Optimal decay for coupled waves with Kelvin–Voigt damping

Energy decay for evolution equations with delay feedbacks

“…Stability results have also been obtained for specific models in the case 𝑘 constant (see e.g. [1,3,4,7,9,13,21,25,26]). See also [14] for a different result in which the presence of a specific time delay can restitute stability to an unstable wave equation.…”

Regularity of the semigroups associated with some damped coupled elastic systems II: A nondegenerate fractional damping case

“…Then, $$$ A $$$ is a densely defined, positive unbounded operator on the Hilbert space $$$ H $$$. Moreover, $$$ V:= D\left({A}^{\frac{1}{2}}\right)={H}_0^1\left(\Omega \right) $$$ and the injections $$$ V\hookrightarrow H\hookrightarrow {V}^{\prime }={H}^{-1}\left(\Omega \right) $$$ are dense and compact.Then, the corresponding semigroup is analytic, according to Theorem 2.1, for $$$ 1/2\le \mu, \theta \le 1 $$$.We note here that the case where $$$ \mu, \theta =\frac{1}{2} $$$ corresponds to a structural damping and $$$ \mu, \theta =1 $$$ corresponds to a Kelvin–Voigt damping; see Oquendo and Pacheco 23 ; the semigroup in either case is analytic. in Gevrey class $$$ \delta >\frac{1}{2\mu } $$$, according to Theorem 3.1, for $$$ \mu \in \left(0,1/2\right),\theta \in \left(0,1\right] $$$ and $$$ \mu \le \theta $$$. Interacting membrane and plate …”

“…Then, the corresponding semigroup is analytic, according to Theorem 2.1, for $$$ 1/2\le \mu, \theta \le 1 $$$.We note here that the case where $$$ \mu, \theta =\frac{1}{2} $$$ corresponds to a structural damping and $$$ \mu, \theta =1 $$$ corresponds to a Kelvin–Voigt damping; see Oquendo and Pacheco 23 ; the semigroup in either case is analytic. in Gevrey class $$$ \delta >\frac{1}{2\mu } $$$, according to Theorem 3.1, for $$$ \mu \in \left(0,1/2\right),\theta \in \left(0,1\right] $$$ and $$$ \mu \le \theta $$$. …”

“…We note here that the case where $$$ \mu, \theta =\frac{1}{2} $$$ corresponds to a structural damping and $$$ \mu, \theta =1 $$$ corresponds to a Kelvin–Voigt damping; see Oquendo and Pacheco 23 ; the semigroup in either case is analytic.…”

A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface

“…Over the past few years, the coupled systems received a vast attention due to their potential applications. The system of coupled wave equations with only one Kelvin-Voigt damping was considered in [31]. The authors considered the damping and the coupling coefficients to be constants and they established a polynomial energy decay rate of type t −1/2 and an optimality result.…”