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


  • Oscar A. Sierra Fonseca
  • George J. Bautista


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). [...]


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Biografia do Autor

Oscar A. Sierra Fonseca

EMAp-FGV, Rio de Janeiro, Brazil.

George J. Bautista

Facultad de Ingeniería, Escuela Profesional de Ingeniería Civil, Universidad Tecnológica de los Andes, Apurímac, Peru.


G. J. Bautista e A. F. Pazoto. “Large-time behavior of a linear Boussinesq system for the water waves”. Em: J. Dynam. Differential Equations 31 (2019), pp. 959–978.

O. S. Fonseca e A. F. Pazoto. “Asymptotic behavior of a linear higher-order BBM-system with damping”. Em: Z. Angew. Math. Phys. 73 (2022), p. 231. doi: 10.1007/s00033- 022-01861-2.

L. Rosier. “On the Benjamin-Bona-Mahony equation with a localized damping”. Em: J. Math. Study 49 (2016), pp. 195–204.