Initial blow-up of solutions of semilinear parabolic inequalities

We study classical nonnegative solutions u(x,t)u(x,t) of the semilinear parabolic inequalities0⩽ut−Δu⩽upin Ω×(0,1) where p is a positive constant and Ω   is a bounded domain in RnRn, n⩾1n⩾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+t→0+ is p⩽1+2/np⩽1+2/n and in this case the bound on u ismaxx∈Ku(x,t)=O(t−n/2)as t→0+.If in addition, Ω is smooth, u   satisfies the boundary condition u=0u=0 on ∂Ω×(0,1)∂Ω×(0,1), and p<1+2/np<1+2/n, then we obtain a bound for u on the entire set Ω   as t→0+t→0+.