Life span of solutions with large initial data for a semilinear parabolic system

We investigate the initial–boundary value problem{ut=Δu+vpin Ω×(0,T),vt=Δv+uqin Ω×(0,T),u(x,t)=v(x,t)=0on ∂Ω×(0,T),u(x,0)=ρφ(x),v(x,0)=ρψ(x)in Ω, where p,q⩾1p,q⩾1 and pq>1pq>1, Ω   is a bounded domain in RNRN with a smooth boundary ∂Ω  , ρ>0ρ>0 is a parameter, φ(x)φ(x) and ψ(x)ψ(x) are nonnegative continuous functions on Ω¯. We show that the life span (or blow-up time) of the solution of this problem approaches the life span of the solution of the ODE system obtained when dropping the diffusion terms as ρ→∞ρ→∞. The proof is based on the comparison principle and Kaplanʼs method.