# 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 and Continuous Dynamical Systems, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [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. [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. [4] N. Berestycki, Stochastic calculus and applications., Available from: , (). [5] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Math. Anal., 10 (2003), 491-513. [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), 1557-1581. doi: 10.1080/03605309808821394. [7] C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977. [8] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002. [9] I. Chueshov, "Monotone Random Systems Theory and Applications," Lect. Notes of Math., Vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [10] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225. [11] H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors, J. Dynamics and Differential Equations, 21 (2009), 233-247. doi: 10.1007/s10884-009-9135-8. [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [13] A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (1997), 473-491. doi: 10.1080/07362999708809490. [14] 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. [15] 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. [16] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Reports, 59 (1996), 21-45. [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), 1087-1097. [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), 124507. doi: 10.1063/1.2038966. [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), 066114. doi: 10.1103/PhysRevE.68.066114. [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), 201-237. doi: 10.1017/S0956792505005930. [21] 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. [22] H. Mahara, et. al, Three-variable reversible Gray-Scott model, J. Chem. Physics, 121 (2004), 8968-8972. [23] 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. [24] D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots, Physica D, 192 (2004), 33-62. doi: 10.1016/j.physd.2003.12.012. [25] B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," Sixth edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [26] J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189. [27] I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems. II, J. Chem. Physics, 48 (1968), 1695-1700. doi: 10.1063/1.1668896. [28] K. R. Schenk-Hoppé, Random attractors-general properties, existence and applications to stochastic bifurcation theory, Disc. Cont. Dyn. Systems, 4 (1998), 99-130. doi: 10.3934/dcds.1998.4.99. [29] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, (1992), 185-192. [30] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [31] 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. [32] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp. [33] 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. [34] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Elec. J. Diff. Eqns., 2012 (2012), 18 pp. [35] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbfR^2$, Stud. Appl. Math., 110 (2003), 63-102. doi: 10.1111/1467-9590.00231. [36] Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947. [37] Y. You, Asymptotic dynamics of Selkov equations, Discrete and Continuous Dynamical Systems, Series S, 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193. [38] Y. You, Asymptotic dynamics of the modified Schnackenberg equations, Discrete and Continuous Dynmical Systems, Dynamical Systems, Differential Equations and Applications, $7^{th}$ AIMS Conference, suppl., (2009), 857-868. [39] Y. You, Dynamics of three-component reversible Gray-Scott model, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 1671-1688. doi: 10.3934/dcdsb.2010.14.1671. [40] Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Analysis, 74 (2011), 1969-1986. doi: 10.1016/j.na.2010.11.004. [41] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, 75 (2012), 3049-3071. doi: 10.1016/j.na.2011.12.002.

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##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [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. [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. [4] N. Berestycki, Stochastic calculus and applications., Available from: , (). [5] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Math. Anal., 10 (2003), 491-513. [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), 1557-1581. doi: 10.1080/03605309808821394. [7] C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977. [8] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002. [9] I. Chueshov, "Monotone Random Systems Theory and Applications," Lect. Notes of Math., Vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [10] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225. [11] H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors, J. Dynamics and Differential Equations, 21 (2009), 233-247. doi: 10.1007/s10884-009-9135-8. [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [13] A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (1997), 473-491. doi: 10.1080/07362999708809490. [14] 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. [15] 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. [16] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Reports, 59 (1996), 21-45. [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), 1087-1097. [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), 124507. doi: 10.1063/1.2038966. [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), 066114. doi: 10.1103/PhysRevE.68.066114. [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), 201-237. doi: 10.1017/S0956792505005930. [21] 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. [22] H. Mahara, et. al, Three-variable reversible Gray-Scott model, J. Chem. Physics, 121 (2004), 8968-8972. [23] 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. [24] D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots, Physica D, 192 (2004), 33-62. doi: 10.1016/j.physd.2003.12.012. [25] B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," Sixth edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [26] J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189. [27] I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems. II, J. Chem. Physics, 48 (1968), 1695-1700. doi: 10.1063/1.1668896. [28] K. R. Schenk-Hoppé, Random attractors-general properties, existence and applications to stochastic bifurcation theory, Disc. Cont. Dyn. Systems, 4 (1998), 99-130. doi: 10.3934/dcds.1998.4.99. [29] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, (1992), 185-192. [30] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [31] 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. [32] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp. [33] 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. [34] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Elec. J. Diff. Eqns., 2012 (2012), 18 pp. [35] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbfR^2$, Stud. Appl. Math., 110 (2003), 63-102. doi: 10.1111/1467-9590.00231. [36] Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947. [37] Y. You, Asymptotic dynamics of Selkov equations, Discrete and Continuous Dynamical Systems, Series S, 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193. [38] Y. You, Asymptotic dynamics of the modified Schnackenberg equations, Discrete and Continuous Dynmical Systems, Dynamical Systems, Differential Equations and Applications, $7^{th}$ AIMS Conference, suppl., (2009), 857-868. [39] Y. You, Dynamics of three-component reversible Gray-Scott model, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 1671-1688. doi: 10.3934/dcdsb.2010.14.1671. [40] Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Analysis, 74 (2011), 1969-1986. doi: 10.1016/j.na.2010.11.004. [41] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, 75 (2012), 3049-3071. doi: 10.1016/j.na.2011.12.002.
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