# Global solutions of continuous coagulation-fragmentation equations with unbounded coefficients

## TL;DR

This paper establishes the existence of global classical solutions for continuous coagulation-fragmentation equations with unbounded coefficients, relaxing previous growth restrictions by analyzing the fragmentation semigroup's analyticity.

## Contribution

It introduces a new approach based on the analyticity of the fragmentation semigroup to prove global solutions without polynomial growth bounds.

## Key findings

- Proved existence of global classical solutions under weaker conditions.
- Showed weak solutions coincide with classical solutions when they exist.
- Extended the understanding of coagulation-fragmentation dynamics with unbounded coefficients.

## Abstract

In this paper we prove the existence of global classical solutions to continuous coagulation-fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation-fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation--fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.