American Institute of Mathematical Sciences

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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:
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L. Arnold, "Random Dynamical Systems", Springer-Verlag, New York and Berlin, 1998.  Google Scholar

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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.  Google Scholar

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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

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J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biology, 81 (1979), 389-400. Google Scholar

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G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.  Google Scholar

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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. Google Scholar

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Y. You, Dynamics of three-component reversible Gray-Scott model, DCDS-B, 14 (2010), 1671-1688.  Google Scholar

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Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, Series A, 75 (2012), 3049-3071.  Google Scholar

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Y. You, Random attractor for stochastic reversible Schnackenberg equations, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 1347-1362.  Google Scholar

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems", Springer-Verlag, New York and Berlin, 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), 845-869.  Google Scholar

[3]

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

[4]

I. Chueshov, "Monotone Random Systems Theory and Applications", Lect. Notes of Math., Vol. 1779, Springer, New York, 2002.  Google Scholar

[5]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  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), 1087-1097. Google Scholar

[7]

P. Martin-Rubio and J.C. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stochastics and Dynamics, 3 (2003), 279-297.  Google Scholar

[8]

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

[9]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. Google Scholar

[10]

J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biology, 81 (1979), 389-400. Google Scholar

[11]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 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), 229-264. Google Scholar

[13]

Y. You, Dynamics of three-component reversible Gray-Scott model, DCDS-B, 14 (2010), 1671-1688.  Google Scholar

[14]

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, Series A, 75 (2012), 3049-3071.  Google Scholar

[15]

Y. You, Random attractor for stochastic reversible Schnackenberg equations, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 1347-1362.  Google Scholar

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