Singular Perturbations of Forward-Backward p-Parabolic Equations

  • S. N. AntontsevEmail author
  • I. V. Kuznetsov
Original Paper


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.

2010 Mathematics Subject Classification

Primary: 35D99, 35K55, 35K65, 35K92 Secondary: 28A33, 35B50, 35R25 

Key words and phrases

entropy solution forward-backward parabolic equation gradient Young measure maximum principle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [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.MathSciNetzbMATHGoogle Scholar
  2. [2]
    M. Chipot, S. Guesmia, A. Sengouga, Singular perturbations of some nonlinear problems, Journal of Mathematical Sciences 176, (2011), no. 6, 828–843.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Chipot: Asymptotic issues for some partial differential equations, ICP, London, 2016.CrossRefzbMATHGoogle Scholar
  4. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    I.V. Kuznetsov, Entropy solutions to differential equations with variable parabolicity direction, Journal of Mathematical Sciences 202, (2014), no. 1, 91–112.MathSciNetCrossRefGoogle Scholar
  9. [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).Google Scholar
  10. [10]
    S.N. Kruzhkov, First order quasi-linear equations in several independent variables, Mathematics of the USSR Sbornik 10, (1970), no. 2, 217–243.CrossRefzbMATHGoogle Scholar
  11. [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.MathSciNetzbMATHGoogle Scholar
  12. [12]
    O.A. Ladyzhenskaya, N.N. Ural’tseva: Linear and quasilinear elliptic equations, Nauka, Moscow, 1973. (in Russian)zbMATHGoogle Scholar
  13. [13]
    M. Borsuk, V. Kondratiev: Elliptic boundary value problems of second order in piecewise smooth domains, North-Holland Mathematical Library, 69, Elsevier, Amsterdam, 2006.Google Scholar
  14. [14]
    J. Simon, Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata 146, (1987), no. 1, 65–96.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Kinderlehrer, P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, The Journal of Geometric Analysis 4, (1994), 59–90Google Scholar
  16. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen: Variational methods in imaging, Applied Mathematical Sciences, 167, Springer, New York, 2009.Google Scholar
  20. [20]
    L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.zbMATHGoogle Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing 2016

Authors and Affiliations

  1. 1.Lavrentyev Institute of Hydrodynamics Siberian Branch of RASNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.CMAF-CIOUniversity of LisbonLisbonPortugal

Personalised recommendations