Harnack-type Inequalities for some Degenerate Parabolic Equations

  • E. Di Benedetto
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 12)


We give here a brief account on some recent results [14] concerning an intrinsic version of the Harnack inequality for nonnegative solutions of degenerate parabolic equations of the type
$$\begin{gathered} {u_t} - div(|Du{|^{p - 2}}Du) = 0,{\text{ }}p > 2{\text{ }}in{\text{ }}D'({\Omega _T}) \hfill \\ {\text{ }}u \in C(0,T;{L^2}(\Omega )) \cap {L^p}(0,T:{W^{1,p}}(\Omega )), \hfill \\ \end{gathered}$$
and of the type
$$\begin{gathered} {u_t} - \Delta {u^m} = 0,{\text{ }}m > 1{\text{ }}in{\text{ }}D'({\Omega _T}) \hfill \\ u \in C(0,T;{L^2}(\Omega ));{u^m} \in {L^2}(0,T;;{W^{1,2}}(\Omega )), \hfill \\ \end{gathered}$$
where Ω is an open set in ℝ N , 0 < T < ∞, Ω T = Ω × (0,T] and D denotes the gradient with respect to the space variables only.


Weak Solution Parabolic Equation Local Behavior Harnack Inequality Nonnegative Solution 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D.G. Aronson and L.A. Caffarelli, The initial trace of a solution of the porous media equation, Trans. AMS 280, No. 1 (1983), 351–366.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    D.G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), 81–123.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Math. Meh. 16 (1952), 67–78.MATHMathSciNetGoogle Scholar
  4. 4.
    M. Bertsch and L.A. Peletier, A positivity property of solutions of nonlinear diffusion equations, Jour. of Diff. Equ. 53, No. 1 (1984), 30–47.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    B.E. Dahlberg and C.E. Kenig, Nonlinear filtration.Google Scholar
  6. 6.
    E. Di Benedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22.CrossRefMathSciNetGoogle Scholar
  7. 7.
    E. Di Benedetto, On the local behavior of solutions of degenerate parabolic equations with measurable coefficients, Ann. Sc. Norm. Sup. Pisa, Ser. IV 13, No. 3 (1986), 487–535.Google Scholar
  8. 8.
    J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo 3, Ser. 2 (1954), 337–346.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    N.V. Krylov and M.V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izvestija 16, No. 1 (1981), 151–164.CrossRefMATHGoogle Scholar
  10. 10.
    J. Moser, A Harnack inequality for parabolic differential equations, comm. Pure Appl. Math. 17 (1964), 101–134.CrossRefMATHGoogle Scholar
  11. 11.
    J. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta Math. 111 (1964), 302–347.CrossRefMathSciNetGoogle Scholar
  12. 12.
    N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic partial differential equations, Comm. Pure Appl. Math 20 (1967), 721–747.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    E. Di Benedetto, Intrinsic Harnack-type inequalities for solutions of certain degenerate parabolic equations, Archive for Rat. Mech. Anal. (to appear).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • E. Di Benedetto
    • 1
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

Personalised recommendations