Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

Benaissa, Abbès; Rafa, Said

doi:10.1002/mana.201800224

Cited by 9 publications

(5 citation statements)

References 34 publications

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“…This system can be viewed as a model of sound propagation under reflection subject to viscoelastic damping at the boundary, and in this case the boundary condition captures memory effects, u t and −∇u are the pressure and velocity of the fluid and Fk, or alternatively the Laplace transform of k, is the acoustic impedance; for further details see [37], where the same model is considered also for higher-dimensional domains. The results in this section are closely related to those obtained independently in [10], where rates of energy decay are investigated for a very similar model. We begin by recasting the problem in the form of an abstract Cauchy problem, (5.3) ż(t) = Az(t), t ≥ 0,…”

Section: Application To a Wave Equation With Viscoelastic Dampingsupporting

confidence: 79%

Optimal rates of decay for operator semigroups on Hilbert spaces

Rozendaal

Seifert²,

Stahn

2019

Advances in Mathematics

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We investigate rates of decay for C0-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty, Chill and Tomilov (J. Eur. Math. Soc., vol. 18(4), pp. 2016). In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.2010 Mathematics Subject Classification. 47D06, 34D05, 34G10 (35B40, 35L05, 26A12).

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Section: Application To a Wave Equation With Viscoelastic Dampingsupporting

confidence: 79%

Optimal rates of decay for operator semigroups on Hilbert spaces

Rozendaal

Seifert²,

Stahn

2019

Advances in Mathematics

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“…Recently, Benaissa and Benkhedda considered the stabilization for the following wave equation with dynamic boundary control of fractional derivative type ( CF ):

({, \begin{matrix} array y_{t t} (x, t) - y_{x x} (x, t) = 0 & array in (0, L) \times (0, + \infty) \\ array y (0, t) = 0 & array in (0, + \infty) \\ array m y_{t t} (L, t) + y_{x} (L, t) = - γ \partial_{t}^{α, η} y (L, t) & array in (0, + \infty) \\ array y (x, 0) = y_{0} (x), y_{t} (x, 0) = y_{1} (x) & array in (0, L) . \end{matrix})

They proved that the decay of the energy is not exponential but polynomial, that is, E ( t ) ≤ C 1/ t (2 − α ) . Very recently, Benaissa and Rafa extended the result of Mbodje to higher‐space dimension and boundary control of diffusive type as in this paper (with m = 0) and established a less precise decay estimate by adopting the multiplier method.…”

Section: Introductionmentioning

confidence: 87%

A general decay result of a wave equation with a dynamic boundary control of diffusive type

Cheheb

Benkhedda

Benaissa

2019

Math Methods in App Sciences

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“…The authors proved that the energy decays polynomially as t −2/(1−α) . Recently, A. Benaissa and S. Rafa [5] studied the well-possedness and asymptotic stability of a similar wave equation with general boundary condition of diffusive type, that is…”

Section: Introductionmentioning

confidence: 99%

Stability of a Schrödinger equation with internal fractional damping

Ibrahim,

Naima,

Abbes

2023

AUCMCS

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In this paper, we are concerned with the stabilization of a linear Schrödinger equation in an n-dimensional open bounded domain under Dirichlet boundary conditions with an internal fractional damping. We reformulate the system into an augmented model and prove the well-posedness of it by using semigroup method. Based on a general criteria of Arendt-Batty, we show that the system is strongly stable. By combining frequency domain method and multiplier techniques, we establish an optimal polynomial energy decay rate.

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Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

Cited by 9 publications

References 34 publications

Optimal rates of decay for operator semigroups on Hilbert spaces

Optimal rates of decay for operator semigroups on Hilbert spaces

A general decay result of a wave equation with a dynamic boundary control of diffusive type

Stability of a Schrödinger equation with internal fractional damping

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