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 with . Although the -norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space with is proved under the assumptions and .
"The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation." Abstr. Appl. Anal. 2012 (SI11) 1 - 15, 2012. https://doi.org/10.1155/2012/872187