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

We investigate the initial-boundary problem { u t = Δ u + f ( u ) in Ω × ( 0 , ∞ ) , u = 0 on ∂ Ω × ( 0 , ∞ ) , u ( x , 0 ) = ρ φ ( x ) in Ω , where Ω is a bounded domain in R N 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 ‖ ∞ ∞ d u 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 | Δ φ | at the maximal points of φ. The proof is based on a careful construction of a supersolution and a subsolution.