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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)

Abstract

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
$$\begin{gathered} {u_t} - div(|Du{|^{p - 2}}Du) = 0,{\text{ }}p > 2{\text{ }}in{\text{ }}D'({\Omega _T}) \hfill \\ {\text{ }}u \in C(0,T;{L^2}(\Omega )) \cap {L^p}(0,T:{W^{1,p}}(\Omega )), \hfill \\ \end{gathered}$$
(1.1)
and of the type
$$\begin{gathered} {u_t} - \Delta {u^m} = 0,{\text{ }}m > 1{\text{ }}in{\text{ }}D'({\Omega _T}) \hfill \\ u \in C(0,T;{L^2}(\Omega ));{u^m} \in {L^2}(0,T;;{W^{1,2}}(\Omega )), \hfill \\ \end{gathered}$$
(1.2)
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.

Keywords

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.

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

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

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