Long-time asymptotics of a linear higher-order water wave model

Abstract

Introducing appropriate damping mechanisms for the associated linear model of the equation (1), we study the asymptotic behavior in time of the corresponding damped models. This is done both in the case of internal and boundary damping. We first address the internal stabilization problem: consider a periodic domain and introduce a localized damping mechanism acting in the equation. More precisely, we study the system ηt + ηx − γ1 ηtxx + γ2 ηxxx + δ1 ηtxxxx + Bη = 0 for x ∈ (0, 2π), t > 0 ∂ r η(t, 0) = ∂xr η(t, 2π), for t > 0, 0 ≤ r ≤ 3, (2) x η(0, x) = η0 (x) for x ∈ (0, 2π), where r is an integer number, γ1 , δ1 > 0 and B : Hps (0, 2π) −→ Hps (0, 2π) is a bounded linear operator and Hps (0, 2π) denotes the Sobolev space of 2π-periodic functions. Let a ∈ Cp∞ (0, 2π) a nonnegative function on (0, 2π) with a(x) > 0 on a given open set Ω ⊂ (0, 2π). [...]