Fast propagation for reaction-diffusion cooperative systems

This paper deals with the spatial propagation for reaction-diffusion cooperative systems. It is well-known that the solution of a reaction-diffusion equation with monostable nonlinearity spreads at a finite speed when the initial condition decays to zero exponentially or faster, and propagates fast when the initial condition decays to zero more slowly than any exponentially decaying function. However, in reaction-diffusion cooperative systems, a new possibility happens in which one species propagates fast although its initial condition decays exponentially or faster. The fundamental reason is that the growth sources of one species come from the other species. Simply speaking, we find a new interesting phenomenon that the spatial propagation of one species is accelerated by the other species. This is a unique phenomenon in reaction-diffusion systems. We present a framework of fast propagation for reaction-diffusion cooperative systems.