# American Institute of Mathematical Sciences

December  2017, 37(12): 6471-6485. doi: 10.3934/dcds.2017280

## Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation

 1 Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China 2 Faculty of Information Technology, Macau University of Science and Technology, Macau, China

1Corresponding author

Received  June 2017 Published  August 2017

Fund Project: This work was partially supported by NNSFC (No.11671407), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004).

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in $H^s (\mathbb{S}), s > \frac{3}{2},$ by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

Citation: Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
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##### References:
 [1] Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643 [2] Alejandro Sarria. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 641-673. doi: 10.3934/dcdsb.2015.20.641 [3] Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084 [4] Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719 [5] Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023 [6] Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347 [7] Jaeho Choi, Nitin Krishna, Nicole Magill, Alejandro Sarria. On the $L^p$ regularity of solutions to the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6349-6365. doi: 10.3934/dcdsb.2019142 [8] Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925 [9] Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 [10] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [11] Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 [12] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [13] István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134 [14] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [15] Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263 [16] Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108 [17] Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011 [18] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [19] Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021093 [20] Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

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