Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements

Abstract

In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary growth at infinity of the relaxation function.

1 Introduction

In this paper, we consider a Bresse system in one-dimensional open bounded interval subjected to homogeneous Dirichlet–Neumann–Neumann boundary conditions and with the presence of one infinite memory acting on the shear angle equation. Precisely, we are concerned with the following problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-k_1 (\varphi _{x}+\psi +lw)_{x} -lk_3 (w_x -l\varphi )=0,\\ \rho _2 \psi _{tt}-k_2 \psi _{xx}+k_1 (\varphi _{x}+\psi +lw)+\displaystyle \int _{0}^{+\infty }g (s)\psi _{xx} (x,t-s)\,\mathrm{d}s =0,\\ \rho _1 w_{tt}-k_3 (w_{x} -l\varphi )_x +lk_1 (\varphi _{x}+\psi +lw)=0,\\ \varphi (0,t)=\psi _x (0,t)=w_x (0,t)=\varphi (L,t)=\psi _x (L,t)=w_x (L,t)=0,\\ \varphi (x ,0)=\varphi _{0} (x),\,\varphi _{t}(x ,0)=\varphi _{1} (x),\\ \psi (x ,-t)=\psi _{0} (x,t),\,\psi _{t}(x ,0)=\psi _{1} (x),\\ w (x,0)=w_0 (x),\,w_t (x,0)=w_1 (x), \end{array} \right. \end{aligned}$$

(1.1)

where $(x,t)\in ]0,L[\times \mathbb {R}_{+}$, $g : \mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is a given function, and $L,\,l,\,\rho _i ,\,i=1,2,$ and $k_j,\,j=1,2,3,$ are positive constants. The integral term in system (1.1) represents the infinite memory, and the state (unknown) is

$$\begin{aligned} (\varphi ,\psi ,w):]0,L[\times ]0,+\infty [ \rightarrow \mathbb {R}^3 . \end{aligned}$$

Our objective is to establish the well posedness and the asymptotic stability of this problem in terms of the growth of g at infinity and the speeds of wave propagations given by

$$\begin{aligned} s_1 ={\sqrt{{\frac{k_1}{\rho _1}}}},\quad s_2 ={\sqrt{{\frac{k_2}{\rho _2}}}}\quad \hbox {and}\quad s_3 ={\sqrt{{\frac{k_3}{\rho _1}}}}. \end{aligned}$$

(1.2)

The Bresse system is known as the circular arch problem and is given by the following equations:

$$\begin{aligned} \rho _{1}\varphi _{tt}=Q_{x}+lN+F_{1} ,\quad \rho _{2}\psi _{tt}=M_{x}-Q+F_{2} ,\quad \rho _{1}w_{tt}=N_{x}-lQ+F_{3}, \end{aligned}$$

with

$$\begin{aligned} N=k_{0}(w_{x}-l\varphi ),\quad Q=k(\varphi _{x}+lw+\psi )\quad \hbox {and} \quad M=b\psi _{x}, \end{aligned}$$

where $\rho _{1},\,\rho _{2},\,l,\,k,\,k_{0}$ and b are positive physical constants, $N,\,Q$ and M denote, respectively, the axial force, the shear force and the bending moment, and $w,\,\varphi$ and $\psi$ represent, respectively, the longitudinal, vertical and shear angle displacements. Here,

$$\begin{aligned} \rho _{1}=\rho A,\quad \rho _{2}=\rho I,\quad k_{0}=EA,\quad k=k^{\prime }GA,\quad b=EI\quad \hbox {and}\quad l=R^{-1}, \end{aligned}$$

such that $\rho$, E, G, $k^{\prime }$, A, I and R are positive constants and denote, respectively, the density, the modulus of elasticity, the shear modulus, the shear factor, the cross-sectional area, the second moment of area of the cross-section and the radius of curvature. Finally, $F_{1},F_{2}$ and $F_{3}$ are the external forces defined in $]0,L[\times ]0,+\infty [$. For more reading about this matter, we refer to Lagnese et al. [18, 19]. It is worth noting that the system considered by Bresse [3] is obtained by taking

$$\begin{aligned} (F_{1},F_{2},F_{3})=(0,-\gamma \psi _{t},0), \end{aligned}$$

(1.3)

with $\gamma >0$.

To stabilize the Bresse system, various dampings have been employed and several decay results have been established. Alabau-Boussouira et al. [1] considered the case (1.3) and proved that the exponential stability is equivalent to

$$\begin{aligned} s_{1}=s_{2}=s_{3}. \end{aligned}$$

(1.4)

When (1.4) is not satisfied, they showed that the norm of solutions decays polynomially to zero with rates depending on the regularity of the initial data. These latter results were extended and improved in [22] by considering a locally distributed dissipation (that is, $\gamma$ in (1.3) is replaced by a non-negative function $a:]0,L[\rightarrow \mathbb {R}_{+}$ which is positive only on a part of ]0, L[). In their work, the authors of [22] obtained a better decay rate when (1.4) does not hold. The exponential stability result of [1] was also established by Soriano et al. [29] for the case of indefinite damping. That is, when $\gamma =a(x),$ where $a:]0,L[\rightarrow \mathbb {R}$ is a function with a positive average on ]0, L[ and such that

$$\begin{aligned} \left\| a-\displaystyle \int _{0}^{L}a(x)\,dx\right\| _{L^{2}(]0,L[)} \end{aligned}$$

is small enough. In such a situation, a may change sign in ]0, L[. Also, some optimal polynomial decay rates for Bresse systems for the case (1.3) were proved in [7] when (1.4) does not hold. Wehbe and Youcef [31] treated the case

$$\begin{aligned} (F_{1},F_{2},F_{3})=(0,-a_{1}(x)\psi _{t},-a_{2}(x)w_{t}), \end{aligned}$$

where $a_{i}:]0,L[\rightarrow \mathbb {R}_{+}$ are non-negative functions which can vanish on some part of ]0, L[, and proved that the exponential stability holds if and only if $s_{1}=s_{2}$. When $s_{1}\ne s_{2}$, a polynomial decay rate depending on the regularity of the initial data was obtained. This rate, in the case of classical solutions, is $t^{-\frac{1}{2}+\epsilon }$.

When only the first and second equations are controlled by means of linear frictional dampings; that is,

$$\begin{aligned} (F_{1},F_{2},F_{3})=(-\gamma _{1}\varphi _{t},-\gamma _{2}\psi _{t},0), \end{aligned}$$

with $\gamma _{i}>0$, the equivalence between the exponential stability and the equality $s_{1}=s_{3}$ was established in [2]. In addition, a polynomial stability was also shown when $s_{1}\ne s_{3}$, where the decay rate depends on the regularity of the initial data. In the particular case of classical solutions, the polynomial decay of [2] is of the rate $t^{-\frac{1}{2}}$ and it is optimal. Soufyane and Said-Houari [30] looked into the case of three frictional dampings in the whole space $\mathbb {R}$ (instead of ]0, L[) and established some polynomial stability estimates. For stabilization via nonlinear frictional dampings, we refer the readers to [4, 28].

Concerning the stabilization via heat effect, one of the earliest results concerning the asymptotic behavior of the Bresse system is due to Liu and Rao [20], where a Bresse system of the form

$$\begin{aligned} \left\{ \begin{array}{l} \rho _{1}\varphi _{tt}-k(\varphi _{x}+\psi +lw)_{x}-lk_{0}(w_{x}-l\varphi )+l\gamma \chi =0, \\ \rho _{2}\psi _{tt}-b\psi _{xx}+k(\varphi _{x}+\psi +lw)+\gamma \theta _{x}=0, \\ \rho _{1}w_{tt}-k_{0}(w_{x}-l\varphi )_{x}+lk(\varphi _{x}+\psi +lw)+\gamma \chi _{t}=0, \\ \rho _{3}\theta _{t}-\theta _{xx}+\gamma \psi _{xt}=0, \\ \rho _{3}\chi _{t}-\chi _{xx}+\gamma (w_{x}-l\varphi )_{t}=0, \end{array} \right. \end{aligned}$$

(1.5)

in a bounded interval, together with initial and boundary conditions, has been considered. In that work, Liu and Rao [20] proved that the norm of solutions decays exponentially if and only if (1.4) holds. Otherwise, the solutions decay polynomially with rates depending on the regularity of the initial data. For the classical solutions, with boundary conditions of Dirichlet–Neumann–Neumann or Dirichlet–Dirichlet–Dirichlet type, these rates are of the form $t^{-\frac{1}{4}+\epsilon }$ or $t^{-\frac{1}{8}+\epsilon },$ respectively, where $\epsilon >0$ is an arbitrary “small” constant. Other results similar to those of [20] were obtained in [8] for the Bresse system (1.5) without $\chi$. The obtained decay for classical solutions when (1.4) is not satisfied is, in general, of the rate $t^{-\frac{1}{6}+\epsilon }$; whereas the rate is $t^{-\frac{1}{3}+\epsilon }$ when $s_{1}\ne s_{2}$ and $s_{1}=s_{3}$. Najdi and Wehbe [21] extended the results of [8] to the case where the thermal dissipation is locally distributed, and improved the polynomial stability estimate to $t^{-\frac{1}{2}}$ when (1.4) is not satisfied. Recently, Keddi et al. [16] studied a thermoelastic Bresse system with Cattaneo’s thermal dissipation of the form

$$\begin{aligned} \left\{ \begin{array}{l} \rho _{1}\varphi _{tt}-k(\varphi _{x}+\psi +lw)_{x}-lk_{0}(w_{x}-l\varphi )=0, \\ \rho _{2}\psi _{tt}-b\psi _{xx}+k(\varphi _{x}+\psi +lw)+\gamma \theta _{x}=0, \\ \rho _{1}w_{tt}-k_{0}(w_{x}-l\varphi )_{x}+lk(\varphi _{x}+\psi +lw)=0, \\ \rho _{3}\theta _{t}+q_{x}+\gamma \psi _{xt}=0, \\ \tau q_{t}+\beta q+\theta _{x}=0, \end{array} \right. \end{aligned}$$

in a bounded interval, where $\varphi \, ,\psi$ and w are, respectively, the vertical, shear angle and longitudinal displacements, $\theta$ and q denote the temperature difference and the heat flux, and $\rho _{1},\,\rho _{2},\,\rho _{3},\,k,\,k_{0},\,b,\,\beta ,\,\gamma$ and $\tau$ are positive constants. Under suitable relations between the constants, the authors of [16] showed exponential and optimal polynomial decay rates. The same system was treated by Said-Houari and Hamadouche [25] in the whole space $\mathbb {R}$, where they showed that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and they proved that the decay rate of the solution in the $L^{2}$-norm is of the rate $t^{-1/12}$. For more problems of thermoelastic Bresse systems, we refer the reader to [24], where a global existence was proved using two heat equations, and to [26, 27], where Cauchy thermoelastic Bresse problems were treated.

Concerning the stability of Bresse systems via memories, there are only very few results. For instance, Guesmia and Kafini [10] discussed, without restrictions on the speeds, the stability issue for the case when the three equations are controlled via infinite memories of the form

$$\begin{aligned} F_{1}= & {} -\displaystyle \int _{0}^{+\infty }g_{1}(s)\varphi _{xx}(x,t-s)\,\mathrm{d}s,\quad F_{2}=-\displaystyle \int _{0}^{+\infty }g_{2}(s)\psi _{xx}(x,t-s)\,\mathrm{d}s, \\ F_{3}= & {} -\displaystyle \int _{0}^{+\infty }g_{3}(s)w_{xx}(x,t-s)\,\mathrm{d}s, \end{aligned}$$

where $g_{i}:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ are differentiable, non-increasing and integrable functions on $\mathbb {R}_{+}$. Their decay estimate depends only on the growth of the relaxation functions $g_{i}$ at infinity, which are allowed to have a decay rate at infinity arbitrary close to $\frac{1}{s}$. The same stability estimate of [10] was later established in [11] when only two infinite memories are considered, that is

$$\begin{aligned}&(F_{1},F_{2},F_{3})=\left( 0,-\displaystyle \int _{0}^{+\infty }g_{2}(s)\psi _{xx}(x,t-s)\,\mathrm{d}s,-\displaystyle \int _{0}^{+\infty }g_{3}(s)w_{xx}(x,t-s)\,\mathrm{d}s\right) , \end{aligned}$$

(1.6)

$$\begin{aligned}&(F_{1},F_{2},F_{3})=\left( -\displaystyle \int _{0}^{+\infty }g_{1}(s)\varphi _{xx}(x,t-s)\,\mathrm{d}s,0,-\displaystyle \int _{0}^{+\infty }g_{3}(s)w_{xx}(x,t-s)\,\mathrm{d}s\right) \end{aligned}$$

(1.7)

or

$$\begin{aligned} (F_{1},F_{2},F_{3})=\left( -\displaystyle \int _{0}^{+\infty }g_{1}(s)\varphi _{xx}(x,t-s)\,\mathrm{d}s,-\displaystyle \int _{0}^{+\infty }g_{2}(s)\psi _{xx}(x,t-s)\,\mathrm{d}s,0\right) , \end{aligned}$$

(1.8)

under the following conditions on the speeds of wave propagations:

$$\begin{aligned} s_{1}=s_{2}\,\, \hbox {in cases } (1.6) \hbox { and } (1.7),\quad s_{1}=s_{3}\,\, \hbox {in case } (1.8). \end{aligned}$$

(1.9)

When (1.9) does not hold, a weak stability estimate was given in [11], where the decay rate depends also on the smoothness of the initial data. Similar results were obtained in [15] when the memory term acts on the longitudinal displacements. However, when the memory term acts on the vertical displacements, it was proved in [14] that the system can not be exponentially stable even if the speeds of wave propagations are equal, but it is still polynomially stable.

To the best of our knowledge, the only known stability results for Bresse systems with only one infinite memory acting on the shear angle displacements are the ones obtained in [6] in case

$$\begin{aligned} (F_{1},F_{2},F_{3})=\left( 0,-\displaystyle \int _{0}^{+\infty }g(s)\psi _{xx}(x,t-s)\,\mathrm{d}s,0\right) , \end{aligned}$$

(1.10)

where $g:\mathbb {R}_+\rightarrow \mathbb {R}_{+}$ is differentiable, non-increasing and integrable function on $\mathbb {R}_{+}$. In [6], it was assumed that g satisfies, for $\alpha _{1},\alpha _{2}>0,$

$$\begin{aligned} -\alpha _{2}g(s)\le g^{\prime }(s)\le -\alpha _{1}g(s),\quad \forall s\in \mathbb {R}_{+}, \end{aligned}$$

(1.11)

and was shown that the exponential stability holds if and only if (1.4) is satisfied. Otherwise, only the polynomial stability with a decay rate of type $t^{-\frac{1}{2}}$ and its optimality were obtained. Notice that the condition (1.11) implies that g converges exponentially to zero at infinity and satisfies

$$\begin{aligned} g(0)\mathrm{e}^{-\alpha _{2}s}\le g(s)\le g(0)\mathrm{e}^{-\alpha _{1}s},\quad \forall s\in \mathbb {R}_{+}. \end{aligned}$$

(1.12)

Our goal in this work is to study the well posedness and asymptotic stability of system (1.1) in terms of the arbitrary growth at infinity of the kernel g and the speeds of wave propagations (1.2). We prove that the systems is well posed and its energy converges to zero when time goes to infinity and provide two general decay estimates: a uniform stability estimate under (1.4), and another weak stability result in general. Our results generalize those of [6] and allow a wider class of relaxation functions. See Remark 3.3 below.

The proof of the well posedness is based on the semigroup theory. For the stability estimates, we use the energy method and an approach introduced by the present authors in [12, 13].

The paper is organized as follows. In Sect. 2, we present our assumptions on the relaxation function g and state and prove the well posedness of (1.1). In Sect. 3, we present our stability results. The proof of our uniform and weak decay estimates are given, respectively, in Sects. 4 and 5.

2 Well posedness

In this section, we discuss the well posedness of (1.1) using the semigroup approach. Following the method of [5], we consider the functional

$$\begin{aligned} \eta (x,t,s)=\psi (x,t)-\psi (x,t-s)\quad \hbox {in}\,\, ]0,L[\times \mathbb {R}_{+}\times \mathbb {R}_{+}. \end{aligned}$$

(2.1)

This functional satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \eta _t +\eta _s -\psi _t =0\quad \quad &{}\hbox {in}\,\,]0,L[\times \mathbb {R}_{+}\times \mathbb {R}_{+} ,\\ \eta _x (0,t,s)=\eta _x (L,t,s)=0\quad \quad &{}\hbox {in}\,\,\mathbb {R}_{+}\times \mathbb {R}_{+},\\ \eta (x,t,0)=0\quad \quad &{}\hbox {in}\,\,]0,L[\times \mathbb {R}_{+} . \end{array} \right. \end{aligned}$$

(2.2)

Let $\eta ^0 (x,s)=\eta (x,0,s)$,

$$\begin{aligned} U^0= & {} \left( \varphi _0 , \psi _0 ,w_0 ,\varphi _1 , \psi _1 ,w_1 , \eta ^0\right) ^T , \end{aligned}$$

(2.3)

$$\begin{aligned} U= & {} (\varphi , \psi ,w,\varphi _t , \psi _t ,w_t , \eta )^T \end{aligned}$$

(2.4)

and

$$\begin{aligned} g^0 = \displaystyle \int _{0}^{+\infty } g (s)\,\mathrm{d}s. \end{aligned}$$

(2.5)

Then, the system (1.1) takes the following abstract form:

$$\begin{aligned} \left\{ \begin{array}{ll} U_t ={{\mathcal {A}}} U ,\\ U (t=0)=U^0 , \end{array} \right. \end{aligned}$$

(2.6)

where ${{\mathcal {A}}}$ is the linear operator defined by

$$\begin{aligned} {{\mathcal {A}}} U =\left( \begin{array}{c} \varphi _t \\ \psi _t \\ w_t \\ {{k_1}\bar{\rho _1}}\varphi _{xx}-{{l^2 k_3}\bar{\rho _1}}\varphi +{{k_1}\bar{\rho _1}}\psi _x+{{l}\bar{\rho _1}}(k_1 +k_3 )w_x \\ -{{k_1}\bar{\rho _2}}\varphi _{x}+{1\bar{\rho _2}}\left( k_2 -g^0\right) \psi _{xx}-{{k_1}\bar{\rho _2}}\psi -{{lk_1}\bar{\rho _2}}w+{1\bar{\rho _2}}\displaystyle \int _0^{+\infty } g \eta _{xx} \,\mathrm{d}s \\ -{{l}\bar{\rho _1}}(k_1 +k_3 )\varphi _{x}-{{lk_1}\bar{\rho _1}}\psi +{{k_3}\bar{\rho _1}}w_{xx}-{{l^2 k_1}\bar{\rho _1}}w \\ \psi _t -\eta _s \end{array} \right) . \end{aligned}$$

Let

$$\begin{aligned} L_2 =\left\{ v:\,\mathbb {R}_{+}\rightarrow H_*^1 (]0,L[),\,\displaystyle \int _0^L \displaystyle \int _0^{+\infty } gv_x^2 \,\mathrm{d}s\,\mathrm{d}x<+\infty \right\} \end{aligned}$$

(2.7)

and

$$\begin{aligned} {{\mathcal {H}}} =H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2 \times L^2 (]0,L[)\times \left( L_*^2 (]0,L[)\right) ^2 \times L_2 , \end{aligned}$$

(2.8)

where

$$\begin{aligned} L_*^2 (]0,L[) =\left\{ v\in L^2 (]0,L[),\,\int _0^L v \,\mathrm{d}x=0\right\} \end{aligned}$$

(2.9)

and

$$\begin{aligned} H^{1}_* (]0,L[) =\left\{ v\in H^{1} (]0,L[),\,\int _0^L v \,\mathrm{d}x=0\right\} . \end{aligned}$$

(2.10)

The domain $D({{\mathcal {A}}})$ of ${{\mathcal {A}}}$ is defined by

$$\begin{aligned} D({{\mathcal {A}}})=&\Bigl \{V=(v_1 ,\ldots ,v_7)^T\in {{\mathcal {H}}} ,\,\,{{\mathcal {A}}} V \in {{\mathcal {H}}} ,\,\, v_7 (0)=0,\,\,\partial _x v_2 (0)=\partial _x v_3 (0)=0, \\&\quad \partial _x v_2 (L)=\partial _x v_3 (L) =0,\,\,\partial _x v_7 (\cdot ,0)=\partial _x v_7 (\cdot ,L)=0\Bigr \};\nonumber \end{aligned}$$

(2.11)

that is, according to the definition of ${{\mathcal {H}}}$ and ${{\mathcal {A}}}$,

$$\begin{aligned}&D({{\mathcal {A}}})=\Bigl \{(v_1 ,\ldots ,v_7)^T\in {{\mathcal {H}}} ,\,(v_1 ,\ldots ,v_6)^T\in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2 \times H_0^1 (]0,L[) \times \left( H_*^1 (]0,L[)\right) ^2 , \\&v_1 ,\,v_3\in H^2 (]0,L[),\,\,\left( k_2 -g^0\right) \partial _{xx} v_2 +\int _{0}^{+\infty }g \partial _{xx} v_7 \,\mathrm{d}s\in L_*^2 (]0,L[),\,\, \partial _s v_7 \in L_2 , \\&v_7 (0)=0,\,\,\partial _x v_2 (0)=\partial _x v_3 (0)=\partial _x v_2 (L)=\partial _x v_3 (L) =0,\,\,\partial _x v_7 (\cdot ,0)=\partial _x v_7 (\cdot ,L)=0\Bigr \}. \end{aligned}$$

More generally, for $n\in \mathbb {N}$,

$$\begin{aligned} D({{\mathcal {A}}}^n)=\left\{ \begin{array}{ll} {{\mathcal {H}}}&{}\quad \hbox {if}\quad \quad \, n=0, \\ D({{\mathcal {A}}})&{}\quad \hbox {if}\quad \quad n=1, \\ \Bigl \{V\in D({{\mathcal {A}}}^{n-1}) , \, {{\mathcal {A}}} V\in D({{\mathcal {A}}}^{n-1})\Bigr \}&{}\quad \hbox {if}\quad \quad n=2,3,\ldots . \end{array} \right. \end{aligned}$$

Remark 2.1

As in [11], by integrating on ]0, L[ the second and third equations in (1.1), and using the boundary conditions, we get

$$\begin{aligned} \partial _{tt}\left( \int _{0}^{L}\psi \,\mathrm{d}x\right) +{\frac{k_1}{{\rho _{2}}}}\int _{0}^{L}\psi \,\mathrm{d}x+{\frac{lk_1}{{\rho _{2}}}}\int _{0}^{L} w\,\mathrm{d}x =0 \end{aligned}$$

(2.12)

and

$$\begin{aligned} \partial _{tt}\left( \int _{0}^{L} w\,\mathrm{d}x\right) +{\frac{l^2 k_1}{{\rho _{1}}}}\int _{0}^{L} w\,\mathrm{d}x+{\frac{lk_1}{{\rho _{1}}}}\int _{0}^{L} \psi \,\mathrm{d}x=0. \end{aligned}$$

(2.13)

Therefore, (2.12) implies that

$$\begin{aligned} \int _{0}^{L} w\,\mathrm{d}x=-{\frac{\rho _2}{lk_1}} \partial _{tt}\left( \int _{0}^{L}\psi \,\mathrm{d}x\right) -{\frac{1}{l}}\int _{0}^{L}\psi \,\mathrm{d}x. \end{aligned}$$

(2.14)

Substituting (2.14) into (2.13), we get

$$\begin{aligned} \partial _{tttt}\left( \int _{0}^{L}\psi \,\mathrm{d}x\right) +\left( {\frac{k_1}{{\rho _{2}}}}+{\frac{l^2 k_1}{{\rho _{1}}}} \right) \partial _{tt}\left( \int _{0}^{L}\psi \,\mathrm{d}x\right) =0. \end{aligned}$$

(2.15)

Let $l_0 ={\sqrt{{\frac{k_1}{{\rho _{2}}}}+{\frac{l^2 k_1}{{\rho _{1}}}}}}$. Then, solving (2.15), we find

$$\begin{aligned} \int _{0}^{L}\psi \, dx={{\tilde{c}}}_1 \cos \,(l_0 t)+{{\tilde{c}}}_2 \sin \,(l_0 t)+{{\tilde{c}}}_3 t +{{\tilde{c}}}_4 , \end{aligned}$$

(2.16)

where ${{\tilde{c}}}_1 ,\ldots ,{{\tilde{c}}}_4$ are real constants. By combining (2.14) and (2.16), we get

$$\begin{aligned} \int _{0}^{L} w\, dx={{\tilde{c}}}_1 \left( \frac{\rho _2 l_0^2 }{lk_1} -\frac{1}{l} \right) \cos \,(l_0 t) +{{\tilde{c}}}_2 \left( \frac{\rho _2 l_0^2}{lk_1}-\frac{1}{l} \right) \sin \, (l_0 t)-\frac{{{\tilde{c}}}_3}{l} t -\frac{{{\tilde{c}}}_4}{l} . \end{aligned}$$

(2.17)

Let

$$\begin{aligned} ({\tilde{\psi }_0} (x),{\tilde{w}_0} (x))=(\psi _0 (x,0),w_0 (x)). \end{aligned}$$

Using the initial data of $\psi$ and w in (1.1), we see that

$$\begin{aligned} \left\{ \begin{array}{ll} {{\tilde{c}}}_1 =\frac{k_1}{\rho _2 l_0^2 } \displaystyle \int _{0}^{L} {\tilde{\psi }_0}\, dx+\frac{lk_1}{\rho _2 l_0^2 } \displaystyle \int _{0}^{L} {\tilde{w}_0}\, dx, \\ {{\tilde{c}}}_2 =\frac{k_1}{\rho _2 l_0^3} \displaystyle \int _{0}^{L} \psi _1\, dx+\frac{lk_1}{\rho _2 l_0^3} \displaystyle \int _{0}^{L} w_1\, dx, \\ {{\tilde{c}}}_3 =\left( 1-\frac{k_1}{\rho _2 l_0^2}\right) \displaystyle \int _{0}^{L} \psi _1\, dx-\frac{lk_1}{\rho _2 l_0^2} \displaystyle \int _{0}^{L} w_1\, dx, \\ {{\tilde{c}}}_4 =\left( 1-\frac{k_1}{\rho _2 l_0^2}\right) \displaystyle \int _{0}^{L} {\tilde{\psi }_0}\, dx-\frac{lk_1}{\rho _2 l_0^2} \displaystyle \int _{0}^{L} {\tilde{w}_0}\, dx. \end{array} \right. \end{aligned}$$

Let

$$\begin{aligned} {\tilde{\psi }} =\psi -\frac{1}{L}\left( {{\tilde{c}}}_1 \cos \,(l_0 t)+{{\tilde{c}}}_2 \sin \, (l_0 t)+{{\tilde{c}}}_3 t +{{\tilde{c}}}_4\right) \end{aligned}$$

(2.18)

and

$$\begin{aligned} {\tilde{w}} =w -\frac{1}{L}\left( {{\tilde{c}}}_1 \left( \frac{\rho _2 l_0^2}{lk_1}-\frac{1}{l} \right) \cos \, (l_0 t) +{{\tilde{c}}}_2 \left( \frac{\rho _2 l_0^2}{lk_1}-\frac{1}{l} \right) \sin \, (l_0 t)-\frac{{{\tilde{c}}}_3}{l} t -\frac{{{\tilde{c}}}_4}{l}\right) . \end{aligned}$$

(2.19)

Then, from (2.16) and (2.17), one can check that

$$\begin{aligned} \displaystyle \int _{0}^{L}{\tilde{\psi }} \,\mathrm{d}x=\displaystyle \int _{0}^{L}{\tilde{w}} \,\mathrm{d}x=0, \end{aligned}$$

(2.20)

and, hence,

$$\begin{aligned} \displaystyle \int _{0}^{L}{\tilde{\eta }} \,\mathrm{d}x=0, \end{aligned}$$

(2.21)

where

$$\begin{aligned} {\tilde{\eta }} (x,t,s)={\tilde{\psi }} (x,t)-{\tilde{\psi }} (x,t-s)\quad \hbox {in}\,\, ]0,L[\times \mathbb {R}_{+}\times \mathbb {R}_{+}. \end{aligned}$$

Therefore, Poincaré’s inequality

$$\begin{aligned} \exists \, c_0 >0 :\,\,\int _{0}^{L}v^{2} \,\mathrm{d}x\le c_{0}\int _{0}^{L} v_{x}^{2} \,\mathrm{d}x,\quad \forall v\in H_{*}^{1}(]0,L[) \end{aligned}$$

(2.22)

is applicable for ${\tilde{\psi }}$, ${\tilde{w}}$ and ${\tilde{\eta }}$, provided that ${\tilde{\psi }},\,{\tilde{w}}\in H^{1} (]0,L[)$. In addition, $(\varphi ,{\tilde{\psi }}, {\tilde{w}})$ satisfies the boundary conditions and the first three equations in (1.1) with initial data

$$\begin{aligned}&\psi _0 -{\frac{1}{L}} ({\tilde{c}}_1 +{\tilde{c}}_4 ),\quad \psi _1 -{\frac{1}{L}}(l_0 {\tilde{c}}_2 +{\tilde{c}}_3 ), \\&w_0 -\frac{1}{L}\left( {{\tilde{c}}}_1 \left( \frac{\rho _2 l_0^2 }{lk_1} -\frac{1}{l} \right) -\frac{{{\tilde{c}}}_4}{l}\right) \quad \hbox {and}\quad w_1 -{\frac{1}{L}}\left( {{\tilde{c}}}_2 l_0\left( \frac{\rho _2 l_0^2}{lk_1} -\frac{1}{l} \right) -\frac{{{\tilde{c}}}_3}{l}\right) \end{aligned}$$

instead of $\psi _0$, $\psi _1$, $w_0$ and $w_1$, respectively. In the sequel, we work with ${\tilde{\psi }}$, ${\tilde{w}}$ and ${\tilde{\eta }}$ instead of $\psi$, w and $\eta$, but, for simplicity of notation, we use $\psi$, w and $\eta$ instead of ${\tilde{\psi }}$, ${\tilde{w}}$ and ${\tilde{\eta }}$, respectively.

Now, to prove the well posedness of (2.6), we make the following hypothesis:

(H1) The function $g :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is differentiable, non-increasing and integrable on $\mathbb {R}_{+}$ such that there exists a positive constant $k_0$ such that, for any

$$\begin{aligned} (\varphi ,\psi , w)^T\in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2, \end{aligned}$$

we have

$$\begin{aligned} k_0 \displaystyle \int _0^L \left( \varphi _{x}^{2} +\psi _{x}^{2}+w_{x}^{2} \right) \,\mathrm{d}x \le \displaystyle \int _0^L \left( \left( k_2 -g^0\right) \psi _{x}^{2}+k_1 (\varphi _{x}+\psi +lw)^{2}+k_3 (w_{x}-l\varphi )^{2}\right) \,\mathrm{d}x. \end{aligned}$$

(2.23)

Moreover, there exists a positive constant $\beta$ such that

$$\begin{aligned} -\beta g (s)\le g' (s),\quad \forall s\in \mathbb {R}_{+} . \end{aligned}$$

(2.24)

Remark 2.2

• 1. It is evident that (2.23) implies that

  $$\begin{aligned} k_0 \displaystyle \int _0^L \left( \varphi _{x}^{2} +\psi _{x}^{2}+w_{x}^{2} \right) \,\mathrm{d}x\le \displaystyle \int _0^L \left( k_2 \psi _{x}^{2}+k_1 (\varphi _{x}+\psi +lw)^{2}+k_3 (w_{x}-l\varphi )^{2}\right) \,\mathrm{d}x. \end{aligned}$$

  (2.25)

  On the other hand, thanks to (2.22) applied for $\psi$ and w, and Poincaré’s inequality

  $$\begin{aligned} \exists \,{{\tilde{c}}}_0 >0 :\,\int _{0}^{L}v^{2}\, dx\le {{\tilde{c}}}_{0}\int _{0}^{L}v_{x}^{2} \,\mathrm{d}x,\quad \forall v\in H_0^{1}(]0,L[) \end{aligned}$$

  (2.26)

  applied for $\varphi$, there exists a positive constant ${{\tilde{k}}}_0$ such that, for any

  $$\begin{aligned} (\varphi ,\psi , w)^T\in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2 , \end{aligned}$$

  we have

  $$\begin{aligned} \displaystyle \int _0^L \left( k_2 \psi _{x}^{2}+k_1 (\varphi _{x}+\psi +lw)^{2}+k_3 (w_{x}-l\varphi )^{2}\right) \,\mathrm{d}x\le {{\tilde{k}}}_0 \displaystyle \int _0^L \left( \varphi _{x}^{2} +\psi _{x}^{2}+w_{x}^{2} \right) \,\mathrm{d}x. \end{aligned}$$

  (2.27)

  Thus, from (2.25) and (2.27), we deduce that the right hand side of the inequality (2.25) defines a norm on $H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2$ equivalent to the natural norm of $\left( H^1 (]0,L[)\right) ^3$.

• 2. As in [11], we conclude from (2.23) that

  $$\begin{aligned} k_0 +g^0 -k_2 \le 0. \end{aligned}$$

  (2.28)

  Indeed, for the choice $\varphi =w=0$, (2.23) gives

  $$\begin{aligned} \left( k_0 +g^0 -k_2 \right) \displaystyle \int _0^L \psi _{x}^{2}\,\mathrm{d}x\le k_1\displaystyle \int _0^L \psi ^{2}\,\mathrm{d}x,\quad \forall \psi \in H_*^1 (]0,L[). \end{aligned}$$

  This inequality implies, for $\psi (x)= \cos \, (\lambda x)-\frac{1}{\lambda L}\sin \,(\lambda L)$ and $\lambda \in ]0,+\infty [$ (notice that $\psi \in H_*^1 (]0,L[)$),

  $$\begin{aligned} \left( k_0 +g^0 -k_2 \right) \left( L-\frac{1}{2\lambda }\sin \,(2\lambda L)\right) \le \frac{k_1}{\lambda ^2} \left( L+\frac{1}{2\lambda }\sin \,(2\lambda L)-\frac{2}{\lambda ^2 L}\sin ^2\,(\lambda L)\right) ,\quad \forall \lambda >0. \end{aligned}$$

  By letting $\lambda$ go to $+\infty$, we deduce (2.28).

According to Remark 2.2, we notice that, under the hypothesis (H1), the sets $L_2$ and ${{\mathcal {H}}}$ are Hilbert spaces equipped, respectively, with the inner products that generate the norms, for $v\in L_2$ and $V=(v_1 ,\ldots ,v_7 )^T \in {{\mathcal {H}}}$,

$$\begin{aligned} \Vert v\Vert _{L_2}^2 =\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g v_x^2 \,\mathrm{d}s\,\mathrm{d}x \end{aligned}$$

(2.29)

and

$$\begin{aligned} \Vert V \Vert _{{{\mathcal {H}}}}^2= & {} \displaystyle \int _0^L \left( \left( k_2 -g^0\right) (\partial _x v_2 )^{2}+k_1 (\partial _x v_1 +v_2 +lv_3)^{2}+k_3 (\partial _x v_3 -lv_1 )^{2}\right) \,\mathrm{d}x \\&+ \displaystyle \int _0^L \left( \rho _1 v_4^{2}+\rho _2 v_5^2 +\rho _1 v_6^2 \right) \,\mathrm{d}x+\Vert v_7\Vert _{L_2}^2 .\nonumber \end{aligned}$$

(2.30)

Now, a simple computation implies that, for any $V=(v_1 ,\ldots ,v_7 )^T\in D({{\mathcal {A}}})$,

$$\begin{aligned} \left\langle {{\mathcal {A}}} V ,V\right\rangle _{{{\mathcal {H}}}} ={1\over 2} \displaystyle \int _0^L \displaystyle \int _0^{+\infty } g^{\prime } (\partial _x v_7 )^2 \,\mathrm{d}s \,\mathrm{d}x. \end{aligned}$$

(2.31)

Since g is non-increasing, we deduce from (2.31) that

$$\begin{aligned} \left\langle {{\mathcal {A}}} V ,V\right\rangle _{{{\mathcal {H}}}}\le 0. \end{aligned}$$

(2.32)

This implies that A is dissipative. Notice that, according to (2.24) and the fact that g is non-increasing, we see that, for $v \in L_{2}$,

$$\begin{aligned} \begin{array}{lll} \left| \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty }g^{\prime } v_x^2 \,\mathrm{d}s\,\mathrm{d}x\right| &{}=&{} -\displaystyle \int _0^L \displaystyle \int _{0}^{+\infty }g^{\prime } v_x^2 \,\mathrm{d}s\,\mathrm{d}x \\ &{}\le &{} \beta \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty }g v_x^2 \,\mathrm{d}s\,\mathrm{d}x \\ &{}\le &{} \beta \Vert v\Vert _{L_2}^2 \\ &{}<&{} +\infty , \end{array} \end{aligned}$$

so the integral in the right hand side of (2.31) is well defined.

Next, we follow the proof given in [11] to prove that $Id-{{\mathcal {A}}}$ is surjective, where Id is the identity operator. Let $F=(f_1,\ldots ,f_7)^T\in {{\mathcal {H}}}$. We seek the existence of $V=(v_1,\ldots ,v_7)^T\in D({{\mathcal {A}}})$, a solution of the equation

$$\begin{aligned} (Id-{{\mathcal {A}}})V=F. \end{aligned}$$

(2.33)

The first three equations in (2.33) take the form

$$\begin{aligned} \left\{ \begin{array}{ll} v_4 =v_1 -f_1 ,\\ v_5 =v_2 -f_2 ,\\ v_6 =v_3 -f_3 . \end{array} \right. \end{aligned}$$

(2.34)

Using (2.34), the last equation in (2.33) is equivalent to

$$\begin{aligned} \partial _s v_7 +v_7 =v_2 +f_7 -f_2 . \end{aligned}$$

(2.35)

By integrating (2.35) and using the fact that $v_7 (0)=0$ (from (2.11)), we get

$$\begin{aligned} v_7 (s)=(1- \mathrm{e}^{-s})(v_2 -f_2 )+\mathrm{e}^{-s}\displaystyle \int _0^s \mathrm{e}^{\tau }f_7 (\tau )\,\mathrm{d}\tau , \end{aligned}$$

(2.36)

We see that, from (2.34), if $(v_1 ,v_2 ,v_3) \in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2$, then $(v_4 ,v_5 ,v_6) \in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2$. On the other hand, using Fubini theorem, Hölder’s inequality and noting that $f_7\in L_2$, we get

$$\begin{aligned} \begin{aligned} \int _0^L \int _0^{+\infty } g (s)&\left( \mathrm{e}^{-s} \int _0^s \mathrm{e}^{\tau }\partial _{x} f_7 (\tau )\,\mathrm{d}\tau \right) ^2 \, ds\,\mathrm{d}x\\&\le \int _{0}^{+\infty } \mathrm{e}^{-2s} g (s)\left( \int _{0}^{s}\mathrm{e}^{\tau } \,\mathrm{d}\tau \right) \int _{0}^{s} \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}s\,\mathrm{d}x\\&\le \int _0^L \int _{0}^{+\infty } \mathrm{e}^{-s} (1-\mathrm{e}^{-s})g (s)\int _{0}^{s} \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}s\,\mathrm{d}x\\&\le \int _0^L \int _{0}^{+\infty } \mathrm{e}^{-s} g (s)\int _{0}^{s} \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}s\,\mathrm{d}x\\&\le \int _0^L \int _{0}^{+\infty } \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \int _{\tau }^{+\infty } \mathrm{e}^{-s} g (s) \,\mathrm{d}s\,\mathrm{d}\tau \,\mathrm{d}x\\&\le \int _0^L \int _{0}^{+\infty } \mathrm{e}^{\tau } g (\tau )(\partial _{x} f_7 (\tau ))^2 \int _{\tau }^{+\infty } \mathrm{e}^{-s} \,\mathrm{d}s\,\mathrm{d}\tau \,\mathrm{d}x\\&\le \int _0^L \int _{0}^{+\infty } g (\tau )(\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}x\\&\le \Vert f_7 \Vert _{L_2}^2\\&< +\infty , \end{aligned} \\ \begin{array}{lll} \displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\left( \mathrm{e}^{-s} \int _0^s \mathrm{e}^{\tau }\partial _{x} f_7 (\tau )\,\mathrm{d}\tau \right) ^2 \, ds\,\mathrm{d}x&{}\le &{} \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty } \mathrm{e}^{-2s} g (s)\left( \displaystyle \int _{0}^{s}\mathrm{e}^{\tau } \,\mathrm{d}\tau \right) \displaystyle \int _{0}^{s} \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}s\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty } \mathrm{e}^{-s} (1-\mathrm{e}^{-s})g (s)\displaystyle \int _{0}^{s} \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}s\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty } \mathrm{e}^{-s} g (s)\displaystyle \int _{0}^{s} \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}s\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty } \mathrm{e}^{\tau } (\partial _{x} f_7 (\tau ))^2 \displaystyle \int _{\tau }^{+\infty } \mathrm{e}^{-s} g (s) \,\mathrm{d}s\,\mathrm{d}\tau \,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty } \mathrm{e}^{\tau } g (\tau )(\partial _{x} f_7 (\tau ))^2 \displaystyle \int _{\tau }^{+\infty } \mathrm{e}^{-s} \,\mathrm{d}s\,\mathrm{d}\tau \,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L \displaystyle \int _{0}^{+\infty } g (\tau )(\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \,\mathrm{d}x\\ &{}\le &{} \Vert f_7 \Vert _{L_2}^2\\ &{}<&{} +\infty , \end{array} \end{aligned}$$

then

$$\begin{aligned} s\mapsto \mathrm{e}^{-s} \int _0^s \mathrm{e}^{\tau } f_7 (\tau )\,\mathrm{d}\tau \in L_2 , \end{aligned}$$

and therefore, (2.36) implies that $v_7\in L_2$. Moreover, $\partial _s v_7\in L_2$ by (2.35). Therefore, to prove that (2.33) admits a solution $V\in D({{\mathcal {A}}})$, it is enough to show that

$$\begin{aligned} \partial _x v_7 (\cdot ,0) =\partial _x v_7 (\cdot ,L) =0 \end{aligned}$$

(2.37)

and $(v_1 ,v_2 ,v_3)$ exists and satisfies the required regularity and boundary conditions in $D({{\mathcal {A}}})$, that is

$$\begin{aligned}&(v_1,v_2,v_3)^T\in \left( H^2 (]0,L[)\cap H_0^1 (]0,L[)\right) \times H_*^1 (]0,L[)\times \left( H^2 (]0,L[)\cap H_*^1 (]0,L[)\right) ^2 , \end{aligned}$$

(2.38)

$$\begin{aligned}&\left( k_2 -g^0\right) \partial _{xx} v_2 +\int _{0}^{+\infty }g \partial _{xx} v_7 \,\mathrm{d}s\in L_*^2 (]0,L[) \end{aligned}$$

(2.39)

and

$$\begin{aligned} \partial _x v_2 (0)=\partial _x v_3 (0)=\partial _x v_2 (L)=\partial _x v_3 (L)=0. \end{aligned}$$

(2.40)

Let us assume that (2.37)–(2.40) hold. Multiplying the fourth, fifth and sixth equations in (2.33) by $\rho _1 {{\tilde{v}}}_1$, $\rho _2 {{\tilde{v}}}_2$ and $\rho _1{{\tilde{v}}}_3$, respectively, integrating their sum over ]0, L[, using the boundary conditions (2.37) and (2.40), and inserting (2.34) and (2.36), we get that $(v_1,v_2,v_3)$ solves the variational problem

$$\begin{aligned} a_1 \left( (v_1,v_2,v_3)^T,({{\tilde{v}}}_1 ,{{\tilde{v}}}_2 ,{{\tilde{v}}}_3)^T\right) ={{\tilde{a}}}_1 \left( ({{\tilde{v}}}_1 ,{{\tilde{v}}}_2 ,{{\tilde{v}}}_3)^T \right) , \end{aligned}$$

(2.41)

for any $({{\tilde{v}}}_1 ,{{\tilde{v}}}_2 ,{{\tilde{v}}}_3)^T \in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2$, where

$$\begin{aligned}&a_1 \left( (v_1,v_2,v_3)^T,({{\tilde{v}}}_1 ,{{\tilde{v}}}_2 ,{{\tilde{v}}}_3)^T\right) \\&\begin{array}{lll} &{}=&{}\displaystyle \int _0^L \left( k_1 (\partial _{x}v_1+v_2 +lv_3)(\partial _{x}{{\tilde{v}}}_1+{{\tilde{v}}}_2 +l{{\tilde{v}}}_3)+k_3 (\partial _{x} v_3-lv_1)(\partial _{x} {{\tilde{v}}}_3-l{{\tilde{v}}}_1)\right) \,\mathrm{d}x \\ &{}&{} +\displaystyle \int _0^L \left( \rho _1 v_1 {{\tilde{v}}}_1 +\rho _2 v_2 {{\tilde{v}}}_2 +\rho _1 v_3 {{\tilde{v}}}_3 +(k_2 -{{\tilde{g}}}^0)\partial _{x}v_2 \partial _{x}{{\tilde{v}}}_2 \right) \,\mathrm{d}x,\nonumber \end{array} \end{aligned}$$

(2.42)

${{\tilde{g}}}^0 =\displaystyle \int _0^{+\infty } \mathrm{e}^{-s}g (s)\,\mathrm{d}s$ and

$$\begin{aligned} \begin{array}{lll} {{\tilde{a}}}_1 \left( ({{\tilde{v}}}_1 ,{{\tilde{v}}}_2 ,{{\tilde{v}}}_3)^T \right) &{}=&{} \displaystyle \int _0^L \left( \rho _1 (f_1 +f_4 ){{\tilde{v}}}_1 +\rho _2 (f_2 +f_5 ) {{\tilde{v}}}_2 +\rho _1 (f_3 +f_6 ){{\tilde{v}}}_3 \right) \,\mathrm{d}x \\ &{}&{} +(g^0 -{{\tilde{g}}}^0 )\displaystyle \int _0^L\partial _{x} f_2 \partial _{x}{{\tilde{v}}}_2 \,\mathrm{d}x \\ &{}&{} -\displaystyle \int _0^L \left( \displaystyle \int _0^{+\infty } \mathrm{e}^{-s} g (s)\int _0^s \mathrm{e}^{\tau }\partial _{x} f_7 (\tau )\,\mathrm{d}\tau \, ds\right) \partial _{x}{{\tilde{v}}}_2 \,\mathrm{d}x. \end{array} \end{aligned}$$

(2.43)

We note that, as before, using again Fubini theorem, Hölder’s inequality and the fact that $f_7\in L_2$,

$$\begin{aligned} \begin{aligned} \int _0^L\left( \int _0^{+\infty } \mathrm{e}^{-s} g (s)\int _0^s\right.&\left. \mathrm{e}^{\tau }\partial _{x} f_7 (\tau )\,\mathrm{d}\tau \, ds\right) ^2 \,\mathrm{d}x\\&\le \int _0^L\left( \int _{0}^{+\infty } \mathrm{e}^{-s} g (s)\int _{0}^{s}\mathrm{e}^{\tau }\vert \partial _{x} f_7 (\tau )\vert \,\mathrm{d}\tau \,\mathrm{d}s\right) ^2\,\mathrm{d}x\\&\le \int _0^L\left( \int _{0}^{+\infty }\mathrm{e}^{\tau }\vert \partial _{x} f_7 (\tau )\vert \int _{\tau }^{+\infty } g (s)\mathrm{e}^{-s}\,\mathrm{d}s\,\mathrm{d}\tau \right) ^2\,\mathrm{d}x\\&\le \int _0^L\left( \int _{0}^{+\infty } \mathrm{e}^{\tau } g (\tau )\vert \partial _{x} f_7 (\tau )\vert \int _{\tau }^{+\infty }\mathrm{e}^{-s} \,\mathrm{d}s\,\mathrm{d}\tau \right) ^2\,\mathrm{d}x\\&\le \int _0^L\left( \int _{0}^{+\infty }g (\tau )\vert \partial _{x} f_7 (\tau )\vert \,\mathrm{d}\tau \right) ^2\,\mathrm{d}x\\&\le \int _0^L\left( \int _{0}^{+\infty }g (\tau )\,\mathrm{d}\tau \right) \left( \int _{0}^{+\infty }g (\tau ) (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \right) \,\mathrm{d}x\\&\le g^0\Vert f_7 \Vert _{L_2}^2\\&< +\infty , \end{aligned} \\ \begin{array}{lll} \displaystyle \int _0^L\left( \displaystyle \int _0^{+\infty } \mathrm{e}^{-s} g (s)\int _0^s \mathrm{e}^{\tau }\partial _{x} f_7 (\tau )\,\mathrm{d}\tau \, ds\right) ^2 \,\mathrm{d}x&{}\le &{} \displaystyle \int _0^L\left( \displaystyle \int _{0}^{+\infty } \mathrm{e}^{-s} g (s)\displaystyle \int _{0}^{s}\mathrm{e}^{\tau }\vert \partial _{x} f_7 (\tau )\vert \,\mathrm{d}\tau \,\mathrm{d}s\right) ^2\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L\left( \displaystyle \int _{0}^{+\infty }\mathrm{e}^{\tau }\vert \partial _{x} f_7 (\tau )\vert \displaystyle \int _{\tau }^{+\infty } g (s)\mathrm{e}^{-s}\,\mathrm{d}s\,\mathrm{d}\tau \right) ^2\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L\left( \displaystyle \int _{0}^{+\infty } \mathrm{e}^{\tau } g (\tau )\vert \partial _{x} f_7 (\tau )\vert \displaystyle \int _{\tau }^{+\infty }\mathrm{e}^{-s} \,\mathrm{d}s\,\mathrm{d}\tau \right) ^2\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L\left( \displaystyle \int _{0}^{+\infty }g (\tau )\vert \partial _{x} f_7 (\tau )\vert \,\mathrm{d}\tau \right) ^2\,\mathrm{d}x\\ &{}\le &{} \displaystyle \int _0^L\left( \displaystyle \int _{0}^{+\infty }g (\tau )\,\mathrm{d}\tau \right) \left( \displaystyle \int _{0}^{+\infty }g (\tau ) (\partial _{x} f_7 (\tau ))^2 \,\mathrm{d}\tau \right) \,\mathrm{d}x\\ &{}\le &{} g^0\Vert f_7 \Vert _{L_2}^2\\ &{}<&{} +\infty , \end{array} \end{aligned}$$

which implies that

$$\begin{aligned} x\mapsto \displaystyle \int _0^{+\infty } \mathrm{e}^{-s} g (s)\int _0^s \mathrm{e}^{\tau }\partial _{x} f_7 (\tau )\,\mathrm{d}\tau \, ds\in L^2 (]0,L[). \end{aligned}$$

On the other hand, ${{\tilde{g}}}^0 \le g^0 <k_2$ (by (2.28)). Then, by virtue of (2.23) and (2.27), we have $a_1$ is a bilinear, continuous and coercive form on

$$\begin{aligned} \left( H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2\right) \times \left( H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2\right) , \end{aligned}$$

and ${{\tilde{a}}}_1$ is a linear and continuous form on $H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2$. Consequently, using the Lax–Milgram theorem, we deduce that (2.41) has a unique solution

$$\begin{aligned} (v_1,v_2,v_3)^T\in H_0^1 (]0,L[)\times \left( H_*^1 (]0,L[)\right) ^2 . \end{aligned}$$

Therefore, using classical elliptic regularity arguments, we conclude that the forth, fifth and sixth equations in (2.33) are satisfied with $(v_1,v_2,v_3)^T$ satisfying (2.38) and (2.40), and, using (2.34) and (2.36), $v_7$ satisfies (2.37) and (2.39). Thus, we deduce that (2.33) admits a unique solution $V\in D({{\mathcal {A}}})$, and then $Id-{{\mathcal {A}}}$ is surjective.

The operator $-{{\mathcal {A}}}$ is then linear maximal monotone, and $D({{\mathcal {A}}})$ is dense in ${{\mathcal {H}}}$. Finally, thanks to the Hille–Yosida theorem (see [23]), we deduce from (2.32) and (2.33) that ${{\mathcal {A}}}$ generates a $C_0$-semigroup of contractions in ${{\mathcal {H}}}$. This gives the following well-posedness results of (2.6) (see [17, 23]).

Theorem 2.3

Assume that (H1) holds. For any $n\in \mathbb {N}$ and $U^0 \in D({{\mathcal {A}}}^n)$, (2.6) has a unique solution

$$\begin{aligned} U \in \bigcap _{k=0}^n C^{n-k} \left( \mathbb {R}_{+} ;D\left( {{\mathcal {A}}}^k\right) \right) . \end{aligned}$$

(2.44)

3 Stability

In this section, we study the stability of (2.6), where the obtained two (uniform and weak) decay rates of solution depend on the speeds of wave propagations (1.2) and the growth of g at infinity characterized by the following additional hypothesis:

(H2) Assume that $g(0)>0$ and there exists a non-increasing differentiable function $\xi :\,\mathbb {R}_{+} \rightarrow \mathbb {R}_{+}^*$ such that

$$\begin{aligned} g^{\prime } (s)\le -\xi (s) g (s),\quad \forall s\in \mathbb {R}_{+} . \end{aligned}$$

(3.1)

We start by considering the case where the speeds of wave propagations (1.2) satisfy (1.4).

Theorem 3.1

Assume that (H1), (H2) and (1.4) are satisfied such that

$$\begin{aligned} l\,\,\hbox {is small enough}. \end{aligned}$$

(3.2)

Let $U^0 \in {{\mathcal {H}}}$ be such that

$$\begin{aligned} \xi \equiv \,\,\hbox {constant}\quad \hbox {or}\quad \sup _{s\in \mathbb {R}_+} \displaystyle \int _0^{L} \left( \eta _x^0 (x,s)\right) ^2 \,\mathrm{d}x<+\infty . \end{aligned}$$

(3.3)

Then, there exist constants $\beta _0\in ]0,1]$ and $\alpha _1 >0$ such that, for all $\alpha _0\in ]0,\beta _0[$, the solution of (2.6) satisfies

$$\begin{aligned} \Vert U (t)\Vert _{{{\mathcal {H}}}}^2\le \alpha _1 \left( 1+\displaystyle \int _{0}^{t}(g(s))^{1-\alpha _0}\,\mathrm{d}s\right) \mathrm{e}^{-\alpha _0 \displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}+\alpha _1 \displaystyle \int _{t}^{+\infty } g(s)\,\mathrm{d}s,\quad \forall t\in \mathbb {R}_+ . \end{aligned}$$

(3.4)

When (1.4) does not hold, we prove the following weaker stability result for (2.6).

Theorem 3.2

Assume that (H1), (H2) and (3.2) are satisfied. Let $U^0 \in D({\mathcal {A}})$ be such that

$$\begin{aligned} \xi \equiv \,\,\hbox {constant}\quad \hbox {or}\quad \sup _{s\in \mathbb {R}_+}\max _{k=0,1}\displaystyle \int _0^{L} \left( \partial _s^k\eta _x^{0} (x,s)\right) ^2 \,\mathrm{d}x <+\infty \end{aligned}$$

(3.5)

and

$$\begin{aligned} s_1 =s_3 . \end{aligned}$$

(3.6)

Then, there exists a positive constant $\alpha _1$ such that

$$\begin{aligned} \Vert U (t)\Vert _{{{\mathcal {H}}}}^2\le \frac{\alpha _1 \left( 1+\displaystyle \int _{0}^{t}\xi (s)\displaystyle \int _{s}^{+\infty } g(\tau )\,\mathrm{d}\tau \,\mathrm{d}s\right) }{\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s},\quad \forall t> 0. \end{aligned}$$

(3.7)

Remark 3.3

• 1. If (3.1) holds with $\xi \equiv \,\,\hbox {constant}$, then (3.4) and (3.7) give, respectively, for some positive constants $d_1$ and $d_2$,

  $$\begin{aligned} \Vert U (t)\Vert _{{{\mathcal {H}}}}^2\le d_1 \mathrm{e}^{-d_2 t},\quad \forall t\in \mathbb {R}_+ \end{aligned}$$

  (3.8)

  and

  $$\begin{aligned} \Vert U (t)\Vert _{{{\mathcal {H}}}}^2\le \frac{d_1}{t},\quad \forall t>0 . \end{aligned}$$

  (3.9)

  Therefore, this particular case includes the results of [6]. The estimates (3.8) and (3.9) give the best decay rates which can be obtained from (3.4) and (3.7), respectively.

• 2. When $\xi \equiv$ constant, condition (3.1) implies that g converges exponentially to zero at infinity. However, when $\xi \ne$ constant, condition (3.1) allows $s\mapsto g(s)$ to have a decay rate arbitrarily close to $\frac{1}{s}$ at infinity, which represents the critical limit, since g is integrable on $\mathbb {R}_+$. To illustrate our general stability estimates, we give here some particular examples of g satisfying (3.1), and show the specific corresponding decay rates given by (3.4) and (3.7).

  • (i) Let $g(t)=d\mathrm{e}^{-(1+t)^{q}}$ with $0<q<1$ and $d>0$ (g converges to zero at infinity faster than any polynomial). Then, (3.1) holds with $\xi (t)=q(1+t)^{q-1}$, and consequently, (3.4) and (3.7) give, respectively, for two positive constants $c_{1}$ and $c_{2}$,

    $$\begin{aligned} E(t)\le c_{1}\mathrm{e}^{-c_{2}(1+t)^{q}},\quad \forall t\in \mathbb {R}_{+} \end{aligned}$$

    and

    $$\begin{aligned} E(t)\le c_{1}(1+t)^{-q},\quad \forall t\in \mathbb {R}_{+} . \end{aligned}$$

  • (ii) Let $g(t)=d(1+t)^{-q}$ with $q>1$ and $d>0$ (g has at most a polynomial decay at infinity). Assumption (3.1) holds with $\xi (t)=q(1+t)^{-1}$, and consequently, (3.4) and (3.7) give, respectively, for two positive constants $c_{1}$ and $c_{2}$,

    $$\begin{aligned} E(t)\le c_{1}(1+t)^{-c_2},\quad \forall t\in \mathbb {R}_{+} \end{aligned}$$

    and

    $$\begin{aligned} E(t)\le c_{1}(\ln (1+t))^{-1},\quad \forall t>0. \end{aligned}$$

To prove (3.4) and (3.7), we will consider suitable multipliers and construct appropriate Lyapunov functionals satisfying some differential inequalities, for any $U^0 \in D({{\mathcal {A}}})$ and $t\in \mathbb {R}_+$; so all the calculations are justified. By integrating these differential inequalities, we get (3.4) and (3.7), for any $U^0 \in D({{\mathcal {A}}})$. By simple density arguments ($D({{\mathcal {A}}})$ is dense in ${{\mathcal {H}}}$), (3.4) remains valid, for any $U^0 \in {{\mathcal {H}}}$.

We will use c, throughout the rest of this paper, to denote a generic positive constant which depends continuously on the initial data $U^0$ and the fixed parameters in (1.1), (2.22) and (2.26), and can be different from step to step. When c depends on some new constants $y_1$, $y_2$, $\ldots$, introduced in the proof, the constant c is noted $c_{y_1}$, $c_{y_1 ,y_2}$, $\ldots$.

Let us consider the energy functional E associated to (2.6) defined by

$$\begin{aligned} E (t)={\frac{1}{2}}\Vert U (t)\Vert _{{{\mathcal {H}}}}^{2}. \end{aligned}$$

(3.10)

From (2.6) and (2.31), we see that

$$\begin{aligned} E_i^{\prime } (t)={1\over 2} \displaystyle \int _0^L \displaystyle \int _0^{+\infty } g^{\prime } \eta _x^2 \,\mathrm{d}s \,\mathrm{d}x. \end{aligned}$$

(3.11)

Recalling that g is non-increasing, (3.11) implies that E is non-increasing, and consequently, (2.6) is dissipative.

4 Proof of uniform decay (3.4)

First, we consider the following functional:

$$\begin{aligned} I (t)=-\rho _{2}\displaystyle \int _{0}^{L}\psi _{t}\displaystyle \int _{0}^{+\infty }g(s)\eta \,\mathrm{d}s\,\mathrm{d}x. \end{aligned}$$

(4.1)

Lemma 4.1

For any $\delta _0 >0$, there exists $c_{\delta _0}>0$ such that

$$\begin{aligned} \begin{array}{lll} I^{\prime }(t) &{}\le &{}-\rho _{2} \left( g^{0}-\delta _0 \right) \displaystyle \int _{0}^{L}\psi _{t}^{2} \,\mathrm{d}x+\delta _0 \displaystyle \int _{0}^{L}\left( \psi _{x}^{2}+(\varphi _{x}+\psi +lw)^{2}\right) \,\mathrm{d}x \\ &{}&{} +c_{\delta _0 }\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }\left( g(s)-g^{\prime }(s)\right) \eta _x^{2} \,\mathrm{d}s\,\mathrm{d}x. \end{array} \end{aligned}$$

(4.2)

Proof

First, we note that

$$\begin{aligned} \begin{array}{lll} \partial _{t}\displaystyle \int _{0}^{+\infty }g(s)\eta \,\mathrm{d}s &{}=&{}\partial _{t}\displaystyle \int _{-\infty }^{t}g(t-s)(\psi (t)-\psi (s))\,\mathrm{d}s \\ &{}=&{}\displaystyle \int _{-\infty }^{t}g^{\prime }(t-s)(\psi (t)-\psi (s))\,\mathrm{d}s+\left( \displaystyle \int _{-\infty }^{t}g(t-s)\,\mathrm{d}s\right) \psi _{t} ; \end{array} \end{aligned}$$

that is

$$\begin{aligned} \partial _{t}\displaystyle \int _{0}^{+\infty }g(s)\eta \,\mathrm{d}s =\displaystyle \int _{0}^{+\infty }g^{\prime }(s)\eta \,\mathrm{d}s+g^{0} \psi _{t} . \end{aligned}$$

(4.3)

Second, using Young’s and Hölder’s inequalities, we get the following inequality: for all $\lambda >0$, there exists $c_{\lambda }>0$ such that, for any $v\in L^2 (]0,L[)$ and ${{\hat{\eta }}} \in \{\eta ,\partial _{x}\eta \}$,

$$\begin{aligned} \left| \displaystyle \int _{0}^{L} v\displaystyle \int _{0}^{+\infty }g(s) {{\hat{\eta }}} \,\mathrm{d}s\,\mathrm{d}x\right| \le \lambda \displaystyle \int _{0}^{L} v^{2} \,\mathrm{d}x+c_{\lambda }\displaystyle \int _{0}^{L} \displaystyle \int _{0}^{+\infty } g(s) {{\hat{\eta }}}^{2}\,\mathrm{d}s\,\mathrm{d}x. \end{aligned}$$

(4.4)

Similarly,

$$\begin{aligned} \left| \displaystyle \int _{0}^{L} v\displaystyle \int _{0}^{+\infty }g^{\prime } (s) {{\hat{\eta }}} \,\mathrm{d}s\,\mathrm{d}x\right| \le \lambda \displaystyle \int _{0}^{L} v^{2} \,\mathrm{d}x-c_{\lambda }\displaystyle \int _{0}^{L} \displaystyle \int _{0}^{+\infty } g^{\prime }(s) {{\hat{\eta }}}^{2}\,\mathrm{d}s\,\mathrm{d}x. \end{aligned}$$

(4.5)

Now, direct computations, using the first equation in (1.1), integrating by parts and using the boundary conditions and (4.3), yield

$$\begin{aligned} \begin{array}{lll} I^{\prime }(t) &{}=&{}-\rho _{2}g^{0} \displaystyle \int _{0}^{L}\psi _{t}^{2} \,\mathrm{d}x +\displaystyle \int _{0}^{L}\left( \displaystyle \int _{0}^{+\infty } g(s) \eta _x \,\mathrm{d}s\right) ^{2}\,\mathrm{d}x \\ &{}&{} +\left( k_{1}-g^0\right) \displaystyle \int _{0}^{L} \psi _x\displaystyle \int _{0}^{+\infty } g(s)\eta _x \,\mathrm{d}s\,\mathrm{d}x \\ &{}&{}+k_{1}\displaystyle \int _{0}^{L}(\varphi _{x}+\psi +lw)\displaystyle \int _{0}^{+\infty } g(s)\eta \, ds\, dx \\ &{}&{} -\rho _{2} \displaystyle \int _{0}^{L}\psi _{t}\displaystyle \int _{0}^{+\infty } g^{\prime }(s)\eta \,\mathrm{d}s\,\mathrm{d}x. \end{array} \end{aligned}$$

Using (4.4) and (4.5) for the last three terms of this equality, Poincaré’s inequality (2.22) for $\eta$, and Hölder’s inequality to estimate

$$\begin{aligned} \left( \displaystyle \int _{0}^{+\infty } g(s)\partial _{x}\eta \,\mathrm{d}s\right) ^{2} , \end{aligned}$$

we get (4.2). $\square$

Lemma 4.2

Let

$$\begin{aligned} \begin{array}{lll} J(t) &{}=&{} \rho _{2}\displaystyle \int _{0}^{L} (\varphi _{x}+\psi +lw)\psi _t \,\mathrm{d}x+\frac{k_2 \rho _{1}}{k_1} \displaystyle \int _{0}^{L} \psi _{x}\varphi _t \,\mathrm{d}x \\ &{}&{} -\frac{\rho _1}{k_1}\displaystyle \int _{0}^{L} \varphi _{t}\displaystyle \int _0^{+\infty }g (s)\psi _x (t-s)\,\mathrm{d}s\,\,\mathrm{d}x. \end{array} \end{aligned}$$

(4.6)

Then, for any $\delta _0 ,\,\epsilon _0 ,\,\epsilon _1 ,\,\epsilon _2 >0$, there exist $c_{\delta _0},\,c_{\epsilon _0}>0$ such that

$$\begin{aligned} \begin{array}{lll} J^{\prime }(t) &{}\le &{} -k_1\displaystyle \int _{0}^{L} (\varphi _{x}+\psi +lw)^2\,\mathrm{d}x +\left( \delta _0 +\frac{lk_2 k_3 \epsilon _1}{2k_1} +\frac{lk_3 g^0 \epsilon _2}{2k_1} \right) \displaystyle \int _{0}^{L} (w_{x}-l\varphi )^2\,\mathrm{d}x \\ &{}&{} +\delta _0\displaystyle \int _{0}^{L} \varphi _t^2\,\mathrm{d}x+\left( \frac{lk_2 k_3 }{2k_1 \epsilon _1} +\frac{lk_3 g^0}{2k_1 \epsilon _2}\right) \displaystyle \int _{0}^{L} \psi _{x}^2\,\mathrm{d}x+\displaystyle \int _{0}^{L} \left( c_{\epsilon _0 }\psi _{t}^2 + \epsilon _0 w_{t}^2 \right) \,\mathrm{d}x\\ &{}&{} +\left( \frac{k_2 \rho _1}{k_1} -\rho _2\right) \displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+c_{\delta _0} \displaystyle \int _0^L \displaystyle \int _0^{+\infty } (g (s) -g' (s))\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.7)

Proof

First, notice that

$$\begin{aligned} \begin{array}{lll} \partial _{t}\displaystyle \int _{0}^{+\infty }g(s)\psi _{x} (t-s)\,\mathrm{d}s &{}=&{}\partial _{t}\displaystyle \int _{-\infty }^{t} g(t-s)\psi _x (s)\,\mathrm{d}s \\ &{}=&{}g(0)\psi _x (t)+\displaystyle \int _{-\infty }^{t}g^{\prime }(t-s)\psi _x (s)\,\mathrm{d}s \\ &{}=&{}-\displaystyle \int _{0}^{+\infty }g^{\prime }(s)\psi _x (t)\,\mathrm{d}s+\displaystyle \int _{0}^{+\infty }g^{\prime }(s)\psi _x (t-s)\,\mathrm{d}s; \end{array} \end{aligned}$$

that is

$$\begin{aligned} \partial _{t}\displaystyle \int _{0}^{+\infty }g(s)\psi _{x} (t-s)\,\mathrm{d}s =-\displaystyle \int _{0}^{+\infty }g^{\prime }(s) \eta _x\,\mathrm{d}s. \end{aligned}$$

(4.8)

Now, by exploiting the first two equations in (1.1), integrating by parts, using (4.8) and the boundary conditions, we get

$$\begin{aligned} \begin{array}{lll} J^{\prime }(t) &{}=&{} -k_1\displaystyle \int _{0}^{L} (\varphi _{x}+\psi +lw)^2\,\mathrm{d}x+\left( \frac{k_2 \rho _1}{k_1}-\rho _2 \right) \displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+\rho _2 \displaystyle \int _{0}^{L} \psi _{t}^2\,\mathrm{d}x \\ &{}&{} +\rho _2 l\displaystyle \int _{0}^{L} \psi _{t} w_t\,\mathrm{d}x+\frac{lk_3}{k_1} \left( k_2 -g^0\right) \displaystyle \int _{0}^{L} (w_{x}-l\varphi )\psi _x \,\mathrm{d}x \\ &{}&{} +\frac{\rho _1}{k_1}\displaystyle \int _0^L \varphi _t\displaystyle \int _0^{+\infty } g' (s)\eta _x \,\mathrm{d}s \,\mathrm{d}x +\frac{lk_3}{k_1}\displaystyle \int _0^L (w_x -l\varphi )\displaystyle \int _0^{+\infty } g (s)\eta _x \,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

By applying (4.4), (4.5) and Young’s inequality for the last four terms of the above equality, we deduce (4.7).

$\square$

Lemma 4.3

Let

$$\begin{aligned} K(t) = -\rho _{1} \displaystyle \int _{0}^{L} (\varphi _{x}+\psi +lw)w_t \,\mathrm{d}x-\frac{k_3 \rho _{1}}{k_1} \displaystyle \int _{0}^{L} (w _{x} -l\varphi )\varphi _t \,\mathrm{d}x . \end{aligned}$$

(4.9)

Then, for any $\epsilon _0 >0$, there exists $c_{\epsilon _0}>0$ such that

$$\begin{aligned} \begin{array}{lll} K^{\prime }(t) &{}\le &{} lk_1 \displaystyle \int _{0}^{L} (\varphi _{x}+\psi +lw)^2\,\mathrm{d}x -\frac{l k_3^2 }{k_1}\displaystyle \int _{0}^{L} (w_{x}-l\varphi )^2\,\mathrm{d}x+c_{\epsilon _0}\displaystyle \int _{0}^{L} \psi _{t}^2 \,\mathrm{d}x \\ &{}&{} +\displaystyle \int _{0}^{L} \left( \frac{l\rho _1 k_3}{k_1}\varphi _{t}^2 +(-l\rho _1 +\epsilon _0 ) w_{t}^2\right) \,\mathrm{d}x+\rho _1\left( \frac{k_3}{k_1} -1\right) \displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.10)

Proof

Using the first and third equations in (1.1), integrating by parts, recalling (4.8) and using the boundary conditions, we find

$$\begin{aligned} \begin{array}{lll} K^{\prime }(t) &{}=&{} lk_1\displaystyle \int _{0}^{L} (\varphi _{x}+\psi +lw)^2\,\mathrm{d}x-\frac{lk_3^2}{k_1} \displaystyle \int _{0}^{L} (w_{x}-l\varphi )^2 \,\mathrm{d}x+\rho _1\left( \frac{k_3}{k_1} -1 \right) \displaystyle \int _{0}^{L} \varphi _{xt} w_{t} \,\mathrm{d}x\\ &{}&{} -l\rho _1 \displaystyle \int _{0}^{L} w_{t}^2\,\mathrm{d}x+ \frac{lk_3 \rho _1}{k_1}\displaystyle \int _{0}^{L} \varphi _{t}^2\,\mathrm{d}x-\rho _1 \displaystyle \int _{0}^{L} \psi _{t} w_t\,\mathrm{d}x . \end{array} \end{aligned}$$

By applying Young’s inequality for the last four term of the above equality, we obtain (4.10).    $\square$

Lemma 4.4

Let

$$\begin{aligned} \begin{array}{lll} P (t) &{}=&{} -\rho _{1}k_3 \displaystyle \int _{0}^{L} (w_x -l\varphi )\displaystyle \int _{0}^{x} w_t (y,t)\,\mathrm{d}y \,\mathrm{d}x \\ &{}&{} -\rho _{1} k_1 \displaystyle \int _{0}^{L} \varphi _t \displaystyle \int _{0}^{x} (\varphi _x +\psi + lw)(y,t)\,\mathrm{d}y \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.11)

Then, for any $\epsilon _0,\,\delta _1 >0$, there exists $c_{\epsilon _0}>0$ such that

$$\begin{aligned} \begin{array}{lll} P^{\prime }(t) &{}\le &{} k_1^2 \displaystyle \int _{0}^{L} (\varphi _x + \psi +lw)^2\,\mathrm{d}x-k_3^2\displaystyle \int _{0}^{L} (w_x -l \varphi )^2\,\mathrm{d}x +c_{\epsilon _0}\displaystyle \int _{0}^{L} \psi _t^2 \,\mathrm{d}x \\ &{}&{} +\left( -\rho _1 k_1 +\epsilon _0 +\frac{l\rho _1 \vert k_3 -k_1\vert \delta _1}{2}\right) \displaystyle \int _{0}^{L} \varphi _t^2\,\mathrm{d}x +\rho _1\left( k_3 +\frac{{{\tilde{c}}}_0 l\vert k_3 -k_1\vert }{2\delta _1}\right) \displaystyle \int _{0}^{L} w_t^2\,\mathrm{d}x. \end{array} \end{aligned}$$

(4.12)

Proof

By exploiting the first and third equations in (1.1), integrating by parts and using (2.20) and the boundary conditions, we get

$$\begin{aligned} \begin{array}{lll} P^{\prime }(t) &{}=&{} -\rho _1 k_3\displaystyle \int _{0}^{L} w_t^2\,\mathrm{d}x+\rho _1 k_1\displaystyle \int _{0}^{L} \varphi _t^2\,\mathrm{d}x -k_1^2 \displaystyle \int _{0}^{L} (\varphi _x + \psi +lw)^2\,\mathrm{d}x \\ &{}&{} +k_3^2\displaystyle \int _{0}^{L} (w_x -l \varphi )^2\,\mathrm{d}x +\rho _1 \displaystyle \int _{0}^{L} \varphi _t \displaystyle \int _{0}^{x} (k_1 \psi _t (y,t)+l(k_1 -k_3)w_t (y,t))\,\mathrm{d}y\,\mathrm{d}x . \end{array} \end{aligned}$$

(4.13)

Noting that the functions

$$\begin{aligned} x\mapsto \displaystyle \int _{0}^{x} \psi _t (y,t)\,\mathrm{d}y\quad \hbox {and}\quad x\mapsto \displaystyle \int _{0}^{x} w_t (y,t)\,\mathrm{d}y \end{aligned}$$

vanish at 0 and L (because of (2.20)), then, applying (2.26), we have

$$\begin{aligned} \displaystyle \int _{0}^{L} \left( \displaystyle \int _{0}^{x} \psi _t (y,t)\,\mathrm{d}y\right) ^2\,\mathrm{d}x\le {{\tilde{c}}}_0 \displaystyle \int _{0}^{L} \psi _t^2\,\mathrm{d}x \end{aligned}$$

(4.14)

and

$$\begin{aligned} \displaystyle \int _{0}^{L} \left( \displaystyle \int _{0}^{x} w_t (y,t)\,\mathrm{d}y\right) ^2\,\mathrm{d}x\le {{\tilde{c}}}_0 \displaystyle \int _{0}^{L} w_t^2\,\mathrm{d}x. \end{aligned}$$

(4.15)

By applying Young’s inequality for the last term in (4.13), and recalling (4.14) and (4.15), we conclude (4.12).    $\square$

Lemma 4.5

Let

$$\begin{aligned} R(t) =-\displaystyle \int _{0}^{L}(\rho _{1} \varphi \varphi _{t} +\rho _{2} \psi \psi _{t}+\rho _{1} w w_{t}) \,\mathrm{d}x. \end{aligned}$$

(4.16)

Then, for any $\delta _0>0$, there exists $c_{\delta _0}>0$ such that

$$\begin{aligned} \begin{array}{lll} R^{\prime }(t) &{}\le &{} \displaystyle \int _{0}^{L}\left( \left( k_2 +\delta _0 -g^0 \right) \psi _{x}^{2} +k_1 (\varphi _{x} +\psi +lw)^{2} +k_3 (w _{x} -l\varphi )^{2} \right) \,\mathrm{d}x\\ &{}&{} -\displaystyle \int _{0}^{L}\left( \rho _{1} \varphi _{t}^{2} +\rho _{2}\psi _{t}^{2} +\rho _{1} w_{t}^{2}\right) \,\mathrm{d}x+c_{\delta _0} \displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.17)

Proof

By exploiting the first three equations in (1.1), integrating by parts and using the boundary conditions, we find

$$\begin{aligned} \begin{array}{lll} R^{\prime }(t) &{}=&{} \displaystyle \int _{0}^{L}\left( \left( k_2 -g^0\right) \psi _{x}^{2} +k_1 (\varphi _{x} +\psi +lw)^{2} +k_3 (w_{x} -l\varphi )^{2} \right) \,\mathrm{d}x\\ &{}&{} -\displaystyle \int _{0}^{L}\left( \rho _{1}\varphi _{t}^{2} +\rho _{2} \psi _{t}^{2} +\rho _{1} w_{t}^{2}\right) \,\mathrm{d}x+\displaystyle \int _0^L \psi _x \displaystyle \int _0^{+\infty } g (s)\eta _x \,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

By applying (4.4) for the last term in this equality, we arrive at (4.17). $\square$

Lemma 4.6

Let

$$\begin{aligned} D(t) =-\rho _{2} \displaystyle \int _{0}^{L} \psi _{x} \displaystyle \int _{0}^{x} \psi _t (y,t)\,\mathrm{d}y \,\mathrm{d}x. \end{aligned}$$

(4.18)

Then, for any $\delta _0 ,\,\delta _2 >0$, there exists $c_{\delta _0}>0$ such that

$$\begin{aligned} \begin{array}{lll} D^{\prime }(t) &{}\le &{} \rho _2 \displaystyle \int _{0}^{L} \psi _t^2\,\mathrm{d}x+\left( \frac{k_1}{2\delta _2} +g^0 +\delta _0 -k_2\right) \displaystyle \int _{0}^{L} \psi _x^2\,\mathrm{d}x\\ &{}&{} +\frac{{{\tilde{c}}}_0 k_1 \delta _2}{2} \displaystyle \int _{0}^{L} (\varphi _x + \psi +lw)^2\,\mathrm{d}x+c_{\delta _0}\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\eta _x^2\,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.19)

Proof

By exploiting the second equation in (1.1), integrating by parts and using the boundary conditions, we find

$$\begin{aligned} \begin{array}{lll} D^{\prime }(t) &{}=&{} \rho _2 \displaystyle \int _{0}^{L} \psi _t^2\,\mathrm{d}x+\left( g^0 -k_2\right) \displaystyle \int _{0}^{L} \psi _x^2\,\mathrm{d}x-\displaystyle \int _0^L \psi _{x}\displaystyle \int _0^{+\infty } g (s)\eta _x\,\mathrm{d}s \,\mathrm{d}x\\ &{}&{} +k_1 \displaystyle \int _{0}^{L} \psi _x \displaystyle \int _{0}^{x} (\varphi _x (y,t)+ \psi (y,t)+lw(y,t))\,\mathrm{d}y\,\mathrm{d}x. \end{array} \end{aligned}$$

(4.20)

Noting that the function

$$\begin{aligned} x\mapsto \displaystyle \int _{0}^{x}(\varphi _x (y,t)+ \psi (y,t)+lw(y,t))\,\mathrm{d}y \end{aligned}$$

vanishes at 0 and L (because of (2.20)), then, applying (2.26), we have

$$\begin{aligned} \displaystyle \int _{0}^{L} \left( \displaystyle \int _{0}^{x} (\varphi _x (y,t)+ \psi (y,t)+lw(y,t))\,\mathrm{d}y\right) ^2\,\mathrm{d}x\le {{\tilde{c}}}_0\displaystyle \int _{0}^{L} (\varphi _x + \psi +lw)^2 \,\mathrm{d}x. \end{aligned}$$

(4.21)

Then, application of Young’s inequality and (4.4) for the last two terms in (4.20), and use of (4.21) yield (4.19).

$\square$

Let $N,\,N_{1} ,\,N_{2} ,\,N_{3} ,\,N_4 ,\,N_5 > 0$ and

$$\begin{aligned} F :=NE +N_{1} I +N_{2} P +N_{3} K +N_{4} R+N_{5} D +J. \end{aligned}$$

(4.22)

Then, by combining (4.2), (4.7), (4.10), (4.12), (4.17) and (4.19), we obtain

$$\begin{aligned} \begin{array}{lll} F^{\prime }(t)&{} \le &{} \displaystyle \int _{0}^{L}\left( l_1 \varphi _{t}^2 +l_2 \psi _{t}^{2} +l_3 w_t^2 +l_4 \psi _{x}^{2} +l_5 (w_{x} -l\varphi )^{2} +l_6 (\varphi _{x} +\psi +lw)^{2}\right) \,\mathrm{d}x\\ &{}&{}+NE^{\prime }(t)+c_{N_1 ,N_4 ,N_5 ,\delta _0}\displaystyle \int _0^L \displaystyle \int _0^{+\infty } \left( g (s) -g^{\prime } (s)\right) \eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\\ &{}&{}+\delta _0 c_{N_1 ,N_4 ,N_5}\displaystyle \int _{0}^{L}\left( \psi _{x}^{2} +(\varphi _{x} +\psi +lw)^{2} +(w_{x} -l\varphi )^{2} +\varphi _{t}^{2} +\psi _{t}^{2} \right) \,\mathrm{d}x \\ &{}&{}+\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+N_3 \rho _1\left( \frac{k_3}{k_1} -1\right) \displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x\\ &{}&{}+\epsilon _0 c_{N_2 ,N_3}\displaystyle \int _{0}^{L}\left( \varphi _{t}^{2} +w_{t}^{2}\right) \,\mathrm{d}x+c_{N_2 ,N_3 ,\epsilon _0}\displaystyle \int _{0}^{L} \psi _{t}^{2} \,\mathrm{d}x, \end{array} \end{aligned}$$

(4.23)

where

$$\begin{aligned} l_1= & {} -\rho _1 k_1 N_2 -\rho _1 N_4 +\frac{l\rho _1 \vert k_3 -k_1\vert \delta _1 N_2}{2} +\frac{l\rho _1 k_3 N_3}{k_1},\\ l_2= & {} -\rho _2 g^0 N_1 -\rho _2 N_4 +\rho _2 N_5 , \\ l_3= & {} -l\rho _1 N_3 -\rho _1 N_4 +\rho _1 \left( k_3 +\frac{l{{\tilde{c}}}_0 \vert k_3 -k_1\vert }{2\delta _1} \right) N_2 , \\ l_4= & {} -\left( k_2 -\frac{k_1}{2\delta _2}\right) N_5 +k_2 N_4 +\frac{lk_2 k_3 }{2k_1 \epsilon _1} +g^0 \left( N_5 -N_4 +\frac{lk_3 }{2k_1 \epsilon _2}\right) , \\ l_5= & {} -k_3^2 N_2 -\frac{lk_3^2 N_3}{k_1} +k_3 N_4 +\frac{lk_2 k_3 \epsilon _1}{2k_1} + \frac{lk_3 g^0\epsilon _2}{2k_1}\\ l_6= & {} -k_1 +k_1^2 N_2 +lk_1 N_3 +k_1 N_4 +\frac{{{\tilde{c}}}_0 k_1 \delta _2 N_5}{2} . \end{aligned}$$

Using (2.23), (2.30), (3.10) and (3.11), we get from (4.23) that

$$\begin{aligned} F^{\prime }(t)\le & {} \displaystyle \int _{0}^{L}\left( l_1 \varphi _{t}^2 +l_2 \psi _{t}^{2} +l_3 w_t^2 +l_4 \psi _{x}^{2} +l_5 (w_{x} -l\varphi )^{2} +l_6 (\varphi _{x} +\psi +lw)^{2} \right) \,\mathrm{d}x \\&\begin{array}{lll} &{}+&{}\delta _0 c_{N_1 ,N_4 ,N_5} E (t)+(N-c_{N_1 ,N_4 ,N_5 ,\delta _0})E^{\prime }(t)+c_{N_1 ,N_4 ,N_5 ,\delta _0}\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\\ &{}+&{}\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+N_3 \rho _1\left( \frac{k_3}{k_1} -1\right) \displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x\\ &{}+&{}\epsilon _0 c_{N_2 ,N_3}\displaystyle \int _{0}^{L}\left( \varphi _{t}^{2} +w_{t}^{2}\right) \,\mathrm{d}x+c_{N_2 ,N_3,\epsilon _0}\displaystyle \int _{0}^{L} \psi _{t}^{2} \,\mathrm{d}x. \end{array}\nonumber \end{aligned}$$

(4.24)

At this point, we choose carefully the constants $N,\,N_i ,\,\delta _i$ and $\epsilon _i$ to get suitable values of $l_i$.

First, let us take

$$\begin{aligned} N_{3}=\delta _{1}=1,\quad \varepsilon _{1}=\frac{k_{3}}{k_{2}},\quad \varepsilon _{2}=\frac{k_{3}}{2g^{0}},\quad \delta _{2}=\frac{k_{1}}{ k_{2}-g^{0}},\quad N_{4}=k_{3}N_{2},\quad N_{5}=4k_{3}N_{2}; \end{aligned}$$

thus, the $l_{i}$’s take the forms

$$\begin{aligned} \left\{ \begin{array}{l} l_{1}=-\rho _{1}(k_{1}+k_{3})N_{2}+l\rho _{1}\left( \frac{|k_{1}-k_{3}|}{2}N_{2}+\frac{k_{3}}{k_{1}}\right) , \\ l_{2}=-\rho _{2}(g^{0}N_{1}-3k_{3}N_{2}), \\ l_{3}=-l\rho _{1}\left( 1-\frac{{{\tilde{c}}}_0 |k_{1}-k_{3}|}{2}N_{2}\right) , \\ l_{4}=-(k_{2}-g^{0})k_{3}N_{2}+\frac{l}{k_{1}}\left( \frac{k_{2}^{2}}{2}+(g^{0})^{2}\right) , \\ l_{5}=-\frac{lk_{3}^{2}}{4k_{1}} <0, \\ l_{6}=-k_{1}\left( 1-\left( k_{1}+k_{3}+\frac{2{{\tilde{c}}}_{0}k_{1}k_{3}}{k_{2}-g^{0}}\right) N_{2}\right) +lk_{1} . \end{array} \right. \end{aligned}$$

Now, we choose $N_{2} >0$ so small that

$$\begin{aligned} 1-{{\tilde{c}}}_0 |k_{1}-k_{3}|N_{2}>0,\quad 1-\left( k_{1}+k_{3}+\frac{2{{\tilde{c}}}_{0}k_{1}k_{3}}{k_{2}-g^{0}}\right) N_{2}>0, \end{aligned}$$

then, take $\varepsilon _{0}=\frac{1}{2c_{N_{2},N_{3}}}l\rho _{1},$ so that we have

$$\begin{aligned} \left\{ \begin{array}{l} {{\tilde{l}}}_{1} =l_{1}+\varepsilon _{0}c_{N_{2},N_{3}}=-\rho _{1}(k_{1}+k_{3})N_{2}+l\rho _{1}\left( \frac{1}{2} +\frac{|k_{1}-k_{3}|}{2}N_{2}+\frac{k_{3}}{k_{1}}\right) ,\\ {{\tilde{l}}}_{2} =l_{2}+c_{N_{2},N_{3},\varepsilon _{0}} ,\\ {{\tilde{l}}}_{3} =l_{3}+\varepsilon _{0}c_{N_{2},N_{3}}=-\frac{l\rho _{1}}{2}\left( 1-{{\tilde{c}}}_0 |k_{1}-k_{3}|N_{2}\right) <0. \end{array} \right. \end{aligned}$$

Next, we recall (3.2) to select $l>0$ small enough such that

$$\begin{aligned} {{\tilde{l}}}_{1}<0,\quad l_{4}<0,\quad l_{6}<0. \end{aligned}$$

After that, we pick $N_{1} >0$ very large so that ${{\tilde{l}}}_{2}<0$. Then, we find that

$$\begin{aligned} {{\hat{l}}}:= 2\max \left\{ \frac{1}{\rho _1}{{\tilde{l}}}_{1},\frac{1}{\rho _2}{{\tilde{l}}}_{2},\frac{1}{\rho _1}{{\tilde{l}}}_{3}, \frac{1}{k_2}l_{4},\frac{1}{k_3}l_{5},\frac{1}{k_1}l_{6}\right\} <0 \end{aligned}$$

and, using (2.30) and (3.10),

$$\begin{aligned} \begin{array}{lll} &{}&{}\displaystyle \int _{0}^{L}\left( {{\tilde{l}}}_{1} \varphi _{t}^2 +{{\tilde{l}}}_{2} \psi _{t}^{2} +{{\tilde{l}}}_{3} w_t^2 +l_4 \psi _{x}^{2} +l_5 (w_{x} -l\varphi )^{2} +l_6 (\varphi _{x} +\psi +lw)^{2} \right) \,\mathrm{d}x+\delta _0 c_{N_1 ,N_4 ,N_5} E (t)\\ &{}&{}\quad \le \frac{{{\hat{l}}}}{2}\displaystyle \int _{0}^{L}\left( \rho _1\varphi _{t}^2 +\rho _2\psi _{t}^{2} +\rho _1 w_t^2 +k_2 \psi _{x}^{2} +k_3 (w_{x} -l\varphi )^{2} +k_1 (\varphi _{x} +\psi +lw)^{2} \right) \,\mathrm{d}x+\delta _0 c_{N_1 ,N_4 ,N_5} E (t)\\ &{}&{}\quad \le ({{\hat{l}}}+\delta _0 c_{N_1 ,N_4 ,N_5})E(t)+\frac{{{\hat{l}}}g^0}{2}\displaystyle \int _{0}^{L} \psi _{x}^{2} \,\mathrm{d}x-\frac{{{\hat{l}}}}{2}\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\\ &{}&{}\quad \le ({{\hat{l}}}+\delta _0 c_{N_1 ,N_4 ,N_5})E(t) -\frac{{{\hat{l}}}}{2}\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.25)

Finally, we take $\delta _{0} >0$ small enough so that

$$\begin{aligned} {{\hat{l}}} +\delta _{0} c_{N_{1},N_{2},N_{5}}<0. \end{aligned}$$

Consequently, we obtain from (4.24) and (4.25), for some positive constants $c,{{\tilde{c}}}_{1},$

$$\begin{aligned} \begin{array}{lll} F^{\prime }(t) &{}\le &{} -{{\tilde{c}}}_1 E (t)+(N-c)E^{\prime }(t)+c\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\\ &{}&{} +\left( \frac{k_2 \rho _1}{k_1} -\rho _2\right) \displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+N_3 \rho _1\left( \frac{k_3}{k_1} -1\right) \displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.26)

Now, we estimate the integral of $g \eta _x^2$ in (4.26).

Case $\xi \equiv$ constant. From (3.1), we have

$$\begin{aligned} \begin{array}{lll} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x &{} = &{} \displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }\xi g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\\ &{} \le &{} -\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }g^{\prime } (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x, \end{array} \end{aligned}$$

then, using (3.11), we find

$$\begin{aligned} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\le -2E^{\prime }(t). \end{aligned}$$

(4.27)

Case $\xi \ne$ constant. Following the arguments of [12] and [13], and using (3.1) and the fact that $\xi$ is non-increasing, we get

$$\begin{aligned} \begin{array}{lll} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{t}g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x &{} \le &{} \displaystyle \int _{0}^{L}\displaystyle \int _{0}^{t }\xi (s)g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\\ &{} \le &{} -\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{t}g^{\prime } (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x, \end{array} \end{aligned}$$

then, recalling (3.11), we obtain

$$\begin{aligned} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{t}g (s)\eta _x^2 \,\mathrm{d}s \,\mathrm{d}x\le -2E^{\prime }(t). \end{aligned}$$

(4.28)

On the other hand, the definition of E, (2.23) and the fact that E is non-increasing imply that

$$\begin{aligned} \displaystyle \int _{0}^{L} \psi _x^{2} (x,t)\,\mathrm{d}x\le cE (0). \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{array}{lll} \displaystyle \int _{0}^{L} \eta _x^2 \,\mathrm{d}x &{} = &{}\displaystyle \int _{0}^{L} \left( \eta _x^0 (x,s-t)+\psi _x (x,t)-\psi _x (x,0)\right) ^2 \,\mathrm{d}x\\ &{} \le &{} c\left( E (0)+\sup _{s\in \mathbb {R}_+} \displaystyle \int _0^{L} \left( \eta _{x}^0 (x,s)\right) ^2 \,\mathrm{d}x\right) . \end{array} \end{aligned}$$

Then, using the boundedness condition on $\eta ^0$ in (3.3), we deduce that

$$\begin{aligned} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{t}^{+\infty }g (s)\eta _x^2 \,\mathrm{d}s\,\mathrm{d}x\le c \xi (t)\displaystyle \int _{t}^{+\infty }g(s)ds. \end{aligned}$$

(4.29)

Hence, by combining (4.28) and (4.29), we find

$$\begin{aligned} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }g (s)\eta _x^2 \,\mathrm{d}s\,\mathrm{d}x\le -2E^{\prime }(t) +c \xi (t)\displaystyle \int _{t}^{+\infty }g(s)ds. \end{aligned}$$

(4.30)

Finally, multiplying (4.26) by $\xi (t)$ and combining with (4.27) and (4.30), we get for the two previous cases, for some ${{\tilde{c}}}_2 >0$,

$$\begin{aligned} \begin{array}{lll} \xi (t)F^{\prime }(t) &{} \le &{} -{{\tilde{c}}}_1\xi (t)E(t)+c\xi (t)\displaystyle \int _{t}^{+\infty }g(s)ds+(N-c)\xi (t)E^{\prime }(t)-{{\tilde{c}}}_2 E^{\prime }(t)\\ &{} &{} +\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \xi (t)\displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+N_3 \rho _1\left( \frac{k_3}{k_1} -1\right) \xi (t)\displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x. \end{array} \end{aligned}$$

(4.31)

On the other hand, from (2.23), (2.30) and (3.10), we deduce that there exists a positive constant $\gamma$ (independent of N) satisfying

$$\begin{aligned} \left| N_{1} I +N_{2} P +N_{3} K +N_{4} R+N_{5} D +J\right| \le \gamma E , \end{aligned}$$

which, combined with (4.22), implies that

$$\begin{aligned} (N -\gamma ) E \le F\le (N +\gamma )E. \end{aligned}$$

(4.32)

Choosing N so that

$$\begin{aligned} N\ge c \quad \hbox {and}\quad N>\gamma , \end{aligned}$$

noting that $E^{\prime } \le 0$ and using (4.31) and (4.32), we deduce that $F \sim E$ and

$$\begin{aligned} \begin{array}{lll} {{\tilde{F}}}^{\prime }(t) &{} \le &{} -{{\tilde{c}}}_1 \xi (t)E (t)+ch(t)+\xi ^{\prime }(t)F (t) \\ &{} &{} +\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \xi (t)\displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+N_3 \rho _1\left( \frac{k_3}{k_1} -1\right) \xi (t)\displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x, \end{array} \end{aligned}$$

(4.33)

where

$$\begin{aligned} {{\tilde{F}}} =\xi F +{{\tilde{c}}}_2 E\quad \hbox {and}\quad h(t)=\xi (t)\displaystyle \int _{t}^{+\infty }g(s)\mathrm{d}s. \end{aligned}$$

From (4.32) and the relation $0\le \xi (t)F (t)\le \xi (0) F (t)$, we see that

$$\begin{aligned} {{\tilde{c}}}_2 E \le {{\tilde{F}}}\le ({{\tilde{c}}}_2 +\xi (0)(N +\gamma ))E. \end{aligned}$$

(4.34)

Therefore, (4.33) implies that, for any $\alpha _0\in ]0,\beta _0[$, where $\beta _0 = \min \left\{ 1,\frac{{{\tilde{c}}}_1}{{{\tilde{c}}}_2 +\xi (0)(N +\gamma )}\right\}$,

$$\begin{aligned} \begin{array}{lll} {{\tilde{F}}}^{\prime }(t) &{} \le &{} -\alpha _0 \xi (t){{\tilde{F}}} (t)+ch(t) \\ &{} &{} +\left( \frac{k_2 \rho _1}{k_1} -\rho _2\right) \xi (t)\displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x+N_3 \rho _1\left( \frac{k_3}{k_1} -1\right) \xi (t)\displaystyle \int _{0}^{L} w_{t} \varphi _{xt} \,\mathrm{d}x, \end{array} \end{aligned}$$

(4.35)

Since the last two terms in (4.35) vanish (thanks to (1.4)), then (4.35) implies that

$$\begin{aligned} \partial _t \left( \mathrm{e}^{\alpha _{0} \displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s} {{\tilde{F}}} (t)\right) \le c\mathrm{e}^{\alpha _{0} \displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}h(t). \end{aligned}$$

Therefore, by integrating over [0, T] with $T\ge 0$, we get

$$\begin{aligned} {{\tilde{F}}} (T)\le \mathrm{e}^{-\alpha _{0}\displaystyle \int _{0}^{T}\xi (s)\,\mathrm{d}s}\left( {{\tilde{F}}} (0)+c\displaystyle \int _{0}^{T}\mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}h(t)\mathrm{d}t\right) , \end{aligned}$$

which implies, according to (4.34), that

$$\begin{aligned} E (T)\le c\mathrm{e}^{-\alpha _{0}\displaystyle \int _{0}^{T}\xi (s)\,\mathrm{d}s}\left( 1+\displaystyle \int _{0}^{T} \mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}h(t)\,\mathrm{d}t\right) . \end{aligned}$$

(4.36)

Since

$$\begin{aligned} \mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}h(t)={\frac{1}{{\alpha _{0}}}}\partial _t\left( \mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}\right) \displaystyle \int _{t}^{+\infty }g(s)\,\mathrm{d}s, \end{aligned}$$

then, by integration by parts, we obtain

$$\begin{aligned} \begin{aligned} \displaystyle \int _{0}^{T}&\mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}h(t)\,\mathrm{d}t\\ {}&={\frac{1}{{\alpha _{0} }}}\left( \mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{T}\xi (s)\,\mathrm{d}s}\displaystyle \int _{T}^{+\infty }g(s)\,\mathrm{d}s-\displaystyle \int _{0}^{+\infty }g(s)\,\mathrm{d}s+\displaystyle \int _{0}^{T}\mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}g(t)\,\mathrm{d}t\right) . \end{aligned} \end{aligned}$$

Consequently, combining with (4.36), we arrive at

$$\begin{aligned} \begin{array}{lll} E (T) &{} \le &{} c\left( \mathrm{e}^{-\alpha _0 \displaystyle \int _0^T \xi (s)\,\mathrm{d}s}+\displaystyle \int _T^{+\infty } g(s)\,\mathrm{d}s\right) \\ &{} &{} +c\mathrm{e}^{-\alpha _0 \displaystyle \int _0^T \xi (s)\,\mathrm{d}s}\displaystyle \int _0^{T} \mathrm{e}^{\alpha _0 \displaystyle \int _0^t \xi (s)\,\mathrm{d}s}g(t)\,\mathrm{d}t. \end{array} \end{aligned}$$

(4.37)

On the other hand, (3.1) implies that

$$\begin{aligned} \partial _t \left( \mathrm{e}^{\alpha _0 \displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}(g(t))^{\alpha _0 }\right) = \alpha _0 (g(t))^{\alpha _0 -1}(\xi (t)g(t)+g'(t))\mathrm{e}^{\alpha _0 \displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}\le 0, \end{aligned}$$

and, hence,

$$\begin{aligned} \mathrm{e}^{\alpha _0 \displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}(g(t))^{\alpha _0} \le (g(0))^{\alpha _0} . \end{aligned}$$

Therefore,

$$\begin{aligned} \displaystyle \int _{0}^{T}\mathrm{e}^{\alpha _{0}\displaystyle \int _{0}^{t}\xi (s)\,\mathrm{d}s}g(t)\,\mathrm{d}t\le (g(0))^{\alpha _{0}}\displaystyle \int _{0}^{T}(g(t))^{1-\alpha _{0}}\,\mathrm{d}t. \end{aligned}$$

(4.38)

Finally, (3.10) and (4.38) give (3.4).

5 Proof of weak decay (3.7)

In this section, we treat the case when (1.4) does not hold but (3.6) holds. In this case, the last term in (4.35) vanishes. Therefore, we need to estimate

$$\begin{aligned} \left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \xi (t)\displaystyle \int _{0}^{L} \psi _{xt} \varphi _{t} \,\mathrm{d}x \end{aligned}$$

using the following system resulting from differentiating (1.1) with respect to time t:

$$\begin{aligned} \left\{ \begin{array}{ll} \rho _1 \varphi _{ttt}-k_1 (\varphi _{xt}+\psi _t +lw_t )_{x} -lk_3 (w_{xt} -l\varphi _t )=0,\\ \rho _2 \psi _{ttt}-k_2 \psi _{xxt}+k_1 (\varphi _{xt}+\psi _t +lw_t )+\displaystyle \int _{0}^{+\infty }g (s)\psi _{xxt} (x,t-s)\,\mathrm{d}s =0,\\ \rho _1 w_{ttt}-k_3 (w_{xt} -l\varphi _t )_x +lk_1 (\varphi _{xt}+\psi _t +lw_t )=0,\\ \varphi _t (0,t)=\psi _{xt} (0,t)=w_{xt} (0,t)=\varphi _t (L,t)=\psi _{xt} (L,t)=w_{xt} (L,t)=0. \end{array} \right. \end{aligned}$$

(5.1)

System (5.1) is well posed for initial data $U^0 \in D({{\mathcal {A}}})$ thanks to Theorem 2.3, where $U_t \in C(\mathbb {R}_{+}; {{\mathcal {H}}})$. Let $U^0 \in D({{\mathcal {A}}})$ and ${{\tilde{E}}}$ be the energy of (5.1) defined by

$$\begin{aligned} {{\tilde{E}}} (t)= \frac{1}{2} \Vert U_t (t)\Vert _{{{\mathcal {H}}}}^2 . \end{aligned}$$

(5.2)

Similarly to (3.11), we have

$$\begin{aligned} {{\tilde{E}}}^{\prime } (t)={1\over 2}\displaystyle \int _0^L \displaystyle \int _0^{+\infty } g^{\prime } \eta _{xt}^2 \,\mathrm{d}s \,\mathrm{d}x\le 0; \end{aligned}$$

(5.3)

so ${{\tilde{E}}}$ is non-increasing. We use an idea introduced in [9] to get the following lemma.

Lemma 5.1

For any $\epsilon >0$, there exists $c_{\epsilon } >0$ such that

$$\begin{aligned} \left| \left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _0^L \psi _{xt} \varphi _{t} \,\mathrm{d}x\right| \le c_{\epsilon } \displaystyle \int _{0}^{L} \displaystyle \int _0^{+\infty } g (s)\eta _{xt}^2\,\mathrm{d}s\,\mathrm{d}x +\epsilon E (t) -c_{\epsilon } E^{\prime } (t). \end{aligned}$$

(5.4)

Proof

We have, by the definition of $\eta$,

$$\begin{aligned} \begin{array}{lll} \left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _0^L \psi _{xt} \varphi _{t} \,\mathrm{d}x &{}=&{} \frac{1}{g^0}\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _0^L \varphi _{t}\displaystyle \int _0^{+\infty } g (s) \eta _{xt}\,\mathrm{d}s\,\mathrm{d}x \\ &{}&{} +\frac{1}{g^0}\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _0^L \varphi _{t} \displaystyle \int _0^{+\infty } g (s)\psi _{xt} (t-s)\,\mathrm{d}s\,\mathrm{d}x. \end{array} \end{aligned}$$

(5.5)

Using (4.4) and (3.10), we get, for all $\epsilon >0$,

$$\begin{aligned} \begin{array}{lll} \left| \frac{1}{g^0}\left( \frac{k_2 \rho _1}{k_1} -\rho _2 \right) \displaystyle \int _0^L \varphi _{t}\int _0^{+\infty } g (s) \eta _{xt}\,\mathrm{d}s\,\mathrm{d}x\right| &{}\le &{} \frac{\epsilon }{2} E (t) \\ &{}&{} +c_{\epsilon } \displaystyle \int _0^L \displaystyle \int _0^{+\infty } g (s) \eta _{xt}^2\,\mathrm{d}s \,\mathrm{d}x. \end{array} \end{aligned}$$

(5.6)

On the other hand, by integrating with respect to s and using the definition of $\eta$, we obtain

$$\begin{aligned} \begin{array}{lll} \displaystyle \int _0^L \varphi _{t} \displaystyle \int _0^{+\infty } g (s)\psi _{xt} (t-s)\,\mathrm{d}s\,\mathrm{d}x &{}=&{} -\displaystyle \int _0^L \varphi _{t} \displaystyle \int _0^{+\infty } g (s)\partial _s (\psi _{x} (t-s))\,\mathrm{d}s\,\mathrm{d}x \\ &{}=&{} \displaystyle \int _0^L \varphi _{t} \left( g (0)\psi _x (t) +\displaystyle \int _0^{+\infty } g^{\prime } (s)\psi _{x} (t-s)\,\mathrm{d}s\right) \,\mathrm{d}x \\ &{}=&{} -\displaystyle \int _0^L \varphi _{t}\int _0^{+\infty } g^{\prime } (s)\eta _x\,\mathrm{d}s\,\mathrm{d}x. \end{array} \end{aligned}$$

Therefore, using (4.5) and (3.11),

$$\begin{aligned} \left| \frac{1}{g^0}\left( -\frac{k_2 \rho _1}{k_1} -\rho _2\right) \displaystyle \int _0^L \varphi _{t}\displaystyle \int _0^{+\infty } g (s)\psi _{xt} (t-s)\,\mathrm{d}s\,\mathrm{d}x\right| \le \frac{\epsilon }{2} E (t)-c_{\epsilon } E^{\prime } (t). \end{aligned}$$

(5.7)

Inserting (5.6) and (5.7) into (5.5), we obtain (5.4). $\square$

Now, using (3.6), combining (4.35) and (5.4), and choosing $\epsilon$ small enough, we find

$$\begin{aligned} \begin{array}{lll} {{\tilde{F}}}^{\prime }(t) &{} \le &{} -c\xi (t)E (t)+ch(t)-c\xi (t)E^{\prime } (t)\\ &{} &{} +c\xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }g (s)\eta _{xt}^2\,\mathrm{d}s\,\mathrm{d}x. \end{array} \end{aligned}$$

(5.8)

On the other hand, using the boundedness condition on $\eta ^0$ in (3.5), we have (as for (4.27) and (4.30))

$$\begin{aligned} \xi (t)\displaystyle \int _{0}^{L}\displaystyle \int _{0}^{+\infty }g (s)\eta _{xt}^{2} \,\mathrm{d}s\,\mathrm{d}x\le -c{\tilde{E}}^{\prime }(t)+ ch(t). \end{aligned}$$

(5.9)

Hence, combining (5.8) and (5.9), we have

$$\begin{aligned} \left( {{\tilde{F}}} (t)+c{\tilde{E}} (t)+c\xi (t)E (t)\right) ^{\prime }\le -c\xi (t)E (t)+ch(t), \end{aligned}$$

(5.10)

since $\xi$ is non-increasing. Therefore, by integrating on [0, T] and using the fact E is non-increasing, we get

$$\begin{aligned} c E (T)\displaystyle \int _{0}^{T}\xi (t)\,\mathrm{d}t\le {{\tilde{F}}}(0)+c{\tilde{E}} (0)+c\xi (0)E (0) +c\displaystyle \int _{0}^{T} h(t) \,\mathrm{d}t, \end{aligned}$$

which gives (3.7), since (3.10).

Comments.

• 1. This work generalizes the results of [6] and allows a wider class of relaxation functions.

• 2. Note that when $w=l=0$, we obtain the Timoshenko system and our results reduce to those of [12].

• 3. It would be very interesting to obtain these general decay results without conditions (3.2) and (3.3).