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Mathematische Annalen

, Volume 350, Issue 4, pp 801–827 | Cite as

Multiple blow-up for a porous medium equation with reaction

  • Noriko MizoguchiEmail author
  • Fernando Quirós
  • Juan Luis Vázquez
Article

Abstract

The present paper is concerned with the Cauchy problem
$$\left\{\begin{array}{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \end{array}\right.$$
with p, m > 1. A solution u with bounded initial data is said to blow up at a finite time T if \({{\lim {\rm sup}_{t \nearrow T}||u(t)||_{L^\infty(\mathbb{R}^N)} =\infty}}\). For N ≥ 3 we obtain, in a certain range of values of p, weak solutions which blow up at several times and become bounded in intervals between these blow-up times. We also prove a result of a more technical nature: proper solutions are weak solutions up to the complete blow-up time.

Mathematics Subject Classification (2000)

35K20 35K55 58K57 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Noriko Mizoguchi
    • 1
    • 2
    Email author
  • Fernando Quirós
    • 3
  • Juan Luis Vázquez
    • 4
  1. 1.Department of MathematicsTokyo Gakugei UniversityKoganeiJapan
  2. 2.Precursory Research for Embryonic Science and Technology (PRESTO)Japan Science and Technology Agency (JST)SaitamaJapan
  3. 3.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  4. 4.Departamento de Matemáticas and ICMATUniversidad Autónoma de MadridMadridSpain

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