On the Global Existence of a Class of Strongly Coupled Parabolic Systems

We establish the existence of strong solutions to a class of cross diffusion systems on $\RR^N$ consists of $m$ equations ($m,N\ge 2$). which generalizes the Shigesada-Kawasaki-Teramoto (SKT) model in population dynamics. We introduce the concept of a {\em strong-weak solution} of the systems and show that their existence can be established under weaker conditions. These {\em strong-weak solutions} coincide with strong solutions so that the existence of strong solutions is proved. The SKT model on planar domains ($N=2$) with cubic diffusions and advections is completely solved.