Well‐posedness, blow‐up phenomena and analyticity for a two‐component higher order Camassa–Holm system

In this paper, the local well‐posedness for the Cauchy problem of a two‐component higher‐order Camassa–Holm system (2HOCH) is established in Besov spaces Bp,rs×Bp,rs−2 with 1≤p,r≤+∞ and s>2+1p,52 (and also in Sobolev spaces Hs×Hs−2=B2,2s×B2,2s−2 with s>5/2), which improves the corresponding re...

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Bibliographic Details
Published in:Mathematische Nachrichten 2018-07, Vol.291 (10), p.1595-1619
Main Author: Zhou, Shouming
Format: Article
Language:English
Subjects:
2HOCH
35G25
35L15
35Q53
35Q58
analyticity
Cauchy problems
Function space
Gevrey regularity
local well‐posedness
Sobolev space
wave breaking
Well posed problems
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Summary:In this paper, the local well‐posedness for the Cauchy problem of a two‐component higher‐order Camassa–Holm system (2HOCH) is established in Besov spaces Bp,rs×Bp,rs−2 with 1≤p,r≤+∞ and s>2+1p,52 (and also in Sobolev spaces Hs×Hs−2=B2,2s×B2,2s−2 with s>5/2), which improves the corresponding results for higher‐order Camassa–Holm in , where the Sobolev index s=3,s>7/2,s≥7/2 is required, respectively. Then the precise blow‐up mechanism and global existence for the strong solutions of 2HOCH are determined in the lowest Sobolev space Hs×Hs−2 with s>5/2. Finally, the Gevrey regularity and analyticity of the 2HOCH are presented.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201600469