The Role of Critical Exponents in Blow-Up Theorems: The Sequel

Abstract In [ 27 ] Fujita showed that for positive solutions, the initial value problem (in R N ) for u t  = Δ u  +  u p with p  > 1 exhibited the following behavior: If p p c  ≡ 1 + 2/ N , then the initial value problem does not have any nontrivial, non-negative solution existing on R N  × [0, ∞) (a global solution), whereas if p  >  p c , there exist global, small data, positive solutions as well as solutions which are non-global. We call such a result a blow-up theorem of Fujita type. In [ 50 ], Levine discussed the various theorems of this type that had appeared in the literature prior to 1990. In this paper we revisit the literature since 1990.

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