Fast-Diffusion Limit for Reaction–Diffusion Equations with Degenerate Multiplicative and Additive Noise

Journal of Dynamics and Differential Equations volume 33, pages577–592(2021)Cite this article

Abstract

In the present work we consider a quite general class of reaction–diffusion equations forced by additive and multiplicative noise. When the diffusion is large, one can approximate the solutions of the stochastic reaction–diffusion equations with polynomial term by the solutions of a stochastic ordinary equations with additive noise. We illustrate our results by applying it to logistic equation and nonlinear heat equation.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. 1.

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

    Google Scholar 

  2. 2.

    Boutet de Monvel, L., Chueshov, I.D., Rezounenko, A.V.: Inertial manifolds for retarded semilinear parabolic equations. Nonlinear Anal. 34(6), 907–925 (1998)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Blömker, D., Mohammed, W.W.: Amplitude equations for SPDEs with cubic nonlinearities. Stoch. Int. J. Probab. Stoch. Process. 85, 181–215 (2013)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Blömker, D., Mohammed, W.W.: Amplitude equations for SPDEs with quadratic nonlinearities. Electron. J. Probab. 14, 2527–2550 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Blömker, D., Wang, W.: Qualitative properties of local random invariant manifolds for SPDEs with quadratic nonlinearity. J. Dyn. Differ. Equ. 22, 677–695 (2010)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Blömker, D., Hairer, M., Pavliotis, G.A.: Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity 20, 1–25 (2007)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Caraballo, T., Langa, J.A., Robinson, J.C.: Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete Contin. Dyn. Syst. 6(4), 875–892 (2000)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Caraballo, T., Langa, J.A., Robinson, J.C.: A stochastic pitchfork bifurcation in a reaction–diffusion equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457, 2041–2061 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chekroun, M.D., Liu, H., Wang, S.: On stochastic parameterizing manifolds: pullback characterization and non-Markovian reduced equations (2013). Preprint, arxiv:1310.3896v1.pdf

  10. 10.

    Robinson, J.C.: Stability of random attractors under perturbation and approximation. J. Differ. Equ. 186(2), 652–669 (2002)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Da Prato, G., Debussche, A.: Construction of stochastic inertial manifolds using backward integration. Stoch. Stoch. Rep. 59(3–4), 305–324 (1996)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Duan, J., Li, X., Sun, X.: An impact of noise on invariant manifolds in dynamical systems. J. Math. Phys. 51, 042702 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Duan, J., Lu, K., Schmalfuß, B.: Invariant manifolds for stochastic partial differential equations. Ann. Prob. 31(4), 2109–2135 (2003)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Flandoli, F.: A stochastic reaction–diffusion equation with multiplicative noise. Appl. Math. Lett. 4, 45–48 (1991)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Funaki, T.: Random motion of strings and related stochastic evolution equations. Nagoya Math. 89, 129–193 (1983)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

    Google Scholar 

  17. 17.

    Kan, X., Duan, J., Kevrekidis, I.G., Roberts, A.J.: Simulating stochastic inertial manifolds by a backward–forward approach. SIAM J. Appl. Dyn. Syst. 12(1), 487–514 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Liu, K.: Stability of Infinite Dimensional Stochastic Differential Equations with Applications, vol. 135. Chapman and Hall /CRC Monographs, Boca Raton (2006)

    Google Scholar 

  19. 19.

    Mohammed, W.W., Blömker, D.: Fast-diffusion limit with large noise for systems of stochastic reaction–diffusion equations. J. Stoch. Anal. Appl. 34(6), 961–978 (2016)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mohammed, W.W., Blömker, D.: Fast diffusion limit for reaction–diffusion systems with stochastic neumann boundary conditions. SIAM J. Math. Anal. 48(5), 3547–3578 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Mohammed, W.W., Blömker, D., Klepel, K.: Multi-scale analysis of SPDEs with degenerate additive noise. J. Evol. Equ. 14(2), 273–298 (2014)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Murray, J.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003)

    Google Scholar 

  23. 23.

    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Google Scholar 

  24. 24.

    Roberts, A.J., Wang, W.: Macroscopic reduction for stochastic reaction–diffusion equations. J. Appl. Math. 78, 1237–1264 (2013)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Walsh, J.B.: An introduction to stochastic partial differential equations. Ecole d’etk de Probabilitks de St. FlourXIV, Lecture Notes in Mathematics, vol. 1180, pp. 266–439 (1986)

Download references

Acknowledgements

The author would like to thank the editor and anonymous reviewers for their precious remarks and suggestions.

Author information

Affiliations

  1. Department of Mathematics, Faculty of Science, Ha’il University, Ha’il, Saudi Arabia

    Wael W. Mohammed

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

    Wael W. Mohammed

Authors
  1. Wael W. Mohammed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wael W. Mohammed.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohammed, W.W. Fast-Diffusion Limit for Reaction–Diffusion Equations with Degenerate Multiplicative and Additive Noise. J Dyn Diff Equat 33, 577–592 (2021). https://doi.org/10.1007/s10884-020-09821-y

Download citation

Keywords

  • Reaction–diffusion equations
  • Fast diffusion limit
  • Additive noise
  • Multiplicative noise

Mathematics Subject Classification

  • 60H10
  • 60H15
  • 35R60