Home Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
Article Open Access

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

  • Adel M. Al-Mahdi EMAIL logo
Published/Copyright: May 23, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study

ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 ,

where the convolution memory function g belongs to a class of L 1 ( 0 , ∞ ) functions that satisfies

g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 ,

where ξ is a positive nonincreasing differentiable function and ϒ is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p and p covers the full admissible range [ 1 , 2 ) . Given this degree of generality, our results improve some of earlier related results in the literature.

Keywords: Timoshenko system; Shear beam models; multiplier method; general and optimal decay; memory
MSC 2010: 35B40; 93D20

1 Introduction

In 1921, Timoshenko [1] presented the following system:

(1.1) ρ 1 φ t t − κ ( φ x + ψ ) x = 0 , in ( 0 , L ) × ( 0 , + ∞ ) , ρ 2 ψ t t − b ψ x x + κ ( φ x + ψ ) = 0 , in ( 0 , L ) × ( 0 , + ∞ ) ,

as a model of a motion of a thick beam, where φ is the transverse displacement; ψ is the rotational angle of the filament of the beam; and ρ 1 , ρ 2 , b , and κ are the fixed positive physical parameters. The issue of stabilization of Timoshenko systems has attracted a great deal of research, and many results concerning the well posedness and long-time behavior of the system have been established. For this matter, various types of dissipation, such as boundary and/or internal feedback, heat or thermoelasticity, memory, and Kelvin-Voigt damping, have been used (see, for example, [2–4]). It is well known that the exponential stability of System (1.1) is achieved in the presence of linear dampings in both equations of (1.1) without imposing any condition on the speeds of wave propagation. But if the damping effect is acting on only one equation, the system is exponentially stable if and only if it has equal speeds of wave propagation, i.e.,

(1.2) κ ρ 1 = b ρ 2 .

The reader is advised to consult the above-cited references for detailed discussions on the stability analysis of Timoshenko systems.

It is worth mentioning that the original Timoshenko (1.1) is characterized by two natural frequencies, which lead to a physical paradox, known as the second spectrum, which was not noted in Timoshenko original work. This required (mathematically) some relation, called the equal-speed propagation. However, this is an unrealistic requirement. To overcome this paradox and to eliminate the anomaly of the second spectrum, Elishakoff [5] proposed the following truncated version of the classical Timoshenko system:

(1.3) ρ 1 φ t t − κ ( φ x + ψ ) x = 0 , − ρ 2 φ t t x − b ψ x x + κ ( φ x + ψ ) = 0 ,

where ( x , t ) ∈ [ 0 , L ] × R + . Model (1.3) has not been well studied, and only a few results regarding the stability have been established. For example, Almeida Júnior et al. [7] considered the following Shear beam model with no rotary inertia:

(1.4) ρ 1 φ t t − κ ( φ x + ψ ) x + μ φ t = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 ,

and proved that the energy of this system has exponential decay without imposing any relationship between the coefficients of the system. The Shear beam model (1.4) has only one finite wave speed for all wave numbers, which is κ ρ 1 . This property is unlike the one in the classical Timoshenko-type system which has two wave speeds κ ρ 1 and b ρ 1 (Table 1).

Table 1

Wave speed for Timoshenko beam models

Beam models Wave speeds Exponential decay Polynomial decay
Classical Timoshenk [1] κ ρ 1 , b ρ 2 κ ρ 1 = b ρ 2 κ ρ 1 ≠ b ρ 2
Truncated Timoshenk [5] b ρ 2 1 + b ρ 1 κ ρ 2 No matter wave speed —
Shear model [6] κ ρ 1 No matter wave speed —

Concerning the stabilization via heat dissipation, only a few works are available in the literature. We point out the work by Apalara et al. [8], where they considered the following system:

(1.5) ρ 1 φ t t − κ ( φ x + ψ ) x = 0 , − ρ 2 φ t t x − b ψ x x + κ ( φ x + ψ ) + γ θ x = 0 , ρ 3 θ t − β θ x x + γ ψ x t t = 0 ,

where θ is the temperature difference, and ρ 3 , β > 0 , and γ ≠ 0 are the capacity, diffusivity, and adhesive stiffness, respectively. The authors discussed briefly the well posedness and proved an exponential decay result irrespective of the coefficients of the system. Recently, Keddi et al. [9] looked into the following thermoelastic Timoshenko system:

(1.6) ρ 1 φ t t − κ ( φ x + ψ ) x = 0 , in ( 0 , 1 ) × ( 0 , ∞ ) , ρ 2 φ t t x − b ψ x x + κ ( φ x + ψ ) + δ θ x = 0 , in ( 0 , 1 ) × ( 0 , ∞ ) , c θ t + q x + δ ψ x t = 0 , in ( 0 , 1 ) × ( 0 , ∞ ) , τ q t + β q + θ x = 0 , in ( 0 , 1 ) × ( 0 , ∞ ) ,

and established the well posedness and an exponential decay result. For more results in this topic, we refer the reader to see [10–27]. Motivated by the aforementioned works and discussion about the wave speeds, we are concerned with the following viscoelastic Shear beam model:

(1.7) ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 ,

(1.8) − b ψ x x + κ ( φ x + ψ ) = 0 ,

where the convolution g ∗ φ x x is defined by

( g ∗ φ x x ) ( t ) = ∫ 0 t g ( t − s ) φ x x ( s ) d s

and ( x , t ) ∈ [ 0 , 1 ] × R + . We consider (1.7)–(1.8) subject to the following initial and boundary conditions:

(1.9) φ ( x , 0 ) = φ 0 , φ t ( x , 0 ) = φ 1 , ψ ( x , 0 ) = ψ 0 , x ∈ [ 0 , 1 ] ,

(1.10) φ x ( 0 , t ) = φ x ( 1 , t ) = ψ ( 0 , t ) = ψ ( 1 , t ) = 0 , ∀ t > 0 .

Our aim is to prove the stability of Systems (1.7)–(1.10). To the best of my knowledge, the stability of (1.7)–(1.10) has not been discussed in the current nature. From the boundary conditions (1.10), Poincaré’s inequality cannot be implemented over φ . So, integration (1.7) over ( 0 , 1 ) and using the boundary conditions (1.10), we obtain

(1.11) d 2 d t 2 ∫ 0 1 φ ( x , t ) d x = 0 .

Now, solving (1.11) and imposing the initial condition (1.9), we obtain

(1.12) ∫ 0 1 φ ( x , t ) d x = ∫ 0 1 φ 0 ( x ) d x + t ∫ 0 1 φ 1 ( x ) d x .

We introduce the following new variable:

φ ˜ ( x , t ) = φ ( x , t ) − ∫ 0 1 φ ( x , t ) d x .

Therefore, we obtain

(1.13) ∫ 0 1 φ ˜ ( x , t ) d x = 0 .

Thus, we can use the application of Poincaré’s inequality on φ ˜ . Thus, we work with ( φ ˜ , ψ ) , instead of ( φ , ψ ) but, for simplicity, we write ( φ , ψ ) .

2 Assumptions

This section introduces some information that is necessary for the validation of our findings. The letter c is used to represent any positive constant in this article. We assume the following for the relaxation function g :

( A 1 ) : The memory kernel g : [ 0 , + ∞ ) → ( 0 , + ∞ ) is a C 1 nonincreasing function satisfying

(2.1) g ( 0 ) > 0 and ℓ ≔ κ − 3 ∫ 0 + ∞ g ( s ) d s > 0 .

In addition, there exists a C 1 function ϒ : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) , which is linear, or it is strictly increasing and strictly convex C 2 function on ( 0 , r ] , r ≤ g ( 0 ) , with ϒ ( 0 ) = ϒ ′ ( 0 ) = 0 , and a positive nonincreasing differentiable function ξ , such that

(2.2) g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 .

( A 2 ) : The coefficients b and κ satisfy

(2.3) b − κ 2 ℓ > 0 .

Remark 2.1

Condition (2.2) was introduced by Alabau-Boussouira et al. [28] and improved by Mustafa [29].

We state without proof the existence result of the weak solutions of Systems (1.7)–(1.8) in the following theorem.

Theorem 2.2

If the initial date ( φ 0 , φ 1 , ψ 0 ) ∈ ℋ , then Systems (1.7)–(1.8) have a weak solution satisfying

φ ∈ L ∞ ( [ 0 , T ) ; H * 1 ( 0 , 1 ) ) , φ t ∈ L ∞ ( [ 0 , T ) ; L 2 ( 0 , 1 ) ) , ψ ∈ L ∞ ( [ 0 , T ) ; H 0 1 ( 0 , 1 ) ) ,

where

ℋ = H * 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) × H 0 1 ( 0 , 1 ) , H * 1 ( 0 , 1 ) = H 1 ( 0 , 1 ) ∩ L * 2 ( 0 , 1 )

and

L * 2 ( 0 , 1 ) = u ∈ L 2 ( 0 , 1 ) : ∫ 0 1 u ( x ) d x = 0 .

The proof of the aforementioned theorem (and for more regularity of the solutions) can be proved using the Faedo-Galerkin method as the proof of Theorem 2.1 established by Almeida Júnior et al. [7].

Now, we introduce the “modified” energy associated with Systems (1.7)–(1.10) as follows:

(2.4) E ( t ) ≔ 1 2 ∫ 0 1 φ t 2 + b ψ x 2 − ∫ 0 t g ( s ) d s φ x 2 + κ ( φ x + ψ ) 2 d x + 1 2 ( g ∘ φ x ) ( t ) .

3 Technical lemmas

In this section, we state and establish several lemmas needed for the proofs of our main results.

Lemma 3.1

The energy E ( t ) satisfies

(3.1) E ′ ( t ) = 1 2 ( g ′ ∘ φ x ) ( t ) − 1 2 g ( t ) ∫ 0 1 φ x 2 d x ≤ 0 .

Proof

By multiplying (1.7) by φ t and (1.8) by ψ t , using integration by parts and adding the results, it is easy to arrive at the proof of (3.1).□

Remark 3.2

To prove that the energy is positive, using the positivity of g and using Poincaré’s inequality, we note that

(3.2) b ψ x 2 − ∫ 0 t g ( s ) d s φ x 2 + κ ( φ x + ψ ) 2 = b ψ x 2 − 3 ∫ 0 t g ( s ) d s + κ φ x 2 + 2 κ φ x ψ + κ ψ 2 + 2 ∫ 0 t g ( s ) d s ≥ b − κ 2 ℓ ψ 2 + ℓ φ x + κ ℓ ψ 2 ≥ 0 ,

where ℓ = κ − 3 ∫ 0 + ∞ g ( s ) d s > 0 (from Assumption ( A 2 ) ).

Lemma 3.3

[29] There exist positive constants ϱ and t 1 such that

(3.3) g ′ ( t ) ≤ − ϱ g ( t ) , ∀ t ∈ [ 0 , t 1 ] .

Lemma 3.4

[29] For any 0 < θ < 1 ,

(3.4) ∫ 0 1 ∫ 0 t g ( t − s ) ∣ φ x ( s ) − φ x ( t ) ∣ d s 2 d x ≤ C θ ( Θ ∘ φ x ) ( t ) ,

where

(3.5) C θ = ∫ 0 ∞ g 2 ( s ) θ g ( s ) − g ′ ( s ) d s a n d Θ ( t ) = θ g ( t ) − g ′ ( t ) .

Lemma 3.5

Assume that ( A 1 ) – ( A 2 ) hold. Then, for all t ∈ R + , the derivative of the functional

Δ 1 ( t ) ≔ − ρ 1 ∫ 0 1 φ x ∫ 0 x φ t ( y ) d y d x − 1 2 ∫ 0 1 φ x 2 d x

satisfies

(3.6) Δ 1 ′ ( t ) ≤ ρ 1 ∫ 0 1 φ t 2 d x − κ − ℓ 4 − ε 1 − ∫ 0 ∞ g ( s ) d s ∫ 0 1 φ x 2 d x + κ 2 ℓ ∫ 0 1 ψ x 2 d x + C ε 1 C θ ( Θ ∘ φ x ) ( t ) .

Proof

Differentiating Δ 1 , then using (1.7)–(1.8), we end up with

(3.7) Δ 1 ′ ( t ) = − ρ 1 ∫ 0 1 φ x t ∫ 0 x φ t ( y ) d y d x − ∫ 0 1 φ x ∫ 0 x κ ( φ y + ψ ) y − ∫ 0 t g ( t − s ) φ y y ( s ) d y .

Integrating by parts and using the boundary conditions (1.10), (3.7) becomes

(3.8) Δ 1 ′ ( t ) = ρ 1 ∫ 0 1 φ t 2 d x − κ ∫ 0 1 φ x 2 d x − κ ∫ 0 1 φ x ψ d x + ∫ 0 1 φ x ∫ 0 t g ( t − s ) φ x ( s ) d s d x ︸ I .

Using Young’s and Poincaré’s inequalities gives, for any ε 1 > 0 ,

(3.9) − κ ∫ 0 1 φ x ψ d x ≤ ℓ 4 ∫ 0 1 φ x 2 d x + κ 2 ℓ ∫ 0 1 ψ x 2 d x .

We observe that

I = ∫ 0 t g ( s ) d s ∫ 0 1 φ x 2 d x + ∫ 0 1 φ x ∫ 0 t g ( t − s ) ( φ x ( s ) − φ x ( t ) ) d s d x ︸ I 2 .

Using Young’s inequality, we obtain

I 2 ≤ ε 1 ∫ 0 1 φ x 2 d x + 1 4 ε 1 ∫ 0 1 ∫ 0 t g ( t − s ) ( φ x ( s ) − φ x ( t ) ) d s 2 d x .

Then, using (3.4), we obtain

1 4 ε 1 ∫ 0 1 ∫ 0 t g ( t − s ) ( φ x ( s ) − φ x ( t ) ) d s 2 d x ≤ C ε 1 C θ ( Θ ∘ φ x ) ( t ) .

So,

I ≤ ε 1 + ∫ 0 + ∞ g ( s ) d s ∫ 0 1 φ x 2 d x + C ε 1 C θ ( Θ ∘ φ x ) ( t ) .

Combining all the above, the proof of the estimate (3.6) is completed.□

Lemma 3.6

Assume that ( A 1 ) – ( A 2 ) hold, the functional

Δ 2 ( t ) = ρ 1 ∫ 0 1 φ t φ d x ,

satisfies, along the solution of Systems (1.7)–(1.10), for any ε 2 > 0 ,

(3.10) Δ 2 ′ ( t ) ≤ ρ 1 ∫ 0 1 φ t 2 d x − b ∫ 0 1 ψ x 2 d x − κ ∫ 0 1 ( φ x + ψ ) 2 d x + ε 2 + ∫ 0 + ∞ g ( s ) d s ∫ 0 1 φ x 2 d x + C ε 2 C θ ( Θ ∘ ψ x ) ( t ) .

Proof

Taking the derivative of Δ 2 ( t ) , using (1.7), and integrating by parts, we obtain

(3.11) Δ 2 ′ ( t ) = ρ 1 ∫ 0 1 φ t 2 d x − κ ∫ 0 1 ( φ x + ψ ) φ x d x − ∫ 0 1 φ ∫ 0 t g ( t − s ) φ x x ( s ) d s d x .

Adding and subtracting some terms, (3.11) becomes

(3.12) Δ 2 ′ ( t ) = ρ 1 ∫ 0 1 φ t 2 d x − κ ∫ 0 1 ( φ x + ψ ) 2 d x + κ ∫ 0 1 ( φ x + ψ ) ψ d x + ∫ 0 1 φ x ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x + ∫ 0 t g ( s ) d s ∫ 0 1 φ x 2 d x .

By multiplying (1.8) by ψ and integrating over ( 0 , 1 ) , we obtain

(3.13) κ ∫ 0 1 ( φ x + ψ ) ψ d x = − b ∫ 0 1 ψ x 2 d x .

Using (3.13), equation (3.12) becomes

(3.14) Δ 2 ′ ( t ) = ρ 1 ∫ 0 1 φ t 2 d x − κ ∫ 0 1 ( φ x + ψ ) 2 d x − b ∫ 0 1 ψ x 2 d x + ∫ 0 1 φ x ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x + ∫ 0 t g ( s ) d s ∫ 0 1 φ x 2 d x .

Using Young’s inequality, we obtain for ε 2 > 0

(3.15) ∫ 0 1 φ x ∫ 0 t g ( t − s ) [ φ x ( t ) − φ x ( s ) ] d s d x ≤ ε 2 ∫ 0 1 φ x 2 d x + 1 4 ε 2 ∫ 0 1 ∫ 0 t g ( t − s ) [ φ x ( s ) − φ x ( t ) ] d s 2 d x ≤ ε 2 ∫ 0 1 φ x 2 d x + C θ ε 2 ( Θ ∘ φ x ) ,

which implies (3.10).□

Lemma 3.7

Assume that ( A 1 ) – ( A 2 ) hold. Then, for any t 1 > 0 , the functional

Δ 3 ( t ) ≔ − ρ 1 ∫ 0 1 φ t ∫ 0 t g ( t − s ) ( φ ( t ) − φ ( s ) ) d s d x

satisfies, for any δ 1 , δ 2 , δ 3 > 0 ,

(3.16) Δ 3 ′ ( t ) ≤ − ρ 1 g 1 2 ∫ 0 1 φ t 2 d x + δ 1 ∫ 0 1 φ x 2 d x + δ 3 ∫ 0 1 ( φ x + ψ x ) 2 d x + c C θ δ 1 + c δ 2 ( 1 + C θ ) + c C θ δ 3 + C θ ( Θ ∘ φ x ) ( t ) ,

where g 1 = ∫ 0 t 1 g ( s ) d s .

Proof

Differentiating Δ 3 and integrating by parts, we obtain

(3.17) Δ 3 ′ ( t ) = − ρ 1 ∫ 0 t g ( s ) d s ∫ 0 1 φ t 2 d x + κ ∫ 0 1 ( φ x + ψ ) ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x + ∫ 0 1 ∫ 0 t g ( t − s ) φ x ( s ) d s ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x − ρ 1 ∫ 0 1 φ t ∫ 0 t g ′ ( t − s ) ( φ ( t ) − φ ( s ) ) d s d x = − ρ 1 ∫ 0 t g ( s ) d s ∫ 0 1 φ t 2 d x + ∫ 0 1 ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s 2 d x + ∫ 0 t g ( s ) d s ∫ 0 1 φ x ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x + κ ∫ 0 1 ( φ x + ψ ) ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x − ρ 1 ∫ 0 1 φ t ∫ 0 t g ′ ( t − s ) ( φ ( t ) − φ ( s ) ) d s d x .

Applying Young’s inequality, we obtain for δ 1 > 0

(3.18) ∫ 0 t g ( s ) d s ∫ 0 1 φ x ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x ≤ δ 1 ∫ 0 1 φ x 2 ( t ) d x + c C θ δ 1 ( Θ ∘ φ x ) ( t ) .

Similarly, we can find, for any δ 2 > 0 ,

− ρ 1 ∫ 0 1 φ t ( t ) ∫ 0 t g ′ ( t − s ) ( φ ( t ) − φ ( s ) ) d s d x = ρ 1 ∫ 0 1 φ t ∫ 0 t Θ ( t − s ) ( φ ( t ) − φ ( s ) ) d s d x − ρ 1 ∫ 0 t φ t ( t ) ∫ 0 t θ g ( t − s ) ( φ ( t ) − φ ( s ) ) d s d x ≤ δ 2 ∫ 0 1 φ t 2 ( t ) d x + ρ 1 2 ∫ 0 t Θ ( s ) d s 2 δ 2 ( Θ ∘ φ ) ( t ) + ρ 1 2 θ 2 2 δ 2 ∫ 0 1 ∫ 0 t g ( t − s ) φ ( t ) − φ ( s ) d s 2 d x ≤ δ 2 ∫ 0 1 φ t 2 ( t ) d x + c δ 2 ( Θ ∘ φ x ) ( t ) + θ 2 C θ δ 2 ( Θ ∘ φ x ) ( t ) ≤ δ 2 ∫ 0 1 φ t 2 ( t ) d x + c δ 2 ( 1 + C θ ) ( Θ ∘ φ x ) ( t ) .

Using Young’s inequality and performing similar calculations as in (3.18), we obtain, for any δ 3 > 0 ,

(3.19) κ ∫ 0 1 ( φ x + ψ ) ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s d x ≤ δ 3 ∫ 0 1 ( φ x + ψ ) 2 d x + c C θ δ 3 ( Θ ∘ φ x ) ( t ) .

Using Lemma 3.4, we find

(3.20) ∫ 0 1 ∫ 0 t g ( t − s ) ( φ x ( t ) − φ x ( s ) ) d s 2 d x ≤ C θ ( Θ ∘ φ x ) ( t ) .

Combining all the aforementioned estimates, we find, for any t ≥ t 1 ,

(3.21) Δ 3 ′ ( t ) ≤ − ρ 1 ∫ 0 t g ( s ) d s − δ 2 ∫ 0 1 φ t 2 d x + δ 1 ∫ 0 1 φ x 2 d x + δ 3 ∫ 0 1 ( φ x + ψ ) 2 d x + c C θ δ 1 + c δ 2 ( 1 + C θ ) + c C θ δ 3 + C θ ( Θ ∘ φ x ) ( t ) .

By taking δ 2 = g 1 2 , we obtain the desired Inequality (3.16).□

Lemma 3.8

Under Assumptions ( A 1 ) – ( A 2 ) , the functional

(3.22) Δ 4 ( t ) ≔ ∫ 0 1 ∫ 0 t Λ ( t − s ) ∣ φ x ( s ) ∣ 2 d s d x

satisfies, along the solution of Systems (1.7)–(1.10), the estimate

(3.23) Δ 4 ( t ) ≤ − 1 2 ( g ∘ φ x ) ( t ) + 3 ( κ − ℓ ) ∫ 0 1 ∣ φ x ( s ) ∣ 2 d x ,

where Λ ( t ) = 3 ∫ t + ∞ g ( s ) d s .

Proof

The proof is similar to the one in [30].□

Lemma 3.9

The functional ℒ defined by

(3.24) ℒ ( t ) = Ξ E ( t ) + Δ 1 ( t ) + Δ 2 ( t ) + Ξ 3 Δ 3 ( t ) ,

satisfies, for suitable choice of Ξ , Ξ 3 > 0 and for all t ≥ t 1 ,

(3.25) ℒ ′ ( t ) ≤ − ( κ − ℓ ) 4 ∫ 0 1 φ x 2 d x − ∫ 0 1 ψ x 2 d x − ∫ 0 1 φ t 2 d x − ∫ 0 1 ( φ x + ψ ) 2 d x + 1 64 ( g ∘ φ x ) ( t ) ,

where t 1 = g − 1 ( r ) as introduced earlier in this article, and the following equivalence relation

(3.26) ℒ ( t ) ∼ E ( t ) .

Proof

By taking the derivative of the functional ℒ and using the aforementioned estimates,

ℒ ′ ( t ) ≤ Ξ θ 2 ( g ∘ φ x ) ( t ) − κ − ℓ 4 − 2 ∫ 0 ∞ g ( s ) d s − ε 1 − ε 2 − δ 1 Ξ 3 ∫ 0 1 φ x 2 d x − ( κ − δ 3 Ξ 3 ) ∫ 0 1 ( φ x + ψ ) 2 d x − b − κ 2 ℓ ∫ 0 1 ψ x 2 d x − c 0 2 Ξ 3 − 2 ρ 1 ∫ 0 1 φ t 2 d x − Ξ 2 − Ξ 3 c C θ δ 1 + c δ 2 ( 1 + C θ ) + c C θ δ 3 + C θ − C θ − C ε C θ ( Θ ∘ φ x ) ( t ) .

By fixing ε 1 = ε 2 = ε , then

ℒ ′ ( t ) ≤ Ξ θ 2 ( g ∘ φ x ) ( t ) − 1 4 4 κ − ℓ − 8 ∫ 0 ∞ g ( s ) d s − 8 ε − 4 δ 1 Ξ 3 ∫ 0 1 φ x 2 d x − ( κ − δ 3 Ξ 3 ) ∫ 0 1 ( φ x + ψ ) 2 d x − b − κ 2 ℓ ∫ 0 1 ψ x 2 d x − c 0 2 Ξ 3 − 2 ρ 1 ∫ 0 1 φ t 2 d x − Ξ 2 − Ξ 3 c C θ δ 1 + c δ 2 ( 1 + C θ ) + c C θ δ 3 + C θ − C θ − C ε C θ ( Θ ∘ φ x ) ( t ) .

Recall that κ − 3 ∫ 0 ∞ g ( s ) = ℓ > 0 . This gives that

3 κ − 9 ∫ 0 ∞ g ( s ) = 3 ℓ > 0 .

By choosing 8 ε = ∫ 0 ∞ g ( s ) . From this, we note that

ℒ ′ ( t ) ≤ Ξ θ 2 ( g ∘ φ x ) ( t ) − 1 4 κ − ℓ + 3 κ − 9 ∫ 0 ∞ g ( s ) d s − 4 δ 1 Ξ 3 ∫ 0 1 φ x 2 d x − ( κ − δ 3 Ξ 3 ) ∫ 0 1 ( φ x + ψ ) 2 d x − b − κ 2 ℓ ∫ 0 1 ψ x 2 d x − c 0 2 Ξ 3 − 2 ρ 1 ∫ 0 1 φ t 2 d x − Ξ 2 − Ξ 3 c C θ δ 1 + c δ 2 ( 1 + C θ ) + c C θ δ 3 + C θ − C θ − c C θ ( Θ ∘ φ x ) ( t ) .

By choosing δ 3 = κ 2 Ξ 3 , δ 1 = 3 ℓ 4 Ξ 3 , and Ξ 3 > 4 c 0 and since θ g 2 ( s ) θ g ( s ) − g ′ ( s ) < g ( s ) , then using the Lebesgue-dominated convergence theorem, we obtain

(3.27) θ C θ = ∫ 0 ∞ δ g 2 ( s ) δ g ( s ) − g ′ ( s ) d s ⟶ 0 , as θ → 0 .

Hence, there exists some 0 < θ * < 1 , such that if θ < θ * , then

(3.28) θ C θ < 1 128 1 + c + Ξ 3 c δ 1 + c δ 2 + c δ 3 + 1 .

By putting δ = 1 32 Ξ and choosing Ξ sufficiently large, we arrive at

Ξ 2 − Ξ 3 c C θ δ 1 + c δ 2 ( 1 + C θ ) + c C θ δ 3 + C θ − C θ − c C θ > 0 .

Therefore, we show that (3.25) holds. Moreover, we can choose Ξ even larger (if needed) so that (3.26) is satisfied, which means that, for some constants μ 1 , μ 2 > 0 , we have

□ μ 1 E ( t ) ≤ ℒ ( t ) ≤ μ 2 E ( t ) .

Lemma 3.10

The energy functional associated with Systems (1.7)–(1.10) satisfies

(3.29) ∫ 0 ∞ E ( s ) d s < ∞ .

Proof

We introduce the following functional ℱ defined by

(3.30) ℱ ( t ) = ℒ ( t ) + 1 16 Δ 4 ( t ) .

Applying Lemma 3.8, we see that there exists a positive constant δ such that

(3.31) ℱ ′ ( t ) ≤ − ( κ − ℓ ) ∫ 0 1 φ x 2 d x − ∫ 0 1 ψ x 2 d x − ∫ 0 1 φ t 2 d x − ∫ 0 1 ( φ x + ψ ) 2 d x − 1 64 ( g ∘ φ x ) ( t ) ≤ − δ E ( t ) .

Then, integrating the result over ( 0 , t ) , we obtain

(3.32) δ ∫ 0 t E ( s ) d s ≤ − ℱ ( t ) + ℱ ( 0 ) ≤ ℱ ( 0 ) .

Hence, the proof of the bound in (3.29) is completed.□

4 Main result

In this section, we state and prove our main results and give some examples to illustrate our theoretical result.

Theorem 4.1

Assume ( A 1 ) – ( A 2 ) hold and the function ϒ is  linear. Then, there exist positive constants c 1 and c 2 such that the energy of Systems (1.7)–(1.10) satisfies, for all t ≥ t 1 ,

(4.1) E ( t ) ≤ c 1 e − c 2 ∫ t 1 t ξ ( s ) d s ,

where t 1 = g − 1 ( r ) .

Proof

To prove Theorem 4.1, combine (2.4) and (3.25), to obtain, for some positive constants c and ϑ ,

ℒ ′ ( t ) ≤ − ϑ E ( t ) + c ∫ 0 t 1 g ( t − s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( s ) ∣ 2 d x d s + c ∫ t 1 t g ( t − s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( s ) ∣ 2 d x d s .

Now, imposing Lemma 3.3 and recalling (3.1), we obtain

(4.2) Γ ′ ( t ) ≤ − ϑ E ( t ) + c ∫ t 1 t g ( t − s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( s ) ∣ 2 d x d s ,

where Γ ( t ) = ℒ ( t ) + c E ( t ) . Since ϒ is linear, by multiplying (4.2) by ξ ( t ) , using the condition ( A 1 ) , and (3.1), we obtain

(4.3) ξ ( t ) Γ ′ ( t ) ≤ − ϑ ξ ( t ) E ( t ) + c ξ ( t ) ∫ t 1 t g ( t − s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( s ) ∣ 2 d x d s ≤ − ϑ ξ ( t ) E ( t ) + c ∫ t 1 t ξ ( s ) g ( t − s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( s ) ∣ 2 d x d s ≤ − ϑ ξ ( t ) E ( t ) − c ∫ t 1 t g ′ ( s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s ≤ − ϑ ξ ( t ) E ( t ) − c E ′ ( t ) .

Using the fact ξ ′ ( t ) ≤ 0 , the functional Γ 1 ≔ ξ Γ + c E satisfies Γ 1 ∼ E and

(4.4) Γ 1 ′ ( t ) ≤ − ϑ ξ ( t ) E ( t ) ≤ − ϑ 1 ξ ( t ) Γ 1 ( t ) , ∀ t ≥ t 1 .

After integrating the last inequality over ( t 1 , t ) , the proof of Theorem (4.1) is completed.□

Theorem 4.2

Assume ( A 1 ) – ( A 2 ) hold and the function ϒ is nonlinear. Then, there exist positive constants c 1 and c 2 such that the energy of Systems (1.7)–(1.10) satisfies, for all t > t 1 ,

(4.5) E ( t ) ≤ c 2 ϒ 1 − 1 c 1 ∫ g − 1 ( r ) t ξ ( s ) d s ,

where ϒ 1 ( t ) = ∫ t r 1 s ϒ ′ ( s ) d s , is strictly decreasing and convex on ( 0 , r ] , with lim t → 0 ϒ 1 ( t ) = + ∞ .

Proof

Thanks to the bound in (3.29), we can find a constant 0 < β < 1 so that the function I defined by

(4.6) 0 < Φ ( t ) ≔ β ∫ t 1 t ∫ 0 1 ∣ φ x ( t ) − φ x ( s ) ∣ 2 d x d s < 1 , ∀ t ≥ t 1 .

Let us define

(4.7) Ψ ( t ) ≔ − ∫ t 1 t g ′ ( s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s .

By (3.1), we see that Ψ ( t ) ≤ − c E ′ ( t ) . Since ϒ is strictly convex on ( 0 , r ] and ϒ ( 0 ) = 0 ,

(4.8) ϒ ( ε s ) ≤ θ ϒ ( s ) , 0 ≤ ε ≤ 1 , and s ∈ ( 0 , r ] .

Using (4.6), Assumption ( A 1 ) , and Jensen’s inequality, we obtain

(4.9) Ψ ( t ) = 1 β Φ ( t ) ∫ t 1 t Φ ( t ) ( − g ′ ( s ) ) β ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s ≥ 1 β Φ ( t ) ∫ t 1 t Φ ( t ) ξ ( s ) ϒ ( g ( s ) ) β ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s ≥ ξ ( t ) β Φ ( t ) ∫ t 1 t ϒ ( Φ ( t ) g ( s ) ) β ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s ≥ ξ ( t ) β ϒ 1 Φ ( t ) ∫ t 1 t Φ ( t ) g ( s ) β ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s = ξ ( t ) β ϒ β ∫ t 1 t g ( s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s = ξ ( t ) β ϒ ¯ β ∫ t 1 t g ( s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s ,

where ϒ ¯ is an extension of ϒ such that ϒ ¯ is strictly increasing and strictly convex C 2 function on ( 0 , ∞ ) . Hence, we obtain

(4.10) ∫ t 1 t g ( s ) ∫ 0 1 ∣ φ x ( t ) − φ x ( t − s ) ∣ 2 d x d s ≤ 1 β ϒ ¯ − 1 β Ψ ( t ) ξ ( t ) .

Therefore, (4.2) becomes

(4.11) Γ ′ ( t ) ≤ − σ 1 E ( t ) + c ϒ ¯ − 1 β Ψ ( t ) ξ ( t ) , ∀ t ≥ t 1 .

Now, for ε 0 < r , let

(4.12) Γ 1 ( t ) ≔ ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) Γ ( t ) + E ( t ) ,

which is equivalent to E . Using the fact that E ′ ≤ 0 , ϒ ¯ ′ > 0 , ϒ ¯ ″ > 0 , then (4.11) converts to

(4.13) Γ 1 ′ ( t ) = ε 0 E ′ ( t ) E ( 0 ) ϒ ¯ ″ ε 0 E ( t ) E ( 0 ) Γ ( t ) + ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) Γ ′ ( t ) + E ′ ( t ) ≤ − σ 1 E ( t ) ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) + c ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) ϒ ¯ − 1 β Ψ ( t ) ξ ( t ) + E ′ ( t ) .

Thanks to the argument given in Arnold [31], we obtain

(4.14) ϒ ¯ * ( s ) = s ( ϒ ¯ ′ ) − 1 ( s ) − ϒ ¯ ( ( ϒ ¯ ′ ) − 1 ( s ) ) , if s ∈ ( 0 , ϒ ¯ ′ ( r ) ] ,

where ϒ ¯ * is the conjugate of ϒ ¯ in the sense of Young [31] and ϒ ¯ * satisfies the following Young’s inequality:

(4.15) A B ≤ ϒ ¯ * ( A ) + ϒ ¯ ( B ) , if A ∈ ( 0 , ϒ ¯ ′ ( r ) ] , B ∈ ( 0 , r ] .

So, with A = ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) and B = ϒ ¯ − 1 β Ψ ( t ) ξ ( t ) , using (3.1) and (4.13)–(4.15), we arrive at

(4.16) Γ 1 ′ ( t ) ≤ − σ 1 E ( t ) ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) + c ϒ ¯ * ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) + c β Ψ ( t ) ξ ( t ) + E ′ ( t ) ≤ − σ 1 E ( t ) ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) + c ε 0 E ( t ) E ( 0 ) ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) + c β Ψ ( t ) ξ ( t ) + E ′ ( t ) .

Using the fact that ε 0 E ( t ) E ( 0 ) < r , then ϒ ¯ ′ ε 0 E ( t ) E ( 0 ) = ϒ ′ ε 0 E ( t ) E ( 0 ) , and multiplying (4.16) by ξ ( t ) , we find

(4.17) ξ ( t ) Γ 1 ′ ( t ) ≤ − σ 1 ξ ( t ) E ( t ) ϒ ′ ε 0 E ( t ) E ( 0 ) + c ε 0 ξ ( t ) E ( t ) E ( 0 ) ϒ ′ ε 0 E ( t ) E ( 0 ) + c β Ψ ( t ) + ξ ( t ) E ′ ( t ) ≤ − σ 1 ξ ( t ) E ( t ) ϒ ′ ε 0 E ( t ) E ( 0 ) + c ε 0 ξ ( t ) E ( t ) E ( 0 ) ϒ ′ ε 0 E ( t ) E ( 0 ) − c E ′ ( t ) .

Hence, by taking ϒ 2 ( t ) = ξ ( t ) Γ 1 ( t ) + c E ( t ) , we find for some positive constants b 1 , b 2 > 0 ,

(4.18) b 1 Γ 2 ( t ) ≤ E ( t ) ≤ b 2 Γ 2 ( t ) .

Consequently, with a suitable choice of ε 0 , we obtain, for a constant Λ > 0 and ∀ t ≥ t 1 ,

(4.19) Γ 2 ′ ( t ) ≤ − Λ ξ ( t ) E ( t ) E ( 0 ) ϒ ′ ε 0 E ( t ) E ( 0 ) = − Λ ξ ( t ) ϒ 2 E ( t ) E ( 0 ) ,

where ϒ 2 ( t ) = t ϒ ′ ( ε 0 t ) . Since ϒ 2 ′ ( t ) = ϒ ′ ( ε 0 t ) + ε 0 t ϒ ″ ( ε 0 t ) , using the convexity of ϒ on ( 0 , r ] , we find that ϒ 2 ′ ( t ) , ϒ 2 ( t ) > 0 on ( 0 , 1 ] . Thus, with

(4.20) Δ ( t ) = b 1 Γ 2 ( t ) E ( 0 ) ∼ E ( t ) .

By (4.19), we find for some positive constant β 1 > 0 ,

(4.21) Δ ′ ( t ) ≤ − β 1 ξ ( t ) ϒ 2 ( Δ ( t ) ) , ∀ t ≥ t 1 .

Then, integration over ( t 1 , t ) yields

(4.22) ∫ t 1 t − Δ ′ ( t ) ϒ 2 ( Δ ( t ) ) d s ≥ β 1 ∫ t 1 t ξ ( s ) d s ,

which gives

(4.23) ∫ ε 0 Δ ( t ) ε 0 Δ ( t 1 ) 1 s ϒ ′ ( s ) d s ≥ β 1 ∫ t 1 t ξ ( s ) d s .

Hence, if ϒ 1 ( t ) = ∫ t r 1 s ϒ ′ ( s ) d s , then using the properties of ϒ , the fact ϒ 1 is strictly decreasing function on ( 0 , r ] and lim t → 0 ϒ 1 ( t ) = + ∞ , we obtain

(4.24) Δ ( t ) ≤ 1 ε 0 ϒ 1 − 1 β 1 ∫ t 1 t ξ ( s ) d s .

Consequently, from (4.20) and (4.24), we obtain the desired result (4.5).□

Example 4.3

  1. If g ( t ) = ℵ e − ζ t , t ≥ 0 , ℵ , ζ > 0 are the constants. ℵ is chosen such that ( A 1 ) is valid, then

    g ′ ( t ) = − ζ ϒ ( g ( t ) ) , with ξ ( t ) = ζ and ϒ ( s ) = s .

    So under the assumptions of Theorem 4.1, we conclude that the solution of (1.7)–(1.10) satisfies, for two constants ϰ 1 , ϰ 2 > 0 , the energy estimate

    E ( t ) ≤ ϰ 1 e − ϰ 2 t , ∀ t > t 1 .

  2. Let g ( t ) = ℵ e − ( 1 + t ) ν , for t ≥ 0 , 0 < ν < 1 . ℵ is chosen such that condition ( A 1 ) is valid, then

    g ′ ( t ) = − ξ ( t ) ϒ ( g ( t ) ) , with ξ ( t ) = ν ( 1 + t ) ν − 1 and ϒ ( s ) = s .

    So under the assumptions of Theorem 4.1, we conclude that the energy (2.4) satisfies, for some constant ϱ > 0 ,

    E ( t ) ≤ ϱ e − ϱ ( 1 + t ) ν , when t is large enough .

  3. For

    g ( t ) = ℵ ( 1 + t ) ν , t ≥ 0 ,

    for ν > 1 and ℵ is chosen so that hypothesis ( A 1 ) remains valid. Then,

    g ′ ( t ) = − ζ ϒ ( g ( t ) ) , with ξ ( t ) = ζ and ϒ ( s ) = s p ,

    where ζ is a fixed constant, p = 1 + ν ν , which satisfies 1 < p < 2 . So under the assumptions of Theorem 4.2, we conclude that the energy (2.4) satisfies, for some constant ϱ > 0 and t 1 > 0 ,

    E ( t ) ≤ ϱ ( 1 + t ) ν , ∀ t > t 1 .

Remark 4.4

From the aforementioned examples, we note that the decay rate of the energy E ( t ) is consistent with the decay rate of the relaxation function g . So, the decay rate of the energy E ( t ) is optimal.

5 Conclusion

In this work, we investigated the asymptotic behavior of solutions of the viscoelastic Shear beam model (no rotary inertia). This model has only one finite wave speed for all wave numbers. We proved that the dissipative Shear beam model has general and optimal decay results. These results improve and generalize some earlier results in the literature.

Acknowledgements

The author would like to express his profound gratitude to King Fahd University of Petroleum and Minerals (KFUPM) for its continuous support. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project No. INCB2403, is also greatly acknowledged.

  1. Funding information: This work was funded by KFUPM, Grant No. INCB2403.

  2. Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation. The author read and approved the final version of the manuscript.

  3. Conflict of interest: The author declares no competing interests.

  4. Data availability statement: No data were used to support this study.

References

[1] S. P. Timoshenko, LXVI. on the correction for shear of the differential equation for transverse vibrations of prismatic bars, London Edinburgh Dublin Philosoph. Magazine J. Sci. 41 (1921), no. 245, 744–746. 10.1080/14786442108636264Search in Google Scholar

[2] A. Guesmia and S. A. Messaoudi, On the control of a viscoelastic damped Timoshenko-type system, Appl. Math. Comput. 206 (2008), no. 2, 589–597. 10.1016/j.amc.2008.05.122Search in Google Scholar

[3] S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dyn. Syst. Appl. 18 (2009), no. 3, 457–468. 10.1063/1.3183537Search in Google Scholar

[4] S. A. Messaoudi, J. H. Hassan, et al., General and optimal decay in a memory-type Timoshenko system, J. Integral Equ. Appl. 30 (2018), no. 1, 117–145. 10.1216/JIE-2018-30-1-117Search in Google Scholar

[5] I. Elishakoff, An equation both more consistent and simpler than the Bresse-Timoshenko equation, in: R. Gilat, L. Banks-Sills (eds.), Advances in Mathematical Modeling and Experimental Methods for Materials and Structures, Solid Mechanics and Its Applications, Vol. 168, Springer, Dordrecht, 2009, pp. 249–254. 10.1007/978-90-481-3467-0_19Search in Google Scholar

[6] S. M. Han, H. Benaroya, and T. Wei, Dynamics of transversely vibrating beams using four engineering theories, J. Sound Vibrat. 225 (1999), no. 5, 935–988. 10.1006/jsvi.1999.2257Search in Google Scholar

[7] D. S. Almeida Júnior, A. J. A. Ramos, and M. M. Freitas, Energy decay for damped shear beam model and new facts related to the classical Timoshenko system, Appl. Math. Lett. 120 (2021), 107324. 10.1016/j.aml.2021.107324Search in Google Scholar

[8] T. A. Apalara, C. A. Raposo, and A. Ige, Thermoelastic Timoshenko system free of second spectrum, Appl. Math. Lett. 126 (2022), 107793. 10.1016/j.aml.2021.107793Search in Google Scholar

[9] A. Keddi, S. A. Messaoudi, and M. Alahyane, On a thermoelastic Timoshenko system without the second spectrum: Existence and stability, J. Thermal Stresses 46 (2023), no. 8, 823–838.10.1080/01495739.2023.2191662Search in Google Scholar

[10] D. S. Almeida Júnior, M. L. Santos, and J. E. Muñoz Rivera, Stability to weakly dissipative timoshenko systems, Math. Meth. Appl. Sci. 36 (2013), no. 14, 1965–1976. 10.1002/mma.2741Search in Google Scholar

[11] I. Elishakoff, F. Hache, and N. Challamel, Variational derivation of governing differential equations for truncated version of Bresse-Timoshenko beams, J. Sound Vibrat. 435 (2018), 409–430. 10.1016/j.jsv.2017.07.039Search in Google Scholar

[12] D. S. Almeida Júnior, A. J. A. Ramos, M. L. Santos, and L. Gutemberg, Asymptotic behavior of weakly dissipative Bresse-Timoshenko system on influence of the second spectrum of frequency, ZAMM Z. Angew. Math. Mech. 98 (2018), no. 8, 1320–1333. 10.1002/zamm.201700211Search in Google Scholar

[13] I. Elishakoff, Who developed the so-called Timoshenko beam theory?, Math. Mechanics Solids 25 (2020), no. 1, 97–116. 10.1177/1081286519856931Search in Google Scholar

[14] D. S. Almeida Júnior, A. J. A. Ramos, A. Soufyane, M. L. Cardoso, and M. L. Santos, Issues related to the second spectrum, Ostrogradsky’s energy and the stabilization of Timoshenko-Eehrenfest-type systems, Acta Mech. 231 (2020), 3565–3581. 10.1007/s00707-020-02730-7Search in Google Scholar

[15] A. J. A. Ramos, D. S. Almeida Júnior, M. M. Freitas, and M. J. Dos Santos, A new exponential decay result for one-dimensional porous dissipation elasticity from second spectrum viewpoint, Appl. Math. Lett. 101 (2020), 106061. 10.1016/j.aml.2019.106061Search in Google Scholar

[16] A. J. A. Ramos, D. S. Almeida Júnior, and L. G. R. Miranda, An inverse inequality for a Bresse-Timoshenko system without second spectrum of frequency, Arch. Math. (Basel) 114 (2020), 709–719. 10.1007/s00013-020-01452-5Search in Google Scholar

[17] B. Feng, D. S. Almeida Júnior, M. J. Dos Santos, and L. G. R. Miranda, A new scenario for stability of nonlinear Bresse-Ttimoshenko type systems with time dependent delay, ZAMM Z. Angew. Math. Mech. 100 (2020), no. 2, e201900160.10.1002/zamm.201900160Search in Google Scholar

[18] D. S. Almeida Júnior, M. L. Santos, and J. E. Muñoz Rivera, Stability to 1-d thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys. 65 (2014), no. 6, 1233–1249. 10.1007/s00033-013-0387-0Search in Google Scholar

[19] T. A. Apalara, S. A. Messaoudi, and A. A. Keddi, On the decay rates of Timoshenko system with second sound, Math. Meth. Appl. Sci. 39 (2016), no. 10, 2671–2684. 10.1002/mma.3720Search in Google Scholar

[20] M. A. Ayadi, A. Bchatnia, M. Hamouda, and S. Messaoudi, General decay in a Timoshenko-type system with thermoelasticity with second sound, Adv. Nonlinear Anal. 4 (2015), no. 4, 263–284. 10.1515/anona-2015-0038Search in Google Scholar

[21] B. Feng, On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors, Discrete Contin. Dyn. Syst. 37 (2017), no. 9, 4729–4751. 10.3934/dcds.2017203Search in Google Scholar

[22] B. Feng and X.-G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Appl. Anal. 96 (2017), no. 4, 606–625. 10.1080/00036811.2016.1148139Search in Google Scholar

[23] A. Guesmia, S. A. Messaoudi, and A. Soufyane, Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-Heat systems, Electron. J. Differential Equations 2012 (2012), no. 193, 1–45. Search in Google Scholar

[24] M. Kafini, S. A. Messaoudi, and M. I. Mustafa, Energy decay result in a Timoshenko-type system of thermoelasticity of type III with distributive delay, J. Math. Phys. 54 (2013), 101503. 10.1063/1.4826102Search in Google Scholar

[25] A. Malacarne and J. E. Muñoz Rivera, Lack of exponential stability to Timoshenko system with viscoelastic Kelvin-Voigt type, Z. Angew. Math. Phys. 67 (2016), 67. 10.1007/s00033-016-0664-9Search in Google Scholar

[26] X. Tian and Q. Zhang, Stability of a Timoshenko system with local Kelvin–Voigt damping, Z. Angew. Math. Phys. 68 (2017), no. 1, 20. 10.1007/s00033-016-0765-5Search in Google Scholar

[27] A. M. Al-Mahdi, M. M. Al-Gharabli, A. Guesmia, and S. A. Messaoudi, New decay results for a viscoelastic-type Timoshenko system with infinite memory, Z. Angew. Math. Phys. 72 (2021), 22. 10.1007/s00033-020-01446-xSearch in Google Scholar

[28] F. Alabau-Boussouira, P. Cannarsa, and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal. 254 (2008), no. 5, 1342–1372. 10.1016/j.jfa.2007.09.012Search in Google Scholar

[29] M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Meth. Appl. Sci. 41 (2018), no. 1, 192–204. 10.1002/mma.4604Search in Google Scholar

[30] A. Youkana, A. M. Al-Mahdi, and S. A. Messaoudi, General energy decay result for a viscoelastic swelling porous-elastic system, Z. Angew. Math. Phys. 73 (2022), 88. 10.1007/s00033-022-01696-xSearch in Google Scholar

[31] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Vol. 60, Springer-Verlag New York, NY, 2013. Search in Google Scholar
Received: 2023-12-01
Revised: 2024-03-24
Accepted: 2024-03-29
Published Online: 2024-05-23

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0011
Keywords for this article
Timoshenko system; Shear beam models; multiplier method; general and optimal decay; memory
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0011/html
Scroll to top button