Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude

In this paper, we study continuity and persistence for a nonlinear evolution equation describing the free surface of shallow water wave with a moderate amplitude, which was proposed by Constantin and Lannes [7]. By the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for Constantin-Lannes equation, which are bounded in the Sobolev space Hs(R) with s>3/2, and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous. Furthermore, the solution map for Constantin-Lannes equation is shown Hölder-continuous in Hr-topology for all 0≤r