Article

Israel Journal of Mathematics

, 98:141

First online:

Critical exponent and critical blow-up for quasilinear parabolic equations

  • Kiyoshi MochizukiAffiliated withDepartment of Mathematics, Tokyo Metropolitan University Email author 
  • , Ryuichi SuzukiAffiliated withDepartment of Mathematics, Tokyo Metropolitan College of Aeronautical Engineering

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Abstract

We consider nonnegative solutions to the Cauchy problem or to the exterior Dirichlet problem for the quasilinear parabolic equationsu tu 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.