Skip to main content
SpringerLink
Log in

Existence of entropy solutions to nonlinear parabolic problems with variable exponent and \(L^1\)-data

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

In the present paper we prove existence results of entropy solutions to a class of nonlinear parabolic p(.)-Laplacian problem with Neumann-type boundary conditions and \(L^1\) data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

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. Hulshof, J.: Bounded weak solutions of an elliptic–parabolic neumann problem. Trans. Am. Math. Soc. 303, 211–227 (1987)

    Article  MathSciNet  Google Scholar 

  2. Andereu, F., Mazón, J.M., De leon, S.S., Teledo, J.: Existence and uniqueness for a degenerate parabolic equation with \(L^1\)-data. Trans. Am. Math. Soc. 351, 285–306 (1999)

    Article  Google Scholar 

  3. El Hachimi, A., Jamea, A.: Nonlinear parabolic problems with Neumann-type boundary conditions and \(L^1\)-data. Electron. J. Qual. Theory Differ. Equ. 27, 1–22 (2007)

    Article  Google Scholar 

  4. El Hachimi, A., Igbida, J., Jamea, A.: Existence result for nonlinear parabolic problems with \(L^1\)-data. Appl. Math. 37, 483–508 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Fan, X.: On the sub-supersolution method for \(p(x)\)-Laplacian equations. J. Math. Anal. Appl. 330, 665–682 (2007)

    Article  MathSciNet  Google Scholar 

  6. Hästö, P.A.: The \(p(x)\)-Laplacian and applications. In: Proceedings of the International Conference on Geometric Function Theory, vol. 15, pp. 53–62 (2007)

  7. Zhang, C.: Entropy solutions for nonlinear elliptic equations with variable exponents. Electron. J. Differ. Equ. 92, 1–14 (2014)

    MathSciNet  Google Scholar 

  8. Wittbold, P., Zimmermann, A.: Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponent and \(L^1\)-data. Nonlinear Anal. Theory Methods Appl. 72, 2990–3008 (2010)

    Article  Google Scholar 

  9. Jamea, Ahmed: Weak solutions to nonlinear parabolic problems with variable exponent. Int. J. Math. Anal. 10, 553–564 (2016)

    Article  Google Scholar 

  10. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)

    Article  MathSciNet  Google Scholar 

  11. Ru̇žička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)

    Book  Google Scholar 

  12. Eden, A., Michaux, B., Rakotoson, J.M.: Semi-discretized nonlinear evolution equations as dynamical systems and error analysis. Indiana Univ. J. 39(3), 737–783 (1990)

    Article  MathSciNet  Google Scholar 

  13. Benzekri, F., El Hachimi, A.: Doubly nonlinear parabolic equations related to the \(p\)-Laplacian operator: semi-discretization. Electron. J. Differ. Equ. 2002, 1–14 (2002)

    MATH  Google Scholar 

  14. Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  Google Scholar 

  15. Bénilan, Ph, Boccardo, L., Gallouet, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa 22, 241–273 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Bonzi, B.K., Nyanquini, I., Ouaro, S.: Existence and uniqueness of weak and entropy solutions for homogeneous neumann boundary-value problems involving variable exponents. Electron. J. Differ. Equ. 2012, 1–19 (2012)

    Article  MathSciNet  Google Scholar 

  17. Fu, Y.: The existence of solutions for elliptic systems with nonuniform growth. Stud. Math. 151, 227–246 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, CRMEF Casablanca-Settat, Casablanca, Morocco

    Ahmed Jamea

  2. LMAI, Faculté des Sciences, Université Chouaib Doukkali, El Jadida, Morocco

    Ahmed Jamea

  3. Département de Mathématiques, CRMEF Fes-Meknes, Fez, Morocco

    Abdelilah Alaoui Lamrani

  4. Faculté des Sciences, Université Mohammed V, Rabat, Morocco

    Abderrahmane El Hachimi

Authors
  1. Ahmed Jamea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelilah Alaoui Lamrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abderrahmane El Hachimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Jamea.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jamea, A., Lamrani, A.A. & El Hachimi, A. Existence of entropy solutions to nonlinear parabolic problems with variable exponent and \(L^1\)-data. Ricerche mat 67, 785–801 (2018). https://doi.org/10.1007/s11587-018-0359-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-018-0359-y

Keywords

Mathematics Subject Classification

Search

Navigation