Large-time approximation of a solution to a reaction-diffusion system with a balance law by its spatial average

A reaction-diffusion system which has a balance law is studied. Homogeneous Neumann boundary conditions are imposed. When a nonnegative global solution uniformly converges to a steady state with a polynomial rate as time goes to infinity, it is proved that the spatial average of the solution to the system describes a large-time approximation of the solution itself with an exponential rate being sharper than those obtained before. The proof is based on properties of a solution for the corresponding system of ordinary differential equations and an L estimate of an analytic semigroup.