On the solutions of a model equation for shallow water waves of moderate amplitude

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This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces Bp,rs, p,r∈[1,∞], s>max{32,1+1p} (which generalize the Sobolev spaces Hs) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with s=32, p=2, r=1) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated.