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Critical exponent and critical blow-up for quasilinear parabolic equations

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Abstract

We consider nonnegative solutions to the Cauchy problem or to the exterior Dirichlet problem for the quasilinear parabolic equationsu t=Δu m+up with 1<m<p. In case of the Cauchy problem, it is well known thatp *m =m+2/N is the critical exponent of blow-up. Namely, ifp<p *m , then all nontrivial solutions blow up in finite time (blow-up case), and ifp>p *m , then there are nontrivial global solutions (global existence case). In this paper we show: (i) For the Cauchy problem,p * m belongs to the blow-up case. (ii) For the exterior Dirichlet problem,p * m also gives the critical exponent of blow-up.

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Author notes

  1. Ryuichi Suzuki

    Present address: Department of Mathematics, Faculty of Arts, Kokushikan University, Machida, 195, Tokyo, Japan

Authors and Affiliations

  1. Department of Mathematics, Tokyo Metropolitan University, Hachioji, 192-03, Tokyo, Japan

    Kiyoshi Mochizuki

  2. Department of Mathematics, Tokyo Metropolitan College of Aeronautical Engineering, Arakawa, 116, Tokyo, Japan

    Ryuichi Suzuki

Authors
  1. Kiyoshi Mochizuki
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  2. Ryuichi Suzuki
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Correspondence to Kiyoshi Mochizuki.

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Mochizuki, K., Suzuki, R. Critical exponent and critical blow-up for quasilinear parabolic equations. Israel J. Math. 98, 141–156 (1997). https://doi.org/10.1007/BF02937331

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  • DOI: https://doi.org/10.1007/BF02937331

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