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.