In this paper, we investigate the N − abc family of Camassa-Holm type equation with (N + 1)-order nonlinearities. Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N − abc family of Camassa-Holm type equation in the energy space H 1 (R) × W 1,2N (R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1 − 1 2N. This result precisely shows how the regularity of solution changes with respect to the the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa-Holm equation (linear wave speed u 1), Novikov equation (quadratic wave speed u 2) and the N − abc family of Camassa-Holm type equation (wave speed u N). This result improves earlier ones in the literatures, such as [1], [8], [45] and [50].
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