. Li, Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.

[27]

J. NagumoS. Arimoto and S. Yosimzawa, An active pulse transimission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.

[28]

J. C. Robinson, Infinite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[29]

B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185-192.

[30]

B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in International Conference on Differential Equations (eds. B. Fiedler, K. Gröger and J. Sprekels), World Sci. Publishing, Singapore, (2000), 684-690.

[31]

B. Tang, Regularity of pullback random attractors for stochasitic FitzHugh-Nagumo system on unbounded domains, Discrete Contin. Dynam. Syst. Ser. A, 35 (2015), 441-466. doi: 10.3934/dcds.2015.35.441.

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[33]

M. J. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Boston, 1988. doi: 10.1007/978-94-009-1423-0.

[34]

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

[35]

B. Wang, Random attractors for non-autonomous stochastic wave euqations with multiplicative noises, Discrete Contin. Dynam. Syst., 34 (2014), 269-330. doi: 10.3934/dcds.2014.34.269.

[36]

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

[37]

B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[38]

Y. WangY. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems, J. Difference Equ. Appl., 14 (2008), 799-817. doi: 10.1080/10236190701859542.

[39]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwanese J. Math., 20 (2016), 589-616. doi: 10.11650/tjm.20.2016.6699.

[40]

W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_ρ^p$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045.

[41]

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

[42]

W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\mathbb{R}^N$, Comput. Math. Appl., 75 (2018), 3801-3824. doi: 10.1016/j.camwa.2018.02.031.

[43]

W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$with an unbounded additive noise, J. Math. Anal. App., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025.

[44]

W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonliner Anal., 152 (2017), 196-219. doi: 10.1016/j.na.2017.01.004.

[45]

W. Zhao, Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise, Discrete Contin. Dynam. Syst. Ser. B. doi: 10.3934/dcdsb.2018065.

[46]

S. Zhou and Z. Wang, Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, J. Math. Anal. Appl., 441 (2016), 648-667. doi: 10.1016/j.jmaa.2016.04.038.

[47]

K. Zhu and F. Zhou, Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105. doi: 10.1016/j.camwa.2016.04.004.

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Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1

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