Available online xxxx Communicated by J. Bourgain MSC: primary 35Q53 secondary 37K10 Keywords: Ovsyannikov theorem for nonlocal equations Well-posedness of Cauchy problem in analytic spaces Integrable Camassa-Holm equations Continuity of solution mapThis work presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution. Then, using this theorem it studies the Cauchy problem for Camassa-Holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line, which is the main goal of this paper. Finally, it studies the continuity of the data-to-solution map in spaces of analytic functions.

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Let (x, t) ∈ R m × R and u ∈ C 2 (R m × R). We study the Gevrey micro-regularity of solutions u of the nonlinear equationis a Gevrey function of order s > 1 and holomorphic in (ζ 0 , ζ ). We show that the Gevrey wave-front set of any C 2 solution u is contained in the characteristic set of the linearized operatorTo achieve this, we study the notion of Gevrey approximate solutions, a concept which we believe is of independent interest and could be applied to much more general situations.

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