Skip to main content
SpringerLink
Log in

A Lyapunov function for fully nonlinear parabolic equations in one spatial variable

  • Published:
São Paulo Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Lyapunov functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano’s method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.

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. Amann, H.: Parabolic evolution equations and nonlinear boundary conditions. J. Differ. Equ. 72, 201–269 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brunovský, P., Fiedler, B.: Connecting orbits in scalar reaction diffusion equations II: the complete solution. J. Differ. Equ. 81, 106–135 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carvalho, A.N., Langa, J.L., Robinson, J.C.: Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, vol. 182. Springer, New York (2013)

    MATH  Google Scholar 

  4. Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. AMS, Providence (2002)

    MATH  Google Scholar 

  5. Fiedler, B.: Do global attractors depend on boundary conditions? Doc. Math. J. DMV 1, 215–228 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Fiedler, B., Gedeon, T.: A Lyapunov function for tridiagonal competitive-cooperative systems. SIAM J. Math. Anal. 30(3), 469–478 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fiedler, B., Grotta-Ragazzo, C., Rocha, C.: An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle. Russian Math. Surv. 69, 27–42 (2014)

    Article  MATH  Google Scholar 

  8. Fiedler, B., Rocha, C.: Heteroclinic orbits of semilinear parabolic equations. J. Differ. Equ. 125, 239–281 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Mathematical Survey, vol. 25. AMS, Providence (1988)

    MATH  Google Scholar 

  10. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981)

    Book  MATH  Google Scholar 

  11. Hu, B.: Blow-Up Theories for Semilinear Parabolic Equations. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  12. Ladyzhenskaya, O.: Attractors for Semi-groups and Evolution Equations. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  13. Lappicy, P.: Sturm attractors for quasilinear parabolic equations. J. Differ. Equ. 265, 4642–4660 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lappicy, P.: Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients (2018). arXiv:1806.04019

  15. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Springer, Basel (1995)

    Book  MATH  Google Scholar 

  16. Matano, H.: Asymptotic behaviour of solutions of semilinear heat equations on \(S^{1}\). In: Ni, W.-M., Peletier, L.A., Serrin, J. (eds.) Nonlinear Diffusion Equations and Their Equilibrium States II, pp. 139–162. Springer, New York (1988)

    Chapter  Google Scholar 

  17. Zelenyak, T.I.: Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Differ. Uravn. 4, 34–45 (1968)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We are indebted to Marek Fila for helpful discussions concerning nonlinear boundary conditions. Phillipo Lappicy was supported by FAPESP, Brasil, Grant No. 2017/07882-0. Bernold Fiedler was partially supported by SFB 910 of the Deutsche Forschungsgemeinschaft, and some generous libations of Cachaça de Jambú.

Author information

Authors and Affiliations

  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida trabalhador são-carlense 400, São Carlos, SP, 13566-590, Brazil

    Phillipo Lappicy

  2. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Bernold Fiedler

Authors
  1. Phillipo Lappicy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernold Fiedler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Phillipo Lappicy.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lappicy, P., Fiedler, B. A Lyapunov function for fully nonlinear parabolic equations in one spatial variable. São Paulo J. Math. Sci. 13, 283–291 (2019). https://doi.org/10.1007/s40863-018-00115-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40863-018-00115-2

Keywords

Search

Navigation