  We establish the global existence of a class of strongly coupled parabolic
systems. The necessary apriori estimates will be obtained via our new approach
to the regularity theory of parabolic scalar equations with integrable data and
new $W^{1,p}$ estimates of their solutions. The key assumption here is that the
$L^p$ norms of solutions are uniformly bounded for some sufficiently large
$p\in (1,\infty)$, an assumption can be easily affirmed for systems with
polynomial growth data. This replaces the usual condition that the solutions
are uniformly bounded which is very hard to be verified because maximum
principles for systems are generally unavailable.
