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

Hideshi Yamane

doi:10.3934/dcds.2020182

Article Contents

2020, Volume 40, Issue 7: 4307-4340. Doi: 10.3934/dcds.2020182 open access

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

Hideshi Yamane

Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japan

Received: June 2019

Revised: December 2019

Published: April 2020

This work was partially supported by JSPS KAKENHI Grant Number 26400127

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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-Misiołek, $ \mu $DP of Lenells-Misiołek-Tiğlay, 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.

Keywords:

Ovsyannikov theorem for nonlocal equations,
Camassa-Holm equations,
analytic Cauchy problem,
global solvability.

Mathematics Subject Classification: Primary: 35R09, 35A10, 35G25; Secondary: 35A01.

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References

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