Electron. J. Diff. Equ., Vol. 2013 (2013), No. 44, pp. 1-27.

Wave-breaking phenomena and global solutions for periodic two-component Dullin-Gottwald-Holm systems

Min Zhu, Junxiang Xu

In this article we study the initial-value problem for the periodic two-component b-family system, including a special case, when b = 2, which is referred to as the two-component Dullin-Gottwald-Holm (DGH) system. We first show that the two-component b-family system can be derived from the theory of shallow-water waves moving over a linear shear flow. Then we establish several results of blow-up solutions corresponding to only wave breaking with certain initial profiles for the periodic two-component DGH system. Moreover, we determine the exact blow-up rate and lower bound of the lifespan for the system. Finally, we give a sufficient condition for the existence of the strong global solution to the periodic two-component DGH system.

Submitted November 14, 2012. Published February 8, 2013.
Math Subject Classifications: 35B30, 35G25.
Key Words: Two-component Dullin-Gottwald-Holm system; periodic two-component b-family system; blow-up; wave-breaking; global solution.

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Min Zhu
Department of Mathematics, Nanjing Forestry University
Nanjing 210037, China
email: zhumin@njfu.edu.cn
  Junxiang Xu
Department of Mathematics, Southeast University
Nanjing 211189, China
email: xujun@seu.edu.cn

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