Persistence property and the local well-posedness of the modified Camassa-Holm equation in critical Besov equation

TL;DR

This paper proves local well-posedness and persistence properties of the modified Camassa-Holm equation in critical Besov spaces, extending previous results and providing a deeper understanding of the equation's behavior in these function spaces.

Contribution

The paper establishes the local well-posedness of the modified Camassa-Holm equation in critical Besov spaces and demonstrates its persistence property, improving upon earlier results.

Findings

Local well-posedness in critical Besov spaces $B^{1/p}_{p,1}$ for $1 \\leq p < \\infty$.

Persistence property of solutions to the modified Camassa-Holm equation.

Significant extension of previous well-posedness results.

Abstract

In this paper, we first establish the local well-posednesss for the Cauchy problem of a modified Camassa-Holm (MOCH) equation in critical Besov spaces $B^{\frac 1 p}_{p,1}$ with $1\leq p<+\infty.$ The obtained results improve considerably the recent result in \cite{Luo1}. Then we show the persiscence property of MOCH.