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 and Continuous Dynamical Systems - S, 2014, 7 (6) : 1347-1362. doi: 10.3934/dcdss.2014.7.1347
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, New York and Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[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]

D. L. Benson, J. A. Sherratt and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. Math. Biology, 55 (1993), 365-384.

[5]

M. L. Campbell, Cell Modeling, Master's Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, 2002.

[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.

[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.

[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, New-York, 2002. doi: 10.1007/b83277.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

J. D. Murray, Mathematical Biology, I and II, 3rd edition, Springer, New-York, 2002.

[19]

B. Øksendal, Stochastic Differential Equations, 6th edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.

[20]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189.

[21]

I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 46 (1967), 1665-1700. doi: 10.1063/1.1841255.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.

[30]

B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Elec. J. Diff. Eqns., (2012), 18 pp.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, New York and Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[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]

D. L. Benson, J. A. Sherratt and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. Math. Biology, 55 (1993), 365-384.

[5]

M. L. Campbell, Cell Modeling, Master's Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, 2002.

[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.

[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.

[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, New-York, 2002. doi: 10.1007/b83277.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

J. D. Murray, Mathematical Biology, I and II, 3rd edition, Springer, New-York, 2002.

[19]

B. Øksendal, Stochastic Differential Equations, 6th edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.

[20]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189.

[21]

I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 46 (1967), 1665-1700. doi: 10.1063/1.1841255.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.

[30]

B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Elec. J. Diff. Eqns., (2012), 18 pp.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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