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The well-posedness of solutions for a generalized shallow water wave equation. (English) Zbl 1242.35191
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 () 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 1s3/2 is proved under the assumptions u 0 H s and ||u 0x || L <.
MSC:
35Q35PDEs in connection with fluid mechanics