  In this work we study the existence of classical solutions for a class of
reaction-diffusion systems with quadratic growth naturally arising in mass
action chemistry when studying networks of reactions of the type $A_i+A_j
\rightleftharpoons A_k$ with Fickian diffusion, where the diffusion
coefficients might depend on time, space and on all the concentrations $c_i$ of
the chemical species. In the case of one single reaction, we prove global
existence for space dimensions $N\leq 5$. In the more restrictive case of
diffusion coefficients of the type $d_i(c_i)$, we use an $L^2$-approach to
prove global existence for $N\leq 9$. In the general case of networks of such
reactions we extend the previous method to get global solutions for general
diffusivities if $N\leq 3$ and for diffusion of type $d_i(c_i)$ if $N\leq 5$.
In the latter quasi-linear case of $d_i(c_i)$ and for space dimensions $N=2$
and $N=3$, global existence holds for more than quadratic reactions. We can
actually allow for more general rate functions including fractional power
terms, important in applications. We obtain global existence under appropriate
growth restrictions with an explicit dependence on the space dimension $N$.
