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

We investigate the initial-boundary problem{ut=Δu+f(u)inΩ×(0,∞),u=0on∂Ω×(0,∞),u(x,0)=ρφ(x)inΩ, where Ω   is a bounded domain in RNRN with a smooth boundary ∂Ω  , ρ>0ρ>0, φ(x)φ(x) is a nonnegative continuous function on Ω¯, f(u)f(u) is a nonnegative superlinear continuous function on [0,∞)[0,∞). We show that the life span (or blow-up time) of the solution of this problem, denoted by T(ρ)T(ρ), satisfies T(ρ)=∫ρ‖φ‖∞∞duf(u)+h.o.t. as ρ→∞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)T(ρ) which depends on the minimal value of |Δφ||Δφ| at the maximal points of φ. The proof is based on a careful construction of a supersolution and a subsolution.