Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

TL;DR

This paper proves the asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, demonstrating exponential convergence to a stable state under broad conditions.

Contribution

It establishes the exponential growth behavior for solutions of the growth-fragmentation equation with unbounded rates, using moment creation and semigroup analysis.

Findings

Proves exponential convergence to a stable state

Demonstrates the creation of moments for solutions

Establishes quasi-compactness of the semigroup

Abstract

The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.