In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt + umx + 2ux m = 0, m =(1- δ2x)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.
In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt+umx+2uxm=0, m=(1??2x)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 di?cult, 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.