The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely$$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$with \(0 < a_{{+}} < a_{{-}}\). We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

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关于临界增长-破碎半群和折射Lévy过程族

生长碎片方程对随着时间推移而生长和分裂的粒子系统进行建模。一个重要的问题涉及其解决方案的长时间渐近性。Doumic和Escobedo（Kinet。Relat。Models，9（2）：251–297，[12]）观察到，当增长是质量的线性函数并且碎片是均匀的时，所谓的马尔萨斯行为就失败了。在这项工作中，我们通过考虑分段线性增长来进一步分析临界情况，即$$ c（x）= \ textstyle \ begin {cases} a _ {{-}} x \ quad x <1 \\ a _ {{+} } x \ quad x \ geq 1，\ end {cases} $$与\（0 <a _ {{+}} <a _ {{-}} \）。我们为系数提供了充要条件，以确保马尔萨斯行为以指数速度收敛到渐近曲线，并提供后者的明确表示。我们的方法主要依赖于在这种情况下自然产生的所谓的折射Lévy过程的属性。