Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion

Min Zhu; Ying Wang

doi:10.3934/dcds.2017027

Article Contents

2017, Volume 37, Issue 1: 645-661. Doi: 10.3934/dcds.2017027

This issue Previous Article Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion Next Article

Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion

Min Zhu^1, and
Ying Wang^2,

1.
Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China
2.
Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 611731, China

Received: March 31, 2016

Revised: April 30, 2016

Published: September 2017

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Abstract

Considered herein is the blow-up mechanism to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. The first one is designed for the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. The second one handles the general situation when the weak linear dispersion is at present. Fortunately, there exist some conserved quantities that bound the $\|u_x\|_{L^4} $ for the periodic generalized modified Camassa-Holm equation, then the breakdown mechanisms are set up for the general case.

Keywords:

Periodic modified generalized Camassa-Holm equation,
blow up,
integrable equation,
sign-changing momentum,
varying linear dispersion.

Mathematics Subject Classification: 35B44, 35G25.

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References

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