Title: Growth-fragmentation model for a population presenting heterogeneity in growth rate: Malthus parameter and long-time behavior Show Chinese title

Title: 具有生长率异质性的种群的生长分裂模型：马尔萨斯参数和长时间行为

Abstract: The goal of the present paper is to explore the long-time behavior of the growth-fragmentation equation formulated in the case of equal mitosis and variability in growth rate, under fairly general assumptions on the coefficients. The first results concern the monotonicity of the Malthus parameter with respect to the coefficients. Existence of a solution to the associated eigenproblem is then stated in the case of a finite set of growth rates thanks to Kre\u{i}n-Rutman theorem and a series of estimates on moments. Afterwards, adapting the classical general relative entropy (GRE) method enables us to ensure uniqueness of the eigenelements and derive the long-time asymptotics of the Cauchy problem. We prove convergence towards the steady state including in the case of individual exponential growth known to exhibit oscillations at large times in absence of variability. A few numerical simulations are eventually performed in the case of linear growth rate to illustrate our monotonicity results and the fact that variability, providing enough mixing in the heterogeneous population, is sufficient to re-establish asynchronicity. Show Chinese abstract

Abstract: 本文的目标是探讨在等分裂和生长速率变化的情况下，生长-分裂方程的长时间行为，在系数上采用相当一般的假设。 第一个结果涉及马尔萨斯参数相对于系数的单调性。 通过Kreĭn-Rutman定理以及对矩的一系列估计，在生长速率有限集的情况下，给出了相关特征问题解的存在性。 随后，通过适应经典的广义相对熵（GRE）方法，我们能够确保特征元的唯一性，并推导出柯西问题的长时间渐进行为。 我们证明了收敛到稳态的情况，包括在个体指数增长的情况下，已知在没有变化性时会在长时间内表现出振荡。 最后，在线性生长率的情况下进行了一些数值模拟，以说明我们的单调性结果以及变化性在异质种群中提供足够的混合时，足以重新建立不同步性。