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We establish several bounds for solutions to elliptic/paraboliccross-diffusion systems of $m$ equations ($m\ge2$) on 2d/3d domains $\Og$. Wesettle the existence and global existence problems in these cases and alsoprovide new counter-examples for the case of general dimensions. Mostimportantly, we prove that when $m=N=3$, the thinness of $\Og$ in $\RR^3$ issufficient and necessary. When $m$ is arbitrary and $N=3$, we establish globalexistence results for nonlinear cross-diffusion systems (the case of the scalarsemilinear equation was considered in the literature but the classical methodsdo not seem to be applicable here).

A priori Bound of Solutions to a Class of Elliptic/Parabolic Cross  Diffusion Systems of $m$ Equations on Two/Three Dimensional Domains.  Existence on thin domains