Initial trace of positive solutions to semilinear parabolic inequalities

Abstract We study the existence and properties of the initial trace, at t = 0, of positive solutions of ut - ∆u + g(x, t, u) ≥ 0 in Ω × (0, T), where g(x, t, u) > 0 for (x, t ) ∈ Ω × (0, T) and u > 0. We prove that, under some general conditions on g , this initial trace is a Borel measure (which may blow up on compact sets). In addition we derive pointwise estimates for the blow up of solutions at initial singular points. Finally these results are applied to the cases g(., t, r) = tα|r|q-1r (α > 0 ) , and g(., t, r ) = e-1/t|r|q-1r. In this last case we exhibit a new type of full initial blow-up.