Non-uniform dependence for the periodic higher dimensional Camassa-Holm equations

In this paper, we investigate the dependence on initial data of solutions to the higher dimensional Camassa-Holm equations with periodic boundary condition in Besov spaces. We show that when s>1+d2(d≥2) and 1≤r≤∞, the solution map is not uniformly continuous from B2,rs(Td) into C([0,T];B2,rs(Td)) for r<∞ or from B2,∞s(Td) into L∞([0,T];B2,∞s(Td)) for r=∞.