  In this paper, we study unique, globally defined uniformly bounded weak
solutions for a class of semilinear reaction-diffusion-advection systems. The
systems under consideration can exhibit a high degree of heterogeneity. Namely,
the coefficients of the differential operators and the initial data are only
required to be measureable 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. These functionals are mixed
polynomials of degree $p$ with well chosen coefficients based on the
intermediate sum condition. Although our primary focus is semilinear systems
with homogenous Dirichlet boundary conditions, our methods are sufficiently
robust to extend to other boundary conditions such as Neumann and Robin types,
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, reactive flows, and a model for the behavioral reaction of individuals
responding to a catastrophic event.
