We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative trait whose dynamics is described by a diffusion process. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show by probabilistic arguments that the semi-group of the stochastic trait dynamics admits a density. We deduce that the diffusion-growth-fragmentation equation admits a function solution with a certain Besov regularity.

我们研究了由退化扩散-生长-破碎方程和微分方程构成的非线性耦合系统解的存在性和唯一性，该系统是对恒化器中细菌生长的建模产生的。这个系统是从一个随机的基于个体的模型中派生出来的，在一个大的人口近似中，每个个体的特征都是一个非负面的特征，其动态由一个扩散过程来描述。突出显示了两个唯一性结果。他们在有关资源对个体特征动态影响的假设方面有所不同，主要困难是由于这种依赖性和扩散系数的退化而导致的非线性。此外，我们通过概率论证证明随机特征动力学的半群具有密度。