General Energy Decay for a Viscoelastic Equation of Kirchhoff Type with Acoustic Boundary Conditions

Maatoug, Abdelkader

doi:10.1007/s00009-017-1038-z

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…After their introduction, acoustic boundary conditions for locally reacting surfaces have been studied in several papers, as for instance [16], [19], [22], [23], [28], [29], [30], [32], [34], [36], [39], [41]. When one dismisses the simplifying assumption that neighboring point do not interact (in the terminology of [40, p.266]) such surfaces are called of extended reaction.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To prove iii) we remark that, using ( 44), (80), ( 31), ( 84) and (39), for all U = (u, v, w, z) ∈ D(A) we have…”

Section: Asymptotic Stabilitymentioning

confidence: 98%

“…Multiplying (85) by v and using (39) we then get ´Γ1 σ|∇ Γ v| 2 Γ + κ|v| 2 = 0. Then ∇ Γ v = 0 so, being Γ 1 connected, v = c 2 ∈ C and, when κ ≡ 0, c 2 = 0.…”

Section: Asymptotic Stabilitymentioning

confidence: 99%

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The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

Barbieri¹,

Vitillaro²

2022

Preprint

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The aim of the paper is to study the problem where Ω is a open domain of R N with uniformly C r boundary (N ≥ 2, r ≥ 1), Γ = ∂Ω, (Γ 0 , Γ 1 ) is a relatively open partition of Γ with Γ 0 (but not Γ 1 ) possibly empty. Here div Γ and ∇ Γ denote the Riemannian divergence and gradient operators on Γ, ν is the outward normal to Ω, the coefficients µ, σ, δ, κ, ρ are suitably regular functions on Γ 1 with ρ, σ and µ uniformly positive, d is a suitably regular function in Ω and c is a positive constant.In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when Ω is bounded, Γ 1 is connected, r = 2, ρ is constant and κ, δ, d ≥ 0.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To prove iii) we remark that, using ( 44), (80), ( 31), ( 84) and (39), for all U = (u, v, w, z) ∈ D(A) we have…”

Section: Asymptotic Stabilitymentioning

confidence: 98%

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The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

Barbieri¹,

Vitillaro²

2022

Preprint

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“…After their introduction, acoustic boundary conditions for locally reacting surfaces have been the subject of several papers. See, for example, [2,10,11,14,17,18,20,[22][23][24][25][26][27]29,31]. When one dismisses the simplifying assumption that neighboring points do not interact, such surfaces, again using the terminology of [30, p. 266]), are called of extended reaction.…”

Section: Presentation Of the Problem And Literature Overviewmentioning

confidence: 99%

Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaces

Vitillaro

2024

J. Evol. Equ.

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The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces.

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“…After their introduction acoustic boundary conditions for locally reacting surfaces have been the subject of several papers. See for example [CFL04, FL06, FG00, GGG03, JG12a, JG12b, JGSH12, Kan16, KT08, LLX18, Maa17,Mug06]. When one dismisses the simplifying assumption that neighboring point do not interact (in the terminology of [MI68,p.266]) such surfaces are called of extended reaction.…”

Section: Presentation Of the Problem And Literature Overviewmentioning

confidence: 99%

The wave equation with acoustic boundary conditions on non-locally reacting surfaces

Mugnolo¹,

Vitillaro²

2021

Preprint

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Chapter 1. Introduction and main results vii 1.1. Presentation of the problem and literature overview vii 1.2. Main results I: well-posedness ix 1.3. Main results II: solutions' behavior xii 1.4. Physical discussion of the model xvi 1.5. The case of Γ 1 disconnected and paper organization xviii Chapter 2. Background and preliminaries xix 2.1. Notation xix 2.2. Assumptions xix 2.3. Geometric preliminaries xxii 2.4. Lebesgue spaces on Γ xxvii Chapter 3. Sobolev spaces on Γ and operators xxix 3.1. Spaces of integer order xxix 3.2. Spaces of real order xxxix 3.3. Operators xli Chapter 4. Well-posedness of problem (1.1) xlv 4.1. Abstract analysis xlv 4.2. Well-posedness for problem (1.1) li Chapter 5. Regularity when r ≥ 2. lvii Chapter 6. Qualitative behavior of solutions when Ω is bounded lxxi 6.1. General discussion lxxi 6.2. The simplest case: Γ 1 connected lxxix 6.3. The general case: Γ 1 possibly disconnected lxxxvi Chapter 7. The physical model xciii 7.1. The Eulerian approach xciii 7.2. The Lagrangian approach and its relation to the Eulerian one xcvi 7.3. Conclusions xcviii Bibliography cv

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General Energy Decay for a Viscoelastic Equation of Kirchhoff Type with Acoustic Boundary Conditions

Cited by 5 publications

References 15 publications

The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaces

The wave equation with acoustic boundary conditions on non-locally reacting surfaces

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