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Low Regularity for the Higher Order Nonlinear Dispersive Equation in Sobolev Spaces of Negative Index.
- Source :
-
Journal of Dynamics & Differential Equations . Mar2019, Vol. 31 Issue 1, p419-433. 15p. - Publication Year :
- 2019
-
Abstract
- In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000): ∂ t u + α ∂ x 7 u + β ∂ x 5 u + γ ∂ x 3 u + μ ∂ x u + λ u ∂ x u = 0 , x ∈ R , t ∈ R , u (x , 0) = u 0 (x) , x ∈ R with the initial data in the Sobolev space H s (R). Benefited from the ideas of Huo and Jia (Z Angew Math Phys 59:634–646, 2008), Zhang et al. (Acta Math Sci 37B(2):385–394, 2017) and Zhang and Huang (Math Methods Appl Sci 39(10):2488–2513, 2016) that is, using Fourier restriction norm method, Tao's [k, Z]-multiplier method and the contraction mapping principle, we prove that IVP is locally well-posed for the initial data u 0 ∈ H s (R) with s ≥ - 5 8 . Moreover, based on the local well-posedness and conservation law, we establish the global well-posedness for the initial data u 0 ∈ H s (R) with s = 0 . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10407294
- Volume :
- 31
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of Dynamics & Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 135715018
- Full Text :
- https://doi.org/10.1007/s10884-018-9669-8