A note on the solution map for the periodic Camassa-Holm equation

In this paper, we study the dependence on initial data of solutions to Camassa-Holm equation with periodic boundary condition in Besov spaces. We show that when s > 3/2 and 1 <= r <= infinity, the solution map is not uniformly continuous from B-2,B-r (S)(T) into C(inverted right perpendicular0, Tinverted left perpendicular; B-2,r(S) (T)) for r < infinity or from B-2,infinity(S)(T) into L-infinity(0, T; B-2,infinity(S)(T)) for r = infinity. Moreover, we prove that if a weaker B-p,r(q)-topology is used, then the solution map becomes Holder continuous in B-p,r(q) (T). It seems that the non-uniform dependence on initial data in periodic Besov spaces has not appeared in the previous literature.