Advances in Differential Equations

Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems

D.C. Antonopoulos, V.A. Dougalis, and D.E. Mitsotakis

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In this paper we consider the one-parameter family of Bona-Smith systems, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a channel. We study three initial-boundary-value problems for these systems, corresponding, respectively, to nonhomogeneous Dirichlet, reflection, and periodic boundary conditions posed at the endpoints of a finite spatial interval, and establish existence and uniqueness of their solutions. We prove that the initial-boundary-value problem with Dirichlet boundary conditions is well posed in appropriate spaces locally in time, while the analogous problems with reflection and periodic boundary conditions are globally well posed under mild restrictions on the initial data.

Article information

Adv. Differential Equations, Volume 14, Number 1/2 (2009), 27-53.

First available in Project Euclid: 18 December 2012

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Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35G25: Initial value problems for nonlinear higher-order equations 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]


Antonopoulos, D.C.; Dougalis, V.A.; Mitsotakis, D.E. Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems. Adv. Differential Equations 14 (2009), no. 1/2, 27--53.

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  • Correction: D.C. Antonopoulos, V.A. Dougalis, D.E. Mitsotakis. Corrigendum to the paper: "Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems". Adv. Differential Equations 14 (2009), no. 9-10, 1019.