Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle

Kazuo Kobayashi, Hiroki Ohwa

    Research output: Contribution to journalArticle

    23 Citations (Scopus)

    Abstract

    We study the comparison principle for anisotropic degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and super-solution, which immediately deduces the L1 contractivity and therefore, uniqueness of entropy solutions. The method used here is based upon the kinetic formulation and the kinetic techniques developed by Lions, Perthame and Tadmor. By adapting and modifying those methods to the case of Dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property. Moreover, in the quasi-isotropic case the existence of entropy solutions is proved.

    Original languageEnglish
    Pages (from-to)137-167
    Number of pages31
    JournalJournal of Differential Equations
    Volume252
    Issue number1
    DOIs
    Publication statusPublished - 2012 Jan 1

    Fingerprint

    Degenerate Parabolic Equation
    Entropy Solution
    Rectangle
    Existence and Uniqueness
    Kinetic Formulation
    Boundary conditions
    Contractivity
    Nonhomogeneous Boundary Conditions
    Supersolution
    Subsolution
    Entropy
    Comparison Principle
    Comparison Theorem
    Boundary Problem
    Hyperbolic Equations
    Dirichlet Problem
    Parabolic Equation
    Immediately
    Deduce
    Uniqueness

    Keywords

    • Anisotropic
    • Comparison theorem
    • Degenerate parabolic equation
    • Dirichlet boundary problem
    • Kinetic formulation
    • Uniqueness and existence

    ASJC Scopus subject areas

    • Analysis

    Cite this

    Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle. / Kobayashi, Kazuo; Ohwa, Hiroki.

    In: Journal of Differential Equations, Vol. 252, No. 1, 01.01.2012, p. 137-167.

    Research output: Contribution to journalArticle

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