# 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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E, 56 (1997), 185. 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, (1992), 185. Google Scholar [24] J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior,, J. Theor. Biology, 81 (1979), 389. 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 (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. doi: 10.1137/0150037. Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, 68 (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. 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. 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). Google Scholar [31] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Disc. Cont. Dyn. Systems, 34 (2014), 269. 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. 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. doi: 10.3934/cpaa.2008.7.947. Google Scholar [34] Y. You, Asymptotic dynamics of Selkov equations,, Disc. Cont. Dyn. Systems, 2 (2009), 193. 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, (2010), 293. Google Scholar [36] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonlinear Analysis, 75 (2012), 3049. 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. doi: 10.3934/dcds.2014.34.301. Google Scholar

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##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (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. 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. 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. Google Scholar [5] M. L. Campbell, Cell Modeling,, Master's Thesis, (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, 21 (2008), 415. 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. doi: 10.1080/03605309808821394. Google Scholar [8] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002). Google Scholar [9] I. Chueshov, Monotone Random Systems Theory and Applications,, Lect. Notes of Math., (1779). doi: 10.1007/b83277. Google Scholar [10] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. 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. 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. 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. 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. 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). 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. 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. doi: 10.1142/S0219493703000772. Google Scholar [18] J. D. Murray, Mathematical Biology, I and II,, 3rd edition, (2002). Google Scholar [19] B. Øksendal, Stochastic Differential Equations,, 6th edition, (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. 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. 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. 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, (1992), 185. Google Scholar [24] J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior,, J. Theor. Biology, 81 (1979), 389. 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 (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. doi: 10.1137/0150037. Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, 68 (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. 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. 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). Google Scholar [31] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Disc. Cont. Dyn. Systems, 34 (2014), 269. 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. 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. doi: 10.3934/cpaa.2008.7.947. Google Scholar [34] Y. You, Asymptotic dynamics of Selkov equations,, Disc. Cont. Dyn. Systems, 2 (2009), 193. 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, (2010), 293. Google Scholar [36] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonlinear Analysis, 75 (2012), 3049. 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. doi: 10.3934/dcds.2014.34.301. Google Scholar
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