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Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2177–2204 | Cite as

Weak Pullback Attractors for Mean Random Dynamical Systems in Bochner Spaces

  • Bixiang WangEmail author
Article

Abstract

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.

Keywords

Mean random attractor Pullback attractor Weak topology Bochner space Reaction–diffusion equation 

Mathematics Subject Classification

Primary 35B40 Secondary 35B41 37L30 

References

  1. 1.
    Adili, A., Wang, B.: Random attractors for stochastic FitzHugh–Nagumo systems driven by deterministic non-autonomous forcing. Contin. Discrete Dyn. Syst. Ser. B 18, 643–666 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bates, P.W., Lu, K., Wang, B.: Random attractors for stochastic reaction–diffusion equations on unbounded domains. J. Differ. Equ. 246, 845–869 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beyn, W.J., Gess, B., Lescot, P., Röckner, M.: The global random attractor for a class of stochastic porous media equations. Commun. Partial Differ. Equ. 36, 446–469 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations I: local theory. Inven. Math. 173, 449–475 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations II: a global existence result. Inven. Math. 173, 477–496 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Caraballo, T., Garrido-Atienza, M.J., Schmalfuss, B., Valero, J.: Non-autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete Contin. Dyn. Syst. 21, 415–443 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Caraballo, T., Real, J., Chueshov, I.D.: Pullback attractors for stochastic heat equations in materials with memory. Discrete Contin. Dyn. Syst. B 9, 525–539 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Caraballo, T., Langa, J.A.: On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10, 491–513 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Caraballo, T., Garrido-Atienza, M.J., Schmalfuss, B., Valero, J.: Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete Contin. Dyn. Syst. Ser. B 14, 439–455 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Caraballo, T., Garrido-Atienza, M.J., Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74, 3671–3684 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Caraballo, T., Langa, J.A., Melnik, V.S., Valero, J.: Pullback attractors for nonautonomous and stochastic multivalued dynamical systems. Set Valued Anal. 11, 153–201 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chueshov, I., Scheutzow, M.: On the structure of attractors and invariant measures for a class of monotone random systems. Dyn. Syst. 19, 127–144 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chueshow, I.: Monotone Random Systems—Theory and Applications. Lecture Notes in Mathematics, vol. 1779. Springer, Berlin (2001)Google Scholar
  14. 14.
    Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Diestel, J., Uhl Jr., J.: Vector Measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  17. 17.
    Duan, J., Schmalfuss, B.: The 3D quasigeostrophic fluid dynamics under random forcing on boundary. Commun. Math. Sci. 1, 133–151 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59, 21–45 (1996)zbMATHGoogle Scholar
  19. 19.
    Garrido-Atienza, M.J., Schmalfuss, B.: Ergodicity of the infinite dimensional fractional Brownian motion. J. Dyn. Differ. Equ. 23, 671–681 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. Dyn. 11, 369–388 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Garrido-Atienza, M.J., Maslowski, B., Schmalfuss, B.: Random attractors for stochastic equations driven by a fractional Brownian motion. Int. J. Bifurc. Chaos 20, 2761–2782 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gess, B., Liu, W., Rockner, M.: Random attractors for a class of stochastic partial differential equations driven by general additive noise. J. Differ. Equ. 251, 1225–1253 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Gess, B.: Random attractors for degenerate stochastic partial differential equations. J. Dyn. Differ. Equ. 25, 121–157 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gess, B.: Random attractors for singular stochastic evolution equations. J. Differ. Equ. 255, 524–559 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Huang, J., Shen, W.: Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discret. Contin. Dyn. Syst. 24, 855–882 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kloeden, P.E., Langa, J.A.: Flattening, squeezing and the existence of random attractors. Proc. R. Soc. Lond. Ser. A 463, 163–181 (2007)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176. American Mathematical Society, Providence (2011)Google Scholar
  28. 28.
    Kloeden, P.E., Lorenz, T.: Mean-square random dynamical systems. J. Differ. Equ. 253, 1422–1438 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lions, J.L.: Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969)zbMATHGoogle Scholar
  30. 30.
    Lv, Y., Wang, W.: Limiting dynamics for stochastic wave equations. J. Differ. Equ. 244, 1–23 (2008)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Prevot, C., Rockner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)zbMATHGoogle Scholar
  32. 32.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  33. 33.
    Schmalfuss, B.: Backward cocycles and attractors of stochastic differential equations. In: International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, pp. 185–192. Dresden (1992)Google Scholar
  34. 34.
    Shen, Z., Zhou, S., Shen, W.: One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation. J. Differ. Equ. 248, 1432–1457 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)zbMATHGoogle Scholar
  36. 36.
    Wang, B.: Random attractors for the Stochastic Benjamin–Bona–Mahony equation on unbounded domains. J. Differ. Equ. 246, 2506–2537 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Wang, B.: Asymptotic behavior of stochastic wave equations with critical exponents on \({\mathbb{R}}^3\). Trans. Am. Math. Soc. 363, 3639–3663 (2011)zbMATHGoogle Scholar
  38. 38.
    Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253, 1544–1583 (2012)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Wang, B.: Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch. Dyn. 14(4), 1450009, 1–31 (2014)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Wang, B.: Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discret. Contin. Dyn. Syst. Ser. A 34, 269–300 (2014)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Wang, B.: Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations. Nonlinear Anal. TMA 103, 9–25 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Wang, B.: Pullback attractors for non-autonomous reaction–diffusion equations on \({\mathbb{R}}^n\). Front. Math. China 4, 563–583 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNew Mexico Institute of Mining and TechnologySocorroUSA

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