Strong Persistence of a Class of Strongly Coupled Parabolic Systems of $m$ Equations

We establish one of the most important assumptions of the strong persistence theory for dynamical systems associated to cross diffusion systems of $m$ equations ($m\ge 2$): the stable sets of semi-trivial steady cannot intersect the interior of the positive cone of $C(\mathrm{\Omega },{\mathbb{R}}^m)$. Many examples will be provided to show the effects of the cross diffusion.