Solutions with peaks for a coagulation-fragmentation equation. Part II: aggregation in peaks

TL;DR

This paper proves that solutions to a specific coagulation-fragmentation equation with peaked initial data tend to stable stationary solutions over time, demonstrating the stability of solutions concentrated in Dirac masses.

Contribution

It establishes the asymptotic stability of peaked stationary solutions for a class of coagulation-fragmentation equations with near-diagonal kernels.

Findings

Solutions with concentrated initial data approach stationary solutions over time

Stability of solutions with Dirac mass concentration is demonstrated

Asymptotic decay of tails of stationary solutions is confirmed

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. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In this paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.