Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space

Abstract

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation, Adv Math (China).45(2016),pp.80–88] as follows:

with initial data in the Sobolev space Benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ; Acta Math Sci.37(2017),pp.385C39], first, we show that the local well-posedness is established for the initial data with () and () respectively. Then,using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data with (). Finally, benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation,Adv Math (China).45(2016),pp.80–88], i.e. using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property (UCP henceforth) for the IVP.

Keywords:

• The generalized Kawahara equation
• Fourier restriction norm method
• low regularity
• well-posedness
• unique continuation property (UCP)

AMS Subject Classifications:

• 35A07
• 35Q53

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Hunan Provincial Natural Science Foundation of China [grant number 2016JJ2061], [grant number 2016JJ4037]; Scientific Research Fund of Hunan Provincial Education Department [grant number 15B102], [grant number 15A077]; NNSF of China [grant number 11671101], [grant number 11271118], the construct program of the key discipline in Hunan province [grant number 201176], the Aid program for Science and Technology Innovative Research Team in Higher Educational Instituions of Hunan Province [grant number 2014207] and the Hunan Provincial Local Cooperation Project of China Scholarship Council . Special Funds of Guangxi Distinguished Experts Construction Engineering.