# American Institute of Mathematical Sciences

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

## Pullback uniform dissipativity of stochastic reversible Schnackenberg equations

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