Stability of thermoelastic Timoshenko system with variable delay in the internal feedback

1.   Introduction

In 1921, S. P. Timoshenko studied the transverse vibration of a beam, and found that the motion could be described by a family of partial differential equations in which the bending moment and shear stress were involved. Based on the constitutive laws in the mathematical elasticity theory, the following system, called the Timoshenko system ^[1], was derived

where $L > 0$ is the length of beam, $\varphi$ is the vertical displacement of beam, $\psi$ denotes the rotation angle of the filament of beam, and $\rho_1, \rho_2, k, b$ are positive constants.

So far, many significative results on the dissipative Timoshenko system have been derived, involving the linear and nonlinear cases, dynamical behavior, stability, and so on. Regarding the exponential stability of the dissipative Timoshenko system, the following equal speed condition is usually important:

To achieve the desired dissipation, damping terms are needed necessarily to system (1.1), and we can refer to ^[2,3,4] and references therein, in which the frictional damping, indefinite damping, and weak damping were involved. In general, the influence of thermal damping was usually considered in research on Timoshenko system ^{[5,6,7,8,9,10,11,12]}. The coupled system, called the thermoelastic Timoshenko system, is always combined with other damping terms, in which the heat conduction is ordinarily under some law ($\tau q_t+ q+\beta\theta_x = 0$), e.g., the Fourier law ($\tau = 0$) and the Cattaneo law ($\tau\neq 0$).

In 2009, H. D. Fernández Sare and R. Racke studied the thermoelastic Timoshenko system with memory under the Fourier law and obtained the exponential stability, which was not true for the case of Cattaneo law ^[13]. The stability of thermoelastic system was also obtained in ^[14], in which the thermal damping was in the shear moment, and it is different from ^[13]. Many good results could be referred in ^{[15,16,17,18]}. Furthermore, the influence of delay appears in economics, physics, and other fields, the delay term could be constant delay, continuous delay, or the distributed delay, which is important to the stability of system; We can refer to ^[19,20,21] for the results on fluid system with delay, and see ^[22,23,24] for the Timoshenko system with delay.

In this paper, under the Fourier law we consider the thermoelastic Timoshenko system subject to memory and variable time delay on $(0, L)\times \mathbb{R}^+$

where $\theta$ is the temperature, $q$ is the heat flux vector, $a, d, \sigma, \rho_3, \tilde{b}, \beta$ are positive constants satisfying $d > \sqrt{2}a$, $(x, t)\in (0, L)\times \mathbb{R}^+$, the delay function $\rho(\cdot):\ \mathbb{R}^+\rightarrow [M_0, h]\ (h > M_0 > 0)$ is $C^1$ continuous and satisfies that

and the positive exponentially decaying kernel function $g$ satisfies for some constant $k_1 > 0$ that

The primary characteristics of this article are different from the previous results, and details are as follows:

1) Different from ^[13], the thermal damping $\sigma\theta_x$ is in the shear moment term of this article, which is similar with the case of ^[14]. Also, the continuous delay terms $a\varphi_t(x, t-\rho(t))$ and $d\varphi_t(x, t)$ are considered in the shear moment term, which bring us some difficulties in deriving the dissipation and stability.

2) Since the operator $A(t)$ generated in (1.3) is nonautonomous, the autonomous operator method in ^[25] is invalid. So, the variable norm technique in ^[26] is used, which is suitable for the case of the nonautonomous operator.

The rest of this article is arranged as follows. In Section 2 we give the equivalent problem of (1.3), and present a useful lemma concerning the perturbation theory and main results in Section 3. The well-posedness of (1.3) is derived by the Lummer-Phillips theorem in Section 4. Four sufficient conditions in Kato's perturbation theory are verified in Section 5, and the exponential stability of (1.3) is proved finally.

2.   Equivalent problem

At first, we introduce a new variable for the delay feedback term

and we have

For the memory term, we present

Using the above transformation, we convert the system (1.3) to the following equivalent form:

where $(x, \kappa, t)\in (0, L)\times(0, 1)\times \mathbb{R}^+$. Also, the new equivalent system (2.4) is equipped with the initial conditions

and the boundary conditions

To solve the problems (2.4)–(2.6), we write

For the delay space $L^2((0, L)\times(0, 1))$, we define the following inner product and norm

where $\xi = \xi(t)$ satisfies

We consider the phase (Hilbert) space

endowed with the norm

and corresponding inner product $(\cdot, \cdot)_\mathscr{H}$ for all $U = (\varphi, \Phi, \psi, \Psi, \theta, \eta, z)^T\in \mathscr{H}$, where $\Phi = \varphi_t$ and $\Psi = \psi_t$. Basing on the equivalent equations, for any $t\in \mathbb{R}^+$, we define the operator $A(t)$ as the following:

with domain

From the definition of $D(A(t))$ we know

and the new system (2.4) can be written as the following abstract form

3.   A useful lemma and main results

To study the nonlinear system (2.9) with nonautonomous operator, we introduce here the perturbation theory (see ^[26]) suitable for the case of the nonautonomous operator.

Lemma 3.1. Suppose that for any $t\in \mathbb{R}^+$, there holds

(P-I) The domain $D(A(t))$ is dense in the phase space $\mathscr{H}$, and $D(A(t)) = D(A(0))$.

(P-II) In space $\mathscr{H}$, the operator $A(t)$ generates a $C_0$ semigroup $\{S_{A(t)}(s)\}_{s\geq 0}$ for any $t\in [0, T]$, where $T > 0$ is some fixed constant.

(P-III) There exist positive constants $K_1, K_2$ independent of $t\in [0, T]$ satisfying

(P-IV) The operator $A(t):\ [0, T]\rightarrow B(D(A(0)), \mathscr{H})$ is essentially bounded and strongly measurable.

Then, for any $U_0\in \mathscr{H}$, the solution $U$ to system (2.9) is unique and satisfies

Theorem 3.1. For any fixed $t\in [0, T]$ and any $U_0\in \mathscr{H}$, the solution $U$ to the system (2.9) exists uniquely, generates a $C_0$ semigroup of contraction $\{S_{A(t)}(s)\}_{s\geq 0}$ in $\mathscr{H}$, whose infinitesimal generator is $A(t)$, and there holds that

To prove Theorem 3.1, we apply the Lummer-Phillips theorem, and need to show for any fixed $t\in [0, T]$ that $0\in \varrho(A(t))$ (spectral set of $A(t)$) and the dissipation of $A(t)$.

Theorem 3.2. For any $U_0\in \mathscr{H}$, there exist constants $C, \varsigma > 0$, being independent of $U_0$, such that for any $s\in \mathbb{R}^+$,

which means the system (2.9) is exponentially stable.

To prove Theorem 3.2, we use the well-known characterization of exponential stability for $C_0$ semigroup as follows, and we can refer to ^[25] for a detailed process of proof.

Theorem 3.3. For any fixed $t\in [0, T]$, let $S_{A(t)}(s) = e^{A(t)s}$ be a $C_0$ semigroup of contraction in space $\mathscr{H}$. Then, the semigroup is exponentially stable if and only if

4.   Proof of Theorem 3.1

Basing on the Lummer-Phillips theorem, we need to derive the following lemmas to get the well-posedness, and we can also see ^[24] for reference.

Lemma 4.1. For any fixed $t\in [0, T]$, the operator $A(t)$ has the property of dissipation.

Proof. For any $U = (\varphi, \Phi, \psi, \Psi, \theta, \eta, z)^T\in D(A(t))$, we have

and it follows that

Lemma 4.2. For any fixed $t\in [0, T]$, $0\in \varrho(A(t))$.

Proof. For any $V = (v_1, v_2, v_3, v_4, v_5, v_6, v_7)^T\in \mathscr{H}$, we study the equations

that is,

By (4.4) and (4.6), we get

From (4.10), we know

It follows that

Here, some equalities could be rewritten as

As to (4.15), we use the conclusion that $\rho_3v_5+\sigma(v_{1x}+v_3)\in L^2(0, L)$ together with the standard elliptic theory, and we obtain that

It follows from (4.13) that

In addition, we use the standard elliptic theory again and the relation $k(\varphi_x+\psi)-\sigma\theta+\rho_2v_4\in L^2(0, L)$, and we derive that

Using the same technique as shown as in ^[13], we can obtain that

In summary, the system (4.3) is solved uniquely and $U = (\varphi, \Phi, \psi, \Psi, \theta, \eta, z)^T\in D(A(t))$. From the above derivation process, we can show that there exists a positive constant $C$ such that

which means $0\in \varrho(A(t))$ for any fixed $t\in \mathbb{R}^+$. According to the Lummer-Phillips Theorem, we finish the proof of Theorem 3.1.

5.   Verification of conditions in Lemma 3.1

In order to be able to use the autonomous operator theory to the nonautonomous case, we need to verify the conditions in the Kato's perturbation theory, and we can also refer to ^[27] and draw inspiration.

Lemma 5.1. $\overline{D(A(0))} = \mathscr{H}$.

Proof. Suppose that $U_1 = (\varphi_1, \Phi_1, \psi_1, \Psi_1, \theta_1, \eta_1, z_1)^T\perp D(A(0))$, then there holds for any $U = (\varphi, \Phi, \psi, \Psi, \theta, \eta, z)^T\in D(A(0))$ that

When $U = (0, 0, 0, 0, 0, 0, z)^T$ and $z\in C_{0}^{\infty}((0, L)\times(0, 1))$, $U\in D(A(0))$, and from (5.1) we have

The fact that $C_{0}^{\infty}((0, L)\times(0, 1))$ is dense in $L^{2}((0, L)\times(0, 1))$ leads to $z_1(x, \kappa, t) = 0$ in $(0, L)\times(0, 1)\times \mathbb{R}^{+}$. Using the same technique, we can show that

It follows that $\overline{D(A(0))} = \mathscr{H}$, and the condition (P-Ⅰ) is verified.

Lemma 5.2. For any $t\in [0, T]$, the operator family $\{A(t)\}$ is stable in $\mathscr{H}$.

Proof. For any $U\in \mathscr{H}$, we write $\|U\|^2_{t} = \|(\varphi, \Phi, \psi, \Psi, \theta, \eta, \sqrt{\rho(t)}z)^T\|^2_{\mathscr{H}}$. From assumptions on $\rho(t)$, we know that for any $t\in [0, T]$ there holds

If we choose $C' = min\{M_0, 1\}, \ C'' = max\{h, 1\}$, then

It follows that $\|U\|^2_{t} = \|(\varphi, \Phi, \psi, \Psi, \theta, \eta, \sqrt{\rho(t)}z)^T\|^2_{\mathscr{H}}$ is an equivalent norm of space $\mathscr{H}$, and we claim here that

In fact, for any $0\leq t_0\leq t \leq T$, there holds

From the properties of $\rho(t)$, we know for any $\vartheta\in (t_0, t)$ there holds

which means

From the fact that $1-e^{\frac{\widetilde{M_1}}{2M_0}(t-t_0)}\leq 0$, we derive the conclusion, which means the condition (P-Ⅲ) is verified.

Lemma 5.3. For any $t\in [0, T]$, the operator $A(t)\in L_{\star}^{\infty}([0, T], B(D(L(0)), \mathscr{H}))$.

Proof. For any $U = (\varphi, \Phi, \psi, \Psi, \theta, \eta, z)^T\in \mathscr{H}$, from (2.8), we know

It follows that

which is bounded for $t\in [0, T]$, and the conclusion is finished, which means the condition (P-Ⅳ) is verified.

To sum up, basing on the result in Theorem 3.1, we confirm the four conditions in Lemma 3.1.

6.   Proof of Theorem 3.2

6.1.   Proof of (3.1)

Assume that $i\mathbb{R}\subset \varrho(A(t))$ is not true. Therefore, there is a constant $C_\ast$, a sequence $\{\beta_n\}$ satisfying $0 < \beta_n\rightarrow C_\ast$ and $i\beta_n\in \varrho(A(t))$, and a sequence of functions

such that

That is,

Lemma 6.1. For any $t\in \mathbb{R}^+$, we have, as $n\rightarrow \infty$,

Proof. Multiplying (6.2) by $U_n$ in $\mathscr{H}$ yields that

which means

and the conclusion holds finally.

Thus, the system (6.2) could be reduced into the following simplified form:

Lemma 6.2. For any $t\in \mathbb{R}^+$, we have, as $n\rightarrow \infty$,

Proof. Multiplying (6.22) by $\overline{z_n}$ and considering the real part, we obtain

which means

and

It follows from (6.12) that the conclusion is finished.

Lemma 6.3. For any $t\in \mathbb{R}^+$, we have, as $n\rightarrow \infty$,

Proof. Multiplying (6.16) and (6.17) by $\rho_1\overline{\Phi_n}$ and $\overline{\varphi_n}$, respectively, we get

and

Adding (6.26) and (6.27), and taking the real part, we have

and from Lemmas (6.1) and (6.2), we derive that

Multiplying (6.18) and (6.19) by $\rho_2\overline{\Psi_n}$ and $\overline{\psi_n}$, respectively, we get

and

Adding (6.30) and (6.31), we have

and

Using (6.29) and (6.32), we derive that

and the result is obtained.

From the Lemma 2.5 in ^[14], we know

Combining (6.2), (6.34), and (6.35), from the fact that $\|U_n\|^2_\mathscr{H} = \rho_1\|\Phi_n\|_{L^2}^2+\rho_2\|\Psi_n\|_{L^2}^2+k\|\varphi_{nx}+\psi_n\|_{L^2}^2+b\|\psi_{nx}\|_{L^2}^2 = 1$, we can show that

Multiplying (6.17) and (6.19) by $\frac{b}{\kappa}{\overline{\psi_{nx}}}$ and $\overline{\varphi_{nx}+\psi_n}$, respectively, we get

and

Adding (6.37) and (6.38), we have

Multiplying (6.16) and (6.18) by $\overline{\Psi_{n}}$ and $i\beta_n\rho_2\overline{\psi_{n}}$, respectively, we get

and

Adding (6.40) and (6.41), we have

Adding (6.39) and (6.42), we have

Multiplying (6.18) by $\rho_2\overline{\Phi_{nx}}$, we get

Adding (6.43) and (6.44), we get

From the equal speed condition, we have

and there holds

This is a contradiction with (6.36), and we prove that $i\mathbb{R}\subset \varrho(A(t))$.

6.2.   Proof of (3.2)

To achieve the goal, for any $V = (v_1, v_2, v_3, v_4, v_5, v_6, v_7)^T\in\mathscr{H}$, we consider the resolvent equation

that is,

Lemma 6.4. For any $t\in \mathbb{R}^+$, there exists a constant $C_0$ independent of $V$ such that

Proof. Multiplying (6.48) by $U$ in $\mathscr{H}$, combining Lemma 6.1, and taking the real part, we derive that

Lemma 6.5. For $\lambda$ large enough and any $t\in \mathbb{R}^+$, there holds that

Proof. Combining (6.49), (6.51), and (6.53), we have

Multiplying (6.58) by $\overline{k(\varphi_{x}+\psi)}$ in $L^2(0, L)$, we have

In the following way, we estimate each term of (6.59). From (6.50), we derive that

Next, from (6.49) and (6.51), we derive

and it follows that

We see that

that is,

and Young's inequality leads to the conclusion finally.

Lemma 6.6. For $\lambda$ large enough, any $t\in \mathbb{R}^+$, and any $\varepsilon > 0$, there exists a constant $C_\varepsilon > 0$ satisfying

Proof. Multiplying (6.50) by $-\overline{\varphi}$ in $L^2(0, L)$ and using (6.49), we have

From Lemma 6.4, we can derive that

and

It follows from Lemma 6.5 and Young's inequality that we can derive the conclusion.

Lemma 6.7. For $\lambda$ large enough, any $t\in \mathbb{R}^+$, and any $\varepsilon > 0$, there exists a constant $C_\varepsilon > 0$ satisfying

Proof. Multiplying (6.52) by $\int_{0}^{\infty}g(s)\overline{\eta(s)}ds$ in $L^2(0, L)$, we have

By (6.54), we obtain

and integration by parts leads to

Using (6.49) and (6.51), we show that

and from the H$\ddot{o}$lder inequality we get

Combining Lemmas 6.4 and 6.6, and (6.70)–(6.75), and using the H$\ddot{o}$lder inequality and the Young inequality, we can derive the conclusion. We can also refer to ^[14] for excellent details.

Lemma 6.8. For $\lambda$ large enough, any $t\in \mathbb{R}^+$, and any $\varepsilon > 0$, there exists a constant $C_\varepsilon > 0$ satisfying

Proof. Multiplying (6.19) by $\overline{\psi}$ in $L^2(0, L)$, we have

From (6.51), we have

and using Lemmas 6.6 and 6.7 leads to

It follows from Lemma 6.4 and the Young inequality that the conclusion is derived.

In summary, combination of Lemmas 6.4–6.8 help us to obtain

and (3.2) is finished finally.

7.   Further research

In this article, we study the exponential stability of thermoelastic Timoshenko with variable delay in the internal feedback. If the variable delay is replaced by the distributed delay, the relating problem is still open, which is our next objective.

Use of AI tools declaration

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

Acknowledgments

This work was partly supported by College Student Innovation and Entrepreneurship Training Program of Henan (Grant No. 5506200119), and the Key Research Project of Institutions of Higher Education of Henan (Grant No. 23B110003).

Author contribution

Conceptualization, methodology, writing-original draft preparation, K. Su; software, writing-review and editing, X. Ge. All authors have read and agreed to the published version of the manuscript.

Conflict of interest

The authors declare there is no conflict of interest.