Life span of solutions with large initial data for a superlinear heat equation

We investigate the initial-boundary problem ut = u +f (u) in Ω ×(0,∞), u =0 on ∂Ω ×(0,∞), u(x, 0) = ρϕ(x) in Ω, where Ω is a bounded domain in RN with a smooth boundary ∂Ω, ρ > 0, ϕ(x) is a nonnegative continuous function on Ω, f (u) is a nonnegative superlinear continuous function on [0,∞). We show that the life span (or blow-up time) of the solution of this problem, denoted by T (ρ), satisfies T (ρ) = ∞ ρ ϕ ∞ du f (u) + h.o.t. as ρ→∞. Moreover, when the maximum of ϕ is attained at a finite number of points in Ω, we can determine the higher-order term of T (ρ) which depends on the minimal value of | ϕ| at the maximal points of ϕ. The proof is based on a careful construction of a supersolution and a subsolution. © 2008 Elsevier Inc. All rights reserved.