Global existence and asymptotic behavior for a Timoshenko system with internal damping and logarithmic source terms

Abstract

This manuscript deals with a Timoshenko system with damping and source. The existence and stability of the solution are analyzed taking into account the competition of the internal damping versus the logarithmic source. We use the potential well theory. For initial data in the stability set created by the Nehari surface, the existence of global solutions is proved using Faedo–Galerkin’s approximation. The exponential decay is given by the Nakao theorem. A numerical approach is presented to illustrate the results obtained.

1 Introduction

The Timoshenko system is a model widely studied in the scientific community for vibrations of elastic beams. Its mathematical formulation is given by a system of two partial differential equations

$$\begin{aligned} \rho A\varphi _{tt}= & {} [S(x,t)]_x +N_1(x,t),\\ \rho I\psi _{tt}= & {} [M(x,t)]_x-S(x,t)+N_2(x,t), \end{aligned}$$

where the functions $\varphi$ and $\psi$ depending upon $(x,t)\in (0,L)\times (0,T)$ model the transverse displacement of a beam with reference configuration $(0,L)\subset {\mathbb {R}}$ are transverse displacement and the rotations in the transverse sections, respectively. The functions M and S represent, respectively, the bending moment and the shear stress and satisfies

$$\begin{aligned} S(x,t)= & {} kAG \left( \varphi _{x}+\psi \right) ,\\ M(x,t)= & {} EI\psi _{x},\\ N_1(x,t)= & {} \mu _1\varphi \;\ln |\varphi |_{{\mathbb {R}}}^{2},\\ N_2(x,t)= & {} \mu _2\psi \; \ln |\psi |_{{\mathbb {R}}}^{2}, \end{aligned}$$

where $\mu _j> 0$, $j= 1, 2$ and $|\cdot |_{{\mathbb {R}}}$ denote the absolute value of a real number.

The constants, $\rho$ is the mass density, A the cross-sectional area, and I the moment of inertia. To simplify the notation, let us denote by $\rho _1=\rho A,$ $\rho _2=\rho I,$ $\kappa =kAG$ and $b=EI$. Under these conditions, we consider the initial-boundary problem for the following logarithmic Timoshenko System:

$$\begin{aligned}&\rho _1\varphi _{tt} - \kappa \left( \varphi _x+\psi \right) _x + \gamma _1\varphi _t = \mu _1\varphi \;\ln |\varphi |^2\,\,\, \text { in } (0,L)\times (0,\infty ), \end{aligned}$$

(1)

$$\begin{aligned}&\rho _2\psi _{tt} -b\psi _{xx}+ \kappa \left( \varphi _x+\psi \right) +\gamma _2\psi _t = \mu _2\psi \;\ln |\psi |^2\,\,\, \text { in } (0,L)\times (0,\infty ), \end{aligned}$$

(2)

$$\begin{aligned}&\varphi (x,0)=\varphi _0(x),\ \ \varphi _t(x,0) = \varphi _1(x),\,\,\, x\in (0,L), \end{aligned}$$

(3)

$$\begin{aligned}&\psi (x,0)=\psi _0(x),\ \ \psi _t(x,0) = \psi _1(x), \,\,\, x\in (0,L), \end{aligned}$$

(4)

$$\begin{aligned}&\varphi (0,t) = \varphi (L,t)=\psi (0,t)=\psi (L,t) = 0, \,\,\, t\ge 0, \end{aligned}$$

(5)

where $\gamma _i>0$, $i= 1, 2.$ Shear deformation effects were first introduced by Rankine [24] in 1858. Rotary inertia effects were apparently discovered independently by Bresse [6] in 1859 and Rayleigh [25] in 1945. One contributor to developing the theory that takes into account both effects was Paul Ehrenfest, who was cited by Timoshenko [27] in the footnote of his book, in Russian, Course in Elasticity (second volume) in 1916. Nowadays, this celebrated theory is often knowledge by Timoshenko’s paper [28] of 1921. For more detailed historic context, see [10,11,12] with references therein.

The internal damping is associated with an oscillating system and produces a loss of energy to overcome external sources that act in the mechanical resistance of the material. Logarithmic non-linearity is a class of nonlinearities distinguished by several interesting physical properties, see [7]. It appears, for instance, in dynamics of Q-ball in theoretical physics [14], theories of quantum gravity [31], inflationary models [4], and quantum mechanics [5].

There are several studies on this competition, that is, stability analysis of the global solution taking account the effect provoked by the presence of both, stabilizing mechanism and source term. Below, we cite a few. [9] studied the existence and exponential stability of the global solution to a Klein–Gordon equation of Kirchhoff–Carrier type with strong damping and logarithmic source term. An extensible beam equation of Kirchhoff type with internal damping and source term was investigated in [22]. Kirchhoff plate equations with internal damping and logarithmic non-linearity were considered in [23]. General decay result for a plate equation with non-linear damping and a logarithmic source term was established in [2]. For global solution and blow-up of logarithmic Klein–Gordon equation, see [29].

Motivated by the above studies, in this paper, we prove the global existence for the problem (1)–(5) by applying the potential well theory introduced by Payne and Sattinger [20] and Sattinger [26]. Furthermore, we obtain the exponential decay of solution for this problem.

This paper is organized as follows: In the next section, we are going to give some preliminaries. Section 3 deals with potential well theory. We introduce the stability set. In Sect. 4, we prove the existence of global solution. In Sect. 5, we study the exponential decay. Finally, Sect. 6 is devoted to the numerical approach.

2 Preliminaries

We denote $L^{2}(0,L)$ the Hilbert’s space of square-integrable function on the interval (0, L),  with the inner product

$$\begin{aligned} (u,v)=\int _0^Luv\;\textrm{d}x,\quad \forall u,v\in L^2(0,L) \end{aligned}$$

and norm

$$\begin{aligned} |u|^2=(u,u) \quad \forall u\in L^2(0,L). \end{aligned}$$

We use Sobolev space notation and properties as in [1]. We denote

$$\begin{aligned} H^1(0,L)=\{u\mid u \in L^2(0,L),\ u_x\in L^2(0,L)\} \end{aligned}$$

and

$$\begin{aligned} H_0^1(0,L)=\left\{ u\in H^1(0,L)\mid \ u(0)=u(L)=0\right\} . \end{aligned}$$

In this section, we present some results needed for the proof of our results. We start defining the energy functional associated with the problem (1)–(5)

$$\begin{aligned} E(t)&=\frac{1}{2}\left( \rho _1|\varphi _t(t)|^{2}+\rho _2|\psi _t(t)|^{2}+\kappa |\varphi _x(t)+\psi (t)|^{2}+b|\psi _x(t)|^{2}+\mu _1|\varphi (t)|^{2}+\mu _2|\psi (t)|^{2}\right. \nonumber \\&\quad \left. -\mu _1\int _{0}^{L}\varphi ^{2}(t)\;\ln |\varphi (t)|_{{\mathbb {R}}}^{2}\;\textrm{d}x-\mu _2\int _{0}^{L}\psi ^{2}(t)\;\ln |\psi (t)|_{{\mathbb {R}}}^{2}\;\textrm{d}x\right) . \end{aligned}$$

(6)

Direct differentiation of (6) gives us

$$\begin{aligned} \dfrac{\textrm{d}}{\textrm{d}t}{E}(t)= -\gamma _1|\varphi _t(t)|^{2}-\gamma _2\left| \psi _t(t)\right| ^{2}. \end{aligned}$$

(7)

Now, consider the following lemmas:

Lemma 2.1

(Sobolev–Poincaré Inequality) Let p be a number with in $2<p<\infty$ if $n=1, 2$ or $2\le p\le \dfrac{2n}{n-2}$ if $n\ge 3,$ then there exist a constant $C>0$, such that

$$\begin{aligned} ||u||_{p}\le C|u_x|, \;\;\;\; \forall \;\; u\in H_0^{1}(0,L). \end{aligned}$$

(8)

Lemma 2.2

(Aubin–Lions compactness Theorem [16], Theorem 5.1) Let $T>0$, $1< p_0, p_1 < \infty$. Consider $B_0 \subset B \subset B_1$ Banach spaces, $B_0, B_1$ reflexives, $B_0$ with compact embedding in B. Define $W=\{u\,\,|\,\,u \in L^{p_0}(0,T; B_0)\,,\,u_t \in L^{p_1}(0,T; B_1) \}$ equipped with the norm $||u||_{W} = ||u||_{L^{p_0}(0,T; B_0)} + ||u_t||_{L^{p_1}(0,T; B_1)}.$ Then, W has compact embedding in $L^{p_0}(0,T; B)$.

Lemma 2.3

(Lions [16], Lemma 1.3 ) Let $Q=\varOmega \times (0,T),\,\, T>0$ a bounded open set of ${\mathbb {R}}^n \times {\mathbb {R}}$ and $g_m, g\,:\, Q \rightarrow {\mathbb {R}}$ functions of $L^p(0,T;L^p(\varOmega ))= L^p(Q), \, 1< p < \infty$, such that $||g_m||_{L^{p}(Q)} \le C,\,\,\,g_m \rightarrow g\,\,\text{ a.e. } \text{ in }\,\,Q$. Then, $g_m \rightharpoonup g \,\,\,\text{ in }\,\,\,L^p(Q) \,\,\text{ as }\,\, m \rightarrow \infty$.

Lemma 2.4

(Nakao’s Lemma) [18] Suppose that $\phi (t)$ is a bounded nonnegative function on ${\mathbb {R}}^+$, satisfying

$$\begin{aligned} \mathop {\text {sup ess}}\limits _{t \le s \le t+1} \phi (s) \le C_0\left[ \phi (t)-\phi (t+1)\right] , \end{aligned}$$

for any $t\ge 0$, where $C_0$ is a positive constant. Then

$$\begin{aligned} \phi (t) \le C e^{-\alpha t}, \forall ~t\ge 0, \end{aligned}$$

where C and $\alpha$ are positive constants.

3 The potential well

In this section, we present the potential well corresponding to the Eqs. (1)–(2). We define the operator $J\;:\;\bigg (H_0^1(0,L)\bigg )^{2}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} J(\varphi ,\psi )&{\mathop {=}\limits ^{\textrm{def}}}&\dfrac{1}{2}\left[ \kappa | \varphi _x+\psi |^2+b| \psi _x |^{2} +\mu _1|\varphi |^{2}+\mu _2|\psi |^{2}- \mu _1\int _{0}^{L} \varphi ^{2}\; \ln \;|\varphi |_{{\mathbb {R}}}^{2}\textrm{d}x-\mu _2\int _{0}^{L} \psi ^{2}\;\ln \;|\psi |_{{\mathbb {R}}}^{2}\textrm{d}x \right] . \end{aligned}$$

For $(\varphi ,\;\psi )\in \bigg (H_0^1(0,L)\bigg )^{2}$ and $\lambda >0$, we have

$$\begin{aligned} J(\lambda \varphi ,\lambda \psi )&{\mathop {=}\limits ^{\textrm{def}}} \dfrac{\lambda ^2}{2}\left[ \kappa | \varphi _x+\psi |^2+b| \psi _x |^{2} +\mu _1|\varphi |^{2}+\mu _2|\psi |^{2}-2\mu _1\ln \;\lambda \int _{0}^{L}\varphi ^{2}\;\textrm{d}x\right. \\&\quad \left. - \mu _1\int _{0}^{L}\varphi ^{2}\ln \;|\varphi |_{{\mathbb {R}}}^{2}\;dx-2\mu _1\ln \;\lambda \int _{0}^{L}\psi ^{2}\;\textrm{d}x-\mu _2\int _{0}^{L}\psi ^{2}\;\ln \;|\psi |_{{\mathbb {R}}}^{2}\;\textrm{d}x\right] . \end{aligned}$$

Associated with J, we have the well-known Nehari Manifold

$$\begin{aligned} {\mathcal {N}} {\mathop {=}\limits ^{def}} \left\{ \big (\varphi ,\psi \big )\in \big (H_0^1(0,L)\big )^{2}/\{0\} ;\;\; \ \ \left[ \dfrac{\textrm{d}}{\textrm{d}\lambda }J(\lambda \varphi ,\lambda \psi )\right] _{\lambda =1} =0 \right\} . \end{aligned}$$

Equivalently

$$\begin{aligned} {\mathcal {N}} = \left\{ \big (\varphi ,\psi \big )\in \left( H_0^1(0,L)\right) ^{2} ;\;\kappa | \varphi _x+\psi |^2+b| \psi _x |^{2} = \mu _1\int _{0}^{L}\varphi ^{2}\;\ln \;|\varphi |_{{\mathbb {R}}}^{2}\; \textrm{d}x +\mu _2\int _{0}^{L}\psi ^{2}\;\ln \;|\psi |_{{\mathbb {R}}}^{2}\; \textrm{d}x \right\} . \end{aligned}$$

We define as in the Mountain Pass theorem due to Ambrosetti and Rabinowitz [3]

$$\begin{aligned} d {\mathop {=}\limits ^{\textrm{def}}} \inf _{\big (\varphi ,\psi \big )\in \big (H_0^1(0,L)\big )^{2}/\{0\} } \sup _{\lambda > 0}J( \lambda u). \end{aligned}$$

According to Willem [30], Theorem 4.2, the depth of the well d is a strictly positive constant given by

$$\begin{aligned} 0<d=\inf _{\varphi ,\psi \;\in {\mathcal {N}}}J(\lambda u). \end{aligned}$$

Now, we introduce

$$\begin{aligned} W = \left\{ \big (\varphi ,\psi \big )\in \big (H_0^1(0,L\big )^{2};\ \ J(\varphi ,\psi )< d \right\} \cup \{0\}, \end{aligned}$$

and partition it into two sets as follows:

$$\begin{aligned} W_1=\left\{ \big (\varphi ,\psi \big )\in W ;\;\kappa | \varphi _x+\psi |^2+b| \psi _x |^{2}> \mu _1\int _{0}^{l}\varphi ^{2}\;\ln \;|\varphi |_{{\mathbb {R}}}^{2}+\mu _2\int _{0}^{l}\psi ^{2}\;\ln \;|\psi |_{{\mathbb {R}}}^{2} \right\} \cup \{0\} \end{aligned}$$

and

$$\begin{aligned} W_2=\left\{ \big (\varphi ,\psi \big )\in W ;\;\kappa | \varphi _x+\psi |^2+b| \psi _x |^{2}< \mu _1\int _{0}^{l}\varphi ^{2}\;\ln \;|\varphi |_{{\mathbb {R}}}^{2}+\mu _2\int _{0}^{l}\psi ^{2}\;\ln \;|\psi |_{{\mathbb {R}}}^{2} \right\} . \end{aligned}$$

Therefore, we define by $W_1$ the set of stability for the problem (1)–(5).

Proposition 3.1

Let $\big (\varphi _0,\psi _0\big )\in W_1$ and $\big (\varphi _1,\psi _1\big )\in \big (L^2(0,L)\big )^{2}$. If $E(0) < d$, then $(\varphi ,\psi ) \in W_{1}.$

Proof

We introduce the functional $I(\varphi ,\psi )$ given by

$$\begin{aligned} I(\varphi ,\psi )&{\mathop {=}\limits ^{\textrm{def}}}&\dfrac{1}{2}\left[ \kappa | \varphi _x+\psi |^2+b| \psi _x |^{2} - \mu _1\int _{\varOmega } \varphi ^{2}\;\textrm{ln} \;|\varphi |_{{\mathbb {R}}}^{2}\textrm{d}x-\mu _2\int _{\varOmega } \psi ^{2}\;\textrm{ln} \;|\psi |_{{\mathbb {R}}}^{2}\textrm{d}x\right] . \end{aligned}$$

Let $T >0$. From (7), we get

$$\begin{aligned} E(t) \le E(0) < d,\,\, \text{ for } \text{ all } \,\, t \in [0,T), \end{aligned}$$

and then

$$\begin{aligned} \dfrac{1}{2}\left[ |\varphi _t(t)|^{2} + |\psi _t(t)|^{2} \right] + J(\varphi (t),\psi (t)) < d,\,\, \text{ for } \text{ all } \,\, t \in [0,T). \end{aligned}$$

(9)

Note that in $W_{1}$, we have $I(\varphi (t),\psi (t)) > 0$ for all $t \in (0,T)$. Arguing by contradiction, we suppose that there exists a first $t_0 \in (0,T)$, such that $I(\varphi (t_0),\psi (t_0) )=0$ and $I(\varphi (t),\psi (t)) > 0$ for all $0 \le t < t_0$, that is

$$\begin{aligned} \dfrac{1}{2}\left[ |\varphi _t(t_0)|^{2} + |\psi _t(t_0)|^{2}\right] + J(u(t_0),v(t_0)) = 0. \end{aligned}$$

From the definition of ${\mathcal {N}}$, we have that $(\varphi (t_0),\psi (t_0)) \in {\mathcal {N}}$, which leads to

$$\begin{aligned} J(\varphi (t_0),\psi (t_0)) \ge \inf _{(\varphi (t),\psi (t)) \in {\mathcal {N}}} J(u(t),v(t))= d. \end{aligned}$$

We deduce

$$\begin{aligned} \dfrac{1}{2}\left[ |\varphi _t(t_0)|^{2} + |\psi _t(t_0)|^{2} \right] + J(\varphi (t_0),\psi (t_0)) \ge d, \end{aligned}$$

which contradicts with (9). Then, $(\varphi (t),\psi (t)) \in W_{1}$ for all $t \in [0,T)$. $\square$

4 Existence of global weak solution

In this section, we prove the existence of global weak solutions.

Theorem 4.1

Let $\big (\varphi _0,\psi _0\big )\in W_1,\; E(0)<d$ and $\big (\varphi _1,\psi _1\big )\in \big (L^2(0,L)\big )^{2}$. Then, the problem (1)–(5) admits a weak solution $\big (\varphi , \psi \big )$ in the class

$$\begin{aligned}&\big (\varphi , \psi \big ) \in \left( L_\textrm{loc}^{\infty }\left( 0,\infty ; H_0^1(0,L)\right) \right) ^{2}&\end{aligned}$$

(10)

$$\begin{aligned}&\big (\varphi _t, \psi _t \big ) \in \left( L_\textrm{loc}^ {\infty }\left( 0,\infty ; L^2(0,L)\right) \right) ^{2} \end{aligned}$$

(11)

satisfying $w,z\in H_0^1(0,L)$

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\big (\rho _1\varphi _{t}(t),w\big ) +\big ( \kappa (\varphi _x+\psi ) (t),w_x) + \big (\gamma _1\varphi _t(t),w\big )-\big (\mu _1\varphi (t)\;\ln |\varphi (t)|_{{\mathbb {R}}}^2 ,w\big )=0, \end{aligned}$$

(12)

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\big (\rho _2\psi _{t}(t),z\big ) +\big ( b\psi _{x} (t),z_x\big ) +\big (\kappa (\varphi _x+\psi )(t),z\big )+ \big (\gamma _2\psi _t(t),z\big )-\big (\mu _2\psi (t)\;\ln |\psi (t)|_{{\mathbb {R}}}^2 ,z\big )=0, \end{aligned}$$

(13)

$$\begin{aligned}&\big (\varphi ,\psi \big )(x,0)=\big (\varphi _0,\psi _0\big ), \end{aligned}$$

(14)

$$\begin{aligned}&\big (\varphi _t,\psi _t\big )(x,0)=\big (\varphi _1,\psi _1\big ), \end{aligned}$$

(15)

in ${\mathcal {D}}'(0,T)$.

Proof

We use the Faedo–Galerkin’s method. The proof of the global existence of solutions will be made in three steps: approximated problem, a priori estimates, and passage to the limit. $\square$

4.1 Approximated problem

Let $(w_\nu )_{\nu \in {\mathbb {N}}}$ be a basis of $H_0^1(0,L)$ from the eigenvectors of the operator $-\varDelta$, and

$$\begin{aligned} V_m=\text {span}\left\{ w_1,\;w_2,\;\ldots ,w_m\right\} . \end{aligned}$$

Consider

$$\begin{aligned} \varphi ^m(t) = \sum _{j=1}^m g_{jm}(t)w_j\;\;\;\; \text{ and }\;\;\;\;\psi ^m(t) = \sum _{j=1}^m h_{jm}(t)w_j\;\; \end{aligned}$$

a solution of the approximated problem

$$\begin{aligned}&\left( \rho _1\varphi _{tt}^m(t),w\right) + \left( \kappa (\varphi ^{m}_x(t)+\psi ^{m}(t)),w_x\right) + (\gamma _1\varphi _t^m(t), w) -\left( \mu _1\varphi ^m(t)\;\ln |\varphi ^m(t)|^2 ,w\right) =0, \end{aligned}$$

(16)

$$\begin{aligned}&(\rho _2\psi _{tt}^m(t),z) + \left( b\psi ^{m}_{x}(t),z_x\right) +\left( \kappa (\varphi ^{m}_x(t)+\psi ^{m}(t)),z\right) + (\gamma _2\psi _t^m(t), z) -\left( \mu _2\psi ^m(t)\;\ln |\psi ^m(t)|^2 ,z\right) =0, \end{aligned}$$

(17)

$$\begin{aligned}&\left( \varphi ^m(0),\psi ^m(0)\right) =( \varphi _{0m},\psi _{0m}) \longrightarrow ( \varphi _{0},\psi _{0}) \text { strongly in } \left( H_0^1(0,l)\right) ^2, \end{aligned}$$

(18)

$$\begin{aligned}&\left( \varphi _t^m(0),\psi _t^m(0)\right) = ( \varphi _{1m},\psi _{1m}) \longrightarrow ( \varphi _{1},\psi _{1}) \text { strongly in } \left( L^2(0,l)\right) ^{2}, \end{aligned}$$

(19)

$\forall ~w,z\in V_m$. By virtue of Carathéodory’s theorem, see [8], the system (16) has a local solution in $[0,t_m)$, $0<t_m \le T$. The extension of the solution to the whole interval [0, T] is a consequence of the following a priori estimates.

4.2 A priori estimates

Let $w=\varphi _t^m(t)$ and $z=\psi _t^m(t)$ in (16) and (17), respectively. Then, we have

$$\begin{aligned}{} & {} \dfrac{1}{2}\dfrac{{\text {d}}}{{\text {d}}t}\left[ \rho _1\left| \varphi _t^m(t)\right| ^2 +\rho _2\left| \psi _t^m(t)\right| ^2+ \kappa \left| \varphi ^{m}_x(t)+\psi ^{m}(t)\right| ^{2}+b\left| \psi _{x}^m(t)\right| ^2+\mu _1\left| \varphi ^m(t)\right| ^2+\mu _2\left| \psi ^m(t)\right| ^2 \right. \\{} & {} \quad \left. -\mu _1\int _{0}^{L}\varphi ^m(t)^2\;\ln \;\left| \varphi ^m(t)\right| _{{\mathbb {R}}}^2 -\mu _2\int _{0}^{L}\psi ^m(t)^2\;\ln \;\left| \psi ^m(t)\right| _{{\mathbb {R}}}^2\;{\text {d}}x \right] +\gamma _1\left| \varphi _t^m(t)\right| ^2 +\gamma _2\left| \psi _t^m(t)\right| ^2= 0. \end{aligned}$$

From (6), we have

$$\begin{aligned} \dfrac{{\text {d}}}{{\text {d}}t}E_m(t)+\gamma _1\left| \varphi _t^m(t)\right| ^2 +\gamma _2\left| \psi _t^m(t)\right| ^2= 0, \end{aligned}$$

(20)

where $E_{m}(t)$ is the approximated energy of the problem (16). Now, integrating (20) from 0 to t, $0\le t \le t_m$, we obtain

$$\begin{aligned} E_{m}(t)+\gamma _1\int _0^t \left| \varphi _t^m(t)\right| ^2\textrm{d}s+\gamma _2\int _0^t \left| \psi _t^m(s)\right| ^2\textrm{d}s {\text {d}}s=E_m(0). \end{aligned}$$

(21)

Thus

$$\begin{aligned}{} & {} E_{m}(t)+\gamma _1\int _0^t |\varphi _t^m(s)|^2{\text {d}}s+\gamma _2\int _0^t \left| \psi _t^m(s)\right| ^2 {\text {d}}s= \rho _1\left| \varphi _{1m}\right| ^2 +\rho _2\left| \psi _{1m}\right| ^2+ \kappa \left| \varphi _{0mx}+\psi _{0m}\right| ^{2}\\{} & {} \quad +b\left| \psi _{0mx}\right| ^2+\gamma _1\left| \varphi _{0m}\right| ^2+\gamma _2\left| \psi _{0m}\right| ^2-\mu _1\int _{0}^{L}\varphi _{0m}^2\;\ln \;\left| \varphi _{0m}\right| _{{\mathbb {R}}}^2-\mu _2\int _{0}^{L}\psi _{0m}^2\;\ln \;\left| \psi _{0m}\right| _{{\mathbb {R}}}^2\;\textrm{d}x, \end{aligned}$$

which gives us the following estimate:

$$\begin{aligned} E_{m}(t)+\gamma _1\int _0^t \left| \varphi _t^m(s)\right| ^2{\text {d}}s+\gamma _2\int _0^t \left| \psi _t^m(s)\right| ^2 {\text {d}}s\le & {} \rho _1\left| \varphi _{1m}\right| ^2 +\rho _2\left| \psi _{1m}\right| ^2+J\left( \varphi _{0m},\psi _{0m}\right) . \end{aligned}$$

We have that $J(\varphi _{0m},\psi _{0m})<d,$ and then, by (16), we get

$$\begin{aligned} E_{m}(t)+\mu _1\int _0^t \left| \varphi _t^m(s)\right| ^2{\text {d}}s+\mu _2\int _0^t \left| \psi _t^m(s)\right| ^2 {\text {d}}s\le & {} C_1, \end{aligned}$$

(22)

where $C_1$ is a positive constant independent of m and t.

These estimates imply that the approximated solution $(\varphi ^{m},\psi ^{m})$ exists globally in $[0,\infty ).$ See [13]. Then, by estimate (22), we have

$$\begin{aligned}&\left( \varphi ^m\right) ,\left( \psi ^m\right) \,\,\,\text {are bounded in }\,\, L_\textrm{loc}^{\infty }\left( 0,T;H_0^1(0,L)\right) \end{aligned}$$

(23)

$$\begin{aligned}&\left( \varphi _{t}^m\right) ,\left( \psi _{t}^m\right) \,\,\,\text {are bounded in } \,\, L_\textrm{loc}^{\infty }\left( 0,T;L^{2}(0,L)\right) . \end{aligned}$$

(24)

Now, by the logarithmic inequality

$$\begin{aligned} \left| t^{2}\;\ln t\right| \le C\left( 1+|t|^{3}\right) , \end{aligned}$$

we get

$$\begin{aligned}{} & {} \mu _1 \int _{0}^{L}\left| \varphi ^{m}(t)\;\ln \left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^2\right| ^{2}\;\textrm{d}x=4\mu _1\int _{0}^{L}\left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^2\;\ln \left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^2\;\textrm{d}x\nonumber \\{} & {} \quad =4\mu _1\int _{x\in (0,L);\;\left| \varphi ^{m}\right| <1}\left| \varphi ^m(t)\right| ^2_{{\mathbb {R}}}\ln \left| \varphi ^m(t)\right| ^2\textrm{d}x+4\mu _1\int _{x\in (0,L);\;\left| \varphi ^{m}\right| \ge 1}\left| \varphi ^m(t)\right| ^2_{{\mathbb {R}}}\;\ln \left| \varphi ^m(t)\right| ^2\textrm{d}x\nonumber \\{} & {} \quad \le 4\mu _1\int _{0}^{L}\left| \varphi ^m(t)\right| ^2_{{\mathbb {R}}}\;\textrm{d}x+4\mu _1\int _{0}^{L}\left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^4\;\ln \left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^2\;\textrm{d}x \le 4\mu _1 \left| \varphi ^m(t)\right| ^2+4\mu _1C\int _{0}^{L}\left( 1+\left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^6\right) \;\textrm{d}x\nonumber \\{} & {} \quad =4\mu _1\left| \varphi ^m(t)\right| ^2+4\mu _1CL+C\left| \varphi ^{m}(t)\right| ^6_2 \le \mu _1\left| \varphi ^m(t)\right| ^2+CL+C\left| \varphi ^{m}(t)\right| ^6 \le {\tilde{C}}_1. \end{aligned}$$

(25)

Analogously, we have

$$\begin{aligned} \mu _2\int _{0}^{L}| \varphi ^{m}(t)\;\ln \left| \varphi ^{m}(t)\right| _{{\mathbb {R}}}^2|^{2}\;\textrm{d}x \le {\tilde{C}}_2, \end{aligned}$$

(26)

where ${\tilde{C}}_1$ and ${\tilde{C}}_2$ are constant independent of m and t. From (25) and (26), we get

$$\begin{aligned}&\varphi ^m \ln |\varphi |_{{\mathbb {R}}}^{2}\,\,\,\text {are bounded in }\,\, L_\textrm{loc}^{2}\left( 0,\infty ;L^2(0,L)\right) , \end{aligned}$$

(27)

$$\begin{aligned}&\psi ^m \ln |\psi |_{{\mathbb {R}}}^{2}\,\,\,\text {are bounded in }\,\, L_\textrm{loc}^{2}\left( 0,\infty ;L^2(0,L)\right) . \end{aligned}$$

(28)

4.3 Passage to the limit

From estimates (23) and (24), there exists a subsequence of $(\varphi ^m)$, $(\psi ^m)$ also denoted by $(\varphi ^m)$, $(\psi ^m)$, such that

$$\begin{aligned}&(\varphi ^m), (\psi ^m) {\mathop {\rightharpoonup }\limits ^{~*~}} \varphi , \psi \text { weakly star in } L_\textrm{loc}^\infty \left( 0,\infty ;H_0^{1}(0,L)\right) , \end{aligned}$$

(29)

$$\begin{aligned}&(\varphi ^m_t), (\psi ^m_t) {\mathop {\rightharpoonup }\limits ^{~*~}}\varphi _t, \psi _t \text { weakly in } L_\textrm{loc}^{\infty };L^{2}(0,L)). \end{aligned}$$

(30)

Applying the Aubin–Lions compactness Theorem (Lemma 2.2), we get from (29) and (30)

$$\begin{aligned} \left( \varphi ^m\right) ,\left( \psi ^m\right) \longrightarrow \varphi ,\psi \text { strongly in }L_\textrm{loc}^{2}\left( 0,\infty ;L^{2}(0,L)\right) ,&\end{aligned}$$

(31)

and for all $T>0$

$$\begin{aligned}&\left( \varphi ^m\right) \longrightarrow \varphi \text { a.e in } (0,L)\times (0,T) \end{aligned}$$

(32)

$$\begin{aligned}&\left( \psi ^m\right) \longrightarrow \psi \text { a.e in } (0,L) \times (0,T). \end{aligned}$$

(33)

Now, since that $f(s)=s\;ln|s|^{2}$ is continuous, we have the convergence

$$\begin{aligned} \mu _1 \varphi ^m \;\ln \left| \varphi ^m\right| _{{\mathbb {R}}}^2\longrightarrow \mu _1\varphi \;\ln |\varphi |_{{\mathbb {R}}}^2\text { a.e in } (0,L)\times (0,T) \end{aligned}$$

(34)

and

$$\begin{aligned} \mu _2\psi ^m \;\ln \left| \psi ^m\right| _{{\mathbb {R}}}^2\longrightarrow \mu _2\psi \;\ln |\psi |_{{\mathbb {R}}}^2\text { a.e in } (0,L)\times (0,T). \end{aligned}$$

(35)

From (27), (28), (34), and (35) using the Lions’s Lemma (Lemma 2.3), we obtain

$$\begin{aligned} \mu _1\varphi ^m \;\ln \left| \varphi ^m\right| _{{\mathbb {R}}}^2\rightharpoonup \mu _1\varphi \;\ln |\varphi |_{{\mathbb {R}}}^2\text { weakly in } L_\textrm{loc}^2\left( 0,\infty ;L^2(0,L)\right) \end{aligned}$$

(36)

and

$$\begin{aligned} \mu _2\psi ^m \;\ln \left| \psi ^m\right| _{{\mathbb {R}}}^2\rightharpoonup \mu _2\psi \;\ln |\psi |_{{\mathbb {R}}}^2\text { weakly in } L_\textrm{loc}^2\left( 0,\infty ;L^2(0,L)\right) . \end{aligned}$$

(37)

By the convergences (23), (24), (34) and (35), we can pass to the limit in the approximate system (16) and (17) and obtain for all $w,z\in H_0^{1}(0,L)$

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\left( \rho _1\varphi _{t}(t),w\right) +\left( \kappa (\varphi _x+\psi ) (t),w_x\right) + \left( \gamma _1\varphi _t(t),w\right) -\left( \mu _1\varphi (t)\;\ln |\varphi (t)|_{{\mathbb {R}}}^2 ,w\right) =0, \end{aligned}$$

(38)

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t}\big (\rho _2\psi _{t}(t),z\big ) +\big ( b\psi _{x} (t),z_x\big ) +\big (\kappa (\varphi _x+\psi )(t),z\big )+ \big (\gamma _2\psi _t(t),z\big )-\big (\mu _2\psi (t)\;\ln |\psi (t)|_{{\mathbb {R}}}^2 ,z\big )=0,\nonumber \\{} & {} \quad \text{ in } \; \ \ {\mathcal {D}}'(0,T). \end{aligned}$$

(39)

The verification of the initial data is obtained in a standard way.

5 Exponential decay

In this section, we provide the exponential decay of the energy associated with the system solution (1)–(5).

Theorem 5.1

Under the hypothesis of Theorem 4.1, the energy associated with problem (1)–(5) satisfies

$$\begin{aligned} E(t)\le C_{0}\; e^{-\alpha t}, \;\;\;\; \forall \; t\ge 0, \end{aligned}$$

where $C_0\;\text{ and }\; \alpha$ are positive constants.

Proof

Let $w=\varphi _{t}(t)$ and $z=\psi _{t}(t)$ in (38) and (39), respectively, and summing up the result, we obtain

$$\begin{aligned}{} & {} \dfrac{1}{2}\dfrac{\textrm{d}}{\textrm{d}t}\left[ \rho _1|\varphi _{t}(t)|^2+\rho _2|\psi (t)|^2+\kappa |\varphi _x(t)+\psi (t)|^2)+b|\psi _{x}(t)|^2+\mu _1|\varphi (t)|^{2}+\mu _2|\psi (t)|^{2}\;\;\;\;\;\right. \nonumber \\{} & {} \quad \left. - \mu _1 \int _{0}^{L}\; |\varphi (t)|_{{\mathbb {R}}}^2\; \ln |\varphi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x - \mu _2 \int _{0}^{L}\; |\psi (t)|_{{\mathbb {R}}}^2\; \ln |\psi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x\right] +\gamma _1|\varphi _{t}(t)|^2+\gamma _2|\psi _{t}(t)|^2=0,\qquad \end{aligned}$$

(40)

that is

$$\begin{aligned} \dfrac{\textrm{d}}{\textrm{d}t}E(t)+\gamma _1|\varphi _{t}(t)|^2+\gamma _2|\psi _{t}(t)|^2\le 0, \end{aligned}$$

(41)

where E(t) is define in (6). Integrating (40) from t to $t+1,$ we obtain

$$\begin{aligned} \int _{t}^{t+1}\; \left[ \gamma _1|\varphi _{t}(s)|^2+\gamma _2|\psi _{t}(s)|^2\right] \;\textrm{d}s\le E(t)- E(t+1)\; {\mathop {:=}\limits ^{\textrm{def}}} F^2(t); \end{aligned}$$

(42)

therefore, there exist $t_1\in \bigg [t,t+\dfrac{1}{4}\bigg ]$ and $t_2\in \bigg [t+\dfrac{3}{4},t+1\bigg ]$, such that

$$\begin{aligned} \gamma _1\left| \varphi _{t}(t_i)\right| ^2+\gamma _2\left| \psi _{t}(t_i)\right| ^2\le 4 F(t_i), \;\;\;\;\;\;\;\; i=1, 2. \end{aligned}$$

(43)

Let $w=\varphi (t)$ and $z=\psi (t)$ in (38) and (39), respectively. Summing the result, we get

$$\begin{aligned}{} & {} b| \psi _{x}(t)|^{2}+\kappa | \varphi _x(t)+\psi (t)|^{2}-\mu _1\int _{0}^{L}\; (\varphi (t))^2\; \ln |\varphi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x-\mu _2\int _{0}^{L}\; (\psi (t))^2\; \ln |\psi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x\nonumber \\{} & {} \quad =-\dfrac{\textrm{d}}{\textrm{d}t}\rho _1\left( \varphi _{t}(t),\varphi (t)\right) +\rho _1|\varphi _{t}(t)|^{2}-\dfrac{\textrm{d}}{\textrm{d}t}\rho _2\left( \psi _{t}(t),\psi (t)\right) +\rho _2\left| \psi _{t}(t)\right| ^{2}-\gamma _1\left( \varphi _{t}(t),\varphi (t)\right) \nonumber \\{} & {} \qquad -\gamma _2\left( \psi _{t}(t),\psi (t)\right) . \end{aligned}$$

(44)

Integrating (44) from $t_1$ to $t_2$, and using (43), we obtain

$$\begin{aligned}{} & {} \int _{t_1}^{t_2}\left[ b\left| \psi _{x}(t)\right| ^{2}+\kappa \left| \varphi _x(t)+\psi (t)\right| ^{2}-\mu _1\int _{0}^{L}\; \left( \varphi (t)\right) ^2\;\ln |\varphi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x-\mu _2\int _{0}^{L}\; (\psi (t))^2\; \ln |\psi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x\right] \;\textrm{d}s\\{} & {} \quad \le \rho _1\left| \varphi _t(t_1)\right| \left| \varphi (t_1)\right| +\rho _1\left| \varphi _t(t_2)\right| \left| \varphi (t_2)\right| +\rho _2\left| \psi _t(t_1)\right| \left| \psi (t_1)\right| +\rho _2\left| \psi _t(t_2)\right| \left| \psi (t_2)\right| \\{} & {} \qquad +\rho _1\int _{t_1}^{t_2}\left| \varphi _t(s)\right| ^{2}\textrm{d}s+\rho _2\int _{t_1}^{t_2}\left| \psi _t(s)\right| ^{2}\textrm{d}s+\gamma _1\int _{t_1}^{t_2}\left| \varphi _t(s)\right| \;\left| \varphi (s)\right| \;\textrm{d}s+\gamma _2\int _{t_1}^{t_2}|\psi _t(s)|\;|\psi (s)|\;\textrm{d}s; \end{aligned}$$

therefore

$$\begin{aligned}{} & {} \int _{t_1}^{t_2}\left[ b| \psi _{x}(t)|^{2}+\kappa \left| \varphi _x(t)+\psi (t)\right| ^{2}-\mu _1\int _{0}^{L}\; (\varphi (t))^2\; \ln |\varphi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x-\mu _2\int _{0}^{L}\; (\psi (t))^2\; \ln |\psi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x\right] \;\textrm{d}s\nonumber \\{} & {} \quad \le C_1\left[ F(t)\mathop {\text {sup ess}}\limits _{t\le s\le t+1}E^{1/2}(s)+\dfrac{1}{4}\mathop {\text {sup ess}}\limits _{t\le s\le t+1}E(s)+F^2(t)\right] {\mathop {:=}\limits ^{\textrm{def}}} G^2(t), \end{aligned}$$

(45)

where $C_1=C_1(\rho _1,\rho _2,\gamma _1,\gamma _2)>0$ is a constant. Now, from (42) and (45), we get

$$\begin{aligned}{} & {} \int _{t_1}^{t_2}\left[ \rho _1|\varphi _{t}(t)|^2+\rho _2|\psi (t)|^2+b| \psi _{x}(t)|^{2}+\kappa | \varphi _x(t)+\psi (t)|^{2}-\mu _1\int _{0}^{L}\; (\varphi (t))^2\; \ln |\varphi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x\right. \nonumber \\{} & {} \quad \left. -\mu _2\int _{0}^{L}\; (\psi (t))^2\; \ln |\psi (t)|_{{\mathbb {R}}}^2\;\textrm{d}x\right] \;\textrm{d}s\le 2\left[ F^2(t)+ G^2(t)\right] ; \end{aligned}$$

(46)

thus, there exists $t^{*}\in [t_{1},t_{2}]$, such that

$$\begin{aligned}{} & {} \rho _1\left| \varphi _{t}(t^{*})\right| ^2+\rho _2\left| \psi (t^{*})\right| ^2+b\left| \psi _{x}(t^{*})\right| ^{2}+\kappa \left| \varphi _x(t^{*})+\psi (t^{*})\right| ^{2}-\mu _1\int _{\varOmega }\; \left( \varphi (t^{*})\right) ^2\; \ln \left| \varphi (t^{*})\right| _{{\mathbb {R}}}^2\;\textrm{d}x\nonumber \\{} & {} \quad -\mu _2\int _{0}^{L}\; \left( \psi (t^{*})\right) ^2\; \ln |\psi (t^{*})|_{{\mathbb {R}}}^2\;\textrm{d}x\le C_2\left[ F^2(t)\;+\; G^2(t)\right] . \end{aligned}$$

(47)

We deduce

$$\begin{aligned} \left| \varphi (t^{*})\right| ^{2}+\left| \psi (t^{*})\right| ^{2}\le C_3\left[ \left| \varphi _x(t^{*})+\psi (t^{*})\right| ^{2}+\left| \psi _{x}(t^{*})\right| ^{2}\right] . \end{aligned}$$

(48)

By (47) and (48), we have

$$\begin{aligned} E(t^{*})\le C_4\left[ F^2(t)+ G^2(t)\right] . \end{aligned}$$

(49)

Since that E(t) is increasing, by (42), (48), and (49), we obtain

$$\begin{aligned} \mathop {\text {sup ess}}\limits _{t\le s\le t+1}\; E(s)\le & {} E(t^{*})+\int _{t}^{t+1}\left[ \gamma _1|\varphi _{t}(s)|^{2}+\gamma _2|\psi _{t}(s)|^{2}\right] \;\textrm{d}s\\\le & {} C_5\left[ F^2(t)\;+\; G^2(t)\right] \\\le & {} C_6\left[ F(t)\mathop {\text {sup ess}}\limits _{t\le s\le t+1}\; E^{1/2}(s)+F^{2}(t)+\dfrac{1}{4}\mathop {\text {sup ess}}\limits _{t\le s\le t+1}\; E(s)\right] \\\le & {} C_7F^2(t)+\dfrac{1}{2}\mathop {\text {sup ess}}\limits _{t\le s\le t+1}\; E(s). \end{aligned}$$

Hence, by Nakao’s Lemma (Lemma 42)

$$\begin{aligned} \mathop {\text {sup ess}}\limits _{t\le s\le t+1}\; E(s)\;\le \;C_8F^2(t)=C_9[E(t)-E(t+1)], \end{aligned}$$

where $C_{i}=1, 2,\ldots , 9$ are positive constants. By Lemma (2.4), we conclude

$$\begin{aligned} E(t)\le C_0e^{-\alpha t},\;\;\;\forall \; t\ge 0, \end{aligned}$$

where $C_0$ and $\alpha$ are positive constants. $\square$

6 Numerical approach

6.1 Variational formulation

Here, we use a representation to the functions $\varphi , \psi$ and logarithmic source terms by component vectorial

$$\begin{aligned} {{\textbf {u}}}=[\varphi , \psi ]^\top \,\,\text { and}\,\, {\mathcal {F}}({{\textbf {u}}})=\left[ \mu _1\varphi \ln |\varphi |^2_{{\mathbb {R}}},\mu _2\psi \ln |\psi |^2_{{\mathbb {R}}}\right] ^{\top }. \end{aligned}$$

Thus, from (1) to (5), we get the following variational problem:

$$\begin{aligned} ({{\textbf {u}}}_{tt}(t),\tilde{{{\textbf {u}}}})+ a_1({{\textbf {u}}}(t),\tilde{{{\textbf {u}}}})+a_2({{\textbf {u}}}_{t}(t),\tilde{{{\textbf {u}}}})= & {} a_3({\mathcal {F}}({{\textbf {u}}}),\tilde{{{\textbf {u}}}}), \end{aligned}$$

(50)

where ${{\textbf {u}}}$ satisfies the initial conditions

$$\begin{aligned} ({{\textbf {u}}}(0),\tilde{{{\textbf {u}}}})=({{\textbf {u}}}_0,\tilde{{{\textbf {u}}}}), \quad ({{\textbf {u}}}_t(0),\tilde{{{\textbf {u}}}})=({{\textbf {u}}}_1,\tilde{{{\textbf {u}}}}). \end{aligned}$$

(51)

Here

$$\begin{aligned} \left( {{\textbf {u}}}_{tt}(t),\tilde{{{\textbf {u}}}}\right)&=\rho _1\left( \varphi _t, u_1\right) +\rho _2\left( \psi _t, u_2\right) ,\\ a\left( {{\textbf {u}}}(t),\tilde{{{\textbf {u}}}}\right)&=\kappa \left( \varphi _x+\psi ,u_{1,x}+u_2\right) +b\left( \psi _x,u_{2,x}\right) ,\\ \left( {{\textbf {u}}}_{t}(t),\tilde{{{\textbf {u}}}}\right)&=\gamma _1\left( \varphi _{t},u_1\right) +\gamma _2\left( \psi _{t},u_{2}\right) ,\\ \left( {\mathcal {F}}({{\textbf {u}}}),\tilde{{{\textbf {u}}}}\right)&=\mu _1\left( \varphi \ln |\varphi |^2_{{\mathbb {R}}},u_1\right) + \mu _2\left( \psi \ln |\psi |^2_{{\mathbb {R}}},u_2\right) . \end{aligned}$$

Here

$$\begin{aligned} a:{\mathbb {U}}\times {\mathbb {U}}\mapsto {\mathbb {R}}, \end{aligned}$$

where

$$\begin{aligned} {\mathbb {U}}=H_0^1(0,L)\times H_0^1(0,L). \end{aligned}$$

6.2 Algorithms

Here, we developed an algorithms to obtain the numerical solutions and verified the properties of theoretical results to the Timoshenko beam’s with Logarithms source. We adopt our approximated solution by Finite-Element Method (FEM), in spatial variable and a finite difference method in the temporal variable with iterative methods. First, we consider a partition $X_h$ over the interval $\varOmega =(0,L),$ that is, $X_h=\left\{ 0=x_0<x_1<\cdots <x_N=L\right\} ,\quad \varOmega _{j+1}=(x_j,x_{j+1}),$ and, $\varOmega _{i}\bigcap \varOmega _j=\text{\O },\ i\ne j\ \text{ and }\ \varOmega =\bigcup _{e=1}^{Ne}{\overline{\varOmega }}_e$ where $N_e$ is the number of the elements obtained of partition. We consider the following finite-dimensional subspaces:

$$\begin{aligned} S^h_1= & {} \left\{ u\in C(0,L); u\Big |_{\varOmega _e}\in P_1(\varOmega _e)\right\} , \\ U^h= & {} \left\{ u^h\in S^h_1; u^h(0)=u^h(L)=0\right\} , \end{aligned}$$

where $P_1$ is the set linear polynomials defined over the element $\varOmega _e.$ We use a representation of the numerical solution ${{\textbf {u}}}^{h}=[\varphi ^h, \psi ^h]^\top$ analogous like in [17], and then, we have ${{\textbf {u}}}^h(t,x)=\sum _{i=1}^{2N}d_i(t)\phi _i(x)$ where 2N is the number total of degrees of freedom of the finite-element approximation, and $\phi _i(x),\ i = 1,\cdots , 2N,$ are the global vector interpolation functions. Therefore, we obtain the following dynamical problem in ${\mathbb {R}}^{2N}$:

$$\begin{aligned} {{\textbf {M}}}\ddot{{{\textbf {d}}}}(t)+{{\textbf {C}}}\dot{{{\textbf {d}}}}(t)+{{\textbf {K}}}{{\textbf {d}}}(t)= & {} {{\textbf {F}}}({{\textbf {d}}}(t)),\\ {{\textbf {d}}}(0)= & {} {{\textbf {d}}}_{0},\\ \dot{{{\textbf {d}}}}(0)= & {} \dot{{{\textbf {d}}}}_1, \end{aligned}$$

where ${{\textbf {M}}}:$ the consistent mass matrix, ${{\textbf {C}}}:$ the damping matrix, ${{\textbf {K}}}:$ the vector of consistent nodal elastic stiffness at time t, and ${{\textbf {F}}}({{\textbf {d}}}(t)):$ the vector of consistent nodal to logarithmic source at time t,  and ${{\textbf {d}}}(t):$ the vector of displacement nodal generalized at time t. Furthermore, ${{\textbf {d}}}_0$ and $\dot{{{\textbf {d}}}}_1$ are displacement and velocities, nodal initial, respectively.

Fig. 1

Evolution of solutions:$\ \varphi ^{h}(x,t),\ \psi ^h(x,t),$ respectively. Numerical energy at time 2.0 s and 5.0 s, respectively

Fig. 2

Evolution of solutions:$\ \varphi ^{h}(x,t),\ \psi ^h(x,t),$ respectively. Numerical Energy at time 2.0 s and 5.0 s, respectively

To solve this system above, we introduce a partition P of the time domain [0, T] into M intervals of length $\varDelta t$, such that $0=t_0<t_1<\cdots <t_M=T,$ with $t_{n+1}-t_n=\varDelta t$ and we use the well-known Newmark’s methods [19]. Since, in our work, we have a non-linear system we need to modify our scheme

$$\begin{aligned} {{\textbf {M}}}\ddot{{{\textbf {d}}}}_{n+1}+{{\textbf {C}}}\dot{{{\textbf {d}}}}_{n+1}+{{\textbf {K}}}{{\textbf {d}}}_{n+1}= & {} {{\textbf {F}}}\left( {{\textbf {d}}}_{n+1}\right) \\ {{\textbf {d}}}_{n+1}= & {} {{\textbf {d}}}_{n}+\varDelta t\dot{{{\textbf {d}}}}_{n}+\frac{\varDelta t^2}{2}\left[ (1-2\beta )\ddot{{{\textbf {d}}}}_{n}+ 2\beta \ddot{{{\textbf {d}}}}_{n+1}\right] \\ \dot{{{\textbf {d}}}}_{n+1}= & {} \dot{{{\textbf {d}}}}_n+\varDelta t\left[ (1-\gamma )\ddot{{{\textbf {d}}}}_n+\gamma \ddot{{{\textbf {d}}}}_{n+1}\right] , \end{aligned}$$

where $\beta ,\ \gamma$ and $\alpha$ are two parameters that govern the stability and accuracy of the methods. In this case

$$\begin{aligned} \displaystyle {{\textbf {M}}}=\bigcup _{e=1}^N{{\textbf {m}}}^{e},\ {{\textbf {C}}}=\bigcup _{e=1}^N{{\textbf {c}}}^{e}\,\,\text {and}\,\, \displaystyle {{\textbf {K}}}=\bigcup _{e=1}^N({{\textbf {k}}}^{e}_b+{{\textbf {k}}}^{e}_s); \end{aligned}$$

for instance, considering linear functions, we have

$$\begin{aligned} \textbf{m}^{\textbf{e}}= & {} \left[ \begin{array}{cccc} \rho _1h/3 &{} 0 &{} \rho _1h/6 &{} 0\\ 0 &{} \rho _2h/3 &{} 0 &{} \rho _2h/6\\ \rho _1h/6 &{} 0 &{} \rho _1h/3&{} 0\\ 0 &{} \rho _2h/6 &{} 0 &{} \rho _2h/3 \end{array} \right] ,\ \ c^e= \left[ \begin{array}{cccc} \gamma _1 h/3 &{} 0 &{} \gamma _1h/6 &{} 0\\ 0 &{} \gamma _2h/3 &{} 0 &{} \gamma _2h/6\\ \gamma _1h/6 &{} 0 &{} \gamma _1h/3&{} 0\\ 0 &{} \gamma _2h/6 &{} 0 &{} \gamma _2h/3 \end{array} \right] \\ \textbf{k}^{\textbf{e}}_{\textbf{b}}= & {} \left[ \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} b/h &{} 0&{} -b/h\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} -b/h &{} 0&{} b/h\\ \end{array} \right] ,\ \ k^e_s=\left[ \begin{array}{cccc} \kappa /h &{} -\kappa /2 &{} -\kappa /h &{} -\kappa /2\\ -\kappa /2&{} \kappa h/3 &{} \kappa /2&{} \kappa h/6\\ -\kappa /h &{} \kappa /2 &{} \kappa /h &{} \kappa /2\\ -\kappa /2 &{} \kappa h/6 &{} \kappa /2&{} \kappa h/3\\ \end{array} \right] . \end{aligned}$$

Due to its non-linearity, we have a vector ${{\textbf {F}}}({{\textbf {d}}}(t))$ with entries for each element of

$$\begin{aligned} {{\textbf {F}}}^{{\textbf {e}}}=\left[ \int _{\varOmega _e}\mu _1({{\textbf {u}}}^h(t))\ln |{{\textbf {u}}}^h(t)|^2 \phi ^{e}_i\ \textrm{d}x, \int _{\varOmega _e}\mu _2({{\textbf {u}}}^h(t))\ln |{{\textbf {u}}}^h(t)|^2 \phi ^{e}_i\ \textrm{d}x\right] ^\top . \end{aligned}$$

These vectorial components are obtained by Gaussian Quadrature using two points.

Remark 6.1

We point out to numerical pathology which occurs in penalized systems the locking problem, in particular, to Timoshenko system, it is the shear locking. It is characterized by the following over-estimation about the coefficient b, given by:

$$\begin{aligned} b=EI\left( 1+\frac{\kappa GAh^2}{12EI}\right) . \end{aligned}$$

It is clear that the numerical alternatives to this problem were performed in the literature, and to more details, we indicate the classical reference by Hughes et al. [15] and Prathap and Bhashyam [21].

Remark 6.2

To get computational results, we use the implemented code in Language C. The graphics were developed using GNUplot.

In the sequel, we realize some numerical experiments to highlight our theoretical results.

6.3 Numerical experiments

In our performed numerical experiments to view the asymptotic properties, we consider an uniform mesh $h=0.01$ m, $\varDelta t=10^{-5}$ s. The parameters Newmark’s rules algorithms are $\gamma =\frac{1}{2},$ $\beta =\frac{1}{4}.$

Experimento 1: (Conservative case: $\gamma _1=\gamma _2=0$)

We consider a rectangular beam with $L= 1.0$ m, thickness 0.09 m, width 0.09 m, $E=69\cdot 10^{7}$N/$\text{ m}^2$ $\rho =2700$ Kg/$\text{ m}^3$,$\ \kappa =5/6,\ r=0.33$ (Poisson ratio). Furthermore, we have $\mu _1=\mu _2=1$ and the following initial conditions:

$$\begin{aligned} \varphi (x,0)=0,\ \varphi _t(x,0)=\sin \displaystyle 3\pi x,\ \psi (x,0)=0,\ \text{ and }\ \psi _t(x,0)=\sin \displaystyle 5\pi x. \end{aligned}$$

Experimento 2: (Dissipative case: $\gamma _1=23, \gamma _2=0.015$)

We consider a rectangular beam with $L= 1.0$ m, thickness 0.09 m, width 0.09 m $E=69\cdot 10^{7}$N/$\text{ m}^2$ $\rho =2700$ Kg/$\text{ m}^3$,$\ \kappa =5/6,\ r=0.33$(Poisson ratio) and $\mu _1=1.0,\ \mu _2=1.0$. and the following initial conditions:

$$\begin{aligned} \varphi (x,0)=0,\ \varphi _t(x,0)=\sin \displaystyle 3\pi x,\ \psi (x,0)=0,\ \text{ and }\ \psi _t(x,0)=\sin \displaystyle 5\pi x. \end{aligned}$$