Unique continuation and time decay for a higher-order water wave model

TL;DR

This paper proves exponential energy decay for a higher-order water wave model on a periodic domain by establishing a unique continuation property through Carleman estimates, advancing understanding of wave damping mechanisms.

Contribution

It introduces a novel approach combining energy estimates and Carleman inequalities to demonstrate decay in a complex higher-order water wave equation.

Findings

Exponential decay of energy for the model

Establishment of a unique continuation property

Development of Carleman estimates for coupled systems

Abstract

This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg -de Vries (KdV)--Benjamin-Bona-Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [11], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.