Analyticity of transmission problem to thermoelastic plates

Abstract

Abstract. In this paper we consider an oscillation model to a plate comprised of two different thermoelastic materials; that is, we study a transmission problem to thermoelastic plates. Our main result is to prove that the corresponding semigroup associated to this problem is of analytic type.

Analyticity and stability results for a plate‐membrane type transmission problem

“…Since 𝒜 𝜂 𝒰 = 𝑖𝜆𝒰 − ℱ ∈ 𝐷(𝒜 𝜂 ), then 𝒰 ∈ 𝐷(𝒜 2 𝜂 ) and thus (4.44) holds changing 𝑤 and 𝑓 by 𝒰 and ℱ, respectively. Hence, ‖(𝑖𝜆 − 𝒜 𝜂 ) −1 ℱ‖ ℋ 𝜂 ≤ 𝐶|𝜆| 24 ‖𝒜 𝜂 ℱ‖ ℋ 𝜂 .…”

“…The goal of this section is to show an analytical result when both the inertial term and the structural damping are not present, but there is Kelvin–Voigt damping on the membrane and temperature on the plate. As a consequence, we will have that $$w(t)=\mathcal T_0(t)w_0\in D(\mathcal {A}_0^\infty )\coloneqq \cap _{k=0}^\infty D(\mathcal {A}_0^k)$$ for $$w_0\in \mathcal {H}_0$$, that is, no matter how irregular the initial data are, always the corresponding solution is of class $$C^\infty$$ (see [24, p. 2] or [42, p. 179]). Other implications of the analyticity of a semigroup can be found in [38].…”

“…There are few analytical results for transmission problems in the literature. In [24], the authors were able to demonstrate the analyticity of the semigroup associated to a transmission problem consisting of two thermoelastic plates. In [43], the authors investigated a transmission problem between a (thermo)viscoelastic system with Kelvin–Voigt damping and a purely elastic system.…”

Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation

“…In addition, transmission problems in the context of thermoelasticity were considered by Muñoz Rivera and Naso [36] and Fernandéz Sare and Muñoz Rivera [18]. Concerning transmission problems with viscoelasticy of Kelvin-Voigt type, we quote the works of Alves et al [1,2].…”