Nonnegative global solutions to a class of strongly coupled reaction-diffusion systems

A class of strongly coupled reaction-diffusion systems is studied. First, under some conditions, it is shown that a nonnegative solution exists globally in time. After that, asymptotic behavior of the nonnegative global solution is considered. Especially, when the solution uniformly converges to a steady state with a polynomial rate as time goes to infinity, large-time approximation of the solution is investigated. By the energy method and analytic semigroup theory, it is proved that a global solution for the corresponding system of ordinary differential equations has the role of an asymptotic solution for the reaction-diffusion system and that the spatial average of the global solution to the reaction-diffusion system gives an asymptotic description.