Bifurcation for a fractional-order Lotka-Volterra predator-Cprey model with delay feedback control

1.   Introduction

The fundamental work of Hansen and Spies ^[4] modeled a two-layer beam with a structural damping due to the interfacial slip through the following system

where $\varphi = \varphi(x, t)$ is the transverse displacement, $\psi = \psi(x, t)$ is the rotation angle, $w = w(x, t)$ is proportional to the amount of slip along the interface, $3w-\psi$ denotes the effective rotation angle. The physical quantities $\rho, I_{\rho}, G, D, \beta$ and $\gamma$ are respectively: the density, mass moment of inertia, shear stiffness, flexural rigidity, adhesive damping and adhesive stiffness. Equation $(1.1)_3$ describes the dynamics of the slip. For $\beta = 0$, system $(1.1)$ describes the coupled laminated beams without structural damping at the interface. In the recent result ^[1], Apalara considered the thermoelastic-laminated beam system without structural damping, namely

where $(x, t)\in (0, 1)\times (0, +\infty)$, $\theta = \theta (x, t)$ is the difference temperature. The positive quantities $\gamma, \beta, k, \lambda$ are adhesive stiffness, adhesive damping, heat capacity and the diffusivity respectively. The author proved that (1.2) is exponential stable provided

When $\beta > 0$, the adhesion at the interface supplies a restoring force proportion to the interfacial slip. But this is not enough to stabilize system (1.1), see for instance ^[2]. To achieve exponential or general stabilization of system (1.1), many authors in literature have used additional damping. In this direction, Gang et al. ^[9] studied the following memory-type laminated beam system

and established a general decay result for more regular solutions and $\frac{G}{\rho}\neq\frac{D}{I_{\rho}}.$ Mustafa ^[15] also considered the structural damped laminated beam system (1.4) and established a general decay result provided $\frac{G}{\rho} = \frac{D}{I_{\rho}}.$ Feng et al. ^[8] investigated the following laminated beam system

and established the well-posedness, smooth global attractor of finite fractal dimension as well as existence of generalized exponential attractors. See also, recent results by Enyi et al. ^[20]. We refer the reader to ^{[5,6,7,11,13,14,17,18]} and the references cited therein for more related results.

In this present paper, we consider a thermoelastic laminated beam problem with a viscoelastic damping

under initial conditions

and boundary conditions

In the system (1.6), the integral represents the viscoelastic damping, and $g$ is the relaxation function satisfying some suitable assumptions specified in the next section. According to the Boltzmann Principle, the viscoelastic damping (see ^[21] for details) is represented by a memory term in the form of convolution. It acts as a damper to reduce the internal/external forces like the beam's weight, heavy loads, wind, etc., that cause undesirable vibrations.

In most of the above works, the authors have established their decay result by including the structural damping along with other dampings. So, the natural question that comes to mind.

Is it possible to obtain general/optimal decay result (decay rates that agrees with that of $g$) to the thermoelastic laminated beam system (1.6)–(1.8), in the absence of the structural damping.

The novelty of this article is to answer this question in a consenting way, by using the ideas developed in ^[10] to establish general and optimal decay results for Problem 1.6. Moreover, we establish a weaker decay result in the case of a non-equal wave of speed propagation. To the best of our knowledge, there is no stability result for the latter in the literature.

The rest of work is organized as follows: In Section 2, we recall some preliminaries and assumptions on the memory term. In Section 3, we state and prove the main stability result for the case equal-speed and in the case of non-equal-speed of propagation. We also give some examples to illustrate our findings. Finally, in Section 4, we give the proofs of the lemmas used our main results.

2.   Preliminaries

In this section, we recall some useful materials and conditions. Through out this paper, $C$ is a positive constant that may change through lines, $\langle., .\rangle$ and $\|.\|_2$ denote respectively the inner product and the norm in $L^2(0, 1)$. We assume the relaxation function $g$ obeys the assumptions:

(G1). $g:[0, +\infty)\longrightarrow (0, +\infty)$ is a non-inecreasing $C^1 -$ function such that

(G2). There exist a $C^1$ function $H:[0, +\infty)\rightarrow (0, +\infty)$ which is linear or is strictly convex $C^2$ function on $(0, \epsilon_0)$, $\epsilon_0 \leq g(0),$ with $H(0) = H'(0) = 0$ and a positive nonincreasing differentiable function $\xi:[0, +\infty)\rightarrow (0, +\infty)$, such that

Remark 2.1. As in ^[10], we note here that, if $H$ is a strictly increasing convex $C^2-$ function on $(0, r]$, with $H(0) = H'(0) = 0,$ then $H$ has an extension $\bar{H}$, which is strictly increasing and strictly convex $C^2$-function on $(0, +\infty).$ For example, $\bar{H}$ can be defined by

Let

The existence and regularity result of problem (1.6) is the following

Theorem 2.1. Let $(w_0, 3s_0-\psi_0, s_0, \theta_0)\in H_{\ast}^1(0, 1)\times \bar{H}_{\ast}^1(0, 1)\times \bar{H}_{\ast}^1(0, 1)\times H_{\ast}^1(0, 1)$ and $(w_1, 3s_1- \psi_1, s_1)\in L^2(0, 1)\times L^2(0, 1)\times L^2(0, 1)$ be given. Suppose (G1) and (G2) hold. Then problem (1.6) has a unique global weak solution $(w, 3s-\psi, s, \theta)$ which satisfies

Furthermore, if $(w_0, (3s_0- \psi_0), s_0, \theta_0)\in H_{\ast}^2(0, 1)\times \bar{H}_{\ast}^2(0, 1)\times \bar{H}_{\ast}^2(0, 1)\times H^2(0, 1)\cap H_{\ast}^1(0, 1)$ and $(w_1, (3s_1-\psi_1), s_1)\in H_{\ast}^1(0, 1)\times \bar{H}_{\ast}^1(0, 1)\times \bar{H}_{\ast}^1(0, 1),$ then the solution of (1.6) satisfies

The proof of Theorem 2.1 can be established using the Galerkin approximation method as in ^[16]. Throughout this paper, we denote by $\diamond$ the binary operator, defined by

We also define $h(t)$ and $C_\alpha$ as follow

The following lemmas will be applied repeatedly throughout this paper

Lemma 2.1. For any function $f\in L^2_{loc}([0, +\infty), L^2(0, 1))$, we have

Lemma 2.2. Let $v\in H_{\ast}^1(0, 1)\ \ or\ \ \bar{H}_{\ast}^1(0, 1)$, we have

where ${\rm C_p > 0}$ is the poincaré constant.

Lemma 2.3. Let $(w, 3s-\psi, s, \theta)$ be the solution of (1.6). Then, for any $0 < \alpha < 1$ we have

Proof. Using Cauchy-Schwarz inequality, we have

Lemma 2.4. ^[12] Let $F$ be a convex function on the close interval $[a, b]$, $f, j:\Omega\rightarrow [a, b]$ be integrable functions on $\Omega,$ such that $j(x)\geq 0$ and $\int_{\Omega}j(x)dx = \alpha_1 > 0.$ Then, we have the following Jensen inequality

In particular if $F(y) = y^{\frac{1}{p}}, \ \ y\geq 0, \ \ p > 1$, then

Lemma 2.5. The energy functional $E(t)$ of the system (1.6)-(1.8) defined by

satisfies

Proof. Multiplying $(1.6)_{1}$, $(1.6)_{2}$, $(1.6)_{3}$ and $(1.6)_{4},$ respectively, by $w_t$, $(3s_t-\psi_t)$, $s_t$ and $\theta$, integrating over $(0, 1)$, and using integration by parts and the boundary conditions (1.7), we arrive at

and

Adding the equations (2.13)–(2.16), taking into account $(G1)$ and $(G2)$, we obtain (2.12) for regular solutions. The result remains valid for weak solutions by a density argument. This implies the energy functional is non-increasing and

3.   Stability results

This section is subdivided into two. In the first subsection, we prove the stability result for equal-wave-speed of propagation, whereas in the second subsection, we focus on the stability result for non-equal-wave-speed of propagation.

3.1.   Equal-wave-speed of propagation

Our aim, in this subsection, is to prove an explicit, general and optimal decay rate of solutions for system (1.6)–(1.8). To achieve this, we define a Lyapunov functional

where $N, \ N_j, \ j = 1, 2, 3, 4, 5, 6$ are positive constants to be specified later and

where

The following lemma is very important in the proof of our stability result.

Lemma 3.1. Suppose $\frac{G}{\rho} = \frac{D}{I_{\rho}}$. Under suitable choice of $t_0, N, \ N_j, \ j = 1, 2, 3, 4, 5, 6$, the Lyapunov functional $L$ satisfies, along the solution of $(1.6)-(1.8)$, the estimate

and the equivalence relation

holds for some $\beta > 0, \ \alpha_1, \ \alpha_2 > 0.$

Proof. By virtue of assumption (3.1) and using $h(t) = \alpha g(t)-g'(t)$, it follows from Lemmas 2.5, 4.1-4.6 (see the Appendix for detailed derivations) that, for all $t\geq t_0 > 0,$

Now, we choose

and select $N_1$ large enough such that

Next, we choose $N_6$ large so that

Also, we select $N_2$ large enough so that

After fixing $N_1, N_2, N_6$, and letting $\epsilon_3 = \dfrac{\mu_1}{2N_3}$, we then select $\epsilon_1, \epsilon_2,$ and $\delta_4$ very small such that

and select $N_3$ large enough so that

Now, we note that $\frac{\alpha g^2(s)}{h(s)} = \frac{\alpha g^2(s)}{\alpha g(s)-g'(s)} < g(s);$ thus the dominated convergence theorem gives

Therefore, we can choose some $0 < \alpha_0 < 1$ such that for all $0 < \alpha\leq \alpha_0,$

Finally, we select $N$ so large enough and take $\alpha = \frac{1}{N}$ So that

Combination of (3.6) - (3.13) yields the estimate (3.2). The equivalent relation (3.3) can be obtain easily by using Young's, Cauchy-Schwarz, and Poincaré's inequalities.

Now, we state and prove our stability result for this subsection.

Theorem 3.1. Assume $\frac{G}{\rho} = \frac{D}{I_{\rho}}$ and (G1) and (G2) hold. Then, there exist positive constants $a_1$ and $a_2$ such that the energy solution (2.11) satisfies

and $H_1$ is a strictly decreasing and strictly convex function on $(0, r],$ with $\lim _{t\rightarrow 0} H_1(t) = +\infty.$

Proof. Using the fact that $g$ and $\xi$ are positive, non-increasing and continuous, and $H$ is positive and continuous, we have that for all $t\in [0, t_0]$

Thus for some constants $a, b > 0$, we obtain

Therefore, for any $t\in [0, t_0]$, we get

and

From (2.12) and (3.15), it follows that

From (3.2) and (3.17), we have

Thus, we get

where $L_1 = L + C E\sim E$ by virtue of (3.3). To finish our proof, we distinct two cases:

Case 1: H(t) is linear. In this case, we multiply (3.18) by $\xi(t)$, keeping in mind (2.12) and $(G2),$ to get

Therefore

Since $\xi$ is non-increasing and $L_1\sim E,$ we have

Thus, from (3.20), we get for some positive constant $\alpha$

Integrating (3.22) over $(t_0, t)$ and recalling (3.21), we obtain

Case 2: H(t) is nonlinear. In this case, we consider the functional $\mathcal{L}(t) = L(t)+ I_7(t).$ From (3.2) and Lemma 4.7 (see the Appendix), we obtain

where $d > 0$ is a positive constant. Therefore,

Hence, we get

Using (3.24), we define $p(t)$ by

where $0 < \eta < 1$ so that

Moreover, we can assume $p(t) > 0$ for all $t\geq t_0$; otherwise using (3.18), we obtain an exponential decay rate. We also define $q(t)$ by

Then $q(t)\leq -CE'(t), \ \forall t\geq t_0.$ Now, we have that $H$ is strictly convex on $(0, r]$ (where $r = g(t_0)$) and $H(0) = 0.$ Thus,

Using (3.26), condition $(G2)$, (3.25), and Jensen's inequality, we get

where $\bar{H}$ is the convex extention of $H$ on $(0, +\infty)$ (see remark $2.1$). From (3.27), we have

Therefore, (3.18) yields

For $r_0 < r$, we define $L_3(t)$ by

since $L_1\sim E$. From (3.28) and using the fact that

we obtain for all $t\geq t_0$

Let us consider the convex conjugate of $\bar{H}$ denoted by $\bar{H}^{\ast}$ in the sense of Young (see ^[3] page 61-64). Thus,

and $\bar{H}^{\ast}$ satisfies the generalized Young inequality

Let $A = \bar{H}'\left(r_0\frac{E(t)}{E(0)}\right)$ and $B = \bar{H}^{-1}\left(\mu\frac{z(t)}{\xi(t)}\right),$ It follows from (2.12) and (3.29)-(3.31) that

Next, we multiply (3.32) by $\xi(t)$ and recall that $r_0\frac{E(t)}{E(0)} < r$ and

we arrive at

Let $L_4(t) = \xi(t) L_3(t) + CE(t)$. Since $L_3\sim E$, it follows that

for some $b_0, b_1 > 0$. Thus (3.33) gives

We select $r_0 < r$ small enough so that $\beta E(0)-Cr_0 > 0$, we get

for some constant $m > 0$ and $H_2(\tau) = \tau H'(r_0 \tau).$ We note here that

thus the strict convexity of $H$ on $(0, r],$ yields $H_2(\tau) > 0, H_2'(\tau) > 0$ on $(0, r].$ Let

From (3.34) and (3.35), we obtain

and

Integrating (3.37) over $(t_0, t),$ we arrive at

This implies

Using the properties of $H$, we see easily that $H_1$ is strictly decreasing function on $(0, r]$ and

Hence, (3.14) follows from (3.36) and (3.39). This completes the proof.

Remark 3.1. The stability result in (3.1) is general and optimal in the sense that it agrees with the decay rate of $g$, see ^[10], Remark $2.3$.

Corollary 3.2. Suppose $\frac{G}{\rho} = \frac{D}{I_{\rho}}$, and $(G_1),$ and $(G_2)$ hold. Let the function $H$ in $(G_2)$ be defined by

then the solution energy $(2.11)$ satisfies

for some positive constants $a_2, a_1$ and $C.$

3.2.   Nonequal-wave-speed of propagation

In this subsection, we establish another stability result in the case non-equal speeds of wave propagation. To achieve this, we consider a stronger solution of (1.6). Let $(w, 3s-\psi, s, \theta)$ be the strong solution of problem (1.6)–(1.8), then differentiation of 1.6 with respect to $t$ gives

where $(x, t)\in (0, 1)\times (0, +\infty)$ and $(3s-\psi)_{xx}(x, 0) = (3s_0-\psi_0)_{xx}.$ The modified energy functional associated to $(3.42)$ is defined by

Lemma 3.2. Let $(w, 3s-\psi, s, \theta)$ be the strong solution of problem (1.6)-(1.8). Then, the energy functional (3.43) satisfies, for all $t\ge0$

and

Proof. The proof of (3.44) follows the same steps as in the proof of Lemma 2.5. From (3.44), it is obvious that

So, using Cauchy-Schwarz inequality, we obtain

This implies

Integrating (3.47) over $(0, t)$ yields

Hence, (3.45) follows.

Remark 3.2. Using Young's inequality, we observe from (3.44) and (3.45) that

for some fixed positive constant $c_1$. Similarly, we obtain

As in the case of equal-wave-speed of propagation, we define a Lyapunov functional

where $\tilde{N}, \ \tilde{ N_j}, \ j = 1, 2, 3, 4, 5, 6,$ are positive constants to be specified later and

Lemma 3.3. Suppose $\frac{G}{\rho}\neq\frac{D}{I_{\rho}}$. Then, under suitable choice of $\tilde{N}, \ \tilde{ N_j}, \ j = 1, 2, 3, 4, 5, 6$, the Lyapunov functional $\tilde{L}$ satisfies, along the solution of $(1.6)$, the estimate

for some positive constants $\tilde{\beta}$ and $c_1.$

Proof. Following the proof of Lemma 3.1, we end up with (3.52).

Lemma 3.4. Suppose assumptions $(G1)$ and $(G2)$ hold and the function $H$ in $(G2)$ is linear. Let $(w, 3s-\psi, s, \theta)$ be the strong solution of problem (1.6)-(1.8). Then,

where $c_1$ is a fixed positive constant.

Proof. Using (3.50) and the fact that $\xi$ is decreasing, we have

Our stability result of this subsection is

Theorem 3.3. Assume (G1) and (G2) hold and $\frac{G}{\rho}\neq\frac{D}{I_{\rho}}$. Then, there exist positive constants $a_1, a_2$ and $t_2 > t_0$ such that the energy solution (2.11) satisfies

Proof. Case 1: $H$ is linear. Multiplying (3.52) by $\xi(t)$ and using $(G1)$, we get

Using the fact that $\xi$ non-increasing, we obtain

for some fixed positive constant $c_2$. This implies

Integrating (3.56) over $(t_0, t)$, using the fact that E is non-increasing and the inequality (3.45), we arrive at

Thus, we have

Case II: $H$ is nonlinear. First, we observe from (3.52) that

From (2.12), (3.16) and (3.50), we have for any $t\geq t_0$

where $c_2$ is a fixed positive constant. Substituting (3.60) into (3.59), we obtain for any $t\geq t_0$

where $c_3$ is a fixed positive constant. Now, we define the functional $\Phi$ by

Using (2.11), (2.12), (3.43) and (3.45), we easily get

This last inequality allows us to choose $0 < \sigma < 1$ such that

Hence forth, we assume $\Phi(t) > 0$, otherwise, we get immediately from (3.61)

Next, we define the functional $\mu$ by

and observe that

where $c_4$ is a fixed positive constant. The fact that $H$ is strictly convex and $H(0) = 0$ implies

Using assumption $(G1)$, (3.67), Jensen’s inequality and (3.64), we get for any $t > t_0$

where

and $\bar{H}$ is the $C^2-$ strictly increasing and convex extension of $H$ on $(0, +\infty).$ This implies

Thus, the inequality (3.61) becomes

Let $\tilde{L}_1(t): = \tilde{L}(t)+C\left(E(t)+ E_1(t) \right).$ Then (3.70) becomes

For $0 < r_1 < r,$ we define the functional $\tilde{L}_2$ by

From (3.71) and the fact that

we obtain, for all $t > t_0,$

Let $\bar{H}^{\ast}$ be the convex conjugate of $\bar{H}$ as in (3.30) and let

Then, (3.30), (3.31) and (3.73) yield, for all $t > t_0,$

By selecting $r_1$ small enough so that $(\tilde{\beta}E(0)-Cr_1) > 0$, we arrive at

for some positive constant $\tilde{\beta}_2.$

Now, multiplying (3.75) by $\xi(t)$ and recalling that $r_1\frac{E(t)}{E(0)} < r$, we arrive at

Since $\frac{r_1}{t-t_0}\longrightarrow 0$ as $t\longrightarrow \infty,$ there exists $t_2 > t_0$ such that $\frac{r_1}{t-t_0} < r_1$, whenever $t > t_2$. Using this fact and observing that $H'$ strictly increasing, and $E$ and $\xi$ are non-decreasing, we get

Using (3.77), it follows from (3.76) that

where $\tilde{L}_3 = (\xi\tilde{L}_2+CE+ CE_1)$ and $c_5 > 0$ is a constant. Using the non-increasing property of $\xi,$ we have

Using the fact that $E$ is non-increasing and $H'' > 0$ we conclude that the map

is non-increasing. Therefore, integrating (3.79) over $(t_2, t)$ yields

Next, we multiply both sides of (3.80) by $\frac{1}{t-t_0}$, for $t > t_2,$ we get

Since $H'$ is strictly increasing, then $H_2(\tau) = \tau H'(\tau)$ is a strictly increasing function. It follows from (3.81) that

for some positive constants $a_1$ and $a_2.$ This completes the proof.

3.3.   Examples

$(1).$ Let $g(t) = ae^{-bt}, \ t\geq 0, \ \ a, \ b > 0$ are constants and $a$ is chosen such that ($G_1)$ holds. Then

Therefore, from (3.14), the energy function (2.11) satisfies

Also, for $H_2(\tau) = \tau$, it follows from (3.55) that, there exists $t_2 > 0$ such that the energy function (2.11) satisfies

for some positive constant $C.$

$(2).$Let $g(t) = ae^{-(1+t)^{b}}, \ t\geq 0, \ \ a > 0, \ 0 < b < 1$ are constants and $a$ is chosen such that ($G_1)$ holds. Then,

where $\xi(t) = b(1+t)^{b-1}$ and $H(t) = t.$ Thus, we get from (3.14) that

Likewise, for $H_2(t) = t$, then estimate (3.55) implies there exists $t_2 > 0$ such that the energy function (2.11) satisfies

for some positive constant $C.$

$(3).$Let $g(t) = \frac{a}{(1+t)^{b}}, \ t\geq 0, \ \ a > 0, \ b > 1$ are constants and $a$ is chosen in such a way that ($G_1)$ holds. We have

where

Hence, we deduce from (3.41) that

Furthermore, for $H_2(t) = qt^q$, estimate (3.55) implies there exists $t_2 > 0$ such that the energy function (2.11) satisfies

for some positive constant $C.$

4.   Appendix

In this section, we prove the functionals $L_i, i = 1\cdots8,$ used in the proof of our stability results.

Lemma 4.1. The functional $I_1(t)$ satisfies, along the solution of $(1.6)-(1.8),$ for all $t\geq t_0 > 0$ and for any $\epsilon_1, \epsilon_2 > 0$, the estimate

where $g_0 = \int_0^{t_0}g(\tau)d\tau\leq \int_0^{t}g(\tau)d\tau.$

Proof. Differentiating $I_1(t)$, using $(1.6)_2$ and integrating by part, we have

where $D(t) = \left(D-\int_0^t g(\tau)d\tau\right).$ Now, we estimate the terms on the right hand-side of (4.2). Exploiting Young's and Poincaré's inequalities, Lemmas 2.1-2.6 and performing similar computations as in (2.8), we have for any $\epsilon_1 > 0$,

and

Also, for $\delta_1 > 0,$ we have

For the last term, we have

Combination of (4.2)-(4.6) lead to

Since $g(0) > 0$ and $g$ is continuous. Thus for any $t\geq t_0 > 0,$ we get

We select $\delta_1 = \dfrac{I_{\rho}g_0}{2}$ to get (4.1).

Lemma 4.2. The functional $I_2(t)$ satisfies, along the solution of $(1.6)-(1.8)$ and for any $\delta_4 > 0$, the estimate

Proof. Differentiation of $I_2(t)$, using $(1.6)_1$ and $(1.6)_3$ and integration by part, leads to

Applying Cauchy-Schwarz and Young's inequalities and (2.5), we get for any $\delta_4 > 0,$

This completes the proof.

Lemma 4.3. The functional $I_3(t)$ satisfies, along the solution of $(1.6)-(1.8)$ and for any $\epsilon_3 > 0,$ the estimate

Proof. Differentiation of $I_3$, using $(1.6)_3$, $(1.6)_4$ and integration by parts, yields

Using Cauchy-Schwarz, Young's and Poincaré's inequalities together with Lemmas 2.1-2.6, we have

We choose $\delta_2 = \dfrac{5I_{\rho}\delta}{2}$ to get (4.10).

Lemma 4.4. The functional $I_4(t)$ satisfies, along the solution of $(1.6)-(1.8)$ and for any $\epsilon_4 > 0,$ the estimate

Proof. Using $(1.6)_1$ and integration by parts, we have

We note that $w_x = -(\psi-w_x)-(3s-\psi)+3s$ to arrive at

It follows from Young's and Poincaré's inequalities that

This completes the proof.

Lemma 4.5. The functional $I_5(t)$ satisfies, along the solution of $(1.6)-(1.8)$ and for any $0 < \alpha < 1$, the estimate

Proof. Differentiating $I_5$, using $(1.6)_2$, we arrive at

Applying Lemmas 2.1-2.6, Cauchy-Schwarz, Young's and Poincaré's inequalities, we obtain any $\delta_3 > 0$

We select $\delta_3 = \dfrac{l_0}{4}$ and obtain the desired result.

Lemma 4.6. The functional $I_6(t)$ satisfies, along the solution of $(1.6)-(1.8)$ and for any for any $\epsilon_1$, the estimate

Proof. Differentiating $I_6(t)$, using $(1.6)_1$ and $(1.6)_3$ and integration by parts, we obtain

Young's and Poincaré's inequalities give

Substituting (4.16) into (4.15), we obtain (4.14). This completes the proof.

Lemma 4.7. The functional $I_7(t)$ satisfies, along the solution of $(1.6)-(1.8),$ the estimate

Proof. Differentiate $I_7(t)$ and use the fact that $J'(t) = -g(t)$ to get

Using Cauchy-Schwarz and (G1), we have

Thus, we get

Since $J$ is decreasing $(J'(t) = -g(t)\leq 0)$, so $J(t)\leq J(0) = D-l_0$. Hence, we arrive at

The next lemma is used only in the proof of the stability result for nonequal-wave-speed of propagation.

Lemma 4.8. Let $(w, 3s-\psi, s, \theta)$ be the strong solution of problem (1.6). Then, for any positive numbers $\sigma_1, \sigma_2, \sigma_3$, the functional $I_8(t)$ satisfies

Proof. Differentiation of $I_8$, using integration by part and the boundary condition give

We note that $w_x = -(\psi-w_x)-(3s-\psi)+3s$ and from $(1.6)_4, \ \lambda\theta_{xx} = k\theta_t +\delta s_{xt}$. So, (4.22) becomes

Using Young's and Poincaré's inequalities, we have for any positive numbers $\sigma_1, \sigma_2, \sigma_3$,

Substituting (4.24) into (4.23), we obtain (4.21).

5.   Conclusions

In this paper, we have established a general and optimal stability estimates for a thermoelastic Laminated system, where the heat conduction is given by Fourier's Law and memory as the only source of damping. Our results are established under weaker conditions on the memory and physical parameters. From our results, we saw that the decay rate is faster provided the wave speeds of the first two equations of the system are equal (see (1.3)). A similar result was established recently in ^[19] when the heat conduction is given by Maxwell-Cattaneo's Law. An interesting case is when the kernel memory term is couple with the first or third equations in system (1.6). Our expectation is that the stability in both cases will depend on the speed of wave propagation.

Acknowledgments

The authors appreciate the continuous support of University of Hafr Al Batin, KFUPM and University of Sharjah. The first and second authors are supported by University of Hafr Al Batin under project #G-106-2020 . The third author is sponsored by KFUPM under project #S B181018.

Conflict of interest

The authors declare no conflict of interest