On the Benjamin-Bona-Mahony equation with a localized damping
Lionel Rosier (CAS)
TL;DR
This paper investigates energy dissipation mechanisms in the Benjamin-Bona-Mahony (BBM) equation, establishing conditions for global well-posedness and stability when applying localized or boundary damping strategies.
Contribution
It introduces new damping mechanisms for the BBM equation and proves their effectiveness in energy dissipation and stability under certain conditions.
Findings
Global well-posedness of the damped BBM system
Convergence to solutions null on a band
Asymptotic stability under the Unique Continuation Property
Abstract
We introduce several mechanisms to dissipate the energy in the Benjamin-Bona-Mahony (BBM) equation. We consider either a distributed (localized) feedback law, or a boundary feedback law. In each case, we prove the global wellposedness of the system and the convergence towards a solution of the BBM equation which is null on a band. If the Unique Continuation Property holds for the BBM equation, this implies that the origin is asymp-totically stable for the damped BBM equation.
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Taxonomy
TopicsStability and Controllability of Differential Equations · Advanced Mathematical Physics Problems · Nonlinear Photonic Systems