June  2017, 22(4): 1683-1717. doi: 10.3934/dcdsb.2017081

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Shengfan Zhou

Received  April 2016 Revised  August 2016 Published  February 2017

Fund Project: The second author is supported by the National Natural Science Foundation of China under Grant No. 11471290, Zhejiang Natural Science Foundation under Grant No. LY14A010012 and Zhejiang Normal University Foundation under Grant No. ZC304014012.

In this paper, we study the long-term dynamical behavior of stochastic Boisso nade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attrac tor for the considered systems. And then we establish the upper semi-continui ty of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

Citation: Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
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V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.   Google Scholar

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

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H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.  doi: 10.1023/A:1022605313961.  Google Scholar

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A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.  doi: 10.1080/07362999708809490.  Google Scholar

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A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988.  doi: 10.1016/S0021-7824(99)80001-4.  Google Scholar

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V. Dufied and J. Boissonade, Dynamics of turing pattern monelayers close to onset, Phys. Rev. E, 53 (1996), 4883-4892.   Google Scholar

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X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741.  Google Scholar

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X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.  Google Scholar

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X. Fan and H. Chen, Attractors for the stochastic reaction-diffusion equation driven by linear multiplicative noise with a variable coefficient, J. Math. Anal. Appl., 398 (2013), 715-728.  doi: 10.1016/j.jmaa.2012.09.027.  Google Scholar

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R. Fitz-Hugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys., 1 (1961), 445-466.   Google Scholar

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

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J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating a nerve axon, Proc. I. R. E., 50 (2007), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

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J. NagumoS. Yoshizawa and S. Arimoto, Bistable Transmission Lines, IEEE Trans. Circuit Theory, CT-12 (2003), 400-412.  doi: 10.1109/TCT.1965.1082476.  Google Scholar

[25]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar

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J. Tu, Global attractors and robustness of the boissonade system, J. Dynam. Differential Equations, 27 (2015), 187-211.  doi: 10.1007/s10884-014-9396-8.  Google Scholar

[29]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[30]

B. Wang, Sufficient 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.  Google Scholar

[31]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1142/S0219493714500099.  Google Scholar

[32]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, 139 (2009), 1-18.   Google Scholar

[33]

Z. Wang and S. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl., 384 (2011), 160-172.  doi: 10.1016/j.jmaa.2011.02.082.  Google Scholar

[34]

G. Wang and Y. Tang, (L2, H1)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Abstr. Appl. Anal. 2013 (2013), Article ID 279509, 23 pages.  Google Scholar

[35]

G. Wang and Y. Tang, Random attractors for stochastic reaction-diffusion equations with multiplicative noise in H01, Math. Nachr., 287 (2014), 1774-1791.  doi: 10.1002/mana.201300114.  Google Scholar

[36]

W. Zhao, H1-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.  Google Scholar

[37]

W. Zhao and Y. Li, (L2, Lp)-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

[38]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin, Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[39]

S. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal., 75 (2012), 2793-2805.  doi: 10.1016/j.na.2011.11.022.  Google Scholar

[40]

S. ZhouY. Tian and Z. Wang, Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput., 276 (2016), 80-95.  doi: 10.1016/j.amc.2015.12.009.  Google Scholar

[41]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.  Google Scholar

[42]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar

[43]

S. ZhouC. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst., 21 (2008), 1259-1277.  doi: 10.3934/dcds.2008.21.1259.  Google Scholar

show all references

References:
[1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.  doi: 10.1007/978-3-662-12878-7.  Google Scholar
[2]

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

[3]

T. CaraballoJ. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin. Dynam. Systems, 6 (2000), 875-892.  doi: 10.3934/dcds.2000.6.875.  Google Scholar

[4]

V. V. Chepyzhov and M. Efendiev, Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example, Comm. Pure Appl. Math., 53 (2000), 647-665.   Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society, Providence, RI, 2002.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.   Google Scholar

[7] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.  doi: 10.1007/b83277.  Google Scholar
[8]

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

[9]

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

[10]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.  doi: 10.1023/A:1022605313961.  Google Scholar

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511666223.  Google Scholar
[12]

A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.  doi: 10.1080/07362999708809490.  Google Scholar

[13]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988.  doi: 10.1016/S0021-7824(99)80001-4.  Google Scholar

[14]

V. Dufied and J. Boissonade, Dynamics of turing pattern monelayers close to onset, Phys. Rev. E, 53 (1996), 4883-4892.   Google Scholar

[15]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741.  Google Scholar

[16]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.  Google Scholar

[17]

X. Fan and H. Chen, Attractors for the stochastic reaction-diffusion equation driven by linear multiplicative noise with a variable coefficient, J. Math. Anal. Appl., 398 (2013), 715-728.  doi: 10.1016/j.jmaa.2012.09.027.  Google Scholar

[18]

R. Fitz-Hugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys., 1 (1961), 445-466.   Google Scholar

[19]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[20]

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.  Google Scholar

[21]

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

[22] J. D. Murry, Random attractors of reaction-diffusion equations with multiplicative noise in Lp, Springer-Verlag, New York, 2002.   Google Scholar
[23]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating a nerve axon, Proc. I. R. E., 50 (2007), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[24]

J. NagumoS. Yoshizawa and S. Arimoto, Bistable Transmission Lines, IEEE Trans. Circuit Theory, CT-12 (2003), 400-412.  doi: 10.1109/TCT.1965.1082476.  Google Scholar

[25]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar

[26] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.   Google Scholar
[27] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, SpringerVerlag, New York, 1997.  doi: 10.1007/978-1-4612-0645-3.  Google Scholar
[28]

J. Tu, Global attractors and robustness of the boissonade system, J. Dynam. Differential Equations, 27 (2015), 187-211.  doi: 10.1007/s10884-014-9396-8.  Google Scholar

[29]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[30]

B. Wang, Sufficient 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.  Google Scholar

[31]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1142/S0219493714500099.  Google Scholar

[32]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, 139 (2009), 1-18.   Google Scholar

[33]

Z. Wang and S. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl., 384 (2011), 160-172.  doi: 10.1016/j.jmaa.2011.02.082.  Google Scholar

[34]

G. Wang and Y. Tang, (L2, H1)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Abstr. Appl. Anal. 2013 (2013), Article ID 279509, 23 pages.  Google Scholar

[35]

G. Wang and Y. Tang, Random attractors for stochastic reaction-diffusion equations with multiplicative noise in H01, Math. Nachr., 287 (2014), 1774-1791.  doi: 10.1002/mana.201300114.  Google Scholar

[36]

W. Zhao, H1-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.  Google Scholar

[37]

W. Zhao and Y. Li, (L2, Lp)-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

[38]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin, Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[39]

S. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal., 75 (2012), 2793-2805.  doi: 10.1016/j.na.2011.11.022.  Google Scholar

[40]

S. ZhouY. Tian and Z. Wang, Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput., 276 (2016), 80-95.  doi: 10.1016/j.amc.2015.12.009.  Google Scholar

[41]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.  Google Scholar

[42]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar

[43]

S. ZhouC. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst., 21 (2008), 1259-1277.  doi: 10.3934/dcds.2008.21.1259.  Google Scholar

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