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 and Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
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[1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.  doi: 10.1007/978-3-662-12878-7.
[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.

[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.

[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. 

[5]

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

[6]

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

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

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

[9]

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

[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.

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

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

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

[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.

[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.

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

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

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

[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.

[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. 

[5]

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

[6]

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

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

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

[9]

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

[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.

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

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

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

[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.

[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.

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

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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