Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa-Holm system

TL;DR

This paper investigates the mathematical properties of a multi-component Camassa-Holm system, establishing conditions for well-posedness, global solutions, and blow-up phenomena, thereby advancing understanding of its integrability and solution behavior.

Contribution

The paper proves local well-posedness, provides continuation criteria, and presents new global existence and blow-up results for specific two-component subsystems of the integrable system.

Findings

Established local well-posedness and continuation criteria.

Derived global existence results for certain subsystems.

Identified conditions leading to solution blow-up.

Abstract

This paper is concerned with a multi-component Camassa-Holm system, which has been proven to be integrable and has peakon solutions. This system includes many one-component and two-component Camassa-Holm type systems as special cases. In this paper, we first establish the local well-posedness and a continuation criterion for the system, then we present several global existence or blow-up results for two important integrable two-component subsystems. Our obtained results cover and improve recent results in \cite{Gui,yan}.