Well-Posedness and Long-Time Dynamics of a Water-Waves Model with Time-Varying Boundary Delay

TL;DR

This paper investigates a nonlinear water-waves model with a time-varying boundary delay, establishing conditions for well-posedness and demonstrating energy decay in the linearized case, advancing understanding of such complex systems.

Contribution

It introduces a novel analysis of a Boussinesq system with a dynamic boundary delay, providing new criteria for well-posedness and energy decay.

Findings

Well-posedness conditions established using Kato's norm and Fixed-Point Theorem

Energy decay demonstrated for the linearized system

Enhanced understanding of water-waves models with boundary delays

Abstract

A higher-order nonlinear Boussinesq system with a time-dependent boundary delay is considered. Sufficient conditions are presented to ensure the well-posedness of the problem by utilizing Kato's variable norm technique and the Fixed-Point Theorem. More significantly, the energy decay for the linearized problem is demonstrated using the energy method.