On the non existence of non negative solutions to a critical Growth-Fragmentation Equation

TL;DR

This paper investigates a growth-fragmentation equation with balanced growth and division rates, demonstrating that under certain conditions, no global non-negative solutions exist, and providing explicit solutions when they do.

Contribution

It proves the non-existence of global non-negative solutions for a specific growth-fragmentation model when the Malthusian hypothesis fails, and offers explicit solutions when local solutions exist.

Findings

No global non-negative solutions under certain conditions.

Non-existence of local solutions with bounded moments.

Explicit expression for local solutions when they exist.

Abstract

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $(M_+)$ of J. Bertoin and A. Watson (2016) is not necessarily satisfied. It is proved that when that happens, and as it was first suggested by these authors, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exist. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.