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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 $$H^s(\mathbb{R})$$ 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 $$||u_{0x}||_{L^\infty} < \infty$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics
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##### References:
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