THE CAUCHY PROBLEM FOR A GENERALIZED CAMASSA-HOLM EQUATION

被引:3
作者
Himonas, A. Alexandrou [1 ]
Holliman, Curtis [2 ]
机构
[1] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA
[2] Univ Alabama Birmingham, Dept Biostat, Birmingham, AL 35294 USA
关键词
SHALLOW-WATER EQUATION; WELL-POSEDNESS; NONUNIFORM CONTINUITY; ILL-POSEDNESS; TRAJECTORIES; DEPENDENCE; KDV;
D O I
暂无
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
This work studies the initial value problem (i.v.p.) for a generalized Camassa-Holm equation with (k + 1)-order nonlinearities (g-kbCH) and containing, as its members, three integrable equations: the Camassa-Holm, the Degasperis-Procesi and the Novikov equations. For s > 3/2, using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces H-s on both the circle and the line in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates.
引用
收藏
页码:161 / 200
页数:40
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