Advanced Search
Article Contents
Article Contents

Random attractors and robustness for stochastic reversible reaction-diffusion systems

Abstract Related Papers Cited by
  • For a typical stochastic reversible reaction-diffusion system with multiplicative white noise, the trimolecular autocatalytic Gray-Scott system on a three-dimensional bounded domain with random noise perturbation proportional to the state of the system, the existence of a random attractor and its robustness with respect to the reverse reaction rates are proved through sharp and uniform estimates showing the pullback uniform dissipation and the pullback asymptotic compactness.
    Mathematics Subject Classification: Primary: 37L30, 37L55; Secondary: 35B40, 35K55, 60H15.


    \begin{equation} \\ \end{equation}
  • [1]

    L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.


    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.


    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.


    N. BerestyckiStochastic calculus and applications. Available from: http://www.statslab.cam.ac.uk/~beresty/teach/StoCal/sc3.pdf.


    T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Math. Anal., 10 (2003), 491-513.


    T. Caraballo, J. A. Langa and J. C. Robibson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.doi: 10.1080/03605309808821394.


    C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977.


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


    I. Chueshov, "Monotone Random Systems Theory and Applications," Lect. Notes of Math., Vol. 1779, Springer-Verlag, Berlin, 2002.doi: 10.1007/b83277.


    H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.doi: 10.1007/BF02219225.


    H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors, J. Dynamics and Differential Equations, 21 (2009), 233-247.doi: 10.1007/s10884-009-9135-8.


    H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.doi: 10.1007/BF01193705.


    A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (1997), 473-491.doi: 10.1080/07362999708809490.


    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.


    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.


    F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Reports, 59 (1996), 21-45.


    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.


    Y. Hayase and H. R. Brand, The Gray-Scott model under the influence of noise: Reentrant spatiotemporal intermittency in a reaction-diffusion system, J. Chem. Phys., 123 (2005), 124507.doi: 10.1063/1.2038966.


    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.


    T. Kolokolnikov and J. Wei, On ring-like solutions for the Gray-Scott model: Existence, instability and self-replicating rings, Euro. J. Appl. Math., 16 (2005), 201-237.doi: 10.1017/S0956792505005930.


    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.


    H. Mahara, et. al, Three-variable reversible Gray-Scott model, J. Chem. Physics, 121 (2004), 8968-8972.


    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.


    D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots, Physica D, 192 (2004), 33-62.doi: 10.1016/j.physd.2003.12.012.


    B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," Sixth edition, Universitext, Springer-Verlag, Berlin, 2003.doi: 10.1007/978-3-642-14394-6.


    J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192.doi: 10.1126/science.261.5118.189.


    I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems. II, J. Chem. Physics, 48 (1968), 1695-1700.doi: 10.1063/1.1668896.


    K. R. Schenk-Hoppé, Random attractors-general properties, existence and applications to stochastic bifurcation theory, Disc. Cont. Dyn. Systems, 4 (1998), 99-130.doi: 10.3934/dcds.1998.4.99.


    B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, (1992), 185-192.


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


    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.


    B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp.


    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.


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


    J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbfR^2$, Stud. Appl. Math., 110 (2003), 63-102.doi: 10.1111/1467-9590.00231.


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


    Y. You, Asymptotic dynamics of Selkov equations, Discrete and Continuous Dynamical Systems, Series S, 2 (2009), 193-219.doi: 10.3934/dcdss.2009.2.193.


    Y. You, Asymptotic dynamics of the modified Schnackenberg equations, Discrete and Continuous Dynmical Systems, Dynamical Systems, Differential Equations and Applications, $7^{th}$ AIMS Conference, suppl., (2009), 857-868.


    Y. You, Dynamics of three-component reversible Gray-Scott model, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 1671-1688.doi: 10.3934/dcdsb.2010.14.1671.


    Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Analysis, 74 (2011), 1969-1986.doi: 10.1016/j.na.2010.11.004.


    Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, 75 (2012), 3049-3071.doi: 10.1016/j.na.2011.12.002.

  • 加载中

Article Metrics

HTML views() PDF downloads(83) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint