Stability of thermoelastic Timoshenko beam with suspenders and time-varying feedback

Abstract

This paper considers a one-dimensional thermoelastic Timoshenko beam system with suspenders, general weak internal damping with time varying coefficient, and time-varying delay terms. Under suitable conditions on the nonlinear terms, we prove a general stability result for the beam model, where exponential and polynomial decay are special cases. We also gave some examples to illustrate our theoretical finding.

Introduction

In this paper, we consider a thermoelastic Timoshenko beam with suspension cables, weak internal damping, and a time-varying delay damping of the form

$$ \textstyle\begin{cases} \rho u_{tt}(x,t)-\alpha u_{xx}(x,t)-\lambda ( \varphi -u )(x,t) \\ \quad {} + \gamma _{1}a(t)g_{1}(u_{t}(x,t))+ \gamma _{2}a(t)g_{2}(u_{t}(x,t- \tau (t)))=0, \\ \rho _{1}\varphi _{tt}(x,t)-k (\varphi _{x} +\psi )_{x}(x,t) +\lambda (\varphi -u )(x,t)+\gamma _{3}\varphi _{t}(x,t)=0, \\ \rho _{2} \psi _{tt}(x,t)-b \psi _{xx}(x,t) + k (\varphi _{x} + \psi )(x,t)- m\theta _{x}(x,t)=0, \\ \rho _{3} \theta _{t}(x,t)-\beta \theta _{xx}(x,t) -m \psi _{xt}(x,t)=0, \end{cases} $$

(1.1)

where $(x,t)\in (0,1)\times (0,\infty )$, φ represents the transverse displacement (in vertical direction) of the beam cross section, ψ is the angle of rotation of a cross-section. The vertical displacement of the vibrating spring (the cable) is represented by the function u, θ depicts the thermal moment of the beam, $\lambda >0$ is the common stiffness of the string, and $\alpha >0$ is the elastic modulus of the string (holding the cable to the deck). The positive constants ρ, $\rho _{1}$, $\rho _{2}$ are the density of the mass material of the cable, the mass density, and the moment of mass inertia of the beam, respectively. Also, b, k, β, m represent the rigidity coefficient of the cross-section, the shear modulus of elasticity, the thermal diffusivity, and the coupling coefficient which depends on the material properties, respectively. The function $\tau (t)>0$ is the time-varying delay, $\gamma _{1}$ and $\gamma _{2}$ are real positive damping constants, $g_{1}$ and $g_{2}$ are the damping functions, and $a(t)$ is a nonlinear weight function. We supplement (1.1) with the boundary conditions

$$ \textstyle\begin{cases} u(0,t)=\varphi _{x}(0,t)=\psi (0,t)=\theta _{x}(0,t)=0, \quad t>0, \\ u(1,t)=\varphi (1,t)=\psi _{x}(1,t)=\theta (1,t)=0, \quad t>0, \end{cases} $$

(1.2)

and the initial data

$$ \textstyle\begin{cases} u(x,0)=u_{0}(x),\qquad \varphi (x,0)= \varphi _{0}(x), \\ \psi (x,0)= \psi _{0}(x), \qquad \theta (x,0)=\theta _{0}(x), & \text{in } (0,1), \\ u_{t}(x,0)=u_{1}(x),\qquad \varphi _{t}(x,0)= \varphi _{1}(x), \\ \psi _{t}(x,0)= \psi _{1}(x), & \text{in } (0,1), \\ u_{t}(x,t-\tau (0))=f_{0}(x,t-\tau (0)),\quad \text{in } (0,1)\times (0, \tau (0)). \end{cases} $$

(1.3)

The stability of the above thermoelastic Timoshenko system with suspension cables would be our penultimate focus in this work. The Timoshenko beam model is arguably very popular and most used when the vibration of a beam exhibits significant transverse shear strain. A model to describe this phenomenon was introduced by Timoshenko [35] in 1921, see also [15, 18]. The nonlinear vibration of suspension bridges have captured the attention of different researchers and a number of research articles were written on the topic. The somewhat unpredictable large oscillations of suspension bridges have been modeled in diverse ways, one may see [1, 14, 25]. In any attempt to adequately describe the complicated dynamics of a suspension bridge, a robust model would be one with a considerable amount of degrees of freedom. Without prejudice, some simplified models have been considered, but do not account for a number of realistic behavior of suspension bridges, e.g., torsional oscillations. Of an advantage is the fact that rigorous mathematical analysis is easily carried out with such simpler models. A typical simplified model is the one-dimensional vibrating beam model, where torsional motion is neglected by ignoring sectional dimensions of the beam when they are negligible compared to length of the beam. The emergence of string-beam systems which model a nonlinear coupling of a beam and main cable (the string) were born out of the pioneering works of Lazer, McKenna, and Walter [23, 25, 26] (see also [7] and its references). Though initially modeled through the classic Euler–Bernoulli beam theory, the Timoshenko beam theory is also proven to perform better in predicting a beam response to vibrations than a model based on the classical Euler–Bernoulli beam theory. Indeed, the Timoshenko beam theory takes into account both rotary inertia and shear deformation effects, these are often neglected when applying Euler–Bernoulli beam theory.

In the Timoshenko beam with suspenders which is modeled by (1.1), the suspenders are cables which are elastic in nature and are attached to the beam with elastic springs. The temperature dissipation here is assumed to be governed by the Fourier law of heat conduction. For $a(t)\equiv 1$, $g_{1}(s)\equiv s$ and $\gamma _{2}\equiv 0$, $g_{2}\equiv 0$ in system (1.1), Bochichio et al. [6] proved a well-posedness and an exponential stability result. A number of works have been done on different thermoelastic Timoshenko models without suspenders (see [10, 12, 16, 17, 28] and references in them). Time delays occur in systems modeling many phenomena in areas such as biosciences, medicine, physics, robotics, economics, chemical, thermal, and structural engineering. These phenomena depend on both present and some past history of occurrences, see [8, 9, 13, 21, 34] and the examples therein. In the case of constant delay and constant weight, the delay term usually accounts for the past history of strain, only up to some finite time $\tau (t)\equiv \tau$.

A step further involves results in the literature about constant weights ($\gamma _{1}a(t)\equiv \gamma _{1}$, $\gamma _{2}a(t)\equiv \gamma _{2}$ constants) and time-varying delay $\tau (t)$. Works presenting the exponential stability result for wave equation with boundary or internal time-varying delay appeared in Nicaise et al. [32, 33]. Enyi and Mukiawa in [11] presented a general decay result for a viscoelastic plate equation under the condition $|\gamma _{2}|<|\gamma _{1}|\sqrt{(1-d)}$. Furthermore, in [4, 24], the authors presented some existence and stability results for wave equation with internal time-varying delay and time-varying weights; and for suspension bridge models, see Mukiawa [3, 27, 29, 30].

Motivated by the works in [3, 6, 29], in the current paper, we are concerned with the stability result for the thermoelastic Timoshenko system with suspension cables, time-varying internal feedback, and time-varying weight given in (1.1)–(1.3). The result in [6] is a particular case of our result in this paper.

We arrange this paper in the following manner. In Sect. 2, we state the needed assumptions. In Sect. 3, we present the proof of some technical and needed lemmas for our main result. In the last Sect. 4, we present and prove our main stability result. Throughout this paper, c and $c_{i}$, $i=1,2,\dots$, are generic positive constants, which are not necessarily the same from line to line.

Functional settings and assumptions

In this section, we state some needed assumptions on the damping coefficients, nonlinear functions, and the time-varying delay. As in [5, 32, 33], we assume the following conditions:

$(A_{1})$:

Function $a:[0,+\infty )\rightarrow (0,+\infty )$ is a nonincreasing $C^{1}$-function such that there exists a positive constant C satisfying

$$ \bigl\vert a'(t) \bigr\vert \leq Ca(t),\qquad \int _{0}^{+\infty}a(t)\,dt=+\infty . $$

(2.1)

$(A_{2})$:

Fuction $g_{1}:\mathbb{R}\rightarrow \mathbb{R}$ is a nondecreasing $C^{0}$-function such that there exist positive constants $C_{1}$, $C_{2}$, r and a convex increasing function $\chi \in C^{1}([0,+\infty ))\cap C^{2}((0,+\infty ))$ satisfying $\chi (0)=0$ or χ is a nonlinear strictly convex $C^{2}$-function on $(0, r]$ with $\chi '(0), \chi ''>0$ such that

$$\begin{aligned}& s^{2}+g_{1}^{2}(s) \le \chi ^{-1}\bigl(s g_{1}(s)\bigr),\quad \text{for all } \vert s \vert \le r, \end{aligned}$$

(2.2)

$$\begin{aligned}& C_{1}s^{2} \le sg_{1}(s)\le C_{2}s^{2}, \quad \text{for all } \vert s \vert \ge r. \end{aligned}$$

(2.3)

Function $g_{2}:\mathbb{R}\rightarrow \mathbb{R}$ is an increasing and odd $C^{1}$-function such that for some positive constants $C_{3}$, $\alpha _{1}$, $\alpha _{2}$,

$$\begin{aligned}& \bigl\vert g_{2}'(s) \bigr\vert \leq C_{3}, \end{aligned}$$

(2.4)

$$\begin{aligned}& \alpha _{1}\bigl(sg_{2}(s)\bigr) \leq G(s) \leq \alpha _{2}\bigl(sg_{1}(s)\bigr), \end{aligned}$$

(2.5)

where

$$ G(s)= \int _{0}^{s}g_{2}(r)\,dr. $$

(2.6)

$(A_{3})$:

There exist $\tau _{0},\tau _{1}>0$ such that

$$\begin{aligned}& 0 < \tau _{0}\leq \tau (t)\leq \tau _{1},\quad \forall t>0, \end{aligned}$$

(2.7)

$$\begin{aligned}& \tau \in W^{2,\infty}(0,T), \quad \forall T>0, \end{aligned}$$

(2.8)

$$\begin{aligned}& \tau '(t) \leq d< 1,\quad \forall t>0. \end{aligned}$$

(2.9)

$(A_{4})$:

The damping coefficients satisfy

$$ \gamma _{2} \alpha _{2}(1-d\alpha _{1})< \alpha _{1}(1-d)\gamma _{1}. $$

(2.10)

Remark 2.1

Using the monotonicity of $g_{2}$ and the mean value theorem for integrals, we deduce that

$$ G(s)= \int _{0}^{s}g_{2}(r)\,dr< sg_{2}(s). $$

(2.11)

It follows from (2.5) that $\alpha _{1}<1$.

Similarly, as in Nicaise and Pignotti [31], we introduce the following change of variable:

$$ z(x,\sigma , t)=u_{t}\bigl(x,t-\tau (t)\sigma \bigr), \quad \text{for } (x, \sigma ,t)\in (0,1)\times (0,1)\times (0,\infty ). $$

(2.12)

It follows that

$$ \tau (t)z_{t}(x,\sigma ,t)+\bigl(1-\tau '(t)\sigma \bigr)z_{\sigma}(x,\sigma ,t)=0. $$

(2.13)

Therefore, system (1.1) becomes

$$ \textstyle\begin{cases} \rho u_{tt}(x,t)-\alpha u_{xx}(x,t)-\lambda ( \varphi -u )(x,t)+ \gamma _{1}a(t)g_{1}(u_{t}(x,t)) \\ \quad {}+ \gamma _{2}a(t)g_{2}(z(x,1,t))=0, \\ \rho _{1}\varphi _{tt}(x,t)-k (\varphi _{x} +\psi )_{x}(x,t) +\lambda (\varphi -u )(x,t)+\gamma _{3}\varphi _{t}(x,t)=0, \\ \rho _{2} \psi _{tt}(x,t)-b \psi _{xx}(x,t) + k (\varphi _{x} + \psi )(x,t)- m\theta _{x}(x,t)=0, \\ \rho _{3} \theta _{t}(x,t)-\beta \theta _{xx}(x,t) -m \psi _{xt}(x,t)=0, \\ \tau (t)z_{t}(x,\sigma ,t)+(1-\tau '(t)\sigma )z_{\sigma}(x,\sigma ,t)=0, \end{cases} $$

(2.14)

subjected to the boundary conditions

$$ \textstyle\begin{cases} u(0,t)=\varphi _{x}(0,t)=\psi (0,t)=\theta _{x}(0,t)=0, \quad t>0, \\ u(1,t)=\varphi (1,t)=\psi _{x}(1,t)=\theta (1,t)=0, \quad t>0, \\ z(x,0,t)=u_{t}(x,t), \quad x\in (0,1), t>0, \end{cases} $$

(2.15)

and initial data

$$ \textstyle\begin{cases} u(x,0)=u_{0}(x),\qquad \varphi (x,0)= \varphi _{0}(x), \\ \psi (x,0)= \psi _{0}(x),\qquad \theta (x,0)=\theta _{0}(x), & \text{in } (0,1), \\ u_{t}(x,0)=u_{1}(x),\qquad \varphi _{t}(x,0)= \varphi _{1}(x), \\ \psi _{t}(x,0)= \psi _{1}(x), & \text{in } (0,1), \\ z(x,\sigma ,0)= u_{t}(x,-\tau (0)\sigma )=f_{0}(x,-\tau (0)\sigma ), & \text{in } (0,1)\times (0,1). \end{cases} $$

(2.16)

We introduce the following spaces:

$$\begin{aligned} & H_{a}^{1}(0,1)=\bigl\{ \phi \in H^{1}(0,1): \phi (0) = 0\bigr\} , \\ & H_{b}^{1}(0,1)= \bigl\{ \phi \in H^{1}(0,1): \phi (1) = 0\bigr\} , \\ & H^{2}_{a}(0,1)=\bigl\{ \phi \in H^{2}(0,1): \phi _{x}\in H_{a}^{1}(0,1)\bigr\} , \\ & H^{2}_{b}(0,1)=\bigl\{ \phi \in H^{2}(0,1): \phi _{x}\in H_{b}^{1}(0,1) \bigr\} . \end{aligned}$$

For completeness, we state without proof the existence and uniqueness result for problem (1.1)–(1.3). The result can be established using the Faedo–Galerkin approximation method, see [5] or standard nonlinear semigroup method, see [19, 20].

Theorem 2.1

Let

$$ \begin{aligned} (u_{0},\varphi _{0},\psi _{0},\theta _{0})&\in H^{2}(0,1)\cap H^{1}_{0}(0,1)\times H^{2}_{a}(0,1)\cap H^{1}_{b}(0,1) \times H^{2}_{b}(0,1) \\ &\quad {}\cap H^{1}_{a}(0,1) \times H^{2}_{a}(0,1)\cap H^{1}_{b}(0,1) \end{aligned} $$

and

$$ (u_{1},\varphi _{1},\psi _{1})\in H^{1}_{0}(0,1)\times H^{1}_{a}(0,1) \times H^{1}_{b}(0,1),\qquad f_{0}\bigl(\cdot ,-\tau (0) \bigr)\in H^{1}_{0} \bigl( (0,1);H^{1}(0,1) \bigr) $$

be given such that

$$ f_{0}(\cdot ,0)=u_{1}. $$

Suppose conditions $(A_{1})$–$(A_{4})$ hold. Then, problem (1.1)–(1.3) has a unique global weak solution in the class

$$\begin{aligned} &u\in L^{\infty} \bigl( [0,+\infty);H^{2}(0,1) \cap H^{1}_{0}(0,1) \bigr), \qquad u_{t}\in L^{\infty} \bigl( [0,+\infty); H^{1}_{0}(0,1) \bigr), \\ & u_{tt} \in L^{\infty} \bigl( (0,+\infty );L^{2}(0,1) \bigr), \\ &\varphi \in L^{\infty} \bigl( [0,+\infty);H^{2}_{a}(0,1) \cap H^{1}_{b}(0,1) \bigr),\qquad \varphi _{t} \in L^{\infty} \bigl( [0,+\infty);H^{1}_{b}(0,1) \bigr), \\ &\varphi _{tt} \in L^{\infty} \bigl( (0,+\infty );L^{2}(0,1) \bigr), \\ &\psi \in L^{\infty} \bigl( [0,+\infty);H^{2}_{b}(0,1) \cap H^{1}_{a}(0,1) \bigr), \qquad \psi _{t} \in L^{\infty} \bigl( [0,+\infty);H^{1}_{a}(0,1) \bigr), \\ &\psi _{tt} \in L^{\infty} \bigl( (0,+\infty );L^{2}(0,1) \bigr), \\ &\theta \in L^{\infty} \bigl( [0,+\infty);H^{2}_{a}(0,1) \cap H^{1}_{b}(0,1) \bigr),\qquad \theta _{t}\in L^{\infty} \bigl( (0,+\infty );L^{2}(0,1) \bigr). \end{aligned}$$

Technical lemmas

In this section, we prove some important lemmas which will be essential in establishing the main result. Let μ̄ be a positive constant satisfying

$$ \frac{\gamma _{2}(1-\alpha _{1})}{\alpha _{1}(1-d)}< \bar{\mu}< \frac{\gamma _{1}-\gamma _{2}\alpha _{2}}{\alpha _{2}} $$

(3.1)

and set

$$ \mu (t)=\bar{\mu}a(t). $$

The energy functional of system (2.14)–(2.16) is defined by

$$ \begin{aligned} E(t)={}& \frac{1}{2} \int _{0}^{1} \bigl[\rho u_{t}^{2} + \rho _{1} \varphi _{t}^{2} + \rho _{2} \psi _{t}^{2} + \alpha u_{x}^{2} + k( \varphi _{x}+\psi )^{2}+b\psi _{x}^{2}+ \lambda (\varphi -u)^{2} \bigr]\,dx \\ &{}+ \frac{1}{2} \int _{0}^{1}\rho _{3}\theta ^{2}\,dx + \mu (t)\tau (t) \int _{0}^{1} \int _{0}^{1} G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx. \end{aligned} $$

(3.2)

Lemma 3.1

Let $(u,\varphi , \psi , \theta ,z)$ be the solution of system (2.14)–(2.16). Then, the energy functional (3.2) satisfies

$$ \begin{aligned} \frac{dE(t)}{dt}\leq {}&-a(t) [ \gamma _{1}-\bar{\mu}\alpha _{2}- \gamma _{2}\alpha _{2} ] \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx \\ &{}-a(t) \bigl[\bar{\mu}\bigl(1-\tau '(t)\bigr)\alpha _{1}- \gamma _{2}(1-\alpha _{1}) \bigr] \int _{0}^{1}z(x,1,t)g_{2}\bigl(z(x,1,t) \bigr)\,dx \\ & {}-\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1} \theta _{x}^{2}\,dx \\ \leq{} &0, \quad \forall t\geq 0. \end{aligned} $$

(3.3)

Proof

Multiplying (2.14)₁ by $u_{t}$, (2.14)₂ by $\varphi _{t}$, (2.14)₃ by $\psi _{t}$, and (2.14)₄ by θ, integrating the outcome over $(0,1)$, and applying integration by parts and the boundary conditions, we get

$$\begin{aligned} \begin{aligned} &\frac {1}{2}\frac {d}{dt} \int _{0}^{1} \bigl[ \rho u^{2}_{t} + \alpha u_{x}^{2}+\lambda (\varphi -u)^{2} \bigr]\,dx \\ & \quad =\lambda \int _{0}^{1}\varphi _{t}(\varphi -u)\,dx- \gamma _{1}a(t) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx- \gamma _{2}a(t) \int _{0}^{1}u_{t} g_{2} \bigl(z(x,1,t)\bigr)\,dx, \end{aligned} \end{aligned}$$

(3.4)

$$\begin{aligned} \begin{aligned} &\frac {1}{2}\frac {d}{dt} \int _{0}^{1} \bigl[ \rho _{1}\varphi _{t}^{2} +k(\varphi _{x}+\psi )^{2} \bigr]\,dx \\ & \quad =-\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \lambda \int _{0}^{1} \varphi _{t}(\varphi -u)\,dx +k \int _{0}^{1} \psi _{t}(\varphi _{x}+ \psi )\,dx, \end{aligned} \end{aligned}$$

(3.5)

$$\begin{aligned} &\frac {1}{2}\frac {d}{dt} \int _{0}^{1} \bigl[ \rho _{2}\psi ^{2}_{t} + b\psi _{x}^{2} \bigr]\,dx= m \int _{0}^{1} \psi _{t}\theta _{x}\,dx-k \int _{0}^{1} \psi _{t}(\varphi _{x}+\psi )\,dx, \end{aligned}$$

(3.6)

$$\begin{aligned} &\frac {1}{2} \int _{0}^{1} \rho _{3}\theta ^{2}\,dx=-\beta \int _{0}^{1} \theta _{x}^{2}\,dx-m \int _{0}^{1}\psi _{t}\theta _{x}\,dx. \end{aligned}$$

(3.7)

Adding (3.4)–(3.7), we arrive at

$$\begin{aligned}& \begin{aligned} &\frac{1}{2} \int _{0}^{1} \bigl( \rho u^{2}_{t} +\alpha u_{x}^{2}+ \lambda (\varphi -u)^{2}+\rho _{1}\varphi _{t}^{2} +k(\varphi _{x}+ \psi )^{2}+\rho _{2}\psi ^{2}_{t} + b\psi _{x}^{2}+\rho _{3} \theta ^{2} \bigr)\,dx \\ &\quad = -\gamma _{1}a(t) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx- \gamma _{2}a(t) \int _{0}^{1}u_{t} g_{2} \bigl(z(x,1,t)\bigr)\,dx \\ &\qquad {}-\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1} \theta _{x}^{2}\,dx. \end{aligned} \end{aligned}$$

(3.8)

Now, multiplying equation (2.14)₅ by $\mu (t)g_{2}(z(x,\sigma ,t))$ and integrating over $(0,1)\times (0,1)$, we obtain

$$ \begin{aligned} &\mu (t)\tau (t) \int _{0}^{1} \int _{0}^{1} z_{t}(x,\sigma ,t) g_{2}\bigl(z(x, \sigma ,t)\bigr)\,d\sigma \,dx \\ &\quad {} +\mu (t) \int _{0}^{1} \int _{0}^{1}\bigl(1-\tau '(t)\sigma \bigr)z_{\sigma}(x, \sigma , t) g_{2}\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx=0. \end{aligned} $$

(3.9)

On account of (2.6), we can write

$$ \frac{\partial }{\partial \sigma} \bigl[ G\bigl(z(x,\sigma ,t)\bigr) \bigr]=z_{ \sigma}(x, \sigma , t) g_{2}\bigl(z(x,\sigma ,t)\bigr). $$

(3.10)

Therefore, (3.9) becomes

$$ \begin{aligned} &\mu (t)\tau (t) \int _{0}^{1} \int _{0}^{1}z_{t}(x,\sigma ,t) g_{2}\bigl(z(x, \sigma ,t)\bigr)\,d\sigma \,dx \\ &\quad =-\mu (t) \int _{0}^{1} \int _{0}^{1}\bigl(1-\tau '(t)\sigma \bigr) \frac{\partial }{\partial \sigma} \bigl[ G\bigl(z(x,\sigma ,t)\bigr) \bigr]\,d\sigma \,dx. \end{aligned} $$

(3.11)

It follows that

$$ \begin{aligned} &\frac{d}{dt} \biggl(\mu (t)\tau (t) \int _{0}^{1} \int _{0}^{1}G\bigl(z(x, \sigma ,t)\bigr)\,d\sigma \,dx \biggr) \\ &\quad =-\mu (t) \int _{0}^{1} \int _{0}^{1}\frac{\partial}{\partial \sigma} \bigl[\bigl(1-\tau '(t)\sigma \bigr)G\bigl(z(x,\sigma ,t)\bigr) \bigr]\,d\sigma \,dx \\ &\qquad {} + \mu '(t)\tau (t) \int _{0}^{l} \int _{0}^{1}G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\ &\quad =\mu (t) \int _{0}^{1} \bigl( G\bigl(z(x,0,t)\bigr)-G \bigl(z(x,1,t)\bigr) \bigr)\,dx +\mu (t) \tau '(t) \int _{0}^{1} G\bigl(z(x,1,t)\bigr)\,dx \\ &\qquad {}+\mu '(t)\tau (t) \int _{0}^{1} \int _{0}^{1}G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\ &\quad =\mu (t) \int _{0}^{1}G\bigl(u_{t}(x,t)\bigr)\,dx-\mu (t) \bigl(1-\tau '(t)\bigr) \int _{0}^{1}G(z(x,1,t)\,dx \\ &\qquad {}+\mu '(t)\tau (t) \int _{0}^{1} \int _{0}^{1}G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx. \end{aligned} $$

(3.12)

Recalling the definition of the energy functional (3.2), and adding (3.8) and (3.12), we obtain

$$ \begin{aligned} \frac{dE(t)}{dt}={}& -\gamma _{1}a(t) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx- \gamma _{2}a(t) \int _{0}^{1}u_{t} g_{2} \bigl(z(x,1,t)\bigr)\,dx \\ &{} +\mu (t) \int _{0}^{1}G\bigl(u_{t}(x,t)\bigr)\,dx-\mu (t) \bigl(1-\tau '(t)\bigr) \int _{0}^{1}G(z(x,1,t)\,dx \\ & {}-\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1} \theta _{x}^{2}\,dx +\mu '(t)\tau (t) \int _{0}^{1} \int _{0}^{1}G\bigl(z(x, \sigma ,t)\bigr)\,d\sigma \,dx. \end{aligned} $$

(3.13)

On the account of $(A_{1})$ and (2.5), we get

$$ \begin{aligned} \frac{dE(t)}{dt}\leq{} &- \bigl( \gamma _{1}a(t)-\mu (t)\alpha _{2} \bigr) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx- \gamma _{2}a(t) \int _{0}^{1}u_{t} g_{2} \bigl(z(x,1,t)\bigr)\,dx \\ &{} -\mu (t) \bigl(1-\tau '(t)\bigr) \int _{0}^{1}G(z(x,1,t)\,dx -\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1}\theta _{x}^{2}\,dx . \end{aligned} $$

(3.14)

Now, we consider the convex conjugate of G defined by

$$ G^{*}(s)=s\bigl(G'\bigr)^{-1}(s)-G \bigl(\bigl(G'\bigr)^{-1}(s)\bigr), \quad \forall s\geq 0, $$

(3.15)

which satisfies the generalized Young inequality (see [2])

$$ AB\leq G^{*}(A)+G(B), \quad \forall A,B >0. $$

(3.16)

Using (2.5) and the definition of G, we get

$$ G^{*}(s)=sg_{2}^{-1}(s)-G \bigl(g_{2}^{-1}(s)\bigr), \quad \forall s\geq 0. $$

(3.17)

Therefore, on account of (2.5) and (3.17), we have

$$ \begin{aligned} G^{*}\bigl(g_{2} \bigl(z(x,1,t)\bigr)\bigr)&=z(x,1,t)g_{2}\bigl(z(x,1,t)\bigr)-G \bigl(z(x,1,t)\bigr) \\ &\leq (1-\alpha _{1})z(x,1,t)g_{2}\bigl(z(x,1,t)\bigr). \end{aligned} $$

(3.18)

A combination of (3.14), (3.16), and (3.18) leads to

$$\begin{aligned} \frac{dE(t)}{dt} \leq &- \bigl( \gamma _{1}a(t)-\mu (t)\alpha _{2} \bigr) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx \\ &{} +\gamma _{2}a(t) \int _{0}^{1}(G(u_{t})+G^{*} \bigl( g_{2}\bigl(z(x,1,t)\bigr)\bigr)\,dx \\ &{} -\mu (t) \bigl(1-\tau '(t)\bigr) \int _{0}^{1}G(z(x,1,t)\,dx -\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1}\theta _{x}^{2}\,dx \\ \leq& - \bigl( \gamma _{1}a(t)-\mu (t)\alpha _{2} \bigr) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx +\gamma _{2}a(t)\alpha _{2} \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx \\ &{} +\gamma _{2}a(t) (1-\alpha _{1}) \int _{0}^{1}z(x,1,t)g_{2}\bigl(z(x,1,t) \bigr)\,dx \\ &{} -\mu (t) \bigl(1-\tau '(t)\bigr) \int _{0}^{1}G(z(x,1,t)\,dx-\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1}\theta _{x}^{2}\,dx \\ \leq& - \bigl( \gamma _{1}a(t)-\mu (t)\alpha _{2}-\gamma _{2}a(t) \alpha _{2} \bigr) \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx \\ &{} - \bigl(\mu (t) \bigl(1-\tau '(t)\bigr)\alpha _{1}- \gamma _{2}a(t) (1- \alpha _{1}) \bigr) \int _{0}^{1}z(x,1,t)g_{2}\bigl(z(x,1,t) \bigr)\,dx \\ &{} -\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1} \theta _{x}^{2}\,dx. \end{aligned}$$

(3.19)

Recalling that $\mu (t)=\bar{\mu}a(t)$, it follows from (3.19) that

$$ \begin{aligned} \frac{dE(t)}{dt}\leq{}& -a(t) [ \gamma _{1}-\bar{\mu}\alpha _{2}- \gamma _{2}\alpha _{2} ] \int _{0}^{1}u_{t} g_{1}(u_{t})\,dx \\ & {}-a(t) \bigl[\bar{\mu}\bigl(1-\tau '(t)\bigr)\alpha _{1}-\gamma _{2}(1- \alpha _{1}) \bigr] \int _{0}^{1}z(x,1,t)g_{2}\bigl(z(x,1,t) \bigr)\,dx \\ & {}-\gamma _{3} \int _{0}^{1}\varphi _{t}^{2}\,dx- \beta \int _{0}^{1} \theta _{x}^{2}\,dx. \end{aligned} $$

(3.20)

Using (2.9) and (3.1), we obtain the desired result. This finishes the proof. □

Lemma 3.2

The functional $F_{1}$, defined by

$$ F_{1}(t):=-\rho _{2}\rho _{3} \int _{0}^{1}\psi _{t} \int _{0}^{x} \theta (y,t)\,dy\,dx, $$

satisfies, along the solution of system (2.14)–(2.16) and for any $\epsilon _{1}, \epsilon _{2}>0$, the estimate

$$ \begin{aligned} F'_{1}(t)\le{}& - \frac {m\rho _{2}}{2} \int _{0}^{1}\psi _{t}^{2}\,dx + \epsilon _{1} \int _{0}^{1}\psi _{x}^{2}\,dx+ \epsilon _{2} \int _{0}^{1}( \varphi _{x}+\psi )^{2}\,dx \\ &{}+c \biggl(1+\frac {1}{\epsilon _{1}} + \frac {1}{\epsilon _{2}} \biggr) \int _{0}^{1}\theta _{x}^{2}\,dx. \end{aligned} $$

(3.21)

Proof

Differentiating $F_{1}$, using (2.14)₃ and (2.14)₄, then integrating by parts and exploiting the boundary conditions lead to

$$ \begin{aligned} F_{1}'(t)={}& b \rho _{3} \int _{0}^{1}\psi _{x}\theta \,dx +k\rho _{3} \int _{0}^{1}(\varphi _{x}+\psi ) \int _{0}^{x}\theta (y,t)\,dy\,dx \\ & {}+ m\rho _{3} \int _{0}^{1}\theta ^{2}\,dx -\rho _{2}\beta \int _{0}^{1} \psi _{t}\theta _{x}\,dx-\rho _{2}m \int _{0}^{1}\psi _{t}^{2}\,dx. \end{aligned} $$

(3.22)

Making use of Cauchy–Schwarz, Young’s, and Poincaré’s inequalities, we get (3.21). □

Lemma 3.3

The functional $F_{2}$, defined by

$$ F_{2}(t):= \int _{0}^{1} \biggl(\rho uu_{t} +\rho _{1}\varphi \varphi _{t} + \rho _{2}\psi \psi _{t}+\frac{\gamma _{3}}{2}\varphi ^{2} \biggr)\,dx, $$

satisfies, along the solution of system (2.14)–(2.16), the estimate

$$ \begin{aligned} F'_{2}(t)\le{}& - \int _{0}^{1} \biggl(\frac{\alpha}{2} u_{x}^{2}+ \lambda (\varphi -u)^{2}+k(\varphi _{x} +\psi )^{2}+\frac {b}{2}\psi _{x}^{2} \biggr)\,dx \\ & {}+ \int _{0}^{1} \bigl(\rho u_{t}^{2}+ \rho _{1}\varphi ^{2}_{t} + \rho _{2} \psi _{t}^{2} \bigr)\,dx +c \int _{0}^{1}\theta _{x}^{2}\,dx \\ &{} +c \int _{0}^{1} \bigl\vert g_{1}(u_{t}) \bigr\vert ^{2}\,dx+ c \int _{0}^{1} \bigl\vert g_{2} \bigl(z(x,1,t)\bigr) \bigr\vert ^{2}\,dx, \quad \forall t\geq 0. \end{aligned} $$

(3.23)

Proof

Directly differentiating $F_{2}$, using (2.14)₁, (2.14)₂, and (2.14)₃, then applying integration by parts and boundary conditions, we obtain

$$ \begin{aligned} F'_{2}(t)=&{}- \int _{0}^{1} \bigl( \alpha u_{x}^{2}+ \lambda (\varphi -u)^{2}+k( \varphi _{x} +\psi )^{2}+b\psi _{x}^{2} \bigr)\,dx \\ &{}+ \int _{0}^{1} \bigl(\rho u_{t}^{2} + \rho _{1}\varphi ^{2}_{t} + \rho _{2} \psi _{t}^{2} \bigr)\,dx +m \int _{0}^{1} \psi \theta _{x}\,dx \\ &{}-\gamma _{1}a(t) \int _{0}^{1} ug_{1}(u_{t})\,dx- \gamma _{2}a(t) \int _{0}^{1}ug_{2}\bigl(z(x,1,t)\bigr)\,dx. \end{aligned} $$

(3.24)

Using $(A_{1})$, Young’s and Poincaré’s inequalities, we obtain (3.23). □

Lemma 3.4

The functional

$$ F_{3}(t):=\bar{\mu}\tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t) \sigma}G\bigl(z(x,\sigma ,t) \bigr)\,d\sigma \,dx, $$

satisfies, along the solution of system (2.14)–(2.16), the estimate

$$ \begin{aligned} F'_{3}(t) \le &-2F_{3}(t)+ \frac{\bar{\mu}\alpha _{2}}{2} \int _{0}^{1} \bigl( u_{t}^{2}+ \bigl\vert g_{1}(u_{t}) \bigr\vert ^{2} \bigr)\,dx, \quad \forall t \geq 0. \end{aligned} $$

(3.25)

Proof

Differentiating $F_{3}$, we get

$$ \begin{aligned} F'_{3}(t) ={}& \bar{\mu}\tau '(t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t) \sigma}G\bigl(z(x,\sigma ,t) \bigr)\,d\sigma \,dx \\ &{} -2\bar{\mu}\tau (t)\tau '(t) \int _{0}^{1} \int _{0}^{1}\sigma e^{-2 \tau (t)\sigma}G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\ & {}+\bar{\mu}\tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t) \sigma}z_{t}(x, \sigma ,t)g_{2}\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx. \end{aligned} $$

(3.26)

Using the last equation in (2.14), we can express the last term on the right hand-side of (3.26) as

$$\begin{aligned}& \tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t)\sigma}z_{t}(x, \sigma ,t)g_{2}\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\& \quad = \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t)\sigma} \bigl(\tau '(t) \sigma -1 \bigr) z_{\sigma}(x,\sigma ,t)g_{2} \bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\& \quad = \int _{0}^{1} \int _{0}^{1}\frac{\partial}{\partial \sigma} \bigl[ e^{-2 \tau (t)\sigma} \bigl(\tau '(t)\sigma -1 \bigr)G\bigl(z(x,\sigma ,t)\bigr) \bigr]\,d\sigma \,dx \\& \begin{aligned}&\qquad {}+2\tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t)\sigma} \bigl(\tau '(t)\sigma -1 \bigr)G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\ &\qquad {}-\tau '(t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t)\sigma}G\bigl(z(x, \sigma ,t) \bigr)\,d\sigma \,dx \end{aligned}\\& \quad =-\bigl(1-\tau '(t)\bigr)e^{-2\tau (t)} \int _{0}^{1}G\bigl(z(x,1,t)\bigr)\,dx + \int _{0}^{l}G(u_{t})\,dx \\& \qquad {}+2\tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t)\sigma} \bigl(\tau '(t)\sigma -1 \bigr)G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx \\& \qquad {}-\tau '(t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t)\sigma}G\bigl(z(x, \sigma ,t) \bigr)\,d\sigma \,dx. \end{aligned}$$

(3.27)

Substituting (3.27) into (3.26), we arrive at

$$ \begin{aligned} F'_{3}(t) ={}&-2 \bar{\mu}\tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2\tau (t) \sigma}G\bigl(z(x,\sigma ,t) \bigr)\,d\sigma \,dx +\bar{\mu} \int _{0}^{1}G(u_{t})\,dx \\ &{} -\bigl(1-\tau '(t)\bigr)e^{-2\tau (t)} \int _{0}^{1}G\bigl(z(x,1,t)\bigr)\,dx. \end{aligned} $$

(3.28)

Using condition (2.5) and Young’s inequality, we obtain (3.26). □

Lemma 3.5

Let $(u,\varphi , \psi ,\theta , z)$ be the solution of system (2.14)–(2.16). Then, for $N, N_{1}, N_{2}>0$ sufficiently large, the Lyapunov functional L, defined by

$$\begin{aligned} L(t):=NE(t)+N_{1}F_{1}(t)+ N_{2}F_{2}(t)+F_{3}(t), \end{aligned}$$

(3.29)

satisfies, for some positive constants $c_{1}$, $c_{2}$, η,

$$ c_{1}E(t) \le L(t) \le c_{2}E(t), \quad \forall t \ge 0, $$

(3.30)

and

$$ \begin{aligned} L'(t)\le -\eta E(t)+c \int _{0}^{1}\bigl(u^{2}_{t}+ \bigl\vert g_{1}(u_{t}) \bigr\vert ^{2} \bigr)\,dx+ c \int _{0}^{1} \bigl\vert g_{2} \bigl(z(x,1,t)\bigr) \bigr\vert ^{2}\,dx, \quad \forall t \ge 0. \end{aligned} $$

(3.31)

Proof

Applying Cauchy–Schwarz, Young’s, and Poincaré’s inequalities, we have

$$\begin{aligned} \bigl\vert L(t)-NE(t) \bigr\vert \leq& N_{1} \biggl\vert -\rho _{2} \int _{0}^{1}\psi _{t} \int _{0}^{x} \theta (y,t)\,dy\,dx \biggr\vert \\ &{} + N_{2} \biggl\vert \int _{0}^{1} \biggl(\rho uu_{t} +\rho _{1} \varphi \varphi _{t} + \rho _{2}\psi \psi _{t}+\frac{\gamma _{3}}{2} \varphi ^{2} \biggr)\,dx \biggr\vert \\ &{} + \biggl\vert \bar{\mu}\tau (t) \int _{0}^{1} \int _{0}^{1} e^{-2 \tau (t)\sigma}G\bigl(z(x,\sigma ,t) \bigr)\,d\sigma \,dx \biggr\vert \\ \leq& \frac{N_{2}\rho}{2} \int _{0}^{1}u_{t}^{2}\,dx + \frac{N_{2}\rho _{1}}{2} \int _{0}^{1}\varphi _{t}^{2}\,dx+ \frac{(N_{1}+N_{2})\rho _{2}}{2} \int _{0}^{1}\psi _{t}^{2}\,dx \\ &{} +\frac{N_{2}\gamma _{3}}{2} \int _{0}^{1}\varphi ^{2}\,dx+ \frac{N_{2}\rho}{2} \int _{0}^{1}u_{x}^{2}\,dx + \frac{N_{2}\rho _{1}}{2} \int _{0}^{1}\varphi _{x}^{2}\,dx \\ &{} + \frac{N_{2}\rho _{2}}{2} \int _{0}^{1}\psi _{x}^{2}\,dx + \frac{ N_{1}\rho _{2}}{2} \int _{0}^{1} \biggl( \int _{0}^{x}\theta (y,t)\,dy \biggr)^{2}\,dx \\ &{} +\bar{\mu}\tau (t) \int _{0}^{1} \int _{0}^{1} G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx. \end{aligned}$$

(3.32)

Using the relations

$$ \begin{aligned} & \int _{0}^{1}\varphi ^{2}\,dx\leq 2 \int _{0}^{l}(\varphi -u)^{2}\,dx + 2 \int _{0}^{1} u_{x}^{2}\,dx, \\ & \int _{0}^{1}\varphi _{x}^{2}\,dx \leq 2 \int _{0}^{1}(\varphi _{x}+ \psi )^{2}\,dx +2 \int _{0}^{1}\psi _{x}^{2}\,dx, \end{aligned} $$

we arrive at

$$ \begin{aligned} \bigl\vert L(t)-NE(t) \bigr\vert \leq{} &\frac{N_{2}\rho}{2} \int _{0}^{1}u_{t}^{2}\,dx + \frac{N_{2}\rho _{1}}{2} \int _{0}^{1}\varphi _{t}^{2}\,dx+ \frac{(N_{1}+N_{2})\rho _{2}}{2} \int _{0}^{1}\psi _{t}^{2}\,dx \\ &{} +N_{2}\gamma _{3} \int _{0}^{l}(\varphi -u)^{2}\,dx+ \biggl( N_{2} \gamma _{3}+\frac{N_{2}\rho}{2} \biggr) \int _{0}^{1}u_{x}^{2}\,dx \\ & {}+N_{2}\rho _{1} \int _{0}^{1}(\varphi _{x}+\psi )^{2}\,dx + \biggl( \frac{N_{2}(\rho _{1}+\rho _{2})}{2} \biggr) \int _{0}^{1}\psi _{x}^{2}\,dx \\ &{} +\frac{ N_{1}\rho _{2}}{2} \int _{0}^{1}\theta ^{2}\,dx+\bar{\mu} \tau (t) \int _{0}^{1} \int _{0}^{1} G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx. \end{aligned} $$

(3.33)

From (3.33), we obtain

$$ \bigl\vert L(t)-NE(t) \bigr\vert \leq \bar{c}E(t). $$

(3.34)

By choosing N large enough such that

$$ c_{1}=N- \bar{c}>0,\qquad c_{2}=N+ \bar{c}>0, $$

(3.35)

estimate (4.14) follows. Next, we establish (3.31). Using Lemmas 3.1–3.4, we get

$$ \begin{aligned} L'(t)\le{}& -\rho \int _{0}^{1}u_{t}^{2}\,dx- [N \gamma _{3}-N_{2} \rho _{1} ] \int _{0}^{1}\varphi _{t}^{2}\,dx- \biggl[N_{1} \frac{m\rho _{2}}{2}-N_{2}\rho _{2} \biggr] \int _{0}^{1}\psi _{t}^{2}\,dx \\ & {}-\frac{N_{2}\alpha}{2} \int _{0}^{1}u_{x}^{2}\,dx-N_{2} \lambda \int _{0}^{1}(\varphi -u)^{2}\,dx- [N_{2}k-N_{1}\epsilon _{2} ] \int _{0}^{1}(\varphi _{x}+\psi )^{2}\,dx \\ &{} - \biggl[N_{2}\frac{b}{2}-N_{1}\epsilon _{1} \biggr] \int _{0}^{1} \psi _{x}^{2}\,dx- \biggl[N\beta -N_{1}c \biggl(1+\frac{1}{\epsilon _{1}} + \frac{1}{\epsilon _{2}} \biggr)-N_{2}c \biggr] \int _{0}^{1}\theta _{x}^{2}\,dx \\ &{} -\frac{2e^{-2\tau _{1}}}{a(0)}\mu (t)\tau (t) \int _{0}^{1} \int _{0}^{1}G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx + \biggl[\rho +N_{2}\rho + \frac{\bar{\mu }\alpha _{2}}{2} \biggr] \int _{0}^{1}u_{t}^{2}\,dx \\ &{} + \biggl[cN_{2}+ \frac{\bar{\mu}\alpha _{2}}{2} \biggr] \int _{0}^{1} \bigl\vert g_{1}(u_{t}) \bigr\vert ^{2}\,dx +cN_{2} \int _{0}^{1} \bigl\vert g_{2} \bigl(z(x,1,t)\bigr) \bigr\vert ^{2}\,dx. \end{aligned} $$

Choosing

$$ N_{2}=1, \qquad \epsilon _{1}=\frac {N_{2}b}{4N_{1}}, \qquad \epsilon _{2}= \frac {N_{2}k}{2N_{1}}, $$

we arrive at

$$ \begin{aligned} L'(t)\le{}& -\rho \int _{0}^{1}u_{t}^{2}\,dx- [N \gamma _{3}-\rho _{1} ] \int _{0}^{1}\varphi _{t}^{2}\,dx- \biggl[N_{1}\frac{m\rho _{2}}{2}- \rho _{2} \biggr] \int _{0}^{1}\psi _{t}^{2}\,dx \\ & {}-\frac{\alpha}{2} \int _{0}^{1}u_{x}^{2}\,dx-\lambda \int _{0}^{1}( \varphi -u)^{2}\,dx- \frac {k}{2} \int _{0}^{1}(\varphi _{x}+\psi )^{2}\,dx \\ & {}-\frac{b}{4} \int _{0}^{1}\psi _{x}^{2}\,dx- \biggl[N\beta -N_{1}c \biggl(1+\frac{4N_{1}}{b} +\frac{2N_{1}}{k} \biggr)-c \biggr] \int _{0}^{1} \theta _{x}^{2}\,dx \\ & {}-\frac{2e^{-2\tau _{1}}}{a(0)}\mu (t)\tau (t) \int _{0}^{1} \int _{0}^{1}G\bigl(z(x,\sigma ,t)\bigr)\,d\sigma \,dx + \biggl[2\rho + \frac{\bar{\mu }\alpha _{2}}{2} \biggr] \int _{0}^{1}u_{t}^{2}\,dx \\ &{} + \biggl[c+ \frac{\bar{\mu }\alpha _{2}}{2} \biggr] \int _{0}^{1} \bigl\vert g_{1}(u_{t}) \bigr\vert ^{2}\,dx +c \int _{0}^{1} \bigl\vert g_{2} \bigl(z(x,1,t)\bigr) \bigr\vert ^{2}\,dx. \end{aligned} $$

(3.36)

Now, we choose $N_{1}$ large such that

$$ N_{1}\frac{m\rho _{2}}{2}-\rho _{2}>0. $$

Next, we select N very large so that (4.14) remains true and

$$ N\gamma _{3}-\rho _{1}>0,\qquad N\beta -N_{1}c \biggl(1+ \frac{4N_{1}}{b} +\frac{2N_{1}}{k} \biggr)-c>0. $$

Therefore, using the energy functional defined by (3.2), we obtain (3.31). □

Stability result

In this section, we are concerned with the main stability result, and is stated as follows.

Theorem 4.1

Let $(u,\varphi , \psi , \theta , z)$ be the solution of system (2.14)–(2.16) and assume (A₁)–(A₄) hold. Then, for some positive constants $\delta _{1}$, $\delta _{2}$, $\delta _{3}$, and $r_{0}$, the energy functional (3.2) satisfies

$$ E(t)\le \delta _{1}\chi _{1}^{-1} \biggl(\delta _{2} \int _{0}^{t} a(s)\,ds +\delta _{3} \biggr),\quad t \ge 0, $$

(4.1)

where

$$ \chi _{1}(t)= \int _{t}^{1}\frac {1}{\chi _{0}(s)}\,ds\quad \textit{and}\quad \chi _{0}(t)=t \chi '(r_{0} t). $$

Proof

We divide the proof into two cases:

Case I: χ is linear. Using (A₂), we get

$$ C_{1} \vert s \vert \le \bigl\vert g_{1}(s) \bigr\vert \le C_{2} \vert s \vert , \quad \forall s\in \mathbb{R}. $$

Thus,

$$\begin{aligned} g_{1}^{2}(s)\le C_{2}sg_{1}(s), \quad \forall s\in \mathbb{R}. \end{aligned}$$

(4.2)

Therefore, multiplying (3.31) by $a(t)$ and using (3.3) and (4.2), we conclude that

$$\begin{aligned} a(t)L'(t)\le &-\eta a(t)E(t)+ca(t) \int _{0}^{1}u_{t}g_{1}(u_{t})\,dx+ca(t) \int _{0}^{1}z(x,1,t)g_{2}\bigl(z(x,1,t) \bigr)\,dx \\ \le & -\eta a(t)E(t)-cE'(t), \quad \forall t\in \mathbb{R}^{+}. \end{aligned}$$

Exploiting (A₂) and (3.30), it follows that

$$ L_{0}(t):= a(t)L(t)+cE(t)\sim E(t) $$

(4.3)

and, for some constant $\eta _{1}>0$, the functional $L_{0}$ satisfies

$$ L'_{0}(t)\le -\eta _{1}a(t)L_{0}(t), \quad \forall t\ge 0. $$

(4.4)

A simple integration of (4.4) over $(0,t)$, using (4.3), yields

$$ E(t)\le \delta _{1} \exp \biggl( -\delta _{2} \int _{0}^{t}a(s)\,ds \biggr)=\delta _{1} \chi _{1}^{-1} \biggl(\delta _{2} \int _{0}^{t} a(s)\,ds \biggr), \quad \forall t\geq 0. $$

(4.5)

Case II: χ is nonlinear on $[0,r]$. Here, as in [22], we select $0< r_{1}\le r$ so that

$$ sg_{1}(s)\le \min \bigl\{ r, \chi (r) \bigr\} , \quad \forall \vert s \vert \le r_{1}. $$

(4.6)

On account of (A₂) and the continuity of $g_{1}$ with the fact that $|g_{1}(s)|>0$, for $s\ne 0$, we conclude that

$$ \textstyle\begin{cases} s^{2}+g_{1}^{2}(s) \le \chi ^{-1}(sg_{1}(s)), \quad \forall \vert s \vert \le r_{1}, \\ C'_{1} \vert s \vert \le \vert g_{1}(s) \vert \le C'_{2} \vert s \vert , \hfill \quad \forall \vert s \vert \ge r_{1}. \end{cases} $$

(4.7)

Now, we introduce the following partitions:

$$\begin{aligned}& I_{1}= \bigl\{ x\in (0, 1): \vert u_{t} \vert \le r_{1} \bigr\} , \qquad I_{2}= \bigl\{ x\in (0, 1): \vert u_{t} \vert >r_{1} \bigr\} ,\\& \tilde{I_{1}}= \bigl\{ x\in (0, 1): \bigl\vert z(x,1,t) \bigr\vert \le r_{1} \bigr\} ,\qquad \tilde{I_{2}}= \bigl\{ x\in (0, 1): \bigl\vert z(x,1,t) \bigr\vert >r_{1} \bigr\} \end{aligned}$$

and the functional h, defined by

$$ h(t)= \int _{I_{1}}u_{t}g_{1}(u_{t})\,dx. $$

Using the fact that $\chi ^{-1}$ is concave and Jensen’s inequality, it follows that

$$ \chi ^{-1}\bigl(h(t)\bigr)\ge c \int _{I_{1}}\chi ^{-1} \bigl(u_{t}g_{1}(u_{t}) \bigr)\,dx. $$

(4.8)

Combining (4.7) and (4.8), we have

$$\begin{aligned} a(t) \int _{0}^{1} \bigl(u_{t}^{2}+g_{1}^{2}(u_{t}) \bigr)\,dx&=a(t) \int _{I_{1}} \bigl(u_{t}^{2}+g_{1}^{2}(u_{t}) \bigr)\,dx+a(t) \int _{I_{2}} \bigl(u_{t}^{2}+g_{1}^{2}(u_{t}) \bigr)\,dx \\ &\le a(t) \int _{I_{1}}\chi ^{-1} \bigl(u_{t}g_{1}(u_{t}) \bigr)\,dx+ca(t) \int _{I_{2}}u_{t}g_{1}(u_{t})\,dx \\ &\le ca(t)\chi ^{-1}\bigl(h(t)\bigr)-cE'(t) . \end{aligned}$$

(4.9)

and

$$ \begin{aligned} a(t) \int _{0}^{1} g_{2}^{2} \bigl(z(x,1,t)\bigr)\,dx={}& a(t) \int _{\bar{I_{1}}}g_{2}^{2}\bigl(z(x,1,t)\bigr)\,dx + a(t) \int _{\bar{I_{2}}}g_{2}^{2}\bigl(z(x,1,t)\bigr)\,dx \\ \leq{}& ca(t) \int _{\bar{I_{1}}}z(x,1,t)g_{2}\bigl(z(x,1,t)\bigr)\,dx \\ &{} +a(t) \int _{\bar{I_{2}}}z(x,1,t)g_{2}\bigl(z(x,1,t)\bigr)\,dx \\ \leq{}& -cE'(t). \end{aligned} $$

(4.10)

Multiplying (3.31) by $a(t)$ and using (4.9) and (4.10), we obtain

$$ a(t)L'(t)+cE'(t)\le -\eta a(t)E(t)+ca(t)\chi ^{-1}\bigl(h(t)\bigr). $$

(4.11)

It follows from (A₁) that

$$ L_{1}'(t)\le -\eta a(t)E(t)+ca(t)\chi ^{-1}\bigl(h(t)\bigr), $$

(4.12)

where

$$ L_{1}(t)=a(t)L(t)+cE(t)\sim E (t)\quad \text{by virtue of (3.30)}. $$

(4.13)

Let $r_{0}< r$ and $\eta _{0}>0$ to be specified later. Then, combining (4.12) and the fact that

$$ E'\le 0,\qquad \chi '>0,\qquad \chi ''>0 \quad \text{on } (0, r], $$

the functional $L_{2}$, defined by

$$ L_{2}(t):=\chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr)L_{1}(t)+\eta _{0}E(t), $$

satisfies

$$ \kappa _{1}L_{2}(t) \le E(t) \le \kappa _{2}L_{2}(t) $$

(4.14)

for some positive constants $\kappa _{1}$, $\kappa _{2}$, and

$$\begin{aligned} L_{2}'(t)&=r_{0} \frac {E'(t)}{E(0)}\chi '' \biggl(r_{0} \frac {E(t)}{E(0)} \biggr)L_{1}(t)+\chi ' \biggl(r_{0} \frac {E(t)}{E(0)} \biggr)L_{1}'(t)+ \eta _{0}E'(t) \\ &\le -\eta a(t) E(t)\chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr)+ \underbrace{ca(t)\chi ' \biggl(r_{0} \frac {E(t)}{E(0)} \biggr)\chi ^{-1} \bigl(h(t) \bigr)}_{A}+ \eta _{0}E'(t). \end{aligned}$$

(4.15)

To estimate the term A in (4.15), we consider the convex conjugate of χ denoted by $\chi ^{*}$, defined by

$$ \chi ^{*}(y)=y\bigl(\chi ' \bigr)^{-1}(y)-\chi \bigl[\bigl(\chi '\bigr)^{-1}(y) \bigr]\le y\bigl(\chi ^{ \prime}\bigr)^{-1}(y),\quad \text{if }y\in (0, \chi '(r)], $$

(4.16)

and which satisfies the generalized Young’s inequality

$$ XY\le \chi ^{*}(X)+\chi (Y), \quad \text{if }X\in \big(0, \chi '(r)\big], Y \in (0, r]. $$

(4.17)

Taking $X=\chi ' (r_{0}\frac {E(t)}{E(0)} )$ and $Y=\chi ^{-1} (h(t) )$ and recalling Lemma 3.1 and (4.6), then (4.15)–(4.17) lead to

$$ \begin{aligned} L_{2}'(t)\le {}& -\eta a(t) E(t) \chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr)+ca(t) \biggl[ \chi * \biggl( \chi ' \biggl(r_{0} \frac {E(t)}{E(0)} \biggr) \biggr) +\chi \bigl( \chi ^{-1} \bigl(h(t) \bigr) \bigr) \biggr] \\ &{}+\eta _{0}E'(t) \\ ={}&-\eta a(t) E(t)\chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr)+ca(t) \chi * \biggl( \chi ' \biggl(r_{0} \frac {E(t)}{E(0)} \biggr) \biggr) \\ &{}+ca(t)h(t)+ \eta _{0}E'(t) \\ \le {}&-\eta a(t) E(t)\chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr)+ cr_{0}a(t) \biggl(\frac{E(t)}{E(0)} \biggr)\chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr) \\ &{}-cE'(t)+ \eta _{0}E'(t) \\ \le {}&-\bigl(\eta E(0)-cr_{0}\bigr)a(t) \biggl(\frac{E(t)}{E(0)} \biggr)\chi ' \biggl(r_{0}\frac {E(t)}{E(0)} \biggr)+(\eta _{0}-c)E'(t). \end{aligned} $$

(4.18)

By choosing $r_{0}=\frac {\eta E(0)}{2c}$, $\eta _{0}=2c$, and recalling that $E'(t)\le 0$, we arrive at

$$\begin{aligned} L_{2}'(t)\le {}&-\eta _{1}a(t) \frac {E(t)}{E(0)}\chi ' \biggl(r_{0} \frac {E(t)}{E(0)} \biggr)=-\eta _{1}a(t)\chi _{0} \biggl( \frac {E(t)}{E(0)} \biggr), \end{aligned}$$

(4.19)

where $\eta _{1}>0$ and $\chi _{0}(t)=t\chi '(r_{0}t)$. Now, since χ is strictly convex on $(0,r]$, we conclude that $\chi _{0}(t)>0$, $\chi '_{0}(t) >0$ on $(0, 1]$. Using (4.14) and (4.19), it follows that the functional

$$ L_{3}(t)=\frac{\kappa _{1}L_{2}(t)}{E(0)} $$

satisfies

$$ L_{3}(t)\sim E(t) $$

(4.20)

and, for some $\delta _{2}>0$,

$$ L_{3}'(t)\le -\delta _{2}a(t) \chi _{0} \bigl(L_{3}(t) \bigr), $$

(4.21)

which yields

$$ \bigl[\chi _{1} \bigl(L_{3}(t) \bigr) \bigr]'\ge \delta _{2}a(t), $$

(4.22)

where

$$ \chi _{1}(t)= \int _{t}^{1}\frac {1}{\chi _{0}(s)}\,ds, \quad t\in (0,1]. $$

Integrating (4.22) over $[0, t]$, keeping in mind the properties of $\chi _{0}$, and the fact that $\chi _{1}$ is strictly decreasing on $(0, 1]$, we obtain

$$ L_{3}(t)\le \chi _{1}^{-1} \biggl( \delta _{2} \int _{0}^{t}a(s)\,ds+ \delta _{3} \biggr), \quad \forall t\in \mathbb{R}^{+}, $$

(4.23)

for some $\delta _{3}>0$. Using (4.20) and (4.23), the proof of Theorem 4.1 is completed. □

Examples

We end this section by giving some examples to illustrate the obtained result.

Let

$$ g_{0}\in C^{2} \bigl( [0,+\infty) \bigr) $$

be a strictly increasing function such that $g_{0}(0)=0$ and, for some positive constants $c_{1}$, $c_{2}$ and r, the function $g_{1}$ satisfies

$$ \begin{aligned} &g_{0}\bigl( \vert s \vert \bigr)\leq \bigl\vert g_{1}(s) \bigr\vert \leq g_{0}^{-1}\bigl( \vert s \vert \bigr), \quad \forall \vert s \vert \leq r, \\ &c_{1}s^{2}\leq sg_{1}(s)\leq c_{2}s^{2}, \quad \forall \vert s \vert \geq r. \end{aligned} $$

(5.1)

We consider the function

$$ \chi (s)= \biggl( \sqrt{\frac{s}{2}} \biggr)g_{0} \biggl(\sqrt{ \frac{s}{2}} \biggr). $$

(5.2)

It follows that χ is a $C^{2}$-strictly convex function on $(0,r]$ when ${\mathrm{g_{0}}}$ is nonlinear and therefore satisfies condition $(A_{2})$. Now, we give some examples of $g_{0}$ such that $g_{1}$ satisfies (5.1) near 0.

• 1. Let $g_{0}(s)= \lambda s$, where $\lambda >0$ a constant, then $\chi (s)= \bar{\lambda} s$, where $\bar{\lambda}=\frac {\lambda}{2}$ satisfies $(A_{2})$ near 0 and from (4.1), we get

  $$ E(t)\leq \bar{\delta}\exp \biggl( -\delta _{2} \int _{0}^{t}a(s)\,ds \biggr), \quad \forall t\geq 0. $$

• 2. Let $g_{0}(s) = \frac {1}{s}e^{-\frac{1}{s^{2}}}$, then $\chi (s)= e^{-\frac{2}{s}}$ satisfies $(A_{2})$ in the neighborhood of 0 and from (4.1), we obtain

  $$ E(t)\leq \delta _{1} \biggl( \ln \biggl(\delta _{2} \int _{0}^{t} a(s)\,ds+ \delta _{3} \biggr) \biggr)^{-1}, \quad \forall t\geq 0. $$

  (5.3)

• 3. Let $g_{0}(s) = e^{-\frac{1}{s}}$, then $\chi (s)= \sqrt{\frac{s}{2}} e^{-\sqrt{\frac{2}{s}}}$ satisfies $(A_{2})$ near 0 and using (4.1), we obtain

  $$ E(t)\leq \delta _{1} \biggl( \ln \biggl(\delta _{2} \int _{0}^{t} a(s)\,ds + \delta _{3} \biggr) \biggr)^{-2}, \quad \forall t\geq 0. $$

  (5.4)

Conclusion

In this work, we obtained some general decay results for a thermoelastic Timoshenko beam system with suspenders, general weak internal damping, time-varying coefficient, and time-varying delay terms. The damping structure in system (2.14)–(2.16) is sufficient enough to stabilize the system without any additional conditions on the coefficient parameters as it is the case with many Timoshenko beam systems in the literature. The result of the present paper generalizes the one established in Bochichio et al. [6] and allows a large class of functions that satisfy condition $(A_{2})$. We also gave some examples to illustrate our theoretical finding.