Global solutions and blow-up phenomena to a shallow water equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:249

Article

Lai, Shaoyong¹  Wu, Yonghong² 

[1] SW Univ Finance & Econ, Dept Appl Math, Chengdu 610074, Peoples R China

[2] Curtin Univ Technol, Dept Math & Stat, Perth, WA 6845, Australia

关键词: Global existence;    Blow-up;    Shallow water model;    Local well-posedness;   

DOI  :  10.1016/j.jde.2010.03.008

来源: Elsevier

【 摘 要 】

A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space H-s(R) with s > 3/2 is developed. Provided that (1 - partial derivative(2)(x))u(0) does not change sign, u(0) is an element of H-s (s > 3/2) and u(0) is an element of L-1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x) is an element of C([0, infinity); H-s(R)) boolean AND C-1 ([0, infinity); Hs-1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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