Well posedness for a higher order modified Camassa-Holm equation. (English) Zbl 1170.35087
Summary: We show that the Cauchy problem for a higher order modification of the Camassa-Holm equation is locally well posed for initial data in the Sobolev space \(H^s(\mathbb R)\) for \(s>s{^{\prime}}\), where \(1/4\leqslant s{^{\prime}}<1/2\) and the value of \(s{^{\prime}}\) depends on the order of equation. Employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35A20 | Analyticity in context of PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
Keywords:
modified Camassa-Holm equation; nonlinear dispersive equations; harmonic analysis; bilinear estimates; well posedness; Cauchy problem; Sobolev spacesReferences:
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