Existence and uniqueness of solutions for coagulation-fragmentation problems with singularity

TL;DR

This paper proves the existence and uniqueness of solutions for a class of coagulation-fragmentation equations with singular and non-singular kernels, broadening the mathematical understanding of particle system evolution.

Contribution

It establishes existence and uniqueness results for coagulation-fragmentation problems with more general kernels, including singular and unbounded cases, under weaker conditions than previous studies.

Findings

Existence of solutions under broad kernel conditions

Uniqueness of solutions for the specified class of problems

Strong convergence results used to establish the theory

Abstract

In this paper, existence and uniqueness of solutions to a non-linear, initial value problem is studied. In particular, we consider a special type of problem which physically represents the time evolution of particle number density resulted due to the coagulation and fragmentation process. The coagulation kernel is chosen from a huge class of functions, both singular and non-singular in nature. On the other hand, fragmentation kernel includes practically relevant non-singular unbounded functions. Moreover, both the kernels satisfy a linear growth rate of particles at infinity. The existence theorem includes lesser restrictions over the kernels as compared to the previous studies. Furthermore, strong convergence results on the sequence of functions are used to establish the existence theory.