# American Institute of Mathematical Sciences

January  2014, 34(1): 301-333. doi: 10.3934/dcds.2014.34.301

## Random attractors and robustness for stochastic reversible reaction-diffusion systems

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

Received  October 2012 Revised  February 2013 Published  June 2013

For a typical stochastic reversible reaction-diffusion system with multiplicative white noise, the trimolecular autocatalytic Gray-Scott system on a three-dimensional bounded domain with random noise perturbation proportional to the state of the system, the existence of a random attractor and its robustness with respect to the reverse reaction rates are proved through sharp and uniform estimates showing the pullback uniform dissipation and the pullback asymptotic compactness.
Citation: Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). 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] N. Berestycki, Stochastic calculus and applications., Available from: , (). Google Scholar [5] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491. Google Scholar [6] T. Caraballo, J. A. Langa and J. C. Robibson, Upper semicontinuity of attractors for small random perturbations of dynamical systems,, Comm. Partial Differential Equations, 23 (1998), 1557. doi: 10.1080/03605309808821394. Google Scholar [7] C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Lecture Notes in Math., (1977). 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, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar [11] H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors,, J. Dynamics and Differential Equations, 21 (2009), 233. doi: 10.1007/s10884-009-9135-8. Google Scholar [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [13] A. Debussche, On the finite dimensionality of random attractors,, Stochastic Analysis and Applications, 15 (1997), 473. doi: 10.1080/07362999708809490. Google Scholar [14] 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 [15] 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 [16] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Reports, 59 (1996), 21. Google Scholar [17] 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 [18] Y. Hayase and H. R. Brand, The Gray-Scott model under the influence of noise: Reentrant spatiotemporal intermittency in a reaction-diffusion system,, J. Chem. Phys., 123 (2005). doi: 10.1063/1.2038966. Google Scholar [19] 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 [20] T. Kolokolnikov and J. Wei, On ring-like solutions for the Gray-Scott model: Existence, instability and self-replicating rings,, Euro. J. Appl. Math., 16 (2005), 201. doi: 10.1017/S0956792505005930. Google Scholar [21] 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 [22] H. Mahara, et. al, Three-variable reversible Gray-Scott model,, J. Chem. Physics, 121 (2004), 8968. Google Scholar [23] 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 [24] D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots,, Physica D, 192 (2004), 33. doi: 10.1016/j.physd.2003.12.012. Google Scholar [25] B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications,", Sixth edition, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar [26] J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. doi: 10.1126/science.261.5118.189. Google Scholar [27] I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems. II,, J. Chem. Physics, 48 (1968), 1695. doi: 10.1063/1.1668896. Google Scholar [28] K. R. Schenk-Hoppé, Random attractors-general properties, existence and applications to stochastic bifurcation theory,, Disc. Cont. Dyn. Systems, 4 (1998), 99. doi: 10.3934/dcds.1998.4.99. Google Scholar [29] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, (1992), 185. Google Scholar [30] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). Google Scholar [31] 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 [32] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electronic Journal of Differential Equations, 2009 (2009). Google Scholar [33] 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 [34] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains,, Elec. J. Diff. Eqns., 2012 (2012). Google Scholar [35] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbfR^2$,, Stud. Appl. Math., 110 (2003), 63. doi: 10.1111/1467-9590.00231. Google Scholar [36] 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 [37] Y. You, Asymptotic dynamics of Selkov equations,, Discrete and Continuous Dynamical Systems, 2 (2009), 193. doi: 10.3934/dcdss.2009.2.193. Google Scholar [38] Y. You, Asymptotic dynamics of the modified Schnackenberg equations,, Discrete and Continuous Dynmical Systems, (2009), 857. Google Scholar [39] Y. You, Dynamics of three-component reversible Gray-Scott model,, Discrete and Continuous Dynamical Systems, 14 (2010), 1671. doi: 10.3934/dcdsb.2010.14.1671. Google Scholar [40] Y. You, Dynamics of two-compartment Gray-Scott equations,, Nonlinear Analysis, 74 (2011), 1969. doi: 10.1016/j.na.2010.11.004. Google Scholar [41] 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

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##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). 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] N. Berestycki, Stochastic calculus and applications., Available from: , (). Google Scholar [5] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491. Google Scholar [6] T. Caraballo, J. A. Langa and J. C. Robibson, Upper semicontinuity of attractors for small random perturbations of dynamical systems,, Comm. Partial Differential Equations, 23 (1998), 1557. doi: 10.1080/03605309808821394. Google Scholar [7] C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Lecture Notes in Math., (1977). 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, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar [11] H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors,, J. Dynamics and Differential Equations, 21 (2009), 233. doi: 10.1007/s10884-009-9135-8. Google Scholar [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [13] A. Debussche, On the finite dimensionality of random attractors,, Stochastic Analysis and Applications, 15 (1997), 473. doi: 10.1080/07362999708809490. Google Scholar [14] 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 [15] 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 [16] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Reports, 59 (1996), 21. Google Scholar [17] 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 [18] Y. Hayase and H. R. Brand, The Gray-Scott model under the influence of noise: Reentrant spatiotemporal intermittency in a reaction-diffusion system,, J. Chem. Phys., 123 (2005). doi: 10.1063/1.2038966. Google Scholar [19] 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 [20] T. Kolokolnikov and J. Wei, On ring-like solutions for the Gray-Scott model: Existence, instability and self-replicating rings,, Euro. J. Appl. Math., 16 (2005), 201. doi: 10.1017/S0956792505005930. Google Scholar [21] 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 [22] H. Mahara, et. al, Three-variable reversible Gray-Scott model,, J. Chem. Physics, 121 (2004), 8968. Google Scholar [23] 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 [24] D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots,, Physica D, 192 (2004), 33. doi: 10.1016/j.physd.2003.12.012. Google Scholar [25] B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications,", Sixth edition, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar [26] J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. doi: 10.1126/science.261.5118.189. Google Scholar [27] I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems. II,, J. Chem. Physics, 48 (1968), 1695. doi: 10.1063/1.1668896. Google Scholar [28] K. R. Schenk-Hoppé, Random attractors-general properties, existence and applications to stochastic bifurcation theory,, Disc. Cont. Dyn. Systems, 4 (1998), 99. doi: 10.3934/dcds.1998.4.99. Google Scholar [29] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, (1992), 185. Google Scholar [30] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). Google Scholar [31] 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 [32] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electronic Journal of Differential Equations, 2009 (2009). Google Scholar [33] 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 [34] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains,, Elec. J. Diff. Eqns., 2012 (2012). Google Scholar [35] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbfR^2$,, Stud. Appl. Math., 110 (2003), 63. doi: 10.1111/1467-9590.00231. Google Scholar [36] 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 [37] Y. You, Asymptotic dynamics of Selkov equations,, Discrete and Continuous Dynamical Systems, 2 (2009), 193. doi: 10.3934/dcdss.2009.2.193. Google Scholar [38] Y. You, Asymptotic dynamics of the modified Schnackenberg equations,, Discrete and Continuous Dynmical Systems, (2009), 857. Google Scholar [39] Y. You, Dynamics of three-component reversible Gray-Scott model,, Discrete and Continuous Dynamical Systems, 14 (2010), 1671. doi: 10.3934/dcdsb.2010.14.1671. Google Scholar [40] Y. You, Dynamics of two-compartment Gray-Scott equations,, Nonlinear Analysis, 74 (2011), 1969. doi: 10.1016/j.na.2010.11.004. Google Scholar [41] 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
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