Stabilization for the Wave Equation with Singular Kelvin–Voigt Damping

Abstract

We consider the wave equation with Kelvin-Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao [18] or Tébou [21] for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose to deal with this geometrical condition by considering a singular Kelvin-Voigt damping which is localized faraway from the boundary. In this particular case it was proved by Liu and Liu [16] the lack of the uniform decay of the energy. However, we show that the energy of the wave equation decreases logarithmically to zero as time goes to infinity. Our method is based on the frequency domain method. The main feature of our contribution is to write the resolvent problem as a transmission system to which we apply a specific Carleman estimate.2010 Mathematics Subject Classification. 35A01, 35A02, 35M33, 93D20. 2 H 1 0 (Ω)×L 2 (Ω) , ∀ t ≥ 0. Simple formal calculations give E(u, 0) − E(u, t) = − d t 0 ω |∇∂ t u(x, s)| 2 dx ds, ∀t ≥ 0,

Optimal polynomial stability of a string with locally distributed Kelvin–Voigt damping and nonsmooth coefficient at the interface

“…However, when the damping region is a neighborhood of the whole or part of the boundary and the damping coefficient is regular enough, in previous studies, 18,19 it was proved that the energy of the system goes exponentially to zero as t goes to infinity for all usual initial data. In 2018, Ammari et al 20 considered the case where the Kelvin-Voigt damping is localized in a subdomain far away from the boundary without geometric conditions. They established a logarithmic energy decay rate for smooth initial data.…”

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

“…Taking into consideration (46), (51), (57) and the fact that g n converges to zero in L 2 (Ω), we deduce that Ω a|∇u n | 2 dx = o (1) which together with the fact that a(x) ≥ a 0 > 0 for almost every x ∈ Ω give the desired estimation (65).…”

“…Well-Posedness of the problem. In this part, by using semigroup theory, we give the Well-Posedness results for Problem (1). For this aim, we introduce the Hilbert energy space H by…”

“…Polynomial stability. In this section, we study the energy decay rate of System (1). Indeed, if the material parameter b is smooth enough at the interface, then the energy of System (1) decays exponentially to zero as t goes to infinity under appropriate geometric conditions imposed on the damped region (see [19], [23], [28], [8]).…”

“…However, they established exponential stability when this coefficient is smooth. Recently, in 2018, K. Ammari, F. Hassine and L. Robbianio considered a wave equation with Kelvin-Voigt damping localized in a subdomain ω faraway from the boundary without geometric conditions (see [1]). They established a logarithmic energy decay rate for smooth initial data.…”