On the decay rate for the wave equation with viscoelastic boundary damping

Stahn, Reinhard

doi:10.1016/j.jde.2018.04.048

Cited by 5 publications

(18 citation statements)

References 15 publications

(33 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, this does not necessarily mean that our result is suboptimal in the multidimensional setting. In fact, the constraints on Ω from compared to our setting are quite different, thus it is hard to say if the better resolvent bound still holds in the setting of our paper.…”

Section: Comments and Future Workmentioning

confidence: 98%

“…1) The topic of our paper is very close to . Taking account of Remark , problem ( P ) becomes

({, \begin{matrix} u_{t t} false(x, t false) - Δ u false(x, t false) = 0 & in Ω \times (0, + \infty), \\ u (x, t) = 0 & on {normalΓ}_{D} \times false(0, + \infty false), \\ 0true \frac{\partial u}{\partial ν} (x, t) = - ζ \int_{0}^{t} h (t - s) u_{t} (x, s) d s & on {normalΓ}_{N} \times false(0, + \infty false), \\ u false(x, 0 false) = u_{0} false(x false), u_{t} false(x, 0 false) = u_{1} false(x false) & on Ω . \end{matrix})

Here h is a positive kernel satisfying certain additional assumptions.…”

Section: Comments and Future Workmentioning

confidence: 99%

“…For the proof of our main result on energy decay rates (Theorem ) we use completely different techniques as compared with . It is interesting to note that the better resolvent bound in is only obtained under a condition on the “acoustic impedance”

\overset{h}{true}

which excludes certain choices of h (depending on Ω) for which does not provide any results. Contrary to this the slightly worse bound from our paper holds for any acoustic impedance.…”

Section: Comments and Future Workmentioning

confidence: 99%

“…In the case > 0 and (Γ ) = 0, Stahn independently obtained related results in [40]. More precisely he looked at the following problem:…”

Section: (P)mentioning

confidence: 99%

“…In the case

η > 0

and

m e a s ({normalΓ}_{D}) = 0

, Stahn independently obtained related results in . More precisely he looked at the following problem:

({, \begin{matrix} p_{t} false(t, x false) + div v false(t, x false) = 0 & (t \in R, x \in Ω), \\ v_{t} false(t, x false) + \nabla p false(t, x false) = 0 & (t \in R, x \in Ω), \\ e * p (t, x) - ν \cdot \nabla v (t, x) = 0 & (t \in R, x \in \partial Ω), \end{matrix})

where Ω is a bounded domain with Lipschitz boundary and

e : R \to [0, \infty)

is an nonzero integrable function, depending on the time‐variable only and vanishing on

(- \infty, 0)

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Benaissa

Rafa

2019

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Section: Comments and Future Workmentioning

confidence: 98%

“…1) The topic of our paper is very close to . Taking account of Remark , problem ( P ) becomes

({, \begin{matrix} u_{t t} false(x, t false) - Δ u false(x, t false) = 0 & in Ω \times (0, + \infty), \\ u (x, t) = 0 & on {normalΓ}_{D} \times false(0, + \infty false), \\ 0true \frac{\partial u}{\partial ν} (x, t) = - ζ \int_{0}^{t} h (t - s) u_{t} (x, s) d s & on {normalΓ}_{N} \times false(0, + \infty false), \\ u false(x, 0 false) = u_{0} false(x false), u_{t} false(x, 0 false) = u_{1} false(x false) & on Ω . \end{matrix})

Here h is a positive kernel satisfying certain additional assumptions.…”

Section: Comments and Future Workmentioning

confidence: 99%

\overset{h}{true}

which excludes certain choices of h (depending on Ω) for which does not provide any results. Contrary to this the slightly worse bound from our paper holds for any acoustic impedance.…”

Section: Comments and Future Workmentioning

confidence: 99%

“…In the case > 0 and (Γ ) = 0, Stahn independently obtained related results in [40]. More precisely he looked at the following problem:…”

Section: (P)mentioning

confidence: 99%

“…In the case

η > 0

and

m e a s ({normalΓ}_{D}) = 0

, Stahn independently obtained related results in . More precisely he looked at the following problem:

({, \begin{matrix} p_{t} false(t, x false) + div v false(t, x false) = 0 & (t \in R, x \in Ω), \\ v_{t} false(t, x false) + \nabla p false(t, x false) = 0 & (t \in R, x \in Ω), \\ e * p (t, x) - ν \cdot \nabla v (t, x) = 0 & (t \in R, x \in \partial Ω), \end{matrix})

where Ω is a bounded domain with Lipschitz boundary and

e : R \to [0, \infty)

is an nonzero integrable function, depending on the time‐variable only and vanishing on

(- \infty, 0)

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Benaissa

Rafa

2019

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Optimal rates of decay for operator semigroups on Hilbert spaces

Rozendaal

Seifert²,

Stahn

2019

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

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).

show abstract

Semi-uniform stability of operator semigroups and energy decay of damped waves

Chill

Seifert

Tomilov

2020

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

Only in the last 15 years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of C 0 -semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems. This article is part of the theme issue ‘Semigroup applications everywhere’.

show abstract

On the decay rate for the wave equation with viscoelastic boundary damping

Cited by 5 publications

References 15 publications

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

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

Optimal rates of decay for operator semigroups on Hilbert spaces

Semi-uniform stability of operator semigroups and energy decay of damped waves

Contact Info

Product

Resources

About