Uniqueness of solution of the unsteady filtration problem in heterogeneous porous media

  • A. LyaghfouriEmail author
  • E. Zaouche
Original Paper


We establish uniqueness of the solution of the unsteady state dam problem in the heterogeneous and rectangular case assuming the dam wet at the bottom and dry near to the top.


Unsteady state dam problem Fluid flow Heterogeneous porous medium Uniqueness of solution 

Mathematics Subject Classification

35A02 35R35 76S05 



The second author is grateful to Prof. J. F. Rodrigues for kindly inviting him to the CMAF where he enjoyed excellent research conditions during the preparation of his Ph.D Thesis.


  1. 1.
    Carrillo, J.: On the uniqueness of the solution of the evolution dam problem. Nonlinear Anal. Theory Methods Appl. 22(5), 573–607 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carrillo, J., Gilardi, G.: La vitesse de propagation dans le problème de la digue. Ann. Fac. Sci. Toulouse Math. 11(3), 7–28 (1990)Google Scholar
  3. 3.
    Carrillo, J., Lyaghfouri, A.: A filtration problem with nonlinear Darcy’s law and generalized boundary conditions. Adv. Differ. Equations 5(4–6), 515–555 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Challal, S., Lyaghfouri, A.: A Filtration Problem through a Heterogeneous Porous Medium. Interfaces Free Bound. 6, 55–79 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Challal, S., Lyaghfouri, A.: On a class of Free Boundary Problems of type \(div(a(X)\nabla u) = -div(H(X)\chi (u))\). Differ. Integral Equations 19(5), 481–516 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Challal, S., Lyaghfouri, A.: The Heterogeneous Dam problem with Leaky Boundary Condition. Commun. Pure Appl. Anal. 10(1), 93–125 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dibenedetto, E., Friedam, A.: Periodic behaviour for the evolutionary dam problem and related free boundary problems. Commun. Part. Differ. Equations 11, 1297–1377 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gilardi, G.: A new approach to evolution free boundary problems. Commun. Part. Differ. Equaitons 4, 1099–1123 (1979) [vol. 5, 983–984 (1980)]Google Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lyaghfouri, A.: The Inhomogeneous Dam Problem with Linear Darcy’s Law and Dirichlet Boundary Conditions. Math. Models Methods Appl. Sci. 8(6), 1051–1077 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lyaghfouri, A.: The evolution dam problem with nonlinear Darcy’s law and Dirichlet boundary conditions. Port. Math. 56(1), 1–38 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lyaghfouri, A.: The evolution dam problem with nonlinear Darcy’s law and leaky boundary conditions. Ricerche di Matematica XLVI I(2), 297–357 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lyaghfouri, A.: A Regularity Result for a Heterogeneous Evolution Dam Problem. Zeitschrift f\(\ddot{u}\)r Anal. und ihre Anwendungen 24(1), 149–166 (2005)Google Scholar
  14. 14.
    Lyaghfouri, A., Zaouche, E.: \(L^p\)-continuity of solutions to parabolic free boundary problems. Electron. J. Differ. Equations 2015(184), 1–9 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Torelli, A.: Existence and uniqueness of the solution of a non steady free boundary problem. Boll. U.M.I. 14-B(5), 423–466 (1977)Google Scholar
  16. 16.
    Zaouche, E.: Existence of a Solution in a Class of Parabolic Free Boundary Problems (2016) submitted Google Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Natural SciencesAmerican University of Ras Al KhaimahRas Al KhaimahUAE
  2. 2.Département de MathématiquesEcole Normale SupérieureAlgiersAlgeria
  3. 3.Département de MathématiquesUniversité d’EL OuedEL OuedAlgeria

Personalised recommendations