# Local and global analyticity for $\mu$-Camassa-Holm equations

## TL;DR

This paper establishes local and global analyticity results for various $mbda$-Camassa-Holm equations, proving local solvability and global existence of solutions in the analytic category, which is a novel achievement for these equations.

## Contribution

It provides the first global-in-time analytic solutions for several $mbda$-Camassa-Holm equations, including $mbda$CH, $mbda$DP, and higher-order variants, with lifespan estimates.

## Key findings

- Proved unique local solvability of the Cauchy problems.
- Established existence of global-in-time analytic solutions for certain $mbda$-Camassa-Holm equations.
- Provided lifespan estimates for solutions.

## Abstract

We solve Cauchy problems for some $\mu$-Camassa-Holm integro-partial differential equations in the analytic category. The equations to be considered are $\mu$CH of Khesin-Lenells-Misio\l{}ek, $\mu$DP of Lenells-Misio\l{}ek-Ti\u{g}lay, the higher-order $\mu$CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for $\mu$CH, $\mu$DP and the higher-order $\mu$CH. The present work is the first result of such a global nature for these equations. AMS subject classification: 35R09, 35A01, 35A10, 35G25