SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION

doi:10.1016/S0252-9602(17)30058-9

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (4): 1061-1082.doi: 10.1016/S0252-9602(17)30058-9

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SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION

Minjie JIANG¹, Wei YAN¹, Yimin ZHANG²

1. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China;
2. School of Science, Wuhan University of Technology, Wuhan 430070, China

Received:2016-02-24 Revised:2016-12-24 Online:2017-08-25 Published:2017-08-25
Contact: Wei YAN,E-mail:yanwei19821115@sina.cn E-mail:yanwei19821115@sina.cn
About author:Minjie JIANG,E-mail:1764915956@qq.com;Yimin ZHANG,E-mail:zhangym802@126.com
Supported by:
This work is supported by Natural Science Foundation of China NSFC (11401180 and 11471330). The second author is also supported by the Young Core Teachers Program of Henan Normal University (15A110033). The third author is also supported by the Fundamental Research Funds for the Central Universities (WUT:2017 IVA 075).

Abstract

Abstract:

This current paper is devoted to the Cauchy problem for higher order dispersive equation
u_{t + ∂_x²ⁿ⁺¹u=∂_x(u∂_xⁿu) + ∂_x^n-1(u_x²), n ≥ 2, n ∈ N⁺.

By using Besov-type spaces, we prove that the associated problem is locally well-posed in H(-n/2 + 3/4,-1/2n)(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H^(s,a)(R) with s < -n/2 + 3/4 and all a ∈ R.}

Key words: Cauchy problem, sharp well-posedness, modified Bourgain spaces

Minjie JIANG, Wei YAN, Yimin ZHANG. SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION[J].Acta mathematica scientia,Series B, 2017, 37(4): 1061-1082.

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References

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SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION

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