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On the linearized Whitham–Broer–Kaup system on bounded domains

Part of: Infinite-dimensional dissipative dynamical systems Spectral theory and eigenvalue problems General higher-order equations and systems Partial differential equations

Published online by Cambridge University Press:  07 September 2023

L. Liverani ,
Y. Mammeri ,
V. Pata  and
R. Quintanilla Open the ORCID record for R. Quintanilla [Opens in a new window]
L. Liverani
Affiliation:
Dipartimento di Matematica e Applicazioni Università di Milano Bicocca, Edificio U5, Via Cozzi 55, Milano, 20125 Italy ([email protected])
Y. Mammeri
Affiliation:
Université Jean Monnet - Institut Camille Jordan CNRS UMR 5208, 23 Rue du Dr Paul Michelon, Saint-Etienne, 42100 France ([email protected])
V. Pata
Affiliation:
Politecnico di Milano - Dipartimento di Matematica, Piazza Leonardo da Vinci, 32, Milano, 20133 Italy ([email protected])
R. Quintanilla
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Colom 11, Terrassa, 08222 Barcelona, Spain ([email protected])
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Abstract

We consider the system of partial differential equations

\[ \begin{cases} \eta_t - \alpha u_{xxx} - \beta \eta_{xx} = 0 \\ u_t + \eta_x + \beta u_{xx} = 0 \end{cases} \]
on bounded domains, known in the literature as the Whitham–Broer–Kaup system. The well-posedness of the problem, under suitable boundary conditions, is addressed, and it is shown to depend on the sign of the number
\[ \varkappa=\alpha-\beta^2. \]
In particular, existence and uniqueness occur if and only if $\varkappa >0$. In which case, an explicit representation for the solutions is given. Nonetheless, for the case $\varkappa \leq 0$ we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.

Keywords

Whitham–Broer–Kaup systemdispersive equationsspectrumlinear semigroups

MSC classification

Primary: 35A01: Existence problems
Secondary: 35G16: Initial-boundary value problems for linear higher-order equations 35P10: Completeness of eigenfunctions, eigenfunction expansions 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 20
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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