Continuity properties of the data-to-solution map for the generalized Camassa–Holm equation

This work studies a generalized Camassa–Holm equation with higher order nonlinearities (g-kbCH). The Camassa–Holm, the Degasperis–Procesi and the Novikov equations are integrable members of this family of equations. g-kbCH is well-posed in Sobolev spaces H s , s > 3 / 2 , on both the line and the circle and its solution map is continuous but not uniformly continuous. In this work it is shown that the solution map is Hölder continuous in H s equipped with the H r -topology for 0 ⩽ r < s , and the Hölder exponent is expressed in terms of s and r.