Well-posedness and analyticity for the Cauchy problem for the generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem for the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev [S. Hakkaev, K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation, Comm. Partial Differential Equations 30 (2005) 761–781]. We establish the local well-posedness for a range of Besov spaces. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extends some results of Danchin [R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988] and Himonas and Misiolek [A. Himonas, G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575–584] to more general equations.