Abstract
In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt+umx+2uxm=0, m=(1-∂x2)2u,u(x,0)=u0(x)∈Hs(ℝ), x∈ℝ, t>0,and show that the solution map is not uniformly continuous in Sobolev spaces HS(ℝ) for s>7/2. Compared with the periodic problem, the non-periodic problem is more difficult, e.g., it depends on the conservation law. 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.
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