Skip to main content

Reaction–diffusion problem in a thin domain with oscillating boundary and varying order of thickness

Zeitschrift für angewandte Mathematik und Physik volume 72, Article number: 5 (2021) Cite this article

Abstract

In this paper, we study a reaction–diffusion problem in a thin domain with varying order of thickness. Motivated by the applications, we assume the oscillating behavior of the boundary and prescribe the Robin-type boundary condition simulating the reaction catalyzed by the upper wall. Using the appropriate functional setting and the unfolding operator method, we rigorously derive lower-dimensional approximation of the governing problem. Five different limit problems have been obtained by comparing the magnitude of the reaction mechanism with the variation in domain’s thickness.

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

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

Fig. 1

References

  1. Arrieta, J.M.: Neumann eigenvalues problems on exterior perturbation of the domain. J. Differ. Equ. 118, 54–103 (1995)

    Article  MathSciNet  Google Scholar 

  2. Arrieta, J.M.: Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case. Trans. Am. Math. Soc. 347, 3503–3531 (1995)

    Article  MathSciNet  Google Scholar 

  3. Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.: Dynamics in dumbbell domains. I. Continuity of the set of equilibria. J. Differ. Equ. 231, 551–597 (2006)

    Article  MathSciNet  Google Scholar 

  4. Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.: Dynamics in dumbbell domains II. The limiting problem. J. Differ. Equ. 247, 174–202 (2009)

    Article  MathSciNet  Google Scholar 

  5. Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.: Dynamics in dumbbell domains III. Continuity of attractors. J. Differ. Equ. 247, 225–259 (2009)

    Article  MathSciNet  Google Scholar 

  6. Arrieta, J.M., Carvalho, A.N., Pereira, M.C., Silva, R.P.: Semilinear parabolic problems in thin domains with a highly oscillatory boundary. Nonlinear Anal. Theory Methods Appl. 74–15, 5111–5132 (2011)

    Article  MathSciNet  Google Scholar 

  7. Arrieta, J.M.,Nakasato, J.C., Pereira, M.C.: The p-Laplacian operator in thin domains: the unfolding approach (2019) arXiv:1803.11318

  8. Arrieta, J.M., Villanueva-Pesqueira, M.: Thin domains with non-smooth oscillatory boundaries. J. Math. Anal. Appl. 446, 130–164 (2017)

    Article  MathSciNet  Google Scholar 

  9. Bonnivard, M., Suarez-Grau, F.J., Tierra, G.: On the influence of wavy riblets on the slip behaviour of viscous fluids. Z. Angew. Math. Phys. 67, 27 (2016)

    Article  MathSciNet  Google Scholar 

  10. Casten, R.G., Holland, C.J.: Instability results for reaction–diffusion equations with Neumann boundary conditions. J. Differ. Equ. 27, 266–273 (1978)

    Article  MathSciNet  Google Scholar 

  11. Ciuperca, I.S.: Reaction–diffusion equations on thin domains with varying order of thinness. J. Differ. Equ. 126, 244–291 (1996)

    Article  MathSciNet  Google Scholar 

  12. Ciuperca, I.S.: Spectral properties of Schrodinger operators on domains with varying order of thinness. J. Dyn. Differ. Equ. 126, 244–291 (1996)

    Article  Google Scholar 

  13. Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40, 1585–1620 (2008)

    Article  MathSciNet  Google Scholar 

  14. Cioranescu, D., Damlamian, A., Donato, P., Griso, G., Zaki, R.: The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44, 718–760 (2012)

    Article  MathSciNet  Google Scholar 

  15. Cioranescu, D., Damlamian, A., Griso, G.: The Periodic Unfolding Method. Theory and Applications to Partial Differential Problems. Springer, Singapore (2018)

    Book  Google Scholar 

  16. Hale, J.K., Raugel, G.: Reaction–diffusion equations on thin domains. J. Math. Pures et Appl. 9, 33–95 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Kishimoto, K., Weinberger, H.F.: The spatial homogeneity of stable equilibria of some reaction–diffusion systems on convex domains. J. Differ. Equ. 58(1), 15–21 (1985)

    Article  MathSciNet  Google Scholar 

  18. Nakasato, J.C., Pažanin, I., Pereira, M.C.: Roughness-induced effects on the convection–diffusion–reaction problem in a thin domain. Appl. Anal. (2019). https://doi.org/10.1080/00036811.2019.1634260

    Article  Google Scholar 

  19. Pažanin, I., Suarez-Grau, F.J.: Homogenization of the Darcy–Lapwood–Brinkman flow through a thin domain with highly oscillating boundaries. Bull. Malays. Math. Sci. Soc. 42, 3073–3109 (2019)

    Article  MathSciNet  Google Scholar 

  20. Tabeling, P.: Introduction to Microfluidics. Oxford University Press, Oxford (2005)

    Google Scholar 

Download references

Acknowledgements

The first author of this work has been supported by CAPES 88887.507830/2020-00. The second author has been supported by the Croatia Science Foundation under the project MultiFM (IP-2019-04-1140). The third author has been partially supported by the Croatia Science Foundation under the Project MultiFM (IP-2019-04-1140) and CNPq 303253/2017-7 (Brazil).

Author information

Authors and Affiliations

  1. Depto. Matemática, ICMC, Univesidade de São Paulo, Avenida Trabalhador São-Carlense, 400, São Carlos, SP, Brazil

    Jean Carlos Nakasato

  2. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička Cesta 30, Zagreb, Croatia

    Igor Pažanin

  3. Depto. Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil

    Marcone C. Pereira

Authors
  1. Jean Carlos Nakasato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Pažanin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcone C. Pereira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Pažanin.

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

Nakasato, J.C., Pažanin, I. & Pereira, M.C. Reaction–diffusion problem in a thin domain with oscillating boundary and varying order of thickness. Z. Angew. Math. Phys. 72, 5 (2021). https://doi.org/10.1007/s00033-020-01436-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-020-01436-z

Keywords

  • Reaction–diffusion equation
  • Robin boundary condition
  • Different orders of compression
  • Oscillating boundary
  • Homogenization

Mathematics Subject Classification

  • 35B25
  • 35B40
  • 35J25