Skip to main content
SpringerLink
Log in

Existence of Random Attractors for 2D-Stochastic Nonclassical Diffusion Equations on Unbounded Domains

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this article, we prove the existence of a random attractor for stochastic nonclassical diffusion equations on unbounded domains, and the asymptotic compactness of the random dynamical system is established by a tail-estimates method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aifantis E.C.: On the problem of diffusion in solids. Acta Mech. 37, 265–296 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lions J.L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  3. Kuttler K., Aifantis E.C.: Existence and uniqueness in nonclassical diffusion. Q. Appl. Math. 45, 549–560 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Kuttler K., Aifantis E.C.: Quasilinear evolution equations in nonclassical diffusion. SIAM J. Math. Anal. 19, 110–120 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aifantis E.C.: Gradient nanomechanics: applications to deformation, fracture, and diffusion in nanopolycrystals. Metall. Mater. Trans. A 42, 2985–2998 (2011)

    Article  Google Scholar 

  6. Kalantarov V.K.: On the attractors for some non-linear problems of mathematical physics. Zap. Nauch. Sem. LOMI 152, 50–54 (1986)

    MathSciNet  MATH  Google Scholar 

  7. Xiao Y.L.: Attractors for a nonclassical diffusion equation. Acta Math. Appl. Sin. 18, 273–276 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sun C.Y., Wang S.Y., Zhong C.K.: Global attractors for a nonclassical diffusion equation. Acta Math. Sin. (English series) 23, 1271–1280 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang S.Y., Li D.S., Zhong C.K.: On the dynamics of a class of nonclassical parabolic equations. J. Math. Anal. Appl. 317, 565–582 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physic. Springer, New York (1997)

    Book  MATH  Google Scholar 

  11. Sun C.Y., Yang M.H.: Dynamics of the nonclassical diffusion equations. Asymptot. Anal. 59, 51–81 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Liu Y.F., MA Q.Z.: Exponential attractors for a nonclassical diffusion equation. Electron. J. Differ. Equ. 9, 1–9 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Wu H.Q., Zhang Z.Y.: Asymptotic regularity for the nonclassical diffusion equation with lower regular forcing term. Dyn. Syst. Int. J. 26(4), 391–400 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pan L.X., Liu Y.F.: Robust exponential attractors for the non-autonomous nonclassical diffusion equation with memory. Dyn. Syst. 28(4), 501–517 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhang, F.H.: Time-dependent global attractor for a class of nonclassical parabolic equations. J. Appl. Math. 748321, 1–6 (2014)

  16. Ma, Q.Z., Liu, Y.F., Zhang, F.H.: Global attractors in \({H^1(\mathbb{R}^N)}\) for nonclassical diffusion equations. Discrete Dyn. Nat. Soc. 2012, article ID 672762 (2012). doi:10.1155/2012/672762

  17. Anh C.T., Bao T.Q.: Dynamics of nonclassical diffusion equations on \({\mathbb{R}^N}\). Commun. Pure Appl. Anal. 11, 1231–1252 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Zhang F.H., Liu Y.F.: Pullback attractors in \({H^1(\mathbb{R}^N)}\) for non-autonomous nonclassical diffusion equations. Dyn. Syst. 29(1), 106–118 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ball J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Caraballo T., Real J.: Attractors for 2D-Navier–Stokes models with delays. J. Differ. Equ. 205(2), 271–297 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang B.X.: Attractors for reaction diffusion equations in unbounded domains. Physica D 128, 41–52 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bates P., Lu K., Wang B.: Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equ. 246, 845–869 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bates P.W., Lisei H., Lu K.: Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 6, 1–21 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Crauel H.: Random point attractors versus random set attractors. J. Lond. Math. Soc. 63, 413–427 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Crauel H., Debussche A., Flandoli F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Flandoli F., Schmalfuss B.: Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative white noise. Stoch. Stoch. Rep. 59, 21–45 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Arnold L.: Random Dynamical Systems. Springer, New York (1998)

    Book  MATH  Google Scholar 

  28. Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Basic Courses, Gansu Institute of Architectural Technology, Lanzhou, 730050, China

    Lihong Bai

  2. Department of Mathematics, Longqiao College of Lanzhou Commercial College, Lanzhou, 730101, China

    Fang-hong Zhang

Authors
  1. Lihong Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang-hong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fang-hong Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bai, L., Zhang, Fh. Existence of Random Attractors for 2D-Stochastic Nonclassical Diffusion Equations on Unbounded Domains. Results. Math. 69, 129–160 (2016). https://doi.org/10.1007/s00025-015-0505-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-015-0505-8

Mathematics Subject Classification

Keywords

Search

Navigation