Initial blow-up of solutions of semilinear parabolic inequalities

• Published in: Journal of Differential Equations 2011-01, Vol.250 (2), p.892-928
• Main Author: Taliaferro, Steven D.
• Format: Article
• Language: English
• Subjects: Initial blow-up; Semilinear parabolic inequalities
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2010.07.033

We study classical nonnegative solutions u ( x , t ) of the semilinear parabolic inequalities 0 ⩽ u t − Δ u ⩽ u p in Ω × ( 0 , 1 ) where p is a positive constant and Ω is a bounded domain in R n , 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 max x ∈ K u ( 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 + .