  In this paper, we study unique, globally defined uniformly bounded weak
solutions for a class of semilinear reaction-diffusion-advection systems. The
coefficients of the differential operators and the initial data are only
required to be measurable and uniformly bounded. The nonlinearities are
quasi-positive and satisfy a commonly called mass control or dissipation of
mass property. Moreover, we assume the intermediate sum condition of a certain
order. The key feature of this work is the surprising discovery that
quasi-positive systems that satisfy an intermediate sum condition automatically
give rise to a new class of $L^p$-energy type functionals that allow us to
obtain requisite uniform a priori bounds. Our methods are sufficiently robust
to extend to different boundary conditions, or to certain quasi-linear systems.
We also show that in case of mass dissipation, the solution is bounded in
sup-norm uniformly in time. We illustrate the applicability of results by
showing global existence and large time behavior of models arising from a
spatio-temporal spread of infectious disease.

Analysis of PDEs