# Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation

## TL;DR

This paper derives an explicit solution for a critical growth-fragmentation equation, revealing detailed asymptotic behavior and periodicity phenomena, especially in the binary fission case.

## Contribution

It provides an explicit formula for the critical growth-fragmentation equation and connects it to previous asymptotic results, clarifying periodicity emergence.

## Key findings

- Explicit solution formula for critical growth-fragmentation equation
- Analysis of asymptotic behavior and periodicity
- Application to binary fission case (α=2)

## Abstract

We give here an explicit formula for the following critical case of the growth-fragmentation equation $$\frac{\partial}{\partial t} u(t, x) + \frac{\partial}{\partial x} (gxu(t, x)) + bu(t, x) = b\alpha^2 u(t, \alpha x), \qquad u(0, x) = u\_0 (x),$$ for some constants $g > 0$, $b > 0$ and $\alpha > 1$ - the case $\alpha = 2$ being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in (Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how periodicity may appear asymptotically.