# Solutions with peaks for a coagulation-fragmentation equation. Part I: stability of the tails

## TL;DR

This paper analyzes the stability of tail behaviors in stationary solutions to a coagulation-fragmentation equation with near-diagonal kernels, demonstrating their asymptotic decay stability.

## Contribution

It constructs a family of stationary solutions with Dirac masses and proves their tail stability under specific kernel assumptions.

## Key findings

- Stationary solutions are concentrated in Dirac masses.
- Tail decay behavior is stable over time.
- Solutions tend to these stationary states for concentrated initial data.

## Abstract

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In a companion paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.