Skip to main content
SpringerLink
Log in

Singular Perturbations of Forward-Backward p-Parabolic Equations

  • Original Paper
  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

In this paper we have proved the existence of entropy measure-valued solutions to forward-backward p-parabolic equations. We have obtained these solutions as singular limits of weak solutions to (p,q)-elliptic regularized boundary-value problems as ε → 0+. When q > 1 and q = 2 we have not defined yet admissible initial and final conditions even in the form of integral inequalities.

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. M. Chipot, S. Guesmia, On the asymptotic behavior of elliptic, anisotropic singular perturbations problems, Communications on Pure and Applied Analysis (CPAA) 8, (2009), no. 1, 179–193.

    MathSciNet  MATH  Google Scholar 

  2. M. Chipot, S. Guesmia, A. Sengouga, Singular perturbations of some nonlinear problems, Journal of Mathematical Sciences 176, (2011), no. 6, 828–843.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Chipot: Asymptotic issues for some partial differential equations, ICP, London, 2016.

    Book  MATH  Google Scholar 

  4. P. Amorim, S. Antontsev, Young measure solutions for the wave equation with p(x,t)-Laplacian: Existence and blow-up, Nonlinear Analysis: Theory, Methods & Applications 92, (2013), 153–167.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Kinderlehrer, P. Pedregal, Weak convergence of integrands and the Young measure representation, SIAM Journal on Mathematical Analysis 23, (1992), no. 1, 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Demoulini, Young measure solutions for a nonlinear parabolic equation of forward-backward type, SIAM Journal on Mathematical Analysis 27, (1996), no. 2, 376–403.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Demoulini, Variational methods for Young measure solutions of nonlinear parabolic evolutions of forward-backward type and of high spatial order, Applicable Analysis 63, (1996), no. 3-4, 363–373.

    Article  MathSciNet  MATH  Google Scholar 

  8. I.V. Kuznetsov, Entropy solutions to differential equations with variable parabolicity direction, Journal of Mathematical Sciences 202, (2014), no. 1, 91–112.

    Article  MathSciNet  Google Scholar 

  9. S.N. Antontsev, I.V. Kuznetsov, Existence of entropy measure-valued solutions for forward-backward p-parabolic equations, submitted to Siberian Electronic Mathematical Reports (SEMR).

  10. S.N. Kruzhkov, First order quasi-linear equations in several independent variables, Mathematics of the USSR Sbornik 10, (1970), no. 2, 217–243.

    Article  MATH  Google Scholar 

  11. F. Otto, Initial-boundary value problem for a scalar conservation law, Comptes Rendus de l’Académie des Sciences - Series I - Mathematics 322, (1996), no. 8, 729–734.

    MathSciNet  MATH  Google Scholar 

  12. O.A. Ladyzhenskaya, N.N. Ural’tseva: Linear and quasilinear elliptic equations, Nauka, Moscow, 1973. (in Russian)

    MATH  Google Scholar 

  13. M. Borsuk, V. Kondratiev: Elliptic boundary value problems of second order in piecewise smooth domains, North-Holland Mathematical Library, 69, Elsevier, Amsterdam, 2006.

  14. J. Simon, Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata 146, (1987), no. 1, 65–96.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Kinderlehrer, P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, The Journal of Geometric Analysis 4, (1994), 59–90

  16. M.A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, Annales de l’Institut Henri Poincare (C) Non Linear Analysis 16, (1999), no. 6, 773–812.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Marcati, R. Natalini, Convergence of the pseudo-viscosity approximation for conservation laws, Nonlinear Analysis: Theory, Methods & Applications 23, (1994), no. 5, 621–628.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Matas, J. Merker, The limit of vanishing viscosity for doubly nonlinear parabolic equations, Electronic Journal of Qualitative Theory of Differential Equations 8, (2014), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  19. O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen: Variational methods in imaging, Applied Mathematical Sciences, 167, Springer, New York, 2009.

  20. L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics Siberian Branch of RAS, av. Ac. Lavrentyev 15, 630090, Novosibirsk, Russia

    S. N. Antontsev

  2. Novosibirsk State University, Pirogov av. 2, 630090, Novosibirsk, Russia

    S. N. Antontsev

  3. CMAF-CIO, University of Lisbon, av. Prof. Gama Pinto 2, 1649-003, Lisbon, Portugal

    I. V. Kuznetsov

Authors
  1. S. N. Antontsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. V. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Antontsev.

Additional information

Research supported by project III.22.4.2 Boundary value problem in dynamics of heterogeneous media.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Antontsev, S.N., Kuznetsov, I.V. Singular Perturbations of Forward-Backward p-Parabolic Equations. J Elliptic Parabol Equ 2, 357–370 (2016). https://doi.org/10.1007/BF03377409

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03377409

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation