Initial blow-up of solutions of semilinear parabolic inequalities
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We study classical nonnegative solutions u(x,t) of the semilinear parabolic inequalities. 0≤ut-δu≤upin ω×(0,1) where p is a positive constant and ω is a bounded domain in Rn, n≤1. We show that a necessary and sufficient condition on p for such solutions u to satisfy a pointwise a priori bound on compact subsets K of ω as t→0+ is p≤1+2/n and in this case the bound on u is. maxx∈Ku(x,t)=O(t-n/2)as t→0+. If in addition, ω is smooth, u satisfies the boundary condition u=0 on ∂ω×(0,1), and p<1+2/n, then we obtain a bound for u on the entire set ω as t→0+. © 2010 Elsevier Inc.
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