Well-posedness and large time behavior of a size-structured growth-coagulation-fragmentation model
Saroj Si, Ankik Kumar Giri
TL;DR
The paper establishes well-posedness for a size-structured growth-coagulation-fragmentation model with a renewal boundary under unbounded coagulation and fragmentation kernels by developing a weak $L^{1}$-compactness framework in the weighted space $Y=L^{1}([0,\infty);(1+u)du)$. The existence result is obtained via truncated problems, uniform moment bounds, tail-control with a de la Vallée-Poussin function, and compactness arguments culminating in a weak solution; uniqueness follows from a stability estimate proved through regularization. In addition, the authors analyze long-time behavior, showing decay of $M_{0}$ and/or $M_{1}$ under diffusion-dominated or coagulation-dominated regimes, depending on kernel structure and fragmentation presence. The results extend prior work by allowing unbounded kernels and providing explicit large-time decay rates, with implications for phytoplankton aggregation and related biological systems. The combination of weak compactness techniques and a careful handling of renewal boundary conditions yields a robust framework for the GCF model on unbounded domains. $M_{0}$ and $M_{1}$ denote zeroth and first moments, respectively, and the framework accommodates physically relevant kernels such as linear/nonlinear shear and gravitational coagulation kernels.
Abstract
The existence and uniqueness of weak solutions to a size-structured growth-coagulation-fragmentation (GCF) equation with a renewal boundary condition are shown for a class of unbounded coagulation and fragmentation kernels. The existence proof is based on a weak compactness framework in the weighted $L^1$-space. This result extends the existence results of Banasiak and Lamb [14] and Ackleh et al. [2,4]. Furthermore, we establish a stability result and derive uniqueness as a direct consequence of it. Moreover, this study explores the large time behavior of weak solutions.