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

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.

where μ j > 0, j = 1, 2 and | · | R denote the absolute value of a real number. The constants, ρ is the mass density, A the cross-sectional area, and I the moment of inertia. To simplify the notation, let us denote by ρ 1 = ρ A, ρ 2 = ρ I, κ = k AG and b = E I . Under these conditions, we consider the initial-boundary problem for the following logarithmic Timoshenko System: where γ 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.

Preliminaries
We denote L 2 (0, L) the Hilbert's space of square-integrable function on the interval (0, L), with the inner product and norm |u| 2 = (u, u) ∀u ∈ L 2 (0, L). We use Sobolev space notation and properties as in [1]. We denote 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) Direct differentiation of (6) gives us d dt Now, consider the following lemmas: Then, W has compact embedding in L p 0 (0, T ; B).
Lemma 2.4 (Nakao's Lemma) [18] Suppose that φ(t) is a bounded nonnegative function on R + , satisfying for any t ≥ 0, where C 0 is a positive constant. Then where C and α are positive constants.

The potential well
In this section, we present the potential well corresponding to the Eqs. (1)- (2). We define the operator J : For (ϕ, ψ) ∈ H 1 0 (0, L) 2 and λ > 0, we have Associated with J , we have the well-known Nehari Manifold We define as in the Mountain Pass theorem due to Ambrosetti and Rabinowitz [3] According to Willem [30], Theorem 4.2, the depth of the well d is a strictly positive constant given by and partition it into two sets as follows: Therefore, we define by W 1 the set of stability for the problem (1)-(5).

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

Approximated problem
Let (w ν ) ν∈N be a basis of H 1 0 (0, L) from the eigenvectors of the operator −Δ, and ∀ w, z ∈ V m . By virtue of Carathéodory's theorem, see [8], the system (16) has a local solution in [0, t m ), 0 < t m ≤ T . The extension of the solution to the whole interval [0, T ] is a consequence of the following a priori estimates.

A priori estimates
Let w = ϕ m t (t) and z = ψ m t (t) in (16) and (17), respectively. Then, we have where E m (t) is the approximated energy of the problem (16). Now, integrating (20) from 0 to t, 0 ≤ t ≤ t m , we obtain Thus which gives us the following estimate: We have that J (ϕ 0m , ψ 0m ) < d, and then, by (16), we get where C 1 is a positive constant independent of m and t. These estimates imply that the approximated solution (ϕ m , ψ m ) exists globally in [0, ∞). See [13]. Then, by estimate (22), we have Now, by the logarithmic inequality Analogously, we have whereC 1 andC 2 are constant independent of m and t. From (25) and (26), we get ψ m ln |ψ| 2 R are bounded in L 2 loc 0, ∞; L 2 (0, L) .

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
where C 0 and α are positive constants.
6 Numerical approach

Variational formulation
Here, we use a representation to the functions ϕ, ψ and logarithmic source terms by component vectorial Thus, from (1) to (5), we get the following variational problem: where u satisfies the initial conditions Here

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 Ω = (0, L), that is, where N e is the number of the elements obtained of partition. We consider the following finite-dimensional subspaces: where P 1 is the set linear polynomials defined over the element Ω e . We use a representation of the numerical solution u h = [ϕ h , ψ h ] analogous like in [17], and then, we have u h (t, where 2N is the number total of degrees of freedom of the finite-element approximation, and φ i (x), i = 1, · · · , 2N , are the global vector interpolation functions. Therefore, we obtain the following dynamical problem in R 2N : where M : the consistent mass matrix, C : the damping matrix, K : the vector of consistent nodal elastic stiffness at time t, and F(d(t)) : the vector of consistent nodal to logarithmic source at time t, and d(t) : the vector of displacement nodal generalized at time t. Furthermore, d 0 andḋ 1 are displacement and velocities, nodal initial, respectively. To solve this system above, we introduce a partition P of the time domain [0, T ] into M intervals of length Δt, such that 0 = t 0 < t 1 < · · · < t M = T, with t n+1 − t n = Δt 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 where β, γ and α are two parameters that govern the stability and accuracy of the methods. In this case for instance, considering linear functions, we have Due to its non-linearity, we have a vector F(d(t)) with entries for each element of 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:   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.