Monatshefte für Mathematik

, Volume 133, Issue 1, pp 1–82 | Cite as

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

  • J. A. Carrillo
  • A. Jüngel
  • P. A. Markowich
  • G. Toscani
  • A. Unterreiter

Abstract.

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.

2000 Mathematics Subject Classification: 35B40, 35K55, 35K65, 35Q35, 35R45 
Key words: Degenerate diffusion, asymptotic behavior, rates of decay, differential inequalities, degenerate parabolic systems, high-order diffusion 

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Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • J. A. Carrillo
    • 1
  • A. Jüngel
    • 2
  • P. A. Markowich
    • 3
  • G. Toscani
    • 4
  • A. Unterreiter
    • 5
  1. 1.Universidad de Granada, SpainES
  2. 2.Universität Konstanz, GermanyDE
  3. 3.Universität Wien, AustriaAT
  4. 4.Universitá di Pavia, ItalyIT
  5. 5.Universität Kaiserslautern, GermanyDE

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