Reaction-Diffusion-Advection Systems with Discontinuous Diffusion and Mass Control

TL;DR

This paper establishes the existence and boundedness of solutions for reaction-diffusion-advection systems with discontinuous coefficients, using novel energy functionals, and applies results to infectious disease models.

Contribution

It introduces a new class of $L^p$-energy functionals for quasi-positive systems satisfying an intermediate sum condition, enabling uniform bounds and global existence results.

Findings

Solutions are globally bounded and unique under minimal regularity assumptions.

Mass dissipation leads to solutions bounded uniformly in time.

Applicable to models of infectious disease spread with complex boundary conditions.

Abstract

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.