Strong Stability for a Viscoelastic Transmission Problem Under a Nonlocal Boundary Control

Abstract

The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui and A. Benaissa. Namely, they focused on a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with an emphasis on rough interfaces and nonlinear dynamic (possibly, nonlocal) boundary conditions along the interface, where a transmission problem in the presence of a boundary control condition of a nonlocal type was investigated in these papers. Owing to the semigroup theory, we prove the question of well-posedness. For the very rare cases, we combined between the frequency domain approach and the Borichev–Tomilov theorem to establish strong stability results.

1. Introduction

The linear theory of viscoelasticity (heredity) has seemingly been comprehensively developed over a century and a half, beginning with the works of Weber, Kohlrausch, Maxwell, Boltzmann, Volterra, and others: in the period from the 1950s to the 1970s (and later), thousands of articles, reports, and monographs were devoted to it. However, strangely enough, despite the rapid development of nonlinear theories, articles and dissertations continue to be published, demonstrating that many mathematical properties of the linear constitutive relation of viscoelasticity—even those directly related to the modeling of classical rheological effects and transmission models—are still little known, not formulated or systematized, the full arsenal of possibilities of the linear theory has not been revealed, the area of its adequacy has not yet been outlined clearly and explicitly enough, in the form of a sufficiently complete system of indicators convenient for testing in experiments, and computer modeling often remains without the necessary foundation. In many works (and in recent years), the general restrictions on creep and relaxation functions are not recognized or even violated, the consequences of the adopted assumptions are not traced, only individual aspects of the behavior of materials and individual necessary conditions for the applicability of the linear theory are isolated (and often presented as sufficient, for example, the independence of the relaxation modulus of the material from the level of deformation or compliance from the level of stress), and the conclusions are formulated in a rather vague form and sometimes contain inaccuracies and misconceptions.

The emergence of the control theory is largely associated with the development of technology and industry, especially when it comes from the generalized Caputo’s fractional derivative. The need to regulate or maintain the current values of some kinematic characteristics of machines or other control objects within the given nonlocal boundary control has led to the creation of a mathematical apparatus of the control theory.

In the control theory, sets of objects (systems) are considered, the behavior of which is described by a certain law. The control problem is the problem of finding a way to change the behavior of a process so as to transfer the system from one given state to another, satisfying additional requirements. As these requirements, we can consider the specified value of the control time T; minimization of control time (speed problem); minimization of some criterion (optimal control problem); and satisfaction with some qualities of the transition process. There is also a stabilization problem that studies the presence of an asymptotically stable solution on an infinite time interval. One of the founders of the classical theory of stability is A. M. Lyapunov, whose fundamental work in this area laid the foundation for rigorous mathematical methods for analyzing the stability of motion, especially for hyperbolic systems. The transmission problem to hyperbolic equations was studied by Dautray and Lions [1], where the existence and regularity of solutions for the linear problem have been proved. The authors in [2] considered the transmission problem of viscoelastic waves, where the damped is taken in the second equation, and studied the wave propagations over materials consisting of elastic and viscoelastic components. It is shown that the viscoelastic part produces exponential decay of the solution, for more details, please see [3,4].

In this paper, where we used more modern methods and considered the following model:

$$\left\{\begin{array}{c}{\partial }_{tt}u(t,x)-{\tau }_1{\partial }_{xx}u(t,x)+{\int }_0^t\mu (t-s){\partial }_{xx}uds=0,t>0,x\in (0,{\ell }_0),{\ell }_0>0,\hfill \\ \\ \rho {\partial }_{tt}v(t,x)-{\tau }_2{\partial }_{xx}v(t,x)=0t>0,x\in ({\ell }_0,L),L>{\ell }_0>0,\hfill \end{array}\right.$$

(1)

where $\rho >0$ represents the density, and ${\tau }_1 \mathrm{and} {\tau }_2>0$ represent, respectively, the tensions of the strings u and v. The function

$$\mu \in C^1({\mathbb{R}}_{+},{\mathbb{R}}_{+}),$$

is assumed to be a positive nonincreasing function defined on ${\mathbb{R}}^{+}$ that satisfies the following:

$${\tau }_1-{\int }_0^t\mu \left(s\right)ds>0,\mu \left(0\right)>0,{\mu }^{\prime }\left(t\right)\le 0,$$

and IC (initial conditions)

$$\left\{\begin{array}{c}u(t,x)=u_0\left(x\right),{\partial }_{t}u(t,x)=u_1\left(x\right)\hfill \\ \\ v(t,x)=v_0\left(x\right),{\partial }_{t}v(t,x)=v_1\left(x\right),\hfill \end{array}\right.$$

(2)

for $t=0$. The TC (transmission condition) is given by

$$\left\{\begin{array}{c}u\left(t,{\ell }_0\right)=v\left(t,{\ell }_0\right),\hfill \\ \\ \rho \left({\tau }_1-{\int }_0^t\mu (t-s)ds\right){\partial }_{x}u\left({\ell }_0\right)={\tau }_2{\partial }_{x}v\left({\ell }_0\right),\forall t>0,\hfill \end{array}\right.$$

(3)

followed by BC (boundary conditions)

$$u(t,0)=0,{\tau }_2{\partial }_{x}v(t,L)+\gamma \rho {\partial }_t^{\beta ,\eta }v(t,L)=0,\forall t\in (0,\infty ),$$

(4)

and CC (conditions of compatibility)

$$\begin{array}{c}\hfill u_0\left({\ell }_0\right)=v_0\left({\ell }_0\right),\\ \hfill u_1\left({\ell }_0\right)=v_1\left({\ell }_0\right),\\ \hfill \rho {\tau }_1{u}_{0x}\left({\ell }_0\right)={\tau }_2{v}_{0x}\left({\ell }_0\right).\end{array}$$

(5)

As in [5], ${\partial }_t^{\beta ,\eta }$ stands for the generalized Caputo’s fractional derivative of order $\beta \in (0,1]$ with respect to the time variable t. It can be given as

$${\partial }_t^{\beta ,\eta }v\left(t\right)=\left\{\begin{array}{cc}{v}_t\left(t\right),\hfill & for\beta =1,\eta \ge 0\hfill \\ \\ {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\int }_0^t{(t-s)}^{-\beta }exp(-\eta (t-s){\displaystyle \frac{dv}{ds}}\left(s\right)ds,\hfill & for\beta \in (0,1),\eta \ge 0.\hfill \end{array}\right.$$

The simple wave equation

$${\partial }_{tt}u={\partial }_{xx}u,x\in (0,L),t>0,$$

is considered in [6] with

$$u(t,0)=0,{\partial }_{x}u(t,L)+\gamma {\partial }_t^{\beta ,\eta }{\partial }_{t}u(t,L)=0.$$

The author investigated the question of the decay rate of the associate energy and found strong asymptotic stability under some conditions, and a polynomial-type decay rate in other conditions.

Next, a Euler–Bernoulli beam equation with boundary dissipation of a nonlocal type is considered in [7]. The non uniform stability is showed by using spectral analysis. The stability results of various damped transmission problems in thermoelasticity are studied from many points of view. Here, we review the works of [8,9,10].

In [11], the stability of a system of weakly coupled hyperbolic equations is studied, based on the frequency domain approach together with the growth of the resolvent on the imaginary axis, and the behavior of energy for smooth initial data are established (see [12]). Also, the Borichev–Tomilov theorem was essential in our analysis; it was initiated in [13] and later pursued in [14].

Lemma 1

([6]). Let ν be the function defined by

$$\nu \left(y\right)={\left|y\right|}^{(2\beta -1)/2},y\in (-\infty ,+\infty ),\beta \in (0,1).$$

Then,

$$\begin{array}{c}{\partial }_t\varpi (t,y)+\left(y^2+\eta \right)\varpi (t,y)-U\left(t\right)\nu \left(y\right)=0,y\in (-\infty ,+\infty ),t>0,\eta \ge 0,\\ \varpi (y,t)=0,fort=0,\end{array}$$

where

$$\begin{array}{c}O\left(t\right)={\left(\pi \right)}^{-1}sin\left(\beta \pi \right){\int }_{-\infty }^{+\infty }\nu \left(y\right)\varpi (t,y)dy,\end{array}$$

is defined by

$$O=I^{1-\beta ,\eta }U,$$

and

$$\left[I^{\beta ,\eta }f\right]\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^t{(t-s)}^{\beta -1}{e}^{-\eta (t-s)}f\left(s\right)ds.$$

A one-dimensional transmission problem of viscoelasticity with arbitrary relaxation and a creep functions is investigated in order to clarify its area of applicability and the range of possibilities for describing the isothermal behavior of viscoelastic-plastic materials. The general properties of the system are analytically studied depending on the characteristics of the relaxation function.

This paper is organized as follows. In Theorem 1, we state the associated energy functional and its derivative to ensure the dissipativity of the system. Then, after we introduce the necessary spaces and the inner product of our problem, the well-posedness of our model is analyzed in Section 2 with Theorem 2 using the semigroup theory. In Section 3, we show the strong stability of the energy when time goes to infinity, which is given in Theorem 3.

2. Well-Posedness

We are going to reformulate the problem (1). By using Lemma 1, it becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{tt}u(t,x)-{\tau }_1{\partial }_{xx}u(t,x)+{\int }_0^t\mu (t-s){\partial }_{xx}uds=0\hfill & x\in \left(0,{\ell }_0\right),t>0,\hfill \\ \rho {\partial }_{tt}v(t,x)-{\tau }_2{\partial }_{xx}v(t,x)=0\hfill & x\in \left({\ell }_0,L\right),t>0,\hfill \\ {\partial }_t\varpi (t,y)+\left(y^2+\eta \right)\varpi (t,y)-{\partial }_{t}v(t,L)\nu \left(y\right)=0\hfill & x\in (-\infty ,\infty ),t>0,\hfill \\ u\left(t,{\ell }_0\right)=v\left(t,{\ell }_0\right),\rho \left({\tau }_1-{\int }_0^t\mu (t-s)ds\right){\partial }_{x}u\left({\ell }_0\right)={\tau }_2{\partial }_{x}v\left({\ell }_0\right)\hfill & \mathrm{on}(0,+\infty ),\hfill \\ u(t,0)=0\hfill & \mathrm{on}(0,+\infty )\hfill \\ {\tau }_2{\partial }_{x}v(t.L)+\zeta \rho {\int }_{-\infty }^{+\infty }\nu \left(y\right)\varpi (t,y)dy=0\hfill & \mathrm{on}(0,+\infty ),\hfill \\ u(t,x)=u_0\left(x\right),{\partial }_{t}u(t,x)=u_1\left(x\right)\hfill & \mathrm{on}\left(0,{\ell }_0\right),fort=0,\hfill \\ v(t,x)=v_0\left(x\right),{\partial }_{t}v(t,x)=v_1\left(x\right)\hfill & \mathrm{on}\left({\ell }_0,L\right),fort=0,\hfill \end{array}\right.\end{array}$$

(6)

where

$$\zeta ={\left(\pi \right)}^{-1}sin\left(\beta \pi \right)\gamma .$$

Theorem 1.

The energy associated with a solution $(u,v,\varpi )$ of (6) is defined as follows:

$$\begin{array}{c}E\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^{{\ell }_0}{\left|{\partial }_{t}u\right|}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{{\ell }_0}^L\left({\left|{\partial }_{t}v\right|}^2+{\displaystyle \frac{{\tau }_2}{\rho }}{\left|{\partial }_{x}v\right|}^2\right)dx+{\displaystyle \frac{\zeta }{2}}{\int }_{-\infty }^{+\infty }{\left|\varpi (t,y)\right|}^{2}dy\\ +{\displaystyle \frac{1}{2}}\left(\mu \circ {\partial }_{x}u\right)\left(t\right)+{\displaystyle \frac{1}{2}}\left({\tau }_1-{\int }_0^t\mu \left(s\right)ds\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dx,\end{array}$$

(7)

and it satisfies

$$E^{\prime }\left(t\right)=-\zeta {\int }_{-\infty }^{+\infty }\left(y^2+\eta \right){\left|\varpi (t,y)\right|}^{2}dy+{\displaystyle \frac{1}{2}}\left({\mu }^{\prime }\circ {\partial }_{x}u\right)\left(t\right)-{\displaystyle \frac{1}{2}}\mu \left(t\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\le 0,$$

(8)

with

$$\left(\mu \circ {\partial }_{x}u\right)\left(t\right)={\int }_0^t\mu (t-s){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(s\right)-{\partial }_{x}u\left(t\right){\mid }^{2}dxds.$$

Proof. 

Via multiplication of ${\left(6\right)}_1$ with ${\partial }_t\overline{u}$, we add its conjugate to obtain

$${\displaystyle \frac{1}{2}}\left[\left({\partial }_{tt}u{\partial }_t\overline{u}+{\partial }_{tt}\overline{v}{\partial }_{t}u\right)-{\tau }_1\left({\partial }_{xx}u{\partial }_t\overline{u}+{\partial }_{xx}\overline{u}{\partial }_{t}u\right)+I\right]=0,$$

(9)

with

$$I={\int }_0^t\mu (t-s){\partial }_{xx}uds{\partial }_t\overline{u}+{\int }_0^t\mu (t-s){\partial }_{xx}\overline{u}ds{\partial }_{t}u.$$

Thanks to integrating by parts in Equation (9), we have

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_0^{{\ell }_0}\mid {\partial }_{t}u{\mid }^{2}dx+{\tau }_1{\int }_0^{{\ell }_0}\mid {\partial }_{x}u{\mid }^{2}dx\right]-{\tau }_1\Re \left({\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)\right)+{\displaystyle \frac{1}{2}}{\int }_0^{{\ell }_0}Idx=0,$$

(10)

and

$$\begin{array}{cc}\hfill {\int }_0^{{\ell }_0}Idx& ={\int }_0^{{\ell }_0}{\int }_0^t\mu (t-s){\partial }_{xx}u\left(s\right){\partial }_t\overline{u}\left(s\right)dsdx+{\int }_0^{{\ell }_0}{\int }_0^t\mu (t-s){\partial }_{xx}\overline{u}\left(s\right){\partial }_{t}u\left(s\right)dsdx\hfill \\ \hfill & ={\int }_0^t\mu (t-s)\left[{\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)-{\int }_0^{{\ell }_0}\left({\partial }_{x}u\left(s\right)-{\partial }_{x}u\left(t\right)\right){\partial }_{tx}\overline{u}\left(t\right)dx\right]ds\hfill \\ \hfill & -{\int }_0^t\mu (t-s){\int }_0^{{\ell }_0}{\partial }_{x}u\left(t\right){\partial }_{tx}\overline{u}\left(t\right)dxds\hfill \\ \hfill & +{\int }_0^t\mu (t-s)\left[{\partial }_x\overline{u}\left({\ell }_0\right){\partial }_{t}u\left({\ell }_0\right)-{\int }_0^{{\ell }_0}\left({\partial }_x\overline{u}\left(s\right)-{\partial }_x\overline{u}\left(t\right)\right){\partial }_{tx}u\left(t\right)dx\right]ds\hfill \\ \hfill & -{\int }_0^t\mu (t-s){\int }_0^{{\ell }_0}{\partial }_x\overline{u}\left(t\right){\partial }_{tx}u\left(t\right)dxds\hfill \\ \hfill & ={\int }_0^t\mu (t-s)\left[2\Re \left({\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)\right)+{\displaystyle \frac{d}{dt}}{\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(s\right)-{\partial }_{x}u\left(t\right){\mid }^{2}dx\right]ds\hfill \\ \hfill & -{\int }_0^t\mu \left(s\right)\left({\displaystyle \frac{d}{dt}}{\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dx\right)ds,\hfill \end{array}$$

(11)

which implies that

$$\begin{array}{cc}\hfill {\int }_0^{{\ell }_0}Idx& =2{\int }_0^t\mu (t-s)\Re \left({\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)\right)ds+{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu (t-s){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(s\right)-{\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]\hfill \\ \hfill & -{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]-{\int }_0^t{\mu }^{\prime }\left(t-s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(s\right)-{\partial }_{x}u\left(t\right){\mid }^{2}dxds\hfill \\ \hfill & +{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\hfill \\ \hfill & =2{\int }_0^t\mu (t-s)\Re \left({\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)\right)ds+{\displaystyle \frac{d}{dt}}\left(\mu \circ {\partial }_{x}u\right)\left(t\right)-{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]\hfill \\ \hfill & -\left({\mu }^{\prime }\circ {\partial }_{x}u\right)\left(t\right)+{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds.\hfill \end{array}$$

(12)

Finally, Equation (9) will be

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_0^{{\ell }_0}\mid {\partial }_{t}u{\mid }^{2}dx+{\tau }_1{\int }_0^{{\ell }_0}\mid {\partial }_{x}u{\mid }^{2}dx\right]\hfill \\ & -{\tau }_1\Re \left({\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)\right)+{\int }_0^t\mu (t-s)\Re \left({\partial }_{x}u\left({\ell }_0\right){\partial }_t\overline{u}\left({\ell }_0\right)\right)ds\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left(\mu \circ {\partial }_{x}u\right)\left(t\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left({\mu }^{\prime }\circ {\partial }_{x}u\right)\left(t\right)+{\displaystyle \frac{1}{2}}{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds=0.\hfill \end{array}$$

(13)

We use the method applied in [15]; thus, ${\left(6\right)}_2$ becomes

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_{{\ell }_0}^L\mid {\partial }_{t}v{\mid }^{2}dx+{\displaystyle \frac{{\tau }_2}{\rho }}{\int }_{{\ell }_0}^L\mid {\partial }_{x}v{\mid }^{2}dx\right]+{\displaystyle \frac{{\tau }_2}{\rho }}\Re \left({\partial }_{x}v\left({\ell }_0\right){\partial }_t\overline{v}\left({\ell }_0\right)\right)-{\displaystyle \frac{{\tau }_2}{\rho }}\Re \left({\partial }_{x}v\left(L\right){\partial }_t\overline{v}\left(L\right)\right)=0.$$

(14)

By adding (13) and (14), we then use the transmission conditions to obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_0^{{\ell }_0}\mid {\partial }_{t}u{\mid }^{2}dx+{\tau }_1{\int }_0^{{\ell }_0}\mid {\partial }_{x}u{\mid }^{2}dx\right]+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_{{\ell }_0}^L\mid {\partial }_{t}v{\mid }^{2}dx+{\displaystyle \frac{{\tau }_2}{\rho }}{\int }_{{\ell }_0}^L\mid {\partial }_{x}v{\mid }^{2}dx\right]\hfill \\ & -{\displaystyle \frac{{\tau }_2}{\rho }}\Re \left({\partial }_{x}v\left(L\right){\partial }_t\overline{v}\left(L\right)\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left(\mu \circ {\partial }_{x}u\right)\left(t\right)-{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left({\mu }^{\prime }\circ {\partial }_{x}u\right)\left(t\right)+{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds=0.\hfill \end{array}$$

From the boundary conditions, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_0^{{\ell }_0}\mid {\partial }_{t}u{\mid }^{2}dx+{\tau }_1{\int }_0^{{\ell }_0}\mid {\partial }_{x}u{\mid }^{2}dx\right]+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_{{\ell }_0}^L\mid {\partial }_{t}v{\mid }^{2}dx+{\displaystyle \frac{{\tau }_2}{\rho }}{\int }_{{\ell }_0}^L\mid {\partial }_{x}v{\mid }^{2}dx\right]\hfill \\ \hfill & +\zeta \Re \left({\partial }_t\overline{v}\left(L\right){\int }_{-\infty }^{+\infty }\nu \left(\epsilon \right)\varpi \left(\epsilon ,t\right)d\epsilon \right)+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d}{dt}}\left(\mu \circ {\partial }_{x}u\right)\left(t\right)-{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left(\left({\mu }^{\prime }\circ {\partial }_{x}u\right)\left(t\right)+{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right)=0.\hfill \end{array}$$

(15)

We apply the method used in [15]; thus, ${\left(6\right)}_3$ becomes

$${\displaystyle \frac{\zeta }{2}}{\displaystyle \frac{d}{dt}}{\parallel \varpi \parallel }_2^2+\zeta {\int }_{-\infty }^{+\infty }\left({\epsilon }^2+\eta \right)\mid \varpi \left(\epsilon ,t\right){\mid }^{2}d\epsilon -\zeta \Re \left({\partial }_{t}v\left(L\right){\int }_{-\infty }^{+\infty }\nu \left(\epsilon \right)\overline{\varpi }\left(\epsilon ,t\right)d\epsilon \right)=0.$$

(16)

By adding (15) and (16), we get

$$\begin{array}{c}{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_0^{{\ell }_0}\mid {\partial }_{t}u{\mid }^{2}dx+{\tau }_1{\int }_0^{{\ell }_0}\mid {\partial }_{x}u{\mid }^{2}dx\right]+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\int }_{{\ell }_0}^L\mid {\partial }_{t}v{\mid }^{2}dx+{\displaystyle \frac{{\tau }_2}{\rho }}{\int }_{{\ell }_0}^L\mid {\partial }_{x}v{\mid }^{2}dx\right]\hfill \\ +{\displaystyle \frac{\zeta }{2}}{\displaystyle \frac{d}{dt}}{\parallel \varpi \parallel }_2^2+\zeta {\int }_{-\infty }^{+\infty }\left({\epsilon }^2+\eta \right)\mid \varpi \left(\epsilon ,t\right){\mid }^{2}d\epsilon -{\displaystyle \frac{1}{2}}\left(\left({\mu }^{\prime }\circ {\partial }_{x}u\right)\left(t\right)+{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right)\hfill \\ +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d}{dt}}\left(\mu \circ {\partial }_{x}u\right)\left(t\right)-{\displaystyle \frac{d}{dt}}\left[{\int }_0^t\mu \left(s\right){\int }_0^{{\ell }_0}\mid {\partial }_{x}u\left(t\right){\mid }^{2}dxds\right]\right)=0.\hfill \end{array}$$

(17)

Consequently, Theorem (1) results from Equation (17). ☐

The well-posedness of (6) will be discussed. To this end, we should introduce the spaces

$$H_{*}^1(0,{\ell }_0)=\left\{w\in H_1(0,{\ell }_0):w\left(t\right)=0,fort=0\right\}.$$

(18)

Then, we write (6) into a semigroup setting. Let

$$\tilde{u}={\partial }_{t}u,$$

$$\tilde{v}={\partial }_{t}v,$$

and set

$$\mathcal{H}=\left\{H_{*}^1(0,{\ell }_0)\times L^2(0,{\ell }_0)\times H^1({\ell }_0,L)\times L^2({\ell }_0,L)\times L^2\left(\mathbb{R}\right):u\left({\ell }_0\right)=v\left({\ell }_0\right)\right\},$$

which is equipped with

$$\begin{array}{c}{⟨U,U_1⟩}_{\mathcal{H}}={\int }_0^{{\ell }_0}\left(\tilde{u}{\overline{\tilde{u}}}_1\right)dx+{\int }_{{\ell }_0}^L\left(\tilde{v}{\overline{\tilde{v}}}_1+{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{x}v{\partial }_x{\overline{v}}_1\right)dx+\zeta {\int }_{-\infty }^{+\infty }\varpi {\overline{\varpi }}_{1}dy\\ +{\displaystyle \frac{1}{2}}\left(\mu \circ ({\partial }_{x}u,{\partial }_{x}v)\right)+{\displaystyle \frac{1}{2}}\left({\tau }_1-{\int }_0^t\mu \left(s\right)ds\right){\int }_0^{{\ell }_0}{\partial }_{x}u{\partial }_x\overline{v}dx,\end{array}$$

(19)

where

$$\left(\mu \circ ({\partial }_{x}u,{\partial }_{x}v)\right)\left(t\right)={\int }_0^t\mu (t-s){\int }_0^{{\ell }_0}\left({\partial }_{x}u\left(s\right)-{\partial }_{x}u\left(t\right)\right)\left({\partial }_x\overline{v}\left(s\right)-{\partial }_x\overline{v}\left(t\right)\right)dxds,$$

for all

$$U={(u,\tilde{u},v,\tilde{v},\varpi )}^T,$$

and

$$U_1={\left(u_1,{\tilde{u}}_1,,v_1,{\tilde{v}}_1,{\varpi }_1\right)}^T.$$

Let

$$U={(u,\tilde{u},v,\tilde{v},\varpi )}^T,$$

and then re-rewrite (6) as

$$\left\{\begin{array}{c}{U}^{\prime }=\mathcal{A}U,\hfill \\ \\ U\left(0\right)=U_0=\left(u_0,u_1,v_0,v_1,{\varpi }_0\right),\hfill \end{array}\right.$$

(20)

where $\mathcal{A}$ is defined as

$$\mathcal{A}\left(\begin{array}{c}u\\ \tilde{u}\\ v\\ \tilde{v}\\ \varpi \end{array}\right)=\left(\begin{array}{c}\tilde{u}\\ {\tau }_1{\partial }_{xx}u-{\int }_0^t\mu (t-s){\partial }_{xx}uds\\ \tilde{v}\\ {\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v\\ -\left(y^2+\eta \right)\varpi +\tilde{v}\left(L\right)\nu \left(y\right)\end{array}\right),$$

(21)

with the domain

$$D\left(\mathcal{A}\right)=\left\{\begin{array}{c}{(u,\tilde{u},v,\tilde{v},\varpi )}^T\mathrm{in}\mathcal{H}:u\in H^2(0,L)\cap H_{*}^1\left(0,{\ell }_0\right),\tilde{u}\in H_{*}^1\left(0,{\ell }_0\right)\hfill \\ v\in H^2\left({\ell }_0,L\right),\tilde{v}\in H^1\left({\ell }_0,L\right),u\left({\ell }_0\right)=v\left({\ell }_0\right),\hfill \\ \rho \left({\tau }_1-{\int }_0^t\mu (t-s)ds\right){\partial }_{x}u\left({\ell }_0\right)={\tau }_2{\partial }_{x}v\left({\ell }_0\right)\hfill \\ \tilde{u}\left({\ell }_0\right)=\tilde{v}\left({\ell }_0\right),-\left(y^2+\eta \right)\varpi +\tilde{v}\left(L\right)\nu \left(y\right)\in L^2\left(\mathbb{R}\right)\hfill \\ {\tau }_2{\partial }_{x}v\left(L\right)+\zeta \rho {\int }_{-\infty }^{+\infty }\nu \left(y\right)\varpi \left(y\right)dy=0\hfill \\ \left|y\right|\varpi \in L^2\left(\mathbb{R}\right)\hfill \end{array}\right\}.$$

(22)

The well-posedness of problem (6) is ensured by the next theorem.

Theorem 2.

1. 

If $U_0\in D\left(\mathcal{A}\right)$, then system $\left(20\right)$ has a unique strong solution

$$U\in C^0\left({\mathbb{R}}_{+},D\left(\mathcal{A}\right)\right)\cap C^1\left({\mathbb{R}}_{+},\mathcal{H}\right).$$

2. 

If $U_0\in \mathcal{H}$, then system $\left(20\right)$ has a unique weak solution

$$U\in C^0\left({\mathbb{R}}_{+},\mathcal{H}\right).$$

Proof. 

We have to show that $\mathcal{A}$ is monotone maximal operator. As a starting point, we have

$$E\left(t\right)={\displaystyle \frac{1}{2}}{⟨U,U⟩}_{\mathcal{H}},$$

and then

$$E^{\prime }\left(t\right)={\displaystyle \frac{1}{2}}\left({⟨U^{\prime },U⟩}_{\mathcal{H}}+\overline{{⟨U^{\prime },U⟩}_{\mathcal{H}}}\right)=\mathcal{R}{⟨\mathcal{A}U,U⟩}_{\mathcal{H}}.$$

On the other hand, we have

$$E^{\prime }\left(t\right)\le 0.$$

(23)

Set

$$G={\left(g_1,g_2,g_3,g_4,g_5\right)}^T\in \mathcal{H},$$

and let

$$U={(u,\tilde{u},v,\tilde{v},\varpi )}^T\in D\left(\mathcal{A}\right)$$

satisfy

$$\lambda U-\mathcal{A}U=G,\lambda >0;$$

thus,

$$\left\{\begin{array}{c}\lambda u-\tilde{u}=g_1,\hfill \\ \lambda \tilde{u}-{\tau }_1{\partial }_{xx}u+{\int }_0^t\mu (t-s){\partial }_{xx}uds=g_2,\hfill \\ \lambda v-\tilde{v}=g_3,\hfill \\ \lambda \tilde{v}-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=g_4,\hfill \\ \lambda \varpi +\left(y^2+\eta \right)\varpi -\tilde{v}\left(L\right)\nu \left(y\right)=g_5.\hfill \end{array}\right.$$

(24)

With the suitable regularity of u and v, we have

$$\begin{array}{cc}\hfill & \tilde{u}=\lambda u-g_1,\hfill \\ \hfill & \tilde{v}=\lambda v-g_3.\hfill \end{array}$$

It is hard to see that

$$\tilde{u}\in H_{*}^1\left(0,{\ell }_0\right),$$

and

$$\tilde{v}\in H^1\left({\ell }_0,L\right).$$

Then, by ${\left(24\right)}_5$, we find $\varpi$ as

$$\varpi ={\displaystyle \frac{g_5\left(y\right)+\nu \left(y\right)\tilde{v}\left(L\right)}{y^2+\eta +\lambda }}.$$

(25)

By ${\left(24\right)}_2$ and ${\left(24\right)}_4$, we have the functions u and v, which satisfy

$$\begin{array}{cc}\hfill & {\lambda }^{2}u-{\tau }_1{\partial }_{xx}u+{\int }_0^t\mu (t-s){\partial }_{xx}uds=g_2+\lambda g_1,\hfill \\ \hfill & {\lambda }^{2}v-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=g_4+\lambda g_3.\hfill \end{array}$$

(26)

Solving $\left(26\right)$ is equivalent to finding

$$u\in H^2\cap H_{*}^1\left(0,{\ell }_0\right),$$

and

$$v\in H^2\left({\ell }_0,L\right),$$

where

$$\begin{array}{cc}\hfill & {\int }_0^{{\ell }_0}\left({\lambda }^{2}u\overline{w}-\left({\tau }_1{\partial }_{xx}u-{\int }_0^t\mu (t-s){\partial }_{xx}uds\right)\overline{w}\right)dx={\int }_0^{{\ell }_0}\left(g_2+\lambda g_1\right)\overline{w}dx\hfill \\ \hfill & {\int }_{{\ell }_0}^L\left({\lambda }^{2}v\overline{\chi }-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v\overline{\chi }\right)dx={\int }_{{\ell }_0}^L\left(g_4+\lambda g_3\right)\overline{\chi }dx.\hfill \end{array}$$

(27)

Then,

$$\begin{array}{c}\hfill {\int }_0^{{\ell }_0}\left({\lambda }^{2}u\overline{w}+({\tau }_1-{\int }_0^t\mu (t-s)ds){\partial }_{x}u{\partial }_x\overline{w}\right)dx+{\int }_{{\ell }_0}^L\left({\lambda }^{2}v\overline{\chi }+{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{x}v{\partial }_x\overline{\chi }\right)dx+\tilde{\zeta }\lambda v\left(L\right)\overline{\chi }\left(L\right)\\ \hfill ={\int }_0^{{\ell }_0}\left(g_2+\lambda g_1\right)\overline{w}dx+{\int }_{{\ell }_0}^L\left(g_4+\lambda g_3\right)\overline{\chi }dx-\zeta {\int }_{-\infty }^{+\infty }{\displaystyle \frac{\nu \left(y\right)}{y^2+\eta +\lambda }}{g}_5\left(y\right)dy\overline{\chi }\left(L\right)+\tilde{\zeta }{g}_3\left(L\right)\overline{\chi }\left(L\right),\end{array}$$

(28)

where

$$\tilde{\zeta }=\zeta {\int }_{-\infty }^{+\infty }{\displaystyle \frac{{\nu }^2\left(y\right)}{y^2+\eta +\lambda }}dy.$$

Then, system (28) is equivalent to

$$a\left(\right(u,v),(w,\chi \left)\right)=L(w,\chi ),$$

where

$$a:{\left[H_{*}^1\left(0,{\ell }_0\right)\times H^1\left({\ell }_0,L\right)\right]}^2\to \mathbb{R},$$

and

$$L:H_{*}^1\left(0,{\ell }_0\right)\times H^1\left({\ell }_0,L\right)\to \mathbb{R},$$

are given by

$$\begin{array}{cc}\hfill & a((u,v),(w,\chi ))={\int }_0^{{\ell }_0}\left({\lambda }^{2}u\overline{w}+({\tau }_1-{\int }_0^t\mu (t-s)ds){\partial }_{x}u{\partial }_x\overline{w}\right)dx\hfill \\ \hfill & +{\int }_{{\ell }_0}^L\left({\lambda }^{2}v\overline{\chi }+{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{x}v{\partial }_x\overline{\chi }\right)dx+\tilde{\zeta }\lambda v\left(L\right)\overline{\chi }\left(L\right),\hfill \end{array}$$

and

$$\begin{array}{cc}L(w,\chi )=\hfill & {\int }_0^{{\ell }_0}\left(g_2+\lambda g_1\right)\overline{w}dx+{\int }_{{\ell }_0}^L\left(g_4+\lambda g_3\right)\overline{\chi }dx\hfill \\ \hfill & -\zeta {\int }_{-\infty }^{+\infty }{\displaystyle \frac{\nu \left(y\right)}{y^2+\eta +\lambda }}{g}_5\left(y\right)dy\overline{\chi }\left(L\right)+\tilde{\zeta }{g}_3\left(L\right)\overline{\chi }\left(L\right).\hfill \end{array}$$

It is now clear that a is coercive and continuous, and that L is continuous. Owing to the Lax–Milgram theorem, we have

$$(w,\chi )\in H_{*}^1\left(0,{\ell }_0\right)\times H^1\left({\ell }_0,L\right);$$

thus, (28) has

$$(u,v)\in H_{*}^1\left(0,{\ell }_0\right)\times H^1\left({\ell }_0,L\right),$$

which is a unique solution. By the classical elliptic regularity on (27), we have

$$(u,v)\in H^2\left(0,{\ell }_0\right)\times H^2\left({\ell }_0,L\right).$$

Therefore, the operator $\lambda I-\mathcal{A}$ is surjective for any $\lambda >0$. At last, the result of Theorem 2 follows from the Hille–Yosida theorem. ☐

3. Strong Stability of the System

A general criteria of Arendt–Batty in [5] is used to show the strong stability of the $C_0$-semigroup $e^{tA}$ associated with (1) in the absence of the compactness of the resolvent of A.

Theorem 3.

The $C_0$-semigroup $e^{tA}$ is strongly stable in $\mathcal{H}$, i.e, for all $U_0\in \mathcal{H}$, the solution of (20) satisfies

$$li{m}_{t\to +\infty }{\Vert e^{tA}{U}_0\Vert }_{\mathcal{H}}=0.$$

Lemma 2.

$\mathcal{A}$ does not have eigenvalues on $i\mathbb{R}$.

Proof. 

We shall make a distinction between $i\lambda =0$ and $i\lambda \ne 0$,

Step 01 Solving for $\mathcal{A}U=0$. From (21), we have

$$\left\{\begin{array}{c}\begin{array}{c}\tilde{u}=0\\ {\tau }_1{\partial }_{xx}u-{\int }_0^t\mu (t-s){\partial }_{xx}uds=0\\ \tilde{v}=0\\ {\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=0\\ -\left(y^2+\eta \right)\varpi +\tilde{v}\left(L\right)\nu \left(y\right)=0.\end{array}\hfill \end{array}\right.$$

(29)

This implies that $\tilde{u}=\tilde{v}=\varpi =0$; moreover, thanks to the boundary conditions in (22), we have $u=v=0$. Hence, $\lambda$ is not an eigenvalue of $\mathcal{A}$.

Step 02 We will argue by contradiction. Let us suppose that we have $\lambda \in {\mathbb{R}}^{*}$ and $U\ne 0$, such that

$$i\lambda U-\mathcal{A}U=0.$$

Thus, we have

$$\left\{\begin{array}{c}i\lambda u-\tilde{u}=0\hfill \\ i\lambda \tilde{u}-{\tau }_1{\partial }_{xx}u+{\int }_0^t\mu (t-s){\partial }_{xx}uds=0\hfill \\ i\lambda v-\tilde{v}=0\hfill \\ i\lambda \tilde{v}-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=0\hfill \\ i\lambda \varpi +\left(y^2+\eta \right)\varpi -\tilde{v}\left(L\right)\nu \left(y\right)=0.\hfill \end{array}\right.$$

(30)

Then, from (8), we have

$$\varpi =0.$$

By using ${\left(30\right)}_5$

$$\tilde{v}\left(L\right)=0.$$

Hence, by ${\left(30\right)}_3$ and ${\left(22\right)}_4$, we have

$$v\left(L\right)=0,{\partial }_{x}v\left(L\right)=0.$$

(31)

By ${\left(30\right)}_3$ and ${\left(30\right)}_4$, we have

$$-{\lambda }^{2}v-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=0.$$

(32)

The solution of (32) is defined by

$$v\left(x\right)=c_{1}cos\left({\displaystyle \frac{\lambda }{r}}x\right)+c_{2}sin\left({\displaystyle \frac{\lambda }{r}}x\right),$$

$$r=\sqrt{{\displaystyle \frac{{\tau }_2}{\rho }}}.$$

By (31), we have

$$v\equiv 0,$$

and by (3), we have

$$u\left({\ell }_0\right)={\partial }_{x}u\left({\ell }_0\right)=0.$$

Similarly, for

$$r=\sqrt{{\tau }_1-{\int }_0^t\mu (t-s)ds}.$$

Then,

$$u\equiv 0.$$

Thus, $U=0$. So, $\mathcal{A}$ does not have purely imaginary eigenvalues. ☐

Lemma 3.

We have the following:

• If $\lambda \ne 0$, the operator $i\lambda I-\mathcal{A}$ is surjective.
• If $\lambda =0$ and $\eta \ne 0$, the operator $i\lambda I-\mathcal{A}$ is surjective.

Proof. 

Case 01: For $\lambda \ne 0$. Let

$$G={\left(g_1,g_2,g_3,g_4,g_5\right)}^T\in \mathcal{H},$$

be defined, and

$$U={(u,\tilde{u},v,\tilde{v},\varpi )}^T\in D\left(\mathcal{A}\right),$$

be where

$$i\lambda U-\mathcal{A}U=F.$$

(33)

Then,

$$\left\{\begin{array}{c}i\lambda u-\tilde{u}=g_1\hfill \\ i\lambda \tilde{u}-{\tau }_1{\partial }_{xx}u+{\int }_0^t\mu (t-s){\partial }_{xx}uds=g_2\hfill \\ i\lambda v-\tilde{v}=g_3\hfill \\ i\lambda \tilde{v}-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=g_4\hfill \\ i\lambda \varpi +\left(y^2+\eta \right)\varpi -\tilde{v}\left(L\right)\nu \left(y\right)=g_5.\hfill \end{array}\right.$$

(34)

By inserting ${\left(34\right)}_1$, ${\left(34\right)}_3$ into ${\left(34\right)}_2$ and ${\left(34\right)}_4$, we get

$$\left\{\begin{array}{c}-{\lambda }^{2}u-{\tau }_1{\partial }_{xx}u+{\int }_0^t\mu (t-s){\partial }_{xx}uds=g_2+i\lambda g_1\hfill \\ -{\lambda }^{2}v-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=g_4+i\lambda g_3.\hfill \end{array}\right.$$

(35)

Solving (35) is equivalent to finding

$$u\in H^2\cap H_{*}^1\left(0,{\ell }_0\right),$$

and

$$v\in H^2\left({\ell }_0,L\right),$$

where

$$\left\{\begin{array}{c}{\int }_0^{{\ell }_0}\left(-{\lambda }^{2}u\overline{w}-\left({\tau }_1{\partial }_{xx}u-{\int }_0^t\mu (t-s){\partial }_{xx}uds\right)\overline{w}\right)dx={\int }_0^{{\ell }_0}\left(g_2+i\lambda g_1\right)\overline{w}dx\hfill \\ {\int }_{{\ell }_0}^L\left(-{\lambda }^{2}v\overline{\chi }-{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v\overline{\chi }\right)dx={\int }_{{\ell }_0}^L\left(g_4+i\lambda g_3\right)\overline{\chi }dx.\hfill \end{array}\right.$$

(36)

We find that the functions u and v satisfy the following system:

$$\begin{array}{c}{\int }_0^{{\ell }_0}\left(-{\lambda }^{2}u\overline{w}+({\tau }_1-{\int }_0^t\mu (t-s)ds){\partial }_{x}u{\partial }_x\overline{w}\right)dx+{\int }_{{\ell }_0}^L\left(-{\lambda }^{2}v\overline{\chi }+{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{x}v{\partial }_x\overline{\chi }\right)dx+\tilde{\zeta }i\lambda v\left(L\right)\overline{\chi }\left(L\right)\hfill \\ ={\int }_0^{{\ell }_0}\left(g_2+i\lambda g_1\right)\overline{w}dx+{\int }_{{\ell }_0}^L\left(g_4+i\lambda g_3\right)\overline{\chi }dx-\zeta {\int }_{-\infty }^{+\infty }{\displaystyle \frac{\nu \left(y\right)}{y^2+\eta +i\lambda }}{g}_5\left(y\right)dy\overline{\chi }\left(L\right)+\tilde{\zeta }{g}_3\left(L\right)\overline{\chi }\left(L\right).\hfill \end{array}$$

(37)

We then rewrite (37) as

$$-{⟨L_{\lambda }U,V⟩}_{H_R^1}+{⟨U,V⟩}_{H_R^1}=l\left(V\right),$$

(38)

where

$$H_R^1\left(0,L\right)=\left\{\left(u,v\right)\in H_{*}^1\left(0,{\ell }_0\right)\times H^1\left({\ell }_0,L\right)/u\left({\ell }_0\right)=v\left({\ell }_0\right)\right\},$$

with

$${⟨U,V⟩}_{H_R^1}={\int }_0^{{\ell }_0}\left({\tau }_1-{\int }_0^t\mu (t-s)ds\right){\partial }_{x}u\overline{w_x}dx+{\displaystyle \frac{{\tau }_2}{\rho }}{\int }_{{\ell }_0}^L{\partial }_{x}v\overline{{\chi }_x}dx+\tilde{y}i\lambda v\left(L\right)\overline{\chi \left(L\right)},$$

and

$${⟨L_{\lambda }U,V⟩}_{H_R^1}={\lambda }^2{\int }_0^{{\ell }_0}u\overline{w}dx+{\lambda }^2{\int }_{{\ell }_0}^{L}v\overline{\chi }dx.$$

Using the compactness embedding from $\left(L^2(0,{\ell }_0)\times L^2({\ell }_0,L)\right)$ into ${\left(H_R^1\left(0,L\right)\right)}^{\prime }$ and from $\left(H_R^1\left(0,L\right)\right)$ into $\left(L^2(0,{\ell }_0)\times L^2({\ell }_0,L)\right)$, we deduce that the operator $L_{\lambda }$ is compact from $L^2(0,{\ell }_0)\times L^2({\ell }_0,L)$ into $L^2(0,{\ell }_0)\times L^2({\ell }_0,L)$. Consequently, with the Fredholm alternative, proving the existence of the U solution of (38) is reduced to proving that 1 is not an eigenvalue of $L_{\lambda }$. Then, if 1 is an eigenvalue, there exists $U\ne 0$, s. t.

$${⟨L_{\lambda }U,V⟩}_{H_R^1}={⟨U,V⟩}_{H_R^1},\forall V\in H_R^1.$$

(39)

For $U=V$, we have

$$\begin{array}{cc}\hfill {\lambda }^2\left[{\Vert u\Vert }_{L^2(0,{\ell }_0)}^2+{\Vert v\Vert }_{L^2({\ell }_0,L)}^2\right]=& \left({\tau }_1-{\int }_0^t\mu (t-s)ds\right)\Vert {\partial }_x{u\Vert }_{L^2(0,{\ell }_0)}^2+{\displaystyle \frac{{\tau }_2}{\rho }}{\Vert {\partial }_{x}v\Vert }_{L^2({\ell }_0,L)}^2\hfill \\ \hfill & +\tilde{y}i\lambda \mid v\left(L\right){\mid }^2.\hfill \end{array}$$

Hence, we have

$$v\left(L\right)=0.$$

(40)

From (39) and (22), we obtain

$${\partial }_{x}v\left(L\right)=0.$$

(41)

Furthermore,

$$\left\{\begin{array}{c}\begin{array}{c}-{\lambda }^{2}u-r_1{\partial }_{xx}u=0,\\ -{\lambda }^{2}v-r_2{\partial }_{xx}v=0,\end{array}\hfill \end{array}\right.$$

(42)

where

$$r_1={\tau }_1-{\int }_0^t\mu (t-s)ds,$$

and

$$r_2={\displaystyle \frac{{\tau }_2}{\rho }}.$$

The solutions of (42) are given by

$$\left\{\begin{array}{c}\begin{array}{c}u\left(x\right)=c_{1}cos\left({\displaystyle \frac{\lambda }{\sqrt{r_1}}}x\right)+c_{2}sin\left({\displaystyle \frac{\lambda }{\sqrt{r_1}}}x\right),\\ v\left(x\right)=c_{3}cos\left({\displaystyle \frac{\lambda }{\sqrt{r_2}}}x\right)+c_{4}sin\left({\displaystyle \frac{\lambda }{\sqrt{r_2}}}x\right).\end{array}\hfill \end{array}\right.$$

(43)

By the (4), we find that $c_1=c_3=c_4=0$. Then, By the (3), we deduce that $c_2=0$. Then, $U=0$. Hence, $i\lambda -\mathcal{A}$ is surjective $\forall \lambda \in {\mathbb{R}}^{*}$. ☐

Case 02 $\lambda =0$ and $\eta \ne 0$. System (34) is reduced to

$$\left\{\begin{array}{c}-\tilde{u}=g_1\hfill \\ -{\tau }_1{\partial }_{xx}u+{\int }_0^t\mu (t-s){\partial }_{xx}uds=g_2\hfill \\ -\tilde{v}=g_3\hfill \\ -{\displaystyle \frac{{\tau }_2}{\rho }}{\partial }_{xx}v=g_4\hfill \\ \left(y^2+\eta \right)\varpi -\tilde{v}\left(L\right)\nu \left(y\right)=g_5.\hfill \end{array}\right.$$

(44)

With the second and fourth equations of (44), we get

$$\left\{\begin{array}{c}u\left(x\right)=-{\displaystyle \frac{1}{r_1}}{\int }_0^x{\int }_0^s{g}_2\left(r\right)drds+Cx,\hfill \\ v\left(x\right)=-{\displaystyle \frac{1}{r_2}}{\int }_{{\ell }_0}^x{\int }_{{\ell }_0}^s{g}_4\left(r\right)drds+C^{\prime }x+C^{″}.\hfill \end{array}\right.$$

(45)

From ${\left(44\right)}_3$ and ${\left(44\right)}_5$, we get

$$-\gamma {\eta }^{\beta -1}{g}_3\left(L\right)+r_2{\partial }_{x}v\left(L\right)+\zeta {\int }_{-\infty }^{+\infty }{\displaystyle \frac{\nu \left(y\right)g_5\left(y\right)}{y^2+\eta }}dy=0.$$

We find

$$C^{\prime }={\displaystyle \frac{1}{r_2}}\left[{\int }_{{\ell }_0}^L{g}_4\left(r\right)dr+\gamma {\eta }^{\beta -1}{g}_3\left(L\right)-\zeta {\int }_{-\infty }^{+\infty }{\displaystyle \frac{\nu \left(y\right)g_5\left(y\right)}{y^2+\eta }}dy\right].$$

From the (3) and (4), we have

$$\begin{array}{c}u\left({\ell }_0\right)=v\left({\ell }_0\right)\Rightarrow {\ell }_{0}C-C^{\prime \prime }={\displaystyle \frac{1}{r_1}}{\int }_0^{{\ell }_0}{\int }_0^s{g}_2\left(r\right)drds+C^{\prime }{\ell }_0.\\ r_1{\partial }_{x}u\left({\ell }_0\right)=r_2{\partial }_{x}v\left({\ell }_0\right)\Rightarrow C{r}_1={\int }_0^{{\ell }_0}{\int }_0^s{g}_2\left(r\right)dr+C^{\prime }{r}_2.\end{array}$$

Then,

$$\begin{array}{cc}& C={\displaystyle \frac{1}{r_1}}\left[{\int }_0^{{\ell }_0}{\int }_0^s{g}_2\left(r\right)dr+C^{\prime }{r}_2\right],\hfill \\ & C^{\prime \prime }={\ell }_0\left(C-C^{\prime }\right)-{\displaystyle \frac{1}{r_1}}{\int }_0^{{\ell }_0}{\int }_0^s{g}_2\left(r\right)dr.\hfill \end{array}$$

Hence, $\mathcal{A}$ is surjective; thus, the proof is completed. For the proof of Theorem 3, by Lemmas 2 and 3 as well as with the closed graph theorem of Banach, it is implied that

$$\sigma \left(\mathcal{A}\right)\cap i\mathbb{R}=\varnothing ,$$

if $\eta >0$ and

$$\sigma \left(\mathcal{A}\right)\cap i\mathbb{R}=\left\{0\right\}$$

as well as if $\eta =0$, which completes the proof.

4. Conclusion and Challenges

In this article, a transmission problem of a viscoelastic wave is considered, with nonlocal boundary control. The non-local wave equation is given by (1), with an initial value of (2). The transmission condition is given by (3), followed by a non-local boundary condition of (4). The well-posed property is proved in Section 2. Equation (1) is reformulated as system (6). The energy dissipation law is derived in Theorem 1. With the help of the energy estimate, the well-posed nature of system (6) is proved in Theorem 2. In addition, the strong stability of the system is established in Section 3.

We can summarize the main contributions and the differences between our work and the articles in the literature in the following points:

• We examined the viscoelastic case and its impact on stability results, where we took an ordinary condition on the relaxation function, which changed the transmission condition.
• We studied the strong stability of the solution and proved it in the same way as in [15], with some differences imposed by the nature of our new model.
• As for the most important difficulties that we encountered in this article, they encompassed extracting the energy terms and finding the appropriate transition condition, as well as the appropriate inner product expression.
• The combination between the frequency domain approach developed in [11] and the Borichev–Tomilov theorem was the main novelty of our analysis. This latter approach was initiated in [13] and later pursued in [14].

For certain non-local parabolic differential equations (with nonlocal operator in space), extensive numerical works have been reported, such as a convergent convex splitting scheme for periodic non-local Cahn–Hilliard equations, second-order convex splitting schemes for periodic non-local Cahn–Hilliard and Allen–Cahn equations, convergence analysis for a stabilized linear semi-implicit numerical scheme for nonlocal Cahn–Hilliard equations, stabilization parameter analysis of a second-order linear numerical scheme for non-local Cahn–Hilliard equations, double stabilizations and convergence analysis of a second-order linear numerical scheme for non-local Cahn–Hilliard equations, etc. The energy stability and convergence analysis have been extensively established in many existing numerical works. Regarding the non-local model studied in our article, a numerical algorithm can be designed, and numerical stability and convergence analysis can be theoretically established, which can be regarded as open problems. Of course, this nonlocal model is much more challenging to analyze than the parabolic one, although some theoretical techniques may be applicable.