The Cauchy problem for a 4-parameter family of equations with peakon traveling waves

The initial value problem for a novel 4-parameter family of evolution equations, which are nonlinear and nonlocal and possess peakon traveling wave solutions, is studied on both the line and the circle. It is proved that this family of equations is well-posed in the sense of Hadamard when the initial data belong to the Sobolev spaces HsHs with s>5/2s>5/2. Also, it is shown that the data-to-solution map is not uniformly continuous. However, if HsHs, s>5/2s>5/2, is equipped with a weaker HrHr norm, 0⩽r