# American Institute of Mathematical Sciences

December  2014, 7(6): 1347-1362. doi: 10.3934/dcdss.2014.7.1347

## Random attractor for stochastic reversible Schnackenberg equations

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

Received  January 2013 Revised  September 2013 Published  June 2014

Asymptotic dynamics of stochastic reversible Schnackenberg equations with multiplicative white noise, which is a typical trimolecular autocatalytic reaction-diffusion system on a three-dimensional bounded domain with Dirichlet boundary condition, is investigated in this paper. The existence of a random attractor is proved through uniform grouping estimates showing the pullback absorbing property and the pullback asymptotic compactness.
Citation: Yuncheng You. Random attractor for stochastic reversible Schnackenberg equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1347-1362. doi: 10.3934/dcdss.2014.7.1347
##### References:
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Systems, Series A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar [32] M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnackenberg model, Stud. Appl. Math., 109 (2002), 229-264. doi: 10.1111/1467-9590.00223.  Google Scholar [33] Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947.  Google Scholar [34] Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Systems, Series S, 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193.  Google Scholar [35] Y. You, Global dissipation and attraction of three-component Schnackenberg systems, in Proceedings of the International Workshop on Nonlinear and Modern Mathematical Physics (eds. W. X. Ma, X. B. Hu and Q. P. Liu), AIP Conf. Proc., 1212, Amer. Inst. Phys., Melville, New York, (2010), 293-311.  Google Scholar [36] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, Series A, 75 (2012), 3049-3071. doi: 10.1016/j.na.2011.12.002.  Google Scholar [37] Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Disc. Cont. Dyn. Systems, 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301.  Google Scholar

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York and Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21. doi: 10.1142/S0219493706001621.  Google Scholar [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Diff. Eqns., 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.  Google Scholar [4] D. L. Benson, J. A. Sherratt and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. Math. Biology, 55 (1993), 365-384. Google Scholar [5] M. L. Campbell, Cell Modeling, Master's Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, 2002. Google Scholar [6] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Disc. Cont. Dyn. Systems, Series A, 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.  Google Scholar [7] T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581. doi: 10.1080/03605309808821394.  Google Scholar [8] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002.  Google Scholar [9] I. Chueshov, Monotone Random Systems Theory and Applications, Lect. Notes of Math., Vol. 1779, Springer, New-York, 2002. doi: 10.1007/b83277.  Google Scholar [10] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar [11] A. Doelman, T. J. Kaper and Paul A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563. doi: 10.1088/0951-7715/10/2/013.  Google Scholar [12] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, The Annals of Probability, 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.  Google Scholar [13] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Reports, 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar [14] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $a+2b\to 3b,b\to c$, Chem. Eng. Sci., 39 (1984), 1087-1097. Google Scholar [15] D. Hochberg, F. Lesmes, F. Morán and J. Pérez-Mercader, Large-scale emergent properties of an autocatalytic reaction-diffusion model subject to noise, Phys. Rev. E, 68 (2003), 066114. doi: 10.1103/PhysRevE.68.066114.  Google Scholar [16] K. J. Lee, W. D. McCormick, Q. Ouyang and H. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194. doi: 10.1126/science.261.5118.192.  Google Scholar [17] P. Martin-Rubio and J. C. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stochastics and Dynamics, 3 (2003), 279-297. doi: 10.1142/S0219493703000772.  Google Scholar [18] J. D. Murray, Mathematical Biology, I and II, 3rd edition, Springer, New-York, 2002.  Google Scholar [19] B. Øksendal, Stochastic Differential Equations, 6th edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [20] J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189.  Google Scholar [21] I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 46 (1967), 1665-1700. doi: 10.1063/1.1841255.  Google Scholar [22] W. Reynolds, J. E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction-diffusion systems, Phys. Rev. E, 56 (1997), 185-198. doi: 10.1103/PhysRevE.56.185.  Google Scholar [23] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, 1992, 185-192. Google Scholar [24] J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biology, 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar [25] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [26] L. J. Shaw and J. D. Murray, Analysis of a model for complex skin patterns, SIAM J. Appl. Math., 50 (1990), 628-648. doi: 10.1137/0150037.  Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [28] D. A. Vasquez, J. W. Wilder and B. F. Edwards, Convective Turing patterns, Physical Review Letters, 71 (1993), 1538-1541. doi: 10.1103/PhysRevLett.71.1538.  Google Scholar [29] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar [30] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Elec. J. Diff. Eqns., (2012), 18 pp.  Google Scholar [31] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disc. Cont. Dyn. Systems, Series A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar [32] M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnackenberg model, Stud. Appl. Math., 109 (2002), 229-264. doi: 10.1111/1467-9590.00223.  Google Scholar [33] Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947.  Google Scholar [34] Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Systems, Series S, 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193.  Google Scholar [35] Y. You, Global dissipation and attraction of three-component Schnackenberg systems, in Proceedings of the International Workshop on Nonlinear and Modern Mathematical Physics (eds. W. X. Ma, X. B. Hu and Q. P. Liu), AIP Conf. Proc., 1212, Amer. Inst. Phys., Melville, New York, (2010), 293-311.  Google Scholar [36] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, Series A, 75 (2012), 3049-3071. doi: 10.1016/j.na.2011.12.002.  Google Scholar [37] Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Disc. Cont. Dyn. Systems, 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301.  Google Scholar
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