  We study the global existence of classical solutions to cross diffusion
systems of $m$ equations on $N$-dimensional domains ($m,N\ge2$). The diffusion
matrix is a triangular block matrix with coupled entries. We establish that the
$W^{1,p}$ norm of solutions for some $p>N$ does not blow up in finite time so
that the results in \cite{Am2} is applicable. We will also show that the
spectral gap condition in \cite{dlebook,dlebook1} can be relaxed via a new
result on BMO norms in \cite{dleBMO}.
