Global existence and blow-up phenomena for a periodic modified Camassa–Holm equation (MOCH)

ABSTRACT

In this paper, we study global existence and blow-up for a periodic modified Camassa–Holm equation in nonhomogeneous Sobolev spaces. Also, we provide a key blow-up criteria to investigate norm inflation and ill-posedness problem for the equation in the critical Sobolev space.

Keywords:

• Periodic modified Camassa–Holm equation (MOCH)
• global existence
• blow up
• ill-posedness

2010 Mathematics Subject Classifications:

• 35Q53
• 35B30
• 35B44

Acknowledgments

This work was partially supported by NNSFC (No. 11671407), FDCT (No. 0091/2018/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A03031104). The author Qiao thanks the 2019–2020 Hunan overseas distinguished professorship (No. 2019014) and the UT President Endowed Professorship (Project # 450000123) for their partial support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by NNSFC [grant number 11671407], FDCT [grant number 0091/2018/A3], Guangdong Special Support Program [grant number 8-2015], and the key project of NSF of Guangdong province [grant number 2016A03031104]. The author Qiao thanks the 2019–2020 Hunan overseas distinguished professorship [grant number 2019014] and the UT President Endowed Professorship (Project # 450000123) for their partial support.