Generalized Entropy Method for the Renewal Equation with Measure Data

TL;DR

This paper extends the analysis of the renewal equation in population dynamics to include measure-valued initial data, using advanced calculus of variations and a refined entropy method to understand long-term behavior.

Contribution

It introduces a generalized entropy approach applicable to measure data, a novel application in the context of the renewal equation.

Findings

Long-time asymptotics established for measure initial data

Refined entropy method adapted for Radon measures

Enhanced understanding of population model dynamics

Abstract

We study the long-time asymptotics for the so-called McKendrick-Von Foerster or renewal equation, a simple model frequently considered in structured population dynamics. In contrast to previous works, we can admit a bounded measure as initial data. To this end, we apply techniques from the calculus of variations that have not been employed previously in this context. We demonstrate how the generalized relative entropy method can be refined in the Radon measure framework.