American Institute of Mathematical Sciences

January  2017, 37(1): 645-661. doi: 10.3934/dcds.2017027

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

 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  April 2016 Revised  May 2016 Published  November 2016

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.

Citation: Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
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