• You, Honglian (Department of Mathematics Binzhou University) ;
  • Yuan, Rong (School of Mathematical Sciences Beijing Normal University)
  • Received : 2013.07.03
  • Published : 2014.09.30


In this paper we are concerned with a class of stochastic partial functional differential equations with infinite delay. Supposing that the linear part is a Hille-Yosida operator but not necessarily densely defined and employing the integrated semigroup and random dynamics theory, we present some appropriate conditions to guarantee the existence of a random attractor.


  1. W. Arendt, Resolvent positive operators and integrated semigroup, Proc. Lond. Math. Soc. 54 (1987), no. 3, 321-349.
  2. L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
  3. P. W. Bates, K. Lu, and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations 246 (2009), no. 2, 845-869.
  4. Z. Brzezniak and Y. Li, Asymptotic compactness and absorbing sets for 2d stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5587-5629.
  5. T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations 205 (2004), no. 2, 271-297.
  6. H. Crauel, A. Debussche, and F. Flandoli, Random attractors, J. Dynam. Differential Equations 9 (1997), no. 2, 307-341.
  7. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), no. 3, 365-393.
  8. X. Ding and J. Jiang, Random attractors for stochastic retarded lattice dynamical systems, Abstr. Appl. Anal. 2012 (2012), Art. ID 409282, 27 pp.
  9. X. Ding and J. Jiang, Random attractors for stochastic retarded reaction-diffusion equations on un-bounded domains, Abstr. Appl. Anal. 2013 (2013), Art. ID 981576, 16 pp.
  10. B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations 25 (2013), no. 1, 121-157.
  11. B. Gess, Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise, Ann. Probab. 42 (2014), no. 2, 818-864.
  12. B. Gess, W. Liu, and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations 251 (2011), no. 4-5, 1225-1253.
  13. J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
  14. H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal. 15 (1989), no. 1, 160-180.
  15. Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations 245 (2008), no. 7, 1775-1800.
  16. P. Marın-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal. 67 (2007), no. 10, 2784-2799.
  17. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
  18. G. Da Prato and E. Sinestrari, Differential operators with non dense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 (1987), no. 2, 285-344.
  19. Z. Shen, S. Zhou, and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations 248 (2010), no. 6, 1432-1457.
  20. H. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations 3 (1990), no. 6, 1035-1066.
  21. B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. 71 (2009), no. 7-8, 2811-2828.
  22. B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on un-bounded domains, Electron. J. Differential Equations 2012 (2012), no. 59, 18 pp.
  23. B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst. (2009), no. Dynamical Systems, Differential Equations and Applications., 7th AIMS Conference, suppl., 800-809.
  24. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.