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Global Existence vs. Blowup in a One-dimensional Smoluchowski-Poisson System

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Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 80))

Abstract

We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one-dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.

Mathematics Subject Classification (2000). 35B44, 35K20, 35K59.

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Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Warsaw University, Banacha 2, PL-02-097, Warszawa, Poland

    Tomasz Cieślak

  2. Institut de Mathématiques de Toulouse CNRS UMR 5219, Université de Toulouse, 118 route de Narbonne, F-31062, Toulouse Cedex 9, France

    Philippe Laurençot

Authors
  1. Tomasz Cieślak
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  2. Philippe Laurençot
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Corresponding author

Correspondence to Tomasz Cieślak .

Editor information

Editors and Affiliations

  1. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Joachim Escher

  2. School of Physical Sciences, Dept. Mathematics, University of California, Irvine, Irvine, 92697-3875, California, USA

    Patrick Guidotti

  3. , Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr 7, Darmstadt, 64289, Germany

    Matthias Hieber

  4. Inst. Applied Mathematics & Mechanics, Warsaw University, ul. Banacha 2, Warszawa, 02-097, Poland

    Piotr Mucha

  5. , FB Mathematik und Informatik, Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, Halle, 06099, Germany

    Jan W. Prüss

  6. 3-4-1, Ohkubo Shinjuku-ku, Tokyo, 169-8555, Japan

    Yoshihiro Shibata

  7. , Deparment of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA

    Gieri Simonett

  8. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Christoph Walker

  9. Fac. Mathematics & Information Sciences, Warsaw University of Technology, pl. Politechniki 1, Warszawa, 00-661, Poland

    Wojciech Zajaczkowski

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Dedicated to Herbert Amann

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Cieślak, T., Laurençot, P. (2011). Global Existence vs. Blowup in a One-dimensional Smoluchowski-Poisson System. In: Escher, J., et al. Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel. https://doi.org/10.1007/978-3-0348-0075-4_6

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