# Critical exponent and critical blow-up for quasilinear parabolic equations

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DOI: 10.1007/BF02937331

- Cite this article as:
- Mochizuki, K. & Suzuki, R. Israel J. Math. (1997) 98: 141. doi:10.1007/BF02937331

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## Abstract

We consider nonnegative solutions to the Cauchy problem or to the exterior Dirichlet problem for the quasilinear parabolic equations*u*_{t}=Δ*u*^{m}+u^{p} with 1<*m*<*p*. In case of the Cauchy problem, it is well known that*p*_{m}^{*}=m+2/N is the critical exponent of blow-up. Namely, if*p<p*_{m}^{*}, then all nontrivial solutions blow up in finite time (blow-up case), and if*p>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.