Global Stabilization for the BBM-KP equations on R2

TL;DR

This paper demonstrates exponential energy decay for the BBM-KP equations on R2 with localized damping, combining theoretical analysis and numerical validation to establish stabilization results.

Contribution

It provides the first analysis of exponential stabilization for the BBM-KP equations with localized damping on R2, including numerical validation.

Findings

Energy decays exponentially with localized damping

Theoretical proof of stabilization on R2

Numerical experiments confirm decay behavior

Abstract

In this paper, we present results on the energy decay of the BBM-KP equations (I and II) posed on $\R^2$ with localized damping. This model offers an alternative to the KP equations, analogous to how the regularized long-wave equation relates to the classical Korteweg-de Vries (KdV) equation. We show that the energy associated with the Cauchy problem decays exponentially when a localized dissipative mechanism is present in a subdomain. Finally, we validate the theoretical results on the exponential stabilization of solutions to the BBM-KP equations with damping through numerical experiments using a spectral-finite difference scheme.