Global solutions and blow-up phenomena to a shallow water equation |
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Authors: | Shaoyong Lai Yonghong Wu |
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Institution: | a Department of Applied Mathematics, Southwestern University of Finance and Economics, 610074, Chengdu, China b Department of Mathematics and Statistics, Curtin University of Technology, WA 6845, Perth, Australia |
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Abstract: | 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 Hs(R) with is developed. Provided that does not change sign, u0∈Hs () and u0∈L1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C(0,∞);Hs(R))∩C1(0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired. |
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Keywords: | 35G25 35L05 |
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