  We prove existence and uniqueness of global solutions for a class of
reaction-advection-anisotropic-diffusion systems whose reaction terms have a
"triangular structure". We thus extend previous results to the case of
time-space dependent anisotropic diffusions and with time-space dependent
advection terms. The corresponding models are in particular relevant for
transport processes inside porous media and in situations in which additional
migration occurs. The proofs are based on optimal $L^p$-maximal regularity
results for the general time-dependent linear operator dual to the one involved
in the considered systems. As an application, we prove global well-posedness
for a prototypical class of chemically reacting systems with mass-action
kinetics, involving networks of reactions of the type $C_1+\ldots+C_{P-1}
\rightleftharpoons C_{P} $. Finally, we analyze how a classical a priori
$L^2$-estimate of the solutions, which holds with this kind of nonlinear
reactive terms, extends to our general anisotropic-advection framework. It does
extend with the same assumptions for isotropic diffusions and is replaced by an
$L^{(N+1)/N}$-estimate in the general situation.
