1 Introduction

In this work, we are going to address the following semilinear non-homogeneous Timoshenko system with localized damping

$$\begin{aligned} \rho _1(x)\varphi _{tt} - [k(x)(\varphi _x + \psi )]_x + f_1(\varphi ) +\alpha _1(x)g_1(\varphi _t)&= 0 ~\hbox { in }~ \Omega \times (0,\infty ), \end{aligned}$$
(1.1)
$$\begin{aligned} \rho _2(x)\psi _{tt} - [b(x)\psi _{x}]_x + k(x)(\varphi _x + \psi )+ f_2(\psi ) + \alpha _2(x)g_2(\psi _t)&= 0~\hbox { in }~ \Omega \times (0,\infty ), \end{aligned}$$
(1.2)

subject to initial-boundary conditions

$$\begin{aligned}&\varphi (0,t)=\varphi (L,t)=\psi (0,t)=\psi (L,t)=0,\ \ t\ge 0, \end{aligned}$$
(1.3)
$$\begin{aligned}&\varphi (x,0)=\varphi _0(x),~\varphi _t(x,0)=\varphi _1(x), ~\psi (x,0)=\psi _0(x),~\psi _t(x,0)=\psi _1(x), \ \ x\in \Omega , \end{aligned}$$
(1.4)

where \( \Omega :=(0,L)\), with \(L>0\) denoting the beam length, \(\varphi =\varphi (x,t)\) and \(\psi =\psi (x,t)\) stand for transversal displacement and rotation angle of a filament of the beam, respectively, and

$$\begin{aligned} \rho _1(x)=\rho (x) A(x), \ \ \rho _2(x)=\rho (x) I(x), \ \ k(x)=k'GA(x), \ \ b(x)=EI(x) \end{aligned}$$

are non-constant positive coefficients whose physical meanings are well known; namely, \(\rho (x)\) is the mass density of the material, A(x) is the area of a cross section , I(x) is second moment of a cross section, \(k'\) is the shear factor coefficient, G is the shear modulus, and E is the modulus of elasticity.

The non-constant coefficients \(\rho _1(x), \rho _2(x), k(x)\) and b(x) are assumed to be smooth functions (\(C^\infty (\Omega )\)) and bounded from below and above by positive constants as well as their derivatives of first order; the localized damping coefficients \(\alpha _1(x)\) and \(\alpha _2(x)\) are supposed to be continuous and nonnegative functions in [0, L], while the nonlinear feedback functions \(g_1\) and \(g_2\) are continuous, monotone increasing and zero at the origin. The nonlinear foundations \(f_1\) and \(f_2\) are assumed to growth polynomially under suitable conditions. All the precise assumptions, as well as the well-posed result, will be stated in Sect. 2.

In what concerns the stabilization of the Timoshenko system with two internal frictional dissipations, there are some results related to variations of the above model (1.1)–(1.4) in the literature. Indeed, Raposo et al. [29] proved the exponential stability for a linear problem related to equations (1.1) and (1.2) with constant coefficients \(\rho _{1},\rho _{2},k,b,\alpha _1,\alpha _2>0\), null sources \(f_1=f_2=0\) and full linear damping \(g_1(\varphi _t)= \varphi _t\), \(g_2(\psi _t)=\psi _t\). In this occasion, since the problem is linear, it is used the semigroup theory to prove their stability result, see, e.g., [29, Theorem 3.1]. In 2014, both Cavalcanti et al. [8] and Santos et al. [30] considered equations (1.1) and (1.2) with constant coefficients \(\rho _{1},\rho _{2},k,b>0\), null sources \(f_1=f_2=0\) and two locally distributed nonlinear damping \(\alpha _1(x)g_1(\varphi _t)\) and \(\alpha _2(x)g_2(\psi _t)\). In both works [8, 30], it is proved that the stability of the corresponding energy is driven by a nonlinear ODE, see, for instance, [8, Theorem 3.1] and [30, Theorem 3.2]. To their proofs, keeping in mind that \(f_1=f_2=0\), it is employed the reduction principle [16] where the problem of decay rates with nonlinear damping is reduced to an appropriate stabilizability inequality for the linear system. In addition, stabilizability inequality for the linearly damped problem is achieved through an observability inequality obtained for the conservative system. Equivalence of these two observability inequalities owns its validity to the fact that the control action is bounded via locally distributed internal feedbacks. This allows the authors in [8] to engage the so called \( B^* L\) methodology presented in [25], whereas in [30] it is used contradiction arguments along with proper cut-off functions, once the observability inequality is obtained for the conservative system in both papers. More recently, Fatori et al. [19] and Ma et al. [27] have been considered problem (1.1)–(1.2) with constant coefficients \(\rho _{1},\rho _{2},k,b,\alpha _1,\alpha _2>0\), linear dissipations \(g_1(\varphi _t)= \varphi _t\) and \(g_2(\psi _t)=\psi _t\), and also nonlinear source and external terms \(f_1(\varphi ,\psi )-h_1\), \(f_2(\varphi ,\psi )-h_2\) in the place of \(f_1(\varphi )\), \(f_2(\psi )\), respectively. Since the damping in the latter is full and linear, the long-time behavior of their problems is mainly achieved by means of standard multipliers and perturbed energy, see, e.g., [19, Lemma 4.6] and [27, Lemma 3.4].

The above chronologically referred papers [8, 19, 27, 29, 30] lead us to the main goal of the present article; namely, our purpose is to obtain general and uniform decay rate estimates for the energy corresponding to the semilinear Timoshenko system (1.1)–(1.4) with minimal support for the damping. As expected, when dealing with damping mechanisms for both displacements of system (1.1)–(1.2), we do not need to assume any relation on the coefficients. This yields a substantial generalization of the works [8, 19, 27, 29, 30] at the levels of result and method. In fact, since \(f_1\ne 0\) and \(f_2\ne 0\), the aforementioned methodologies employed by [8, 29, 30] are no longer valid here. Also, once we are dealing with locally distributed nonlinear damping \(\alpha _1(x)g_1(\varphi _t)\) and \(\alpha _2(x)g_2(\psi _t)\), the multiplier technique explored in [19, 27] fails as well because of the terms in the energy estimates that cannot be directly absorbed (this is not the case when the damping functions are supported on the whole domain as, for example, in [19, 27, 29]). When \(f_1=f_2=0\), in order to overcome this latter situation, special cut-off functions can be introduced in order to eliminate undesirable terms of higher order as considered previously in the works [8, 30]. In addition, since the coefficients \(\rho _1(x), \rho _2(x), k(x)\) and b(x) may depend on the x-variable, a new method is required and, for this purpose, the microlocal analysis (see, for instance, [5, 6, 20, 21, 26] and references therein) seems to be a very useful tool to prove a desired nonlinear observability inequality to problem (1.1)-(1.4) and, consequently, uniform decay rates estimates for the corresponding energy.

It seems to be the first time that microlocal analysis is invoked to stabilize a semilinear Timoshenko system where other methods can not be applied at a first glance. The next table summarizes the new contribution of the present paper at the levels of generality of problem (1.1)-(1.4) and the methodology for stability, when compared with [8, 19, 27, 29, 30].

Summary of Timoshenko systems with fully internal frictional damping

Paper

Nonlinear sources

Variable coefficients

Localized damping coefficients

Nonlinear damping functions

Methodology

Raposo et al. [29]

NO

NO

NO

NO

Linear Semigroup

Cavalcanti et al. [8]

NO

NO

YES

YES

Observability, \( B^* L\) method

Santos et al. [30]

NO

NO

YES

YES

Observability, cut-off mult.

Fatori et al. [19]

YES

NO

NO

NO

Perturbed Energy

Ma et al. [27]

YES

NO

NO

NO

Perturbed Energy

Present Article

YES

YES

YES

YES

Observability, microlocal analysis

In conclusion, our main goal in the present paper is to work in a more general situation than [8, 29, 30] and still considering the minimum amount of supported damping to approach the localized semilinear Timoshenko problem (1.1)–(1.4). To do so, we follow the same spirit of [8, 30] with the difference that here, for the first time, we are going to exploit the microlocal analysis for 1-D systems of Timoshenko type.

There is another scenario in Timoshenko systems involving partially damped problems. In this case, it is necessary to transfer dissipation from one displacement (equation) to another and, therefore, the uniform stability of the whole system will require an assumption on the coefficients usually called equal speeds of wave propagation. However, such types of partially damped systems represent a different situation and comparisons with them will not be given herein.

Our main results (Theorem 3.2 and Proposition 3.3) will be stated in Sect. 3, and their proofs will be given in Sect. 4 right after. In Sect. 5, we finish this paper with some remarks on the geodesics that can be built for the Timoshenko system.

2 Assumptions and well-posedness

Let us start by considering the following phase spaces

$$\begin{aligned} {\mathcal {H}}=H_0^1(\Omega )\times L^2(\Omega )\times H_0^1(\Omega )\times L^2(\Omega ) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {V}}=[H_0^1(\Omega )\cap H^2(\Omega )]\times H_0^1(\Omega )\times [H_0^1(\Omega )\cap H^2(\Omega )]\times H_0^1(\Omega ). \end{aligned}$$

Under the assumptions on the functions \(\rho _{1},\rho _{2},b,k\), then \({\mathcal {H}}\) is a Hilbert space with norm given by

$$\begin{aligned} \Vert (u,v,w,z)\Vert ^2_{{\mathcal {H}}}=\Vert \sqrt{\rho _1}v\Vert ^2+\Vert \sqrt{\rho _2}z\Vert ^2+ \Vert \sqrt{b}w_x\Vert ^2+ \Vert \sqrt{k}(u_x+w)\Vert ^2, \end{aligned}$$

for all \((u,v,w,z)\in {\mathcal {H}},\) where \(\left\Vert \cdot \right\Vert \) stands for the usual \(L^2(\Omega )\)-norm.

Denoting by \(v=\varphi _t\), \(z=\psi _t\), \(U= (\varphi ,v,\psi ,z)\) and \(U_0=(\varphi _0,\varphi _1,\psi _0,\psi _1)\), then system (1.1)–(1.4) can be rewritten as follows

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\mathrm{d}U}{\mathrm{d}t}=AU+BU+F U, \ \ t>0,\\ U(0)=U_0, \end{array} \right. \end{aligned}$$
(2.1)

where the operators A, B and F are defined by

$$\begin{aligned}&AU=\left( v,\frac{1}{\rho _1}[k(u_x+w)]_x,z,\frac{1}{\rho _2}[bw_x]_{x}-\frac{k}{\rho _2}(u_x+w)\right) , \ \ U\in D(A)={\mathcal {V}}, \end{aligned}$$
(2.2)
$$\begin{aligned}&BU=\left( 0,-\frac{\alpha _1}{\rho _1}g_1(v),0,-\frac{\alpha _2}{\rho _2}g_2(z)\right) ,\ \ U\in D(B)={\mathcal {H}},\end{aligned}$$
(2.3)
$$\begin{aligned}&FU=\left( 0,-\frac{1}{\rho _1}f_1(\varphi ),0,-\frac{1}{\rho _2}f_2(\psi )\right) ,\ \ U\in D(F)={\mathcal {H}}. \end{aligned}$$
(2.4)

For the sake of completeness, we provide the notion of solutions for the Cauchy problem (2.1) according to Barbu [2]. This is already recalled for Timoshenko systems in [8, 30].

Definition 2.1

One says that \(U:[0,\infty [\rightarrow {\mathcal {H}}\) is a strong solution for (2.1) if U is continuous on \([0,\infty [\) and Lipschitz on every compact subset of \(]0,\infty [\), U(t) is differentiable a.e. on \(]0,\infty [\) and

$$\begin{aligned} \dfrac{\mathrm{d}U}{\mathrm{d}t}(t)=AU(t)+BU(t)+ F U(t)~\hbox { for a.e. }~ t\in ]0,\infty [. \end{aligned}$$

Definition 2.2

One says that \(U:[0,\infty [\rightarrow {\mathcal {H}}\) is an integral solution for (2.1) if U is continuous on \([0,\infty [\), \(U(0)=U_0\) and the following inequality holds

$$\begin{aligned} \frac{1}{2}\left\Vert U(t)-\Psi \right\Vert _{{\mathcal {H}}}^2\le \frac{1}{2}\left\Vert U(s)-\Psi \right\Vert _{{\mathcal {H}}} ^2+\int \limits _s^t{(A\Psi +BU(r)+ F U(r),U(r)-\Psi )_{{\mathcal {H}}}}dr, \end{aligned}$$

for every \(\Psi \in D(A)\) and \(0\le s\le t<\infty \), where \((\cdot ,\cdot )_{\mathcal {H}}\) stands for the inner product of \({\mathcal {H}}\).

Now, we provide the assumptions on the localized damping coefficients, the dissipative feedback functions and the nonlinear source terms as follows.

Assumption 2.1

Let us consider nonnegative functions \(\alpha _1,\alpha _2\in C[0,L]\) such that

$$\begin{aligned} \alpha _i(x)\ge \alpha _0>0,~x\in I_i,i=1,2, \end{aligned}$$
(2.5)

where \(I_1,I_2\) are open intervals contained in [0, L] satisfying \(\omega :=I_1\cap I_2\ne \emptyset \).

Assumption 2.2

In addition to Assumption 2.1, we suppose that \(\omega \) geometrically controls \(\Omega =(0,L)\), that is, there exists \(T_{0} >0\) such that every geodesic of the metric \(G_i(x)\), \(i=1,2\), where \(G_1=\left( \frac{k}{\rho _1}\right) ^{-1}\),  \(G_2=\left( \frac{b}{\rho _2}\right) ^{-1}\), traveling with speed 1 and issued at \(t = 0\), enters the set \(\omega \) in a time \(t < T_{0}\).

Assumption 2.3

The functions \(g_1,g_2\) are continuous, monotone increasing and satisfy

$$\begin{aligned}&g_i( s)s>0 \ \ \text{ for } \ \ s\ne 0,\end{aligned}$$
(2.6)
$$\begin{aligned}&k_i\,s^{2}\le g_i\left( s\right) s \le K_i\,s^{2} \ \ \text{ for } \ \ \left| s\right| >1, \end{aligned}$$
(2.7)

for some positive constants \(k_i\) and \(K_i\), \(i=1,2.\)

Assumption 2.4

The nonlinear terms \(f_1,f_2 \in C^2({\mathbb {R}})\) satisfy

$$\begin{aligned} \begin{aligned} ~f_i(0)=0, \ \ \ |f_i^{(j)}(s)| \le k_{0}(1 + |s|)^{p-j},~i=1,2, ~j=1,2, ~p>1,\ \forall s \in {\mathbb {R}}, \end{aligned} \end{aligned}$$
(2.8)

and their primitives \(F_i(s) = \int \limits _{0}^{s} f_i(\tau )d\tau \), \(i=1,2,\) verify

$$\begin{aligned} \begin{aligned} 0 \le F_i(s) \le f_i(s)s, \ \ \forall \, s \in {\mathbb {R}}, i=1,2. \end{aligned} \end{aligned}$$
(2.9)

Assumption 2.5

For every \(T > 0\), the only solutions \((\eta , \xi )\) lying in the space \(C(]0,T[;\,{\mathcal {H}})\,\cap \,C(]0,T[;\,{\mathcal {V}}^\prime ),\) to system

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x)\eta _{tt} - (k(x)(\eta _x + \zeta ))_x + V_1(x,t)=0~\hbox { in }~ (0,L)\times (0,T),\\&\rho _2(x)\zeta _{tt} - (b(x)\zeta _{x})_x + k(x)(\eta _x + \zeta )+ V_2(x,t)= 0~\hbox { in }~ (0,L)\times (0,T),\\&\eta =\zeta =0 ~\hbox { in }~ \omega \times (0,T) \end{aligned} \right. \end{aligned}$$
(2.10)

where \(V_i(x,t)\in \,L^\infty (]0,T[;\,L^1((0,L)), i=1,2\) are the trivial solutions \((\eta ,\xi )=(0,0).\)

Under the above assumptions, we make some comments as follows.

Remark 2.1

(a) Assumptions 2.1 and 2.3 are motivated by [8, 30].

(b) Assumption 2.4 is relatively standard for perturbation of wave-like systems. In particular, from (2.8) and the mean value theorem, there exist constants \(C_i>0\) such that

$$\begin{aligned} \begin{aligned} \left| f_i(r)-f_i(s) \right| \le C_i\left( 1+ |r|^{p-1} + |s|^{p-1} \right) |r-s|, \ \ \forall \, r,s\in {\mathbb {R}}, i=1,2. \end{aligned} \end{aligned}$$
(2.11)

(c) For \(V_i(x,t)\in \,L^\infty (]0,T[;\,L^n((0,L))\) Assumption 2.5 follows from the pioneer work of Ruiz [28]. According to Koch and Tataru [22] (see Theorem 8.15), in the more general case where \(V_i(x,t)\in \,L^{\frac{n+1}{2}}(]0,T[;\,L^{\frac{n+1}{2}}(0,L))\), the unique continuation result follows locally. Hence, under the conditions specified in [22], Assumption 2.5 is fulfilled.

(d) Assumption 2.2 is the so-called geometric control condition (G.C.C.) and it will be only used for the stability result. It is well-known that it is a necessary and sufficient for stabilization and control of the linear wave equation (see [1, 7, 10, 11, 18] and references therein). Since in the present paper we do not have any control of the geodesics because of the inhomogeneous medium, we assume that such an assumption must be considered, namely:

For all geodesic \(t\in I \mapsto x(t) \in \Omega \) of the metric \(G_1=\left( \frac{k}{\rho _1}\right) ^{-1}\)   (or \(G_2=\left( \frac{b}{\rho _2}\right) ^{-1}\)), with \(0\in I\), there exists \(t\ge 0\) such that \(\alpha _1(x(t))>0\) (or \(\alpha _2(x(t))>0\)).

We also observe that if \(G_1(x)=G_2(x)=constant\), the above condition holds true, once the bicharacteristics \(t\mapsto \left( t, x(t), \tau , -\tau G(x(t)){\dot{x}}(t)\right) \) (or its projection on \(\Omega \times (0,T)\)) are straight lines. On the other hand, for a general metric the above condition cannot hold if, for instance, G admits trapped bicharacteristics. See Fig. 1.

Fig. 1
figure 1

At the left side the bicharacteristic (or its projection on \(\Omega \times (0,T)\)) is a straight line which enters the region \(\omega \times (0,T)\). At the right side it is illustrated a trapped bicharacteristic that does not reach the damped area \(\omega \times (0,T)\)

Full size image

Under the above notations, assumptions and remarks, we are going to consider the existence and uniqueness result for the Cauchy problem (2.1), under the light of nonlinear semigroup method.

Theorem 2.3

Let us suppose that Assumptions  2.1, 2.3 and  2.4 hold. Then, if \(U_0 \in {\mathcal {H}}\), problem (2.1) has a unique integral solution. Moreover, if \(U_0\in {\mathcal {V}}\), the solution is strong one.

Proposition 2.4

Under the assumptions of Theorem 2.3, let us take \(U_0 \in {\mathcal {H}}\) and consider \(U\in C([0,\infty [;{\mathcal {H}})\) the respective unique integral solution of (2.1). Then, there exists a sequence of strong solutions \(U_n\) of (2.1) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }U_n=U~\hbox { in }~C([0,T];{\mathcal {H}}), \ \ \forall \, T>0. \end{aligned}$$

Remark 2.2

To prove Theorem 2.3 and Proposition 2.4, we follow the same idea as presented in [8, 30], by including the nonlinear source terms whose Assumption  2.4 leads them to locally Lipschitz perturbations. Indeed, from the definitions of operators A and B in (2.2)–(2.3) and using Assumptions 2.1 and 2.3, it is not so difficult to prove that A and B satisfy the hypotheses of [4, Theorems 3.1]. See also [2, 3] for integral solutions. In addition, under Assumption  2.4 (see also (2.11)), one proves that operator F set in (2.4) is a locally Lipschitz perturbation (see Theorems 7.1 and 7.2 in [15]). Therefore, the existence and uniqueness of integral and strong solutions follow from the general theory given by [2,3,4] as well as the statement that every integral solution can be obtained as limit of strong solutions. Summarizing, problem (1.1)–(1.4) is well posed.

3 Asymptotic behavior: main results

3.1 Energy relation

We first introduce the energy functional associated to solutions of system (1.1)–(1.4) for all \(t\ge 0\), namely,

$$\begin{aligned} E_{U}(t) =\int \limits _0^L\left\{ \frac{\rho _1}{2}\varphi _t^2(t)+\frac{\rho _2}{2}\psi _t^2(t)+\frac{b}{2}\psi _x^2(t)+ \frac{k}{2}(\varphi _x+\psi )^2(t)+ F_1(\varphi (t))+ F_2(\psi (t))\right\} \mathrm{d}x. \end{aligned}$$
(3.1)

As a consequence of Assumptions 2.1, 2.3 and  2.4, one can conclude without difficulties that the energy (3.1) is a monotone non-increasing (and nonnegative) function, which is stated in the next result.

Lemma 3.1

Let \(U=(\varphi ,\varphi _t,\psi ,\psi _t)\) be an integral (or strong) solution of (1.1)–(1.4). Then, the energy satisfies

$$\begin{aligned} \dfrac{\mathrm{d}E_{U}}{\mathrm{d}t}(t)=-\int \limits _0^L\Big \{\alpha _1(x)g_1(\varphi _t(t))\varphi _t(t)+\alpha _2(x)g_2(\psi _t(t))\psi _t(t)\Big \} \mathrm{d}x\le 0, \ \ \forall \, t>0. \end{aligned}$$
(3.2)

As a consequence, it follows that

$$\begin{aligned} E_{U}(t)=E_{U}(0)-\int \limits _0^t\int \limits _0^L\Big \{\alpha _1(x)g_1(\varphi _t(s))\varphi _t(s)+\alpha _2(x)g_2(\psi _t(s))\psi _t(s)\Big \}\mathrm{d}x ds,~t\ge 0. \end{aligned}$$
(3.3)

Proof

Let us first consider a strong solution \(U=(\varphi ,\varphi _t,\psi ,\psi _t)\) of (1.1)–(1.4). Thus, multiplying (1.1) by \(\psi _t\), (1.2) by \(\varphi _{t}\), integrating by parts on \(\Omega \) and adding the resulting expressions, we easily get (3.2) and, consequently, (3.3) after integrating on (0, t). From this and Proposition 2.4, the conclusion also holds true for the integral solution. \(\square \)

3.2 Nonlinear ODE relation

Before stating our main result, let us first introduce a nonlinear ODE that shall drive the energy stability. Such a constructive methodology was firstly introduced by Lasiecka and Tataru [23] and, subsequently, by some authors. See, for instance, [9, 13, 14] for wave models and [8, 30] for the technique adapted to Timoshenko systems, from where we follow the same reasoning lines.

Assumption 2.3 allows us, according to [23] (see also [8, 9, 13, 14, 30]), to consider a function \(h:{\mathbb {R}}\rightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} h(s)=h_1(s)+h_2(s), \ \ s\in {\mathbb {R}}, \end{aligned}$$

where \(h_1, h_2\) are continuous, concave, strictly increasing real functions satisfying

$$\begin{aligned} h_i(g_i(s)s)\ge s^2+g^2_i(s), \ \ \left|s\right|\le 1, \ \ i=1,2. \end{aligned}$$
(3.4)

In this way, h has the same properties as its composing functions \(h_1\) and \(h_2\). In addition, we define the auxiliary function r by

$$\begin{aligned} r(s)=h\left( \frac{s}{LT}\right) , \ \ s\in {\mathbb {R}}, \ \ T>0, \end{aligned}$$
(3.5)

and so it is not difficult to see that \(cI+r\) is invertible for any constant \(c\ge 0\), once r is a monotone increasing function. Thus, for nonnegative constants c and M we define

$$\begin{aligned} p(s)=(cI+r)^{-1}(Ms), \ \ s\in {\mathbb {R}}, \end{aligned}$$
(3.6)

and observe that it is a continuous, positive and strictly increasing function with \(p(0)=0\). Keeping the above construction in mind, we finally define the nonlinear function

$$\begin{aligned} q(s)=s-(I+p)^{-1}(s), \ \ s\in {\mathbb {R}}, \end{aligned}$$

and introduce the ODE driven by q as follows

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\mathrm{d}S}{\mathrm{d}t}(t)+q(S(t))=0, \ \ t>0 \\&S(0)=s_0.\\ \end{aligned}\right. \end{aligned}$$
(3.7)

Under the above construction, and relying on the results of [23], if p satisfies \(p(s)>0\) for \(s>0\), then it possible to prove the following stabilization limit for S:

$$\begin{aligned} \lim _{t\rightarrow \infty }S(t) = 0. \end{aligned}$$

3.3 Main results

Under the above notations and previous hypotheses, we are now in position to state our main result on decay rates of the energy associated with problem (1.1)–(1.4). More precisely, we have:

Theorem 3.2

(Asymptotic Behavior) Let us suppose that Assumptions 2.1, 2.2, 2.3 and 2.4 hold and let \(K>0\) be any positive constant such that the initial energy \(E_U(0)\le K\). Then, there exists a time \(T_0>0\) such that

$$\begin{aligned} E_U(t) \le {S}\bigg (\frac{t}{T_0}-1\bigg ), \quad \forall \ t>T_0, \end{aligned}$$
(3.8)

where S(t) is the solution of problem (3.7) with \(s_0=E_U(0)\) and \(\displaystyle \lim _{t\rightarrow \infty }S(t)=0\).

Remark 3.1

We observe that although the decay rate (3.8) depends on the size of the initial energy, there are several examples of functions \(g_i, i=1,2,\) satisfying Assumption 2.3. Consequently, there are several explicit decay rates to solutions S of (3.7) as well as to the energy \(E_U\) by means of the inequality (3.8), see, for instance, [13, Section 8] from where we can similarly achieve such concrete decay rates. See also [8, 16, 24, 30].

We mention that the approach to the proof of Theorem 3.2 is different from that presented in [8, 30]. Here, the guiding idea is the one presented in [12] adapted to the context of Timoshenko systems where to prove the stabilization (3.8) reduces itself to prove an observability inequality, whose proof is now based on contradiction arguments in combination with microlocal analysis, see, e.g., Burq and Gérard [6]. Such an idea has also been considered previously in [17, 18] for different models. The result on observability inequality associated with problem (1.1)-(1.4) reads as follows:

Proposition 3.3

(Observability Inequality) Under the assumptions of Theorem 3.2, there exist constants \(T_0>0\) and \(C=C(K,T)>0\) such that

$$\begin{aligned} E_U(0)\le C \int \limits _0^T{\int \limits _0^L \Big \{\alpha _1(x)\big [\varphi _t^2(t) + g^2_1(\varphi _t(t))\big ]+ \alpha _2(x)\big [\psi _t^2(t) + g^2_2(\psi _t(t))\big ]\Big \}\,\mathrm{d}x}\mathrm{d}t, \end{aligned}$$
(3.9)

for all \(T>T_0\).

The proofs of Proposition 3.3 and Theorem 3.2 are given in the next section.

4 Proofs

4.1 Proof of Proposition 3.3

Let us assume that (3.9) does not hold. Then, there exists \(T>0\) (large enough) and a sequence \(U^n=(\varphi ^n, \varphi _t^n, \psi ^n, \psi _t^n)\) of weak solutions of problem (1.1)–(1.4) in \((0,L)\times (0, T)\), that is,

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x)\varphi _{tt}^n - (k(x)(\varphi _x^n + \psi ))_x + f_1(\varphi ^n) +\alpha _1(x)g_1(\varphi _t^n)=0,\\&\rho _2(x)\psi _{tt}^n - (b(x)\psi _{x}^n)_x + k(x)(\varphi _x^n + \psi )+ f_2(\psi ^n) + \alpha _2(x)g_2(\psi _t^n) = 0,\\&\varphi ^n(0,\cdot )=\varphi ^n(L,\cdot )=\psi ^n(0,\cdot )=\psi ^n(L,\cdot )=0,\\&\varphi ^n(\cdot ,0)=\varphi _0^n,~ \varphi _t^n(\cdot ,0)=\varphi _1^n,~ \psi ^n(\cdot ,0)=\psi _0^n, ~\psi _t^n(\cdot ,0)=\psi _1^n, \end{aligned} \right. \end{aligned}$$
(4.1)

such that \(E_{U^n}(0) \le K\) and

$$\begin{aligned} \lim _{n\rightarrow +\infty } \frac{E_{U^n}(0)}{\int \limits _0^T\int \limits _0^L \left( \alpha _1(x)((\varphi _t^n)^2 + (g_1(\varphi _t^n))^2)+ \alpha _2(x)((\psi _t^n)^2 + (g_2(\psi _t^n))^2)\right) \,\mathrm{d}x\mathrm{d}t }=+\infty . \end{aligned}$$
(4.2)

Thus, (4.2) yields

$$\begin{aligned} \lim _{n\rightarrow +\infty } \frac{\int \limits _0^T\int \limits _0^L \left( \alpha _1(x)((\varphi _t^n)^2 + (g_1(\varphi _t^n))^2)+ \alpha _2(x)((\psi _t^n)^2 + (g_2(\psi _t^n))^2)\right) \,\mathrm{d}x\mathrm{d}t}{E_{U^n}(0)}=0. \end{aligned}$$
(4.3)

Now, we define:

$$\begin{aligned} \alpha _n:= \left[ E_{U^n}(0)\right] ^{1/2},\quad {\tilde{\varphi }}^n:= \frac{\varphi ^n}{\alpha _n},\quad {\tilde{\psi }}^n= \frac{\psi ^n}{\alpha _n}. \end{aligned}$$
(4.4)

From (4.1) and (4.4), we can consider the following sequence of normalized problems in \((0,L)\times (0, T)\):

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x){\tilde{\varphi }}_{tt}^n - (k(x)({\tilde{\varphi }}_x^n + {\tilde{\psi }}^n))_x + \frac{1}{\alpha _n}f_1(\alpha _n{\tilde{\varphi }}^n) + \frac{1}{\alpha _n}\alpha _1(x)g_1(\alpha _n {\tilde{\varphi }}_t^n)=0, \\&\rho _2(x){\tilde{\psi }}_{tt}^n - (b(x){\tilde{\psi }}_{x}^n)_x + k(x)({\tilde{\varphi }}_x^n + {\tilde{\psi }}^n)+ \frac{1}{\alpha _n} f_2(\alpha _n {\tilde{\psi }}^n) + \frac{1}{\alpha _n}\alpha _2(x)g_2(\alpha _n {\tilde{\psi }}_t^n) = 0,\\&{\tilde{\varphi }}^n(0)=\frac{\varphi _0^n}{\alpha _n},~ {\tilde{\varphi }}_t^n(0)=\frac{\varphi _1^n}{\alpha _n},~{\tilde{\psi }}^n(0)=\frac{\psi _0^n}{\alpha _n}, ~{\tilde{\psi }}_t^n(0)=\frac{\psi _1^n}{\alpha _n}, \end{aligned} \right. \end{aligned}$$
(4.5)

with the same boundary conditions for \({\tilde{\varphi }}^n\) and \({\tilde{\psi }}^n.\) The functional energy associated with problem (4.5) is given by

$$\begin{aligned}&E_{{\tilde{U}}^n}(t)=\nonumber \\&\int \limits _0^L\left\{ \frac{\rho _1}{2}({\tilde{\varphi }}_t^n)^2+\frac{\rho _2}{2}({\tilde{\psi }}_t^n)^2+\frac{b}{2}({\tilde{\psi }}_x^n)^2+ \frac{k}{2}({\tilde{\varphi }}_x^n+{\tilde{\psi }}^n)^2+ \frac{1}{\alpha _n^2}F_1(\alpha _n{\tilde{\varphi }}^n)+ \frac{1}{\alpha _n^2}F_2(\alpha _n {\tilde{\psi }}^n)\right\} \mathrm{d}x, \end{aligned}$$
(4.6)

for \(t\ge 0\). From (4.4), a straightforward computation shows that

$$\begin{aligned} E_{{\tilde{U}} ^n}(t)= \frac{1}{\alpha _n^2} E_{U^n}(t) \ \ \text{ for } \text{ all } \ \ t\ge 0, \ n\in {\mathbb {N}}, \end{aligned}$$

which implies, in particular, that \(E_{{\tilde{U}} ^n}(0)=1\) for all \(n\in {\mathbb {N}}\). In order to achieve a contradiction, we are going to prove in what follows that \(E_{{\tilde{U}} ^n}(0)\) converges to zero as n goes to infinity.

Firstly, taking (4.3) and (4.4) into account we deduce

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int \limits _0^T\int \limits _0^L \bigg \{\alpha _1(x)\Big [({\tilde{\varphi }}_t^n)^2 +\frac{g_1^2(\alpha _n {\tilde{\varphi }}_t^n)}{\alpha _n^2} \Big ]+ \alpha _2(x)\Big [({\tilde{\psi }}_t^n)^2 + \frac{g_2^2(\alpha _n {\tilde{\psi }}_t^n)}{\alpha _n^2} \Big ]\bigg \}\mathrm{d}x\mathrm{d}t=0. \end{aligned}$$
(4.7)

Further, from the boundedness \(E_{{\tilde{U}} ^n}(0)\le 1\) we also deduce, for an eventual subsequence of \(\{{\tilde{U}}^n\}\), that

$$\begin{aligned}&({\tilde{\varphi }}_t^n, {\tilde{\psi }}_t^n ) \rightharpoonup ({\tilde{\varphi }}_t,{\tilde{\psi }}_t)&\hbox { weakly star in }L^\infty (0,T; (L^2(0,L))^2),\end{aligned}$$
(4.8)
$$\begin{aligned}&({\tilde{\varphi }}^n, {\tilde{\psi }}^n ) \rightharpoonup ({\tilde{\varphi }},{\tilde{\psi }})&\hbox { weakly star in }L^\infty (0,T; (H_0^1(0,L))^2),\end{aligned}$$
(4.9)
$$\begin{aligned}&({\tilde{\varphi }}^n, {\tilde{\psi }}^n ) \rightarrow ({\tilde{\varphi }},{\tilde{\psi }})&\hbox { strongly in } L^\infty (0,T; (C[0,L])^2). \end{aligned}$$
(4.10)

We observe that \(\alpha _n\) is bounded and, in addition, that \(\alpha _n \rightarrow \alpha \in [0,K^{1/2}]\). We shall divide our proof in two cases: \(\alpha > 0\) or \(\alpha = 0\).

Case (i): \(\alpha > 0\).  Passing to the limit in (4.5) and using (4.7) along with assumptions on \(g_i, \, i=1,2,\) we arrive at

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x){\tilde{\varphi }}_{tt} - (k(x)({\tilde{\varphi }}_x + {\tilde{\psi }}))_x + \frac{1}{\alpha }f_1(\alpha {\tilde{\varphi }})=0 ~\hbox { in } ~ (0,L)\times (0,T),\\&\rho _2(x){\tilde{\psi }}_{tt} - (b(x){\tilde{\psi }}_{x})_x + k(x)({\tilde{\varphi }}_x + {\tilde{\psi }})+ \frac{1}{\alpha } f_2(\alpha {\tilde{\psi }})= 0~\hbox { in }~ (0,L)\times (0,T),\\&{\tilde{\varphi }}_t ={\tilde{\psi }}_t=0 ~\hbox { in }~\omega \times (0,T), \end{aligned} \right. \end{aligned}$$
(4.11)

and for \({\tilde{\varphi }}_{t}=\eta \) and \({\tilde{\psi }}_{t}=\zeta \), it holds in the distributional sense

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x)\eta _{tt} - (k(x)(\eta _x + \zeta ))_x + f_1'(\alpha {\tilde{\varphi }})\eta =0~\hbox { in }~ (0,L)\times (0,T),\\&\rho _2(x)\zeta _{tt} - (b(x)\zeta _{x})_x + k(x)(\eta _x + \zeta )+ f_2'(\alpha {\tilde{\psi }})\zeta = 0~\hbox { in }~ (0,L)\times (0,T),\\&\eta =\zeta =0 ~\hbox { in }~ \omega \times (0,T). \end{aligned} \right. \end{aligned}$$
(4.12)

By noting that \(f_1'(\alpha {\tilde{\varphi }}),~ f_2'(\alpha {\tilde{\psi }})\in L^\infty (\Omega \times (0,T))\), then from Assumption 2.5it results that \((\eta ,\zeta )=(0,0),\) which implies that \(({\tilde{\varphi }}_{t},{\tilde{\psi }}_t)=(0,0)\). Returning to (4.11), we infer

$$\begin{aligned} \left\{ \begin{aligned}&- (k(x)({\tilde{\varphi }}_x + {\tilde{\psi }}))_x + \frac{1}{\alpha }f_1(\alpha {\tilde{\varphi }})=0~\hbox { in }~ (0,L)\times (0,T),\\&- (b(x){\tilde{\psi }}_{x})_x + k(x)({\tilde{\varphi }}_x + {\tilde{\psi }})+ \frac{1}{\alpha } f_2(\alpha {\tilde{\psi }})= 0~\hbox { in }~ (0,L)\times (0,T), \end{aligned} \right. \end{aligned}$$

and since \(f_i(s) s\ge 0\) for \(i=1,2\), we obtain

$$\begin{aligned} \left. \begin{aligned} \int \limits _0^L k(x) ({\tilde{\varphi }}_x + {\tilde{\psi }})^2\,\mathrm{d}x + \int \limits _0^L b(x) ({\tilde{\psi }}_x)^2\,\mathrm{d}x\le 0, \end{aligned} \right. \end{aligned}$$

from which we conclude that \(({\tilde{\varphi }}, {\tilde{\psi }})=(0,0)\).

Case (ii): \(\alpha = 0\). In this case, we have \(\alpha _n \rightarrow 0\). From the assumptions on \(f_1\), we note that we can write

$$\begin{aligned} f_1(s) = f_1'(0)s + R_1(s), ~\hbox { where } |R_1(s)| \le C \left( |s|^2 + |s|^p \right) . \end{aligned}$$

Thus,

$$\begin{aligned}&\frac{1}{\alpha _n} f_1(\alpha _n {\tilde{\varphi }}^n) = f_1'(0) {\tilde{\varphi }}^n + \frac{R(\alpha _n {\tilde{\varphi }}^n)}{\alpha _n},\\&\frac{R_1(\alpha _n {\tilde{\varphi }}^n)}{\alpha _n} \le C \left( \alpha _n |{\tilde{\varphi }}^n|^2 + |\alpha _n|^{p-1}|{\tilde{\varphi }}^n|^p \right) . \end{aligned}$$

We observe that

$$\begin{aligned} \alpha _n |{\tilde{\varphi }}^n|^2 + |\alpha _n|^{p-1}|{\tilde{\varphi }}^n|^p \rightarrow 0~\hbox { in }~ L^2(0,T;L^2(0,L)), \end{aligned}$$

since \(\alpha _n \rightarrow 0\) and \(\{\varphi ^n\}\) is bounded in \(L^\infty (0,T;H_0^1(0,L))\). As a consequence, from the compactness Aubin–Lions–Simon Lemma,

$$\begin{aligned} \frac{1}{\alpha _n} f_1(\alpha _n {\tilde{\varphi }}^n) \rightarrow f_1'(0) {\tilde{\varphi }} ~\hbox { strongly in } L^2(0,T;L^2(0,L)). \end{aligned}$$
(4.13)

Proceeding analogously for \(f_2\), we deduce

$$\begin{aligned} \frac{1}{\alpha _n} f_2(\alpha _n {\tilde{\psi }}^n) \rightarrow f_2'(0) {\tilde{\psi }} ~\hbox { strongly in } L^2(0,T;L^2(0,L)). \end{aligned}$$
(4.14)

Passing to the limit in (4.5) and using (4.13)–(4.14), we arrive at

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x){\tilde{\varphi }}_{tt} - (k(x)({\tilde{\varphi }}_x + {\tilde{\psi }}))_x + f_1'(0) {\tilde{\varphi }}=0~\hbox { in }~ (0,L)\times (0,T),\\&\rho _2(x){\tilde{\psi }}_{tt} - (b(x){\tilde{\psi }}_{x})_x + k(x)({\tilde{\varphi }}_x + {\tilde{\psi }})+ f_2'(0) {\tilde{\psi }}= 0~\hbox { in }~ (0,L)\times (0,T),\\&{\tilde{\varphi }}_t ={\tilde{\psi }}_t=0 ~\hbox { in }~\omega \times (0,T), \end{aligned} \right. \end{aligned}$$
(4.15)

and for \({\tilde{\varphi }}_{t}=\eta \) and \({\tilde{\psi }}_{t}=\zeta \), it yields in the distributional sense

$$\begin{aligned} \left\{ \begin{aligned}&\rho _1(x)\eta _{tt} - (k(x)(\eta _x + \zeta ))_x + f_1'(0)\eta =0~\hbox { in }~ (0,L)\times (0,T),\\&\rho _2(x)\zeta _{tt} - (b(x)\zeta _{x})_x + k(x)(\eta _x + \zeta )+ f_2'(0)\zeta = 0~\hbox { in }~ (0,L)\times (0,T)\\&\eta =\zeta =0 ~\hbox { in }~ \omega \times (0,T), \end{aligned} \right. \end{aligned}$$
(4.16)

which implies from Assumption 2.5that \((\eta ,\zeta )=(0,0)\), therefore \(({\tilde{\varphi }}_{t},{\tilde{\psi }}_t)=(0,0)\). Returning to (4.15) and proceeding verbatim what we have done before we deduce that \(({\tilde{\varphi }}, {\tilde{\psi }})=(0,0)\). As a consequence, one has

$$\begin{aligned}&({\tilde{\varphi }}_t^n, {\tilde{\psi }}_t^n ) \rightharpoonup (0,0)~\hbox { weakly star in }L^\infty (0,T; (L^2(0,L))^2),\end{aligned}$$
(4.17)
$$\begin{aligned}&({\tilde{\varphi }}^n, {\tilde{\psi }}^n ) \rightharpoonup (0,0)~\hbox { weakly star in }L^\infty (0,T; (H_0^1(0,L))^2),\end{aligned}$$
(4.18)
$$\begin{aligned}&({\tilde{\varphi }}^n, {\tilde{\psi }}^n )\rightarrow (0,0)~\hbox { strongly in } L^\infty (0,T; (C([0,L])^2). \end{aligned}$$
(4.19)

Let us keep in mind that our objective is to prove that \(E_{U^n}(0)\) converges to zero. For this purpose, let \(P_1\) the wave operator defined by

$$\begin{aligned} P_1:= \rho _1(x) \partial _t^2 - \partial _x(k(x) \partial _{x})=-\frac{1}{\alpha _n}\,f_1(\alpha _n\,{\widetilde{\varphi }}^n)-\frac{1}{\alpha _n}\,\alpha _1(x)\,g_1(\alpha _n\,{\widetilde{\varphi }}^n_t). \end{aligned}$$

First, we will prove that

$$\begin{aligned}&{\tilde{\varphi }}_t^{n} \rightarrow 0 \hbox { (strongly) in }L^2((0,L)\times (0,T)). \end{aligned}$$

Indeed, from above convergence (4.17) we know that

$$\begin{aligned} \begin{aligned} {\tilde{\varphi }}^{n}_t\rightharpoonup 0~~\text{(weakly) } \text{ in }~~L^2((0,L)\times ]0,T[).\\ \end{aligned} \end{aligned}$$
(4.20)

So, let us consider \(\mu _{{\tilde{\varphi }}}\) be the microlocal defect measure (m.d.m.) associated to \(\{{\tilde{\varphi }}_t^{n}\}\) (which is assured by Theorem 5.5 in Burq–Gérard [6], see also [20]). Then, by (4.7), (4.13) and (4.19), we deduce

$$\begin{aligned} \begin{aligned} \partial _t P_1{\tilde{\varphi }}^n= P_1 {\tilde{\varphi }}_t^n \rightarrow 0 ~~\text{(strongly) } \text{ in }~~H^{-1}_{loc}((0,L)\times ]0,T[). \end{aligned} \end{aligned}$$
(4.21)

Analogously, defining

$$\begin{aligned} P_2:= \rho _2(x) \partial _t^2 - \partial _x(b(x) \partial _{x}), \end{aligned}$$

we also infer (similarly as above) that

$$\begin{aligned} \begin{aligned} \partial _t P_2{\tilde{\psi }}^n= P_2 {\tilde{\psi }}_t^n \rightarrow 0 ~~\text{(strongly) } \text{ in }~~H^{-1}_{loc}((0,L)\times ]0,T[). \end{aligned} \end{aligned}$$
(4.22)

Taking into account that \(\omega \) geometrically controls \(\Omega \), we deduce two facts:

(i):

The \(\hbox {supp}(\mu _{{\tilde{\varphi }}})\) is contained in the characteristic set of the wave equation \( \{\tau ^2=\frac{k(x)}{\rho _1(x)} |\xi |^2\}, \) where \(p_1(t,x,\tau ,\xi )=\frac{1}{2}\left( -\tau ^2 + \frac{k(x)}{\rho _1(x)}\xi ^2\right) \) denotes the principal symbol of \(P_1.\)

(ii):

The m.d.m. \(\mu _{{\tilde{\varphi }}}\) propagates along the bicharacteristic flow of this operator, which signifies, particularly, that if some point \(\omega _0=(t_0,{x}_0,\tau _0, {\xi }_0)\) does not belong to \(\hbox {supp}(\mu _{{\tilde{\varphi }}})\), then the whole bicharacteristic issued from \(\omega _0\) is out of \(\hbox {supp}(\mu _{{\tilde{\varphi }}}).\)

Indeed, from (4.21) and Theorem 5.6 in Burq and Gérard [6] we deduce item \(\mathbf{(i)}\). Furthermore, from Proposition 6.2 and Theorem 6.1 found in Burq and Gérard [6], we deduce that \(\hbox {supp}(\mu _{{\tilde{\varphi }}})\) in \((\Omega \times (0,T))\times S^1\), (\(\Omega :=(0,L)\)) is a union of curves like

$$\begin{aligned} t \in I\cap (0,\infty ) \mapsto m\pm (t)=\left( t, x(t), \frac{\pm 1}{\sqrt{1+|G_1(x(t))\dot{x(t)}|}},\frac{\pm G_1(x(t)) \dot{x(t)}}{\sqrt{1+|G_1(x(t))\dot{x(t)}|}} \right) , \end{aligned}$$
(4.23)

where \(t\in I \mapsto x(t)\in \Omega \) is a geodesic associated to the metric \(G_1=(\frac{k}{\rho _1})^{-1}\).

Since by (4.7), we have \({\tilde{\varphi }}_t^{n} \rightarrow 0\) strongly in \(L^2(\omega \times (0,T))\) then, from Remark 5.15 in Burq and Gérard [6], we have that \(\mu _{{\tilde{\varphi }}}=0\) in \(\omega \times (0,T)\) and, consequently,

$$\begin{aligned} \hbox {supp}(\mu _{{\tilde{\varphi }}}) \subset (\Omega \backslash \omega )\times (0,T). \end{aligned}$$

On the other hand, let \(t_0 \in (0,+\infty )\) and let x be a geodesic of \(G_1\) defined near \(t_0\). Once the geodesics inside \(\Omega \backslash \omega \) enter necessarily in the region \(\omega \), then for any geodesic of the metric \(G_1\), with \(0\in I\) there exists \(t >0 \) such that \(m\pm (t)\) does not belong to \(\hbox {supp}(\mu _{\varphi })\), so that \(m\pm (t_0)\) does not belong as well and item \(\mathbf{(ii)}\) follows.

Once the time \(t_0\) and the geodesic x were taken arbitrarily, we conclude that \(\hbox {supp}(\mu _{{\tilde{\varphi }}})\) is empty. Therefore, employing again [6, Remark 5.15], we have

$$\begin{aligned} {\tilde{\varphi }}_t^{n} \rightarrow 0 \hbox { (strongly) in }L_{\text {loc}}^2((0,L)\times (0,T))\,. \end{aligned}$$
(4.24)

Moreover, from (4.7) we claim that

$$\begin{aligned} {\tilde{\varphi }}_t^{n} \rightarrow 0 \hbox { (strongly) in }L^2((0,L)\times (0,T))\,. \end{aligned}$$
(4.25)

In fact, first of all, we observe that

$$\begin{aligned} \begin{aligned} \int \limits _0^T\int \limits _0^L |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t&=\int \limits _0^T\int \limits _\omega |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t+\int \limits _0^T\int \limits _{(0,L)\setminus \omega } |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t\\&:=L_1+L_2\,. \end{aligned} \end{aligned}$$
(4.26)

From (4.7), we have that \(L_1\rightarrow \,0\) when \( k\rightarrow \infty \,. \) For \( L_2 \) consider the following decomposition:

$$\begin{aligned} \begin{aligned} L_2&=\int \limits _0^\varepsilon \int \limits _{(0,L)\setminus \omega } |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t+\int \limits _\varepsilon ^{T-\varepsilon }\int \limits _{(0,L)\setminus \omega } |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t+\int \limits _{T-\varepsilon }^T\int \limits _{(0,L)\setminus \omega } |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t\\&:=J_1+J_2+J_3\,. \end{aligned} \end{aligned}$$
(4.27)

Note that

$$\begin{aligned} J_1=\int \limits _0^\varepsilon \int \limits _{(0,L)\setminus \omega } |{\tilde{\varphi }}_t^{n}|\,\mathrm{d}x\,\mathrm{d}t \le 2\,{\rho _1}^{-1}\,\int \limits _0^\varepsilon E_{{\widetilde{U}}_n}(t)\,\mathrm{d}t \le 2\,{\rho _1}^{-1}\,\varepsilon \,E_{{\widetilde{U}}_n}(0) \le 2\,{\rho _1}^{-1}\,\varepsilon \end{aligned}$$

since \(E_{{\widetilde{U}}_n}(0) \le 1. \) Therefore,

$$\begin{aligned} \lim _{n\rightarrow +\infty }J_1\le 2\,{\rho _1}^{-1}\,\varepsilon \quad \text {for all }T>\varepsilon >0\,. \end{aligned}$$

Since \( \varepsilon >0 \) is arbitrary, it follows that \(\displaystyle \lim _{n\rightarrow +\infty }J_1=0\,.\) Proceeding in the same way, we show that \(\displaystyle \lim _{n\rightarrow +\infty }J_3=0\,.\) Finally, from (4.24) we deduce that \(\displaystyle \lim _{n\rightarrow +\infty }J_2=0\) as \(n\rightarrow \,\infty \). Then, these facts guarantee us the statement given in (4.25), that is, the kinetic part of the energy with respect to the displacement goes to zero.

Proceeding analogously as above, we can also conclude that the kinetic part of the energy with respect to the rotation angle goes to zero, namely,

$$\begin{aligned} {\tilde{\psi }}_t^{n} \rightarrow 0 \hbox { (strongly) in }L^2((0,L)\times (0,T)). \end{aligned}$$
(4.28)

In what follows, we are going to recover the convergence of the potential part of the energy. To do so, let consider the equipartition of the energy; that is, let us first consider \(\theta \in C_{0}^{\infty }(0,T); 0\le \theta \le 1\) and \(\theta =1 ~~\text{ in }~~(\epsilon ,T-\epsilon )\).

Now, let us take the multipliers \({\tilde{\varphi }}^n \theta (t)\) and \({\tilde{\psi }}^n \theta (t)\) in the first and second equations of (4.5), respectively. Then, adding the resulting expression, we get

$$\begin{aligned}&- \int \limits _0^T\theta '(t) \int \limits _{0}^{L} \rho _1 {\tilde{\varphi }}^n_t {\tilde{\varphi }}^n \,\mathrm{d}x\mathrm{d}t - \int \limits _0^T \int \limits _0^L \rho _1 |{\tilde{\varphi }}_t^n|^2\,\mathrm{d}x\mathrm{d}t\nonumber \\&- \int \limits _0^T\theta '(t) \int \limits _0^L \rho _2 {\tilde{\psi }}^n_t {\tilde{\psi }}^n \,\mathrm{d}x\mathrm{d}t - \int \limits _0^T \int \limits _0^L\rho _2 |{\tilde{\psi }}_t^n|^2\,\mathrm{d}x\mathrm{d}t\nonumber \\&+ \int \limits _0^T \theta (t) \int \limits _0^L b(x) |{\tilde{\psi }}_x^n|^2\,\mathrm{d}x\mathrm{d}t + \int \limits _0^T \theta (t) \int \limits _0^L k(x) ({\tilde{\varphi }}_x^n + {\tilde{\psi }}^n)^2\,\mathrm{d}x\mathrm{d}t\nonumber \\&+ \frac{1}{\alpha _n} \int \limits _0^T \theta (t) \int \limits _0^L f_1(\alpha _n {\tilde{\varphi }}^n) {\tilde{\varphi }}^n\,\mathrm{d}x\mathrm{d}t + \frac{1}{\alpha _n} \int \limits _0^T \theta (t) \int \limits _0^L f_2(\alpha _n {\tilde{\psi }}^n) {\tilde{\psi }}^n \, \mathrm{d}x\mathrm{d}t\nonumber \\&+ \frac{1}{\alpha _n} \int \limits _0^T \theta (t) \int \limits _0^L \alpha _1 g_1(\alpha _n {\tilde{\varphi }}_t^n) {\tilde{\varphi }}^n\,\mathrm{d}x\mathrm{d}t + \frac{1}{\alpha _n} \int \limits _0^T \theta (t) \int \limits _0^L \alpha _2 g_2(\alpha _n {\tilde{\psi }}_t^n) {\tilde{\psi }}^n \, \mathrm{d}x\mathrm{d}t. \end{aligned}$$
(4.29)

Taking (4.7), (4.17), (4.18), (4.19), (4.25), (4.28) and (4.29) into account, yields

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L|{\tilde{\varphi }}_x^n|^2\mathrm{d}x\mathrm{d}t= \lim _{n\rightarrow +\infty }\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L| {\tilde{\psi }}_x^n|^2\mathrm{d}x\mathrm{d}t= 0, \end{aligned}$$
(4.30)

and then

$$\begin{aligned} \begin{aligned} {}&\lim _{n\rightarrow +\infty }\frac{1}{\alpha _n}\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L f_1(\alpha _n{\tilde{\varphi }}^n){\tilde{\varphi }}^n \mathrm{d}x\mathrm{d}t=\lim _{n\rightarrow +\infty }\frac{1}{\alpha _n^2}\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L f_1(\varphi ^n)\varphi ^n\,\mathrm{d}x\mathrm{d}t=0,\\ {}&\lim _{n\rightarrow +\infty }\frac{1}{\alpha _n}\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L f_2(\alpha _n{\tilde{\psi }}^n){\tilde{\psi }}^n \mathrm{d}x\mathrm{d}t=\lim _{n\rightarrow +\infty }\frac{1}{\alpha _n^2}\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L f_2(\psi ^n)\psi ^n \,\mathrm{d}x\mathrm{d}t=0. \end{aligned} \end{aligned}$$
(4.31)

Moreover, from (4.31) and regarding assumption (2.9), we conclude

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{1}{\alpha _n^2}\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L F_1(\varphi ^n)\mathrm{d}x\mathrm{d}t=\lim _{n\rightarrow +\infty }\frac{1}{\alpha _n^2}\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L F_2(\psi ^n)\mathrm{d}x\mathrm{d}t=0, \end{aligned}$$

which implies, jointly with all the above convergences, that

$$\begin{aligned} \int \limits _{\epsilon }^{T-\epsilon } E_{{\tilde{U}}^n}(t) \rightarrow 0~~\text{ as } ~~n \rightarrow +\infty . \end{aligned}$$

Since the energy is non-increasing, we obtain

$$\begin{aligned} (T-2\epsilon )E_{{\tilde{U}}^n}(T-\epsilon )\rightarrow 0. \end{aligned}$$

From this and using the energy identify (see (3.3))

$$\begin{aligned}&E_{{\tilde{U}}^n}(T-\epsilon )-E_{{\tilde{U}}^n}(\epsilon )\\&=-\frac{1}{\alpha _n}\bigg [\int \limits _{\epsilon }^{T-\epsilon }\int \limits _0^L\left( \alpha _1(x)\left( |{\tilde{\varphi }}^n_t|^2+g_1(\alpha _n{\tilde{\varphi }}^n_t){\tilde{\varphi }}^n_t\right) + \alpha _2(x)\left( |{\tilde{\psi }}_t^n|^2 + {\tilde{g}}_2(\alpha _n{\tilde{\psi }}_t^n){\tilde{\psi }}_t^n\right) \right) \mathrm{d}x\mathrm{d}t\bigg ] \end{aligned}$$

along with the convergence (4.7), we arrive at

$$\begin{aligned} E_{{\tilde{U}}^n}(\epsilon )\rightarrow 0. \end{aligned}$$
(4.32)

Thus, from (4.32) and using again the energy identify and the limit in (4.7), we finally conclude that

$$\begin{aligned} \begin{aligned} E_{{\tilde{U}}^n}(0)=&-[E_{{\tilde{U}}^n}(\epsilon )-E_{{\tilde{U}}^n}(0)]+ E_{{\tilde{U}}^n}(\epsilon )\\ =&\int \limits _{0}^{\epsilon }\int \limits _0^L\left( \alpha _1(x)\left( |{\tilde{\varphi }}^n_t|^2+g_1(\alpha _n{\tilde{\varphi }}^n_t){\tilde{\varphi }}^n_t\right) + \alpha _2(x)\left( |{\tilde{\psi }}_t^n|^2 + {\tilde{g}}_2(\alpha _n{\tilde{\psi }}_t^n){\tilde{\psi }}_t^n\right) \right) \,\mathrm{d}x\mathrm{d}t \\&+ E_{{\tilde{U}}^n}(\epsilon )\longrightarrow 0. \end{aligned} \end{aligned}$$

Hence, \(E_{{\tilde{U}}^n}(0)\rightarrow 0 \text{ when } n\rightarrow +\infty ,\) which concludes the desired contradiction.

This completes the proof of (3.9) as desired and finishes the proof of Proposition 3.3. \(\square \) The conclusion of the proof of Theorem 3.2 follows the verbatim the same arguments as presented in the references [8, 30]. For the sake of completeness, we also provide it here as follows.

4.2 Proof of Theorem 3.2

Let \(T>T_0\), where \(T_0\) comes from the observability inequality. From (3.9) we have

$$\begin{aligned} E_U(0)\le {\overline{C}}_T \int \limits _0^T\int \limits _0^L \Big \{\alpha _1(x)\big [\varphi _t^2 + g_1(\varphi _t)^2\big ]+ \alpha _2(x)\big [\psi _t^2 + g_2(\psi _t)^2\big ]\Big \}\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$
(4.33)

for some constant \({\overline{C}}_T>0\).

Now, given a strong solution \( U=(\varphi ,\varphi _t,\psi ,\psi _t)\) of (2.1), we define the following sets

$$\begin{aligned} \Sigma _{\varphi }&= \{(x,t)\in \Omega \times (0,T); |\varphi _t(x,t)|>1\} \text{ and } \Gamma _{\varphi } = \Omega \times (0,T)\setminus \Sigma _{\varphi },\\ \Sigma _{\psi }&= \{(x,t)\in \Omega \times (0,T); |\psi _t(x,t)|>1\} \text{ and } \Gamma _{\psi } = \Omega \times (0,T)\setminus \Sigma _{\psi }. \end{aligned}$$

Our strategy here is to estimate the integrals at the right side of (4.33). First, note that

$$\begin{aligned} \int \limits _0^T\int \limits _0^L{\alpha _1(x)(\varphi _t^2 +g_1(\varphi _t)^2)}\mathrm{d}x\mathrm{d}t =&\int \limits _{\Sigma _{\varphi }}\alpha _1(x)(\varphi _t^2+g_1(\varphi _t)^2)\mathrm{d}x\mathrm{d}t \nonumber \\&+\int \limits _{\Gamma _{\varphi }} \alpha _1(x)(\varphi _t^2+g_1(\varphi _t)^2)\mathrm{d}x\mathrm{d}t. \end{aligned}$$
(4.34)

From Assumption 2.1, we have

$$\begin{aligned} \int \limits _{\Sigma _{\varphi }}\alpha _1(x)(\varphi _t^2+g_1(\varphi _t)^2)\mathrm{d}x\mathrm{d}t \le (k^{-1}_1+K_1)\int \limits _{\Sigma _{\varphi }}\alpha _1(x)g_1(\varphi _t)\varphi _t \mathrm{d}x\mathrm{d}t. \end{aligned}$$
(4.35)

Now, using (3.4) we obtain

$$\begin{aligned} \int \limits _{\Gamma _{\varphi }}\alpha _1(x)(\varphi _t^2+g_1(\varphi _t)^2)\mathrm{d}x\mathrm{d}t&\le \int \limits _{\Gamma _{\varphi }}\alpha _1(x)h_1\bigg (g_1(\varphi _t)\varphi _t\bigg )\mathrm{d}x\mathrm{d}t\nonumber \\&\le \int \limits _{\Gamma _{\varphi }}(1+||\alpha _1||_{\infty })h_1\bigg (\frac{\alpha _1}{1+||\alpha _1||_{\infty }}g_1(\varphi _t)\varphi _t\bigg )\mathrm{d}x\mathrm{d}t\nonumber \\&\le (1+||\alpha _1||_{\infty })\int \limits _{\Gamma _{\varphi }} h_1\bigg (\alpha _1(x)g_1(\varphi _t)\varphi _t\bigg )\mathrm{d}x\mathrm{d}t\nonumber \\&\le (1+||\alpha _1||_{\infty })LTh_1\bigg (\frac{1}{LT}\int \limits _0^T\int \limits _0^L{\alpha _1(x)g_1(\varphi _t)\varphi _t} \mathrm{d}x\mathrm{d}t\bigg ) \end{aligned}$$
(4.36)

where the last inequality is obtained using the Jensen’s inequality. Therefore, using (4.35) and (4.36),

$$\begin{aligned} \int \limits _0^T\int \limits _0^L{\alpha _1(x)(\varphi _t^2+g_1(\varphi _t)^2)}\mathrm{d}x\mathrm{d}t&\le (k^{-1}_1+K_1)\int \limits _0^T\int \limits _0^L{\alpha _1(x)g_1(\varphi _t)\varphi _t}\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +\,(1+||\alpha _1||_{\infty })LTh_1\bigg (\frac{1}{LT}\int \limits _0^T\int \limits _0^L{ \alpha _1(x)g_1(\varphi _t)\varphi _t}\mathrm{d}x\mathrm{d}t\bigg ). \end{aligned}$$
(4.37)

Analogously, we can conclude that

$$\begin{aligned} \int \limits _0^T\int \limits _0^L{\alpha _2(x)(\psi _t^2+g_1(\psi _t)^2)}\mathrm{d}x\mathrm{d}t&\le (k^{-1}_2+K_2)\int \limits _0^T\int \limits _0^L{\alpha _2(x)g_2(\psi _t)\psi _t}\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +\,(1+||\alpha _2||_{\infty })LTh_2\bigg (\frac{1}{LT}\int \limits _0^T\int \limits _0^L{\alpha _2(x)g_2(\psi _t)\psi _t} \mathrm{d}x\mathrm{d}t\bigg ). \end{aligned}$$
(4.38)

Since each \(h_i\) is an increasing function, then combining the energy identity, (4.37) and (4.38), we have

$$\begin{aligned} E_ U(T)&\le {\overline{C}}\sum _{i=1}^{2}(k^{-1}_i+K_i)\int \limits _0^T\int \limits _0^L\Big \{\alpha _1(x)g_1(\varphi _t)\varphi _t + \alpha _2(x)g_2(\psi _t)\psi _t\Big \}\mathrm{d}x\mathrm{d}t\nonumber \\&+{\overline{C}}LT\sum _{i=1}^{2}(1+||\alpha _i||_{\infty })r\bigg (\int \limits _0^T\int \limits _0^L{ \alpha _1(x)g_1(\varphi _t)\varphi _t+\alpha _2(x)g_2(\psi _t)\psi _t}\mathrm{d}x\mathrm{d}t\bigg ), \end{aligned}$$

where r was defined in (3.5). Setting the constants

$$\begin{aligned} M=\frac{1}{{\overline{C}}LT\sum _{i=1}^{2}(1+||\alpha _i||_{\infty })} \; \text{ and } \; c=\frac{\sum _{i=1}^{2}(k^{-1}_i+K_i)}{LT\sum _{i=1}^{2}(1+||\alpha _i||_{\infty })}, \end{aligned}$$

and using (3.2), we arrive at

$$\begin{aligned} M E_ U(T)&\le c\int \limits _0L T\int \limits _0^L\Big \{\alpha _1(x)g_1(\varphi _t)\varphi _t + \alpha _2(x)g_2(\psi _t)\psi _t\Big \}\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +r\bigg (\int \limits _0^T\int \limits _0^L{\alpha _1(x)g_1(\varphi _t)\varphi _t+\alpha _2(x)g_2(\psi _t)\psi _t} \mathrm{d}x\mathrm{d}t\bigg )\nonumber \\&=(cI+r)\bigg (E_ U(0)-E_ U(T)\bigg ). \end{aligned}$$

Using the notation introduced in (3.6), the previous inequality can be rewritten as

$$\begin{aligned} p(E_ U(T))\le E_ U(0)-E_ U(T). \end{aligned}$$
(4.39)

To finish the proof, we replace T by \((m+1)T\) (respectively, 0 by mT) in (4.39), \(m\in {\mathbb {N}}\), in order to obtain

$$\begin{aligned} E_ U\big ((m +1)T\big ) +p\big (E_ U\big ((m+1)T\big )\big )\le E_ U(mT), \text{ for } m = 0,1,\cdots . \end{aligned}$$

Thus, using [23, Lemma 3.3] with \(s_m = E_ U(mT)\), we can conclude that

$$\begin{aligned} E_ U(mT )\le S(m), \ \ m= 0, 1,\cdots . \end{aligned}$$

Finally, observing that for every \(t>T\) we can find \(m\in {\mathbb {N}}\) and \(\tau \in [0,T]\) such that \(t = mT +\tau \), then

$$\begin{aligned} E_U(t)\le E_U(mT)\le S(m)\le S\big (\frac{t-\tau }{T}\big )\le S\big (\frac{t}{T}-1\big ) ~\hbox { for all } ~ t>T, \end{aligned}$$

where we have used the fact that the solution S of (3.7) is dissipative.

Therefore, the proof of Theorem 3.2 is complete. \(\square \)

5 Further remarks

We finish this work by giving some examples of 1-D metrics

$$\begin{aligned} G_1=(k/\rho _1)^{-1} \ \ \text{ or } \ \ G_2=(b/\rho _2)^{-1} \end{aligned}$$

that satisfy the geometric control condition. It is enough to choose one them, for example \(G_1\). To this purpose, let us remember that

$$\begin{aligned} p_1(t,x,\tau ,\xi )=\frac{1}{2} \left( \frac{k(x)}{\rho _1(x)}\xi ^2 - \tau ^2\right) \end{aligned}$$

is the principal symbol of the wave operator \(P_1= \rho _1 \partial _t^2 - [k(x)\partial _x]_x\). In order to obtain the Hamiltonian flow, keeping in mind that

$$\begin{aligned} H_{p_1}(t,x,\tau ,\xi )=\left( \frac{\partial p_1}{\partial \tau }, \frac{\partial p_1}{\partial \xi }, -\frac{\partial p_1}{\partial t}, -\frac{\partial p_1}{\partial x} \right) , \end{aligned}$$

one has in our case

$$\begin{aligned} \frac{\partial p_1}{\partial \tau }=-\tau ,~\frac{\partial p_1}{\partial \xi }=\frac{k(x)}{\rho _1(x)}\xi , ~-\frac{\partial p_1}{\partial t}=0,~-\frac{\partial p_1}{\partial x}=-\frac{1}{2} \left( \frac{k(x)}{\rho _1(x)}\right) '\xi ^2, \end{aligned}$$

where the subscript \('\) denotes the x-derivative of the function \(\frac{k(x)}{\rho _1(x)}\). Thus,

$$\begin{aligned} {\dot{t}}(s)=-\tau (x(s)),~{\dot{x}}(s) = \frac{k(x)}{\rho _1(x)}\xi (s),~{\dot{\tau }}(s)=0, ~ {\dot{\xi }}(s) = -\frac{1}{2} \left( \frac{k(x)}{\rho _1(x)}\right) '\xi ^2(s), \end{aligned}$$

where \(\dot{}\) stands for the time derivative.

We observe that once \({\dot{\tau }}(s)=0\), then \(\tau (s)=\tau _0=constant\). Now, being \(p=0\) on each null bicharacteristic, yields

$$\begin{aligned} \frac{k(x)}{\rho _1(x)}\xi ^2(s) = \tau ^2(s) = \tau _0^2, \ \ \forall \, s\in I. \end{aligned}$$

On the other hand from the above relationships, we deduce that

$$\begin{aligned} \left( \frac{\rho _1}{k} \right) (x(s))({\dot{x}}(s))^2 = \left( \frac{k}{\rho _1} \right) (x(s))\xi ^2(s)= \tau _0^2, \ \ \forall \, s\in I, \end{aligned}$$

and from the second identity on the right hand side of the last identity, we obtain

$$\begin{aligned} \xi (s) = \pm \sqrt{\left( \frac{\rho _1}{k}\right) (x(s))}\tau _0, \end{aligned}$$

from which we deduce that

$$\begin{aligned} {\dot{x}}(s) = \pm \sqrt{\left( \frac{k}{\rho _1}(x(s))\right) }\tau _0. \end{aligned}$$
(5.1)

Analogously, one can conclude

$$\begin{aligned} {\dot{x}}(s) = \pm \sqrt{\left( \frac{b}{\rho _2}(x(s))\right) }\tau _0. \end{aligned}$$
(5.2)

The above formulas (5.1) and (5.2) allow us to deduce concrete examples where the geometric control condition (GCC) holds true; that is, Assumption 2.2 is not empty.

Example 5.1

Let us first consider three cases where the GCC is verified.

\(\mathbf{(a)}\):

Assuming \(\frac{k}{\rho _1}(x)=c_0>0\), then \(x(t) = x(0) \pm c_0^{1/2}\tau _0 t\), for some \(x(0)\in (0,L)\) and \(t\in I \cap (0,\infty )\). Hence, the GCG is ensured.

\(\mathbf{(b)}\):

Considering \(\frac{k}{\rho _1}(x)=x\), then we deduce easily that \(x(t) = \left( (x(0))^2 \pm \frac{\tau _0}{2} t\right) ^2\) and, therefore, the CGC also holds true.

\(\mathbf{(c)}\):

If \(\left( \frac{k}{\rho _1}(x)\right) =x^2\) we infer that \(x(t) = \pm |x(0)| e^{\pm \tau _0\,t}\) and the GCG is also verified.

Finally, we consider an example where GCG fails.

Example 5.2

Let us consider a family of circles centering in the x-axis

$$\begin{aligned} f(s,x(s),\lambda ):=(s-2\,\lambda )^2+x(s)^2 - \lambda ^2=0 . \end{aligned}$$
(5.3)

Performing differentiation with respect to s, we have

$$\begin{aligned} \dfrac{s+x(s)\,{\dot{x}}(s)}{2}=\lambda \,. \end{aligned}$$
(5.4)

Substituting (5.4) in (5.3), it follows that

$$\begin{aligned} 3\,x(s)\left[ {\dot{x}}(s)\right] ^2-2\,s\,x(s)\,{\dot{x}}(s)+4\,x^2(s)-s^2=0\,. \end{aligned}$$
(5.5)

Looking (5.5) as a second-order equation in the variable \( {\dot{x}}(s) \), it follows that its roots are

$$\begin{aligned} {\dot{x}}(s)=\frac{2 s {x}(s)\pm \sqrt{-48 {{{x}(s)}^{3}}+4 {{s}^{2}}\, {{{x}(s)}^{2}}+12 {{s}^{2}} {x}(s)}}{6 {x}(s)}. \end{aligned}$$
(5.6)

On the other hand, recalling (5.1), one gets

$$\begin{aligned} {\dot{x}}(s)=\pm \sqrt{\left( \frac{k}{\rho _1}\,(x(s))\right) }\tau _0, \end{aligned}$$

and combining it with (5.6), we infer

$$\begin{aligned} \frac{k}{\rho _1}\,(x(s))=\left( \frac{2\, s\, {x}(s)\pm \sqrt{-48\, {{{x}(s)}^{3}}+4 {{s}^{2}}\, {{{x}(s)}^{2}}+12 {{s}^{2}}\, {x}(s)}}{6 {x}(s)\,\tau _0}\right) ^2\,. \end{aligned}$$
(5.7)

Now, using computational analysis to solve the above nonlinear algebraic equationFootnote 1, we obtain that the solutions of the ODE are of the form:

$$\begin{aligned} x(s)=\pm \sqrt{\dfrac{\pm \,4 i\, s\, \sinh (3\, c_1) - 4\, i\, s \cosh (3\, c_1) + \sinh (6\, c_1) + \cosh (6\, c_1) - 3\, s^2}{{3}}}, \end{aligned}$$
(5.8)

for some real constant \(c_1.\) Since

$$\begin{aligned} x_0=x(0)=\pm \sqrt{\dfrac{ \sinh (6\, c_1) + \cosh (6\, c_1) }{{3}}}=\pm \sqrt{\dfrac{ \mathrm {e}^{6\,c_1}}{3}}, \end{aligned}$$

then

$$\begin{aligned} c_1=\dfrac{\ln (3\,x_0^2)}{6}. \end{aligned}$$
(5.9)

We observe that there are cases for \( \lambda \) where the circle (in the family (5.3)) intercepts the damped area \( \omega \times (0,T) \). Our next goal is to choose suitable values of \( \lambda \) where this situation does not occur. Indeed, fixing \( \lambda \) such that

$$\begin{aligned} |x-2\,\lambda | > \lambda ,\, \forall \,x\in \,\omega , \end{aligned}$$
(5.10)

that is, \( \lambda >x \) or \( \displaystyle \lambda <\frac{x}{3}, \) for all \( x\in \,\omega , \) we conclude that the trapped bicharacteristics associated with the curves (5.3), which are solutions given in (5.8)–(5.9), do not reach the damped area \( \omega \times (0,T) \), taking into account the restriction (5.10) on \( \lambda ,\) see the illustrative case in Fig. f2. Therefore, the GCG fails.

Fig. 2
figure 2

The solutions of trapped bicharacteristics associated with the curves (5.3) do not reach the damped area \( \omega \times (0,T), \) where \( \lambda \) satisfies (5.10)

Full size image