Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth

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.