Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatial analyticity

Global analytic solution in both the time and the space variables is proved for the Cauchy problem of a generalized CH equation, which contains as its members two integrable equations, namely the Camassa-Holm and the Novikov equations. The main assumptions are that the initial datum u0(x) is analytic on the line, it has uniform radius of analyticity r(0)>0, and is such that the McKean quantity m0(x)≐(1−∂x2)u0(x) does not change sign. Furthermore, an explicit lower bound on the radius of space analyticity at later times is obtained, which is of the form L3exp⁡(−L1exp⁡(L2t)), where L1,L2 and L3 are appropriate positive constants.