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Zbl 1193.35108
Escher, Joachim; Yin, Zhaoyang
Initial boundary value problems for nonlinear dispersive wave equations. (English)
[J] J. Funct. Anal. 256, No. 2, 479-508 (2009). ISSN 0022-1236

Summary: We study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the $b$-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities, we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.

MSC 2000:: *35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35L30 Higher order hyperbolic equations, initial value problems
35L35 Higher order hyperbolic equations, boundary value problems
35G25 Initial value problems for nonlinear higher-order PDE
35B44

Keywords: Camassa-Holm equation; rod equation; Degasperis-Procesi equation; $b$-equation; local well-posedness; global existence; one space dimension; symmetry preserving properties

Cited in: Zbl 1241.35170

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