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Degenerate and Singular Porous Medium Type Equations with Measure Data

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Elliptic and Parabolic Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 119))

Abstract

We consider the inhomogeneous porous medium equation

$$\partial_t u - \Delta u^m = \mu, \qquad m>\frac{(N-2)_+}{N},$$

and more general equations of porous medium type with a non-negative Radon measure μ on the right-hand side. In a first step, we prove a priori estimates for weak solutions in terms of a linear Riesz potential of the right-hand side measure, which takes exactly the same form as the one for the classical heat equation. Then, we give an optimal criterium for the continuity of weak solutions, again in terms of a Riesz potential. Finally, we prove the existence of non-negative very weak solutions, and show that these constructed very weak solutions satisfy the same estimates. We deal with both the degenerate case \(m>1\), and the singular case \(\frac{(N-2)_+}{N}<m<1\).

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Authors and Affiliations

  1. “F. Casorati” Department of Mathematics, University of Pavia, 27100, Pavia, Italy

    Ugo Gianazza

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  1. Ugo Gianazza
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Correspondence to Ugo Gianazza .

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Editors and Affiliations

  1. Institute for Applied Mathematics, Leibniz University of Hannover, Hannover, Germany

    Joachim Escher

  2. Institute for Analysis, Leibniz University of Hannover, Hannover, Germany

    Elmar Schrohe

  3. Department of Mathematics, University of Torino, Torino, Torino, Italy

    Jörg Seiler

  4. Institute for Applied Mathematics, Leibniz University of Hannover, Hannover, Germany

    Christoph Walker

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Gianazza, U. (2015). Degenerate and Singular Porous Medium Type Equations with Measure Data. In: Escher, J., Schrohe, E., Seiler, J., Walker, C. (eds) Elliptic and Parabolic Equations. Springer Proceedings in Mathematics & Statistics, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-12547-3_6

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