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Type of Document Dissertation Author Olson, Erika A. URN etd-04232007-114754 Title The initial value problem for two nonlinear evolution equations Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title Thomas Cosimano Committee Chair Alex Himonas Committee Member Bei Hu Committee Member Gerard Misiolek Committee Member Zhiliang Xu Committee Member Keywords
- hyperelastic rod equation
Date of Defense 2007-04-10 Availability restricted Abstract We consider the initial value problem for two nonlinear evolution equations,first, the hyperelastic rod equation, which, under a certain choice of parameter,
coincides with the Camassa-Holm equation and second, a higher-order modification
of the Camassa-Holm equation. For the hyperelastic rod equation, we show that
solutions to the periodic initial value problem do not depend uniformly continuously
on initial data in Sobolev spaces of index s equal to 1 or s greater than or equal to 2. For the higher-order modification of the Camassa-Holm equation under consideration, we show that the
non-periodic initial value problem is locally well posed for initial data in Sobolev
spaces of index s greater than s' where s' is greater than or equal to 1/4 and less than 1/2 and the value of s' depends on the order of equation.
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