Positivity and long-term behaviour of a diffusion model with measure-valued nonlocal reaction term

TL;DR

This paper analyzes the long-term positivity and convergence of solutions to a diffusion equation with a nonlocal, measure-valued reaction term, using Laplace transform techniques to identify parameter regimes ensuring stability and positivity.

Contribution

It establishes conditions under which solutions remain positive and converge to steady states in a diffusion model with singular, nonlocal reactions, extending understanding of such systems.

Findings

Identifies parameter regimes ensuring solution positivity for all time

Provides conditions for convergence to constant steady states

Uses Laplace transform methods for analysis

Abstract

The behaviour is investigated of solutions to a diffusion equation on the real line with nonlocal and singular reaction term, i.e., given by a Dirac source or sink at the origin. It gives a simplified representation of for example a control system that senses concentration at a distance, but "intervenes" at the origin. Positivity of solutions (for positive initial conditions) cannot be guaranteed for all parameter settings in the model. We determine a parameter regime and conditions on the positive initial condition in terms of monotonicity and symmetry, that do allow us to conclude the positivity of the solution for all time. In addition, we provide conditions that ensure convergence of the system to a constant steady state (pointwise), outside the region of observation. Technically, we extensively use Laplace transform arguments to achieve these results.