Decay for thermoelastic laminated beam with nonlinear delay and nonlinear structural damping

1.   Introduction

In the current work, we study the following thermoelastic laminated beam along with nonlinear structural damping and nonlinear delay

where

with the following initial and boundary conditions

Here, $u, \; \varphi, \; v$, and $\theta$ stand for the transverse displacement, the rotation angle, the amount of slip along the interface, and the difference temperature, respectively. $\varrho, \; G, \; I_\varrho, \; D, \; \delta,$ and $\beta$ are positive parameters representing the density, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness, and adhesive damping, respectively. We denote by $\varrho_3, \; k, \; \gamma$ the positive physical coefficients from thermoelasticity theory.

Herein $\varsigma > 0$ is the time delay, and the positive parameter $\mu$ is the delay weight.

The laminated beam is considered an interesting research subject owed to the broad business applicability of these materials across many industries, thus attracting the attention of researchers. Hansen and Spies in ^[1] proposed the following beam with two layers

The model has a comparable character to the well-known classical Timoshenko system because its equations of movement were contrived based on the concepts of the Timoshenko beam theory. The impacts of the kinetics of interfacial slip are depicted by a third equation which interlocks with the first two ones. Due to their importance, these kind of problems are now highly regarded within the scientific community, and a revived resurgence of interest in examining the asymptotic behavior of the solution of diverse thermoelastic laminated beams has flourished nowadays, see ^[2,3,4,5]. For the readers, the background and the newest works on the qualitative properties related to this topic can be found, especially for the laminated beam, in ^[6,7]. Recent studies have shown that delay may result in instability unless specific conditions are taken into account, and it can also lead to solutions that vary from those obtained in previous studies. Ensuring the stability of systems with delays is of utmost importance; hence, the studies of time delays has emerged as a critical and impactful field of research. Regarding the nonlinear delay, Mpungu and Apalara in ^[8] made a study worth mentioning, in which they took into account system (1.3) and incorporated nonlinear delay and nonlinear structural damping, specifically in the third equation. With the help of convenient conditions on both weight delay and wave speeds, the authors were able to establish a general energy decay rates of the solutions.

Djilali et al. ^[9] integrated a nonlinear delay into a viscoelastic Timoshenko beam problem and managed to demonstrate a global existence result, as well as asymptotic behavior of the solutions while presuming that a certain relation among the weight of the term with no delay and the weight of delay is maintained.

Concerning the researches on boundary stabilization. The work by Wang et al. in ^[10] was the first to present results and to prove an exponential decay result, the authors considered system (1.3) with mixed homogeneous, boundary conditions and unequal wave speeds. Many authors improved upon the work of ^[10], under the assumption that $\varrho G < I_\varrho,$ to establish a similar exponential decay result ^[11,12].

Recently, Fayssal in ^[13], examined a thermoelastic laminated beam with structural damping and proved it to be exponentially stable when the condition below is valid:

The remaining sections of the paper are organized as follows: In Section 2, we exhibit the study's major results after providing its necessary materials. In Section 3, we prove necessary lemmas that will support the proof of our results. In Section 4, once we go by the multiplier technique, our intended stability results are established.

2.   Preliminaries

In this section, we give required assumptions and resources for our study, then we highlight our major results.

We start by setting the necessary assumptions as in ^[14]:

● (A$_1$) The function $\mathfrak{g}_1: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and of class $C^0$. Moreover, there exist constants $\lambda_1, \; \lambda_2, \; \varepsilon > 0$ and a function $\mathfrak{X} \in C^1([0, +\infty)),$ being strictly increasing, fulfilling $\mathfrak{X}(0) = 0,$ and the latter is linear on $[0, \varepsilon]$ or strictly convex of class $C^2$ on $(0, \varepsilon],$ in a way that we have

● (A$_2$) The function $\mathfrak{g}_2 : \mathbb{R} \rightarrow \mathbb{R}$ is odd, increasing, and belongs to $C^1(\mathbb{R})$. In addition, there exist positive constants $\vartheta_*, \; \vartheta_1, \; \vartheta_2,$ such that

and

where

and

Remark 2.1. Exploiting assumption (A$_1$), one can see that

It follows by (A$_2$) and the monotonicity of $\mathfrak{g}_2,$ with the mean value theorem (for integrals) that

therefore

To deal with the nonlinearity of the delay, we shall present a constant $\kappa$ that is positive and fulfilling

As in ^[15], to begin, we introduce

Thus, $\mathcal{S}$ satisfies

Therefore, we obtain the following new system equivalent to the previous one (1.1)

with

Establishing the existence and uniqueness result is achievable by pursuing the reasoning behind the Faedo Galerkin approach, as expounded in ^[16]. To maintain simplicity, we will use $\mathcal{S}(p)$ to represent $\mathcal{S}(x, p, t).$

Now, we shall present our energy of the system (2.8)-(2.9) by

and right after, we exhibit the stability result.

Theorem 2.1. Let $(u, \varphi, v, \theta, \mathcal{S})$ be the solution of (2.8)-(2.9). Suppose that (A$_1$), (A$_2$), and (1.4) hold, then, there exist positive constants $\alpha_0, \; \alpha_1, \; \alpha_2,$ and $\varepsilon_0$ such that

where

and

Prior researches have given examples related to our stability result and assumptions; see ^[8].

3.   Technical lemmas

The lemmas necessary to back up our proof of stability results will be established in this part. To achieve our stability result's proof, a specific method named the multiplier technique will be employed and a generic constant $K^* > 0$ will be used for the sake of simplicity. Note that $K^*$ may change from line to line or in the same line.

Lemma 3.1. Let $(u, \varphi, v, \theta, \mathcal{S})$ be the solution of (2.8)-(2.9), then, the energy functional satisfies

where $\mathcal{M}_0$ and $\mathcal{M}_1$ are positive constants.

Proof. To begin, let us multiply (2.8)$_1$–(2.8)$_4$ by $u_t$, $(3v_t - \varphi_t), v_t$, and $\theta$, respectively, then integrate over $(0, 1)$ and use integration by parts to get

After that, we multiply Eq (2.8)$_5$ by $\kappa\mathfrak{g}_2(\mathcal{S}(p)),$ integrate over $(0, 1) \times(0, 1),$ and notice that $\mathcal{S}(0) = v_t,$ to find

hence,

which, together with both (3.2) and (2.2), gives us

Let us now define the conjugate function of $\xi$ by

thus, $\xi^*$ is the Legendre transformation of $\xi,$ and it is given as

In this way, the following relation is valid (see ^[14,17])

Exploiting (3.5), along with the definition of $\xi$, leads to

The use of (3.7) together with (2.2) yields

Therefore, taking advantage of (3.6), (3.8), and (2.2), we can write

Finally, combining (3.9) and (3.4) with the help of (2.5) and (2.3), the estimate (3.1) is established. □

Lemma 3.2. Consider the functional

then, it satisfies

Proof. We first take $\mathcal{I}_1',$ then exploit Eq $(2.8)_1$–$(2.8)_3,$ integration by parts and $u_x = -(\varphi-u_x)+\varphi,$ to reach

Since $u_x = 3 v-(\varphi-u_x)-(3v-\varphi)$ and hypothesis (1.4) holds, we find

To continue, it is convenient to consider (2.4), along with (3.6) and (3.8), to obtain

We next apply (3.14) and Young and Poincaré's inequalities to finally get (3.11). □

Lemma 3.3. Consider functional

which satisfies, for any $\epsilon_2 > 0,$

Proof. We begin by differentiating $\mathcal{I}_2$, then we take advantage of both (2.8)$_{1, 3},$ and integration by parts. We get

Since $\varphi_t = 3v_t-(3v_t-\varphi_t),$ and by (1.4), we have

Now, with the help of (3.14) and Young and Poincaré's inequalities, one can write

and for any $\epsilon_2 > 0,$

The combination of (3.19)–(3.23) and (3.18) gives us (3.16). □

Lemma 3.4. Consider the functional

then for any $\epsilon_3 > 0$, it satisfies

Proof. We first find the derivative of $\mathcal{I}_3,$ then exploit Eq $(2.8)_{1, 3}$ along with integration by parts, to get

By (3.14) and Young and Poincaré's inequalities, the proof is accomplished. □

Lemma 3.5. Consider the functional

then, it satisfies

Proof. Taking $\mathcal{I}'_4$, Eq (2.8)$_1,$ integration by parts and by $u_x = -(3v-\varphi)-(\varphi-u_x)+3v$, we get

By Young and Poincaré's inequalities, we establish (3.27).

□

Lemma 3.6. Consider the functional

then it satisfies

Proof. By direct calculations, once we consider (2.8)$_2$ and integration by parts, we obtain

By Young and Poincaré's inequalities, we reach

Hence, the combination of (3.32) and (3.31) gives us (3.30). □

Lemma 3.7. Consider the functional

then it satisfies

Proof. Taking both $\mathcal{I}'_6$ and Eq (2.8)$_5$, then exploiting $\mathcal{S}(0) = v_t,$ we get

By using both (2.2) and $e^{-\varsigma} \leq e^{-p\varsigma}\leq 1, \; p \in (0, 1),$ we then prove (3.34). □

4.   Stability result

Our intended stability results are established here based on the previously stated lemmas.

Proof of Theorem 2.1. To begin, we consider a Lyapunov functional

where the constants $N, \; N_i > 0, \; i = 1 \cdots 6,$ will be chosen later.

According to (4.1), we write

By Young, Cauchy-Schwarz, and Poincaré's inequalities, we have

i.e.,

To continue, we take $\mathcal{K}'(t)$ and employ (3.1), (3.11), (3.16), (3.25), (3.27), (3.30), and (3.34), then we set

to get

We then select coefficients in (4.3), to make them all (with the exception of the last two) negative. By taking $N_2$ big enough such that

we can choose $N_3$ fairly large, so

We set $N$ big enough, to get (4.2) and

These choices, with Poincaré's inequality, lead to

In the context of our demonstration, we have two cases to treat:

Case 1. Suppose that $\mathfrak{X}$ is linear on $[0, \varepsilon]$. By hypothesis (A$_1$), we have

which, when combined with (4.4), results in

By merging (3.1) and (4.5), we find

We will now proceed by presenting

Once considering (4.2), we see that

Consequently, when we consider (4.7) and (4.8), we get

Finally, we conclude by simply integrating (4.9) and employing (4.8), to prove that

Case 2. Suppose that $\mathfrak{X}$ is nonlinear on $(0, \varepsilon].$ We take as in ^[18], $0 < \varepsilon_1\leq \varepsilon,$ to have

It is helpful to consider the continuous function $\mathfrak{g}_1,$ with (A$_1$) and to note that $\vert\mathfrak{g}_1(\mathfrak{r})\vert > 0, \; \mathfrak{r}\neq 0,$ to have

Now, we need to work on estimating

To this end, we consider, as in ^[19], the partitions below

The combination of the Jensen's inequality with the concavity of $\mathfrak{X}^{-1},$ results in

where

If we take (3.1), (4.11), and (4.12), we get

We then present the functional

Relation (4.2) implies that

Thus, once we merge (4.13) and (4.4) and exploit (4.14), we conclude that

Let us now consider the functional below

Combining (4.15) and the fact that

we get

Additionally, relation (4.16) yields

Let us set

Similar to what we did earlier with (3.5), we shall now estimate $\mathcal{Q}$ by letting $\mathfrak{X}^*$ be the convex conjugate of $\mathfrak{X}$ given by

Additionally, the use of the general Young's inequality, indicates

We set

By (4.20), (4.21), and

we have

The replacement of (4.22) into (4.19) leads to

Now, selecting $\varepsilon_0 = \frac{\vartheta_3 \mathcal{E}(0) }{2\lambda_7}$ and $\gamma_0 = 2\lambda_8$ gives us

and provided that $\mathcal{E}'(t) \leq 0,$ we get

where $\mathfrak{X}_0(\mathfrak{r}) = \mathfrak{r}\mathfrak{X}'(\varepsilon_0 \mathfrak{r}).$

Now, $\mathfrak{X}$ being strictly convex on $(0, \varepsilon],$ implies that $\mathfrak{X}_0(\mathfrak{r}), \; \mathfrak{X}_0'(\mathfrak{r}) > 0$ on $(0, 1].$ Hence, letting

we find that

Furthermore, the employment of (4.24), results in

In addition, if we take into account (4.26) and that $\mathfrak{X}_0$ is increasing, we achieve

According to (4.27), we have

where

If we integrate (4.28) over $(0, t),$ we get

Since $\mathfrak{X}^{-1}_1$ is a decreasing function, we deduce

We exploit relation (4.26) to ultimately achieve

where $\alpha_0 = \frac{1}{\tilde{a}_1}.$ The proof is then concluded. □

5.   Conclusions

A class of thermoelastic laminated beams is considered. In addition to the impact of thermoelasticity, we are interested here in the interaction between the weights of two terms with delay and without delay given in nonlinear forms. We have shown explicit and general energy decay rates of the solution by using the properties of convex functions and employing the multiplier technique.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by King Khalid University through large research project under grant number R.G.P.2/252/44.

Conflict of interest

The authors declare that there are no conflicts of interest.