# American Institute of Mathematical Sciences

December  2016, 36(12): 7235-7256. doi: 10.3934/dcds.2016115

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

 1 Department of Mathematics, Nanjing Forestry University, Nanjing 210036 2 Department of Mathematics, Southwest University, Chongqing 400715

Received  December 2015 Revised  January 2016 Published  October 2016

Considered herein is the blow-up mechanism to the periodic modified Camassa-Holm equation with varying linear dispersion. We first consider 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. Using the continuity of the solutions and the right transformation, we then obtain this blow-up criterion to the case with negative linear dispersion and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that when the linear dispersion is non-negative, formation of singularity can be induced by an initial datum with a sufficiently steep profile.
Citation: Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
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##### References:
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