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Non-uniform dependence on initial data for the generalized Degasperis–Procesi equation on the line

Yanggeng Fu

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, P. R. China

E-mail Address: fuyanggeng@hqu.edu.cn

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Zanping Yu

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, P. R. China

E-mail Address: yu3423191@yahoo.com.cn

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Jianhe Shen

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, P. R. China

E-mail Address: jhshen@fjnu.edu.cn

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https://doi.org/10.1142/S0129167X16500269Cited by:0 (Source: Crossref)

Abstract

In this paper, we show that the solution map of the generalized Degasperis–Procesi (gDP) equation is not uniformly continuous in Sobolev spaces $H^{s} (ℝ)$ for $s > 3 / 2$ . Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part. It also exploits the fact that the gDP equation conserves a quantity which is equivalent to the $L^{2}$ norm.

Keywords:

AMSC: 35A07, 35B30, 35Q53

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