Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates

TL;DR

This paper investigates the long-term behavior of growth-fragmentation equations with variable drift, providing detailed eigenfunction estimates and proving exponential convergence to equilibrium under general power-law conditions.

Contribution

It offers new precise asymptotic estimates for eigenfunctions and establishes a spectral gap result for a broad class of growth-fragmentation models.

Findings

Eigenfunctions have detailed asymptotics near zero and infinity.

Solutions decay exponentially to equilibrium.

Results apply to general power-law growth and fragmentation rates.

Abstract

We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [Caceres, Canizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.