A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective

Abstract

In this work, we analyze a truncated version for the Timoshenko beam model with thermal and mass diffusion effects derived by Aouadi et al. (Z Angew Math Phys 70:117, 2019). In particular, we study some issues related to the second spectrum of frequency according to a procedure due to Elishakoff (in: Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, 2010). In Aouadi et al. (2019), the lack of exponential stability for the classical Timoshenko beam with thermodiffusion effects without assuming the nonphysical condition of equal wave speeds has be proved. By using the classical Faedo–Galerkin method combined with the a priori estimates, we prove the existence and uniqueness of a global solution of the truncated version of this problem. Then we prove that this solution is exponentially stable without assuming the condition of equal wave speeds.

1 Introduction

Recently, Aouadi et al. [5] introduced a new Timoshenko beam model with thermal and mass diffusion effects given by

$$\begin{aligned} \rho _1 \varphi _{tt}-\kappa (\varphi _x+\psi )_x= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \end{aligned}$$

(1.1)

$$\begin{aligned} \rho _2\psi _{tt}-\alpha \psi _{xx} +\kappa (\varphi _x+\psi )-\gamma _{1}\theta _{x}-\gamma _{2}P_{x}= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \end{aligned}$$

(1.2)

$$\begin{aligned} c\theta _{t}+dP_{t}-K\theta _{xx}-\gamma _{1}\psi _{xt}= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \end{aligned}$$

(1.3)

$$\begin{aligned} d\theta _{t}+rP_{t}-\hbar P_{xx}-\gamma _{2}\psi _{xt}= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \end{aligned}$$

(1.4)

where $\varphi$ is the transverse displacement, $\psi$ is the rotation of the neutral axis due to bending, $\theta$ is the temperature, and P is the chemical potential. The constants $\rho _{1}$, $\rho _{2}$, $\kappa$, $\alpha$, $\gamma _{1}$, $\gamma _{2}$, c, r, d, $\hbar$, and K are physical positive parameters. They showed, without assuming the well-known equal wave speeds condition $\chi :=\kappa /\rho _{1}-b/\rho _{2}=0$, the lack of exponential stability for the problem. Based on [5] and the recent studies due to Almeida Júnior et al. [1,2,3,4], we consider the truncated version given by

$$\begin{aligned} \rho _1 \varphi _{tt}-\kappa (\varphi _x+\psi )_x= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \qquad \quad \end{aligned}$$

(1.5)

$$\begin{aligned} -\rho _2\varphi _{xtt}-\alpha \psi _{xx} +\kappa (\varphi _x+\psi )-\gamma _{1}\theta _{x}-\gamma _{2}P_{x}= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \qquad \quad \end{aligned}$$

(1.6)

$$\begin{aligned} c\theta _{t}+dP_{t}-K\theta _{xx}-\gamma _{1}\psi _{xt}= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [, \qquad \quad \end{aligned}$$

(1.7)

$$\begin{aligned} d\theta _{t}+rP_{t}-\hbar P_{xx}-\gamma _{2}\psi _{xt}= & {} 0 \ \ \text{ in } \ \ ]0, L[ \times ]0,\infty [,\qquad \quad \end{aligned}$$

(1.8)

with the initial conditions

$$\begin{aligned}&\displaystyle \varphi (x,0) = \varphi _0(x), \ \varphi _t( x,0) = \varphi _1(x), \ \varphi _{tt}(x,0) = \varphi _2(x), \quad x\in (0,L), \nonumber \\&\displaystyle \ \psi (x,0) = \psi _0(x), \ \theta (x,0)=\theta _{0}(x), \ P(x,0)=P_{0}(x), \quad x\in (0,L), \end{aligned}$$

(1.9)

and boundary conditions of Dirichlet-Neumann-type

$$\begin{aligned}&\displaystyle \varphi (0,t) = \varphi (L,t) = \psi _x(0,t) = \psi _x(L,t) =0, \quad t\ge 0&\nonumber \\&\displaystyle \theta (0,t)=\theta (L,t)=P(0,t)=P(L,t)= 0, \quad t\ge 0.&\end{aligned}$$

(1.10)

The truncated version (1.5)–(1.10) is obtained by following the procedure of Elishakoff [7] which involves replacing the term $\psi _{tt}$ in (1.2) by $-\varphi _{xtt}$ based on d’Alembert’s principle for dynamic equilibrium. This eliminates the second spectrum of frequency and its damaging consequences for wave propagation speed (see the first results in [1] and also in [9]). Therefore, the goal of this work is to prove the well-posedness of problem (1.5)–(1.10) and the exponential stability of solutions without assuming the nonphysical condition of equal wave speeds.

In order to derive the dissipative nature of the system (1.5)–(1.10), we define its functional energy of solutions

$$\begin{aligned}&\displaystyle E(t):=\frac{\rho _{1}}{2}\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{t}|^2dx+ \displaystyle \frac{\rho _{1}\rho _{2}}{2\kappa }\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{tt}|^2dx+ \displaystyle \frac{\rho _{2}}{2}\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{xt}|^2dx \nonumber \\&\quad \displaystyle + \displaystyle \frac{\alpha }{2}\displaystyle \mathop \int \limits _{0}^{L}|\psi _{x}| ^2dx+\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^2dx+\frac{1}{2}(c-d^2/r)\mathop \int \limits _{0}^{L}|\theta |^{2}dx \nonumber \\&\quad \displaystyle +\displaystyle \frac{1}{2}\displaystyle \mathop \int \limits _{0}^{L} \bigg |\frac{d}{\sqrt{r}}\theta +\sqrt{r}P\bigg |^2dx, \end{aligned}$$

(1.11)

which preserves its positivity property for

$$\begin{aligned} cr-d^2>0. \end{aligned}$$

(1.12)

2 Well-posedness

In this section, the existence and uniqueness of weak and strong solutions to (1.5)–(1.10) will be proved. To this end, we will use the Faedo–Galerkin approximations and pass to the limit by using compactness arguments (see also [6]).

We introduce the phase space

$$\begin{aligned} {\mathcal {H}}:=H_0^1(0,L)\times H_0^1(0,L)\times L^2(0,L)\times H_*^1(0,L)\times L^2(0,L)\times L^2(0,L), \end{aligned}$$

and

$$\begin{aligned} {\mathcal {H}}_1:=(H^2(0,L)\cap H_0^1(0,L))^2\times H_0^1(0,L)\times H_*^2(0,L)\times H_0^1(0,L)\times H_0^1(0,L), \end{aligned}$$

where

$$\begin{aligned} L_*^2(0,L):=\left\{ u\in L^2(0,L):\mathop \int \limits _{0}^{L}u(x)dx=0\right\} , \end{aligned}$$

and

$$\begin{aligned} H_*^1(0,L):=H^1(0,L)\cap L_*^2(0,L),\quad H_*^2(0,L):=H^2(0,L)\cap H_*^1(0,L). \end{aligned}$$

In order to state our main result, we begin with a precise definition of a weak solution to (1.5)–(1.10).

Definition 2.1

Given initial data $(\varphi _0,\varphi _1,\varphi _2,\psi _0,\theta _0,P_0)\in {\mathcal {H}}$, a function $U=(\varphi ,\varphi _t,\varphi _{tt},\psi ,\theta ,P)\in C(0,T;{\mathcal {H}})$ is said to be a weak solution of (1.5)–(1.10) if for almost every $t\in [0,T]$,

$$\begin{aligned}&\rho _1\frac{d}{dt}(\varphi _{t},u) +\kappa (\varphi _x+\psi ,u_x)=0, \end{aligned}$$

(2.1)

$$\begin{aligned}&\rho _1\frac{d}{dt}(\varphi _{tt},v) +\kappa \frac{d}{dt}(\varphi _{x}+\psi ,v_x)=0,\end{aligned}$$

(2.2)

$$\begin{aligned}&\rho _2\frac{d}{dt}(\varphi _{t},w_x) \!+\!\alpha (\psi _x,w_x)\!+\!\kappa (\varphi _x\!+\!\psi ,w)\!+\!(\gamma _1\theta +\gamma _2 P,w_x)=0 ,\end{aligned}$$

(2.3)

$$\begin{aligned}&\frac{d}{dt}(c\theta +d P,\xi )+K(\theta _x,\xi _x)+\gamma _1\frac{d}{dt}(\psi ,\xi _x)=0,\end{aligned}$$

(2.4)

$$\begin{aligned}&\frac{d}{dt}(d\theta +r P,\zeta )+\hbar (P_x,\zeta _x)+\gamma _2\frac{d}{dt}(\psi ,\zeta _x)=0, \end{aligned}$$

(2.5)

for all $u,v,\xi ,\zeta \in H_0^1(0,L),$ $w\in H_*^1(0,L)$, and

$$\begin{aligned} \big (\varphi (0),\varphi _t(0),\varphi _{tt}(0),\psi (0),\theta (0),P(0)\big )=\big (\varphi _0,\varphi _1,\varphi _2,\psi _0,\theta _0,P_0\big ). \end{aligned}$$

Theorem 2.2

Suppose that condition (1.12) holds. Then we have:

(i) If the initial data $(\varphi _{0}, \varphi _{1},\varphi _{2},\psi _0,\theta _0,P_0) \in {\mathcal {H}}$, then problem (1.5)–(1.10) has a weak solution satisfying

$$\begin{aligned} \varphi \in L^\infty \big (0,T;H^{1}_{0}(0,L)\big ),\ \psi \in L^\infty \big (0,T;H^{1}_{*}(0,L)\big ), \\ \varphi _{t}\in L^\infty \big (0,T;H^{1}_0(0,L)\big ),\ \varphi _{tt}\in L^\infty \big (0,T;L^{2}(0,L)\big ), \\ \theta \in L^\infty \big (0,T;L^2(0,L)\big ), \ P \in L^\infty \big (0,T;L^2(0,L)\big ). \end{aligned}$$

(ii) If the initial data $(\varphi _{0}, \varphi _{1},\varphi _{2}, \psi _0,\theta _0,P_0) \in {\mathcal {H}}_1$, then problem (1.5)–(1.10) has a unique stronger weak solution satisfying

$$\begin{aligned}&\varphi \in L^\infty \big (0,T;H^2(0,L)\cap H^{1}_{0}(0,L)\big ), \ \psi \in L^\infty \big (0,T;H^2_*(0,L)\big ), \\&\varphi _{t}\in L^\infty \big (0,T;H^2(0,L)\cap H^{1}_{0}(0,L)\big ), \\&\varphi _{tt}\in L^\infty \big (0,T;H^{1}_0(0,1)\big ), \ \theta \in L^\infty \big (0,T;H^1_0(0,L)\big ), \\&P \in L^\infty \big (0,T;H^1_0(0,L)\big ). \end{aligned}$$

(iii) In both cases, the solution $(\varphi ,\varphi _{t},\varphi _{tt},\psi ,\theta , P)$ depends continuously on the initial data in ${\mathcal {H}}.$ In particular, problem (1.5)–(1.10) has a unique weak solution.

Proof

The proof is given by the Faedo–Galerkin method. We only briefly present the main (six) steps.

Step 1 – Approximate problem. Let us consider the initial data $(\varphi _{0}, \varphi _{1},\varphi _{2},$ $\psi _0,\theta _0,P_0) \in {\mathcal {H}}$. Let $\{\omega _j\}^\infty _{j=1}$ and $\{\mu _j\}^\infty _{j=1}$ be orthogonal bases for $H^2(0,L)\cap H^1_0(0,L)$ and $H^2_*(0,L)$, respectively, which are both orthonormal in $L^2(0,L)$. Now we denote the finite-dimensional subspaces, for any integer $n\in {\mathbb {N}}$, by

$$\begin{aligned} H_n=span\{\omega _1,\omega _2,...,\omega _n\},\quad V_n=span\{\mu _1,\mu _2,...,\mu _n\}. \end{aligned}$$

We will find an approximate solution of the form

$$\begin{aligned} \varphi ^n(x,t)= & {} \sum ^n_{j=1}a_{j,n}\omega _j(x), \ \ \psi ^n(x,t)=\sum ^n_{j=1}b_{j,n}\mu _j(x), \end{aligned}$$

(2.6)

$$\begin{aligned} \theta ^n(x,t)= & {} \sum ^n_{j=1}c_{j,n}\omega _j(x), \ \ P^n(x,t)=\sum ^n_{j=1}d_{j,n}\omega _j(x), \end{aligned}$$

(2.7)

to the following approximate problem

$$\begin{aligned}&\rho _1(\varphi _{tt}^n,u) +\kappa (\varphi _x^n+\psi ^n,u_x)=0, \end{aligned}$$

(2.8)

$$\begin{aligned}&\rho _1(\varphi _{ttt}^n,v) +\kappa (\varphi _{xt}^n+\psi _t^n,v_x)=0, \end{aligned}$$

(2.9)

$$\begin{aligned}&\rho _2(\varphi _{tt}^n,w_x) \!+\!\alpha (\psi _x^n\!,w_x)\!+\!\kappa (\varphi _x^n\!+\!\psi ^n\!,w)\!+\!(\gamma _1\theta ^n\!+\!\gamma _2 P^n\!,\!w_x)=0, \end{aligned}$$

(2.10)

$$\begin{aligned}&(c\theta _t^n+d P_t^n,\xi )+K(\theta _x^n,\xi _x)+\gamma _1(\psi _t^n,\xi _x)=0, \end{aligned}$$

(2.11)

$$\begin{aligned}&(d\theta _t^n+r P_t^n,\zeta )+\hbar (P_x^n,\zeta _x)+\gamma _2(\psi _t^n,\zeta _x)=0, \end{aligned}$$

(2.12)

for all $u,v,\xi ,\zeta \in H_n$, $w\in V_n$ with initial conditions

$$\begin{aligned} (\varphi ^n(0),\varphi _t^n(0),\varphi _{tt}^n(0),\psi ^n(0),\theta ^n(0),P^n(0))=(\varphi _0^n,\varphi _1^n,\varphi _2^n,\psi _0^n,\theta _0^n,P_0^n)\qquad \end{aligned}$$

(2.13)

satisfying

$$\begin{aligned} (\varphi _0^n,\varphi _1^n,\varphi _2^n,\psi _0^n,\theta _0^n,P_0^n)\rightarrow (\varphi _0,\varphi _1,\varphi _2,\psi _0,\theta _0,P_0)\ \text{ strongly } \text{ in } \ {\mathcal {H}}. \end{aligned}$$

From the application of the standard ODE theory, we can obtain a local solution $\big (\varphi ^n(t),\varphi ^n_t(t),\varphi ^n_{tt}(t),\psi ^{n}(t),\theta ^n(t)$, $P^n(t)\big )$ on the maximal interval $[0,t_n)$ with $0<t_n\le T$ for every $n\in {\mathbb {N}}$.

Step 2 – A priori estimate. Replacing u by $\varphi ^n_{t}$ in (2.8), v by $\varphi _{tt}^n$ in (2.9), w by $\psi ^n_t$ in (2.10), $\xi$ by $\theta ^n$ in (2.11), and $\zeta$ by $P^n$, we obtain

$$\begin{aligned} \displaystyle \frac{d}{dt}E^n(t)+K\mathop \int \limits _{0}^{L}|\theta ^n_{x}|^2dx+\hbar \mathop \int \limits _{0}^{L}|P^n_{x}|^2dx=0, \end{aligned}$$

(2.14)

where

$$\begin{aligned}&\displaystyle E^n(t):=\frac{\rho _{1}}{2}\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{t}^n|^2dx+ \displaystyle \frac{\rho _{1}\rho _{2}}{2\kappa }\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{tt}^n|^2dx+ \displaystyle \frac{\rho _{2}}{2}\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{xt}^n|^2dx \\&\quad \displaystyle +\frac{\alpha }{2}\displaystyle \mathop \int \limits _{0}^{L}|\psi _{x}^n|^2dx+\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}^n+\psi ^n|^2dx+\frac{1}{2}(c-d^2/r)\mathop \int \limits _{0}^{L}|\theta ^n|^{2}dx \\&\quad \displaystyle +\displaystyle \frac{1}{2} \displaystyle \mathop \int \limits _{0}^{L}\bigg |\frac{d}{\sqrt{r}}\theta ^n+\sqrt{r}P^n\bigg |^2dx. \end{aligned}$$

Then integrating (2.14) from 0 to $t<t_n$, we obtain from our choice of initial data that for all $t\in [0,T]$ and for every $n\in {\mathbb {N}}$,

$$\begin{aligned} E^n(t)+K\mathop \int \limits _{0}^{t}\mathop \int \limits _{0}^{L}|\theta ^n_{x}(s)|^2dxds+\hbar \mathop \int \limits _{0}^{t}\mathop \int \limits _{0}^{L}|P^n_{x}(s)|^2dxds\le C_1, \end{aligned}$$

(2.15)

where $C_1$ is a positive constant depending on the initial data. Thus, approximate solutions are defined on the whole range [0, T].

Step 3 – Passing to the limit. From (2.15) and definition of $E^n(t)$, we deduce that

$$\begin{aligned} \left\{ \begin{array}{ll} \{\varphi ^n\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T; H^1_0(0,L)\big ), \\ \{\varphi ^n_t\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;H^1_0(0,L)\big ), \\ \{\varphi ^n_{tt}\}\, \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;L^2(0,L)\big ), \\ \{\psi ^n\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;H^1_*(0,L)\big ), \\ \{\theta ^n\} \, \ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;L^2(0,L)\big )\cap L^2\big (0,T;H^1_0(0,L)\big ), \\ \{P^n\}\, \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;L^2(0,L)\big )\cap L^2\big (0,T;H^1_0(0,L)\big ). \end{array} \right. \end{aligned}$$

Then we can extract a subsequence of $\{\varphi ^n\}$, $\{\psi ^n\}$, $\{\theta ^n\}$, and $\{P^n\}$ still denoted by $\{\varphi ^n\}$, $\{\psi ^n\}$, $\{\theta ^n\}$, and $\{P^n\}$, such that

$$\begin{aligned} \left\{ \begin{array}{ll} \varphi ^n \rightarrow \varphi \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T; H^1_0(0,L)\big ), \\ \varphi ^n_t\rightarrow \varphi _t\ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T; H^1_0(0,L)\big ), \\ \varphi ^n_{tt}\rightarrow \varphi _{tt}\ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;L^2(0,L)\big ), \\ \psi ^n\rightarrow \psi \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;H^1_*(0,L)\big ), \\ \theta ^n\rightarrow \theta \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;L^2(0,L)\big ), \\ \theta ^n\rightarrow \theta \ \ \text{ weakly }\ \text{ in }\ L^2\big (0,T;H_0^1(0,L)\big ), \\ P^n \rightarrow P \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;L^2(0,L)\big ), \\ P^n \rightarrow P \ \ \text{ weakly }\ \text{ in }\ L^2\big (0,T;H_0^1(0,L)\big ). \end{array} \right. \end{aligned}$$

Therefore the above limits allow us to pass to the limit in the approximate problem (2.8)–(2.12) to get a weak solution satisfying

$$\begin{aligned} \varphi \in L^\infty \big (0,T;H^{1}_{0}(0,L)\big ),\ \psi \in L^\infty \big (0,T;H^{1}_{*}(0,L)\big ), \\ \varphi _{t}\in L^\infty \big (0,T;H^{1}_0(0,L)\big ),\ \varphi _{tt}\in L^\infty \big (0,T;L^{2}(0,L)\big ), \\ \theta \in L^\infty \big (0,T;L^2(0,L)\big ), \ P \in L^\infty \big (0,T;L^2(0,L)\big ). \end{aligned}$$

Step 4 – Initial data. By using Aubin-Lions lemma, see [8], we arrive at

$$\begin{aligned}&\varphi ^n\rightarrow \varphi \ \text{ strongly }\ \text{ in }\ C(0,T;L^2(0,L)), \end{aligned}$$

(2.16)

$$\begin{aligned}&\varphi ^n_t\rightarrow \varphi _t\ \text{ strongly }\ \text{ in }\ C(0,T;L^2(0,L)). \end{aligned}$$

(2.17)

Consequently

$$\begin{aligned} \big (\varphi (0),\varphi _{t}(0)\big )=\big (\varphi _0,\varphi _{1}\big ). \end{aligned}$$

Now, we multiply (2.9) by a test function

$$\begin{aligned} \eta \in H^1(0,T),\quad \eta (0)=1, \quad \eta (T)=0, \end{aligned}$$

and integrate the result over [0, T] to obtain

$$\begin{aligned} -\rho _1(\varphi _{2}^n,v)-\rho _1\mathop \int \limits _{0}^{T}(\varphi _{tt}^n,v)\eta _t dt+\kappa \mathop \int \limits _{0}^{T}\frac{d}{dt}(\varphi _{x}^n+\psi ^n,v_x)\eta dt=0 \end{aligned}$$

for all $v\in H_0^1(0,L)$. Taking the limit $n\rightarrow \infty$, we obtain

$$\begin{aligned} -\rho _1(\varphi _{2},v)-\rho _1\mathop \int \limits _{0}^{T}(\varphi _{tt},v)\eta _t dt+\kappa \mathop \int \limits _{0}^{T}\frac{d}{dt}(\varphi _{x}+\psi ,v_x)\eta dt=0 \end{aligned}$$

(2.18)

for all $v\in H_0^1(0,L)$. On the other hand, multiplying (2.2) by $\eta$ and integrating the result over [0, T], we obtain

$$\begin{aligned} -\rho _1(\varphi _{tt}(0),v)-\rho _1\mathop \int \limits _{0}^{T}(\varphi _{tt},v)\eta _t dt+\kappa \mathop \int \limits _{0}^{T}\frac{d}{dt}(\varphi _{x}+\psi ,v_x)\eta dt=0 \end{aligned}$$

(2.19)

for all $v\in H_0^1(0,L)$. Combining (2.18) and (2.19), we conclude that $\varphi _{tt}(0)=\varphi _2$. Analogously, we obtain

$$\begin{aligned} \big (\psi (0),\theta (0),P(0)\big )=\big (\psi _0,\theta _0,P_0\big ). \end{aligned}$$

Step 5—Stronger solutions. Suppose that the initial data in the approximate problem (2.8)–(2.12) satisfies $(\varphi _{0}, \varphi _{1},\varphi _{2}, \psi _0,\theta _0,P_0) \in {\mathcal {H}}_1$ and

$$\begin{aligned} (\varphi _0^n,\varphi _1^n,\varphi _2^n,\psi _0^n,\theta _0^n,P_0^n)\rightarrow (\varphi _0,\varphi _1,\varphi _2,\psi _0,\theta _0,P_0)\ \text{ strongly } \text{ in } \ {\mathcal {H}}_1. \end{aligned}$$

(2.20)

Replacing u by $-\varphi ^n_{xxt}$ in (2.8), v by $-\varphi ^n_{xxtt}$ in (2.9), w by $-\psi _{xxt}^n$ in (2.10), $\xi$ by $-\theta _{xx}^n$ in (2.11), and $\zeta$ by $-P_{xx}$ in (2.12), we see that

$$\begin{aligned} \frac{d}{dt}F^n(t)+K\mathop \int \limits _{0}^{L}|\theta ^n_{xx}|^2dx+\hbar \mathop \int \limits _{0}^{L}|P^n_{xx}|^2dx=0, \end{aligned}$$

(2.21)

where

$$\begin{aligned}&\displaystyle F^n(t):=\frac{\rho _{1}}{2}\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{xt}^n|^2dx+ \displaystyle \frac{\rho _{1}\rho _{2}}{2\kappa }\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{xtt}^n|^2dx+ \displaystyle \frac{\rho _{2}}{2}\displaystyle \mathop \int \limits _{0}^{L}|\varphi _{xxt}^n|^2dx \\&\quad \displaystyle +\frac{\alpha }{2}\displaystyle \mathop \int \limits _{0}^{L}|\psi _{xx}^n|^2dx+\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{xx}^n+\psi _x^n|^2dx+\frac{1}{2}(c-d^2/r)\mathop \int \limits _{0}^{L}|\theta _x^n|^{2}dx \\&\quad \displaystyle \displaystyle +\displaystyle \frac{1}{2} \displaystyle \mathop \int \limits _{0}^{L}\bigg |\frac{d}{\sqrt{r}}\theta _x^n+\sqrt{r}P_x^n\bigg |^2dx. \end{aligned}$$

Then from (2.21), we obtain that for all $t\in [0,T]$, $n\in {\mathbb {N}}$,

$$\begin{aligned} F^n(t)+K\mathop \int \limits _{0}^{t}\mathop \int \limits _{0}^{L}|\theta ^n_{xx}(s)|^2dxds+\hbar \mathop \int \limits _{0}^{t}\mathop \int \limits _{0}^{L}|P^n_{xx}(s)|^2dxds\le C_2, \end{aligned}$$

(2.22)

where $C_2$ is a positive constant independent of t and n but depending on the initial data. From (2.22), we deduce that

$$\begin{aligned} \left\{ \begin{array}{ll} \{\varphi ^n\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T; H^2(0,L)\cap H^1_0(0,L)\big ), \\ \{\varphi ^n_t\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T; H^2(0,L)\cap H^1_0(0,L)\big ), \\ \{\varphi ^n_{tt}\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;H^1_0(0,L)\big ), \\ \{\psi ^n\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;H^2_*(0,L)\big ), \\ \{\theta ^n\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;H^1_0(0,L)\big )\cap L^2\big (0,T;H^2(0,L)\cap H^1_0(0,L)\big ), \\ \{P^n\}\ \ \text{ is }\ \text{ bounded }\ \text{ in }\ L^\infty \big (0,T;H^1_0(0,L)\big )\cap L^2\big (0,T;H^2(0,L)\cap H^1_0(0,L)\big ). \end{array} \right. \end{aligned}$$

This implies that

$$\begin{aligned} \left\{ \begin{array}{ll} \varphi ^n \rightarrow \varphi \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T; H^2(0,L)\cap H^1_0(0,L)\big ), \\ \varphi ^n_t\rightarrow \varphi _t\ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T; H^2(0,L)\cap H^1_0(0,L)\big ), \\ \varphi ^n_{tt}\rightarrow \varphi _{tt}\ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;H^1_0(0,L)\big ), \\ \psi ^n\rightarrow \psi \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;H^2_*(0,L)\big ), \\ \theta ^n\rightarrow \theta \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;H_0^1(0,L)\big ), \\ \theta ^n\rightarrow \theta \ \ \text{ weakly }\ \text{ in }\ L^2\big (0,T;H^2(0,L)\cap H_0^1(0,L)\big ), \\ P^n \rightarrow P \ \ \text{ weakly } \text{ star }\ \text{ in }\ L^\infty \big (0,T;H_0^1(0,L)\big ), \\ P^n \rightarrow P \ \ \text{ weakly }\ \text{ in }\ L^2\big (0,T;H^2(0,L)\cap H_0^1(0,L)\big ). \end{array} \right. \end{aligned}$$

From the above limits, we conclude that $(\varphi ,\varphi _t,\varphi _{tt},\psi ,\theta ,P)$ is a stronger weak solution satisfying

$$\begin{aligned}&\varphi \in L^\infty \big (0,T;H^2(0,L)\cap H^{1}_{0}(0,L)\big ), \ \psi \in L^\infty \big (0,T;H^2_*(0,L)\big ), \\&\varphi _{t}\in L^\infty \big (0,T;H^2(0,L)\cap H^{1}_{0}(0,L)\big ), \\&\varphi _{tt}\in L^\infty \big (0,T;H^{1}_0(0,1)\big ), \ \theta \in L^\infty \big (0,T;H^1_0(0,L)\big ), \\&P \in L^\infty \big (0,T;H^1_0(0,L)\big ). \end{aligned}$$

Step 6 – Continuous dependence. Firstly, we consider the case of stronger solutions. Let $U(t)=(\varphi ,\varphi _t,\varphi _{tt},\psi , \theta , P)$ and $V(t)=({\tilde{\varphi }},{\tilde{\varphi }}_t,{\tilde{\varphi }}_{tt},{\tilde{\psi }},{\tilde{\theta }}, {\tilde{P}})$ be the stronger weak solutions of the problem (1.5)–(1.8) corresponding to the initial data $U(0)=(\varphi _0,\varphi _1,\varphi _2,\psi _0,$ $\theta _0,P_0)$, $V(0)=({\tilde{\varphi }}_0,\tilde{\varphi _1},\tilde{\varphi _2},{\tilde{\psi }}_0,{\tilde{\theta }}_{0},{\tilde{P}}_{0})\in {\mathcal {H}}_1$, respectively. Then $(\Phi ,\Phi _t,\Phi _{tt},\Psi ,\Theta , \Upsilon )=U(t)-V(t)$ satisfies the following equations:

$$\begin{aligned} \rho _1\Phi _{tt}-\kappa (\Phi _x+\Psi )_x= & {} 0, \end{aligned}$$

(2.23)

$$\begin{aligned} -\rho _2\Phi _{xtt}-b\Psi _{xx}+\kappa (\Phi _x+\Psi )-\gamma _{1}\Theta _{x}-\gamma _{2}\Upsilon _{x}= & {} 0,\end{aligned}$$

(2.24)

$$\begin{aligned} c\Theta _{t}+d\Upsilon _{t}-K\Theta _{xx}-\gamma _{1}\Psi _{xt}= & {} 0,\end{aligned}$$

(2.25)

$$\begin{aligned} d\Theta _{t}+r\Upsilon _{t}-\hbar \Upsilon _{xx}-\gamma _{2}\Psi _{xt}= & {} 0, \end{aligned}$$

(2.26)

with initial data $(\Phi (0),\Phi _t(0),\Phi _{tt}(0), \Psi (0), \Theta (0),\Upsilon (0))=U(0)-V(0)$.

We multiply (2.23) by $\Phi _t$, (2.24) by $\Psi _t$, (2.25) by $\Theta$, and (2.26) by $\Upsilon$ and integrate the result over (0, L) to derive

$$\begin{aligned} \frac{d}{dt}{\widehat{E}}(t)=-K\mathop \int \limits _{0}^{L}|\Theta ^n_{x}|^2dx-\hbar \mathop \int \limits _{0}^{L}|\Upsilon ^n_{x}|^2dx, \end{aligned}$$

(2.27)

where ${\widehat{E}}(t)$ is the energy corresponding to $U(t)-V(t)$ defined by

$$\begin{aligned}&\displaystyle \widehat{E}(t)=\frac{\rho _1\rho _2}{2\kappa }\mathop \int \limits ^L_0\Phi _{tt}^2dx+\frac{\rho _2}{2}\mathop \int \limits ^L_0\Phi _{xt}^2dx+\frac{\rho _1}{2}\mathop \int \limits ^L_0\Phi _t^2dx \\&\displaystyle +\frac{\kappa }{2}\mathop \int \limits ^L_0(\Phi _x+\Psi )^2dx+\frac{\alpha }{2}\mathop \int \limits ^L_0\Psi _x^2dx+\frac{1}{2}(c-d^2/r)\mathop \int \limits _{0}^{L}|\Theta ^n|^{2}dx \\&\displaystyle +\displaystyle \frac{1}{2} \mathop \int \limits _{0}^{L}\bigg |\frac{d}{\sqrt{r}}\Theta ^n+\sqrt{r}\Upsilon ^n\bigg |^2dx. \end{aligned}$$

Integrating (2.27) over (0, t), we get that there exists a constant $C_T>0$ such that for any $t\in [0,T]$,

$$\begin{aligned} \widehat{E}(t)\le C_T\widehat{E}(0), \end{aligned}$$

which implies the continuous dependence of stronger weak solutions on the initial data. Then we know that the stronger weak solution of problem (1.5)–(1.10) is unique. The continuous dependence and uniqueness for weak solutions can be proved by using density arguments (weak solutions are limits of stronger weak solutions). Combining the above analysis, we complete the proof of Theorem 2.2. $\square$

3 Exponential decay

In this section, we use the energy method to prove that E(t), the energy of system (1.5)–(1.10) given by (1.11), decays exponentially. For this, we assume that $(\varphi , \varphi _{t},\varphi _{tt},\psi ,\theta , P)$ is a solution of the system (1.5)–(1.10) with the regularity stated in Theorem 2.2 and we suppose that condition (1.12) holds true.

Theorem 3.1

Suppose that the hypotheses of Theorem 2.2 hold. Then there exist two positive constants M and $\eta$ such that

$$\begin{aligned} E(t)\le M E(0)e^{-\eta t}, \quad \forall t\ge 0. \end{aligned}$$

(3.1)

The proof of Theorem 3.1 will be established through several lemmas. We have the first lemma regarding the dissipative nature of the energy.

Lemma 3.2

The energy E(t) of the system (1.5)–(1.10) satisfies the energy dissipation law given by

$$\begin{aligned} \displaystyle \frac{d}{dt}E(t)=-K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx, \quad t\ge 0. \end{aligned}$$

(3.2)

Proof

Multiplying Eq. (1.5) by $\varphi _t$, (1.6) by $\psi _t$, (1.7) by $\theta$, (1.8) by P, and integrating the result over [0, L], we obtain the desired result. $\square$

We set

$$\begin{aligned} {\mathcal {F}}_{1}(t):=-\rho _{1}\mathop \int \limits _{0}^{L}\varphi _{t}\varphi dx. \end{aligned}$$

(3.3)

Lemma 3.3

Suppose that the hypotheses of Theorem 2.2 hold. Then we have

$$\begin{aligned} \frac{d}{dt}{\mathcal {F}}_{1}(t)\le -\rho _1\mathop \int \limits _{0}^{L}|\varphi _{t}|^2dx+ \kappa c_{p}\mathop \int \limits _{0}^{L}|\psi _x|^{2}\,dx+\frac{3\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _x+\psi |^{2}dx, \end{aligned}$$

(3.4)

where $c_p>0$ is the Poincaré constant.

Proof

Multiplying Eq. (1.5) by $\varphi$, integrating over [0, L] using integration by parts, and taking into account the boundary conditions (1.10), we have

$$\begin{aligned} \rho _1 \mathop \int \limits _{0}^{L}\varphi _{tt}\varphi \, dx+\kappa \mathop \int \limits _{0}^{L}(\varphi _x+\psi )\varphi _x\,dx=0. \end{aligned}$$

(3.5)

Taking into account the identity $\varphi _{tt}\varphi =\displaystyle \frac{\partial }{\partial t}(\varphi _{t}\varphi )-|\varphi _{t}|^2$ and Young’s inequality, we arrive at

$$\begin{aligned}&\displaystyle -\frac{d}{dt}\bigg (\rho _{1}\mathop \int \limits _{0}^{L}\varphi _{t}\varphi \, dx\bigg )\le -\rho _1\mathop \int \limits _{0}^{L}|\varphi _{t}|^2dx+\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _x+\psi |^{2}dx&\nonumber \\&\displaystyle +\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _x|^{2}\,dx.&\end{aligned}$$

(3.6)

Moreover, we consider the inequality given by $\mathop \int \nolimits _{0}^{L}|\varphi _{x}|^2dx\le 2\mathop \int \nolimits _{0}^{L}|\varphi _{x}+\psi |^2dx+2c_{p}\mathop \int \nolimits _{0}^{L}|\psi _x|^2dx,$ we complete the proof. $\square$

Lemma 3.4

Suppose that the hypotheses of Theorem 2.2 hold. Then we have

$$\begin{aligned}&\displaystyle \frac{d}{dt}{\mathcal {F}}_{2}(t)\le -\frac{\rho _1\rho _2}{\kappa } \mathop \int \limits _{0}^{L}|\varphi _{tt}|^{2}dx -\frac{\alpha }{2}\mathop \int \limits _{0}^{L}|\psi _{x}|^2dx+\rho _2\mathop \int \limits _{0}^{L}|\varphi _{xt}|^{2}\,dx&\nonumber \\&\displaystyle +\frac{3\kappa ^{2}c_{p}}{2\alpha }\mathop \int \limits _{0}^{L}|\varphi _x+\psi |^{2}\,dx+\frac{3\gamma _{1}^2c_p}{2\alpha }\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx +\frac{3\gamma _{2}^2c_p}{2\alpha }\mathop \int \limits _{0}^{L}|P_{x}|^2dx, \end{aligned}$$

(3.7)

where

$$\begin{aligned} {\mathcal {F}}_{2}(t):=\rho _2\mathop \int \limits _{0}^{L}\varphi _{xt}\varphi _{x}\,dx. \end{aligned}$$

(3.8)

Proof

Multiplying Eq. (1.6) by $\psi$ and integrating by parts, we obtain

$$\begin{aligned}&\displaystyle \rho _2 \mathop \int \limits _{0}^{L}\varphi _{tt}\psi _{x} dx+\alpha \mathop \int \limits _{0}^{L}|\psi _{x}|^2dx+\kappa \mathop \int \limits _{0}^{L}(\varphi _x+\psi )\psi \,dx-\gamma _{1}\mathop \int \limits _{0}^{L}\theta _{x}\psi \,dx \nonumber \\&\quad \displaystyle -\gamma _{2}\mathop \int \limits _{0}^{L}P_{x}\psi \,dx=0. \end{aligned}$$

(3.9)

It follows from Eq. (1.5) that $\psi _{x}=\displaystyle \frac{\rho _{1}}{\kappa }\varphi _{tt}-\varphi _{xx}$. Then, substituting $\psi _x$ into (3.9), we obtain

$$\begin{aligned}&\displaystyle \frac{d}{dt}\bigg (\rho _2\mathop \int \limits _{0}^{L}\varphi _{xt}\varphi _{x}\,dx\bigg )-\rho _2\mathop \int \limits _{0}^{L}|\varphi _{xt}|^{2}\,dx +\frac{\rho _1\rho _2}{\kappa } \mathop \int \limits _{0}^{L}|\varphi _{tt}|^{2}dx \\&\quad \displaystyle +\alpha \mathop \int \limits _{0}^{L}|\psi _{x}|^{2}\,dx+\kappa \mathop \int \limits _{0}^{L}(\varphi _x+\psi )\psi \,dx -\gamma _{1}\mathop \int \limits _{0}^{L}\theta _{x}\psi \,dx-\gamma _{2}\mathop \int \limits _{0}^{L}P_{x}\psi dx=0, \end{aligned}$$

and using Young’s and Poincare’s inequalities, we arrive at the desired result.

$\square$

Let us introduce one more functional which is given by

$$\begin{aligned}&\displaystyle {\mathcal {F}}_{3}(t):=\frac{\alpha \rho _{1}}{\kappa }\mathop \int \limits _{0}^{L}\psi _{x}\varphi _{t}dx-\frac{C_{1}}{\gamma _{1}}c\mathop \int \limits _{0}^{L}\theta \varphi _{t}dx-\frac{C_{1}}{\gamma _{1}}d\mathop \int \limits _{0}^{L}P\varphi _{t}dx \nonumber \\&\quad \displaystyle -\rho _{2}\mathop \int \limits _{0}^{L}\varphi _{xt}(\varphi _{x}+\psi )dx. \end{aligned}$$

(3.10)

Lemma 3.5

Suppose that the hypotheses of Theorem 2.2 the hold. Then for all $\varepsilon >0$, there are constants $C_i>0$ $(i=1,2,3)$ such that

$$\begin{aligned}&\displaystyle \frac{d}{dt}{\mathcal {F}}_{3}(t)\le -\frac{\rho _{2}}{2}\mathop \int \limits _{0}^{L}|\varphi _{xt}|^{2}dx-\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^{2}dx+C_{2}\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx \nonumber \\&\quad \displaystyle +C_{3}\mathop \int \limits _{0}^{L}|P_{x}|^2dx+\frac{C_{1}}{\gamma _{1}}(c+d)\varepsilon \mathop \int \limits _{0}^{L}|\varphi _{tt}|^{2}dx. \end{aligned}$$

(3.11)

Proof

Multiplying Eq. (1.6) by $(\varphi _{x}+\psi )$, integrating over [0, L], and using integration by parts, we have

$$\begin{aligned}&\displaystyle -\rho _{2}\mathop \int \limits _{0}^{L}\varphi _{xtt}(\varphi _{x}+\psi )\,dx+\alpha \mathop \int \limits _{0}^{L}\psi _{x}(\varphi _{x}+\psi )_{x}\,dx+\kappa \mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^{2}dx \nonumber \\&\quad \displaystyle -\gamma _{1}\mathop \int \limits _{0}^{L}\theta _{x}(\varphi _{x}+\psi )\,dx-\gamma _{2}\mathop \int \limits _{0}^{L}P_{x}(\varphi _{x}+\psi )\,dx=0. \end{aligned}$$

(3.12)

Now, using Young’s inequality, it follows that

$$\begin{aligned}&\displaystyle -\rho _{2}\mathop \int \limits _{0}^{L}\varphi _{xtt}(\varphi _{x}+\psi )\,dx+\alpha \mathop \int \limits _{0}^{L}\psi _{x}(\varphi _{x}+\psi )_{x}\,dx\le -\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^{2}dx \nonumber \\&\quad \displaystyle +\frac{\gamma _{1}^2}{\kappa }\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx+\frac{\gamma _{2}^2}{\kappa }\mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

(3.13)

On the other hand, it follows from Eq. (1.5) that $(\varphi _{x}+\psi )_{x}=\displaystyle \frac{\rho _{1}}{\kappa }\varphi _{tt}$ and then we can rewrite the above inequality as

$$\begin{aligned}&\displaystyle -\rho _{2}\mathop \int \limits _{0}^{L}\varphi _{xtt}(\varphi _{x}+\psi )\,dx+\frac{\alpha \rho _{1}}{\kappa }\mathop \int \limits _{0}^{L}\varphi _{tt}\psi _{x}\,dx\le -\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^{2}dx \\&\quad \displaystyle +\frac{\gamma _{1}^2}{\kappa }\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx+\frac{\gamma _{2}^2}{\kappa }\mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

Moreover, we consider the identity given by $\varphi _{xtt}(\varphi _{x}+\psi )=\displaystyle \frac{\partial }{\partial t}\big [\varphi _{xt}(\varphi _{x}+\psi )\big ]-\varphi _{xt}(\varphi _{x}+\psi )_{t}$ from where we obtain

$$\begin{aligned}&\displaystyle -\frac{d}{dt}\bigg (\rho _{2}\mathop \int \limits _{0}^{L}\varphi _{xt}(\varphi _{x}+\psi )\,dx+\rho _{2}\mathop \int \limits _{0}^{L}\varphi _{t}\psi _{x}\,dx\bigg )\le -\rho _{2}\mathop \int \limits _{0}^{L}|\varphi _{xt}|^{2}dx \nonumber \\&\quad \displaystyle -C_{1}\mathop \int \limits _{0}^{L}\varphi _{tt}\psi _{x}\,dx-\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^{2}dx+\frac{\gamma _{1}^2}{\kappa }\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx \nonumber \\&\quad \displaystyle +\frac{\gamma _{2}^2}{\kappa }\mathop \int \limits _{0}^{L}|P_{x}|^2dx, \end{aligned}$$

(3.14)

where $C_{1}\displaystyle :=\alpha \big (\rho _{1}/\kappa +\rho _{2}/\alpha \big )$. On the other hand, multiplying Eq. (1.7) by $C_{1}\gamma _{1}^{-1}\varphi _{t}$, we have

$$\begin{aligned} \frac{C_{1}}{\gamma _{1}}c\mathop \int \limits _{0}^{L}\theta _{t}\varphi _{t}dx+\frac{C_{1}}{\gamma _{1}}d\mathop \int \limits _{0}^{L}P_{t}\varphi _{t}dx+\frac{C_{1}}{\gamma _{1}}K\mathop \int \limits _{0}^{L}\theta _{x}\varphi _{xt}dx-C_{1}\mathop \int \limits _{0}^{L}\psi _{xt}\varphi _{t}dx=0. \end{aligned}$$

Taking into account the identities $\displaystyle \theta _{t}\varphi _{t}=\frac{\partial }{\partial t}(\theta \varphi _{t})-\theta \varphi _{tt}$, $\displaystyle P_{t}\varphi _{t}=\frac{\partial }{\partial t}(P\varphi _{t})-P\varphi _{tt}$ and $\displaystyle \psi _{xt}\varphi _{t}=\frac{\partial }{\partial t}(\psi _{x}\varphi _{t})-\psi _{x}\varphi _{tt}$, we have

$$\begin{aligned}&\displaystyle \frac{d}{dt}\bigg (C_{1}\mathop \int \limits _{0}^{L}\psi _{x}\varphi _{t}dx-\frac{C_{1}}{\gamma _{1}}c\mathop \int \limits _{0}^{L}\theta \varphi _{t}dx-\frac{C_{1}}{\gamma _{1}}d\mathop \int \limits _{0}^{L}P\varphi _{t}dx\bigg ) \nonumber \\&\quad = \displaystyle -\frac{C_{1}}{\gamma _{1}}c\mathop \int \limits _{0}^{L}\theta \varphi _{tt}dx-\frac{C_{1}}{\gamma _{1}}d\mathop \int \limits _{0}^{L}P\varphi _{tt}dx+\frac{C_{1}}{\gamma _{1}}K\mathop \int \limits _{0}^{L}\theta _{x}\varphi _{xt}dx \nonumber \\&\qquad \displaystyle +\,C_{1}\mathop \int \limits _{0}^{L}\psi _{x}\varphi _{tt}dx. \end{aligned}$$

(3.15)

Adding (3.14) and (3.15), we obtain

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}{\mathcal {F}}_{3}(t)\le -\rho _{2}\mathop \int \limits _{0}^{L}|\varphi _{xt}|^{2}dx-\frac{\kappa }{2}\mathop \int \limits _{0}^{L}|\varphi _{x}+\psi |^{2}dx+\frac{\gamma _{1}^2}{\kappa }\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx \\ {}&\quad \displaystyle +\frac{\gamma _{2}^2}{\kappa }\mathop \int \limits _{0}^{L}|P_{x}|^2dx-\frac{C_{1}}{\gamma _{1}}c\mathop \int \limits _{0}^{L}\theta \varphi _{tt}dx-\frac{C_{1}}{\gamma _{1}}d\mathop \int \limits _{0}^{L}P\varphi _{tt}dx +\frac{C_{1}}{\gamma _{1}}K\mathop \int \limits _{0}^{L}\theta _{x}\varphi _{xt}dx \end{aligned} \end{aligned}$$

and using again Young’s inequality, we arrive at the desired result. $\square$

Lemma 3.6

Suppose that the hypotheses of Theorem 2.2 hold. Then there are constants $\lambda , \, N_{0}, \, \delta >0$ such that

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )\le -\frac{1}{2}\Big (c-\frac{d^2}{r}\Big )\delta \mathop \int \limits _{0}^{L}|\theta |^2dx \\&\quad \displaystyle -\frac{1}{2}\delta \mathop \int \limits _{0}^{L}\bigg |\frac{d}{\sqrt{r}}\theta +\sqrt{r}P\bigg |^2dx-N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx -N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

Proof

First, we define the constant $\lambda _{0}:=\max \big \{d^2c_p/Kr, \ rc_{p} / \hbar \big \}$. Then, choosing $\lambda >\lambda _{0}$ and using (3.2), we have

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )=-\lambda K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-\lambda \hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx \nonumber \\&\quad \displaystyle -N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

(3.16)

Then, using the Poincaré inequality, we get

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )\le -\frac{\lambda K}{2c_{p}}\mathop \int \limits _{0}^{L}|\theta |^2dx-\frac{\lambda \hbar }{2c_{p}}\mathop \int \limits _{0}^{L}|P|^2dx-\frac{\lambda K}{2c_{p}}\mathop \int \limits _{0}^{L}|\theta |^2dx \\&\quad \displaystyle -\frac{\lambda \hbar }{2c_{p}}\mathop \int \limits _{0}^{L}|P|^2dx-N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

Since $\frac{d^{2}\lambda K}{2rcc_{p}}\mathop \int \nolimits _{0}^{L}|\theta |^2dx>0$ and $-\frac{\lambda \hbar }{2c_{p}}\mathop \int \nolimits _{0}^{L}|P|^2dx<0$, we get the following estimate

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )\le -\big (c-\frac{d^2}{r}\big )\frac{\lambda K}{2cc_{p}}\mathop \int \limits _{0}^{L}|\theta |^2dx-\frac{\lambda K}{2c_{p}}\mathop \int \limits _{0}^{L}|\theta |^2dx \nonumber \\&\quad \displaystyle -\frac{\lambda \hbar }{2c_{p}}\mathop \int \limits _{0}^{L}|P|^2dx-N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

(3.17)

Next, we add the term $\frac{d^2}{2r}\mathop \int \nolimits _{0}^{L}|\theta |^2dx>0$

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )\le -N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx \nonumber \\&\quad \displaystyle -\frac{1}{2}\bigg (\frac{\lambda \hbar }{rc_{p}}-1\bigg )r \mathop \int \limits _{0}^{L}|P|^2dx-\frac{1}{2}\bigg (\frac{d^2}{r}\mathop \int \limits _{0}^{L}|\theta |^2dx+r \mathop \int \limits _{0}^{L}|P|^2dx\bigg ) \nonumber \\&\quad \displaystyle -\big (c-\frac{d^2}{r}\big )\frac{\lambda K}{2cc_{p}}\mathop \int \limits _{0}^{L}|\theta |^2dx-\frac{1}{2}\bigg (\frac{\lambda Kr}{d^2c_{p}}-1\bigg )\frac{d^2}{r}\mathop \int \limits _{0}^{L}|\theta |^2dx. \end{aligned}$$

(3.18)

Since $\lambda >\lambda _{0}=\max \big \{d^2c_p/Kr, \ rc_{p} / \hbar \big \}$, we have $\zeta _{1}:=\frac{\lambda Kr}{d^2c_{p}}-1>0 \quad \text{ and } \quad \zeta _{2}:=\frac{\lambda \hbar }{rc_{p}}-1>0.$ Using Young’s inequality, we have

$$\begin{aligned} 2d\mathop \int \limits _{0}^{L}\theta Pdx\le \frac{d^2}{r}\mathop \int \limits _{0}^{L}|\theta |^2dx+r \mathop \int \limits _{0}^{L}|P|^2dx. \end{aligned}$$

(3.19)

Replacing (3.19) in (3.18), we have

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )\le -\big (c-d^2/r\big )\frac{\lambda K}{2cc_{p}}\mathop \int \limits _{0}^{L}|\theta |^2dx-\frac{\zeta _{1}}{2}\frac{d^2}{r}\mathop \int \limits _{0}^{L}|\theta |^2dx \\&\quad \displaystyle -\frac{\zeta _{2}}{2}r \mathop \int \limits _{0}^{L}|P|^2dx-d\mathop \int \limits _{0}^{L}\theta Pdx-N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

Therefore,

$$\begin{aligned}&\displaystyle \frac{d}{dt}\Big (\lambda E(t)+N_{0}E(t)\Big )\le -\frac{1}{2}\big (c-\frac{d^2}{r}\big )\delta \mathop \int \limits _{0}^{L}|\theta |^2dx \\&\quad \displaystyle -\frac{1}{2}\delta \mathop \int \limits _{0}^{L}\bigg |\frac{d}{\sqrt{r}}\theta +\sqrt{r}P\bigg |^2dx-N_{0}K\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx-N_{0}\hbar \mathop \int \limits _{0}^{L}|P_{x}|^2dx, \ \ \forall \, t\ge 0, \end{aligned}$$

where $\displaystyle \delta :=\min \{1, \ \lambda K/cc_{p}, \,\zeta _{1}, \, \zeta _{2}\}$. $\square$

Now we are ready to prove the main result of this paper.

Proof of Theorem 3.1

We consider the following Lyapunov functional defined by

$$\begin{aligned} {\mathcal {L}}(t):=(\lambda +N_{0})E(t)+{\mathcal {F}}_{1}(t)+N_{2}{\mathcal {F}}_{2}(t)+N_{3}{\mathcal {F}}_{3}(t), \end{aligned}$$

(3.20)

where $\lambda >\lambda _{0}:=\max \big \{d^2c_p/Kr, \ rc_{p} / \hbar \big \}$ and $N_{i},\ i=0,2,3$, are positive constants to be fixed later. Moreover, the coefficients $N_0$ and $\lambda$ will be chosen large enough such that ${\mathcal {L}}(t)$ and E(t) are equivalent. Indeed, from Young’s and Poincaré’s inequalities, we infer that there exists a constant $0<c<N_0+\lambda$ such that

$$\begin{aligned} |{\mathcal {L}}(t)-(N_0+\lambda )E(t)|\le |{\mathcal {F}}_{1}(t)|+N_{2}|{\mathcal {F}}_{2}(t)| +N_{3}|{\mathcal {F}}_{3}(t)|\le c E(t),\ \forall t\ge 0. \end{aligned}$$

Consequently,

$$\begin{aligned} (N_0+\lambda -c)E(t) \le {\mathcal {L}}(t)\le (N_0+\lambda +c)E(t),\ \forall t\ge 0.\end{aligned}$$

(3.21)

Substituting the results of Lemmas 3.3, 3.4, and 3.5 in the time derivative of ${\mathcal {L}}(t)$, we obtain after selecting $\varepsilon :=\frac{\gamma _{1}\rho _{1}\rho _{2}}{2\kappa (c+d) C_{1}N_{3}}$,

$$\begin{aligned}&\displaystyle \frac{d}{dt}{\mathcal {L}}(t)\le - \rho _{1}\mathop \int \limits _{0}^{L}|\varphi _{t}|^2dx -\Big (2N_{2}-1\Big )\frac{\rho _1\rho _2}{2\kappa }\mathop \int \limits _{0}^{L}|\varphi _{tt}|^{2}dx \\&\quad \displaystyle -\Big (N_{3}-2N_{2} \Big )\frac{\rho _2}{2}\mathop \int \limits _{0}^{L}|\varphi _{xt}|^2dx-\bigg (N_{2}-\frac{2c_{p}\kappa }{\alpha } \bigg )\frac{\alpha }{2}\mathop \int \limits _{0}^{L}|\psi _{x}|^2dx \\&\quad \displaystyle -\bigg (N_{3}-3-\frac{3\kappa c_{p}}{\alpha }N_{2}\bigg )\frac{\kappa }{2}\mathop \int \limits _{0}^{L}| \varphi _{x}+\psi |^2dx-\frac{1}{2}\Big (c-d^2/r\Big )\delta \mathop \int \limits _{0}^{L}|\theta |^2dx \\&\quad \displaystyle -\frac{1}{2}\delta \mathop \int \limits _{0}^{L}\bigg |\frac{d}{\sqrt{r}}\theta +\sqrt{r}P\bigg |^2dx-\bigg (KN_{0}-\frac{3\gamma _{1}^{2}c_{p}}{2\alpha }N_{2}-C_{2}N_{3}\bigg )\mathop \int \limits _{0}^{L}|\theta _{x}|^2dx \\&\quad \displaystyle -\bigg (\hbar N_{0}-\frac{3\gamma _{2}^{2}c_{p}}{2\alpha }N_{2}-C_{3}N_{3}\bigg )\mathop \int \limits _{0}^{L}|P_{x}|^2dx. \end{aligned}$$

By choosing $N_{2}>\max \Big \{1/2, \, 2c_{p}\kappa /\alpha \Big \}$, $N_{3}>\max \Big \{2N_{2}, \, 3+3\kappa c_{p}N_{2}/\alpha \Big \}$ and $N_{0}$ large enough such that $N_{0}>\max \Big \{\frac{3\gamma _{1}^{2}c_{p}}{2K\alpha }N_{2}+\frac{C_{2}}{K}N_{3}, \, \frac{3\gamma _{2}^{2}c_{p}}{2\hbar \alpha }N_{2}+\frac{C_{3}}{\hbar }N_{3}\Big \},$ one can obtain that $\xi _{1}:=2N_{2}-1>0$, $\xi _{2}:=N_{3}-2N_{2}>0$, $\xi _{3}:=N_{2}-\frac{2c_{p}\kappa }{\alpha }>0$, $\xi _{4}:=N_{3}-3-\frac{3\kappa c_{p}}{\alpha }N_{2}>0$, $\xi _{5}:=KN_{1}-\frac{3\gamma _{1}^{2}c_{p}}{2\alpha }N_{2}-C_{2}N_{3}>0$, $\xi _{6}:=\hbar N_{1}-\frac{3\gamma _{2}^{2}c_{p}}{2\alpha }N_{2}-C_{3}N_{3}>0$.

Now, we can conclude that there exists a positive constant $\omega :=\min \Big \{2,$ $\, \xi _{1}, \, \xi _{2}, \, \xi _{3}, \, \xi _{4}, \, \delta \Big \}>0$ such that

$$\begin{aligned} \frac{d}{dt}{\mathcal {L}}(t)\le -\omega E(t), \ \ \forall t\ge 0.\end{aligned}$$

(3.22)

Combining (3.21) with (3.22), we obtain

$$\begin{aligned} \frac{d }{dt}{\mathcal {L}}(t)\le -\frac{\omega }{N_0+\lambda +c}{\mathcal {L}}(t),\qquad \forall t \ge 0.\end{aligned}$$

(3.23)

From this and using (3.21) again, we have after integrating over (0, t),

$$\begin{aligned} {{E}}(t)\le \frac{N_0+\lambda +c}{N_0+\lambda -c} {{E}}(0)e^{-\frac{\beta t}{N_0+\lambda +c}},\qquad \forall t \ge 0\end{aligned}$$

(3.24)

which gives (3.1) with $M:=\frac{N_0+\lambda +c}{N_0+\lambda -c}$ and $\eta :=\frac{\beta }{N_0+\lambda +c}$. This completes the proof.$\square$