A new perspective of exponential stability for Timoshenko systems under history and thermal effects 1

Abstract

We address a Timoshenko system with memory in the history context and thermoelasticity of type III for heat conduction. Our main goal is to prove its uniform (exponential) stability by illustrating carefully the sensitivity of the heat and history couplings on the Timoshenko system. This investigation contrasts previous insights on the subject and promotes a new perspective with respect to the stability of the thermo-viscoelastic problem carried out, by combining the whole strength of history and thermal effects.

Keywords

Timoshenko system exponential stability history type III thermoelasticity

1. Introduction

In the present article, our main goal is to study the uniform stability of the following Timoshenko beam model with thermoelasticity of type III and memory with history $\begin{array}{l} (1.1) & ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} + σ θ_{t x} = 0 in (0, l) \times (0, \infty), \\ (1.2) & ρ_{2} ψ_{t t} - β ψ_{x x} + k (φ_{x} + ψ) - \int_{0}^{\infty} g (s) η_{x x} (s) d s - σ θ_{t} = 0 in (0, l) \times (0, \infty), \\ (1.3) & ρ_{3} θ_{t t} - δ θ_{x x} - γ θ_{x x t} + σ {(φ_{x} + ψ)}_{t} = 0 in (0, l) \times (0, \infty), \\ (1.4) & η_{t} + η_{s} - ψ_{t} = 0 in (0, l) \times {(0, \infty)}^{2}, \end{array}$ subject to initial conditions $\begin{matrix} (1.5) & \begin{matrix} (φ, φ_{t}, ψ, ψ_{t}, θ, θ_{t}) (x, 0) = (φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, θ_{0}, θ_{1}) (x), x \in (0, l), \\ η (x, 0, s) = η_{0} (x, s), η (x, t, 0) = 0, x \in (0, l), s > 0, t ⩾ 0, \end{matrix} \end{matrix}$ and either boundary conditions the full Dirichlet case $\begin{array}{l} (1.6a) & \begin{aligned} φ (0, t) = φ (l, t) = ψ (0, t) = ψ (l, t) = θ (0, t) = θ (l, t) = 0, t ⩾ 0, \\ η (0, t, s) = η (l, t, s) = 0, t ⩾ 0, s > 0, \end{aligned} \end{array}$ or the mixed Neumann–Dirichlet one $\begin{array}{l} (1.6b) & \begin{aligned} φ_{x} (0, t) = φ_{x} (l, t) = ψ (0, t) = ψ (l, t) = θ (0, t) = θ (l, t) = 0, t ⩾ 0, \\ η (0, t, s) = η (l, t, s) = 0, t ⩾ 0, s > 0, \end{aligned} \end{array}$ where $ρ_{1}$ , $ρ_{2}$ , $ρ_{3}$ , k, b, δ, γ, σ, and $β : = b - \int_{0}^{\infty} g (s) d s$ are positive constants whose physical meanings are very well described, the unknown functions $φ = φ (x, t)$ , $ψ = ψ (x, t)$ , $θ = θ (x, t)$ , and $η = (x, t, s)$ are, respectively, related to the transversal displacement, the rotation angle, the temperature, and the relative displacement history of a beam with length $l > 0$ . The physical details around problem (1.1)–(1.6a)–(1.6b) will be clarified in Section 2.

In order to prove the uniform stabilization of (1.1)–(1.6a)–(1.6b), we consider the standard exponential assumption on the memory kernel g as follows.

Assumption 1.1.
Let us suppose that $g \in C^{1} (0, \infty) \cap L^{1} (0, \infty)$ satisfies $\begin{matrix} (1.7) & 0 < g (0) < \infty, 0 < \int_{0}^{\infty} g (s) d s < b, and 0 < k_{1} g (s) ⩽ - g^{'} (s), s \in (0, \infty), \end{matrix}$ for some $k_{1} > 0$ .

Under Assumption 1.1, we shall prove that model (1.1)–(1.6a)–(1.6b) is always exponentially stable, independently of any relationship among the coefficients and the boundary conditions ((1.6a) or (1.6b)) taken into account (Theorem 3.2). This fact seems to be, somehow, a surprising result with respect to the possible roles of history and heat conduction in type III thermoelasticity since it strongly contrasts earlier expectations on the stability of the PDE system (1.1)–(1.4). Indeed, as stated without proof in [36, Section 7] an initial-boundary value problem related to (1.1)–(1.6a)–(1.6b) is not exponentially stable in general. The authors claimed that the stability finally depends upon the equal speeds of wave propagation $\begin{matrix} (1.8) & \frac{k}{ρ_{1}} = \frac{b}{ρ_{2}}, \end{matrix}$ by suggesting that the model has an optimal polynomial decay rate when ( 1.8 ) does not hold. More precise details on this statement will be clarified in Remark 3.12 right after the proofs of our main result (Theorem 3.2). Therefore, by means of our main result, a new perspective of stability to (1.1)–(1.6a)–(1.6b) is provided. Surprisingly or not, we advance that the memory and temperature components produce enough dissipation in order to stabilize the whole system. In other words, one can say that problem (1.1)–(1.6a)–(1.6b) is, indeed, fully damped. This fact will be physically clearer at the end of Section 2 and will be mathematically proved in Section 3. Under these statements, a simple question so arises.

Why does our result contrast the statement given in Section 7 of [36]?

The main reason is behind the thermal and viscoelastic couplings on the canonical Timoshenko system. In order to give a deeper response to this question, let us consider another close Timoshenko system (but not the same) with thermoelasticity of type III and memory with history $\begin{array}{l} (1.9) & ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} = 0 in (0, l) \times (0, \infty), \\ (1.10) & ρ_{2} ψ_{t t} - β ψ_{x x} + k (φ_{x} + ψ) - \int_{0}^{\infty} g (s) η_{x x} (s) d s + σ θ_{t x} = 0 in (0, l) \times (0, \infty), \\ (1.11) & ρ_{3} θ_{t t} - δ θ_{x x} - γ θ_{x x t} + σ ψ_{x t} = 0 in (0, l) \times (0, \infty), \\ (1.12) & η_{t} + η_{s} - ψ_{t} = 0 in (0, l) \times {(0, \infty)}^{2} . \end{array}$

System (1.9)–(1.12) was addressed in [30] with proper initial-boundary conditions, see (2.1)–(2.4) therein. Summarizing, for exponential kernels g as in (1.7), the authors prove: (i) under the assumption (1.8), the energy goes to zero exponentially when t goes to infinity (cf. [30, Thm. 2.1]); (ii) Otherwise, if (1.8) does not hold, then the energy is only semi-uniformly stable with polynomial decay-type only for regular initial data (cf. [30, Thm. 3.1]). A complete characterization of the stability for (1.9)–(1.12) would be provided if the authors showed lack of exponential stability in case (1.8) fails, although the authors do not consider this part. In conclusion, problem (1.9)–(1.12) is only partially dissipative, unlike (1.1)–(1.4). Why?

As a matter of fact, instead of answering the previous question $Q 1$ , the aforementioned statements drive us to another intriguing one, as contextualized below. Formally speaking, both problems (1.1)–(1.4) and (1.9)–(1.12) have the same couple of variables, the same number of equations, the same quantity of damping terms, the same energy, and also the same energy derivative which can be achieved with standard computations. Thus, one can ask:

Why is problem (1.1)–(1.4) fully damped while (1.9)–(1.12) is partially dissipative?

In order to give the precise answers to $Q 1$ and $Q 2$ , we must go back to the governing equations for Timoshenko beams along with thermo-(visco-)elastic constitutive laws on the forces of the system. The latter are given by the bending moment and the shear force. Proceeding in this way, we can provide an explicit formulation for both problems by showing that the first one is obtained with thermal and viscoelastic couplings on different forces of the system whereas the second one is reached by considering history and heat couplings on the same force. The full description of the latter statement is presented in Section 2 and answers question $Q 2$ , at least physically. To this end, we follow the modeling provided in [2,3]. Additionally, the mathematical (and technical) answers to questions $Q 1$ – $Q 2$ concerning problem (1.1)–(1.4) are fully provided in Section 3. To this purpose, we employ a refined resolvent analysis in combination with auxiliary results given in Appendix x and well-known results in linear semigroup theory.

To sum up briefly, our contributions in the present paper are:
to bring new perspectives in what concerns the role of memory with history and heat conduction in type III thermoelasticity for Timoshenko-type systems;

to provide the correct uniform stability result with respect to problem (1.1)–(1.4) subject to initial-boundary conditions (1.5)–(1.6a)–(1.6b).

2. Rise of the model with history and thermal effects

Let us initially consider the governing equations for Timoshenko beams (cf. [39,40]): $\begin{matrix} (2.1) & \{\begin{matrix} ρ_{1} φ_{t t} - S_{x} = 0 & in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - M_{x} + S = 0 & in (0, l) \times (0, \infty), \end{matrix} \end{matrix}$ where $ρ_{1}, ρ_{2} > 0$ are physically well-known constants, $φ = φ (x, t)$ , $ψ = ψ (x, t)$ , $S = S (x, t)$ , and $M = S (x, t)$ stand for the transversal displacement, rotation angle, shear force, and bending moment, respectively. In the classical elastic case, the following constitutive laws are in place $\begin{matrix} (2.2) & \{\begin{matrix} S = k (φ_{x} + ψ), \\ M = b ψ_{x}, \end{matrix} \end{matrix}$ where $k, b > 0$ are again well-known constants coming from physical concerns. Therefore, replacing (2.2) in (2.1) we obtain the classical conservative model in differential equations for vibrations of thin beams as originated in Timoshenko’s works.

Now, on the one hand, for materials containing hereditary (history) properties, the Boltzmann theory for aging materials states the stress depends not only on the instantaneous strain but also on the strain history. Under this premise, and following classical Timoshenko’s assumptions on a beam filament of length $l > 0$ , it is rigorously deducted in [3, Section 2] the following viscoelastic constitutive laws $\begin{matrix} (2.3) & \{\begin{matrix} S = k (φ_{x} + ψ) - \int_{0}^{\infty} μ (s) (φ_{x} + ψ) (t - s) d s, \\ M = b ψ_{x} - \int_{0}^{\infty} g (s) ψ_{x} (t - s) d s, \end{matrix} \end{matrix}$ where μ, g are non-negative relaxation functions, so-called memory kernels. The difference here, when compared to [3], relies on the fact that we are considering the constitutive laws (2.3) in the history framework, whereas in [3, Section 2] the analysis is done with null history (see equations (2.7)–(2.8) therein). Some authors also say finite memory to design memory without history. Hence, with identities (2.3) in hand, one can obtain at least three different viscoelastic Timoshenko systems, depending on where we consider the viscoelastic couplings (2.3) in (2.1), namely, on the shear force only (cf. [3] with respect to null history) and then a partially damped system emerges; or just on the bending moment (see [6,31] for both treatments with or without history) still yielding a partially damped system; or else viscoelastic coupling on both the shear and bending forces by producing a fully damped system (we refer to [20] for a slightly modified problem with history).

On the other hand, when the beam model is subject to unknown temperature distribution, then the principles in thermoelasticity state the stress depends not only on the elastic strain but also on the thermal strain. By following up this setting, and still assuming the Timoshenko hypotheses for thin beams, one can find in [2, Section 2] a precise justification of the following thermoelastic constitutive laws $\begin{matrix} (2.4) & \{\begin{matrix} S = k (φ_{x} + ψ) - σ υ, \\ M = b ψ_{x} - ς ϑ, \end{matrix} \end{matrix}$ where $σ, ς > 0$ are coefficients related to the thermal expansion, and $υ = υ (x, t)$ , $ϑ = ϑ (x, t)$ are temperature components standing for temperature deviations from a reference state along the longitudinal and vertical directions. Therefore, in possession of the thermal identities in (2.4), we can also set at least three distinct thermoelastic Timoshenko systems depending now where we regard the thermal couplings (2.4) in (2.1). For instance, with thermal coupling only on the bending moment, a partially damped system arises (cf. [32] where Fourier’s law is taken into account); under the thermal component solely on the shear force we still obtain a partially damped system (as proposed in [1] again under Fourier’s law); and last by invoking the thermal coupling on both the bending moment and the shear force we arrive at a fully damped thermoelastic Timoshenko system (as studied in [2]). All these possibilities are illustrated in [2, Section 2], as well as the existing literature dealing with other thermal laws for the heat flux of conduction is provided therein (e.g. Gurtin-Pipkin, Maxwell–Cattaneo, Coleman-Gurtin, type III).

Under the above statements, one sees that the stability of (2.1) ultimately depends upon damping’s feedback provided by the viscoelastic coupling (2.3) or the thermoelastic coupling (2.4) on both (or not) forces of the system. We refer, for instance, the stability results in [1–6,9,11,20,31,32,36,37] just to name few. Additionally, by laying down possible hybrid dissipative models generated by (2.3) and (2.4) simultaneously, that is, models featured by mixed damping feedback in thermo-(visco-)elasticity, then new perspectives pop up in what concerns the stability of Timoshenko systems with temperature and memory terms, as one can see, for example, in [7,13,15,16,24,28,29,36,38]. In this way, among all possibilities, we are going to take into account the following two ones thermo-(visco-)elastic laws: $\begin{matrix} (2.5) & I. Bending and Shear Coupling: \{\begin{matrix} S = k (φ_{x} + ψ) - σ υ, \\ M = b ψ_{x} - \int_{0}^{\infty} g (s) ψ_{x} (t - s) d s, \end{matrix} \end{matrix}$ or $\begin{matrix} (2.6) & II. Bending Coupling Only: \{\begin{matrix} S = k (φ_{x} + ψ), \\ M = b ψ_{x} - \int_{0}^{\infty} g (s) ψ_{x} (t - s) d s - σ υ . \end{matrix} \end{matrix}$

In what follows, let us first work with (2.5), and then we check what happens to (2.6).

Case I. Replacing (2.5) in (2.1) we arrive at $\begin{matrix} (2.7) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} + σ υ_{x} = 0 in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - b ψ_{x x} + \int_{0}^{\infty} g (s) ψ_{x x} (t - s) d s + k (φ_{x} + ψ) - σ υ = 0 in (0, l) \times (0, \infty) . \end{matrix} \end{matrix}$ Concerning the temperature deviation, we must provide an equation for the heat flux of conduction. Since the coupling for this variable is given on the shear force (2.5)₁, then relying on the facts presented in [2, Section 2] we can derived the following motion equation $\begin{matrix} (2.8) & ρ_{3} υ_{t} = - q_{x} - σ {(φ_{x} + ψ)}_{t} in (0, l) \times (0, \infty), \end{matrix}$ where $ρ_{3} > 0$ is a constant related to the heat capacity and $q = q (x, t)$ stands for the heat flux. The Fourier and Maxwell–Cattaneo laws for the heat flux are considered in the recent paper [24]. Here, we follow the Green and Naghdi theory, cf. [21,22], to consider the so-called type III thermoelasticity for the heat flux of conduction, namely, $\begin{matrix} (2.9) & q = - δ p_{x} - γ p_{x t} with p_{t} = υ, \end{matrix}$ where $δ, γ > 0$ are constants related to the thermal conductivity, and p stands for the thermal displacement whose time derivative is empirically the temperature as in (2.9). Combining (2.8) and (2.9) and plugging the resulting expression into (2.7), we obtain the following type III thermoelastic Timoshenko system with long memory (history) $\begin{matrix} (2.10) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} + σ υ_{x} = 0 in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - b ψ_{x x} + \int_{0}^{\infty} g (s) ψ_{x x} (t - s) d s + k (φ_{x} + ψ) - σ υ = 0 in (0, l) \times (0, \infty), \\ ρ_{3} υ_{t t} - δ υ_{x x} - γ υ_{x x t} + σ {(φ_{x} + ψ)}_{t t} = 0 in (0, l) \times (0, \infty) . \end{matrix} \end{matrix}$

For the sake of completeness, we consider (2.10) with initial conditions $\begin{matrix} (2.11) & \{\begin{matrix} φ (x, 0) = φ_{0} (x), φ_{t} (x, 0) = φ_{1} (x), \\ ψ (x, s) = ψ_{0} (x, s), ψ_{t} (x, 0) = \partial_{t} ψ (x, t) |_{t = 0} : = ψ_{1} (x), s ⩽ 0, \\ υ (x, 0) = υ_{0} (x), υ_{t} (x, 0) = υ_{1} (x), x \in (0, l), \end{matrix} \end{matrix}$ and either boundary conditions $\begin{matrix} (2.12a) & \{\begin{matrix} φ (x, t) = υ (x, t) = 0 & for x = 0, l; t ⩾ 0, \\ ψ (x, t) = 0 & for x = 0, l; t \in R, \end{matrix} \end{matrix}$ or $\begin{matrix} (2.12b) & \{\begin{matrix} φ_{x} (x, t) = υ (x, t) = 0 & for x = 0, l; t ⩾ 0, \\ ψ (x, t) = 0 & for x = 0, l; t \in R . \end{matrix} \end{matrix}$

Remark 2.1.
As one can see from the above construction up to achieve (2.10), the thermoelastic damping feedback comes from the coupling on the shear force, whereas the viscoelastic dissipative mechanism is hidden by the memory component coupled on the bending moment, as conducted by (2.5). It means that we will be able to extract the strength of both dissipations when dealing with the stability of the solution to (2.10). In conclusion, we have a fully damped system not only from the physical point of view but also from the mathematical one, as will be also proved in Section 3.

In what follows, we introduce two new variables in order to see problem (2.10)–(2.12a)–(2.12b) in a dissipative and autonomous scenario, namely, as given in (1.1)–(1.6a)–(1.6b).

Auxiliary temperature variable. By following a similar idea as introduced in [41, Section 1] (see also [14,36]) we set the new variable concerning the temperature distribution $\begin{matrix} (2.13) & θ (x, t) : = \int_{0}^{t} υ (x, s) d s + \frac{1}{δ} z (x), \end{matrix}$ where $z \in H_{0}^{1} (0, l)$ is the solution of the Cauchy problem $\begin{matrix} (2.14) & \{\begin{matrix} z_{x x} = ρ_{3} υ_{1} - γ υ_{0, x x} + σ (φ_{1, x} + ψ_{1}), & x \in (0, l), \\ z (x) = 0, & x = 0, l . \end{matrix} \end{matrix}$

Note that we can formally write down (2.10)₃ as $\begin{matrix} - δ {\int_{0}^{t} υ_{x x} (\cdot, s) d s + \frac{1}{δ} [ρ_{3} υ_{1} - γ υ_{0, x x} + σ (φ_{1, x} + ψ_{1})]} + ρ_{3} υ_{t} - γ υ_{x x} + σ {(φ_{x} + ψ)}_{t} = 0, \end{matrix}$ and from (2.13)–(2.14) the latter turns into $\begin{matrix} (2.15) & ρ_{3} θ_{t t} - δ θ_{x x} - γ θ_{x x t} + σ {(φ_{x} + ψ)}_{t} = 0 . \end{matrix}$

Relative displacement history. Now, as introduced by Dafermos [10] (see also [19, Section 2]) we consider the relative displacement history with respect to the angle rotation $\begin{matrix} (2.16) & η = η (x, t, s) : = ψ (x, t) - ψ (x, t - s), x \in (0, l), t ⩾ 0, s > 0 . \end{matrix}$ Thus, through (2.16) the equation (2.10)₂ can be rewritten as $\begin{matrix} (2.17) & ρ_{2} ψ_{t t} - (b - \int_{0}^{\infty} g (s) d s) ψ_{x x} - \int_{0}^{\infty} g (s) η_{x x} (s) d s + k (φ_{x} + ψ) - σ υ = 0 . \end{matrix}$ From (2.17) one sees that a supplementary equation with respect to η is necessary to draw up the whole problem. To this purpose, we use again (2.16) and initial-boundary conditions (2.11)–(2.12a)–(2.12b) with respect to ψ to derive formally the next identities $\begin{matrix} (2.18) & \{\begin{matrix} η_{t} + η_{s} = ψ_{t} & in (0, l) \times (0, \infty) \times (0, \infty), \\ η (x, t, s) = 0 & for x = 0, l; t ⩾ 0, s > 0, \\ η (x, t, 0) : = {lim}_{s \to 0} η (x, t, s) = 0 & for x \in (0, l), t ⩾ 0, \\ η (x, 0, s) = ψ_{0} (x) - ψ_{0} (x, - s) : = η_{0} (x, s) & for x \in (0, l), s > 0 . \end{matrix} \end{matrix}$

Therefore, using (2.13)–(2.18), and denoting $β : = b - \int_{0}^{\infty} g (s) d s > 0$ , we can rewrite (2.10)–(2.12a)–(2.12b) as the following equivalent IBVP $\begin{matrix} (2.19) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} + σ θ_{t x} = 0 & in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - β ψ_{x x} - \int_{0}^{\infty} g (s) η_{x x} (s) d s \\ + k (φ_{x} + ψ) - σ θ_{t} = 0 & in (0, l) \times (0, \infty), \\ ρ_{3} θ_{t t} - δ θ_{x x} - γ θ_{x x t} + σ {(φ_{x} + ψ)}_{t} = 0 & in (0, l) \times (0, \infty), \\ η_{t} + η_{s} - ψ_{t} = 0 & in (0, l) \times (0, \infty) \times (0, \infty), \\ (φ, φ_{t}, ψ, ψ_{t}, θ, θ_{t}) (\cdot, 0) = (φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, θ_{0}, θ_{1}) (\cdot) & in (0, l), \\ η (\cdot, 0, s) = η_{0} (\cdot, s), η (\cdot, t, 0) = 0 & in (0, l), s > 0, t ⩾ 0, \\ φ = ψ = θ = η (\cdot, \cdot, s) = 0 or \\ φ_{x} = ψ = θ = η (\cdot, \cdot, s) = 0 & on {0, l} \times [0, \infty), s > 0, \end{matrix} \end{matrix}$ where the relationship between $(θ_{0}, θ_{1})$ and $(υ_{0}, υ_{1})$ is assumed as follows $\begin{matrix} θ_{1} = υ_{0} and ρ_{3} υ_{1} = δ θ_{0, x x} + γ θ_{1, x x} - σ (φ_{1, x} + ψ_{1}) . \end{matrix}$

Finally, we observe that (2.19) corresponds precisely to problem (1.1)–(1.6a)–(1.6b), which is the main object of study in this article.

Case II. In this case, we replace (2.6) in (2.1) to obtain $\begin{matrix} (2.20) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} = 0 in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - b ψ_{x x} + \int_{0}^{\infty} g (s) ψ_{x x} (t - s) d s + k (φ_{x} + ψ) + σ υ_{x} = 0 in (0, l) \times (0, \infty) . \end{matrix} \end{matrix}$ With respect to the heat flux equation, instead of (2.8) we have now (cf. [2, Section 2]) $\begin{matrix} (2.21) & ρ_{3} υ_{t} = - q_{x} - σ ψ_{x t} in (0, l) \times (0, \infty) . \end{matrix}$

Combining (2.21) with (2.9) and substituting the resulting expression in (2.20), we obtain the other type III thermoelastic Timoshenko system with history $\begin{matrix} (2.22) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} = 0 in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - b ψ_{x x} + \int_{0}^{\infty} g (s) ψ_{x x} (t - s) d s + k (φ_{x} + ψ) + σ υ_{x} = 0 in (0, l) \times (0, \infty), \\ ρ_{3} υ_{t t} - δ υ_{x x} - γ υ_{x x t} + σ ψ_{x t t} = 0 in (0, l) \times (0, \infty) . \end{matrix} \end{matrix}$

We note that system (2.22) is precisely the one studied in [30], see problem (1.8) therein.

Last, proceeding similarly (with minor modifications) as in (2.13)–(2.18), one can formally convert (2.22) into the following autonomous equivalent system $\begin{matrix} (2.23) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} = 0 & in (0, l) \times (0, \infty), \\ ρ_{2} ψ_{t t} - β ψ_{x x} - \int_{0}^{\infty} g (s) η_{x x} (s) d s \\ + k (φ_{x} + ψ) + σ θ_{t x} = 0 & in (0, l) \times (0, \infty), \\ ρ_{3} θ_{t t} - δ θ_{x x} - γ θ_{x x t} + σ ψ_{x t} = 0 & in (0, l) \times (0, \infty), \\ η_{t} + η_{s} - ψ_{t} = 0 & in (0, l) \times (0, \infty) \times (0, \infty), \end{matrix} \end{matrix}$ with proper initial-boundary conditions, which corresponds exactly to problem (1.9)–(1.12). Remark 2.2.
It is worth mentioning that the above construction to reach (2.22), and consequently (2.23), reveals us that the thermal and viscoelastic couplings are both given on the bending moment, here conducted by (2.6). It means that we can no longer expect a fully dissipative mechanism. On the contrary, both thermal and viscoelastic damping terms now propagate to the bending moment, and only to the shear force by means of the equal wave speeds assumption (1.8), as already aforementioned in the results of [30]. In conclusion, (2.23) represents a partially damped system which is very different, from the stability point of view, when compared to the fully damped problem (2.19), although close to it in terms of variables and other stuff. The coming mathematical results certify these formal statements that have been physically built.

3. Main result on stability

In this section, our main goal is to prove the exponential stability of solutions to problem (1.1)–(1.6a)–(1.6b). Before doing so, let us first introduce the semigroup setting that will be useful hereafter.

3.1. Semigroup solution

For each boundary condition in (1.6a)–(1.6b), we need to take different phase spaces. Here, we consider $\begin{matrix} H_{1} = H_{0}^{1} (0, l) \times L^{2} (0, l) \times H_{0}^{1} (0, l) \times L^{2} (0, l) \times H_{0}^{1} (0, l) \times L^{2} (0, l) \times M for (1.6a), \end{matrix}$ and $\begin{matrix} H_{2} = H_{*}^{1} (0, l) \times L_{*}^{2} (0, l) \times H_{0}^{1} (0, l) \times L^{2} (0, l) \times H_{0}^{1} (0, l) \times L^{2} (0, l) \times M for (1.6b), \end{matrix}$ where $H_{*}^{1} (0, l) = H^{1} (0, l) \cap L_{*}^{2} (0, l)$ , $L_{*}^{2} (0, l) = {u \in L^{2} (0, l); \frac{1}{l} \int_{0}^{l} u (x) d x = 0}$ and $\begin{matrix} M = {η : (0, \infty) \to H_{0}^{1} (0, l); \int_{0}^{\infty} g (s) {‖ η (s) ‖}_{H_{0}^{1} (0, l)}^{2} d s < \infty} . \end{matrix}$

It is well-known that $H_{j}$ , for each $j = 1, 2$ , is a Hilbert space endowed with norm $\begin{array}{l} ‖ U ‖_{H_{j}}^{2} = & \int_{0}^{l} [ρ_{1} | Φ |^{2} + ρ_{2} | Ψ |^{2} + ρ_{3} | Θ_{x} |^{2} + β | ψ_{x} |^{2} + k | φ_{x} + ψ |^{2} + δ | θ_{x} |^{2} \\ + \int_{0}^{\infty} g (s) {| η_{x} (s) |}^{2} d s] d x, \end{array}$ for $U = (φ, Φ, ψ, Ψ, θ, Θ, η) \in H_{j}$ , and respective scalar product ${(\cdot, \cdot)}_{H_{j}}$ .

Under the above notation and setting $Φ : = φ_{t}$ , $Ψ : = ψ_{t}$ , $Θ : = θ_{t}$ , we can convert the particular system (1.1)–(1.6a)–(1.6b) into the following abstract problem $\begin{matrix} (3.1) & \{\begin{matrix} U_{t} = A_{j} U, t > 0, \\ U (0) = (φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, θ_{0}, θ_{1}, η_{0}) : = U_{0}, \end{matrix} \end{matrix}$ where $A_{j} : D (A_{j}) \subset H_{j} \to H_{j}$ is defined by $\begin{matrix} (3.2) & A_{j} U = (\begin{matrix} Φ \\ \frac{k}{ρ_{1}} {(φ_{x} + ψ)}_{x} - \frac{σ}{ρ_{1}} Θ_{x} \\ Ψ \\ \frac{1}{ρ_{2}} {(β ψ + \int_{0}^{\infty} g (s) η (s) d s)}_{x x} - \frac{k}{ρ_{2}} (φ_{x} + ψ) + \frac{σ}{ρ_{2}} Θ \\ Θ \\ \frac{1}{ρ_{3}} {(δ θ + γ Θ)}_{x x} - \frac{σ}{ρ_{3}} (Φ_{x} + Ψ) \\ Ψ - η_{s} \end{matrix}), \end{matrix}$ for any $U = (φ, Φ, ψ, Ψ, θ, Θ, η) \in D (A_{j})$ , with domain $\begin{array}{l} D (A_{1}) = & {U \in H_{1} | Φ, Ψ, Θ \in H_{0}^{1} (0, l), η_{s} \in M, η (0) = 0, \\ φ, δ θ + γ Θ, β ψ + \int_{0}^{\infty} g (s) η (s) d s \in H^{2} (0, l)} for (1.6a) \end{array}$ and $\begin{array}{l} D (A_{2}) = & {U \in H_{2} | Φ \in H_{*}^{1} (0, l), φ_{x}, Ψ, Θ \in H_{0}^{1} (0, l), η_{s} \in M, η (0) = 0, \\ φ, δ θ + γ Θ, β ψ + \int_{0}^{\infty} g (s) η (s) d s \in H^{2} (0, l)} for (1.6b) . \end{array}$

Under the above construction, it is easy to verify that $0 \in ρ (A_{j})$ , $j = 1, 2$ , and also $A_{j}$ is dissipative on $H_{j}$ with $\begin{array}{rcl} Re {(A U, U)}_{H_{j}} & = & - γ \int_{0}^{l} | Θ_{x} |^{2} d x \\ (3.3) & + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) \int_{0}^{l} {| η_{x} (s) |}^{2} d x d s, \forall U \in D (A_{j}), j = 1, 2 . \end{array}$ To perform integration by parts in the last term of (3.3), we proceed analogously to [19], see pages 162–163 therein.

Therefore, relying on Pazy’s book [34], we have that $A_{j}$ is the infinitesimal generator of a $C_{0}$ -semigroup of contractions $T (t) = e^{A_{j} t}$ on $H_{j}$ , and existence and uniqueness result reads as follows.

Theorem 3.1.
Under the above notations and Assumption 1.1 we have for $j = 1, 2$ :
If $U_{0} \in H_{j}$ , then problem $(3.1)$ has a unique mild solution $U \in C^{0} ([0, \infty), H_{j})$ given by $\begin{matrix} U (t) = e^{A_{j} t} U_{0}, t ⩾ 0 . \end{matrix}$

If $U_{0} \in D (A_{j})$ , then problem $(3.1)$ has a unique regular solution $\begin{matrix} U \in C^{0} ([0, \infty), D (A_{j})) \cap C^{1} ([0, \infty), H_{j}) . \end{matrix}$

If $U_{0} \in D ({A_{j}}^{n})$ , $n ⩾ 2$ integer, then the solution is more regular $\begin{matrix} U \in ⋂_{ν = 0}^{n} C^{n - ν} ([0, \infty), D (A^{ν})) . \end{matrix}$

3.2. Exponential stability

Our main result in this section is given below.

Theorem 3.2 (Exponential Stability).

Under the above notations and Assumption 1.1 , there exist constants $M, m > 0$ independent of $U_{0} \in H_{j}$ such that the semigroup solution $U (t) = e^{A_{j} t} U_{0}$ decays as $\begin{matrix} {‖ U (t) ‖}_{H_{j}} ⩽ M e^{- m t} ‖ U_{0} ‖_{H_{j}}, t > 0, j = 1, 2 . \end{matrix}$ In other words, the thermo-viscoelastic system ( 1.1 )–( 1.6a )–( 1.6b ) is exponentially stable.

The proof of Theorem 3.2 is based on the well-known characterization of exponential stability for $C_{0}$ -semigroups of contractions, cf. [17,23,35]. See also [27, Thm. 1.3.2]. Accordingly, we need to proof the following properties: $\begin{matrix} (3.4) & i R \subset ρ (A_{j}) and \underset{| λ | \to \infty}{lim sup} {‖ {(i λ I_{d} - A_{j})}^{- 1} ‖}_{L (H_{j})} < \infty, j = 1, 2, \end{matrix}$ where $A_{j}$ is defined in (3.2).

3.2.1. Resolvent set

Let us assume by contradiction that $i R ⊄ ρ (A_{j})$ , $j = 1, 2$ . Appealing to Proposition A.3, there exist a constant $ω \in (0, ℓ]$ , $ℓ = ‖ {(- A_{j})}^{- 1} ‖_{L (H_{j})}^{- 1}$ , a sequence $ξ_{n} \in R$ , with $ξ_{n} \to w$ and $| ξ_{n} | < w$ , and a sequence of vector functions $\begin{matrix} (3.5) & U_{n} = (φ_{n}, Φ_{n}, ψ_{n}, Ψ_{n}, θ_{n}, Θ_{n}, η_{n}) \in D (A_{j}) with ‖ U_{n} ‖_{H_{j}} = 1, \end{matrix}$ such that $\begin{matrix} (3.6) & i ξ_{n} U_{n} - A_{j} U_{n} \to 0 in H_{j}, j = 1, 2 . \end{matrix}$ Using the expression for $A_{j}$ given in (3.2), then (3.6) can be rewritten in terms of its components as follows $\begin{matrix} (3.7) & \{\begin{matrix} i ξ_{n} φ_{n} - Φ_{n} \to 0 & in H_{0}^{1} (0, l) or H_{*}^{1} (0, l), \\ i ξ_{n} ρ_{1} Φ_{n} - k {(φ_{n, x} + ψ_{n})}_{x} + σ Θ_{n, x} \to 0 & in L^{2} (0, l) or L_{*}^{2} (0, l), \\ i ξ_{n} ψ_{n} - Ψ_{n} \to 0 & in H_{0}^{1} (0, l), \\ i ξ_{n} ρ_{2} Ψ_{n} + k (φ_{n, x} + ψ_{n}) \\ - {(β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s)}_{x x} - σ Θ_{n} \to 0 & in L^{2} (0, l), \\ i ξ_{n} θ_{n} - Θ_{n} \to 0 & in H_{0}^{1} (0, l), \\ i ξ_{n} ρ_{3} Θ_{n} - {(δ θ_{n} + γ Θ_{n})}_{x x} + σ (Φ_{n, x} + Ψ_{n}) \to 0 & in L^{2} (0, l), \\ i ξ_{n} η_{n} + η_{n, s} - Ψ_{n} \to 0 & in M . \end{matrix} \end{matrix}$

Now, our purpose is to prove that $\begin{matrix} (3.8) & ‖ U_{n} ‖_{H_{j}}^{2} \to 0, j = 1, 2, \end{matrix}$ which provides the desired contradiction with (3.5) and, therefore, the proof of $i R \subset ρ (A_{j})$ , $j = 1, 2$ , is complete.

In what follows, the proof of (3.8) will be done through some lemmas, where due to the nature of the boundary conditions (1.6a) and (1.6b) we first conclude it for $U_{n} \in D (A_{2})$ , $j = 2$ , and then for $U_{n} \in D (A_{1})$ , $j = 1$ .

Lemma 3.3.
Under the assumptions of Theorem 3.2 and the above notations, let $U_{n}$ be the sequence satisfying ( 3.6 ). Then,
$Θ_{n}, θ_{n} \to 0$ in $H_{0}^{1} (0, l)$ , as $n \to \infty$ ;

$\int_{0}^{\infty} [- g^{'} (s)] ‖ η_{n, x} (s) ‖_{L^{2}}^{2} d s \to 0$ , as $n \to \infty$ ;

$η_{n} \to 0$ in $M$ , as $n \to \infty$ .

Proof.
From (3.3) and (3.6) we promptly get $\begin{matrix} γ ‖ Θ_{n, x} ‖_{L^{2}}^{2} + \frac{1}{2} \int_{0}^{\infty} [- g^{'} (s)] {‖ η_{n, x} (s) ‖}_{L^{2}}^{2} d s = Re {(i ξ_{n} U_{n} - A_{j} U_{n}, U_{n})}_{H_{j}} \to 0 . \end{matrix}$ From this, (3.7)₅ and condition (1.7), the three statements of Lemma 3.3 are fulfilled. □

Under the limits of Lemma 3.3, the convergences of (3.7) can be reduced into the next ones $\begin{array}{l} (3.9a) & i ξ_{n} φ_{n} - Φ_{n} \to 0 in H_{0}^{1} (0, l) or H_{}^{1} (0, l), \\ (3.9b) & i ξ_{n} ρ_{1} Φ_{n} - k {(φ_{n, x} + ψ_{n})}_{x} \to 0 in L^{2} (0, l) or L_{}^{2} (0, l), \\ (3.9c) & i ξ_{n} ψ_{n} - Ψ_{n} \to 0 in H_{0}^{1} (0, l), \\ (3.9d) & i ξ_{n} ρ_{2} Ψ_{n} + k (φ_{n, x} + ψ_{n}) - {(β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s)}_{x x} \to 0 in L^{2} (0, l), \\ (3.9e) & i ξ_{n} η_{n} + η_{n, s} - Ψ_{n} \to 0 in M . \end{array}$
Lemma 3.4.
Under the assumptions of Theorem 3.2 and the above notations, let $U_{n}$ be the sequence satisfying ( 3.6 ). Then, $\begin{matrix} (3.10) & Ψ_{n}, ψ_{n} \to 0 in H_{0}^{1} (0, l) . \end{matrix}$
Proof.
Firstly, from (3.9c) we see that $\begin{matrix} (3.11) & i ξ_{n} {(ψ_{n, x}, Ψ_{n, x})}_{L^{2}} - ‖ Ψ_{n, x} ‖_{L^{2}}^{2} \to 0 . \end{matrix}$ Using Cauchy–Schwarz and Young’s inequalities, we get $\begin{matrix} (3.12) & ‖ Ψ_{n, x} ‖_{L^{2}}^{2} ⩽ 2 | i ξ_{n} {(ψ_{n, x}, Ψ_{n, x})}_{L^{2}} - ‖ Ψ_{n, x} ‖_{L^{2}}^{2} | + ξ_{n}^{2} ‖ ψ_{n, x} ‖_{L^{2}}^{2} . \end{matrix}$ Thus, from (3.11)–(3.12), since $ξ_{n}$ is bounded and $‖ ψ_{n, x} ‖_{L^{2}}^{2} ⩽ \frac{1}{β} ‖ U_{n} ‖_{H_{j}}^{2}$ , we obtain that ${(‖ Ψ_{n, x} ‖_{L^{2}})}_{n \in N}$ is bounded.

On the other hand, since $η_{n} \in M$ , we have $g ‖ η_{x} (\cdot) ‖_{L^{2}}^{2} \in L^{1} ((0, \infty))$ and also (see again [19]) $\begin{matrix} (3.13) & lim_{z \to \infty} g (z) {‖ η_{n, x} (z) ‖}_{L^{2}}^{2} = 0 . \end{matrix}$

We can see that $s \mapsto \frac{1}{ξ_{n}} Ψ_{n} \in M$ happens for any $n \in N$ . Now, taking the multiplier $\frac{1}{ξ_{n}^{2}} g (s) {\overline{Ψ}}_{n}$ in (3.9e) and integrating on $(0, l) \times (0, \infty)$ , we have $\begin{matrix} (3.14) & i {(η_{n} (\cdot), \frac{Ψ_{n}}{ξ_{n}^{2}})}_{M} + \underset{: = S_{n}}{\underset{︸}{\frac{1}{ξ_{n}^{2}} {(η_{n, s} (\cdot), Ψ_{n})}_{M}}} - \frac{1}{ξ_{n}^{2}} {(Ψ_{n}, Ψ_{n})}_{M} \to 0 . \end{matrix}$ We claim that $S_{n} \to 0$ . Indeed, let us fix $n \in N$ . Integrating $S_{n}$ by parts with respect to s, using Lemma 3.3, the fact that ${(Ψ_{n})}_{n \in N}$ is bounded in $H_{0}^{1} (0, l)$ , (3.13), and Fubini’s Theorem, we get $\begin{array}{rcl} | S_{n} | & = & | - \frac{1}{ξ_{n}^{2}} \int_{0}^{\infty} g (s) {(η_{n, s} (s), Ψ_{n})}_{H_{0}^{1}} d s | \\ = & | - \frac{1}{ξ_{n}^{2}} \int_{0}^{l} \overline{Ψ_{n, x}} (g (s) η_{n, x} (s) |_{0}^{\infty} - \int_{0}^{\infty} g^{'} (s) η_{n, x} (s) d s) d x | \\ ⩽ & \frac{1}{ξ_{n}^{2}} ‖ Ψ_{n, x} ‖_{L^{2}} {‖ \int_{0}^{\infty} g^{'} (s) η_{n, x} (s) d s ‖}_{L^{2}} \\ ⩽ & \frac{1}{ξ_{n}^{2}} {(\int_{0}^{\infty} [- g^{'} (s)] d s)}^{\frac{1}{2}} {(\int_{0}^{\infty} [- g^{'} (s)] {‖ η_{n, x} (s) ‖}_{L^{2}}^{2} d s)}^{\frac{1}{2}} ‖ Ψ_{n, x} ‖_{L^{2}} \to 0 . \end{array}$ Thus, since $g (0) < \infty$ , (3.14) and Lemma 3.3 imply $Ψ_{n} \to 0$ in $H_{0}^{1} (0, l)$ and, consequently, (3.9c) yields $ψ_{n} \to 0$ in $H_{0}^{1} (0, l)$ , which proves (3.10) as desired. □

The next result still holds for both boundary conditions at the same time.
Lemma 3.5.
Under the assumptions of Theorem 3.2 and the above notations, let $U_{n}$ be the sequence satisfying ( 3.6 ). Then, $\begin{matrix} (3.15) & - ρ_{1} ‖ Φ_{n} ‖_{L^{2}}^{2} + k ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} \to 0, \end{matrix}$ and $\begin{matrix} (3.16) & - (β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s) \cdot \overline{φ_{n, x}} |_{0}^{l} + k ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} \to 0 . \end{matrix}$
Proof.
On the one hand, taking the multipliers $ρ_{1} \overline{Φ_{n}}$ in (3.9a) and $\overline{φ_{n}}$ in (3.9b) and adding the resulting expressions, we obtain $\begin{matrix} i ρ_{1} ξ_{n} [{(φ_{n}, Φ_{n})}_{L^{2}} + {(Φ_{n}, φ_{n})}_{L^{2}}] - ρ_{1} ‖ Φ_{n} ‖_{L^{2}}^{2} - k {({(φ_{n, x} + ψ_{n})}_{x}, φ_{n})}_{L^{2}} \to 0 . \end{matrix}$ Integrating by parts and taking the real part, we have $\begin{matrix} (3.17) & - ρ_{1} ‖ Φ_{n} ‖_{L^{2}}^{2} + k Re {(φ_{n, x} + ψ_{n}, φ_{n, x})}_{L^{2}} \to 0 . \end{matrix}$

On the other hand, taking the multiplier $ρ_{2} Ψ_{n}$ in (3.9c) and $\overline{ψ_{n}}$ in (3.9d), and adding the resulting convergences, we get $\begin{array}{l} i ρ_{2} ξ_{n} [{(ψ_{n}, Ψ_{n})}_{L^{2}} + {(Ψ_{n}, ψ_{n})}_{L^{2}}] - {({(β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s)}_{x x}, ψ_{n})}_{L^{2}} \\ + k {(φ_{n, x} + ψ_{n}, ψ_{n})}_{L^{2}} - ρ_{2} ‖ Ψ_{n} ‖_{L^{2}}^{2} \to 0 . \end{array}$ Performing integration by parts and using Lemmas 3.3 and 3.4, we infer $\begin{matrix} (3.18) & k Re {(φ_{n, x} + ψ_{n}, ψ_{n})}_{L^{2}} \to 0 . \end{matrix}$ Computing (3.17) + (3.18), we arrive at (3.15).

Now, taking the multiplier $k ({\overline{φ}}_{n, x} + {\overline{ψ}}_{n})$ in (3.9d), we can write $\begin{array}{l} i β_{n} ρ_{2} k {(Ψ_{n}, φ_{n, x} + ψ_{n})}_{L^{2}} - {({(β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s)}_{x x}, k (φ_{n, x} + ψ_{n}))}_{L^{2}} \\ + k^{2} ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} \to 0 . \end{array}$ Performing integration by parts, using the limits $η_{n} \to 0$ in $M$ , $ψ_{n}, Ψ_{n} \to 0$ in $H_{0}^{1} (0, l)$ and the boundedness of ${(φ_{n, x} + ψ_{n})}_{n \in N}$ in $L^{2} (0, l)$ , we get $\begin{array}{l} - k (β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s) \cdot (\overline{φ_{n, x}} + \overline{ψ_{n}}) |_{0}^{l} + k^{2} ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} \\ (3.19) & + {(β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s, k {(φ_{n, x} + ψ_{n})}_{x})}_{L^{2}} \to 0 . \end{array}$ On the other hand, taking the multiplier $β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s$ in (3.9b), we see $\begin{array}{l} - i ρ_{1} ξ_{n} {(β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s, Φ_{n})}_{L^{2}} \\ - {(β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s, k {(φ_{n, x} + ψ_{n})}_{x})}_{L^{2}} \to 0 . \end{array}$ Using that ${(‖ Φ_{n} ‖_{L^{2}})}_{n \in N}$ is bounded and $‖ β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s ‖_{L^{2}} \to 0$ , we have from the previous limit that $\begin{matrix} (3.20) & - {(β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s, k {(φ_{n, x} + ψ_{n})}_{x})}_{L^{2}} \to 0 . \end{matrix}$ Computing (3.19) + (3.20), we obtain (3.16), which completes the proof of Lemma 3.5. □

We are finally in the position to conclude the proof of (3.8). We proceed in two cases as follows.

Proof of ( 3.8 ) for $j = 2$ . In this case, we are dealing with boundary condition (1.6b). Thus, from (3.16) we directly have $‖ φ_{n, x} + ψ_{n} ‖_{L^{2}} \to 0$ , which in turn combined with (3.15) leads to $ρ_{1} ‖ Φ_{n} ‖_{L^{2}}^{2} \to 0$ . From these limits and Lemmas 3.3–3.5, we conclude $\begin{matrix} ‖ U_{n} ‖_{H_{2}}^{2} \to 0 . \end{matrix}$

Proof of ( 3.8 ) for $j = 1$ . In this case, we are under the boundary condition (1.6a) and more computations are necessary. Initially, we observe that Lemmas 3.3–3.5 imply that (3.9a)–(3.9e) can be rewritten as $\begin{array}{l} (3.21a) & i ξ_{n} φ_{n} - Φ_{n} \to 0 in H_{0}^{1} (0, l) or H_{}^{1} (0, l), \\ (3.21b) & i ξ_{n} ρ_{1} Φ_{n} - k φ_{n, x x} \to 0 in L^{2} (0, l) or L_{}^{2} (0, l), \\ (3.21c) & k φ_{n, x} - {(β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s)}_{x x} \to 0 in L^{2} (0, l), \\ (3.21d) & η_{n, s} (s) \to 0 in M . \end{array}$

Taking the multiplier $(x - \frac{l}{2}) (β \overline{ψ_{n, x}} + \int_{0}^{\infty} g (s) \overline{η_{n, x}} d s)$ in (3.21c), we have $\begin{array}{l} {(k φ_{n, x} (x - \frac{l}{2}), β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s)}_{L^{2}} \\ (3.22) & - {({[β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s]}_{x x} (x - \frac{l}{2}), β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s)}_{L^{2}} \to 0 . \end{array}$

Using, in (3.22), that $‖ β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s ‖_{L^{2}} \to 0$ and ${(‖ φ_{n, x} ‖_{L^{2}})}_{n \in N}$ is bounded, and taking the real part in resulting convergence, we obtain $\begin{matrix} - Re {({[β ψ_{n} + \int_{0}^{\infty} g (s) η_{n} (s) d s]}_{x x} (x - \frac{l}{2}), β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s)}_{L^{2}} \to 0 . \end{matrix}$ Now, using $\frac{d}{d x} | u |^{2} = 2 Re (\overline{u} u_{x})$ and performing integration by parts, we get $\begin{array}{l} \frac{l}{2} ({| β ψ_{n, x} (0) + \int_{0}^{\infty} g (s) η_{n, x} (0, s) d s |}^{2} + {| β ψ_{n, x} (l) + \int_{0}^{\infty} g (s) η_{n, x} (l, s) d s |}^{2}) \\ - {‖ β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s ‖}_{L^{2}}^{2} \to 0 . \end{array}$ Again, using $‖ β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s ‖_{L^{2}} \to 0$ , we arrive at $\begin{matrix} (3.23) & β ψ_{n, x} (0) + \int_{0}^{\infty} g (s) η_{n, x} (0, s) d s, β ψ_{n, x} (l) + \int_{0}^{\infty} g (s) η_{n, x} (l, s) d s \to 0 . \end{matrix}$

On the other hand, taking the multiplier $(x - \frac{l}{2}) (φ_{n, x} + ψ_{n})$ in (3.21b), we infer $\begin{matrix} (3.24) & ρ_{1} i ξ_{n} {(Φ_{n} (x - \frac{l}{2}), φ_{n, x} + ψ_{n})}_{L^{2}} - k {({(φ_{n, x} + ψ_{n})}_{x} (x - \frac{l}{2}), φ_{n, x} + ψ_{n})}_{L^{2}} \to 0 . \end{matrix}$ From convergences (3.9a) and (3.9c), we deduce $\begin{matrix} (3.25) & - i ξ_{n} \overline{(φ_{n, x} + ψ_{n})} - \overline{(Φ_{n, x} + Ψ_{n})} \to 0 in L^{2} (0, l) . \end{matrix}$ Taking the multiplier $ρ_{1} Φ_{n} (x - \frac{l}{2})$ in (3.25), we obtain $\begin{matrix} (3.26) & - ρ_{1} i ξ_{n} {(Φ_{n} (x - \frac{l}{2}), φ_{n, x} + ψ_{n})}_{L^{2}} - ρ_{1} {(Φ_{n} (x - \frac{l}{2}), Φ_{n, x} + Ψ_{n})}_{L^{2}} \to 0 . \end{matrix}$ Computing (3.24) + (3.26), we get $\begin{matrix} (3.27) & k \underset{: = R_{n}}{\underset{︸}{{({(φ_{n, x} + ψ_{n})}_{x} (x - \frac{l}{2}), φ_{n, x} + ψ_{n})}_{L^{2}}}} + ρ_{1} {(Φ_{n} (x - \frac{l}{2}), Φ_{n, x} + Ψ_{n})}_{L^{2}} \to 0 . \end{matrix}$ Integrating by parts, we obtain $\begin{array}{rcl} Re {(Φ_{n} (x - \frac{l}{2}), Φ_{n, x})}_{L^{2}} & = & \frac{1}{2} \int_{0}^{l} (x - \frac{l}{2}) \frac{d}{d x} | Φ_{n} |^{2} d x \\ (3.28) & = & \frac{l}{4} ({| Φ_{n} (0) |}^{2} + {| Φ_{n} (l) |}^{2}) - \frac{1}{2} ‖ Φ_{n} ‖_{L^{2}}^{2} . \end{array}$ Similarly, we get $\begin{matrix} (3.29) & Re (R_{n}) = \frac{l}{4} ({| φ_{n, x} (0) + ψ_{n} (0) |}^{2} + {| φ_{n, x} (l) + ψ_{n} (l) |}^{2}) - \frac{1}{2} ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} . \end{matrix}$ Taking the real part in (3.27) and using the identities given in (3.28)–(3.29), we obtain $\begin{array}{l} \frac{ρ_{1} l}{4} ({| Φ_{n} (0) |}^{2} + {| Φ_{n} (l) |}^{2}) - \frac{ρ_{1}}{2} ‖ Φ_{n} ‖_{L^{2}}^{2} + \frac{k l}{4} ({| φ_{n, x} (0) + ψ_{n} (0) |}^{2} + {| φ_{n, x} (l) + ψ_{n} (l) |}^{2}) \\ (3.30) & - \frac{k}{2} ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} \to 0 . \end{array}$

Combining (3.5) and Lemmas 3.3–3.5, we deduce $\begin{matrix} (3.31) & \frac{k}{2} ‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} + \frac{ρ_{1}}{2} ‖ Φ_{n} ‖_{L^{2}}^{2} \to \frac{1}{2} . \end{matrix}$ Computing (3.30) + (3.31), we conclude $\begin{matrix} (3.32) & \frac{l}{2} (ρ_{1} {| Φ_{n} (0) |}^{2} + ρ_{1} {| Φ_{n} (l) |}^{2} + k {| φ_{n, x} (0) + ψ_{n} (0) |}^{2} + k {| φ_{n, x} (l) + ψ_{n} (l) |}^{2}) \to 1 . \end{matrix}$

We claim that ${(φ_{n, x} (0))}_{n \in N}$ and ${(φ_{n, x} (l))}_{n \in N}$ are bounded. Indeed, let us consider $p \in [0, l]$ . Due to continuous embedding $H_{0}^{1} (0, l) \subset C ([0, l])$ , we infer $| ψ_{n} (p) | ⩽ C ‖ ψ_{n, x} ‖_{L^{2}}$ . Thus, from Lemma 3.4 we have $| ψ_{n} (0) |, | ψ_{n} (l) | \to 0$ . Further, since $| φ_{n, x} (p) |^{2} ⩽ 2 | φ_{n, x} (p) + ψ_{n} (p) |^{2} + 2 | ψ_{n} (p) |^{2}$ in particular for $p = 0$ or $p = l$ , the desired follows from (3.32). Therefore, from convergences in (3.23), we obtain $\begin{matrix} (β ψ_{n, x} + \int_{0}^{\infty} g (s) η_{n, x} (s) d s) \overline{φ_{n, x}} |_{0}^{l} \to 0 . \end{matrix}$ Going back to (3.16) we obtain $‖ φ_{n, x} + ψ_{n} ‖_{L^{2}}^{2} \to 0$ and, from Lemma 3.5, we have $ρ_{1} ‖ Φ_{n} ‖_{L^{2}}^{2} \to 0$ . From this and again from Lemmas 3.3–3.5, we finally concluded $\begin{matrix} ‖ U_{n} ‖_{H_{1}}^{2} \to 0 . \end{matrix}$
3.2.2. Resolvent’s upper bound

In order to prove the second property in (3.4), it is enough to prove that $\begin{matrix} (3.33) & ‖ U ‖_{H_{j}} = {‖ {(i λ I_{d} - A_{j})}^{- 1} F ‖}_{H_{j}} ⩽ C ‖ F ‖_{H_{j}}, as | λ | \to \infty, j = 1, 2, \end{matrix}$ for any $F \in H_{j}$ . Thus, given $F = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}) \in H_{j}$ , let us start by considering the resolvent equation $\begin{matrix} (3.34) & i λ U - A_{j} U = F, j = 1, 2, \end{matrix}$ with $U = (φ, Φ, ψ, Ψ, θ, Θ, η) \in D (A_{j})$ , where we recall that $A_{j}$ is defined in (3.2). In addition, equation (3.34) written in terms of its components turns into $\begin{array}{l} (3.35) & i λ φ - Φ = f_{1}, \\ (3.36) & i λ ρ_{1} Φ - k {(φ_{x} + ψ)}_{x} + σ Θ_{x} = ρ_{1} f_{2}, \\ (3.37) & i λ ψ - Ψ = f_{3}, \\ (3.38) & i λ ρ_{2} Ψ + k (φ_{x} + ψ) - β ψ_{x x} - \int_{0}^{\infty} g (s) η_{x x} (s) d s - σ Θ = ρ_{2} f_{4}, \\ (3.39) & i λ θ - Θ = f_{5}, \\ (3.40) & i λ ρ_{3} Θ - δ θ_{x x} - γ Θ_{x x} + σ (Φ_{x} + Ψ) = ρ_{3} f_{6}, \\ (3.41) & i λ η + η_{s} - Ψ = f_{7} . \end{array}$

The proof of (3.33) will be done as a consequence of some lemmas provided below in combination with the observability result, more precisely Corollary A.6. Hereafter, we simplify the notation by using the same parameter $C > 0$ to denote different constants. Besides, Hölder and Poincaré’s inequalities, as well as $| λ | > 1$ large enough, will be used several times, possibly with no mention.

Lemma 3.6.

Under the assumptions of Theorem 3.2 and the above notations, there exists a constant $C > 0$ such that $\begin{matrix} (3.42) & \int_{0}^{\infty} [- g^{'} (s)] {‖ η_{x} (s) ‖}_{L^{2}}^{2} d s, ‖ Θ_{x} ‖_{L^{2}}^{2}, ‖ η ‖_{M}^{2} ⩽ C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}}, j = 1, 2 . \end{matrix}$

Proof.

Taking the inner product of U with (3.34) in $H_{j}$ , and using (3.3), we have $\begin{matrix} (3.43) & γ ‖ Θ_{x} ‖_{L^{2}}^{2} + \frac{1}{2} \int_{0}^{\infty} [- g^{'} (s)] {‖ η_{x} (s) ‖}_{L^{2}}^{2} d s ⩽ ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}}, \end{matrix}$ from where it follows the first two estimates in (3.42). Also, from (1.7) and (3.43), one easily concludes that $\begin{matrix} ‖ η ‖_{M}^{2} ⩽ \frac{1}{k_{1}} \int_{0}^{\infty} [- g^{'} (s)] {‖ η_{x} (s) ‖}_{L^{2}}^{2} d s ⩽ \frac{2}{k_{1}} ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} . \end{matrix}$ □

Lemma 3.7.

Under the assumptions of Theorem 3.2 and the above notations, there exists a constant $C > 0$ such that $\begin{matrix} (3.44) & ‖ θ_{x} ‖_{L^{2}}^{2}, ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{2} ⩽ C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}}, j = 1, 2 . \end{matrix}$

Proof.

Deriving (3.39) and taking the multiplier $\overline{θ_{x}}$ in the resulting equation, we have $\begin{matrix} i λ \int_{0}^{l} | θ_{x} |^{2} d x = \int_{0}^{l} Θ_{x} \overline{θ_{x}} d x + \int_{0}^{l} f_{5, x} \overline{θ_{x}} d x . \end{matrix}$ From Young’s inequality and (3.42), we deduce the first estimate in (3.44), with $| λ | > 1$ large enough. The estimate for $δ θ_{x} + γ Θ_{x}$ follows from the first part and (3.42). □

Lemma 3.8.

Under the assumptions of Theorem 3.2 and the above notations, there exists a constant $C > 0$ such that $\begin{matrix} (3.45) & ρ_{2} ‖ Ψ ‖_{L^{2}}^{2} ⩽ C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + C ‖ η ‖_{M} (\frac{‖ Φ ‖_{L^{2}}}{| λ |} + ‖ ψ_{x} ‖_{L^{2}}), j = 1, 2 . \end{matrix}$

Proof.

Taking the multiplier $\int_{0}^{\infty} g (s) \overline{η (s)} d s$ in (3.38) and performing integration by parts, we get $\begin{array}{rcl} ρ_{2} \int_{0}^{l} \int_{0}^{\infty} g (s) f_{4} \overline{η (s)} d s d x & = & \underset{: = R_{1}}{\underset{︸}{i λ ρ_{2} \int_{0}^{l} \int_{0}^{\infty} g (s) Ψ \overline{η (s)} d s d x}} - σ \int_{0}^{l} \int_{0}^{\infty} g (s) Θ \overline{η (s)} d s d x \\ + β \int_{0}^{l} \int_{0}^{\infty} g (s) \overline{η_{x} (s)} ψ_{x} d s d x + \int_{0}^{l} {| \int_{0}^{\infty} g (s) η_{x} (s) d s |}^{2} d x \\ + k \int_{0}^{l} \int_{0}^{\infty} g (s) (φ_{x} + ψ) \overline{η (s)} d s d x . \end{array}$ Using (3.41) in $R_{1}$ , we obtain $\begin{array}{l} ρ_{2} \int_{0}^{l} \int_{0}^{\infty} g (s) | Ψ |^{2} d s d x \\ = ρ_{2} \int_{0}^{l} \int_{0}^{\infty} g (s) Ψ \overline{f_{5}} d s d x - \underset{: = R_{2}}{\underset{︸}{ρ_{2} \int_{0}^{l} \int_{0}^{\infty} g (s) Ψ \overline{η_{s} (s)} d s d x}} \\ - β \int_{0}^{l} \int_{0}^{\infty} g (s) \overline{η_{x} (s)} ψ_{x} d s d x - \underset{: = R_{3}}{\underset{︸}{\int_{0}^{l} {| \int_{0}^{\infty} g (s) η_{x} (s) d s |}^{2} d x}} \\ - \underset{: = R_{4}}{\underset{︸}{k \int_{0}^{l} \int_{0}^{\infty} g (s) (φ_{x} + ψ) \overline{η (s)} d s d x}} + ρ_{2} \int_{0}^{l} \int_{0}^{\infty} g (s) f_{4} \overline{η (s)} d s d x \\ (3.46) & + σ \int_{0}^{l} \int_{0}^{\infty} g (s) Θ \overline{η (s)} d s d x . \end{array}$ Performing integration by parts in $R_{2}$ , applying Cauchy–Schwarz and Young’s inequalities, we have $\begin{array}{rcl} (3.47) & | R_{2} | ⩽ ρ_{2} \sqrt{b_{0}} ‖ Ψ ‖_{L^{2}} {(- \int_{0}^{\infty} g^{'} (s) {‖ η (s) ‖}_{L^{2}}^{2} d s)}^{\frac{1}{2}}, \end{array}$ where we have used the notation $b_{0} : = \int_{0}^{\infty} g (s) d s$ (and then $β = b - b_{0}$ ). Moreover, applying Hölder inequality, we get $\begin{matrix} (3.48) & | R_{3} | ⩽ \int_{0}^{l} {(\int_{0}^{\infty} g (s) | η_{x} (s) | d s)}^{2} d x ⩽ b_{0} ‖ η ‖_{M}^{2} . \end{matrix}$ Also, equations (3.35) and (3.37) yield $\begin{array}{rcl} R_{4} & = & - \frac{i k}{λ} \int_{0}^{l} \int_{0}^{\infty} g (s) \overline{η_{x} (s)} d s Φ d x - \frac{i k}{λ} \int_{0}^{l} \int_{0}^{\infty} g (s) \overline{η_{x} (s)} d s f_{1} d x \\ + \frac{i k}{λ} \int_{0}^{l} \int_{0}^{\infty} g (s) \overline{η (s)} d s f_{3} d x + \frac{i k}{λ} \int_{0}^{l} \int_{0}^{\infty} g (s) \overline{η (s)} d s Ψ d x . \end{array}$ Using Poincaré’s inequality, we get $\begin{matrix} (3.49) & | R_{4} | ⩽ C ‖ η ‖_{M} ‖ Φ ‖_{L^{2}} + C ‖ η ‖_{M} ‖ f_{1, x} + f_{3} ‖_{L^{2}} + \frac{C}{| λ |} ‖ η ‖_{M} ‖ f_{3, x} ‖_{L^{2}} + \frac{C}{| λ |} ‖ η ‖_{M} ‖ Ψ ‖_{L^{2}} . \end{matrix}$ Applying Hölder’s inequality in (3.46) and estimates (3.47)–(3.49), we can estimate $\begin{array}{rcl} ρ_{2} ‖ Ψ ‖_{L^{2}}^{2} & ⩽ & C ‖ Ψ ‖_{L^{2}} ‖ f_{5} ‖_{M} + C ‖ Ψ ‖_{L^{2}}^{2} + C \int_{0}^{\infty} [- g^{'} (s)] {‖ η_{x} (s) ‖}_{L^{2}}^{2} d s + C ‖ η ‖_{M}^{2} \\ + C ‖ ψ_{x} ‖_{L^{2}} ‖ η ‖_{M} + C ‖ η ‖_{M} ‖ f_{4} ‖_{L^{2}} + \frac{C}{| λ |} ‖ η ‖_{M} ‖ Φ ‖_{L^{2}} + C ‖ η ‖_{M} ‖ Θ ‖_{L^{2}} \\ + \frac{C}{| λ |} ‖ η ‖_{M} ‖ f_{1, x} + f_{3} ‖_{L^{2}} + \frac{C}{| λ |} ‖ η ‖_{M} ‖ f_{3, x} ‖_{L^{2}} + \underset{: = R_{5}}{\underset{︸}{\frac{C}{| λ |} ‖ η ‖_{M} (‖ Ψ ‖_{L^{2}})}} . \end{array}$ Applying Young’s inequality in $R_{5}$ its follow that $\begin{matrix} ρ_{2} ‖ Ψ ‖_{L^{2}}^{2} ⩽ C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + C ‖ η ‖_{M} ‖ ψ_{x} ‖_{L^{2}} + \frac{C}{| λ |} ‖ η ‖_{M} ‖ Φ ‖_{L^{2}} + C ‖ η ‖_{M} ‖ Θ ‖_{L^{2}} . \end{matrix}$ From this and Lemma 3.6, we finally arrive at (3.45) as desired. □

Lemma 3.9.

Under the assumptions of Theorem 3.2 and the above notations, there exists a constant $C > 0$ such that $\begin{matrix} (3.50) & β ‖ ψ_{x} ‖_{L^{2}}^{2} ⩽ C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + \frac{C}{| λ |} ‖ φ_{x} + ψ ‖_{L^{2}} ‖ Ψ ‖_{L^{2}} + C ‖ Ψ ‖_{L^{2}}^{2}, j = 1, 2 . \end{matrix}$

Proof.

Taking the multiplier $\overline{ψ}$ in (3.38) we obtain $\begin{array}{l} \underset{: = S_{1}}{\underset{︸}{i λ ρ_{2} \int_{0}^{l} Ψ \overline{ψ} d x}} + β \int_{0}^{l} | ψ_{x} |^{2} d x + \int_{0}^{l} \int_{0}^{\infty} g (s) η_{x} (s) \overline{ψ_{x}} d s d x + k \int_{0}^{l} φ_{x} \overline{ψ} d x \\ + k \int_{0}^{l} | ψ |^{2} d x - σ \int_{0}^{l} Θ \overline{ψ} d x = ρ_{2} \int_{0}^{l} f_{4} \overline{ψ} d x . \end{array}$ Replacing (3.37) in $S_{1}$ , we get $\begin{array}{rcl} β \int_{0}^{l} | ψ_{x} |^{2} d x & = & - \int_{0}^{l} \int_{0}^{\infty} g (s) η_{x} (s) \overline{ψ_{x}} d s d x - \underset{: = S_{2}}{\underset{︸}{k \int_{0}^{l} (φ_{x} + ψ) \overline{ψ} d x}} \\ + ρ_{2} \int_{0}^{l} f_{4} \overline{ψ} d x + ρ_{2} \int_{0}^{l} Ψ \overline{f_{3}} d x + ρ_{2} \int_{0}^{l} | Ψ |^{2} d x \\ + σ \int_{0}^{l} Θ \overline{ψ} d x . \end{array}$ Inserting now (3.37) in $S_{2}$ , we obtain $\begin{array}{rcl} β \int_{0}^{l} | ψ_{x} |^{2} d x & = & - \int_{0}^{l} \int_{0}^{\infty} g (s) η_{x} (s) \overline{ψ_{x}} d s d x + \frac{i k}{λ} \int_{0}^{l} (φ_{x} + ψ) \overline{Ψ} d x + σ \int_{0}^{l} Θ \overline{ψ} d x \\ + \frac{i k}{λ} \int_{0}^{l} (φ_{x} + ψ) \overline{f_{3}} d x + ρ_{2} \int_{0}^{l} f_{4} \overline{ψ} d x + ρ_{2} \int_{0}^{l} Ψ \overline{f_{3}} d x + ρ_{2} \int_{0}^{l} | Ψ |^{2} d x . \end{array}$ From the above identity we can conclude (3.50) for some constant $C > 0$ and $| λ | > 1$ large enough. □

Corollary 3.10.

Given $ε > 0$ , there exists $C_{ε} > 0$ such that $\begin{matrix} (3.51) & ‖ ψ_{x} ‖_{L^{2}}^{2} ⩽ ε ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2}, j = 1, 2 . \end{matrix}$

Proof.

Combining Lemmas 3.8 and 3.9, we get $\begin{array}{rcl} ‖ ψ_{x} ‖_{L^{2}}^{2} & ⩽ & \frac{C}{| λ |} ‖ φ_{x} + ψ ‖_{L^{2}} [‖ F ‖_{H_{j}}^{\frac{1}{2}} ‖ U ‖_{H_{j}}^{\frac{1}{2}} + ‖ η ‖_{M}^{\frac{1}{2}} (\frac{‖ Φ ‖_{L^{2}}^{\frac{1}{2}}}{| λ |^{\frac{1}{2}}} + ‖ ψ_{x} ‖_{L^{2}}^{\frac{1}{2}})] \\ + C ‖ η ‖_{M} (\frac{‖ Φ ‖_{L^{2}}}{| λ |} + ‖ ψ_{x} ‖_{L^{2}}) + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} \\ ⩽ & C ‖ U ‖_{H_{j}}^{\frac{3}{2}} ‖ F ‖_{H_{j}}^{\frac{1}{2}} + C ‖ U ‖_{H_{j}}^{\frac{3}{2}} ‖ η ‖_{M}^{\frac{1}{2}} + C ‖ U ‖_{H_{j}} ‖ η ‖_{M} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}}, \end{array}$ for some $C > 0$ . Therefore, from Young’s inequality and Lemma 3.6 along with Young’s inequality once more, one can conclude (3.51). □

In the next result, in order to keep the estimates for both cases $j = 1$ and $j = 2$ , we are going to work with local estimates by means of cut-off functions to avoid point-wise terms. This procedure has been used e.g. in [2,5] and here we adopt the same strategy.

Let us consider $l_{0} \in (0, l)$ and $δ_{0} > 0$ arbitrary numbers such that $(l_{0} - δ_{0}, l_{0} + δ_{0}) \subset (0, l)$ , and a function $s \in C^{2} (0, l)$ satisfying $\begin{matrix} (3.52) & supp s \subset (l_{0} - δ_{0}, l_{0} + δ_{0}), 0 ⩽ s (x) ⩽ 1, x \in (0, l), \end{matrix}$ and $\begin{matrix} (3.53) & s (x) = 1 for x \in [l_{0} - δ_{0} / 2, l_{0} + δ_{0} / 2] . \end{matrix}$

Lemma 3.11.

Under the assumptions of Theorem 3.2 and the above notations, there exists a constant $C > 0$ such that $\begin{array}{l} \int_{l_{0} - \frac{δ_{0}}{2}}^{l_{0} + \frac{δ_{0}}{2}} (| φ_{x} + ψ |^{2} + | Φ |^{2}) d x \\ ⩽ \frac{C}{| λ |^{\frac{3}{2}}} (‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{\frac{1}{2}} ‖ U ‖_{H_{j}}^{\frac{1}{2}} + ‖ U ‖_{H_{j}}^{\frac{1}{2}} ‖ F ‖_{H_{j}}^{\frac{1}{2}}) ‖ U ‖_{H_{j}} \\ + \frac{C}{| λ |^{\frac{4}{3}}} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{\frac{1}{2}} ‖ U ‖_{H_{j}}^{\frac{4}{3}} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} \\ (3.54) & + \frac{C}{| λ |} ‖ F ‖_{H_{j}}^{2} + \frac{C}{| λ |} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}}, j = 1, 2 . \end{array}$ In addition, given $ϵ > 0$ there exists a constant $C_{ϵ} > 0$ such that $\begin{matrix} (3.55) & \int_{l_{0} - \frac{δ_{0}}{2}}^{l_{0} + \frac{δ_{0}}{2}} (| φ_{x} + ψ |^{2} + | Φ |^{2}) d x ⩽ ϵ ‖ U ‖_{H_{j}}^{2} + C_{ϵ} ‖ F ‖_{H_{j}}^{2}, j = 1, 2 . \end{matrix}$

Proof.

Computing ${(3.35)}_{x} + (3.37)$ and inserting the resulting expression in (3.40), we get $\begin{matrix} (3.56) & i λ ρ_{3} Θ - {(δ θ + γ Θ)}_{x x} + i λ σ (φ_{x} + ψ) = ρ_{3} f_{6} + σ (f_{1, x} + f_{3}) . \end{matrix}$ Taking the multiplier $s k \overline{[φ_{x} + ψ]}$ in (3.56), integrating on $(0, l)$ and performing integration by parts, we have $\begin{array}{l} \underset{: = I_{1}}{\underset{︸}{i λ ρ_{3} \int_{0}^{l} Θ k s \overline{(φ_{x} + ψ)} d x}} - \underset{: = I_{2}}{\underset{︸}{\int_{0}^{l} {(δ θ_{x} + γ Θ_{x})}_{x} k s \overline{(φ_{x} + ψ)} d x}} \\ = - i λ σ \int_{0}^{l} (φ_{x} + ψ) k s \overline{(φ_{x} + ψ)} d x + ρ_{3} \int_{0}^{l} f_{6} k s \overline{(φ_{x} + ψ)} d x \\ (3.57) & + σ \int_{0}^{l} (f_{1, x} + f_{3}) k s \overline{(φ_{x} + ψ)} d x . \end{array}$ From equations (3.35) and (3.37), we achieve $\begin{matrix} I_{1} = ρ_{3} \int_{0}^{l} {(k s Θ)}_{x} \overline{Φ} d x - ρ_{3} \int_{0}^{l} Θ k s \overline{Ψ} d x - ρ_{3} \int_{0}^{l} Θ k s \overline{(f_{1, x} + f_{3})} d x . \end{matrix}$ Performing again integration by parts, we have $\begin{array}{rcl} I_{2} & = & - \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) k s_{x} \overline{(φ_{x} + ψ)} d x \underset{: = I_{3}}{\underset{︸}{- \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) k s \overline{{(φ_{x} + ψ)}_{x}} d x}} . \end{array}$ Using (3.36) one gets $\begin{matrix} I_{3} = - \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) s \overline{(i λ ρ_{1} Φ)} d x - \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) σ s \overline{Θ_{x}} d x + \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) ρ_{1} s \overline{f_{2}} d x . \end{matrix}$ Replacing the identity provided by $I_{3}$ in $I_{2}$ , and then replacing the identities provided in $I_{2}$ and $I_{1}$ in (3.57), we obtain $\begin{array}{l} i λ \int_{0}^{l} (φ_{x} + ψ) k s \overline{(φ_{x} + ψ)} d x = & ρ_{3} \int_{0}^{l} f_{6} k s \overline{(φ_{x} + ψ)} d x + σ \int_{0}^{l} (f_{1, x} + f_{3}) k s \overline{(φ_{x} + ψ)} d x \\ - \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) k s_{x} \overline{(φ_{x} + ψ)} d x \\ - \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) σ s \overline{Θ_{x}} d x + \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) s ρ_{1} \overline{f_{2}} d x \\ - ρ_{3} \int_{0}^{l} {(k s Θ)}_{x} \overline{Φ} d x + ρ_{3} \int_{0}^{l} Θ k s \overline{Ψ} d x \\ + ρ_{3} \int_{0}^{l} Θ k s \overline{(f_{1, x} + f_{3})} d x - \int_{0}^{l} (δ θ_{x} + γ Θ_{x}) i λ s ρ_{1} \overline{Φ} d x . \end{array}$ Going back to (3.57) and using condition (3.52) on s, we conclude $\begin{array}{l} | λ | \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | φ_{x} + ψ |^{2} d x \\ ⩽ C | λ | ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} {(\int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x)}^{\frac{1}{2}} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} \\ (3.58) & + C ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} (‖ F ‖_{H_{j}} + ‖ U ‖_{H_{j}} + ‖ Θ_{x} ‖_{L^{2}}) + C ‖ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} . \end{array}$ Moreover, applying Young’s inequality and Lemmas 3.6–3.7 in (3.58), we obtain $\begin{array}{l} \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | φ_{x} + ψ |^{2} d x \\ ⩽ C ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} {(\int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x)}^{1 / 2} + \frac{C}{| λ |} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} \\ (3.59) & + \frac{C}{| λ |} ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + \frac{C}{| λ |} ‖ F ‖_{H_{j}}^{2} + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} . \end{array}$

In what follows, we are going to estimate the term $\int_{0}^{l} s | Φ |^{2} d x$ . Indeed, taking the multiplier $- s \overline{φ}$ in (3.36), performing integration by parts and applying (3.35), we get $\begin{array}{l} (3.60) & ρ_{1} \int_{0}^{l} s | Φ |^{2} d x = k \int_{0}^{l} s | φ_{x} + ψ |^{2} d x - k \int_{0}^{l} s (φ_{x} + ψ) \overline{ψ} d x + I_{4} + I_{5}, \end{array}$ where $\begin{matrix} I_{4} = \frac{i σ}{λ} \int_{0}^{l} s Θ_{x} \overline{[Φ + f_{1}]} d x - ρ_{1} \int_{0}^{l} s [Φ \overline{f_{1}} + f_{2} \overline{φ}] d x and I_{5} = k \int_{0}^{l} s^{'} (φ_{x} + ψ) \overline{φ} d x . \end{matrix}$ It is easy to see that $\begin{matrix} | I_{4} | ⩽ \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ F ‖_{H_{j}} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}}, \end{matrix}$ for some constant $C > 0$ . In addition, from equations (3.35) and (3.37), it follows that $\begin{matrix} | Re I_{5} | ⩽ \frac{C}{| λ |} ‖ U ‖_{H_{j}}^{2} + \frac{C}{| λ |} ‖ F ‖_{H_{j}}^{2}, \end{matrix}$ for some constant $C > 0$ . Thus, taking the real part in (3.60) and using (3.52), we have $\begin{array}{l} \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x ⩽ & C \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | φ_{x} + ψ |^{2} d x \\ + C \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | φ_{x} + ψ | | ψ | d x + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} \\ + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ F ‖_{H_{j}} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + \frac{C}{| λ |} ‖ U ‖_{H_{j}}^{2} + \frac{C}{| λ |} ‖ F ‖_{H_{j}}^{2} \\ ⩽ & C \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | φ_{x} + ψ |^{2} d x + C {(\int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | φ_{x} + ψ |^{2} d x)}^{1 / 2} ‖ ψ ‖_{L^{2}} \\ + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ F ‖_{H_{j}} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + \frac{C}{| λ |} ‖ U ‖_{H_{j}}^{2} \\ + \frac{C}{| λ |} ‖ F ‖_{H_{j}}^{2} . \end{array}$ From estimate (3.59), Young’s inequality, (3.42) and (3.44), it follows that $\begin{array}{l} \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x ⩽ & C ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} {(\int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x)}^{1 / 2} \\ + \frac{C}{| λ |} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} \\ + \frac{C}{| λ |^{1 / 2}} (‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{1 / 2} ‖ U ‖_{H_{j}}^{1 / 2} + ‖ U ‖_{H_{j}}^{1 / 2} ‖ F ‖_{H_{j}}^{1 / 2} + ‖ F ‖_{H_{j}}) ‖ ψ ‖_{L^{2}} \\ + C ‖ F ‖_{H_{j}}^{2} + \frac{C}{| λ |^{2}} ‖ U ‖_{H_{j}}^{2} + C ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{1 / 2} {(\int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x)}^{1 / 4} ‖ ψ ‖_{L^{2}} \\ + \frac{C}{| λ |^{1 / 2}} ‖ Θ_{x} ‖_{L^{2}}^{1 / 2} ‖ U ‖_{H_{j}}^{1 / 2} ‖ ψ ‖_{L^{2}} + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} + \frac{C}{| λ |} ‖ Θ_{x} ‖_{L^{2}} ‖ F ‖_{H_{j}} \\ + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} . \end{array}$ Using again Young inequality, (3.42), (3.44), and then equation (3.37), we get $\begin{array}{l} \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x ⩽ & \frac{C}{| λ |^{3 / 2}} (‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{1 / 2} ‖ U ‖_{H_{j}}^{1 / 2} + ‖ U ‖_{H_{j}}^{1 / 2} ‖ F ‖_{H_{j}}^{1 / 2} + ‖ F ‖_{H_{j}}) (‖ U ‖_{H_{j}} + ‖ F ‖_{H_{j}}) \\ + \frac{C}{| λ |^{4 / 3}} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{2 / 3} (‖ U ‖_{H_{j}}^{4 / 3} + ‖ F ‖_{H_{j}}^{4 / 3}) + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} \\ + C ‖ F ‖_{H_{j}}^{2} + \frac{C}{| λ |} ‖ U ‖_{H_{j}}^{2} . \end{array}$ Applying once more Young’s inequality and estimate (3.44), we obtain $\begin{array}{rcl} \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s | Φ |^{2} d x & ⩽ & \frac{C}{| λ |^{3 / 2}} (‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{1 / 2} ‖ U ‖_{H_{j}}^{1 / 2} + ‖ U ‖_{H_{j}}^{1 / 2} ‖ F ‖_{H_{j}}^{1 / 2}) ‖ U ‖_{H_{j}} \\ + \frac{C}{| λ |^{4 / 3}} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{2 / 3} ‖ U ‖_{H_{j}}^{4 / 3} + \frac{C}{| λ |} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}} ‖ U ‖_{H_{j}} \\ (3.61) & + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + C ‖ F ‖_{H_{j}}^{2} + \frac{C}{| λ |} ‖ U ‖_{H_{j}}^{2} . \end{array}$ Therefore, combining (3.59) with (3.61), using again Young’s inequality and estimate (3.44), we conclude $\begin{array}{l} \int_{l_{0} - δ_{0}}^{l_{0} + δ_{0}} s (| φ_{x} + ψ |^{2} + | Φ |^{2}) d x \\ ⩽ \frac{C}{| λ |^{3 / 2}} (‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{1 / 2} ‖ U ‖_{H_{j}}^{1 / 2} + ‖ U ‖_{H_{j}}^{1 / 2} ‖ F ‖_{H_{j}}^{1 / 2}) ‖ U ‖_{H_{j}} \\ + \frac{C}{| λ |^{4 / 3}} ‖ δ θ_{x} + γ Θ_{x} ‖_{L^{2}}^{2 / 3} ‖ U ‖_{H_{j}}^{4 / 3} + C ‖ U ‖_{H_{j}} ‖ F ‖_{H_{j}} + C ‖ F ‖_{H_{j}}^{2}, \end{array}$ from where (3.54)–(3.55) follow by noting the properties of s in (3.52)–(3.53), Young’s inequality, and (3.44). □

We are now ready to prove (3.33) as follows.

Completion of the proof of ( 3.33 ) for $j = 1, 2$ . Let $ε > 0$ be given. Combining the estimates (3.51) and (3.55) there exists a constant $C_{ε}$ such that $\begin{matrix} (3.62) & \int_{l_{0} - \frac{δ_{0}}{2}}^{l_{0} + \frac{δ_{0}}{2}} (| φ_{x} |^{2} + | Φ |^{2}) d x ⩽ ε ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2} . \end{matrix}$ This is the precise moment where we can take advantage of Corollary A.6 to extend the local estimate (3.62) to the whole interval $(0, l)$ . Indeed, through the first two components (3.35)–(3.36) of the resolvent equation, we observe that $V : = (φ, Φ)$ is a solution of (A.4)–(A.5) with $\begin{matrix} g_{1} : = f_{1} and g_{2} : = ρ_{1} f_{2} - {(σ Θ)}_{x} + k ψ_{x}, \end{matrix}$ and (3.62) implies that (A.9) is verified with $\begin{matrix} b_{1} : = l_{0} - \frac{δ_{0}}{2}, b_{2} : = l_{0} + \frac{δ_{0}}{2}, and Λ : = ε ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2} . \end{matrix}$ Thus, Corollary A.6, Lemma 3.6, Corollary 3.10, and Young’s inequality imply $\begin{matrix} (3.63) & \int_{0}^{l} (| φ_{x} |^{2} + | Φ |^{2}) d x ⩽ ε C ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2} + C ‖ ψ_{x} ‖_{L^{2}}^{2} + C ‖ F ‖_{H_{j}}^{2}, \end{matrix}$ for some constants $C, C_{ε} > 0$ . Using Young and Poincaré’s inequalities again, it is easy to see $\begin{matrix} (3.64) & ‖ φ_{x} + ψ ‖_{L^{2}}^{2} ⩽ 2 ‖ φ_{x} ‖_{L^{2}}^{2} + 2 L^{2} ‖ ψ_{x} ‖_{L^{2}}^{2}, \end{matrix}$ and combining (3.63)–(3.64), we have $\begin{matrix} (3.65) & \int_{0}^{l} (| φ_{x} + ψ |^{2} + | Φ |^{2}) d x ⩽ ε ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2} + C ‖ ψ_{x} ‖_{L^{2}}^{2}, \end{matrix}$ for some constants $C, C_{ε} > 0$ . Adding the estimates from (3.65), and Lemmas 3.8 and 3.9, and using once again Young’s inequality, we get $\begin{array}{l} \int_{0}^{l} (| φ_{x} + ψ |^{2} + β | ψ_{x} |^{2} + | Φ |^{2} + ρ_{2} | Ψ |^{2}) d x \\ ⩽ ε ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2} + C ‖ ψ_{x} ‖_{L^{2}}^{2} \\ (3.66) & + C ‖ F ‖_{H_{j}}^{2} + \frac{ε}{| λ |^{2}} ‖ Φ ‖_{L^{2}}^{2} + C_{ε} ‖ η ‖_{M}^{2} . \end{array}$ Finally, from (3.66), Lemmas 3.6 and 3.7, and Corollary 3.10, we achieve $\begin{matrix} ‖ U ‖_{H_{j}}^{2} ⩽ ε ‖ U ‖_{H_{j}}^{2} + C_{ε} ‖ F ‖_{H_{j}}^{2} + \frac{ε}{| λ |^{2}} ‖ Φ ‖_{L^{2}}^{2} . \end{matrix}$ Therefore, taking $ε > 0$ small enough and $| λ | > 1$ sufficiently large, we conclude that (3.33) holds true. This finishes the proof of Theorem 3.2.

Remark 3.12.

(a) In the authors’ opinion, Theorem 3.2 seems to correct the insight claimed in [36, Section 7] with respect to the stability of a problem related to system (1.1)–(1.6a)–(1.6b), where it is stated (with no computations therein) that it is not exponentially stable in general. Indeed, on p. 670 the authors claim “the model has the optimal polynomial decay rate when $\frac{ρ_{1}}{k} \neq \frac{ρ_{2}}{b}$ , and that it is of the form $t^{- \frac{1}{2}}$ . Again, computations are required”. In the latter statement, the authors refer to problem (7.1)–(7.3) therein, which in turn corresponds to (1.1)–(1.6a)–(1.6b) unless mixed boundary conditions. Nevertheless, through the technical results employed in the proofs of the present paper, Theorem 3.2 shows a different (and new) perspective to this assertion, namely, the thermo-viscoelastic system (1.1)–(1.6a)–(1.6b) is exponentially stable, and such uniform stability is independent of any relationship among the coefficients and boundary conditions addressed.

(b) The physical reason for such uniform stability is already highlighted in Remarks 2.1–2.2. Below (see Table 1) we provide a diagram that clarifies the state of the art in the propagation of dissipativity along the solution according to the proofs of Lemmas 3.6–3.11. It illustrates the strength of the thermo-(visco-)elastic damping feedback in problem (1.1)–(1.6a)–(1.6b), by stressing the mathematical viewpoint of the damping propagation.

Table 1

State of the art of damping propagation

The damping propagation over the bending moment is provided by the history

According to Lemmas 3.6, 3.8, and 3.9.
The dissipation propagates to the shear force through the thermal component

According to Lemmas 3.6, 3.7, and 3.11.

Footnotes

Acknowledgements

The authors would like to express their gratitude to the anonymous referee for all remarks on a previous version as well as to the handling editor that allowed reconstructing this paper under a new perspective and improve itself with respect to the applied part.

M.A. Jorge Silva was partially supported by the CNPq, Grant #301116/2019-9. S.B. Pinheiro was supported by the CAPES, Finance Code 001 (Master and Ph.D. Scholarships).

Auxiliary results

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