A General Stability for a Modified Type III Thermoelastic Bresse System via the Longitudinal Displacement

Abstract

This paper studies a one-dimensional thermoelastic Bresse beam model, in which thermal effects governed by the Green–Naghdi theory of heat conduction are coupled specifically with the longitudinal displacement. Assuming appropriate conditions on the memory kernel and by defining a key stability parameter, we prove a unified decay estimate for the system’s energy. This general result includes both exponential and polynomial decay rates as special instances, offering a comprehensive framework for characterizing the system’s long-term behavior.

1. Introduction

The Bresse system [1] consists of three coupled hyperbolic equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=F_1,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)=F_2,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)=F_3,\hfill \end{array}\right.\end{array}$$

(1)

where $F_i,(i=1:3)$ denote the external forces, $\phi , \omega$, and $\psi$ represent, respectively, the vertical, longitudinal, and shear angle displacements. The coefficients ${\rho }_1, {\rho }_2, k, b, k_0$, and l stand for ${\rho }_1=\rho A, {\rho }_2=\rho I, k=k^{\prime }GA, b=EI, k_0=EA$, and $l=R^{-1}$, where $\rho$ represents the density, A is the cross-sectional area, I is the second moment of area of the cross-section, G is the shear modulus, $k^{\prime }$ is the shear factor, E is the elastic modulus, and R is the radius of curvature of the beam. Here, we assume that all the above coefficients are positive constants.

This system extends the Timoshenko beam model by incorporating a curvature parameter l, which links longitudinal and transverse motions. It is designed to model the vibrations and deformations of arches and curved beams, accounting simultaneously for axial, shear, and bending effects. If the longitudinal displacement $\omega$ is ignored (i.e., when $l=0$), the Bresse system simplifies to the standard Timoshenko beam model. Owing to its capacity to represent more nuanced mechanical responses, the Bresse system has attracted considerable attention in the research literature, especially concerning issues of well-posedness, stability, and long-term behavior under diverse damping mechanisms and boundary conditions. In [2], the authors studied the Bresse system with a frictional dissipation acting only on the angular displacement. They proved that the system is exponentially stable when the wave propagation velocities are equal, while in the case of unequal velocities, exponential stability fails, and only a polynomial decay holds. Soriano et al. [3] considered a Bresse-type system with a nonlinear damping acting on the shear angle displacement and nonlinear localized damping in the remaining equations, and they established the asymptotic stability of the system without imposing the equal-wave-speed condition. In [4], the authors analyzed the Bresse system with a past-history term acting on the shear angle displacement. They showed that the solution decays exponentially if and only if the wave speeds are equal; otherwise, the system exhibits polynomial stability with an optimal decay rate. In [5], Guesmia and Kirane examined a Bresse system with two infinite memory terms acting on two equations and established well-posedness, along with two distinct decay estimates, depending on the wave speeds, the regularity of the initial data, and the growth properties of the relaxation functions. Messaoudi et al. [6] obtained new decay results for a memory-type Bresse system, considering both equal and non-equal wave speeds. Copetti et al. [7] investigated a contact problem between a viscoelastic Bresse beam and a deformable obstacle, establishing the existence and uniqueness of solutions together with the exponential decay of the associated energy. By introducing a Kelvin–Voigt damping, the authors in [8] proved the well-posedness and exponential stability of the Bresse system.

We recall here some relevant literature concerning the thermoelastic Bresse system. Liu et al. [9] studied the thermoelastic Bresse system where the heat flux is governed by Fourier’s law:

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)+l\gamma {\theta }_1=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_x=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_{1x}=0,\hfill \\ {\rho }_3{\theta }_t-{\theta }_{xx}+\gamma {\psi }_{tx}=0,\hfill \\ {\rho }_3{\theta }_{1t}-{\theta }_{1xx}+\gamma \left({\omega }_{tx}-l{\phi }_t\right)=0,\hfill \end{array}\right.$$

where ${\rho }_3$ is a positive constant, and they proved that the system is exponentially stable if and only if $\frac{k}{{\rho }_1}=\frac{b}{{\rho }_2}$ and $k=k_0$. Moreover, when the wave speeds are not equal, they obtained a polynomial type decay. In [10], by considering a single dissipative mechanism through one temperature (i.e., ${\theta }_1=0$ in the previous system), the authors obtained results similar to those in [9]. Next, Dell’Oro [11] examined a thermoelastic Bresse system with Gurtin–Pipkin thermal dissipation:

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_x=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)=0,\hfill \\ {\rho }_3{\theta }_t-k_1{\displaystyle {\int }_0^{\infty }g(t-s){\theta }_{xx}(x,s)ds}{\theta }_{xx}+\gamma {\psi }_{tx}=0,\hfill \end{array}\right.$$

and he established the exponential stability if and only if

$$\left({\displaystyle \frac{{\rho }_1}{{\rho }_{3}k}}-{\displaystyle \frac{1}{g\left(0\right)k_1}}\right)\left({\displaystyle \frac{{\rho }_1}{k}}-{\displaystyle \frac{{\rho }_2}{b}}\right)-{\displaystyle \frac{1}{g\left(0\right)k_1}}{\displaystyle \frac{{\rho }_1{\gamma }^2}{{\rho }_{3}bk}}=0\mathrm{and}k=k_0,$$

In [12], Santos studied a thermoelastic Bresse system where the heat conduction is given by the Green and Naghdi theory of type III:

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)+k{\theta }_{xt}=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)-k{\theta }_t=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)-kl{\theta }_t=0,\hfill \\ {\rho }_3{\theta }_{tt}-\delta {\theta }_{xx}+k{\left({\phi }_x+\psi +l\omega \right)}_t-\gamma {\theta }_{xxt}=0,\hfill \end{array}\right.$$

and proved that the previous system is exponentially stable if and only if ${\displaystyle \frac{{\rho }_1}{k}}={\displaystyle \frac{{\rho }_2}{b}}$ and $k=k_0$. Apart from that, the author showed that the solution decays polynomially, and he proved that the rate of decay is optimal. Later, Keddi et al. [13] considered a thermoelastic Bresse system where the heat flux q is governed by Cattaneo’s law:

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_x=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)=0,\hfill \\ {\rho }_3{\theta }_t+q_x+\gamma {\psi }_{tx}=0,\hfill \\ \tau q_t+\beta q+{\theta }_x=0.\hfill \end{array}\right.$$

(2)

They proved that the system is exponentially stable, depending on the parameters of the system. More precisely, the exponential decay holds if

$$\left(1-{\displaystyle \frac{\tau k{\rho }_3}{{\rho }_1}}\right)\left({\displaystyle \frac{{\rho }_1}{k}}-{\displaystyle \frac{{\rho }_2}{b}}\right)-{\displaystyle \frac{\tau {\gamma }^2}{b}}=0\mathrm{and}k=k_0;$$

otherwise, the authors proved a polynomial decay result. Recently, in [14], the authors studied system (2) with a modified thermoelasticity of type III, that is,

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_{tx}=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)=0,\hfill \\ {\rho }_3{\theta }_{tt}-\delta {\theta }_{xx}+\gamma {\psi }_{tx}+{\displaystyle {\int }_0^{t}g(t-s){\theta }_{xx}(x,s)ds}=0.\hfill \end{array}\right.$$

By imposing appropriate conditions on the memory kernel, the authors showed that the damping induced by the memory term is sufficient to stabilize the system exponentially if and only if

$$\left({\displaystyle \frac{{\rho }_1}{k}}-{\displaystyle \frac{{\rho }_2}{b}}\right)\left({\displaystyle \frac{\delta }{{\rho }_3}}-{\displaystyle \frac{k}{{\rho }_1}}\right)-{\displaystyle \frac{{\gamma }^2}{b{\rho }_3}}=0\mathrm{and}k=k_0.$$

For all the above stability results, at least the shear angle displacement $\psi$ was damped via the heat conduction. Guesmia [15] studied the effect of heat conduction of types I (Fourier’s law) and III on the decay rate of the Bresse system acting only on the longitudinal displacement; more precisely, he analyzed

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_x=0,\hfill \\ {\rho }_3{\theta }_t-\beta {\theta }_{xx}+\gamma {\omega }_{tx}=0,\hfill \end{array}\right.$$

(3)

and

$$\left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=0,\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)=0,\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_{tx}=0,\hfill \\ {\rho }_3{\theta }_{tt}-\delta {\theta }_{xx}+\gamma {\omega }_{tx}-\delta {\theta }_{xxt}=0.\hfill \end{array}\right.$$

(4)

He demonstrated that exponential stability for both systems occurs if and only if the three wave propagation speeds are identical. Furthermore, when at least two of these speeds differ, he showed that each system exhibits polynomial stability, with decay rates influenced by the regularity (smoothness) of the initial conditions. Afilal et al. [16] considered system (3) with Cattaneo’s law instead of Fourier’s law. They established a range of stability results, including exponential, non-exponential, and polynomial decay, depending on the system’s coefficients and the smoothness of the initial data. For additional damping mechanisms and further results on thermoelastic Bresse systems, readers are referred to [17,18,19,20,21] and the references therein. We also mention the works [22,23,24,25], which investigated various thermoelastic systems with microtemperature effects, and [26,27,28,29,30], which dealt with viscoelastic systems, and related references therein.

Motivated by all previous works, we are concerned with the Bresse system (4) with modified thermoelasticity of type III; i.e., we note that the term ${\theta }_{txx}$ in (4) represents a strong damping; for this purpose, we will replace it by a thermo-viscoelastic (weak) damping of the form ${\int }_0^{t}g(t-s){\theta }_{xx}(·,s)ds$. From a physical viewpoint, the memory term ${\displaystyle {\int }_0^{t}g(t-s){\theta }_{xx}(·,s)ds}$ models the delayed response of the material to past thermal gradients, which is consistent with non-Fourier heat conduction laws, such as the Cattaneo–Vernotte and Guyer–Krumhansl models. These models have been validated experimentally in various materials, including metals, polymers, biological tissues, and thermoelectric sensors, where transient thermal responses and nonlocal temperature gradients have been observed.

In particular, recent thermoelectric and pyroelectric experiments confirm that non-Fourier effects and thermal memory play a significant role in nanoscale and biological systems, supporting the physical relevance of the adopted formulation. Therefore, the use of a memory kernel in our Bresse-type system not only generalizes the classical theory mathematically but also captures more realistic thermal behaviors that align with experimental evidence. Thus, the problem we will discuss is

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\rho }_1{\phi }_{tt}-k{\left({\phi }_x+\psi +l\omega \right)}_x-l{k}_0\left({\omega }_x-l\phi \right)=0,\hfill & \mathrm{in}(0,1)\times (0,\infty ),\hfill \\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+k\left({\phi }_x+\psi +l\omega \right)=0,\hfill & \mathrm{in}(0,1)\times (0,\infty ),\hfill \\ {\rho }_1{\omega }_{tt}-k_0{\left({\omega }_x-l\phi \right)}_x+lk\left({\phi }_x+\psi +l\omega \right)+\gamma {\theta }_{tx}=0,\hfill & \mathrm{in}(0,1)\times (0,\infty ),\hfill \\ {\rho }_3{\theta }_{tt}-\delta {\theta }_{xx}+\gamma {\omega }_{tx}+{\displaystyle {\int }_0^{t}g(t-s){\theta }_{xx}(x,s)ds}=0,\hfill & \mathrm{in}(0,1)\times (0,\infty ),\hfill \end{array}\right.\end{array}$$

(5)

We should mention that the fourth equation in Equation (5) above represents a heat conduction problem observed in real materials and incorporates thermal history effects in the sense of Green and Naghdi. It models the dependence on the history of the temperature gradient. Our system is complemented by the following initial conditions:

$$\begin{array}{cc}\hfill & \phi (x,0)={\phi }_0\left(x\right),{\phi }_t(x,0)={\phi }_1\left(x\right),\psi (x,0)={\psi }_0\left(x\right),{\psi }_t(x,0)={\psi }_1\left(x\right),x\in (0,1),\hfill \\ \hfill & \omega (x,0)={\omega }_0\left(x\right),{\omega }_t(x,0)={\omega }_1\left(x\right),\theta (x,0)={\theta }_0\left(x\right),{\theta }_t(x,0)={\theta }_1\left(x\right),x\in (0,1),\hfill \end{array}$$

(6)

and boundary conditions

$$\begin{array}{cc}\hfill & \phi (0,t)={\psi }_x(0,t)={\omega }_x(0,t)=\theta (0,t)=0,t>0,\hfill \\ \hfill & \phi (1,t)={\psi }_x(1,t)={\omega }_x(1,t)=\theta (1,t)=0,t>0.\hfill \end{array}$$

(7)

The main question that can be asked here is the following: Is the memory term strong enough to stabilize system (5)–(7) exponentially when the thermal coupling is taken via the longitudinal displacement? The aim of the present paper is to give a positive answer to the question by establishing a general stability result for system (5)–(7) if and only if

$$\begin{array}{c}\hfill \chi =\left({\rho }_3-{\displaystyle \frac{{\rho }_1\delta }{k}}\right)\left(k-k_0\right)-{\gamma }^2=0\mathrm{and}{\displaystyle \frac{{\rho }_1}{k}}={\displaystyle \frac{{\rho }_2}{b}}.\end{array}$$

(8)

This work is divided into three sections. In Section 2, we present some notations and assumptions needed in our work and state without proof the existence of solutions for the problem (5)–(7). In Section 3, we use the multiplier method to establish a general decay of the energy, depending on the wave speeds of the system and $\chi$ defined by (8).

2. Preliminaries

In this section, we present our hypotheses needed in the proof of our result and state without proof a global existence result for system (5)–(7). The relaxation function g satisfies the following assumptions:

(A1) $g:[0,\infty )\to (0,\infty )$ is a differentiable function satisfying

$$g\left(0\right)>0,\delta -{\int }_0^{\infty }g\left(s\right)ds=\lambda >0.$$

(A2) There exists a non-increasing differentiable function $\eta :[0,\infty )\to [0,\infty )$ satisfies

$$g^{\prime }\left(t\right)\le -\eta \left(t\right)g\left(t\right),\forall t\ge 0.$$

We denote by ∗ the usual convolution product:

$$\left(g\ast h\right)\left(t\right)={\int }_0^{t}g(t-s)h\left(s\right)ds.$$

To facilitate our calculations, we introduce the following notations:

$$\left(g\diamond h\right)\left(t\right)={\int }_0^{t}g(t-s)\left[h\left(t\right)-h\left(s\right)\right]ds,$$

$$\left(g\circ h\right)\left(t\right)={\int }_0^{t}g(t-s){\left[h\left(t\right)-h\left(s\right)\right]}^{2}ds.$$

Lemma 1. 

Under assumptions  (A1)  and  (A2), we have

$$\begin{array}{c}\hfill \left(g\ast h\right)\left(t\right)·h_t\left(t\right)={\displaystyle \frac{1}{2}}\left(g^{\prime }\circ h\right)\left(t\right)-{\displaystyle \frac{1}{2}}g\left(t\right)|h\left(t\right){|}^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[\left({\int }_0^{t}g\left(s\right)ds\right)|h\left(t\right){|}^2-\left(g\circ h\right)\left(t\right)\right].\end{array}$$

Proof. 

See [31]. □

Lemma 2. 

Assume that (A1) holds. Then 

$$\begin{array}{c}\hfill {\int }_0^1{\left(g\diamond h\right)}^{2}dx\le \left({\int }_0^{t}g\left(s\right)ds\right){\int }_0^1\left(g\circ h\right)dx\le c_1{\int }_0^1\left(g\circ h_x\right)dx.\end{array}$$

 Moreover

$$\begin{array}{c}\hfill {\int }_0^1{\left(-g^{\prime }\diamond h\right)}^{2}dx\le -c_1{\int }_0^1\left(g^{\prime }\circ h_x\right)dx,\end{array}$$

where $c_1$, here and throughout this paper, denotes a generic positive constant.

Proof. 

See [31]. □

We consider the following space:

$$\begin{array}{c}\hfill H_{\ast }^1(0,1)=H^1(0,1)\cap L_{\ast }^2(0,1),\mathrm{where}{L}_{\ast }^2(0,1)=\left\{h\in L^2(0,1)\mid {\int }_0^{1}h\left(x\right)dx=0\right\}.\end{array}$$

The well-posedness of (5)–(7) is stated in the following proposition.

Proposition 1 

([14]). Let $\left({\phi }_0,{\psi }_0,{\omega }_0,{\theta }_0\right)\in H_0^1(0,1)\times {\left(H_{\ast }^1(0,1)\right)}^2\times H_0^1(0,1)$ and $\left({\phi }_1,{\psi }_1,{\omega }_1,{\theta }_1\right)\in L^2(0,1)\times {\left(L_{\ast }^2(0,1)\right)}^2\times L^2(0,1)$ be given and assume that g satisfies (A1) and (A2). Then, there exist a unique global solution for problem (5)–(7), which satisfies

$$\begin{array}{c}\hfill \left(\phi ,\psi ,\omega ,\theta \right)\in \left(\begin{array}{c}C\left({\mathbb{R}}_{+};H_0^1(0,1)\right)\cap C^1\left({\mathbb{R}}_{+};L^2(0,1)\right)\\ \times {\left(C\left({\mathbb{R}}_{+};H_{\ast }^1(0,1)\right)\cap C^1\left({\mathbb{R}}_{+};L_{\ast }^2(0,1)\right)\right)}^2\\ \times C\left({\mathbb{R}}_{+};H_0^1(0,1)\right)\cap C^1\left({\mathbb{R}}_{+};L^2(0,1)\right)\end{array}\right).\end{array}$$

Moreover, if $\left({\phi }_0,{\psi }_0,{\omega }_0,{\theta }_0\right)\in H^2(0,1)\cap H_0^1(0,1)\times {\left(H_{\ast }^2(0,1)\cap H_{\ast }^1(0,1)\right)}^2\times H^2(0,1)\cap H_0^1(0,1)$ and $\left({\phi }_1,{\psi }_1,{\omega }_1,{\theta }_1\right)\in H_0^1(0,1)\times {\left(H_{\ast }^1(0,1)\right)}^2\times H_0^1(0,1)$, then the solution satisfies

$$\begin{array}{c}\hfill \left(\phi ,\psi ,\omega ,\theta \right)\in \left(\begin{array}{c}C\left({\mathbb{R}}_{+};H^2(0,1)\cap H_0^1(0,1)\right)\cap C^1\left({\mathbb{R}}_{+};H_0^1(0,1)\right)\cap C^2\left({\mathbb{R}}_{+};L^2(0,1)\right)\\ \times {\left(C\left({\mathbb{R}}_{+};H_{\ast }^2(0,1)\cap H_{\ast }^1(0,1)\right)\cap C^1\left({\mathbb{R}}_{+};H_{\ast }^1(0,1)\right)\cap C^2\left({\mathbb{R}}_{+};L_{\ast }^2(0,1)\right)\right)}^2\\ \times C\left({\mathbb{R}}_{+};H^2(0,1)\cap H_0^1(0,1)\right)\cap C^1\left({\mathbb{R}}_{+};H_0^1(0,1)\right)\cap C^2\left({\mathbb{R}}_{+};L^2(0,1)\right)\end{array}\right),\end{array}$$

where

$$\begin{array}{c}\hfill H_{\ast }^2(0,1)=\left\{h\in H^2(0,1)\mid h_x\left(0\right)=h_x\left(1\right)=0\right\}.\end{array}$$

Remark 1. 

The proof can be established using standard method such as Galerkin method.

Remark 2 

([11]). The choice of the spaces of zero-mean functions for the variables ψ and ω and their derivatives is consistent. Indeed, setting

$$\Psi \left(t\right)={\int }_0^1\psi (x,t)dx\mathit{and}W\left(t\right)={\int }_0^1\omega (x,t)dx,$$

and integrating the second and the third equations in (5) on $(0,1)$, using boundary conditions, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_2\ddot{\Psi }\left(t\right)+k\Psi \left(t\right)+klW\left(t\right)=0,\hfill \\ {\rho }_1\ddot{W}\left(t\right)+k{l}^{2}W\left(t\right)+kl\Psi \left(t\right)=0.\hfill \end{array}\right.\end{array}$$

Thus, if $\Psi \left(0\right)=\dot{\Psi }\left(0\right)=W\left(0\right)=\dot{W}\left(0\right)=0,$ it follows that $\Psi \left(t\right)\equiv W\left(t\right)\equiv 0.$

3. General Stability

In this section, we state and prove our stability result for the energy of the system (5)–(7) using the multiplier technique. To achieve our goal, we introduce a sequence of auxiliary lemmas that progressively contribute to the construction of an appropriate Lyapunov functional. Each lemma provides a differential inequality in which the time derivative of a suitably designed auxiliary functional is shown to be bounded above by the negative of one or more components of the total energy. The purpose of these auxiliary functionals is to control the intricate coupling effects arising among the variables $(\phi ,\psi ,\omega ,\theta )$ and their derivatives. In particular, Lemma 3 establishes the intrinsic dissipativity of the system, whereas the subsequent lemmas introduce cross terms that compensate for non-dissipative interactions between the system components. This hierarchical structure ultimately leads to the construction of a Lyapunov functional equivalent to the natural energy, from which the desired general decay result follows.

Lemma 3. 

Let $\left(\phi ,\psi ,\omega ,\theta \right)$ be the solution of (5)–(7). Then, the energy functional defined by

$$\begin{array}{cc}\hfill E\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^1& [{\rho }_1{\phi }_t^2+k_0{\left({\omega }_x-l\phi \right)}^2+{\rho }_2{\psi }_t^2+b{\psi }_x^2+{\rho }_1{\omega }_t^2+k{\left({\phi }_x+\psi +l\omega \right)}^2\hfill \\ \hfill & +{\rho }_3{\theta }_t^2+\left(\delta -{\int }_0^{t}g\left(s\right)ds\right){\theta }_x^2+\left(g\circ {\theta }_x\right)]dx,\forall t\ge 0\hfill \end{array}$$

(9)

satisfies

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}E\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx-{\displaystyle \frac{1}{2}}g\left(t\right){\int }_0^1{\theta }_x^{2}dx\le 0,t\ge 0.\end{array}$$

(10)

Proof. 

Equation (10) follows from multiplying the four equations of system (5) by ${\phi }_t$, ${\psi }_t$, ${\omega }_t$, and ${\theta }_t$, respectively, integrating by parts over $(0,1)$, using the boundary conditions (7), summing up and using Lemma 1. □

Lemma 4. 

The functional $F_1$ defined by

$$\begin{array}{c}\hfill F_1\left(t\right)={\rho }_2{\int }_0^1{\psi }_t\psi dx,\end{array}$$

(11)

satisfies, along with the solution of (5)–(7), the estimate

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_1\left(t\right)\le & -{\displaystyle \frac{b}{2}}{\int }_0^1{\psi }_x^{2}dx+{\rho }_2{\int }_0^1{\psi }_t^{2}dx+c_1{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx.\hfill \end{array}$$

(12)

Proof. 

Through direct computations, using Equation ${\left(5\right)}_2$ and integrating by parts, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{F}_1\left(t\right)=-b{\int }_0^1{\psi }_x^{2}dx+{\rho }_2{\int }_0^1{\psi }_t^{2}dx-k{\int }_0^1\psi \left({\phi }_x+\psi +l\omega \right)dx.\end{array}$$

Then, (12) easily follows owing to Young’s and Poincaré’s inequalities. □

Lemma 5. 

The functional

$$\begin{array}{c}\hfill F_2\left(t\right)=-{\rho }_1{\int }_0^1\left({\omega }_x-l\phi \right)\left({\int }_0^x{\omega }_t\left(y\right)dy\right)dx,\end{array}$$

(13)

satisfies, for any ${\epsilon }_1>0$, the estimate

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_2\left(t\right)\le & -{\displaystyle \frac{k_0}{2}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+{\epsilon }_1{\int }_0^1{\phi }_t^{2}dx+c_1{\int }_0^1{\theta }_t^{2}dx\hfill \\ \hfill & +{\displaystyle \frac{k^2{l}^2}{k_0}}{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx+c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_1}}\right){\int }_0^1{\omega }_t^{2}dx.\hfill \end{array}$$

(14)

Proof. 

A differentiation of $F_2$ with the use of Equation ${\left(5\right)}_3$, integration by parts, give

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_2\left(t\right)=& {\int }_0^1\left({\omega }_x-l\phi \right)\left[-k_0\left({\omega }_x-l\phi \right)+lk{\int }_0^x\left({\phi }_y+\psi +l\omega \right)\left(y\right)dy+\gamma {\theta }_t\right]dx\hfill \\ \hfill & +{\rho }_1{\int }_0^1{\omega }_t^{2}dx+{\rho }_{1}l{\int }_0^1{\phi }_t\left({\int }_0^x{\omega }_t\left(y\right)dy\right)dx\hfill \\ \hfill =& -k_0{\int }_0^1{\left({\omega }_x-l\phi \right)}^2+lk{\int }_0^1\left({\omega }_x-l\phi \right)\left({\int }_0^x\left({\phi }_y+\psi +l\omega \right)\left(y\right)dy\right)dx\hfill \\ \hfill & +\gamma {\int }_0^1\left({\omega }_x-l\phi \right){\theta }_{t}dx+{\rho }_1{\int }_0^1{\omega }_t^{2}dx+{\rho }_{1}l{\int }_0^1{\phi }_t\left({\int }_0^x{\omega }_t\left(y\right)dy\right)dx.\hfill \end{array}$$

We then use Cauchy–Schwarz and Young’s inequalities with ${\epsilon }_1>0$ to obtain (14). □

Remark 3. 

For any $l\ne {\displaystyle \frac{1}{\sqrt[4]{2}}}$, using Poincaré’s inequality, it is clear that

$$\begin{array}{c}\hfill {\int }_0^1{\phi }_x^{2}dx\le 2{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx+2{\int }_0^1{\psi }_x^{2}dx+2l^2{\int }_0^1{\omega }_x^{2}dx,\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_0^1{\omega }_x^{2}dx\le 2{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+l^2{\int }_0^1{\phi }_x^{2}dx.\end{array}$$

That is,

$$\begin{array}{c}\hfill {\int }_0^1{\omega }_x^{2}dx\le {\displaystyle \frac{2l^2}{1-2l^4}}{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx+{\displaystyle \frac{2}{1-2l^4}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+{\displaystyle \frac{2l^2}{1-2l^4}}{\int }_0^1{\psi }_x^{2}dx.\end{array}$$

(15)

Lemma 6. 

Let $\left(\phi ,\psi ,\omega ,\theta \right)$ be the solution of (5)–(7). Then, the functional

$$F_3\left(t\right)={\rho }_3{\int }_0^1{\theta }_t\theta dx+\gamma {\int }_0^1\theta {\omega }_{x}dx$$

(16)

satisfies, for any $l\ne {\displaystyle \frac{1}{\sqrt[4]{2}}},{\epsilon }_2>0$, the following estimate:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_3\left(t\right)\le & -{\displaystyle \frac{\lambda }{2}}{\int }_0^1{\theta }_x^{2}dx+{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx+{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{\int }_0^1{\psi }_x^{2}dx\hfill \\ \hfill & +{\displaystyle \frac{2}{1-2l^4}}{\epsilon }_2{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}\right){\int }_0^1{\theta }_t^{2}dx+c_1{\int }_0^1\left(g\circ {\theta }_x\right)dx.\hfill \end{array}$$

(17)

Proof. 

Direct computations, using Equation ${\left(5\right)}_4$ and boundary conditions, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_3\left(t\right)=& {\rho }_3{\int }_0^1{\theta }_t^{2}dx-\delta {\int }_0^1{\theta }_x^{2}dx+\gamma {\int }_0^1{\theta }_t{\omega }_{x}dx+{\int }_0^1{\theta }_x\left(g\ast {\theta }_x\right)dx.\hfill \end{array}$$

So, for any ${\epsilon }_2,{\delta }_1>0$, we have

$$\begin{array}{c}\hfill \gamma {\int }_0^1{\theta }_t{\omega }_{x}dx\le {\epsilon }_2{\int }_0^1{\omega }_x^{2}dx+{\displaystyle \frac{c_1}{{\epsilon }_2}}{\int }_0^1{\theta }_t^{2}dx,\end{array}$$

$$\begin{array}{cc}\hfill {\int }_0^1{\theta }_x\left(g\ast {\theta }_x\right)dx& ={\int }_0^{t}g\left(s\right)ds{\int }_0^1{\theta }_x^{2}dx-{\int }_0^1{\theta }_x\left(g\diamond {\theta }_x\right)dx\hfill \\ \hfill & \le \left({\int }_0^{t}g\left(s\right)ds+{\delta }_1\right){\int }_0^1{\theta }_x^{2}dx+{\displaystyle \frac{1}{4{\delta }_1}}\left({\int }_0^{t}g\left(s\right)ds\right){\int }_0^1\left(g\circ {\theta }_x\right)dx,\hfill \end{array}$$

using the fact that

$$\begin{array}{c}\hfill {\int }_0^{t}g\left(s\right)ds\le \delta -\lambda >0.\end{array}$$

(18)

Then, we arrive at

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_3\left(t\right)\le & -\left(\lambda -{\delta }_1\right){\int }_0^1{\theta }_x^{2}dx+{\epsilon }_2{\int }_0^1{\omega }_x^{2}dx+c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}\right){\int }_0^1{\theta }_t^{2}dx+{\displaystyle \frac{c_1}{{\delta }_1}}{\int }_0^1\left(g\circ {\theta }_x\right)dx,\hfill \end{array}$$

which gives estimate (17) by using (15) and choosing ${\delta }_1={\displaystyle \frac{\lambda }{2}}$. □

Lemma 7. 

The functional

$$\begin{array}{c}\hfill F_4\left(t\right)=-{\displaystyle \frac{{\rho }_1}{k}}{\int }_0^1{\phi }_t{\psi }_{x}dx-{\displaystyle \frac{{\rho }_2}{b}}{\int }_0^1{\psi }_t\left({\phi }_x+\psi +l\omega \right)dx\end{array}$$

(19)

satisfies, along with the solution of (5)–(7), for any ${\epsilon }_3>0$, the estimate

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_4\left(t\right)\le & -{\displaystyle \frac{{\rho }_2}{2b}}{\int }_0^1{\psi }_t^{2}dx+{\epsilon }_3{\int }_0^1{\psi }_x^{2}dx+{\displaystyle \frac{k}{b}}{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx\hfill \\ \hfill & +c_1{\int }_0^1{\omega }_t^{2}dx+{\displaystyle \frac{c_1}{{\epsilon }_3}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+\left({\displaystyle \frac{{\rho }_1}{k}}-{\displaystyle \frac{{\rho }_2}{b}}\right){\int }_0^1{\psi }_t{\phi }_{tx}dx.\hfill \end{array}$$

(20)

Proof. 

Differentiating $F_4\left(t\right)$, using Equations ${\left(5\right)}_1$ and ${\left(5\right)}_2$ and integrating by parts, yield

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_4\left(t\right)=& -{\displaystyle \frac{{\rho }_2}{b}}{\int }_0^1{\psi }_t^{2}dx-{\displaystyle \frac{l{\rho }_2}{b}}{\int }_0^1{\psi }_t{\omega }_{t}dx+{\displaystyle \frac{k}{b}}{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx\hfill \\ \hfill & -{\displaystyle \frac{l{k}_0}{k}}{\int }_0^1{\psi }_x\left({\omega }_x-l\phi \right)dx+\left({\displaystyle \frac{{\rho }_1}{k}}-{\displaystyle \frac{{\rho }_2}{b}}\right){\int }_0^1{\psi }_t{\phi }_{tx}dx.\hfill \end{array}$$

(21)

By virtue of Young’s inequality, for ${\epsilon }_3>0$, we obtain

$$\begin{array}{c}\hfill -{\displaystyle \frac{l{\rho }_2}{b}}{\int }_0^1{\psi }_t{\omega }_{t}dx\le {\displaystyle \frac{{\rho }_2}{2b}}{\int }_0^1{\psi }_t^{2}dx+c_1{\int }_0^1{\omega }_t^{2}dx,\end{array}$$

(22)

$$\begin{array}{c}\hfill -{\displaystyle \frac{l{k}_0}{k}}{\int }_0^1{\psi }_x\left({\omega }_x-l\phi \right)dx\le {\epsilon }_3{\int }_0^1{\psi }_x^{2}dx+{\displaystyle \frac{c_1}{{\epsilon }_3}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx.\end{array}$$

(23)

This yields the desired result (20) by inserting (22) and (23) into (21). □

Lemma 8. 

Let $\left(\phi ,\psi ,\omega ,\theta \right)$ be the solution of (5)–(7). Then, the functional

$$\begin{array}{cc}\hfill F_5\left(t\right)=& -{\displaystyle \frac{{\rho }_1{\rho }_3}{\gamma }}{\int }_0^1{\theta }_t\left({\int }_0^x{\omega }_t\left(y\right)dy\right)dx\hfill \end{array}$$

(24)

satisfies, for ${\epsilon }_4,{\epsilon }_5>0$, the estimate

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_5\left(t\right)\le & -{\displaystyle \frac{{\rho }_1}{2}}{\int }_0^1{\omega }_t^{2}dx+{\epsilon }_4{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+{\epsilon }_5{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx\hfill \\ \hfill & +c_1{\int }_0^1{\theta }_x^{2}dx+c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_4}}+{\displaystyle \frac{1}{{\epsilon }_5}}\right){\int }_0^1{\theta }_t^{2}dx+c_1{\int }_0^1\left(g\circ {\theta }_x\right)dx.\hfill \end{array}$$

(25)

Proof. 

Differentiating $F_5$, using the last two equations in (5) and integrating by parts, yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_5\left(t\right)=& -{\rho }_1{\int }_0^1{\omega }_t^{2}dx+{\displaystyle \frac{{\rho }_1\delta }{\gamma }}{\int }_0^1{\theta }_x{\omega }_{t}dx-{\displaystyle \frac{{\rho }_1}{\gamma }}{\int }_0^1{\omega }_t\left(g\ast {\theta }_x\right)dx+{\rho }_3{\int }_0^1{\theta }_t^{2}dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_3{k}_0}{\gamma }}{\int }_0^1{\theta }_t\left({\omega }_x-l\phi \right)dx+{\displaystyle \frac{lk{\rho }_3}{\gamma }}{\int }_0^1{\theta }_t\left({\int }_0^x\left({\phi }_y+\psi +l\omega \right)\left(y\right)dy\right)dx.\hfill \end{array}$$

Then, inequality (25) appears by using Cauchy–Schwarz and Young’s inequalities with ${\epsilon }_4,{\epsilon }_5>0$. □

Lemma 9. 

Let $\left(\phi ,\psi ,\omega ,\theta \right)$ be the solution of (5)–(7). Then, the functional

$$\begin{array}{cc}\hfill F_6\left(t\right)=& -{\rho }_3{\int }_0^1{\theta }_t\left(g\diamond \theta \right)dx\hfill \end{array}$$

(26)

satisfies, for some fixed $t_0>0$ and for any ${\epsilon }_6,{\epsilon }_7>0$, the following estimate

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_6\left(t\right)\le & -{\displaystyle \frac{{\rho }_3{g}_0}{2}}{\int }_0^1{\theta }_t^{2}dx+{\epsilon }_6{\int }_0^1{\theta }_x^{2}dx+{\epsilon }_7{\int }_0^1{\omega }_t^{2}dx\hfill \\ \hfill & +c_1\left({\displaystyle \frac{1}{{\epsilon }_6}}+{\displaystyle \frac{1}{{\epsilon }_7}}\right){\int }_0^1\left(g\circ {\theta }_x\right)dx-c_1{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx,\hfill \end{array}$$

(27)

where ${\displaystyle g_0={\int }_0^{t_0}g\left(s\right)ds}$.

Proof. 

Differentiating $F_6$, taking into account ${\left(5\right)}_4$, integrating by parts together with the boundary conditions, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_6\left(t\right)=& -{\rho }_3{\int }_0^1{\theta }_t{\left(g\diamond \theta \right)}_{t}dx+\delta {\int }_0^1{\theta }_x\left(g\diamond {\theta }_x\right)dx-{\int }_0^1\left(g\ast {\theta }_x\right)\left(g\diamond {\theta }_x\right)dx\hfill \\ \hfill & -\gamma {\int }_0^1{\omega }_t\left(g\diamond {\theta }_x\right)dx\hfill \\ \hfill =& -\left({\rho }_3{\int }_0^{t}g\left(s\right)ds\right){\int }_0^1{\theta }_t^{2}dx-{\rho }_3{\int }_0^1{\theta }_t\left(g^{\prime }\diamond \theta \right)dx+\delta {\int }_0^1{\theta }_x\left(g\diamond {\theta }_x\right)dx\hfill \\ \hfill & -{\int }_0^1\left(g\ast {\theta }_x\right)\left(g\diamond {\theta }_x\right)dx-\gamma {\int }_0^1{\omega }_t\left(g\diamond {\theta }_x\right)dx.\hfill \end{array}$$

(28)

Using Young’s, Cauchy–Schwarz and Poincaré’s inequalities, for any ${\delta }_2,{\epsilon }_6,{\epsilon }_7>0$, we obtain

$$\begin{array}{cc}\hfill -{\rho }_3{\int }_0^1{\theta }_t\left(g^{\prime }\diamond \theta \right)dx& \le {\rho }_3{\delta }_2{\int }_0^1{\theta }_t^{2}dx+{\displaystyle \frac{{\rho }_3}{4{\delta }_2}}{\int }_0^1\left({\int }_0^t-g^{\prime }\left(s\right)ds\right)\left(-g^{\prime }\circ {\theta }_x\right)dx\hfill \\ \hfill & \le {\rho }_3{\delta }_2{\int }_0^1{\theta }_t^{2}dx-{\displaystyle \frac{c_1}{{\delta }_2}}{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx,\hfill \end{array}$$

(29)

$$\begin{array}{c}\hfill \delta {\int }_0^1{\theta }_x\left(g\diamond {\theta }_x\right)dx\le {\displaystyle \frac{{\epsilon }_6}{2}}{\int }_0^1{\theta }_x^{2}dx+{\displaystyle \frac{c_1}{{\epsilon }_6}}{\int }_0^1\left(g\circ {\theta }_x\right)dx,\end{array}$$

(30)

$$\begin{array}{cc}\hfill -{\int }_0^1\left(g\ast {\theta }_x\right)\left(g\diamond {\theta }_x\right)dx=& {\int }_0^1{\left(g\diamond {\theta }_x\right)}^{2}dx-{\int }_0^{t}g\left(s\right)ds{\int }_0^1{\theta }_x\left(g\diamond {\theta }_x\right)dx\hfill \\ \hfill \le & {\displaystyle \frac{{\epsilon }_6}{2}}{\int }_0^1{\theta }_x^{2}dx+{\displaystyle \frac{c_1}{{\epsilon }_6}}{\int }_0^1\left(g\circ {\theta }_x\right)dx,\hfill \end{array}$$

(31)

$$\begin{array}{c}\hfill -\gamma {\int }_0^1{\omega }_t\left(g\diamond {\theta }_x\right)dx\le {\epsilon }_7{\int }_0^1{\omega }_t^{2}dx+{\displaystyle \frac{c_1}{{\epsilon }_7}}{\int }_0^1\left(g\circ {\theta }_x\right)dx.\end{array}$$

(32)

Since g is positive, continuous function and $g\left(0\right)>0$, we have

$${\int }_0^{t}g\left(s\right)ds\ge {\int }_0^{t_0}g\left(s\right)ds=g_0,\forall t\ge t_0>0.$$

(33)

By inserting (29)–(32) into (28), bearing in mind (33), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_6\left(t\right)\le & -{\rho }_3\left[g_0-{\delta }_2\right]{\int }_0^1{\theta }_t^{2}dx+{\epsilon }_6{\int }_0^1{\theta }_x^{2}dx+{\epsilon }_7{\int }_0^1{\omega }_t^{2}dx\hfill \\ \hfill & +c_1\left({\displaystyle \frac{1}{{\epsilon }_6}}+{\displaystyle \frac{1}{{\epsilon }_7}}\right){\int }_0^1\left(g\circ {\theta }_x\right)dx-{\displaystyle \frac{c_1}{{\delta }_2}}{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx,\hfill \end{array}$$

(34)

for all $t\ge t_0$. By taking ${\delta }_2={\displaystyle \frac{g_0}{2}}$, we obtain estimate (27). □

Lemma 10. 

Let $\left(\phi ,\psi ,\omega ,\theta \right)$ be the solution of (5)–(7). Then, the functional

$$\begin{array}{cc}\hfill F_7\left(t\right)=& {\displaystyle \frac{{\rho }_1}{l}}{\int }_0^1{\omega }_t\left({\phi }_x+\psi +l\omega \right)dx+{\displaystyle \frac{{\rho }_1{k}_0}{kl}}{\int }_0^1{\phi }_t\left({\omega }_x-l\phi \right)dx\hfill \\ \hfill & +{\displaystyle \frac{{\rho }_1{\rho }_3}{kl\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_t{\phi }_{t}dx+{\displaystyle \frac{{\rho }_1\delta }{kl\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_x\left({\phi }_x+\psi +l\omega \right)dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1}{kl\gamma }}\left(k_0-k\right){\int }_0^1\left(g\ast {\theta }_x\right)\left({\phi }_x+\psi +l\omega \right)dx\hfill \end{array}$$

(35)

satisfies, for any ${\epsilon }_8>0$, the following estimate

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_7\left(t\right)\le & -{\displaystyle \frac{k}{2}}{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx-{\displaystyle \frac{{\rho }_1{k}_0}{k}}{\int }_0^1{\phi }_t^{2}dx+{\epsilon }_8{\int }_0^1{\psi }_t^{2}dx\hfill \\ \hfill & +{\displaystyle \frac{2k_0^2}{k}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_8}}\right){\int }_0^1{\omega }_t^{2}dx+c_1{\int }_0^1{\theta }_t^{2}dx\hfill \\ \hfill & +c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_8}}\right){\int }_0^1{\theta }_x^{2}dx+c_1\left(1+{\displaystyle \frac{1}{{\epsilon }_8}}\right){\int }_0^1\left(g\circ {\theta }_x\right)dx\hfill \\ \hfill & -c_1{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx+{\displaystyle \frac{\chi }{l\gamma }}{\int }_0^1{\theta }_{tx}\left({\phi }_x+\psi +l\omega \right)dx.\hfill \end{array}$$

(36)

Proof. 

A simple differentiation of $F_7$ gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_7\left(t\right)=& \underset{:=I_1}{\underbrace{{\displaystyle \frac{{\rho }_1}{l}}{\int }_0^1{\omega }_{tt}\left({\phi }_x+\psi +l\omega \right)dx}}+\underset{:=I_2}{\underbrace{{\displaystyle \frac{{\rho }_1}{l}}{\int }_0^1{\omega }_t{\left({\phi }_x+\psi +l\omega \right)}_{t}dx}}+\underset{:=I_3}{\underbrace{{\displaystyle \frac{{\rho }_1{k}_0}{kl}}{\int }_0^1{\phi }_{tt}\left({\omega }_x-l\phi \right)dx}}\hfill \\ \hfill & +\underset{:=I_4}{\underbrace{{\displaystyle \frac{{\rho }_1{k}_0}{kl}}{\int }_0^1{\phi }_t{\left({\omega }_x-l\phi \right)}_{t}dx}}+{\displaystyle \frac{{\rho }_1\left(k_0-k\right)}{kl\gamma }}·\underset{:=I_5}{\underbrace{{\rho }_3{\int }_0^1{\theta }_{tt}{\phi }_{t}dx}}+{\displaystyle \frac{{\rho }_3\left(k_0-k\right)}{kl\gamma }}·\underset{:=I_6}{\underbrace{{\rho }_1{\int }_0^1{\theta }_t{\phi }_{tt}dx}}\hfill \\ \hfill & +{\displaystyle \frac{{\rho }_1\delta }{kl\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_{tx}\left({\phi }_x+\psi +l\omega \right)dx+{\displaystyle \frac{{\rho }_1\delta }{kl\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_x{\left({\phi }_x+\psi +l\omega \right)}_{t}dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1}{kl\gamma }}\left(k_0-k\right)\underset{:=I_7}{\underbrace{{\int }_0^1{\left(g\ast {\theta }_x\right)}_t\left({\phi }_x+\psi +l\omega \right)dx}}\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1}{kl\gamma }}\left(k_0-k\right){\int }_0^1\left(g\ast {\theta }_x\right){\left({\phi }_x+\psi +l\omega \right)}_{t}dx.\hfill \end{array}$$

(37)

Now, using the equations of system (5) and integrating by parts, we have

$$\begin{array}{cc}\hfill I_1=& {\displaystyle \frac{k_0}{l}}{\int }_0^1{\left({\omega }_x-l\phi \right)}_x\left({\phi }_x+\psi +l\omega \right)dx-k{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx\hfill \\ \hfill & -{\displaystyle \frac{\gamma }{l}}{\int }_0^1{\theta }_{tx}\left({\phi }_x+\psi +l\omega \right)dx,\hfill \end{array}$$

(38)

$$\begin{array}{c}\hfill I_2=-{\displaystyle \frac{{\rho }_1}{l}}{\int }_0^1{\omega }_{tx}{\phi }_{t}dx+{\displaystyle \frac{{\rho }_1}{l}}{\int }_0^1{\omega }_t{\psi }_{t}dx+{\rho }_1{\int }_0^1{\omega }_t^{2}dx,\end{array}$$

(39)

$$\begin{array}{c}\hfill I_3=-{\displaystyle \frac{k_0}{l}}{\int }_0^1{\left({\omega }_x-l\phi \right)}_x\left({\phi }_x+\psi +l\omega \right)dx+{\displaystyle \frac{k_0^2}{k}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx,\end{array}$$

(40)

$$\begin{array}{c}\hfill I_4={\displaystyle \frac{{\rho }_1{k}_0}{kl}}{\int }_0^1{\omega }_{tx}{\phi }_{t}dx-{\displaystyle \frac{{\rho }_1{k}_0}{k}}{\int }_0^1{\phi }_t^{2}dx,\end{array}$$

(41)

$$\begin{array}{cc}\hfill I_5=& -\delta {\int }_0^1{\theta }_x{\left({\phi }_x+\psi +l\omega \right)}_{t}dx+\delta {\int }_0^1{\theta }_x{\psi }_{t}dx+\delta l{\int }_0^1{\theta }_x{\omega }_{t}dx-\gamma {\int }_0^1{\omega }_{tx}{\phi }_{t}dx\hfill \\ \hfill & +{\int }_0^1\left(g\ast {\theta }_x\right){\left({\phi }_x+\psi +l\omega \right)}_{t}dx-{\int }_0^1\left(g\ast {\theta }_x\right){\psi }_{t}dx-l{\int }_0^1\left(g\ast {\theta }_x\right){\omega }_{t}dx,\hfill \end{array}$$

(42)

$$\begin{array}{c}\hfill I_6=-k{\int }_0^1{\theta }_{tx}\left({\phi }_x+\psi +l\omega \right)dx+l{k}_0{\int }_0^1{\theta }_t\left({\omega }_x-l\phi \right)dx,\end{array}$$

(43)

$$\begin{array}{c}\hfill I_7=g\left(t\right){\int }_0^1{\theta }_x\left({\phi }_x+\psi +l\omega \right)dx-{\int }_0^1\left(g^{\prime }\diamond {\theta }_x\right)\left({\phi }_x+\psi +l\omega \right)dx.\end{array}$$

(44)

Then, inserting (38)–(44) into (37), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{F}_7\left(t\right)=& -k{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx-{\displaystyle \frac{{\rho }_1{k}_0}{k}}{\int }_0^1{\phi }_t^{2}dx+{\displaystyle \frac{k_0^2}{k}}{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx+{\rho }_1{\int }_0^1{\omega }_t^{2}dx\hfill \\ \hfill & +{\displaystyle \frac{{\rho }_1}{l}}{\int }_0^1{\omega }_t{\psi }_{t}dx+{\displaystyle \frac{{\rho }_1\delta }{kl\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_x{\psi }_{t}dx+{\displaystyle \frac{{\rho }_1\delta }{k\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_x{\omega }_{t}dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1\left(k_0-k\right)}{kl\gamma }}{\int }_0^1\left(g\ast {\theta }_x\right){\psi }_{t}dx-{\displaystyle \frac{{\rho }_1\left(k_0-k\right)}{k\gamma }}{\int }_0^1\left(g\ast {\theta }_x\right){\omega }_{t}dx\hfill \\ \hfill & +{\displaystyle \frac{{\rho }_3{k}_0}{k\gamma }}\left(k_0-k\right){\int }_0^1{\theta }_t\left({\omega }_x-l\phi \right)dx-{\displaystyle \frac{{\rho }_1\left(k_0-k\right)}{kl\gamma }}g\left(t\right){\int }_0^1{\theta }_x\left({\phi }_x+\psi +l\omega \right)dx\hfill \\ \hfill & +{\displaystyle \frac{{\rho }_1\left(k_0-k\right)}{kl\gamma }}{\int }_0^1\left(g^{\prime }\diamond {\theta }_x\right)\left({\phi }_x+\psi +l\omega \right)dx+{\displaystyle \frac{\chi }{l\gamma }}{\int }_0^1{\theta }_{tx}\left({\phi }_x+\psi +l\omega \right)dx.\hfill \end{array}$$

Therefore, the desired result (36) follows easily using Lemma 2 and Young’s inequality. □

Now, we state and prove the main result of this section.

Theorem 1. 

Let

$$\begin{array}{c}\hfill \left({\phi }_0,{\phi }_1,{\psi }_0,{\psi }_1,{\omega }_0,{\omega }_1,{\theta }_0,{\theta }_1\right)\in H_0^1(0,1)\times L^2(0,1)\times {\left(H_{\ast }^1(0,1)\times L_{\ast }^2(0,1)\right)}^2\times H_0^1(0,1)\times L^2(0,1)\end{array}$$

be given, assuming that (8) holds and g satisfies assumptions (A1) and (A2). Then, for small l, the energy functional (9) satisfies

$$\begin{array}{c}\hfill E\left(t\right)\le {\tau }_1{e}^{-{\tau }_2{\int }_{t_0}^t\eta \left(s\right)ds},\forall t\ge t_0,\end{array}$$

(45)

where ${\tau }_1$ and ${\tau }_2$ are positive constants.

Proof. 

We introduce, for $N,N_i>0(i=1:7)$ to be properly chosen later, the Lyapunov functional $\mathcal{L}$ by

$$\begin{array}{c}\hfill \mathcal{L}\left(t\right)=NE\left(t\right)+\sum _{i=1}^7{N}_i{F}_i\left(t\right).\end{array}$$

(46)

Direct computations, using (10), (12), (14), (17), (20), (25), (27), and (36) and setting

$$\begin{array}{c}\hfill {\epsilon }_1={\displaystyle \frac{{\rho }_1{k}_0{N}_7}{2k{N}_2}},{\epsilon }_3={\displaystyle \frac{b{N}_1}{4N_4}},{\epsilon }_4={\displaystyle \frac{k_0{N}_2}{4N_5}},{\epsilon }_5={\displaystyle \frac{k{N}_7}{4N_5}},\\ \hfill {\epsilon }_6={\displaystyle \frac{\lambda N_3}{4N_6}},{\epsilon }_7={\displaystyle \frac{{\rho }_1{N}_5}{4N_6}},{\epsilon }_8={\displaystyle \frac{{\rho }_2{N}_4}{4b{N}_7}},\end{array}$$

give

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\mathcal{L}\left(t\right)\le & -\left[{\displaystyle \frac{b}{4}}{N}_1-{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{N}_3\right]{\int }_0^1{\psi }_x^{2}dx-\left[{\displaystyle \frac{{\rho }_2}{4b}}{N}_4-{\rho }_2{N}_1\right]{\int }_0^1{\psi }_t^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{{\rho }_1}{4}}{N}_5-c_1{N}_2\left(1+{\displaystyle \frac{N_2}{N_7}}\right)-c_1{N}_4-c_1{N}_7\left(1+{\displaystyle \frac{N_7}{N_4}}\right)\right]{\int }_0^1{\omega }_t^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{k}{4}}{N}_7-{\displaystyle \frac{k^2{l}^2}{k_0}}{N}_2-{\displaystyle \frac{k}{b}}{N}_4-c_1{N}_1-{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{N}_3\right]{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{{\rho }_3{g}_0}{2}}{N}_6-c_1{N}_2-c_1{N}_3\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}\right)-c_1{N}_5\left(1+{\displaystyle \frac{N_5}{N_2}}+{\displaystyle \frac{N_5}{N_7}}\right)-c_1{N}_7\right]{\int }_0^1{\theta }_t^{2}dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1{k}_0}{2k}}{N}_7{\int }_0^1{\phi }_t^{2}dx-\left[{\displaystyle \frac{k_0}{4}}{N}_2-{\displaystyle \frac{2}{1-2l^4}}{\epsilon }_2{N}_3-c_1{\displaystyle \frac{N_4^2}{N_1}}-{\displaystyle \frac{2k_0^2}{k}}{N}_7\right]{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{\lambda }{4}}{N}_3-c_1{N}_5-c_1{N}_7\left(1+{\displaystyle \frac{N_7}{N_4}}\right)\right]{\int }_0^1{\theta }_x^{2}dx+\left[{\displaystyle \frac{N}{2}}-c_1{N}_6-c_1{N}_7\right]{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx\hfill \\ \hfill & +\left[c_1{N}_3+c_1{N}_5+c_1{N}_6\left({\displaystyle \frac{N_6}{N_3}}+{\displaystyle \frac{N_6}{N_5}}\right)+c_1{N}_7\left(1+{\displaystyle \frac{N_7}{N_4}}\right)\right]{\int }_0^1\left(g\circ {\theta }_x\right)dx.\hfill \end{array}$$

Furthermore, by setting $N_1=1$ and ${\displaystyle N_7=\frac{9k{l}^2}{2k_0}{N}_2}$, we arrive at

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\mathcal{L}\left(t\right)\le & -\left[{\displaystyle \frac{b}{4}}-{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{N}_3\right]{\int }_0^1{\psi }_x^{2}dx-\left[{\displaystyle \frac{{\rho }_2}{4b}}{N}_4-{\rho }_2\right]{\int }_0^1{\psi }_t^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{{\rho }_1}{4}}{N}_5-c_1{N}_4-c_1{N}_2\left(1+{\displaystyle \frac{N_2}{N_4}}\right)\right]{\int }_0^1{\omega }_t^{2}dx-{\displaystyle \frac{9{\rho }_1{l}^2}{4}}{N}_2{\int }_0^1{\phi }_t^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{k^2{l}^2}{8k_0}}{N}_2-{\displaystyle \frac{k}{b}}{N}_4-c_1-{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{N}_3\right]{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{{\rho }_3{g}_0}{2}}{N}_6-c_1{N}_2-c_1{N}_3\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}\right)-c_1{N}_5\left(1+{\displaystyle \frac{N_5}{N_2}}\right)\right]{\int }_0^1{\theta }_t^{2}dx\hfill \\ \hfill & -\left[k_0\left({\displaystyle \frac{1}{4}}-9l^2\right)N_2-{\displaystyle \frac{2}{1-2l^4}}{\epsilon }_2{N}_3-c_1{N}_4^2\right]{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{\lambda }{4}}{N}_3-c_1{N}_5-c_1{N}_2\left(1+{\displaystyle \frac{N_2}{N_4}}\right)\right]{\int }_0^1{\theta }_x^{2}dx+\left[{\displaystyle \frac{N}{2}}-c_1{N}_6-c_1{N}_2\right]{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx\hfill \\ \hfill & +\left[c_1{N}_3+c_1{N}_5+c_1{N}_6\left({\displaystyle \frac{N_6}{N_3}}+{\displaystyle \frac{N_6}{N_5}}\right)+c_1{N}_2\left(1+{\displaystyle \frac{N_2}{N_4}}\right)\right]{\int }_0^1\left(g\circ {\theta }_x\right)dx.\hfill \end{array}$$

At this point, we choose $N_4$ large enough so that

$$\begin{array}{c}\hfill {\alpha }_1={\displaystyle \frac{{\rho }_2}{4b}}{N}_4-{\rho }_2>0.\end{array}$$

Then, for small l satisfies ${\displaystyle l<\frac{1}{6}}$, we choose $N_2$ large enough such that

$$\begin{array}{c}\hfill {\beta }_1={\displaystyle \frac{k^2{l}^2}{8k_0}}{N}_2-{\displaystyle \frac{k}{b}}{N}_4-c_1>0\mathrm{and}{\beta }_2=k_0\left({\displaystyle \frac{1}{4}}-9l^2\right)N_2-c_1{N}_4^2>0.\end{array}$$

Also, we select $N_5$ large enough so that

$$\begin{array}{c}\hfill {\alpha }_2={\displaystyle \frac{{\rho }_1}{4}}{N}_5-c_1{N}_4-c_1{N}_2\left(1+{\displaystyle \frac{N_2}{N_4}}\right)>0.\end{array}$$

Next, we choose $N_3$ large so that

$$\begin{array}{c}\hfill {\alpha }_3={\displaystyle \frac{\lambda }{4}}{N}_3-c_1{N}_5-c_1{N}_2\left(1+{\displaystyle \frac{N_2}{N_4}}\right)>0;\end{array}$$

we then pick ${\epsilon }_2$ small enough such that

$$\begin{array}{c}\hfill {\epsilon }_2<min\left({\displaystyle \frac{b\left(1-2l^4\right)}{8N_3{l}^2}},{\displaystyle \frac{{\beta }_1\left(1-2l^4\right)}{2N_3{l}^2}},{\displaystyle \frac{{\beta }_2\left(1-2l^4\right)}{2N_3}}\right).\end{array}$$

Consequently, we obtain

$$\begin{array}{c}\hfill {\alpha }_4={\displaystyle \frac{b}{4}}-{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{N}_3>0,{\alpha }_5={\beta }_1-{\displaystyle \frac{2l^2}{1-2l^4}}{\epsilon }_2{N}_3>0,{\alpha }_6={\beta }_2-{\displaystyle \frac{2}{1-2l^4}}{\epsilon }_2{N}_3>0.\end{array}$$

After that, we select $N_6$ large enough so that

$$\begin{array}{c}\hfill {\alpha }_7={\displaystyle \frac{{\rho }_3{g}_0}{2}}{N}_6-c_1{N}_2-c_1{N}_3\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}\right)-c_1{N}_5\left(1+{\displaystyle \frac{N_5}{N_2}}\right)>0.\end{array}$$

Finallay, we choose N very large enough such that

$$\begin{array}{c}\hfill {\alpha }_8={\displaystyle \frac{N}{2}}-c_1{N}_6-c_1{N}_2>0.\end{array}$$

So, we arrive at

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\mathcal{L}\left(t\right)\le & -{\alpha }_4{\int }_0^1{\psi }_x^{2}dx-{\alpha }_1{\int }_0^1{\psi }_t^{2}dx-{\alpha }_2{\int }_0^1{\omega }_t^{2}dx-{\alpha }_{10}{\int }_0^1{\phi }_t^{2}dx\hfill \\ \hfill & -{\alpha }_5{\int }_0^1{\left({\phi }_x+\psi +l\omega \right)}^{2}dx-{\alpha }_7{\int }_0^1{\theta }_t^{2}dx-{\alpha }_6{\int }_0^1{\left({\omega }_x-l\phi \right)}^{2}dx\hfill \\ \hfill & -{\alpha }_3{\int }_0^1{\theta }_x^{2}dx+{\alpha }_8{\int }_0^1\left(g^{\prime }\circ {\theta }_x\right)dx+{\alpha }_9{\int }_0^1\left(g\circ {\theta }_x\right)dx.\hfill \end{array}$$

(47)

On the other hand, exploiting (9) and (46), together with Cauchy–Schwarz, Young’s, and Poincaré’s inequalities, it is easy to verify that there exist two positive constants ${\varpi }_1$ and ${\varpi }_2$ such that

$${\varpi }_{1}E\left(t\right)\le \mathcal{L}\left(t\right)\le {\varpi }_{2}E\left(t\right),\forall t\ge 0.$$

(48)

Then, for positive constant ${\mu }_1$, inequality (47) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\mathcal{L}\left(t\right)\le -{\mu }_{1}E\left(t\right)+{\alpha }_9{\int }_0^1\left(g\circ {\theta }_x\right)dx,\forall t\ge 0.\end{array}$$

(49)

By multiplying (49) by $\eta \left(t\right)$ and using (A2) and (10), we arrive at

$$\begin{array}{cc}\hfill \eta \left(t\right){\displaystyle \frac{d}{dt}}\mathcal{L}\left(t\right)\le & -{\mu }_1\eta \left(t\right)E\left(t\right)-2{\alpha }_9{\displaystyle \frac{d}{dt}}E\left(t\right),\forall t\ge t_0,\hfill \end{array}$$

which can be rewritten as

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left[\eta \left(t\right)\mathcal{L}\left(t\right)+2{\alpha }_{9}E\left(t\right)\right]-\mathcal{L}\left(t\right){\displaystyle \frac{d}{dt}}\eta \left(t\right)\le & -{\mu }_1\eta \left(t\right)E\left(t\right),\forall t\ge t_0,\hfill \end{array}$$

using the fact that ${\displaystyle \frac{d}{dt}\eta \left(t\right)\le 0}$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left[\eta \left(t\right)\mathcal{L}\left(t\right)+2{\alpha }_{9}E\left(t\right)\right]\le & -{\mu }_1\eta \left(t\right)E\left(t\right),\forall t\ge t_0.\hfill \end{array}$$

By exploiting (48), it can easily be shown that

$$\begin{array}{c}\hfill \mathcal{M}\left(t\right)=\eta \left(t\right)\mathcal{L}\left(t\right)+2{\alpha }_{9}E\left(t\right)\sim E\left(t\right).\end{array}$$

(50)

Consequently, for some positive constant ${\tau }_2$, we obtain

$${\displaystyle \frac{d}{dt}}\mathcal{M}\left(t\right)\le -{\tau }_2\eta \left(t\right)\mathcal{M}\left(t\right),\forall t\ge t_0.$$

(51)

Integrating (51) over $(t_0,t)$, yield

$$\mathcal{M}\left(t\right)\le \mathcal{M}\left(t_0\right)e^{-{\tau }_2{\int }_{t_0}^t\eta \left(s\right)ds},\forall t\ge t_0.$$

(52)

Again, using (50) for some positive constant ${\tau }_1$, we obtain

$$E\left(t\right)\le {\tau }_1{e}^{-{\tau }_2{\int }_{t_0}^t\eta \left(s\right)ds},\forall t\ge t_0,$$

which completes the proof. □

Example 1. 

The following two examples illustrate our result:

1. 

Let $g\left(t\right)=a{e}^{-b(1+t)}$, where $b>0$ and $a>0$ are small enough so that (A1) is satisfied. Then, $g^{\prime }\left(t\right)=-\eta \left(t\right)g\left(t\right)$, where $\eta \left(t\right)=b.$ Therefore, by using (45) we get

$$E\left(t\right)\le C_0{e}^{-b{\tau }_{2}t},\forall t\ge t_0.$$

2. 

Let $g\left(t\right)={\displaystyle \frac{d}{{(1+t)}^p}}$, where $p>1$ and d is chosen so that hypothesis (A1) remains valid. Then, $g^{\prime }\left(t\right)=-\eta \left(t\right)g\left(t\right)$, where $\eta \left(t\right)={\displaystyle \frac{d}{{(1+t)}^p}}$. Then, (45) gives

$$E\left(t\right)\le {\displaystyle \frac{C_1}{{(1+t)}^{{\tau }_2}}},\forall t\ge t_0.$$

4. Conclusions

This paper focuses on the general stability of solutions for a one-dimensional thermoelastic Bresse beam model, in which the thermal effects are governed by the Green–Naghdi theory of heat conduction and are coupled with the longitudinal displacement. The present analysis provides a rigorous framework for understanding the dissipative mechanisms arising from thermoelastic interactions. Future work may extend this framework by incorporating fractional time-delay effects or fractional damping mechanisms in place of the viscoelastic damping.