Long-time Asymptotics for Nonlinear Growth-fragmentation Equations

TL;DR

This paper investigates the long-term behavior of nonlinear growth-fragmentation equations, demonstrating convergence to steady states or periodic solutions using entropy methods, stability analysis, and bifurcation theory.

Contribution

It introduces a novel approach combining entropy methods and bifurcation analysis to characterize asymptotic behaviors of nonlinear growth-fragmentation equations.

Findings

Convergence to steady states demonstrated in certain cases.

Existence of periodic solutions proved via bifurcation methods.

Reduction of PDE to ODE system for asymptotic analysis.

Abstract

We are interested in the long-time asymptotic behavior of growth-fragmentation equations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Thanks the General Relative Entropy method applied to well chosen self-similar solutions, we show that the equation can "asymptotically" be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and existence of periodic solutions are proved thanks to the Poincar\'e-Bendixon theorem or by Hopf bifurcation.