The well-posedness of solutions for a generalized shallow water wave equation (Q437654)

Summary: 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 \(H^s(\mathbb{R})\) with \(s > 3/2\). Although the \(H^1\)-norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space \(H^s\) with \(1 \leq s \leq 3/2\) is proved under the assumptions \(u_0 \in H^s\) and \(||u_{0x}||_{L^\infty} < \infty\).