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, New York and Berlin, 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), 1-21. 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), 845-869. 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), 365-384. Google Scholar

[5]

M. L. Campbell, Cell Modeling, Master's Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, 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, Series A, 21 (2008), 415-443. 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), 1557-1581. doi: 10.1080/03605309808821394.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002.  Google Scholar

[9]

I. Chueshov, Monotone Random Systems Theory and Applications, Lect. Notes of Math., Vol. 1779, Springer, New-York, 2002. doi: 10.1007/b83277.  Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. 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), 523-563. 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), 2109-2135. 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), 21-45. 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), 1087-1097. 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), 066114. 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), 192-194. 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), 279-297. doi: 10.1142/S0219493703000772.  Google Scholar

[18]

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

[19]

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

[20]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. 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), 1665-1700. 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), 185-198. 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, Dresden, 1992, 185-192. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

L. J. Shaw and J. D. Murray, Analysis of a model for complex skin patterns, SIAM J. Appl. Math., 50 (1990), 628-648. doi: 10.1137/0150037.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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.  Google Scholar

[30]

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

[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.  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), 229-264. doi: 10.1111/1467-9590.00223.  Google Scholar

[33]

Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947.  Google Scholar

[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.  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, X. B. Hu and Q. P. Liu), AIP Conf. Proc., 1212, Amer. Inst. Phys., Melville, New York, (2010), 293-311.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  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), 845-869. 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), 365-384. Google Scholar

[5]

M. L. Campbell, Cell Modeling, Master's Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, 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, Series A, 21 (2008), 415-443. 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), 1557-1581. doi: 10.1080/03605309808821394.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002.  Google Scholar

[9]

I. Chueshov, Monotone Random Systems Theory and Applications, Lect. Notes of Math., Vol. 1779, Springer, New-York, 2002. doi: 10.1007/b83277.  Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. 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), 523-563. 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), 2109-2135. 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), 21-45. 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), 1087-1097. 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), 066114. 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), 192-194. 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), 279-297. doi: 10.1142/S0219493703000772.  Google Scholar

[18]

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

[19]

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

[20]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. 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), 1665-1700. 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), 185-198. 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, Dresden, 1992, 185-192. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

L. J. Shaw and J. D. Murray, Analysis of a model for complex skin patterns, SIAM J. Appl. Math., 50 (1990), 628-648. doi: 10.1137/0150037.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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.  Google Scholar

[30]

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

[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.  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), 229-264. doi: 10.1111/1467-9590.00223.  Google Scholar

[33]

Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947.  Google Scholar

[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.  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, X. B. Hu and Q. P. Liu), AIP Conf. Proc., 1212, Amer. Inst. Phys., Melville, New York, (2010), 293-311.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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]

Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060

[4]

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

[5]

Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2699-2772. doi: 10.3934/dcdsb.2020028

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

[17]

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

[18]

Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094

[19]

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

[20]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]