Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation

TL;DR

This paper develops a generalized relative entropy inequality for the growth-fragmentation equation, demonstrating convergence to steady-state solutions for measure-valued initial data, thus extending entropy methods to broader structured population models.

Contribution

It introduces a generalized entropy inequality for measure solutions of the growth-fragmentation equation, broadening the applicability of entropy methods in structured population dynamics.

Findings

Proves asymptotic convergence to steady state for measure initial data.

Extends entropy methods to growth-fragmentation equations with measure solutions.

Abstract

The aim of this study is to generalise recent results of the two last authors on en-tropy methods for measure solutions of the renewal equation to other classes of structured population problems. Specifically, we develop a generalised relative entropy inequality for the growth-fragmentation equation and prove asymptotic convergence to a steady-state solution, even when the initial datum is only a non-negative measure.