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Existence of weak solutions to doubly degenerate diffusion equations

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Abstract

We prove existence of weak solutions to doubly degenerate diffusion equations

$\dot u = \Delta _p u^{m - 1} + f(m,p \geqslant 2)$

by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains Ω ⊂ ℝn with Dirichlet or Neumann boundary conditions. The function f can be an inhomogeneity or a nonlinearity involving terms of the form f(u) or div(F(u)). In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.

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Authors and Affiliations

  1. Department of Mathematics, University of West-Bohemia, Univerzitní 22, 306 14, Pilsen, Czech Republic

    Aleš Matas

  2. Institut für Mathematik, Universität Rostock, Universitätsplatz 1, 180 51, Rostock, Germany

    Jochen Merker

Authors
  1. Aleš Matas
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  2. Jochen Merker
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Corresponding author

Correspondence to Aleš Matas.

Additional information

The authors were supported by the Ministry of Education, Youth, and Sports of the Czech Republic, Research Plan # MSM4977751301, and by the German DAAD.

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Matas, A., Merker, J. Existence of weak solutions to doubly degenerate diffusion equations. Appl Math 57, 43–69 (2012). https://doi.org/10.1007/s10492-012-0004-0

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  • DOI: https://doi.org/10.1007/s10492-012-0004-0

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