关键词: 耦合抛物方程组, 生命周期, 比较原理, Kaplan方法

Abstract:

We are concerned with the life span of solutions of the initial-boundary value problem 
u_t-Δu=e^{av,(x,t)∈Ω×(0,T),
v_t-Δv=ebu,(x,t)∈Ω×(0,T),u(x,t)=v(x,t)=0,(x,t)∈∂Ω×(0,T),
u(x,t)=ρφ(x),v(x,t)=ρψ(x),(x,t)∈Ω×{t=0},
Here a>0, b>0 are constants, Ω is a bounded domain in RN with smooth boundary ∂Ω, ρ>0 is a parameter, and φ(x) and ψ(x) are nonnegative continuous functions on Ω. To this end, we first deduce a asymptotic lower bound on its life span by constructing a upper solution of the above initial-boundary value problem which is based on the analysis of a new ODE system, then by the comparison principle and Kaplan's method[3], we can further show that such a lower bound is indeed a asymptotic upper bound and thus we obtain the asymptotic expression of the life span of the solutions for the problem concerned.}

Key words: Coupled parabolic system, Life span, Comparison principle, Kaplan's method