On the Cauchy problem for a class of shallow water wave equations with (k+1)-order nonlinearities

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:445

Article

Zhang, Lei¹  Liu, Bin¹ 

[1] Huazhong Univ Sci & Technol, Sch Math & Stat, Wuhan 430074, Hubei, Peoples R China

关键词: Well-posedness;    Shallow water wave equations;    (k+1)-order nonlinearities;    Blow-up criteria;    Gevrey regularity;    Analyticity;   

DOI  :  10.1016/j.jmaa.2016.07.056

来源: Elsevier

【 摘 要 】

This paper considers the Cauchy problem for a class of shallow water wave equations with (k + 1)-order nonlinearities in the Besov spaces partial derivative(t)u - partial derivative(t)partial derivative(2)(x)u = u(k)partial derivative(3)(x)u + bu(k-1)partial derivative(x)u partial derivative(2)(x)u - (b + 1)u(k)partial derivative(x)u, which involves the Camassa-Holm, the Degasperis-Procesi and the Novikov equations as special cases. Firstly, by means of the transport equation and the Littlewood-Paley theory, we obtain the local well-posedness of the equations in the nonhomogeneous Besov space B-p,r(s)(s > max{1 + 1/p, 3/2} and p, r is an element of [1, +infinity]). Secondly, we consider the local well-posedness in B-2,r(s) with the critical index s = 3/2, and show that the solutions continuously depend on the initial data. Thirdly, the blow-up criteria and the conservative property for the strong solutions are derived. Finally, with the help of a new Ovsyannikov theorem, we investigate the Gevrey regularity and analyticity of the solutions. Moreover, we get a lower bound of the lifespan and the continuity of the data-to-solution mapping. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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