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May  2016, 36(5): 2757-2779. doi: 10.3934/dcds.2016.36.2757

Random attractor of stochastic Brusselator system with multiplicative noise

Junyi Tu 1, and  Yuncheng You 1,

1. 

Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, United States

Received  December 2014 Revised  August 2015 Published  October 2015

Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an $(L^2\times L^2,H^1\times H^1)$ random attractor.
Keywords: Random attractor, stochastic Brusselator system, pullback asymptotic compactness, flattening property., multiplicative noise.
Mathematics Subject Classification: 37L30, 35B40, 35B41, 35K55, 80A32, 92B0.
Citation: Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
References:
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P. W. Bates, K. 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

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T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proc. R. Soc. Lond. A, 457 (2001), 2041-2061. doi: 10.1098/rspa.2001.0819.  Google Scholar

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L. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 48 (1968), 1695-1700. Google Scholar

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

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J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

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R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

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B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar

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Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonl. Anal. A, 75 (2012), 3049-3071. doi: 10.1016/j.na.2011.12.002.  Google Scholar

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Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301.  Google Scholar

[18]

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

show all references

References:
[1]

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

[2]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups of evolutionary equations, J. Math. Pures Appl., 62 (1983), 441-491.  Google Scholar

[3]

P. W. Bates, K. 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

[4]

T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proc. R. Soc. Lond. A, 457 (2001), 2041-2061. doi: 10.1098/rspa.2001.0819.  Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002.  Google Scholar

[6]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.  Google Scholar

[7]

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

[8]

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

[9]

M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Modles in Biology, Chemistry and Population Genetics, Springer, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.  Google Scholar

[10]

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

[11]

L. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 48 (1968), 1695-1700. Google Scholar

[12]

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

[13]

J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[14]

R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[15]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar

[16]

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonl. Anal. A, 75 (2012), 3049-3071. doi: 10.1016/j.na.2011.12.002.  Google Scholar

[17]

Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301.  Google Scholar

[18]

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

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