The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation

A nonlinear partial differential equation containing the famous Camassa-Holm and Degasperis-Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well-posedness of solutions for the equation in the Sobolev space Hs(R) with s>3/2. Although the H1-norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space Hs with 1≤s≤3/2 is proved under the assumptions u0∈Hs and ∥u0x∥L∞<∞