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Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 15
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Articles

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

ORCID Icon, , &
Pages 2655-2685 | Received 14 May 2017, Accepted 19 Sep 2017, Published online: 06 Oct 2017
 

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.

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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.

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