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Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

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Nonlinear Evolution Equations and Related Topics

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Abstract

We consider the problem \( b\left( u \right) - \Delta u + div F\left( u \right) = f\) in a smooth bounded domain \( \Omega \subset {\mathbb{R}^N}\), as well as the corresponding evolution equation \( b{\left( u \right)_t} - \Delta u + div F\left( u \right) = f\), \( b\left( {u\left( {0,.} \right)} \right) = {b^0}.\). For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L 1-contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L 1 -contraction principle for the associated evolution equation.

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Authors and Affiliations

  1. Centre de Mathématiques et Informatique, Université de Provence, 39, rue F. Joliot-Curie, Marseille, 13453, France

    Boris P. Andreianov

  2. Laboratoire de Mathématiques, Université de Franche-Comte, 16, route de Gray, 25030, Besançon, France

    Fouzia Bouhsiss

Authors
  1. Boris P. Andreianov
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  2. Fouzia Bouhsiss
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Editor information

Editors and Affiliations

  1. Department of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Wolfgang Arendt

  2. Analyse Numérique, Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    Haïm Brézis

  3. Department of Mathematics Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brézis

  4. Campus de Ker Lann, Antenne de Bretagne de l’ENS Cachan, 35170, Bruz, France

    Michel Pierre

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Dedicated to the memory of Philippe Bénilan

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Andreianov, B.P., Bouhsiss, F. (2004). Uniqueness for an elliptic-parabolic problem with Neumann boundary condition. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_37

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  • DOI: https://doi.org/10.1007/978-3-0348-7924-8_37

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-7107-4

  • Online ISBN: 978-3-0348-7924-8

  • eBook Packages: Springer Book Archive

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