Life Span of Solutions for a Semilinear Parabolic Problem with Small Diffusion

This paper is concerned with the initial boundary value problem  ut = ε u + u p−1u in × 0 ∞ , u x t =0 on ∂ × 0 ∞ , u x 0 = ϕ x in , where p > 1, ε > 0, is a bounded domain in RN, and ϕ is a continuous function on .It is shown that the blowup time T ε of the solution of this problem satisfies T ε → 1 p−1 ϕ 1−p ∞ as ε → 0.Moreover, when the maximum of ϕ x is attained at one point, we determine the higher order term of T ε which reflects the pointedness of the peak of ϕ .The proof is based on a careful construction of super- and subsolutions.