Well-posedness and stabilization of a type three layer beam system with Gurtin-Pipkin's thermal law

1.   Introduction

In the present work, we consider the Rao-Nakra (three layer) beam system, where the top and the bottom layers of the beam are subjected to Gurtin-Pipkin's thermal law, namely

with the following boundary conditions:

and the initial data

The relaxation functions $g_1$ and $g_2$ are positive non-increasing functions to be specified later. The stabilization of Rao-Nakra beam systems has gathered much interest from researchers recently, and a great number of results have been established. The Rao-Nakra beam model is a beam system that takes into account the motion of two outer face plates (assumed to be relatively stiff) and a sandwiched compliant inner core layer, see ^[1,2,3,4,5] for Rao-Nakra, Mead-Markus and multilayer plates or sandwich models. The basic equations of motion of the Rao-Nakra model are derived thanks to the Euler-Bernoulli beam assumptions for the outer face plate layers, the Timoshenko beam assumptions for the sandwich layer and a "no slip" assumption for the motion along the interface. Suppose $h(j)$, $j = 1, 2, 3$ is the thickness of each layer in the beam of length $\pi$, see Figure 1 and $h = h(1)+h(2)+h(3)$ the total thickness of the beam.

Assuming the Kirchhoff hypothesis is imposed on the outer layers of beam and in addition, there is a continuous, piecewise linear displacements through the cross-sections, Liu et al. ^[6] gave a detailed derivation of following laminated beam system:

where $x\in (0, \pi), t > 0$, $(u, y^1), (v, y^3)$ represent longitudinal displacement and shear angle of the bottom and top layers plates. The transverse displacement of the beam is represented by $w$, and $\tau$ is the shear stress of the core layer. Also, for $j = 1, 2, 3$ (from bottom to top layer), $E_j, G_j, I_j, \rho_j > 0$ are Young's modulus, shear modulus, moments of inertia and density respectively for each layer. Moreover, in $(1.4)_3$, we have that $\rho h = \rho_1h_1+\rho_2h_2+\rho_3h_3$ and $EI = E_1I_1+E_3I_3$. By neglecting the rotary inertia in top and bottom layers of the beam, we obtain $\rho_1I_1 = \rho_3I_3 = 0$ in (1.4)$_4$ and (1.4)$_5$. Furthermore, if we neglect the transverse shear, this leads to the Euler-Bernoulli hypothesis $w_x+y^1 = w_x+y^3 = 0$. Assuming that the core layer consists of a material that is linearly elastic with the stress-strain relationship $\tau = 2G_2\varepsilon$, where the shear strain $\varepsilon$ is defined by

Thus, we arrive at the following Rao-Nakra beam model ^[1], given by

where $k = \dfrac{G_2}{h_2}$, $G_2 = \dfrac{E_2}{2(1+\nu)}$ and $-1 < \nu < \dfrac{1}{2}$ is the Poisson ratio. Furthermore, when the extensional motion of the outer layers is neglected, system (1.4) takes the form of the two-layer laminated beam system derived by Hansen and Spies ^[7]. Li et al. ^[8] showed that system (1.5) is unstable if only one of the equations is damped. When two of the three equations in (1.5) were damped, the authors in ^[8] proved a polynomial stability. For recent results in literature, Méndez et al. ^[9] considered (1.5) with with Kelvin-Voigt damping and studied the well-posedness, lack of exponential decay and polynomial decay. Feng and Özer ^[10] looked at a nonlinearly damped Rao-Nakra beam system and established the global attractor with finite fractal dimension. Feng et al. ^[11] studied the stability of Rao-Nakra sandwich beam with time-varying weight and time-varying delay. Mukiawa et al. ^[12] considered (1.5) with viscoelastic damping on the first equation and heat conduction govern by Fourier's law and proved the well-posedness and a general decay result. Also, Raposo et al. ^[13] coupled (1.5) with Maxwell-Cattaneo heat conduction established the well-posedness. For more results related Rao-Nakra beam system with frictional, delay or thermal damping, see ^{[14,15,16,17,18,19,20]} and the references therein.

An interesting tool used by Mathematician in stabilizing beam models such as the Laminated and Timoshenko beam systems is the Gurtin-Pipkin's thermal law, see ^[21], with constitutive equation

where $\theta = \theta(x, t)$ is the temperature difference, $q = q(x, t)$ is the heat flux, $\beta$ is a coupling constant coefficient and the relaxation $g$ is a summable convex $L^1([0, +\infty))$ function with unit mass. For results related to (1.6), Dell'Oro and Pata ^[22] studied

and proved an exponential stability result if and only if $\chi_h = 0,$ where

For similar results with Gurtin-Pipkin's thermal law, see ^{[23,24,25,26,27,28]} and references therein. As clearly elaborated in ^[22], the Fourier's and Cattaneo's (second sound) thermal law can be recovered from (1.6) by defining the memory function $g$ in (1.6) as

and

respectively. A closely related thermal law to the Gurtin-Pipkin's thermal law is the Coleman-Gurtin's heat conduction law, see ^[29], with constitutive equation given by

where $\eta = 1$ and $\eta = 0$ correspond to the Gurtin-Pipkin's and Fourier thermal laws, respectively. This entails replacing $(1.7)_3$ with

We should note here that systems govern by Coleman-Gurtin's thermal lawa (1.10) gain additional dissipation from the term $-\frac{(1-\eta)}{\beta}\theta_{xx}$ and thus less difficult to handle compare to systems with Gurtin-Pipkin's thermal law (1.6).

Our main focus of this paper is to investigate the well-posedness and the asymptotic behavior of solutions of system (1.1)–(1.3). We mote here that, the rotational inertia term $w_{xxtt}$ which should be in $(1.1)_3$ of the original models is neglected in the present model. However, the result in this paper is not affected by the absent of this term. Also, since the thermal coupling in system (1.1)–(1.3) is not strong enough to achieve exponential stability, a viscous damping term $w_t$ is added to $(1.1)_3$. The rest of work is organized as follows: In Section 2, we state some assumptions and set up our problem (1.1)–(1.3) in appropriate spaces. In Section 3, we prove the existence and uniqueness result for the system (1.1)–(1.3). In Section 4, we study the asymptotic behavior of solution of system (1.1)–(1.3).

2.   Assumptions, problem transformation and functional setting

2.1.   Assumptions on the kernels

For the relaxation functions $g_1$ and $g_2$, we assume the following:

Assumption $(A_0)$:

($a_0$) $g_1, g_2:[0, +\infty)\longrightarrow (0, +\infty)$ are non-increasing $C^2([0, +\infty))$ and convex summable functions satisfying

($b_0$) There exists $\xi_i > 0, \ i = 1, 2$ such that

By setting

assumption $(A_0)$ ensues the following:

Assumption $(A_1)$:

($a_1$) $\mu_1, \mu_2:[0, +\infty)\longrightarrow (0, +\infty)$ are non-increasing $C^1([0, +\infty))$ and convex summable functions satisfying

($b_1$) There exists $\xi_i > 0, \ i = 1, 2$ such that

2.2.   Problem transformation

Due to the work of Dafermos ^[30], we define new functions for the relative past history of $\theta$ and $\vartheta$ as follows:

define by

On account of the boundary conditions (1.2), we have

and routine calculation gives

where $\sigma_0$ and $\zeta_0$ represent the history of $\theta$ and $\vartheta$ respectively. Also, using direct computations, we have

Similarly, we get

On account of (2.6)–(2.9), system (1.1)–(1.3) takes the form

with the boundary conditions

and the initial data

Setting $\Psi = (u, \varphi, v, \psi, w, \phi, \theta, \sigma, \vartheta, \zeta)^T$, with $\varphi = u_t$, $\psi = v_t$ and $\phi = w_t$. Then, the semi-group formulation of system (2.10)–(2.12) is given by the Cauchy problem

where $\Psi_0 = (u_0, u_1, v_0, v_1, w_0, w_1, \theta_0, \sigma_0, \vartheta_0, \zeta_0)^T$ and the linear operator $\mathcal{A}$ is defined by

2.3.   Functional spaces

Let $\langle, \rangle$ and $\|.\|$ denote the inner product and the norm in $L^2(0, \pi)$ respectively and we define following Sobolev spaces:

where $H^2_*$ is equip with the inner product

and norm

It is easy to check that $(H^2_*, \|.\|^2_{H^2_*})$ is a Banach space and the norm $\|.\|^2_{H^2_* }$ is equivalent to the usual norm in $H^2(0, \pi).$ Next, we introduce the weighted-Hilbert space of $H_a^{1}(0, \pi)$-real valued functions on $(0, +\infty)$ by

where

and equip them with the inner product

and norm

Also, we define

Now, we introduce the phase space of our problem given by

and equipped it with the inner product

and norm

for any $\Phi = (w, \varphi, v, \psi, u, \phi, \theta, \sigma, \vartheta, \zeta)^T \in \mathcal{H}$.

The domain of the linear operator $\mathcal{A}$ in (2.13) is defined as follows:

Remark 2.1. (1) Due to (2.5), we can deduce that

(2) Using Hölder's and Young's inequalities, we have that

3.   Well-posedness

In this section, we establish the existence and uniqueness of global weak solution to the system (2.10)–(2.12).

3.1.   Needed lemmas for well-posedness

Lemma 3.1. The linear operator $\mathcal{A}:\mathcal{D}(\mathcal{A})\subset\mathcal{H}\to \mathcal{H}$ defined in (2.13) is monotone.

Proof. Let $\Psi = (u, \varphi, v, \psi, w, \phi, \theta, \sigma, \vartheta, \zeta) \in\mathcal{D}(\mathcal{A})$, then using integration by parts and the boundary conditions (2.11), we have

From (2.5) and (2.6), we obtain

Therefore,

Therefore, $\mathcal{A}$ is monotone. □

Lemma 3.2. The linear operator $\mathcal{A}:\mathcal{D}(\mathcal{A})\subset\mathcal{H}\to \mathcal{H}$ defined in (2.13) maximal, that is $\mathfrak{R}(I+\mathcal{A}) = \mathcal{H}.$

Proof. Given $F = (k^1, k^2, k^3, k^4, k^5, k^6, k^7, k^8, k^9, k^{10})\in\mathcal{H},$ we look for a unique solution

such that $\Psi$ solves the stationary problem

System (3.1) is equivalent to

By multiplying $(3.2)_8$ and $(3.2)_{10}$ by $e^{r}$ and integrating the results over $(0, s),$ we arrive at

From $(3.2)_1$, $(3.2)_3$ and $(3.2)_5$, we get

respectively. Substituting (3.4) and (3.3) into $(3.2)_2$, $(3.2)_4, (3.2)_6, (3.2)_7$ and $(3.2)_9$ leads to

where

Now, we observe that last terms in $f^4$ and $f^ 5$ are in $H^{-1}(0, \pi)$. Indeed, since $k^8\in L^2_{\mu_1}$, we have for any

that

In the same way, we get that

Next, we consider the Banach space $\mathbb{H}: = H^1_b \times H^1_b \times H^2_*\times L^2 \times L^2$ and equip it with the norm

On the account of the weak formulation of (3.5), we consider the bilinear form $\mathcal{B}$ on $\mathbb{H}\times\mathbb{H}$ and linear form $\mathcal{L}$ on $\mathbb{H}$, define as follows:

and

for every $(u, v, w, \theta, \vartheta), \ (u^*, v^*, w^*, \theta^*, \vartheta^*)\in \mathbb{H}$. Routine computations, using Cauchy-Schwarz, Young's and Poincaré's inequalities shows that $\mathcal{B}$ is a bounded and coercive bilinear form on $\mathbb{H}\times\mathbb{H}$, and $\mathcal{L}$ is a bounded linear form on $\mathbb{H}$. Therefore, using Lax-Milgram theorem, there exists a unique $(u, v, w, \theta, \vartheta)\in \mathbb{H}$ such that

From $(3.4)$, it follows that

Then, using standard regularity theory, it follows from $(3.5)$, that

Since $u, v\in H^1_b, w, k^6\in H^2_*$ and $k^6\in L^2$, it easy to see from $(3.5)_3$ that $w$ satisfy

Also, from (3.3), substituting $\theta$ and $\vartheta$, we see that

Finally, from $(3.2)_7$ and $(3.2)_9$, using regularity theory, we get that

Thus, $\Psi = (u, \varphi, v, \psi, w, \phi, \theta, \sigma, \vartheta, \zeta) \in\mathcal{D}(\mathcal{A})$ and satisfies (3.1). That is, the operator $\mathcal{A}$ is maximal. □

3.2.   Well-posedness Result

Theorem 3.1. Suppose $\Psi_0 = (u_0, u_1, v_0, v_1, w_0, w_1, \theta_0, \sigma_0, \vartheta_0, \zeta_0)\in\mathcal{H}$ is given and condition $(A_1)$ holds, then the Cauchy problem (2.13) has a unique weak global solution

Furthermore, if $\Psi_0 = (u_0, u_1, v_0, v_1, w_0, w_1, \theta_0, \sigma_0, \vartheta_0, \zeta_0)\in\mathcal{D}(\mathcal{A})$, then the solution is in the class

Proof. On account of Lemmas 3.1 and 3.2 applying the Hille-Yosida theorem, we have that $\mathcal{A}$ is a generator of a $C_0$-semigroup of contractions $\mathcal{S}(t) = e^{\mathcal{A}t}, \ t\geq 0,$ on $\mathcal{H}.$ By the semigroup theory for linear operators (Pazy ^[31]), we get that

on $\mathcal{H}$ is a unique solution satisfying problem (2.13). □

4.   Stability result

In this section, we study the stability of solution of (2.10)–(2.12). The energy functional associated to the solution $\Psi = (u, u_t, v, v_t, w, w_t, \theta, \sigma, \vartheta, \zeta)$ of system (2.10)–(2.12) is defined by

4.1.   Needed lemmas for stability

Lemma 4.1. Under the conditions of Theorem 3.1, the energy functional (4.1) satisfies

Proof. Multiplication in $L^2(0, \pi)$ the Eq $(2.10)_1, (2.10)_2, (2.10)_3$, $(2.10)_4$ and $(2.10)_6$ by $u_t, v_t, w_t$, $\theta$ and $\vartheta$ respectively, follow by multiplying $(2.10)_5$ and $(2.10)_7$ by $\sigma$ and $\zeta$ in $L^2_{\mu_1}$ and $L^2_{\mu_2}$ respectively, then using integration by parts and the boundary conditions (2.11), we have

and

Addition of (4.3)–(4.9) leads to

Therefore, the energy $\mathcal{E}$ is non-increasing and bounded above by $\mathcal{E}(0)$. Also, the computations here are done for regular solution. However, the result remains true for weak solution by density argument. □

Lemma 4.2. Let $\Psi = (u, u_t, v, v_t, w, w_t, \theta, \sigma, \vartheta, \zeta)\in \mathcal{H}$ be the solution of system (2.10)–(2.12) given by Theorem 3.1, then the functional $G_1$ defined by

satisfies the estimate

Proof. Differentiation of $G_1$ gives

Using Eq $(2.10)_1$, $(2.10)_2$ and $(2.10)_3$, then applying integration by parts over $(0, \pi)$ and making use of the boundary conditions (2.11) leads to

Applying Young's and Poincaré's inequalities, we obtain

□

Lemma 4.3. Let $\Psi = (u, u_t, v, v_t, w, w_t, \theta, \sigma, \vartheta, \zeta)\in \mathcal{H}$ be the solution of system (2.10)–(2.12) given by Theorem 3.1, then the functional $G_2$ defined by

satisfies, for any $\epsilon_1 > 0$ and $\epsilon_2 > 0,$ the the estimate

Proof. Differentiation of $G_2$, using $(2.10)_1$ and $(2.10)_4$, integration by parts and boundary conditions (2.11), we arrive at

Using Cauchy-Schwarz, Young's and Poincaré's inequalities yields

Thus, we obtain (4.12). □

Lemma 4.4. Let $\Psi = (u, u_t, v, v_t, w, w_t, \theta, \sigma, \vartheta, \zeta)\in \mathcal{H}$ be the solution of system (2.10)–(2.12) given by Theorem 3.1, then the functional $G_3$ defined by

satisfies, for any $\epsilon_3 > 0$ and $\epsilon_4 > 0,$ the estimate

Proof. Differentiation of $G_3$, using $(2.10)_2$ and $(2.10)_5$, integration by parts and boundary conditions (2.11), we arrive at

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

Hence, we get (4.13). □

Lemma 4.5. Let $\Psi = (u, u_t, v, v_t, w, w_t, \theta, \sigma, \vartheta, \zeta)\in \mathcal{H}$ be the solution of system (2.10)–(2.12) given by Theorem 3.1, then the functional $G_4$ defined by

satisfies, for any $\epsilon_5 > 0$ and $\epsilon_6 > 0,$ the estimate

Proof. Differentiating $G_4$ with respect to $t$, using $(2.10)_4$ and $(2.10)_5$, integration by parts and the boundary conditions (2.11) and recalling (2.4), we get

Making use of Cauchy-Schwarz and Young's inequalities, we have

Also, using integration by parts with respect to $s$, we get

On account of (4.16)–(4.19), we obtain

□

Lemma 4.6. Let $\Psi = (u, u_t, v, v_t, w, w_t, \theta, \sigma, \vartheta, \zeta)\in \mathcal{H}$ be the solution of system (2.10)–(2.12) given by Theorem 3.1, then the functional $G_5$ defined by

satisfies for any $\epsilon_7 > 0$, the estimate

Proof. Differentiation of $G_5$ with respect to $t$, using $(2.10)_6$ and $(2.10)_7$, integration by parts and the boundary conditions (2.11), and recalling (2.4), we get

Using similar estimations as in (4.16)–(4.19) leads to (4.20). □

4.2.   Main stability result

The main stability result of this work is the following:

Theorem 4.1. Let $\Psi_0 = (u_0, u_1, v_0, v_1, w_0, w_1, \theta_0, \sigma_0, \vartheta_0, \zeta_0)\in\mathcal{D}(\mathcal{A})$ be given. Suppose condition $(A_1)$ holds, then the energy functional $\mathcal{E}(t)$ defined in (4.1) decays exponentially. That is, there exists positive constants $M$ and $\lambda$ such that

Proof. We set

for some $N, N_1, N_2, N_3, N_4, N_5 > 0$ to be specified later. Direct computations, applying Young's, Cauchy-Schwarz and Poincaré's inequalities gives

for some positive constants $\tilde{b}_1$ and $\tilde{b}_2.$ Now, using Lemmas 4.1 and 4.2$-$4.6, we get

From (2.5), we have that

Also, by choosing

then (4.24) takes the form

Next, we specified the rest of the parameters. First, we choose $N_2$ large such that

Second, we select $N_3$ large enough such that

Thirdly, we choose $N_4$ and $N_5$ large enough such that

and

Finally, we choose $N$ very large so that (4.23) remain valid and

Thus, we obtain

for some $\gamma_0 > 0.$ Recalling (4.1), it follows from (4.26) that

for some $\gamma_1 > 0.$ Using (4.23), we obtain

for some $\gamma_2 > 0.$ Integrating (4.28) over $(0, t)$ yields for some $\gamma_3 > 0$

Hence, the exponential estimate of the energy functional $\mathcal{E}(t)$ in $(4.21)$ follows from (4.29) by using (4.23). This completes the proof. □

5.   Conclusions

In this work, we investigated the the effect of Gurtin-Pipkin's thermal law on the outer layers of the Rao-Nakra beam model. Using standard semi-group theory for linear operators and the multiplier method, the well-posedness and a stability result of solutions of the triple beam system have been established.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

The author acknowledges the technical and financial support from the Ministry of education and the University of Hafr Al Batin, Saudi Arabia. This research work was funded by Institutional fund projects $\#$ IFP-A-2022-2-1-04.

Conflict of interest

The author declares no potential conflict of interest.