Discrete fragmentation equations with time-dependent coefficients

TL;DR

This paper studies a model of cluster fragmentation with time-varying coefficients, proving existence and uniqueness of solutions using advanced mathematical frameworks.

Contribution

It introduces a novel approach to analyze discrete fragmentation equations with time-dependent coefficients via non-autonomous evolution equations.

Findings

Established existence of solutions for time-dependent fragmentation models.

Proved uniqueness of classical solutions under certain conditions.

Extended the mathematical theory to non-autonomous systems in weighted spaces.

Abstract

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such discrete-size fragmentation models, we allow the fragmentation coefficients to vary with time. By formulating the initial-value problem for the system as a non-autonomous abstract Cauchy problem, posed in an appropriately weighted $\ell^1$ space, and then applying results from the theory of evolution families, we prove the existence and uniqueness of physically relevant, classical solutions for suitably constrained coefficients.