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Existence for Parabolic–Elliptic Degenerate Diffusion Problems

Part of the Lecture Notes in Mathematics book series (LNM,volume 2049)

Abstract

In this chapter we are concerned with the study of some boundary value problems with initial data formulated for parabolic–elliptic degenerate diffusion equations with advection, focusing especially on the fast diffusion case which involves a free boundary problem (case (a) in Introduction).

Keywords

  • Weak Solution
  • Cauchy Problem
  • Volumetric Water Content
  • Free Boundary Problem
  • Lebesgue Point

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. J.P. Aubin, Un théorème de compacité. C. R. Acad. Sci. Paris Sér. I 256, 5042–5044 (1963)

    MathSciNet  MATH  Google Scholar 

  2. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces (Editura Academiei-Noordhoff International Publishing, Bucureşti-Leyden, 1976)

    CrossRef  MATH  Google Scholar 

  3. V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems (Academic, New York, 1993)

    MATH  Google Scholar 

  4. V. Barbu, Partial Differential Equations and Boundary Value Problems (Kluwer Academic, Dordrecht, 1998)

    MATH  Google Scholar 

  5. V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces (Springer, New York, 2010)

    CrossRef  Google Scholar 

  6. V. Barbu, A. Favini, Periodic problems for degenerate differential equations. Rend. Istit. Mat. Univ. Trieste Suppl. XXVIII, 29–57 (1997)

    Google Scholar 

  7. V. Barbu, A. Favini, Periodic solutions to degenerate second order differential equations in Hilbert spaces. Commun. Appl. Anal. 2, 19–29 (1998)

    MathSciNet  MATH  Google Scholar 

  8. I. Borsi, A. Farina, R. Gianni, M. Primicerio, Continuous dependence on the constitutive functions for a class of problems describing fluid flow in porous media. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl. 20(1), 1–24 (2009)

    Google Scholar 

  9. H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert (North-Holland, Amsterdam, 1973)

    MATH  Google Scholar 

  10. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, Berlin, 2011)

    MATH  Google Scholar 

  11. P. Broadbridge, I. White, Constant rate rainfall infiltration: A versatile nonlinear model, 1. Analytic solution. Water Resour. Res. 24(1), 145–154 (1988)

    CrossRef  Google Scholar 

  12. C. Ciutureanu, G. Marinoschi, Convergence of the finite difference scheme for a fast diffusion equation in porous media. Numer. Func. Anal. Optim. 29, 1034–1063 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. COMSOL Multiphysics v3.5a. Floating Network License 1025226, Comsol Sweden (2007)

    Google Scholar 

  14. A. Favini, G. Marinoschi, Existence for a degenerate diffusion problem with a nonlinear operator. J. Evol. Equ. 7, 743–764 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. A. Favini, G. Marinoschi, Periodic behavior for a degenerate fast diffusion equation. J. Math. Anal. Appl. 351(2), 509–521 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. A. Granas, J. Dugundji, Fixed Point Theory (Springer, New York, 2003)

    CrossRef  MATH  Google Scholar 

  17. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (Dunod, Paris, 1969)

    MATH  Google Scholar 

  18. J. L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I (Springer, Berlin, 1972)

    CrossRef  Google Scholar 

  19. G. Minty, Monotone (nonlinear) operators in Hilbert spaces. Duke Math. J. 29, 341–346 (1962)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. G. Marinoschi, Functional approach to nonlinear models of water flow in soils. Mathematical Modelling: Theory and Applications, vol. 21 (Springer, Dordrecht, 2006)

    Google Scholar 

  21. G. Marinoschi, Periodic solutions to fast diffusion equations with non Lipschitz convective terms. Nonlinear Anal. R. World Appl. 10, 1048–1067 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Favini, A., Marinoschi, G. (2012). Existence for Parabolic–Elliptic Degenerate Diffusion Problems. In: Degenerate Nonlinear Diffusion Equations. Lecture Notes in Mathematics, vol 2049. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28285-0_1

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