On the time decay for a thermoelastic laminated beam with microtemperature e ﬀ ects, nonlinear weight, and nonlinear time-varying delay

: This article examines the joint impacts of microtemperature, nonlinear structural damping, along with nonlinear time-varying delay term, and time-varying coe ﬃ cient on a thermoelastic laminated beam, where, the equation representing the dynamics of slip is a ﬀ ected by the last three mentioned terms. A general decay result was established regarding the system concerned given equal wave speeds and particular assumptions related to nonlinear terms.

Herein ς(t) > 0 is the time-varying delay and µ denotes a positive damping constant, while the function b(t) stands for the nonlinear weight.
Structural beams play a crucial role in numerous engineering applications, as some machines are relying on a multitude of them, making them indispensable.As they need to withstand diverse challenges, and adapt to various scenarios, these beams have evolved into a sophisticated technology, embodying cutting-edge engineering concepts.Researchers have proposed various theories to explain their behavior, including the popular Euler-Bernoulli beam theory and the Timoshenko beam theory, which excels in dealing with thick beams under the influence of shear forces and rotatory inertia.
Frequently, the unwanted vibrations of these beams are caused by internal or external forces, which compel scientists to find efficient ways to rapidly mitigate these vibrations.To achieve this objective, numerous types of dampers have been developed.
Time delays can result in lags among input and output processing as well as in achieving or restoring the stability of the coveted system, after internal or external perturbations.The presence of these lags is due to the nature of transportation and processing of information of control systems.Delay differential equations are the most efficient method for explicitly analyzing the impact of delays on stability in control systems.Even though including delays may support system control in some cases, as indicated in [1], researches suggest that delays can also cause instability and degrade the system efficiency.Regarding the time-varying delay along with nonlinear weight, we should invoke the research of Mukiawa et al. [2], in which a thermoelastic Timoshenko beam with suspenders together with timevarying delay and nonlinear weight was considered, and a general stability result was demonstrated, with convenient assumptions regarding incorporated nonlinear terms.
When it comes to boundary stabilization study, Wang et al. were the pioneers in providing results.They demonstrated an exponential decay result for a laminated beams with structural damping, mixed homogeneous boundary conditions, and unequal wave speeds in their study [3].Later on, Tatar enhanced upon the work of [3] in [4] by also proving a similar exponential decay result, but supposing that G < I .
In the matter of microtemperature effects, we bring up the study of Khochemane [5], where he investigated a theromelastic porous problem, together with microtemperature effects.When the thermal conductivity equals zero, he managed to establish that the dissipation due solely to microtemperature is adequate to stabilize the system exponentially, regardless of the system's wave velocities, and any possible assumption concerning the coefficients.
Newly, a thermoelastic laminated beam along with structural damping was examined by Fayssal in [6] and he came to the conclusion that an exponential stability result is achievable if 3) The coupled system we've described involves several complex physical phenomena, including thermoelasticity, laminated beams, microtemperature effects, nonlinear structural damping, and nonlinear time-varying delay.Let's break down each component and its physical background: Thermoelastic laminated beam: A laminated beam consists of multiple layers of different materials bonded together.Thermoelasticity refers to the combined behavior of thermal and elastic effects in a material.When the beam is subjected to temperature changes or thermal gradients, it experiences thermal expansion/contraction, which induces mechanical stresses and deformations due to the elastic properties of the materials.
Microtemperature effects: This refers to the consideration of temperature variations at a very small scale, such as at the microstructural level of the materials.At this scale, temperature gradients can lead to localized effects, such as material phase changes, microstructural alterations, or thermal stresses, which can influence the overall behavior of the coupled system.
Nonlinear structural damping: Damping is a phenomenon that dissipates energy from a vibrating system.Nonlinear damping implies that the damping force is not linearly proportional to the velocity of the system.This can occur due to various reasons, such as material hysteresis, contact friction, or fluid-structure interactions.Nonlinear damping can significantly affect the dynamic response of the system.
Nonlinear time-varying delay: A time delay occurs when an effect is not instantaneous and takes some time to propagate through a system.Nonlinear and time-varying delays mean that the delay itself changes based on the current state of the system, and this delay may also have nonlinear effects on the overall behavior.Time delays can lead to instability, oscillations, or even chaos in dynamic systems.The physical background of this coupled system involves the intricate interplay of these phenomena.It requires a sophisticated mathematical and computational approach to model and analyze the system's behavior accurately.Researchers and engineers studying such systems aim to understand how these factors interact and influence each other to predict the system's response to different inputs, boundary conditions, and environmental changes.Such analyses are crucial in various fields, including material science, structural engineering, and advanced manufacturing, where a deep understanding of complex coupled systems is essential for designing reliable and efficient systems.
The remnant of the article is arranged in the following manner: In Section 2, we give necessary assumptions and resources for our study, then bring out our major results.In Section 3, we present useful lemmas, which are indispensable later in the proof.In Section 4, we establish, by means of the energy approach our coveted stability results.

Preliminaries
This section is devoted to revealing our major results and setting the necessary assumptions supporting the proof later.
To address problem Eq (2.12) properly, we shall consider the following positive constant: along with ν(t) = νb(t), furthermore, Y(p) will serve to denote Y(x, p, t).
We present the energy of the concerned system Eqs (2.12) and (2.13) by (2.15) We then give the ensuing stability result.
For more details, the existence and uniqueness of the solution of our problem can be established by continuing the arguments of the Faedo-Galerkin method as in reference [12].

Technical lemmas
Establishing the practical lemmas necessary to support our stability results proof is the primary goal of this section.We use a particular method known as the multiplier technique, the latter allows us to demonstrate the stability result of problem Eq (2.12).To make matters simpler, we will utilize χ, Υ * > 0 to symbolize a generic constants that may vary from one line to another (including within the same line).Lemma 3.1.Consider (ψ, u, φ, θ, r, Y) the solution of Eqs (2.12) and (2.13), then, the energy functional satisfies Proof.To start with, we multiply the first five equations of system Eq (2.12) by ψ t , (3φ t − u t ), φ t , θ, and r, respectively.After that, we integrate over (0, 1) and employ integration by parts along with boundary conditions Eq (2.13), to establish Then, we need to multiply Eq (2.12) 6 by ν(t)h 2 (Y(p)), and integrate over (0, 1) × (0, 1), to achieve Thereby, which accompanied with Eq (3.2), (A 2 ) and Eq (2. 2), leads to We shall now define the convex conjugate function of ξ, This makes, the relation listed below valid by means of the general Young's inequality (see [13,14]): We employ the definition of ξ as well as Eq (2. 3), to obtain and the simple combination of Eqs (3.7) and (2.2) results in Then, we benefit of Eqs (3.4), (3.6) and (3.8), to be in position to write We finally prove estimate Eq (3.1), with the aid of Eqs (2.14) and (2.6).
Lemma 3.5.Consider functional Then, it satisfies Proof.To start with, we differentiate I 4 , and consider Eq (2.12) 1 with integration by parts, to achieve and observing that ψ x = −(u − ψ x ) − (3φ − u) + 3φ, we obtain By means of Young and Poincaré's inequalities, we have and once we replace Eqs (3.20) and (3.21) into Eq (3.19), the estimate (3.17) is easily proved.
Lemma 3.6.Consider the functional Then, it satisfies Proof.We advance by differentiating I 5 , employing Eq (2.12) 2 accompanied with integration by parts, which results in We terminate our proof, once Young and Poincaré's inequalities are used.Then, it satisfies Proof.We take here the derivative of I 6 , to find which together with Eq (3.27) leads to To prove Eq (3.26), it is convenient to consider Young's inequality accompanied by Eq (2.2).

Stability result
Here, we exploit lemmas from Section 3 to demonstrate our stability results.
Proof of Theorem 2.1.We advance by introducing a Lyapunov functional where constants N, N i > 0, i = 1 • • • 6, will be fixed later.By Eq (4.1), we are in position to write Subsequently, we select coefficients in Eq (4. 3) such that all of them (excluding the final three) turn negative.To this end, we opt to take N 2 to be sufficiently large so that which makes us opt to take N 3 enough large to have and we finish by taking N to be fairly huge to obtain both Eq (4.2) and Now, it is convenient to consider definition Eq (2.15) along with the above selection of constants, and Poincaré's inequality, to find As a part of this proof, we shall distinguish two cases: Case 1: Suppose that T is linear.Hypothesis (A 1 ) enables us to write where Employing the above estimates, we are in position to write b(t)