The local well-posedness, blow-up and global solution of a new integrable system in Besov spaces

TL;DR

This paper investigates the mathematical properties of a new integrable system, establishing conditions for well-posedness, blow-up, and global solutions in Besov spaces, supported by simulations.

Contribution

It introduces a new integrable system and provides a comprehensive analysis of its well-posedness, blow-up criteria, and global solutions within Besov spaces, including numerical simulations.

Findings

Established local well-posedness in critical and supercritical Besov spaces.

Derived a precise blow-up criterion for the system.

Proved global existence under a sign condition.

Abstract

In this paper, we first establish the local well-posednesss for the Cauchy problem of a $N$-peakon system in the sense of Hadamard in both critical Besov spaces and supercritical Besov spaces. Second, we gain a blow-up criterion. According to blow-up criterion wemreach a precise blow-up criterion. Under a sign condition, we reach the existence of global solution. Finally, based on the first-order difference method, we give a simulation example of the blow-up of the equation and the properties of the global solution.