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

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• 作者：Shaoyong Lai ; Yonghong Wu
• 关键词：Global existence ; Blow-up ; Shallow water model ; Local well-posedness
• 刊名：Journal of Differential Equations
• 出版年：2010
• 出版时间：1 August 2010
• 年：2010
• 卷：249
• 期：3
• 页码：693-706
• 全文大小：194 K

文摘

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 is developed. Provided that does not change sign, u₀H^s () and u₀L¹(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)C([0,∞);H^s(R))∩C¹([0,∞);H^s−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.