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:
[1]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (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. 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. 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.

[5]

M. L. Campbell, Cell Modeling,, Master's Thesis, (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, 21 (2008), 415. 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. doi: 10.1080/03605309808821394.

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002).

[9]

I. Chueshov, Monotone Random Systems Theory and Applications,, Lect. Notes of Math., (1779). doi: 10.1007/b83277.

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. 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. 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. 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. 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.

[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.

[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.

[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.

[18]

J. D. Murray, Mathematical Biology, I and II,, 3rd edition, (2002).

[19]

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

[20]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. 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. 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. 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, (1992), 185.

[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.

[25]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, 143 (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. doi: 10.1137/0150037.

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, 68 (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. doi: 10.1103/PhysRevLett.71.1538.

[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.

[30]

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

[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.

[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.

[33]

Y. You, Global attractor of the Gray-Scott equations,, Comm. Pure Appl. Anal., 7 (2008), 947. doi: 10.3934/cpaa.2008.7.947.

[34]

Y. You, Asymptotic dynamics of Selkov equations,, Disc. Cont. Dyn. Systems, 2 (2009), 193. 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, (2010), 293.

[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.

[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.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (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. 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. 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.

[5]

M. L. Campbell, Cell Modeling,, Master's Thesis, (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, 21 (2008), 415. 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. doi: 10.1080/03605309808821394.

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002).

[9]

I. Chueshov, Monotone Random Systems Theory and Applications,, Lect. Notes of Math., (1779). doi: 10.1007/b83277.

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. 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. 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. 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. 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.

[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.

[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.

[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.

[18]

J. D. Murray, Mathematical Biology, I and II,, 3rd edition, (2002).

[19]

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

[20]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. 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. 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. 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, (1992), 185.

[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.

[25]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, 143 (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. doi: 10.1137/0150037.

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, 68 (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. doi: 10.1103/PhysRevLett.71.1538.

[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.

[30]

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

[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.

[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.

[33]

Y. You, Global attractor of the Gray-Scott equations,, Comm. Pure Appl. Anal., 7 (2008), 947. doi: 10.3934/cpaa.2008.7.947.

[34]

Y. You, Asymptotic dynamics of Selkov equations,, Disc. Cont. Dyn. Systems, 2 (2009), 193. 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, (2010), 293.

[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.

[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.

[1]

Yuncheng You. Pullback uniform dissipativity of stochastic reversible Schnackenberg equations. Conference Publications, 2015, 2015 (special) : 1134-1142. doi: 10.3934/proc.2015.1134

[2]

Yuncheng You. Asymptotical dynamics of the modified Schnackenberg equations. Conference Publications, 2009, 2009 (Special) : 857-868. doi: 10.3934/proc.2009.2009.857

[3]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143

[4]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[5]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060

[6]

Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525

[7]

Giuseppe Da Prato, Arnaud Debussche. Asymptotic behavior of stochastic PDEs with random coefficients. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1553-1570. doi: 10.3934/dcds.2010.27.1553

[8]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106

[9]

Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855

[10]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1

[11]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[12]

Felix X.-F. Ye, Yue Wang, Hong Qian. Stochastic dynamics: Markov chains and random transformations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2337-2361. doi: 10.3934/dcdsb.2016050

[13]

Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models with random perturbation. Mathematical Biosciences & Engineering, 2014, 11 (4) : 1003-1025. doi: 10.3934/mbe.2014.11.1003

[14]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

[15]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

[16]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441

[17]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051

[18]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081

[19]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[20]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]