  We establish several bounds for solutions to elliptic/parabolic
cross-diffusion systems of $m$ equations ($m\ge2$) on 2d/3d domains $\Og$. We
settle the existence and global existence problems in these cases and also
provide new counter-examples for the case of general dimensions. Most
importantly, we prove that when $m=N=3$, the thinness of $\Og$ in $\RR^3$ is
sufficient and necessary. When $m$ is arbitrary and $N=3$, we establish global
existence results for nonlinear cross-diffusion systems (the case of the scalar
semilinear equation was considered in the literature but the classical methods
do not seem to be applicable here).
