Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model

1.   Introduction

Lord Shulman's thermoelasticity has also garnered a lot of interest among scientists in the past few years and there is a broad amount of contributions to explaining this theory. This theory is based on the study of a set of four hyperbolic equations with heat dissipation. In this instance, the heat conduction is also equivalent and hyperbolic in contrast to the one produced for Fourier's law. For more information and explanation about this theory and other theories, see ^[1,2]. Green & Naghdi ^[3,4] created a thermoelasticity model that incorporates thermal displacement gradient and temperature gradient among the constitutive variables and proposed a heat conduction law.

There are numerous results from the coupling of the Fourier law of heat conduction and various systems, which has been discussed by researchers. For example, the Timoshenko system has been studied in ^[5,6], the Bresse system combined with the Cattaneo law of heat conduction (Bresse-Cattaneo) in ^[7], the Bresse system (Bresse Fourier) has been studied in ^[8,9,10] and MGT problem in ^[11]. The following papers are recommended to the reader for more information ^[12,13].

Initially, the basic evolution equations for the one-dimensional Timoshenko thermo-elasticity theories with microtemperature and temperature ^{[14,15,16,17,18]} provided by

Here $E, H, T, q, \zeta, Q, P^{*}, T_{0}, G$ denote the first moment of energy, the equilibrated stress, the stress, the heat flux vector, the entropy, the mean heat flux, the first heat flux moment, the reference temperature and the equilibrated body force, $K, \rho, I_{\rho}, I \& E$ represent the shear modulus, the density, the polar moment of inertia of a cross section, the moment of inertia of a cross section and Young's modulus of elasticity respectively. In order to make calculations simple, let $T_0 = \rho = I_{\rho} = K = 1$ and $EI = a^{2} > 0$.

In this work, the natural counterpart to the Lord-Shulman theory's microtemperatures is taken into consideration ^[1,2]. In this case, we must modify the constitutive equations to take the following form.

where

Here, the functions $\hbar$ and $\Im$ denote the elastic material displacement, $\varpi$ is the microtemperature vector and the function $\theta$ is the temperature difference, $\kappa > 0$ is the relaxation parameter. $\beta_1, \beta_{2} > 0$, the coefficients $\gamma_{1}, k, \gamma_{0}$ denote, the coupling between the volume fraction and the temperature, the thermal conductivity, the coupling between the displacement and the temperature respectively.

As coupling is considered the coefficients $k_1, k_2, k_3, \gamma_{2}, \mu_{1} > 0$ and satisfy the inequalities

The main objective of this study is the thermal effects, let the heat capacity $\beta_{1} = \beta > 0$, and for more excitement in posing the problem, we do not assume the microtemperature effect, and $\beta_{2} = k_1 = k_2 = k_3 = \gamma_2 = 0$.

Now, by substituting (1.2) and (1.3) into (1.1), we obtain that:

where

with initial conditions

We categorise this paper as follows: We utilise our preliminary findings in the remainder of this section to help us understand our major decay conclusion. The Lyapunov functional is built and the estimate for the Fourier image is discovered in the following section utilising the energy approach in Fourier space. The final portion is devoted to the conclusion.

This, to our knowledge, is one of the first studies to look at this issue in the Fourier space. We require the following Lemma to support our main finding.

Lemma 1.1. For any $k, \alpha\geq 0, c > 0$, there exist a constant $C > 0$ in such a way that $\forall \ \ t\geq0$ the below stated estimate hold

2.   Energy method and decay estimates

This section provide the decay estimate of the Fourier image of the solution for problems (1.5) and (1.6). This approach enables us to give the decay rate of the solution in the energy space by utilising Plancherel's theorem along with some integral estimates, such as Lemma 1.1. We create the appropriate Lyapunov functionals and apply the energy method in Fourier space to this problem. We conclude by demonstrating our main finding.

2.1.   The energy method in the Fourier space

Let us introduce the new variables in order to construct the Lyapunov functional in the Fourier space

Then, the system (1.5) can also be written as

where initial condition are

with

Hence, the problems (2.2) and (2.3) can be written as

with $F = (f, j, b, m, \zeta, \varpi)^{T}, \quad F_{0} = (f_{0}, j_{0}, b_{0}, m_{0}, \zeta_{0}, \varpi_{0})$ and

Utilizing the Fourier transform on (2.4), we obtain that:

where $\widehat{F}(\wp, t) = (\widehat{f}, \widehat{j}, \widehat{b}, \widehat{m}, \widehat{\zeta}, \widehat{\varpi})^{T}(\wp, t)$. The quation $(2.5)_{1}$ can be written as

Lemma 2.1. Let $\widehat{F}(\wp, t)$ be a solution of (2.5). Then the energy functional $\widehat{E}(\wp, t)$, given by

satisfies

Proof. Firstly, multiplying $(2.6)_{1, 2, 3, 4}$ by $\overline{\widehat{f}}, \overline{\widehat{j}}, \overline{\widehat{b}}$ and $\overline{\widehat{m}}$ respectively, and multiplying $(2.6)_{5, 6}$ by $\overline{\widehat{\zeta}}, k\overline{\widehat{\varpi}}$, adding these equalities and taking the real part, we obtain that

we find

Hence, we obtain (2.7) and (2.8) ($\widehat{E}$ is non-increasing function). □

Now, to obtain the main result, we need the following lemmas.

Lemma 2.2. The functional

satisfies,

Proof. Differentiating $\mathcal{D}_{1}$ and using (2.6), we get

We evaluate the terms in the RHS of (2.12), utilizing Young's inequality, for any $\delta_{1}, \delta_{2} > 0$ we get

Inserting the above estimates (2.13) into (2.12) and by letting $\delta_{1} = \frac{1}{2}, \delta_{2} = \frac{a}{6}$, we get (2.11). □

Lemma 2.3. The functional

holds, for any $\varepsilon_{1} > 0$

Proof. Differentiating $\mathcal{D}_{2}$ and using (2.6), we get

Similarly, we evaluate the terms in the RHS of (2.16), by applying Young's inequality, for any $\varepsilon_{1}, \delta_{3} > 0$ we find

By substituting (2.17) into (2.16) and letting $\delta_{3} = \frac{\beta}{2}$, we obtain (2.15). □

Next, we will discuss the the below mentioned lemmas.

Lemma 2.4. The functional

satisfies:

(1) For $a = 1$. Then, for any $\varepsilon_{2} > 0$

(2) For $a\neq 1$. Then, for any $\varepsilon_{2}, \varepsilon_{3} > 0$

Proof. Firstly, differentiating $\mathcal{D}_{3}$ and using (2.6), we get

Now we will discuss two cases:

$\mathbf{Case 1.}$ $(a = 1)$.

In this case, by applying the Young's inequality to the terms on the RHS of (2.21), for any $\varepsilon_{2}, \delta_{4} > 0$ we get

Substituting the estimates (2.22) in (2.21) and by letting $\delta_{4} = \frac{1}{4}$, we find (2.19).

$\mathbf{Case 2.}$ $(a\neq 1)$.

In this case, using the Young's inequality to the last terms on the RHS of (2.21), gives for any $\varepsilon_{3} > 0$

Inserting (2.23) and (2.22) in (2.21), we get (2.20). Which completes the proof of Lemma 2.4. □

At this stage, we define the Lyapunov functional for the two cases by

where $N, N_{i}, i = 1, 2, 3$ are positive constants which will be selected later.

Lemma 2.5. There exist $\mu_{2}, \mu_{3}, \mu_{4} > 0$ such that the functional $\mathcal{H}(\wp, t)$ stated by (2.24) satisfies

where

Proof. Firstly, for the case $a = 1$, by differentiating (2.24) and using (2.8), (2.11), (2.15) and (2.19), with the fact that $\frac{\wp^{2}}{1+\wp^{2}}\leq\min\{1, \wp^{2}\}$ and $\frac{1}{1+\wp^{2}}\leq1$, we find

By setting

we obtain

Similarly, for the case $a\neq1$, by differentiating (2.24) and using (2.8), (2.11), (2.15) and (2.20), with the fact that $\frac{\wp^{2}}{1+\wp^{2}}\leq\min\{1, \wp^{2}\}$ and $\frac{1}{1+\wp^{2}}\leq1$, we find

By setting

we obtain

Next, for the two cases (2.28) and (2.30), we fixed $N_{1}$ and choosing $N_{3}$ large enough such that

then we select $N_2$ large enough such as

Hence, for the two cases we arrive at

Additionally, we have

Utilizing Young's inequality and the fact that $\frac{\wp^{2}}{1+\wp^{2}}\leq\min\{1, \wp^{2}\}$ and $\frac{1}{1+\wp^{2}}\leq1$, we find

Therefore, we get

Now, we pick $N$ large enough such as

and exploiting (2.7), estimates (2.31) and (2.32), respectively, there exists a positive constant $\alpha > 0$, $\forall \; t > 0$ and $\forall \; \wp \in \mathbb{R}$, such that

and

then

Consequently, for some positive constant $\mu_{4} = \frac{\lambda_{1}}{\mu_{3}} > 0$, we get

where $\rho(\wp) = \frac{\wp^{4}}{(1+\wp^{2})^{4}}$, for some $\lambda_{1}, \mu_{i} > 0, i = 2, 3, 4$. The proof of the first result (2.25) is completed.

□

The pointwise estimates of the functional $\widehat{E}(\wp, t)$ is stated by the following Proposition.

Proposition 2.1. For any $t\geq0$ and $\wp\in\mathbb{R}$, positive constants $d_{1} > 0$ exist in such a way that the energy functional given by (2.7) satisfies

where $\rho(\wp) = \dfrac{\wp^{4}}{(1+\wp^{2})^{4}}$.

Proof. From $(2.25)_{2}$, we have by integration over $(0, t)$

Hence, by according of (2.25) and (2.38), we establish (2.37). □

2.2.   Decay estimates

Theorem 2.1. Let $s$ be a nonnegative integer, and $F_{0}\in H^{s}(\mathbb{R})\cap L^{1}(\mathbb{R})$. Then, the solution $F$ of problems (2.2) and (2.3) holds, $\forall \; t\geq0$ the following decay estimates

where $\ell$ and $k$ are nonnegative integers such that $k+\ell\leq s$ and $C > 0$ is a positive constant.

Proof. From (2.7), we get $\vert\widehat{F}(\wp, t)\vert^{2}\sim\widehat{E}(\wp, t)$, then by utilizing the Plancherel theorem and exploiting $(2.37)$, we achieve that

Now, we estimate $R_{1}, R_{2}$, the low-frequency part $\vert\wp\vert\leq1$ and the high-frequency part $\vert\wp\vert\geq1$ respectevly. Firstly, we have $\rho(\wp)\geq\frac{1}{16}\wp^{4}$, for $\vert\wp\vert\leq1$. Then

by utilizing Lemma 1.1, we get that

Secondly, we have $\rho(\wp)\geq\frac{1}{16}\wp^{-4}$, for $\vert\wp\vert\geq1$. Then

Expoiting the inequality

we get that

Substituting (2.42) and (2.45) into (2.40), we find (2.39).

□

3.   Conclusions

The investigation of the general decay estimate of solutions to a one-dimensional Lord-Shulman Timoshenko system with thermal effect and damping term is the goal of this study, we prove some optimal decay results for the $L^2$-norm of the solution. More precisely, we prove that the decay rate of the solution is of the form $(1 + t)^{-1/8}$.

To prove our results, we used the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. In system (1.5) the presence of the mechanical damping $\mu_{1}\Im_{t}$ seems to be necessary for our treatment. It is an interesting problem to prove the same result for $\mu_{1} = 0$.

In the upcoming study, we will attempt to utilize the same methodology in the same systems, but with the different types of the memory and the delay terms, we believe that we will obtain similar results.

Use of AI tools declaration

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

Acknowledgments

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.

Conflict of interest

The authors declare no conflicts of interest.