American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 1134-1142. doi: 10.3934/proc.2015.1134

Pullback uniform dissipativity of stochastic reversible Schnackenberg equations

Yuncheng You 1,

1. 

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

Received  September 2014 Revised  January 2015 Published  November 2015

Asymptotic dynamics of stochastic reversible Schnackenberg equations with multiplicative white noise on a three-dimensional bounded domain is investigated in this paper. The pullback uniform dissipativity in terms of the existence of a common pullback absorbing set with respect to the reverse reaction rate of this typical autocatalytic reaction-diffusion system is proved through decomposed grouping estimates.
Keywords: pullback absorbing set, Reaction-diffusion system, random attractor., pullback uniform dissipativity, stochastic Schnackenberg equations.
Mathematics Subject Classification: Primary: 37L30, 37L55; Secondary: 35B40, 35K55, 60H1.
Citation: Yuncheng You. Pullback uniform dissipativity of stochastic reversible Schnackenberg equations. Conference Publications, 2015, 2015 (special) : 1134-1142. doi: 10.3934/proc.2015.1134
References:
[1]

L. Arnold, "Random Dynamical Systems",, Springer-Verlag, (1998). Google Scholar

[2]

P.W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Diff. Eqns., 246 (2009), 845. Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", AMS Colloquium Publications, 49 (2002). Google Scholar

[4]

I. Chueshov, "Monotone Random Systems Theory and Applications",, Lect. Notes of Math., (1779). Google Scholar

[5]

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

[6]

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

[7]

P. Martin-Rubio and J.C. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stochastics and Dynamics, 3 (2003), 279. Google Scholar

[8]

J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, 3rd edition, (2003). Google Scholar

[9]

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

[10]

J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior,, J. Theor. Biology, 81 (1979), 389. Google Scholar

[11]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). Google Scholar

[12]

M.J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnackenberg model,, Stud. Appl. Math., 109 (2002), 229. Google Scholar

[13]

Y. You, Dynamics of three-component reversible Gray-Scott model,, DCDS-B, 14 (2010), 1671. Google Scholar

[14]

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonlinear Analysis, 75 (2012), 3049. Google Scholar

[15]

Y. You, Random attractor for stochastic reversible Schnackenberg equations,, Discrete and Continuous Dynamical Systems, 7 (2014), 1347. Google Scholar

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems",, Springer-Verlag, (1998). Google Scholar

[2]

P.W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Diff. Eqns., 246 (2009), 845. Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", AMS Colloquium Publications, 49 (2002). Google Scholar

[4]

I. Chueshov, "Monotone Random Systems Theory and Applications",, Lect. Notes of Math., (1779). Google Scholar

[5]

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

[6]

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

[7]

P. Martin-Rubio and J.C. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stochastics and Dynamics, 3 (2003), 279. Google Scholar

[8]

J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, 3rd edition, (2003). Google Scholar

[9]

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

[10]

J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior,, J. Theor. Biology, 81 (1979), 389. Google Scholar

[11]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). Google Scholar

[12]

M.J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnackenberg model,, Stud. Appl. Math., 109 (2002), 229. Google Scholar

[13]

Y. You, Dynamics of three-component reversible Gray-Scott model,, DCDS-B, 14 (2010), 1671. Google Scholar

[14]

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonlinear Analysis, 75 (2012), 3049. Google Scholar

[15]

Y. You, Random attractor for stochastic reversible Schnackenberg equations,, Discrete and Continuous Dynamical Systems, 7 (2014), 1347. Google Scholar

[1]

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
[2]

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
[3]

Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114
[4]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343
[5]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307
[6]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[7]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
[8]

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
[9]

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
[10]

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
[11]

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
[12]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[13]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253
[14]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
[15]

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
[16]

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
[17]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179
[18]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[19]

Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 187-208. doi: 10.3934/dcds.2018009
[20]

Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151

 Impact Factor: 

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences