Skip to main content
SpringerLink
Log in

Asymptotic dynamics of stochastic delay nonclassical diffusion equations on unbounded domains

  • Original Paper
  • Published:
Banach Journal of Mathematical Analysis Aims and scope Submit manuscript

Abstract

The goal of this paper is to investigate three kinds of stability of pullback random attractors (PRAs) for stochastic nonclassical diffusion equations with distributed delay and constant delay perturbed by operator-type noise defined on \({\mathbb{R}}^{n}.\) We first prove the existence, uniqueness, backward compactness and backward longtime stability of PRAs for this equation. We then establish the zero-memory stability of PRAs. Finally, we study the asymptotically autonomous stability of PRAs. Due to the problem of the non-compactness of Sobolev embeddings on \({\mathbb{R}}^{n},\) we use the backward tail-estimates method and spectrum decomposition technique to prove the backward asymptotic compactness of solutions.

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

    Article  Google Scholar 

  3. Anh, C.T., Bao, T.Q.: Pullback attractors for a class of non-autonomous nonclassical diffusion equations. Nonlinear Anal. 73, 399–412 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  5. Bai, L., Zhang, F.: Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains. Results Math. 69, 129–60 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caraballo, T., Langa, J.A., Robinson, J.C.: Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commun. Partial Differ. Equ. 23, 1557–1581 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caraballo, T., Langa, J.A.: On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10, 491–513 (2003)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Caraballo, T., Marín-Rubio, P., Valero, J.: Autonomous and non-autonomous attractors for differential equations with delays. J. Differ. Equ. 208, 9–41 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caraballo, T., Márquez-Durán, A.M.: Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay. Dyn. Partial Differ. Equ. 10, 267–281 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caraballo, T., Márquez-Durán, A.M., Rivero, F.: Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay. Int. J. Bifurc. Chaos Appl. Sci. Eng. 25, 1540021 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caraballo, T., Guo, B., Tuan, N.H., Wang, R.: Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains. Proc. R. Soc. Edinb. Sect. A 151, 1700–1730 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Crauel, H., Kloeden, P.E., Yang, M.: Random attractors of stochastic reaction–diffusion equations on variable domains. Stoch. Dyn. 11, 301–314 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cui, H., Kloeden, P.E., Wu, F.: Pathwise upper semi-continuity of random pullback attractors along the time axis. Physica D 374(375), 21–34 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cui, H., Langa, J.A., Li, Y.: Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems. J. Dyn. Differ. Equ. 30, 1873–1898 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. García-Luengo, J., Marín-Rubio, P., Real, J.: Pullback attractors for 2D Navier–Stokes equations with delays and their regularity. Adv. Nonlinear Stud. 13, 331–357 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Harraga, H., Yebdri, M.: Pullback attractors for a class of semilinear nonclassical diffusion equations with delay. Electron. J. Differ. Equ. 1–33, 2016 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Hu, Z., Wang, Y.: Pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay. J. Math. Phys. 53, 072702 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kloeden, P.E.: Upper semi continuity of attractors of retarded delay differential equations in the delay. Bull. Aust. Math. Soc. 73, 299–306 (2006)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Lee, J., Toi, V.M.: Attractors for nonclassical diffusion equations with dynamic boundary conditions. Nonlinear Anal. 195, 111737 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, Y., Gu, A., Li, J.: Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations. J. Differ. Equ. 258, 504–534 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, Y., Wang, R.: Random attractors for 3D Benjamin–Bona–Mahony equations derived by a Laplace-multiplier noise. Stoch. Dyn. 18, 1850004 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, Y., Yang, S.: Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Commun. Pure Appl. Anal. 18, 1155–1175 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Toan, N.D.: Uniform attractors of nonclassical diffusion equations lacking instantaneous damping on \({\mathbb{R} }^{N}\) with memory. Acta Appl. Math. 170, 789–822 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253, 1544–1583 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, B.: Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete Contin. Dyn. Syst. 34, 269–300 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, B.: Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch. Dyn. 14, 1450009 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, R., Wang, B.: Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on \({\mathbb{R} }^N\). Nonlinearity 32, 4524–4556 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, R., Li, Y., Wang, B.: Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete Contin. Dyn. Syst. 39, 4091–126 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, S., Li, Y.: Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations. Physica D 382(383), 46–57 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, X., Lu, K., Wang, B.: Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing. SIAM J. Appl. Dyn. Syst. 14, 1018–1047 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang, S., Li, Y.: Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evol. Equ. Control Theory 9, 581–604 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yang, S., Li, Y.: Asymptotic autonomous attractors for a stochastic lattice model with random viscosity. J. Differ. Equ. Appl. 26, 540–560 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, F., Han, W.: Pullback attractors for nonclassical diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms. Electron. J. Differ. Equ. 1–28, 2016 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Zhang, Q., Li, Y.: Double stabilities of pullback random attractors for stochastic delayed p-Laplacian equations. Math. Methods Appl. Sci. 43, 8406–8433 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhao, W., Song, S.: Dynamics of stochastic nonclassical diffusion equations on unbounded domains. Electron. J. Differ. Equ. 1–22, 2015 (2015)

    MathSciNet  Google Scholar 

  42. Zhu, K., Sun, C.: Pullback attractors for nonclassical diffusion equations with delays. J. Math. Phys. 56, 092703 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express his sincere thanks to the reviewers for valuable comments and suggestions. This work was supported by Natural Science Foundation of Shandong Province (Grant no. ZR2022QA054).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Heze University, Heze, 274015, China

    Qiangheng Zhang

Authors
  1. Qiangheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiangheng Zhang.

Additional information

Communicated by Maria Alessandra Ragusa.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Q. Asymptotic dynamics of stochastic delay nonclassical diffusion equations on unbounded domains. Banach J. Math. Anal. 16, 72 (2022). https://doi.org/10.1007/s43037-022-00225-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43037-022-00225-2

Keywords

Mathematics Subject Classification

Search

Navigation