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The Properties of Solutions for a Generalized b-Family Equation with Peakons

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Abstract

This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y t +u m+1 y x +bu m u x y=0, where b is a constant, \(m\in \mathbb{N}\), and we have the notation \(y:= (1-\partial_{x}^{2}) u\) , which includes the famous Camassa–Holm equation, the Degasperis–Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces \(H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\) is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the \(H^{1}(\mathbb{R})\)-norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces \(H^{s}(\mathbb{R})\) with \(1<s<\frac{3}{2}\) is established, under the assumption \(u_{0}(x)\in H^{s}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})\). Finally, the global weak solution and peakon solution for the equation are also given.

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Acknowledgements

The authors are very grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSF of P.R. China (11071266) and in part by the found of Chongqing Normal University (13XLB006) and in part by Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.

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Authors and Affiliations

  1. College of Mathematics Science, Chongqing Normal University, Chongqing, 400047, China

    Shouming Zhou

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Chunlai Mu

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  1. Shouming Zhou
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Correspondence to Shouming Zhou.

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Communicated by D.D. Holm.

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Zhou, S., Mu, C. The Properties of Solutions for a Generalized b-Family Equation with Peakons. J Nonlinear Sci 23, 863–889 (2013). https://doi.org/10.1007/s00332-013-9171-8

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