1 Introduction

Generally speaking, the vibrations of flexible structures in practice are described by nonlinear models of strings, beams, plates, and so on. The linearized vibrations of flexible structures are usually governed by partial differential equations, in particular, by the second-order wave equation and the fourth-order Euler-Bernoulli beam equation [1]. Up till now, there are many results concerning the stability of wave and plate equations by adding some types of damping, for example, internal damping, boundary damping, thermal damping, and so on, most of which can be found in the literature. For general decay results on the wave equation, here we mention the works by Cao and Yao [2], Guesmia and Messaoudi [3], Said-Houari, Messaoudi, and Guesmia [4], Messaoudi [5–7], Messaoudi and Al-Gharabli [8, 9], Messaoudi and Soufyane [10], Mustafa and Messaoudi [11], Tatar [12], and Wu [13]. When the body vibrates, the balance of linear momentum reads

$$mu_{tt}-\sigma_{x}=f, $$

where σ is the stress. If the body is nonuniform, that is,

$$\sigma=\sigma(u_{x},u_{xt})=p(x)u_{x}+2 \delta(x)u_{xt}, $$

then we can derive the equation of longitudinal vibrations of a flexible structure (see, e.g., Liu and Liu [14])

$$m(x)u_{tt}-\bigl(p(x)u_{x}+2\delta(x)u_{xt} \bigr)_{x}=f, $$

where \(u(x,t)\) represents the longitudinal displacement of a particle. The functions \(m(x)\), \(p(x)\), and \(\delta(x)\) denote the mass per unit length of the structure, the coefficient of internal material damping, and a positive function related to the stress acting on the body, respectively. For this equation, Gorain [15] established the exponential stability of the solution.

For the thermal effect in flexible structures, that is,

$$\theta_{t}+q_{x}-\kappa u_{tx}=0, $$

where \(q(x,t)\) denotes the heat flux vector, and \(\theta(x,t)\) is the temperature difference, we we can find the physical background in Carlson [16]. If we assume that the heat flux satisfies a different thermal law, we can obtain flexible structures with different thermal effects. For example, if the heat flux satisfies

$$q(t)+(1-\alpha)\theta_{x}(t)+\alpha \int^{\infty}_{0}g(s)\theta _{x}(t-s)\,ds=0, \quad\alpha\in(0,1), $$

then we get the flexible structures with the Coleman-Gurtin law. The limit cases \(\alpha=0\) and \(\alpha=1\) correspond to the Fourier and Gurtin-Pipkin cases. Misra et al. [17] studied the thermoelastic flexible structure with Fourier law:

$$\textstyle\begin{cases} m(x)u_{tt}-(p(x)u_{x}+2\delta(x)u_{xt})_{x}+\kappa\theta_{x}=f, \\ \theta_{t}-\theta_{xx}+\kappa u_{xt}=0, \end{cases} $$

and proved the global well-posedness of the system. In addition, they established the exponential decay of energy. It is a remarkable fact that the assumption of Fourier’s law causes an unrealistic property that a sudden disturbance at some point will be felt instantly everywhere else in the material. Green and Naghdi developed a damped model, called thermoelasticity of type III (see, e.g., [18–23]),

$$\textstyle\begin{cases} u_{tt}-\alpha u_{xx}++\beta\theta_{x}=f, \\ \theta_{tt}-\delta\theta_{xx}+\gamma u_{xtt}-\kappa\theta_{xtt}=0. \end{cases} $$

For decay results of this system, we refer to Quintanilla and Racke [24] and Zhang and Zuazua [25] (see also [26]). For flexible structures with thermal effect, we also mention the work of Alves et al. [27], where the authors studied the thermoelastic flexible structure with second sound and proved the well-posedness and stability of the system.

Taking into account all considerations mentioned, in this paper, we consider the following longitudinal vibrations of a thermoelastic beam equation with past history:

$$\begin{aligned}& m(x)u_{tt}-\bigl(p(x)u_{x}+2 \delta(x)u_{xt}\bigr)_{x}+\kappa\theta_{xt}=0, \end{aligned}$$
(1.1)
$$\begin{aligned}& \theta_{tt}-\alpha\theta_{xx}- \int^{\infty}_{0}g(s)\theta _{xx}(t-s)\,ds+ \kappa u_{xt}=0, \end{aligned}$$
(1.2)

where \(x\in[0,L]\), and the constant κ is the coupling coefficient. The boundary conditions are given by

$$\begin{aligned} u(0,t)=u(L,t)=0,\qquad \theta(0,t)=\theta(L,t)=0,\quad t\geq0, \end{aligned}$$
(1.3)

and the initial conditions are

$$\begin{aligned} u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \qquad \theta(x,t)|_{t\leq0}=\theta_{0}(x,t),\qquad \theta_{t}(x,0)=\theta_{1}(x). \end{aligned}$$
(1.4)

Our goal in this paper is to establish a general decay of solutions to problem (1.1)-(1.4) with exponential and polynomial decays as only particular cases. We use the multiplier method and some properties of convex functions to establish a general decay result.

To deal with the memory, motivated by Dafermos [28] and Giorgi et al. [29, 30], we define the new variable \(\eta=\eta^{t}(x,s)\) by

$$\begin{aligned} \eta^{t}(x,s)=\theta(x,t)-\theta(x,t-s),\quad (t,s)\in [0,\infty)\times\mathbb{R}^{+}. \end{aligned}$$
(1.5)

Then

$$\eta_{t}+\eta_{s}=\theta_{t},\quad (x,t,s)\in \mathbb{R}^{n}\times \mathbb{R}^{+}\times\mathbb{R}^{+}. $$

It follows from (1.5) that

$$\int^{\infty}_{0}g(s)\theta_{xx}(t-s)\,ds= \int^{\infty}_{0}g(s)\,ds\theta_{xx}- \int ^{\infty}_{0}g(s)\eta^{t}_{xx}(s) \,ds. $$

Assuming for simplicity that \(\alpha-\int^{\infty}_{0}g(s)=1\), problem (1.1)-(1.4) is transformed into the new problem

$$\begin{aligned}& m(x)u_{tt}-\bigl(p(x)u_{x}+2 \delta(x)u_{xt}\bigr)_{x}+\kappa\theta _{xt}=0, \end{aligned}$$
(1.6)
$$\begin{aligned}& \theta_{tt}-\theta_{xx}+ \int^{\infty}_{0}g(s)\eta_{xx}(s)\,ds+\kappa u_{xt}=0, \end{aligned}$$
(1.7)
$$\begin{aligned}& \eta^{t}_{t}+\eta^{t}_{s}= \theta_{t}, \end{aligned}$$
(1.8)
$$\begin{aligned}& u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \qquad\eta ^{t}(x,0)=0,\quad x\in[0,L], \end{aligned}$$
(1.9)
$$\begin{aligned}& \theta(x,t)|_{t\leq0}=\theta_{0}(x,t),\qquad \theta_{t}(x,0)=\theta_{1}(x),\quad x\in[0,L], \end{aligned}$$
(1.10)
$$\begin{aligned}& \eta^{0}(x,s)=\eta_{0}(x,s),\quad (x,s)\in [0,L]\times\mathbb{R}^{+}, \end{aligned}$$
(1.11)
$$\begin{aligned}& u(0,t)=u(L,t)=0,\qquad\theta(0,t)=\theta(L,t)=0,\quad t\geq0. \end{aligned}$$
(1.12)

The plan of the paper is as follows. In Section 2, we give some assumptions and our main results. The proof of the general decay result is given in Section 3.

2 Assumptions and main results

By \(L^{q}(0,L)\) (\(1\le q \le\infty\)) and \(H^{1}(0,L)\) we denote the standard Lebesgue integral and Sobolev spaces. The norm in a space B is denoted by \(\Vert \cdot \Vert _{B}\). For simplicity, we use \(\Vert u \Vert \) instead of \(\Vert u \Vert _{2}\) when \(q=2\).

Now we give some assumptions used in this paper.

(A1):

The functions \(m(x),p(x),\delta(x):[0,L]\rightarrow \mathbb{R}^{+}\) are functions of class \(C^{1}(0,L)\), and there exist positive constants \(m_{1}\), \(m_{2}\), \(p_{1}\), \(p_{2}\), \(\delta_{1}\), and \(\delta_{2}\) such that

$$\begin{aligned} m_{1}\leq m(x)\leq m_{2},\qquad p_{1} \leq p(x)\leq p_{2},\quad\mbox{and}\quad \delta_{1}\leq \delta(x)\leq\delta_{2}. \end{aligned}$$
(2.1)
(A2):

The function \(g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) is a \(C^{1}\) a nonincreasing function satisfying

$$\begin{aligned} g(0)>0,\qquad 1- \int^{\infty}_{0}g(s)\,ds=l>0, \end{aligned}$$
(2.2)

and there exists a positive constant \(a_{0}< m_{2}\) such that, for any \(u\in H^{1}_{0}(0,L)\),

$$\begin{aligned} a_{0} \int^{L}_{0}u_{x}^{2}\,dx\leq \int ^{L}_{0}m(x)u_{x}^{2}\,dx- \int^{L}_{0} \biggl( \int^{\infty}_{0}g(s)\,ds \biggr)u_{x}^{2} \,dx. \end{aligned}$$
(2.3)

In addition, there exists an increasing strictly convex function \(G:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) of class \(C^{1}(\mathbb{R}^{+})\cap C^{2}(\mathbb{R}^{+})\) satisfying

$$\begin{aligned} G(0)=G'(0)=0,\qquad \lim_{t\rightarrow\infty}G'(t)= \infty, \end{aligned}$$
(2.4)

and

$$\begin{aligned} \int^{\infty}_{0}\frac{g(s)}{G^{-1}(-g'(s))}\,ds+\sup _{s\in \mathbb{R}^{+}}\frac{g(s)}{G^{-1}(-g'(s))}< \infty. \end{aligned}$$
(2.5)

To consider the new variable η, we define the weighted \(L^{2}\)-spaces

$$\mathcal {M}=L^{2}_{g}\bigl(\mathbb{R}^{+},H^{1}_{0}(0,L) \bigr)= \biggl\{ \eta:\mathbb{R}^{+}\rightarrow H^{1}_{0}: \int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds< \infty \biggr\} , $$

which is a Hilbert space endowed with inner product and norm

$$(\eta,\zeta)_{\mathcal {M}}= \int^{\infty}_{0} g(s) \bigl(\eta_{x}(s), \zeta_{x}(s)\bigr)\,ds\quad \mbox{and}\quad \Vert \eta \Vert _{\mathcal {M}}^{2}= \int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds. $$

Now we define the phase space

$$\mathcal{H}=H^{1}_{0}(0,L)\times L^{2}(0,L)\times H^{1}_{0}(0,L)\times L^{2}(0,L)\times\mathcal{M} $$

equipped with the norm

$$\bigl\Vert (u,v,\theta,\vartheta,\eta) \bigr\Vert ^{2}_{\mathcal {H}}= \Vert u_{x} \Vert ^{2}+ \Vert v \Vert ^{2}+ \Vert \theta_{x} \Vert + \Vert \vartheta \Vert + \Vert \eta \Vert ^{2}_{\mathcal {M}}. $$

Using semigroup theory, we can easily prove the existence of solutions to problem (1.6)-(1.12); see, for example, Alves et al. [27].

Theorem 2.1

Assume that (2.1)-(2.2) hold. Let \(U(t)=(u,u_{t},\theta,\theta_{t},\eta)\). If the initial data \(U_{0}\in \mathcal{H}\), then problem (1.6)-(1.12) has a unique mild solution \(U(t)\in C([0,\infty),\mathcal{H})\) with \(U(0)=U_{0}\).

Define the energy \(E(t)\) of problem (1.6)-(1.12) by

$$\begin{aligned} E(t)=\frac{1}{2} \biggl[ \int^{L}_{0}m(x)u_{t}^{2}\,dx+ \int^{L}_{0}p(x)u_{x}^{2}\,dx + \int^{L}_{0}\theta_{t}^{2}\,dx+ \int^{L}_{0}\theta_{x}^{2}\,dx+ \Vert \eta \Vert ^{2}_{\mathcal {M}} \biggr]. \end{aligned}$$
(2.6)

The general decay result can be given in the following theorem.

Theorem 2.2

Assume that (2.1)-(2.5) hold. Let \((u_{0},u_{1},\theta_{0}(\cdot,0),\theta_{1},\eta_{0})\in\mathcal{H}\) be given. Suppose that there exists a constant \(C\geq0\) such that, for any \(s>0\),

$$\begin{aligned} \Vert \eta_{0x} \Vert \leq C. \end{aligned}$$
(2.7)

Then there exist positive constants \(k_{1}\), \(k_{2}\), \(\epsilon_{0}\) such that, for any \(t\in \mathbb{R}^{+}\),

$$\begin{aligned} E(t)\leq k_{1}H^{-1}(k_{2}t), \end{aligned}$$
(2.8)

where

$$H(s)= \int^{1}_{s}\frac{1}{\tau G'(\epsilon_{0}\tau)}\,d\tau. $$

3 General decay

In this section, to prove Theorem 2.2, we establish a general decay of solutions to problem (1.6)-(1.12). We need the following technical lemmas.

Lemma 3.1

For the energy \(E(t)\) defined in (2.6), there exists a constant \(c_{1}>0\) such that, for any \(t>0\),

$$\begin{aligned} E'(t)\leq -2 \int^{L}_{0}\delta(x)u_{xt}^{2} \,dx+ \int^{L}_{0}g'(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds\leq0. \end{aligned}$$
(3.1)

Proof

Multiplying (1.6) by \(u_{t}\) and (1.7) by \(\theta_{t}\) and then integrating the result over \((0,L)\), we easily get the desired estimate (3.1). □

Lemma 3.2

Under the assumptions of Theorem  2.2, for the functional \(\phi(t)\) defined by

$$\phi(t)= \int^{L}_{0}m(x)u(t)u_{t}(t)\,dx+\kappa \int^{L}_{0}\theta_{x}(t)u(t)\,dx, $$

there exists a positive constant \(c_{1}\) such that, for any \(t>0\),

$$\begin{aligned} \phi'(t)\leq -\frac{1}{2} \int^{L}_{0}p(x)u_{x}^{2}\,dx+ \int^{L}_{0}m(x)u_{x}^{2}\,dx+ \frac{\kappa}{2} \int ^{L}_{0}\theta_{x}^{2} \,dx+c_{1} \int^{L}_{0}u_{xt}^{2}\,dx, \end{aligned}$$
(3.2)

where \(c_{1}=\frac{\kappa}{2\lambda_{1}}\), and \(\lambda_{1}>0\) is the Poincaré constant.

Proof

Taking the derivative of \(\phi(t)\) with respect to t and using (1.6), we get

$$ \begin{aligned}[b] \phi'(t)&=-2 \int^{L}_{0}\delta(x)u_{x}u_{xt} \,dx- \int^{L}_{0}p(x)u_{x}^{2}\,dx+ \int ^{L}_{0}m(x)u_{t}^{2}\,dx \\ &\quad {}+\kappa \int^{L}_{0}\theta_{x}u_{t} \,dx.\end{aligned} $$
(3.3)

Young’s inequality, Poincaré’s inequality, and (2.1) give us

$$-2 \int^{L}_{0}\delta(x)u_{x}u_{xt} \,dx\leq \frac{1}{2} \int^{L}_{0}p(x)u_{x}^{2}\,dx+ \frac{4\delta_{2}^{2}}{p_{1}} \int^{L}_{0}u_{xt}^{2}\,dx $$

and

$$\kappa \int^{L}_{0}\theta_{x}u_{t} \,dx \leq\frac{\kappa}{2} \int^{L}_{0}\theta _{x}^{2}\,dx+ \frac{\kappa}{2\lambda_{1}} \int^{L}_{0}u_{xt}^{2}\,dx, $$

which, together with (3.3), give us (3.2). □

Lemma 3.3

Under the assumptions of Theorem  2.2, the functional \(\psi(t)\) defined by

$$\psi(t)= \int^{L}_{0}\theta(t)\theta_{t}(t)\,dx+ \kappa \int^{L}_{0}u_{x}(t)\theta(t)\,dx $$

satisfies

$$\begin{aligned} \psi'(t)\leq -\frac{1}{2} \int^{L}_{0}\theta_{x}^{2}\,dx+ \biggl(\frac{\kappa}{2}+1 \biggr) \int ^{L}_{0}\theta_{t}^{2}\,dx+ \frac{\kappa}{2\lambda_{1}} \int^{L}_{0}u_{xt}^{2}\,dx+ \frac{1-l}{2} \Vert \eta \Vert ^{2}_{\mathcal{M}} \end{aligned}$$
(3.4)

for all \(t>0\).

Proof

It follows from (1.7) that

$$\begin{aligned} \psi'(t)=- \int^{L}_{0}\theta_{x}^{2}\,dx+ \int^{L}_{0}\theta_{t}^{2}\,dx- \int^{L}_{0}\theta _{x}(t) \int^{\infty}_{0}g(s)\eta_{x}(s)\,ds+\kappa \int^{L}_{0}u_{x}\theta_{t} \,dx. \end{aligned}$$
(3.5)

Using Young’s and Poincaré’s inequalities, we infer that

$$- \int^{L}_{0}\theta_{x}(t) \int^{\infty}_{0}g(s)\eta_{x}(s)\,ds \,dx\leq \frac{1}{2} \int^{L}_{0}\theta_{x}^{2}\,dx+ \frac{1-l}{2} \Vert \eta \Vert ^{2} $$

and

$$\kappa \int^{L}_{0}u_{x}\theta_{t} \,dx \leq \frac{\kappa}{2} \int^{L}_{0}\theta_{t}^{2}\,dx+ \frac{\kappa}{2\lambda_{1}} \int ^{L}_{0}u_{xt}^{2}\,dx, $$

which, along with (3.5), implies (3.4). □

Lemma 3.4

Under the assumptions of Theorem  2.2, for the functional \(\chi(t)\) defined by

$$\chi(t)=- \int^{L}_{0}\theta_{t}(t) \int^{\infty}_{0}g(s)\eta(s)\,ds \,dx, $$

there exists a positive constant \(c_{2}\) such that, for any \(t>0\),

$$\begin{aligned} \chi'(t) \leq& -\frac{1-l}{2} \int^{L}_{0}\theta_{t}^{2}\,dx+ \frac{\kappa^{2}}{2\lambda_{1}} \int ^{L}_{0}u_{xt}^{2}\,dx+ 2(1-l) \Vert \eta \Vert ^{2}_{\mathcal {M}} \\ &{}-c_{2} \int^{\infty}_{0}g'(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.6)

Proof

Direct differentiation using (1.7) implies

$$ \begin{aligned} \chi'(t)&=\underbrace{ \int^{L}_{0}\theta_{x} \biggl( \int^{\infty}_{0}g(s)\eta _{x}(s)\,ds \biggr) \,dx}_{:=I_{1}} \underbrace{{}+ \int^{L}_{0} \biggl( \int^{\infty}_{0}g(s)\eta_{x}(s)\,ds \biggr)^{2}\,dx}_{:=I_{2}} \\ &\quad {}\underbrace{{}-\kappa \int^{L}_{0} u_{t} \int^{\infty}_{0}g(s)\eta_{x}(s)\,ds \,dx}_{:=I_{3}}{\underbrace{{}- \int^{L}_{0}\theta _{t} \int^{\infty}_{0}g(s)\eta_{t}(s)\,ds \,dx}_{:=I_{4}}}. \end{aligned} $$
(3.7)

It follows from Hölder’s and Young’s inequalities that

$$I_{1}\leq\frac{1}{2} \int^{L}_{0}\theta_{x}^{2}\,dx+ \frac{1-l}{2} \Vert \eta \Vert ^{2}_{\mathcal {M}},\qquad I_{2}\leq(1-l) \Vert \eta \Vert ^{2}_{\mathcal{M}}, $$

and

$$I_{3}\leq \frac{\kappa^{2}}{2\lambda_{1}} \int^{L}_{0}u_{xt}^{2}\,dx+ \frac{1-l}{2} \Vert \eta \Vert ^{2}_{\mathcal {M}}. $$

Noting (1.8) and using Young’s inequality, we have

$$\begin{aligned} I_{4} =&-(1-l) \int^{L}_{0}\theta_{t}^{2}\,dx- \int^{L}_{0}\theta_{t} \int ^{\infty}_{0}g'(s)\eta(s)\,ds \,dx \\ \leq&-\frac{(1-l)}{2} \int^{L}_{0}\theta_{t}^{2}\,dx- \frac{l_{1}}{2(1-l)\lambda _{1}} \int^{\infty}_{0}g'(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds, \end{aligned}$$

where \(l_{1}=-\int^{\infty}_{0}g'(s)\,ds\). Inserting these estimates into (3.7), we get (3.6) with

$$c_{2}=\frac{l_{1}}{2(1-l)\lambda_{1}}. $$

The proof is completed. □

We further define the Lyapunov functional \(\mathcal{L}(t)\) by

$$\mathcal{L}(t)=NE(t)+N_{1}\phi(t)+N_{2}\psi(t)+N_{3} \chi(t), $$

where N, \(N_{1}\), \(N_{2}\), \(N_{3}\) are positive constants to be chosen later. First, it is easy to verify that there exists two positive constants \(\beta_{1}\) and \(\beta_{2}\) such that

$$\begin{aligned} \beta_{1}E(t)\leq\mathcal{L}(t)\leq\beta_{2}E(t). \end{aligned}$$
(3.8)

Lemma 3.5

For suitable constants \(N,N_{1},N_{2},N_{3}>0\), there exist two positive constants \(c_{4}\) and \(c_{5}\) such that

$$\begin{aligned} \mathcal {L}'(t)\leq-c_{4}E(t)+c_{5} \int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.9)

Proof

Combining (3.1)-(3.2), (3.4), and (3.6) and using (2.1), we can obtain that, for any \(t>0\),

$$\begin{aligned} \mathcal {L}'(t) =&NE'(t)+N_{1} \phi'(t)+N_{2}\psi'(t)+N_{3} \chi'(t) \\ \leq&- \biggl[2N\delta_{1}- \biggl(\frac{m_{2}}{\lambda_{1}}+c_{1} \biggr)N_{1}-\frac {\kappa}{2\lambda_{1}}N_{2}-\frac{\kappa^{2}}{2\lambda_{1}}N_{3} \biggr] \int^{L}_{0}u_{xt}^{2}\,dx \\ &{}-\frac{N_{1}p_{1}}{2} \int^{L}_{0}u_{x}^{2}\,dx- \biggl(\frac{N_{2}}{2}-\frac{\kappa }{2}N_{1} \biggr) \int^{L}_{0}\theta_{x}^{2} \,dx+c_{3} \Vert \eta \Vert ^{2}_{\mathcal {M}} \\ &{}- \biggl[\frac{1-l}{2}N_{3}- \biggl(\frac{\kappa}{2}+1 \biggr)N_{2} \biggr] \int ^{L}_{0}\theta_{t}^{2} \,dx+(N-c_{2}N_{3}) \int^{\infty}_{0}g'(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds, \end{aligned}$$

where \(c_{3}=N_{1}\frac{1-l}{2}+2N_{3}(1-l)\).

At this point, we first choose \(N_{1},N_{2},N_{3}>0\) that satisfy

$$N_{3}>\frac{\kappa+2}{2(1-l)}N_{2},\qquad N_{2}>\kappa N_{1}, $$

which implies

$$\frac{N_{2}}{2}-\frac{\kappa}{2}N_{1}>0,\qquad \frac{1-l}{2}N_{3}- \biggl(\frac{\kappa}{2}+1 \biggr)N_{2}>0. $$

Then we get that there exist constants \(\gamma>0\) and \(\mu>0\) such that

$$\begin{aligned} \mathcal {L}'(t) \leq&- \biggl[2N\delta_{1}- \biggl(\frac{m_{2}}{\lambda_{1}}+c_{1} \biggr)N_{1}- \frac{\kappa}{2\lambda_{1}}N_{2}-\frac{\kappa^{2}}{2\lambda_{1}}N_{3}- \frac {\mu}{\lambda_{1}} \biggr] \int^{L}_{0}u_{xt}^{2}\,dx \\ &{}-\gamma E(t)+c_{3} \Vert \eta \Vert ^{2}_{\mathcal {M}}+(N-c_{2}N_{3}) \int^{\infty}_{0}g'(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.10)

Finally, for fixed \(N_{1},N_{2},N_{3}>0\), we take \(N>0\) large enough so that

$$2N\delta_{1}- \biggl(\frac{m_{2}}{\lambda_{1}}+c_{1} \biggr)N_{1}-\frac{\kappa }{2\lambda_{1}}N_{2}-\frac{\kappa^{2}}{2\lambda_{1}}N_{3}- \frac{\mu}{\lambda_{1}}>0, \qquad N-c_{2}N_{3}>0, $$

which, together with (3.10), gives us (3.9). □

Lemma 3.6

Under the assumptions of Theorem  2.2, there exists a positive constant \(\gamma_{1}>0\) such that, for any \(\epsilon_{0}>0\),

$$\begin{aligned} G'\bigl(\epsilon_{0}E(t)\bigr) \int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds\leq-\gamma_{1}E'(t)+ \gamma_{1}\epsilon_{0}E(t)G'\bigl( \epsilon_{0}E(t)\bigr). \end{aligned}$$
(3.11)

Proof

We prove this lemma by using the method developed by Guesmia [31]. It follows from (2.3), (2.7), and (3.1) that

$$\begin{aligned} \int^{L}_{0}\eta_{x}^{2}\,dx \leq&2 \int^{L}_{0}\bigl(u_{x}^{2}(x,t)+u_{x}^{2}(x,t-s) \bigr)\,dx \\ \leq&4\sup_{t\geq0} \int^{L}_{0}u_{x}^{2}(t)\,dx+2 \sup_{s>0} \int ^{L}_{0}u^{2}_{0x}(x,s)\,dx \\ \leq&\frac{8}{a_{0}}E(0)+4\sup_{s>0} \int^{L}_{0}\bigl(\bigl(\eta _{x}^{0} \bigr)^{2}(x,s)+u^{2}_{0x}(x,0)\bigr)\,dx \\ \leq&\frac{16}{a_{0}}E(0)+4C:= C_{0}, \end{aligned}$$
(3.12)

where \(C_{0}\) is a positive constant depending on \(E(0)\), \(a_{0}\), and C.

First, if \(g'(s_{0})=0\) for some \(s_{0}\geq0\), then from (2.5) we infer that \(g(s_{0})=0\). Since \(g(t)\) is nonincreasing and nonnegative, we know that \(g(s)=0\) for any \(s\geq s_{0}\). Therefore

$$\int^{\infty}_{0}g(s)\eta_{x}^{2}(s) \,ds= \int^{s_{0}}_{0}g(s)\eta_{x}^{2}(s) \,ds. $$

Without loss of generality, we can assume that \(g'(s)<0\) for \(s\in\mathbb{R}^{+}\).

For \(s\in\mathbb{R}^{+}\), we define \(K(s)=\frac{s}{G^{-1}(s)}\). By the properties of G we know that \(K(0)=G'(0)=0\), and hence the function \(K(s)\) is nondecreasing. Taking into account (3.12), we obtain that, for any \(s_{1}>0\),

$$K\bigl(-s_{1}g'(s) \Vert \eta_{x} \Vert ^{2}\bigr)\leq K\bigl(-C_{0}s_{1}g'(s) \bigr). $$

Therefore, for \(\epsilon_{0},\tau_{1}>0\),

$$\begin{aligned}& \int^{L}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds \\& \quad =\frac{1}{\tau_{1}G'(\epsilon_{0}E(t))} \int^{\infty}_{0}G^{-1}\bigl(-s_{1}g(s) \Vert \eta_{x} \Vert ^{2}\bigr)\frac{\tau_{1}G'(\epsilon _{0}E(t))g(s)}{-s_{1}g(s)}K \bigl(-s_{1}g(s) \Vert \eta_{x} \Vert ^{2} \bigr)\,ds \\& \quad \leq \frac{1}{\tau_{1}G'(\epsilon_{0}E(t))} \int^{\infty}_{0}G^{-1}\bigl(-s_{1}g(s) \Vert \eta_{x} \Vert ^{2}\bigr) \frac{\tau_{1}G'(\epsilon_{0}E(t))g(s)}{-s_{1}g(s)}K \bigl(-C_{0}s_{1}g'(s)\bigr)\,ds \\& \quad \leq \frac{1}{\tau_{1}G'(\epsilon_{0}E(t))} \int^{\infty}_{0}G^{-1}\bigl(-s_{1}g(s) \Vert \eta_{x} \Vert ^{2}\bigr) \frac{C_{0}\tau_{1}G'(\epsilon_{0}E(t))g(s)}{G^{-1}(-C_{0}s_{1}g'(s))} \,ds. \end{aligned}$$
(3.13)

Now we denote the conjugate function of the convex function G by \(G^{*}\) (see, e.g., Arnold [32], Daoulatli et al. [33], Lasiecka and Doundykov [34], and Lasiecka and Tataru [35]), that is,

$$G^{*}(s)=\sup_{t\in \mathbb{R}^{+}}\bigl(st-G(t)\bigr). $$

Then,

$$G^{*}(s)=s\bigl(G'\bigr)^{-1}(s)-G\bigl[ \bigl(G'\bigr)^{-1}(s)\bigr] $$

for \(s\geq0\) is the Legendre transform of G, which satisfies

$$t_{1}t_{2}\leq G(t_{1})+G^{*}(t_{2}). $$

for \(t_{1},t_{2}\geq0\). Denoting

$$t_{1}=G^{-1}\bigl(-s_{1}g(s) \Vert \eta_{x} \Vert ^{2}\bigr),\qquad t_{2}= \frac{C_{0}\tau_{1}G'(\epsilon_{0}E(t))g(s)}{G^{-1}(-C_{0}s_{1}g'(s))}, $$

from (3.13) we see that

$$\begin{aligned} \int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds \leq& \frac{-s_{1}}{\tau_{1}G'(\epsilon_{0}E(t))} \int^{\infty}_{0}g'(s) \bigl\Vert \eta _{x}(s) \bigr\Vert ^{2}\,ds \\ &{}+\frac{1}{\tau_{1}G'(\epsilon_{0}E(t))} \int^{\infty}_{0} G^{*} \biggl(\frac{C_{0}\tau_{1}G'(\epsilon _{0}E(t))g(s)}{G^{-1}(-C_{0}s_{1}g'(s))} \biggr) \,ds, \end{aligned}$$

which, together with (3.1) and the inequality \(G^{*}(s)\leq s(G')^{-1}(s)\), gives

$$ \begin{aligned}[b] &\int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds\\ &\quad \leq \frac{-2s_{1}}{\tau_{1}G'(\epsilon_{0}E(t))}E'(t) \\ &\qquad {}+C_{0} \int^{\infty}_{0}\frac{g(s)}{G^{-1}(-C_{0}s_{1}g'(s))}\bigl(G' \bigr)^{-1} \biggl(\frac {C_{0}\tau_{1}G'(\epsilon_{0}E(t))g(s)}{G^{-1}(-C_{0}s_{1}g'(s))} \biggr)\,ds. \end{aligned} $$
(3.14)

Using (2.5), we denote

$$\sup_{s>0}\frac{g(s)}{G^{-1}(-g'(s))}=b< +\infty. $$

By the properties of G we get the function \((G')^{-1}\) is nondecreasing. Then taking \(s_{1}=\frac{1}{C_{0}}\), we conclude from (3.14) that

$$\begin{aligned} \int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds \leq& \frac{-2}{C_{0}\tau_{1}G'(\epsilon_{0}E(t))}E'(t) \\ &{}+C_{0}\bigl(G'\bigr)^{-1} \bigl(C_{0}b\tau_{1}G'\bigl(\epsilon_{0}E(t) \bigr)\bigr) \int^{\infty}_{0}\frac {g(s)}{G^{-1}(-g'(s))}\,ds. \end{aligned}$$
(3.15)

Similarly, denoting

$$\int^{\infty}_{0}\frac{g(s)}{G^{-1}(-g'(s))}\,ds=d< +\infty $$

and picking \(\tau_{1}=\frac{1}{C_{0}b}\), we infer from (3.15) that

$$\int^{\infty}_{0}g(s) \bigl\Vert \eta_{x}(s) \bigr\Vert ^{2}\,ds\leq \frac{-2b}{G'(\epsilon_{0}E(t))}E'(t)+C_{0} \,d\epsilon_{0}E(t), $$

which gives (3.11). □

Proof of Theorem 2.2

Multiplying (3.9) by \(G'(\epsilon_{0}E(t))\) and using (3.11), we obtain

$$\begin{aligned} G'\bigl(\epsilon_{0}E(t)\bigr)\mathcal {L}'(t)\leq-(c_{4}-c_{5}\gamma_{1} \epsilon_{0})E(t)G'\bigl(\epsilon_{0}E(t)\bigr)- \gamma _{1}c_{5}E'(t). \end{aligned}$$

Taking \(\epsilon_{0}\) small enough so that \(c_{4}-c_{5}\gamma_{1}\epsilon_{0}>0\), we get that there exists a constant \(c_{6}>0\) such that

$$\begin{aligned} G'\bigl(\epsilon_{0}E(t)\bigr)\mathcal {L}'(t)+\gamma_{1}c_{5}E'(t) \leq-c_{6}E(t)G'\bigl(\epsilon_{0}E(t)\bigr). \end{aligned}$$
(3.16)

Define

$$\mathcal{E}(t)=\tau\bigl(G'\bigl(\epsilon_{0}E(t)\bigr) \mathcal {L}(t)+\gamma_{1}c_{5}E(t)\bigr) $$

with \(\tau>0\). Noting that \(G'(\epsilon_{0}E(t))\) is nonincreasing, we can easily get from (3.8) that

$$\begin{aligned} \mathcal {E}'(t)\leq-c_{6}\tau G'\bigl(\epsilon_{0}E(t)\bigr)E(t). \end{aligned}$$
(3.17)

Using (3.8) and the inequality \(G'(\epsilon_{0}E(t))\leq G'(\epsilon _{0}E(0))\), we can get that there exist two positive constants \(\beta_{3}\) and \(\beta_{4}\) such that

$$\beta_{3} E(t)\leq\mathcal{E}(t)\leq\beta_{4}E(t). $$

Choose \(\tau>0\) small enough so that

$$\begin{aligned} \mathcal {E}(t)\leq E(t),\qquad \mathcal{E}(0)\leq1. \end{aligned}$$
(3.18)

Noting that \(s\mapsto sG'(\epsilon_{0}s)\) is nondecreasing, from (3.17) we obtain

$$\begin{aligned} \mathcal {E}'(t)\leq-c_{7} \mathcal{E}(t)G'\bigl(\epsilon_{0}\mathcal {E}(t)\bigr) \end{aligned}$$
(3.19)

with \(c_{7}=c_{6}\tau\). This shows that \((H(\mathcal{E}(t)))'\geq c_{7}\), where

$$H(t)= \int^{1}_{t}\frac{1}{sG'(\epsilon_{0}s)}\,ds. $$

Integrating it over \([0,t]\), we see that

$$H\bigl(\mathcal {E}(t)\bigr)\geq c_{7}t+H\bigl(\mathcal{E}(0)\bigr), $$

which, together with (3.18) and \(H(1)=0\), implies

$$H\bigl(\mathcal{E}(t)\bigr)\geq c_{7}t. $$

Since \(H^{-1}\) is decreasing, we get

$$\mathcal{E}(t)\leq H^{-1}(c_{7}t). $$

Then (2.8) follows from the equivalence \(\mathcal{E}(t)\sim E(t)\). □

Remark 3.7

Condition (2.5) allows g to have a decay rate close to \(\frac{1}{t}\), and the rate of energy decay (2.8) depends on g.

Finally, we give three examples to illustrate several rates of energy decay, which can be found in Guesmia [31].

Example 1

Let \(g(t)=\frac{\mu}{(1+t)^{p}}\) with \(p>1\) and \(\mu>0\) small enough so that (2.3) holds. Condition (2.5) is satisfied for \(G(t)=t^{1+\frac{1}{q}}\) with \(q\in(0,\frac{p-1}{2})\). Then from (2.8) we get that there exists a constant \(\beta_{5}>0\) such that, for any \(q\in(0,\frac{p-1}{2})\),

$$E(t)\leq\frac{\beta_{5}}{(1+t)^{q}}. $$

Example 2

Let \(g(t)=\mu e^{-(\ln(2+t))^{p}}\) with \(p>1\) and \(\mu>0\) small enough so that (2.3) holds. For

$$G(t)= \int^{t}_{0}(-\ln s)^{1-\frac{1}{q}}e^{-(-\ln s)^{\frac{1}{q}}} \,ds, $$

when t is near zero, (2.5) holds with with \(q\in(1,p)\). Then from (2.8) we get that there exist two constants \(\beta_{5}>0\) and \(\beta_{6}>0\) such that, for any \(q\in(1,p)\),

$$E(t)\leq\beta_{5}e^{-\beta_{6}(\ln(1+t))^{q}}. $$

Example 3

Let \(g(t)=\mu e^{-(1+t)^{p}}\) with \(p\in(0,1)\) and \(\mu>0\) small enough so that (2.3) holds. For

$$G(t)= \int^{t}_{0}(-\ln s)^{1-\frac{1}{q}}\,ds, $$

when t is near zero, (2.5) holds with \(q\in(1,\frac{p}{2})\). Then from (2.8) we get that there exist two constants \(\beta_{5}>0\) and \(\beta_{6}>0\) such that, for any \(q\in(1,\frac{p}{2})\),

$$E(t)\leq\beta_{5}e^{-\beta_{6}t^{q}}. $$

4 Conclusions

In this work, we consider a model to the longitudinal vibrations of a beam equation with thermoviscoelastic damping. Motivated by Dafermos [28], we introduce a new variable, and the system is transformed into a new system. We give the global existence of solutions without proof. The main result is the energy decay of solutions. Under suitable assumptions, we established a general decay result of energy for the initial value problem by using the energy perturbation method and some properties of convex functions. Finally, we give three examples, which can be found in Guesmia [31], to illustrate several rates of energy decay.