Non-uniform dependence on initial data for the two-component fractional shallow water wave system

This paper focuses on continuity of a two-component high-order Camassa-Holm system, which was proposed by Escher and Lyons (2015). We know that its solutions depend continuously on their initial data from the local well-posedness results. In present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces H-s1 (R) x H-s2 (R) with s(1) > r + 1/2 and 1/2 < s(2) <= s(1) - 1 <= s(2) + 2r - 2, r is an element of Z(+), which improves the corresponding results for higher-order Camassa-Holm in Tang and Liu (2015) and Wang and Li (2019) to two-component and the lowest Sobolev Space. (C) 2019 Elsevier Ltd. All rights reserved.