Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks

TL;DR

This paper investigates the well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, demonstrating the system's stability with damping at the central node using frequency domain methods.

Contribution

It establishes the well-posedness and asymptotic stabilization of the BBM equation on star-shaped networks with damping, a novel analysis for such networked PDE systems.

Findings

Proved well-posedness of the BBM system on star-shaped networks.

Achieved asymptotic stabilization using frequency domain techniques.

Demonstrated damping effectiveness at the central node.

Abstract

We study the stabilization issue of the Benjamin-Bona-Mahony (BBM) equation on a finite star-shaped network with a damping term acting on the central node. In a first time, we prove the well-posedness of this system. Then thanks to the frequency domain method, we get the asymptotic stabilization result.