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

Abstract

We investigate the initial–boundary value problem { u t = Δ u + v p in Ω × ( 0 , T ) , v t = Δ v + u q in Ω × ( 0 , T ) , u ( x , t ) = v ( x , t ) = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = ρ φ ( x ) , v ( x , 0 ) = ρ ψ ( x ) in Ω , where p , q ⩾ 1 and p q > 1 , Ω is a bounded domain in R N with a smooth boundary ∂ Ω, ρ > 0 is a parameter, φ ( x ) and 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 → ∞ . The proof is based on the comparison principle and Kaplanʼs method.