  We consider the Neumann and Cauchy problems for positivity preserving
reaction-diffusion systems of $m$ equations enjoying the mass and entropy
dissipation properties. We show global classical existence in any space
dimension, under the assumption that the nonlinearities have at most quadratic
growth. This extends previously known results which, in dimensions $n\ge 3$,
required mass conservation and were restricted to the Cauchy problem. Our proof
is also simpler.
