Initial blow-up of solutions of semilinear parabolic inequalities

We study classical nonnegative solutions u(x,t) of the semilinear parabolic inequalities where p is a positive constant and Ω is a bounded domain in , 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 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+.