Skip to main content
Log in

On nonlinear systems consisting of different types of differential equations

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. R. A. Adams, Sobolev spaces, Academic Press, New York-San Francisco-London, 1975.

    MATH  Google Scholar 

  2. J. Berkovits and V. Mustonen, Topological degreee for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems, Rend. Mat. Ser. VII Roma, 12 (1992), 597–621.

    MATH  MathSciNet  Google Scholar 

  3. Á. Besenyei, Existence of solutions of a nonlinear system modelling fluid flow in porous media, Electron. J. Differential Equations, 153 (2006), 1–19.

    MathSciNet  Google Scholar 

  4. Á. Besenyei, Stabilization of solutions to a nonlinear system modelling fluid flow in porous media, Annales Univ. Sci. Budapest, to appear.

  5. M. Chipot and L. Molinet, Asymptotic behavior of some nonlocal diffusion problems, Applicable Analysis, 80 (2001), 279–315.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Chipot and B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. Advances in quenching, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8 (2001), 35–51.

    MATH  MathSciNet  Google Scholar 

  7. S. Cinca, Diffusion und Transport in porösen Medien bei veränderlichen Porosität, Diplomawork, Univ. Heidelberg, 2000.

    Google Scholar 

  8. J. I. Díaz and G. Hetzer, A quasilinear functional reaction-diffusion equation arising in climatology, PDEs and Applications, Dunod, Paris, 1998.

    Google Scholar 

  9. M. Farkas, Two ways of modelling cross-diffusion, Nonlin. Analysis, TMA, 30 (1997), 1225–1233.

    Article  MATH  MathSciNet  Google Scholar 

  10. U. Hornung and W. Jäger, Diffusion, absorption and reaction of chemicals in porous media, J. Differential Equations, 92 (1991), 199–225.

    Article  MATH  MathSciNet  Google Scholar 

  11. U. Hornung, W. Jäger and A. Mikelic, Reactive transport through an array of cells with semi-permeable membrans, Math. Modelling Num. Anal., 28 (1994), 59–94.

    MATH  Google Scholar 

  12. W. Jäger and L. Simon, On a system of quasilinear parabolic functional differential equations, Acta Math. Hungar., 112 (2006), 39–55.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. L. Lions, Quelques métodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.

    Google Scholar 

  14. J. D. Logan, M. R. Petersen and T. S. Shores, Numerical study of reactionmineralogy-porosity changes in porous media, Appl. Math. Comput., 127 (2002), 149–164.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Simon, On different types of nonlinear parabolic functional differential equations, Pure Math. Appl., 9 (1998), 181–192.

    MATH  MathSciNet  Google Scholar 

  16. L. Simon and W. Jäger, On non-uniformly parabolic functional differential equations, Studia Sci. Math. Hungar., to appear.

  17. L. Simon, On a system with a singular parabolic equation, Folia FSN Univ. Masarykianae Brunensis, Math., 16 (2007), 149–156.

    Google Scholar 

  18. L. Simon, On qualitative properties of a system of partial functional equations, Electron. J. Qual. Theory Differ. Equ., submitted.

  19. E. Zeidler, Nonlinear functional analysis and its applications II A and II B, Springer, New York, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to László Simon.

Additional information

Dedicated to the memory of Professor Miklós Farkas

Supported by the Hungarian NFSR under grant OTKA T 049819.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Simon, L. On nonlinear systems consisting of different types of differential equations. Period Math Hung 56, 143–156 (2008). https://doi.org/10.1007/s10998-008-5143-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-008-5143-3

Mathematics subject classification numbers

Key words and phrases

Navigation