Optimal memory-type boundary control of the Bresse system

Abstract

In this paper, we study the stability of a Bresse system with memory-type boundary conditions. For a wider class of kernel functions, we establish an optimal explicit energy decay result. Our stability result improves many earlier results in the literature. Finally, we also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme.

Keywords

Bresse system Timoshenko viscoelastic damping general decay convexity

1. Introduction

In this paper we are concerned with the following Bresse system in $(0, 1) \times R^{+}$ , $\begin{array}{l} (1.1) & \{\begin{matrix} ρ_{1} φ_{t t} - k_{1} {(φ_{x} + ψ + l ω)}_{x} - l k_{3} (ω_{x} - l φ) = 0, \\ ρ_{2} ψ_{t t} - k_{2} ψ_{x x} + k_{1} (φ_{x} + ψ + l ω) + β ψ_{t} = 0, \\ ρ_{1} ω_{t t} - k_{3} {(ω_{x} - l φ)}_{x} + l k_{1} (φ_{x} + ψ + l ω) = 0, \end{matrix} \end{array}$ together with the following boundary conditions $\begin{array}{l} (1.2) & \{\begin{matrix} φ (0, t) = ψ_{x} (0, t) = ω_{x} (0, t) = ω (1, t) = 0, t > 0, \\ φ (1, t) = - \int_{0}^{t} g_{1} (t - s) (φ_{x} + ψ + l ω) (1, s) d s, t > 0, \\ ψ (1, t) = - \int_{0}^{t} g_{2} (t - s) ψ_{x} (1, s) d s, t > 0, \end{matrix} \end{array}$ and initial conditions $\begin{array}{l} (1.3) & \{\begin{matrix} φ (x, 0) = φ_{0} (x), ψ (x, 0) = ψ_{0} (x), ω (x, 0) = ω_{0} (x), & in (0, 1), \\ φ_{t} (x, 0) = φ_{1} (x), ψ_{t} (x, 0) = ψ_{1} (x), ω_{t} (x, 0) = ω_{1} (x), & in (0, 1) . \end{matrix} \end{array}$ It is well-known that the Bresse system, which is also called circular arch problem, is given by the following evolution equations (see [10,27]), $\begin{array}{l} ρ_{1} φ_{t t} - S_{x} - l N = F_{1}, \\ (1.4) & ρ_{2} ψ_{t t} - M_{x} + S = F_{2}, \\ ρ_{1} ω_{t t} - N_{x} + l S = F_{3}, \end{array}$ where $\begin{array}{l} S = k_{1} (φ_{x} + ψ + l ω), M = k_{2} ψ_{x}, N = k_{3} (ω_{x} - l φ) \end{array}$ denote the shear force, the bending moment and the axial force, respectively. Here, $ρ_{1} = ρ A$ , $ρ_{2} = ρ I$ , $k_{1} = k^{'} G A$ , $k_{2} = E I$ , $k_{3} = E A$ and $l = R^{- 1}$ , such that ρ denotes the density, A represents the cross-sectional area, I denotes the second moment of area of the cross-section, $k^{'}$ is the shear factor, G is the shear modulus, E represents the elastic modulus, and R denotes the radius of curvature of the beam. In this paper we assume that all the coefficients are positive constants. In the recent decades, the qualitative properties of the Bresse system have been investigated by many authors using various dissipative mechanism. We first mention the contribution of Liu and Rao [30], in which the authors considered a linear thermoelastic Bresse system and established that the exponential decay of solutions can be obtained if and only if the following hypothesis of equal-speed waves of propagation holds $\begin{array}{l} (1.5) & \frac{ρ_{1}}{ρ_{2}} = \frac{k_{1}}{k_{2}}, and k_{1} = k_{3} . \end{array}$ Otherwise, they showed that the energy decays polynomially. After that, the relation (1.5) has been used by many authors to establish exponential decay rates, see for example, [9,13,14,26,39]. If two frictional dissipations are added to the first two equations in (1.4), the exponential stability is obtained if and only if $k_{1} = k_{3}$ , see [5], where the authors investigated the optimality of polynomial decay rate. Recently, Afilal et. al. [2] considered the bresse system with only one damping acting only on the third equation and they proved the exponential and polynomial decays depending on a new relation between the coefficients of the model. But if the frictional dissipations were added to each equation in (1.4), the exponential stability is obtained without any conditions on the coefficients, see for instance [11]. We also mention the work of Alves et al. [6], Benaissa and Kasmi [8], Muñoz Rivera and Naso [34] and Rifo et al. [36], where the authors considered the Bresse system with some classes of boundary dissipations. Recently, Messaoudi and Hassan [32] considered a Bresse system with finite memory in the internal feedback. For a wide class of relaxation functions, they established a more general decay results. Regarding results on energy solution decay of Bresse system with infinite memories, we refer to Fatoti et al. [12], Guesmia [18], Guesmia and Kafini [19] and Guesmia and Kirane [20]. For more results on thermoelastic Bresse system, one can see [1,3,16] among others. We note that if $l \to 0$ the Bresse system reduces to Timoshenko beam equations, see [17,27,31], which have been studied by many authors. For Timoshenko system, it is well-known that if one adds the damping term in one of the two equations, then the system is exponentially stable under the “equal wave speed” assumption. But if the damping terms are added in the two equations, the system decays exponentially regardless to the wave speed assumption. So many results in this direction can be found in the literature. Here we only refer the reader to [33,35,38], where the authors studied Timoshenko system with memory-type boundary conditions and established some general decay results. To the best of our knowledge, there is no result on Bresse system with memory-type boundary conditions. Then system (1.1)–(1.3) describes the Bresse system (1.1) subjected to the boundary viscoelastic dampings. In this paper, we study the probem (1.1)–(1.3). And we impose the assumptions on the kernels are more general than the previous works, such as [33,35,38] for Timoshenko beams. We establish optimal explicit and general energy decay results to system (1.1)–(1.3). In particular, the energy results hold for $M (s) = s^{p}$ (see assumptions below) where p allows to be taken the full admissible range $[1, 2)$ instead of $[1, 3 / 2)$ . The arguments in this paper mainly rely on Lyapunov functional method and some properties of convex function developed Alabau-Boussouira and Cannarsa [4] initiated in Lasiecka and Tataru [28]. The paper is organized as follows. Section 2 is devoted to the assumptions. In Section 3, we state our main results. The proof of stability result will be given in Section 4. Numerical approximations using Finite volume techniques is introduced in Section 5. Concluding remarks and open questions are given in Section 6.

Throughout this work, the variable c denotes a positive constant that may change from one line to the other or within the same line.

2. Preliminaries

In this section, we state our assumptions, make some transformations and state and prove a useful lemma. For this purpose, we introduce the following spaces $\begin{array}{l} H_{*}^{1} (0, 1) = {u | u \in H^{1} (0, 1), u (0) = 0} and {\tilde{H}}_{*}^{1} (0, 1) = {u | u \in H^{1} (0, 1), u (1) = 0}, \end{array}$ and the energy space by $\begin{array}{l} H = H_{*}^{1} (0, 1) \times L^{2} (0, 1) \times {\tilde{H}}_{*}^{1} (0, 1) \times L^{2} (0, 1) \times {\tilde{H}}_{*}^{1} (0, 1) \times L^{2} (0, 1), \end{array}$ equipped with the norm $\begin{array}{l} ‖ U ‖_{H}^{2} = \int_{0}^{1} [{\tilde{φ}}^{2} + {\tilde{ψ}}^{2} + {\tilde{ω}}^{2} + φ_{x}^{2} + ψ_{x}^{2} + ω_{x}^{2}] d x, \end{array}$ where $U = (φ, \tilde{φ}, ψ, \tilde{ψ}, ω, \tilde{ω})$ . We also define the l-dependent norm $\begin{array}{l} ‖ U ‖_{H_{l}}^{2} = \int_{0}^{1} [ρ_{1} {\tilde{φ}}^{2} + ρ_{2} {\tilde{ψ}}^{2} + ρ_{1} {\tilde{ω}}^{2} + k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} + k_{3} {(ω_{x} - l φ)}^{2}] d x . \end{array}$ We know that the l-dependent norm is equivalent to the standard norm if l is small. In fact, it is easy to verify that there exists $a_{1} > 0$ such that $‖ U ‖_{H_{l}}^{2} ⩽ a_{1} ‖ U ‖_{H}^{2}$ . On the other hand, by a contradiction argument, there exists $a_{2} > 0$ (dependent on l) such that $‖ U ‖_{H}^{2} ⩽ a_{2} ‖ U ‖_{H_{l}}^{2}$ . Hence we have there exists $a_{3} > 0$ such that $\begin{array}{l} \int_{0}^{1} [φ_{x}^{2} + ψ_{x}^{2} + ω_{x}^{2}] d x ⩽ a_{3} \int_{0}^{1} [k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} + k_{3} {(ω_{x} - l φ)}^{2}] d x . \end{array}$

Taking the derivative of (1.2)₂ and (1.2)₃ with respect to t, we get $\begin{array}{l} (φ_{x} + ψ + l ω) (1, t) = - \frac{1}{g_{1} (0)} [φ_{t} (1, t) + g_{1}^{'} * (φ_{x} + ψ + l ω) (1, t)], \end{array}$ and $\begin{array}{l} ψ_{x} (1, t) = - \frac{1}{g_{2} (0)} [ψ_{t} (1, t) + g_{2}^{'} * ψ (1, t)], \end{array}$ where $(g * u) (t)$ is the convolution product operator $\begin{array}{l} (g * u) (t) = \int_{0}^{t} g (t - s) u (s) d s . \end{array}$ In view of the Volterra inverse operator, we have $\begin{array}{l} (φ_{x} + ψ + l ω) (1, t) = - \frac{1}{g_{1} (0)} [φ_{t} (1, t) + σ_{1} * (φ_{x} + ψ + l ω) (1, t)], \end{array}$ and $\begin{array}{l} ψ_{x} (1, t) = - \frac{1}{g_{2} (0)} [ψ_{t} (1, t) + σ_{2} * ψ_{t} (1, t)], \end{array}$ where $σ_{i}$ ( $i = 1, 2$ ) are the resolvent kernels of $(- g_{i}^{'} / g_{i} (0))$ satisfying for $t ⩾ 0$ , $\begin{array}{l} σ_{i} (t) + \frac{1}{g_{i} (0)} (g_{i}^{'} * σ_{i}) (t) = - \frac{1}{g_{i} (0)} g_{i}^{'} (t) . \end{array}$ Denoting $α_{i} = \frac{1}{g_{i} (0)}$ $(i = 1, 2)$ and assuming $φ_{0} (1) = ψ_{0} (1) = 0$ in this paper, we see that $\begin{array}{l} (2.1) & (φ_{x} + ψ + l ω) (1, t) = - α_{1} [φ_{t} (1, t) + σ_{1} (0) φ (1, t) + σ_{1}^{'} * φ (1, t)], \end{array}$ and $\begin{array}{l} (2.2) & ψ_{x} (1, t) = - α_{2} [ψ_{t} (1, t) + σ_{2} (0) ψ (1, t) + σ_{2}^{'} * ψ (1, t)] . \end{array}$ So, we use boundary conditions (2.1)–(2.2) instead of (1.2)₂ and (1.2)₃ and give some assumptions on the resolvent kernel $σ_{i}$ $(i = 1, 2)$ .

(A1) The functions $σ_{i} : R^{+} \to R^{+}$ are twice differentiable nonincreasing functions satisfying for any $t ⩾ 0$ , $\begin{array}{l} (2.3) & σ_{i} (0) > 0, σ_{i}^{'} (t) ⩽ 0 . \end{array}$ (A2) There exist two $C^{1}$ functions $M_{i} : R^{+} \to R^{+}$ , with $M_{i} (0) = M_{i}^{'} (0) = 0$ , which are linear or are strictly increasing and strictly convex functions of class $C^{2} (R^{+})$ on $(0, r]$ , $r ⩽ - σ_{i}^{'} (0)$ , such that $\begin{array}{l} (2.4) & σ_{i}^{″} (t) ⩾ η_{i} (t) M_{i} (- σ_{i}^{'} (t)), \forall t ⩾ 0, \end{array}$ where $η_{i} (t)$ are nonincreasing non-negative continuous functions of class $C^{1}$ .

Then we can get the following lemma.

Lemma 2.1.
For $i = 1, 2$ , assume $σ_{i}$ satisfies (A1)–(A2) and, further, $\begin{array}{l} lim_{t \to \infty} σ_{i} (t) = 0 . \end{array}$ Then there exists some $t_{1} ⩾ 0$ such that for any $t \in [0, t_{1}]$ , $\begin{array}{l} (2.5) & σ_{i}^{″} (t) ⩾ - d σ_{i}^{'} (t), \end{array}$ where d is a positive constant.
Proof.
For $i = 1, 2$ , since ${lim}_{t \to + \infty} σ_{i} (t) = 0$ and $(- σ_{i}^{'} (t))$ is nonnegative and nonincreasing, we easily see ${lim}_{t \to + \infty} (- σ_{i}^{'} (t)) = 0$ . Therefore, there exists some $t_{1} ⩾ 0$ so large that $\begin{array}{l} - σ_{1}^{'} (t_{1}) = r \Rightarrow - σ_{i}^{'} (t) ⩽ r, \forall t ⩾ t_{1} . \end{array}$ Noting that $(- σ_{i}^{'})$ is nonincreasing, $- σ_{i}^{'} (0) > 0$ and $- σ_{i}^{'} (t_{1}) > 0$ , then we get $- σ_{i}^{'} (t) > 0$ for any $t \in [0, t_{1}]$ and, $\begin{array}{l} 0 < - σ_{i}^{'} (t_{1}) ⩽ - σ_{i}^{'} (t) ⩽ - σ_{i}^{'} (0), 0 < η_{i} (t_{1}) ⩽ η_{i} (t) ⩽ η_{i} (0), \forall t \in [0, t_{1}] . \end{array}$ Hence, for any $t \in [0, t_{1}]$ , there exist some constants $a_{i} > 0$ and $b_{i} > 0$ , $\begin{array}{l} a_{i} ⩽ η_{i} (t) M_{i} (- σ_{i}^{'} (t)) ⩽ b_{i}, \forall t \in [0, t_{1}] . \end{array}$ This gives for any $t \in [0, t_{1}]$ , $\begin{array}{l} σ_{i}^{″} (t) ⩾ η_{i} (t) M_{i} (- σ_{i}^{'} (t)) ⩾ a_{i} = \frac{a_{i}}{σ_{i}^{'} (0)} σ_{i}^{'} (0) ⩾ \frac{a_{i}}{σ_{i}^{'} (0)} σ_{i}^{'} (t) . \end{array}$ Then we know that there exists a positive constant d, $\begin{array}{l} σ_{i}^{″} (t) ⩾ - d σ_{i}^{'} (t), \forall t \in [0, t_{1}], \end{array}$ which ends the proof. □

3. Main results

For completeness, we give the global existence and regularity result in the following theorem.

Theorem 3.1.
Assume that (A1) holds. Let $(φ_{0}, φ_{1}) \in H_{}^{1} (0, 1) \times L^{2} (0, 1)$ , $(ψ_{0}, ψ_{1}), (ω_{0}, ω_{1}) \in {\tilde{H}}_{}^{1} (0, 1) \times L^{2} (0, 1)$ , then problem ( 1.1 )–( 1.3 ) has a unique weak solution satisfying $\begin{array}{l} φ \in C (R^{+}; H_{}^{1} (0, 1)) \cap C^{1} (R^{+}; L^{2} (0, 1)), \end{array}$ and* $\begin{array}{l} ψ, w \in C (R^{+}; {\tilde{H}}_{}^{1} (0, 1)) \cap C^{1} (R^{+}; L^{2} (0, 1)), \end{array}$ In addition, if* $(φ_{0}, φ_{1}) \in H_{}^{2} (0, 1) \times H_{}^{1} (0, 1)$ and $(ψ_{0}, ψ_{1}), (ω_{0}, ω_{1}) \in {\tilde{H}}_{}^{2} (0, 1) \times {\tilde{H}}_{}^{1} (0, 1)$ with the following compatibility conditions $\begin{array}{l} \{\begin{matrix} (φ_{0 x} + ψ_{0} + l ω_{0}) (1) = - α_{1} φ_{1} (1), \\ ψ_{0 x} (1) = - α_{2} ψ_{1} (1), \end{matrix} \end{array}$ then the solution is strong and satisfies $\begin{array}{l} φ \in C (R^{+}; H_{}^{2} (0, 1)) \cap C^{1} (R^{+}; H_{}^{1} (0, 1)) \cap C^{2} (R^{+}; L^{2} (0, 1)), \end{array}$ and $\begin{array}{l} ψ, w \in C (R^{+}; {\tilde{H}}_{}^{2} (0, 1)) \cap C^{1} (R^{+}; {\tilde{H}}_{}^{1} (0, 1)) \cap C^{2} (R^{+}; L^{2} (0, 1)), \end{array}$ where $H_{}^{2} (0, 1) = H^{2} (0, 1) \cap H_{}^{1} (0, 1)$ and ${\tilde{H}}_{}^{2} (0, 1) = H^{2} (0, 1) \cap {\tilde{H}}_{}^{1} (0, 1)$ .
Remark 3.1.
The proof of Theorem 3.1 can be established by repeating the same steps of the proofs of existence results of [37] and [40]. In [37] and [40], the authors considered a single semilinear wave equation with a memory condition at the boundary and the Kirchhoff thermal plates equations with memory boundary conditions, respectively. The authors proved the global existence and uniqueness of solutions of the two systems by using the Galerkin method.

The total energy of the system is given by $\begin{array}{l} E (t) = & \frac{1}{2} \int_{0}^{1} [ρ_{1} φ_{t}^{2} + ρ_{2} ψ_{t}^{2} + ρ_{1} ω_{t}^{2} + k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} \\ + k_{3} (ω_{x} - l φ)] d x + \frac{α_{1}}{2} k_{1} [σ_{1} (t) φ^{2} (1, t) - σ_{1}^{'} \circ φ (1, t)] \\ (3.1) & + \frac{α_{2}}{2} k_{2} [σ_{2} (t) ψ^{2} (1, t) - σ_{2}^{'} \circ ψ (1, t)], \end{array}$ where $\begin{array}{l} (σ \circ u) (t) = \int_{0}^{t} σ (t - s) {[u (t) - u (s)]}^{2} d s . \end{array}$

The following lemma is proved in Santos [38].
Lemma 3.2.
If $σ, u \in C^{1} (R^{+})$ , we have $\begin{array}{l} (3.2) & (σ * u) u_{t} = - \frac{1}{2} σ (t) {| u (t) |}^{2} + \frac{1}{2} σ^{'} \circ u - \frac{1}{2} \frac{d}{d t} (σ \circ u - (\int_{0}^{t} σ (s) d s) {| u (t) |}^{2}) . \end{array}$

The stability result is stated in the following theorem.
Theorem 3.3.
Assume (A1)–(A2) hold and assume further ( 1.5 ) and $\begin{array}{l} lim_{t \to + \infty} σ_{i} (t) = 0, i = 1, 2 . \end{array}$ Then for l small enough, the energy $E (t)$ satisfies $\begin{array}{l} (3.3) & E (t) ⩽ μ_{2} M_{4}^{- 1} (μ_{1} \int_{K^{- 1} (r)}^{t} η (s) d s), \forall t > K^{- 1} (r), \end{array}$ where $μ_{1}$ , $μ_{2} > 0$ are the positive constant, $\begin{array}{l} M_{4} (t) = \int_{t}^{r} \frac{1}{s M_{0} (s)} d s, M_{0} (t) = min {M_{1}^{'} (t), M_{2}^{'} (t)}, \end{array}$ and $η (t) = min {η_{1} (t), η_{2} (t)}$ , $K (t) = max {- σ_{1}^{'} (t), - σ_{2}^{'} (t)}$ .

In particular, when $M_{i} (t) = c t^{p_{i}}$ $(i = 1, 2)$ , then the total energy $E (t)$ satisfies that for any $t > 0$ , $\begin{array}{l} (3.4) & E (t) ⩽ \{\begin{matrix} c_{1} e^{- c_{2} \int_{0}^{t} η (s) d s}, & if p = 1, \\ c_{3} {(1 + \int_{0}^{t} η (s) d s)}^{- \frac{1}{p - 1}}, & if 1 < p < 2, \end{matrix} \end{array}$ where $c_{1}, c_{3}$ and $c_{2}$ are positive constants, and $p = max {p_{1}, p_{2}}$ .
Remark 3.2.
Our result is obtained under a much larger class of kernels that guarantee the optimal decay rates of the energy. In particular, the energy result holds for $M (t) = c t^{p}$ with the full admissible range $[1, 2)$ instead of $[1, 3 / 2)$ . We should point out that, if the memory terms are as internal feedback, Xiao and Liang [41] firstly provided the proof for optimal decay rates of wave equation with memory in the full admissible range $[1, 2)$ . And then Lasiecka and Wang [29] extended the result for an abstract second order equation with memory. One can find more recent results on the stabilization of wave-type systems with memory in [22,24,25], etc.

Before proving our main theorem, we give some examples to illustrate our decay rates of the energy. Example 1.
Let $σ_{1}^{'} (t) = - a e^{- α t}$ , $α > 0$ , and $σ_{2}^{'} (t) = - \frac{b}{{(1 + t)}^{μ}}$ , $μ > 1$ , a and b are two positive constants. Then there exists a certain $c_{1} > 0$ such that for $t > t_{1}$ , $\begin{array}{l} E (t) ⩽ \frac{c_{1}}{{(1 + t)}^{μ}} . \end{array}$
Example 2.
Consider $σ_{1}^{'} (t) = - a e^{- α t}$ , $α > 0$ , and $σ_{2}^{'} (t) = - b e^{- {(1 + t)}^{μ}}$ , $0 < μ < 1$ , a and b are two positive constants. Then there exists two positive constants $c_{1}$ and $c_{2}$ such that for $t > t_{1}$ , $\begin{array}{l} E (t) ⩽ c_{1} e^{- c_{2} {(1 + t)}^{μ}} . \end{array}$
Example 3.
Consider $σ_{1}^{'} (t) = σ_{2}^{'} (t) = - e^{- t^{q}}$ with $0 < q < 1$ , then $σ_{i}^{″} (t) = M_{i} (- σ_{i}^{'} (t))$ ( $i = 1, 2$ ) where $M_{1} (t) = M_{2} (t) = \frac{q t}{{[ln (1 / t)]}^{\frac{1}{q} - 1}}$ . As $\begin{array}{l} M_{1}^{'} (t) = M_{2}^{'} (t) = \frac{(1 - q) + q ln (\frac{1}{t})}{{[ln (\frac{1}{t})]}^{\frac{1}{q}}} and M_{1}^{″} (t) = M_{2}^{″} (t) = \frac{(1 - q) [ln (\frac{1}{t}) + \frac{1}{q}]}{{[ln (\frac{1}{t})]}^{\frac{1}{q} + 1}}, \end{array}$ then the functions $M_{1}$ and $M_{2}$ satisfies (A2) on $(0, r]$ ( $0 < r < 1$ ), and consequently we have $\begin{array}{l} E (t) ⩽ c_{1} e^{- c_{2} t^{q}} . \end{array}$
Example 4.
Let us choose $σ_{1}^{'} (t) = σ_{2}^{'} (t) = \frac{- 1}{(t + e) {[ln (t + e)]}^{p}}$ with $p > 1$ , then $σ_{1}^{″} (t) = σ_{2}^{″} (t) = \frac{[ln (t + e) + p]}{{(t + e)}^{2} {[ln (t + e)]}^{p + 1}}$ . It is easy to get $\begin{array}{l} σ_{i}^{″} (t) = \frac{[ln (t + e) + p]}{(t + e) ln (t + e)} [- σ_{i}^{'} (t)] . \end{array}$ From (3.4)₁ we infer that $\begin{array}{l} E (t) ⩽ c_{1} exp (- c_{2} \int_{0}^{t} \frac{[ln (t + e) + p]}{(t + e) ln (t + e)} d s) = \frac{c_{1}}{{((t + e) {[ln (t + e)]}^{p})}^{c_{2}}} . \end{array}$ Because $c_{2} ⩽ 1$ , this is a slower rate of decay than that of $[- σ_{i}^{'} (t)]$ . On the other hand, since $\begin{array}{l} σ_{i}^{″} (t) = \frac{[ln (t + e) + p]}{{(t + e)}^{1 - \frac{1}{p}}} {(- σ_{i}^{'} (t))}^{1 + \frac{1}{p}} . \end{array}$ Then it follows, from (3.4)₂, that for large t, $\begin{array}{l} E (t) ⩽ c_{3} {(1 + \int_{0}^{t} \frac{ln (t + e) + p}{{(t + e)}^{1 - \frac{1}{p}}} d s)}^{- p} ⩽ \frac{c_{3}}{(t + e) {[ln (t + e)]}^{p}}, \end{array}$ which is the same rate of decay of $[- σ_{i}^{'} (t)]$ .

4. Proof of main result

To prove Theorem 3.3, we firstly need the following lemmas.

4.1. Technical lemmas

Lemma 4.1.
The total energy $E (t)$ satisfies for any $t ⩾ 0$ , $\begin{array}{l} E^{'} (t) ⩽ & - 4 β \int_{0}^{1} ψ_{t}^{2} d x - \frac{α_{1}}{2} k_{1} [2 φ_{t}^{2} (1, t) + σ_{1}^{″} \circ φ (1, t)] \\ (4.1) & - \frac{α_{2}}{2} k_{2} [2 ψ_{t}^{2} (1, t) + σ_{2}^{″} \circ ψ (1, t)] ⩽ 0 . \end{array}$
Proof.
We multiply (1.1)₁ by $φ_{t}$ , (1.1)₂ by $ψ_{t}$ and (1.1)₃ by $ω_{t}$ , and then integrate over by parts over $(0, 1)$ and we use (2.1) and (2.2), to derive $\begin{array}{l} \frac{1}{2} \int_{0}^{1} [ρ_{1} φ_{t}^{2} + ρ_{2} ψ_{t}^{2} + ρ_{1} ω_{t}^{2} + k_{1} {(φ_{x} + ψ + l ω)}_{x} + k_{2} ψ_{x} + k_{3} (ω_{x} - l φ)] d x \\ = - β \int_{0}^{1} ψ_{t}^{2} d x + k_{1} φ_{t} (1, t) (φ_{x} + ψ + l ω) (1, t) + k_{2} ψ_{t} (1, t) ψ_{x} (1, t) \\ = - β \int_{0}^{1} ψ_{t}^{2} d x - α_{1} k_{1} φ_{t}^{2} (1, t) - α_{1} k_{1} σ_{1} (0) φ (1, t) φ_{t} (1, t) - α_{1} k_{1} φ_{t} (1, t) σ_{1}^{'} * φ (1, t) \\ (4.2) & - α_{2} k_{2} ψ_{t}^{2} (1, t) - α_{2} k_{2} σ_{2} (0) ψ (1, t) ψ_{t} (1, t) - α_{2} k_{2} ψ_{t} (1, t) σ_{2}^{'} * ψ (1, t) . \end{array}$ From (3.2), we have $\begin{array}{l} \frac{α_{i}}{2} \frac{d}{d t} [σ_{i} (t) u^{2} (1, t) - σ_{i}^{'} \circ u (1, t)] \\ = \frac{α_{i}}{2} σ_{i}^{'} (t) u^{2} (1, t) + α_{i} σ_{i} (0) u (1, t) u_{t} (1, t) - \frac{α_{i}}{2} σ_{i}^{″} \circ u (1, t) + α_{i} u_{t} (1, t) σ_{i}^{'} * u (1, t), \end{array}$ which, along with (4.2) gives $\begin{array}{l} E^{'} (t) = & - β \int_{0}^{1} ψ_{t}^{2} d x - α_{1} k_{1} φ_{t}^{2} (1, t) + \frac{α_{1}}{2} k_{1} σ_{1}^{'} (t) φ^{2} (1, t) - \frac{α_{1}}{2} k_{1} σ_{1}^{″} \circ φ (1, t) \\ (4.3) & - α_{2} k_{2} ψ_{t}^{2} (1, t) + \frac{α_{2}}{2} k_{2} σ_{2}^{'} (t) ψ^{2} (1, t) - \frac{α_{2}}{2} k_{2} σ_{2}^{″} \circ ψ (1, t) . \end{array}$ Then we get (4.1) from (4.3) and the fact that $σ_{i}^{'} (t) ⩽ 0 (i = 1, 2)$ . □
Lemma 4.2.
We define $I_{1} (t)$ by $\begin{array}{l} I_{1} (t) : = - \int_{0}^{1} (ρ_{1} φ φ_{t} + ρ_{2} ψ ψ_{t} + ρ_{1} ω ω_{t}) d x - \frac{β}{2} \int_{0}^{1} ψ^{2} d x . \end{array}$ Under the assumptions of Theorem 3.3 , the functional $I_{1} (t)$ satisfies for any $ε_{1}^{'} > 0$ , $\begin{array}{l} I_{1}^{'} (t) ⩽ & - ρ_{1} \int_{0}^{1} φ_{t}^{2} d x - ρ_{1} \int_{0}^{1} ω_{t}^{2} d x + \frac{c}{ε_{1}^{'}} \int_{0}^{1} ψ_{t}^{2} d x \\ + (k_{1} + c ε_{1}^{'} + \frac{c}{ε_{1}^{'}} σ_{1}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x \\ + (k_{2} + c ε_{1}^{'} + \frac{c}{ε_{1}^{'}} σ_{1} + \frac{c}{ε_{1}^{'}} σ_{2}) \int_{0}^{1} ψ_{x}^{2} d x \\ + (k_{3} + c ε_{1}^{'} + \frac{c}{ε_{1}^{'}} σ_{1}) \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x \\ (4.4) & + \frac{c}{ε_{1}^{'}} [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}] + \frac{c}{ε_{1}^{'}} [ψ_{t}^{2} (1, t) + {(- σ_{2}^{'} ⊙ ψ (1, t))}^{2}], \end{array}$ where $\begin{array}{l} (σ ⊙ u) (t) = \int_{0}^{t} σ (t - s) [u (t) - u (s)] d s . \end{array}$
Proof.
In view of (1.1)₁ and (1.2)₂, $\begin{array}{l} I_{1}^{'} (t) = & - ρ_{1} \int_{0}^{1} φ_{t}^{2} d x - ρ_{2} \int_{0}^{1} ψ_{t}^{2} d x - ρ_{1} \int_{0}^{1} ω_{t}^{2} d x - k_{1} φ (1, t) (φ_{x} + ψ + l ω) (1, t) \\ (4.5) & + k_{1} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x - k_{2} ψ (1, t) ψ_{x} (1, t) + k_{3} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x . \end{array}$ By using $(σ^{'} * u) (t) = (- σ^{'} ⊙ u) (t) + [σ (t) - σ (0)] u (t)$ , then (2.1) yields $\begin{array}{l} (4.6) & {(φ_{x} + ψ + l ω)}^{2} (1, t) ⩽ c [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2} + σ_{1}^{2} (t) φ^{2} (1, t)] . \end{array}$ By (1.2) we easily see that $\begin{array}{l} φ^{2} (1, t) & = {(\int_{0}^{1} φ_{x} d x)}^{2} ⩽ \int_{0}^{1} φ_{x}^{2} d x ⩽ \int_{0}^{1} {[(φ_{x} + ψ + l ω) - ψ - l ω]}^{2} \\ ⩽ 2 \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + 2 \int_{0}^{1} {(ψ - l ω)}^{2} d x \\ (4.7) & ⩽ c \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + c \int_{0}^{1} ψ_{x}^{2} d x + c \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x . \end{array}$ Therefore, Young’s inequality gives, for any $ε_{1} > 0$ $\begin{array}{l} - k_{1} φ (1, t) (φ_{x} + ψ + l ω) (1, t) \\ ⩽ ε_{1} φ^{2} (1, t) + \frac{c}{ε_{1}} {(φ_{x} + ψ + l ω)}^{2} (1, t) \\ ⩽ c ε_{1} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + c ε_{1} \int_{0}^{1} ψ_{x}^{2} d x + c ε_{1} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x \\ + \frac{c}{ε_{1}} [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2} + σ_{1}^{2} (t) φ^{2} (1, t)] \\ ⩽ (c ε_{1} + \frac{c}{ε_{1}} σ_{1}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + (c ε_{1} + \frac{c}{ε_{1}} σ_{1}) \int_{0}^{1} [ψ_{x}^{2} + {(ω_{x} - l φ)}^{2}] d x \\ (4.8) & + \frac{c}{ε_{1}} [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}] . \end{array}$ Similarly for any $ε_{1} > 0$ , $\begin{array}{l} (4.9) & - k_{2} ψ (1, t) ψ_{x} (1, t) ⩽ (c ε_{1} + \frac{c}{ε_{1}} σ_{2}) \int_{0}^{1} ψ_{x}^{2} d x + \frac{c}{ε_{1}} [ψ_{t}^{2} (1, t) + {(- σ_{2}^{'} ⊙ ψ (1, t))}^{2}] . \end{array}$ Inserting (4.8) and (4.9) in (4.5), estimate (4.4) is established. □
Lemma 4.3.
Assume that (A1)–(A2) hold. Then the functional $I_{2} (t)$ , defined by $\begin{array}{l} I_{2} (t) : = ρ_{2} \int_{0}^{1} x ψ_{t} ψ_{x} d x, \end{array}$ satisfies, for any $ε_{2}, δ_{2} > 0$ , $\begin{array}{l} I_{2}^{'} (t) ⩽ & - (\frac{k_{2}}{2} - ε_{2} - \frac{k_{1}}{8 δ_{2}} - c σ_{2}) \int_{0}^{1} ψ_{x}^{2} d x + 2 c k_{1} δ_{2} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x \\ (4.10) & + \frac{c}{ε_{2}} \int_{0}^{1} ψ_{t}^{2} d x + c [ψ_{t}^{2} (1, t) + {(- σ_{2}^{'} ⊙ ψ (1, t))}^{2}] . \end{array}$
Proof.
Taking the derivative of $I_{2} (t)$ and using (1.1)₂, we have $\begin{array}{l} I_{2}^{'} (t) = & \frac{k_{2}}{2} \int_{0}^{1} x \frac{d ψ_{x}^{2}}{d x} d x - k_{1} \int_{0}^{1} (φ_{x} + ψ + l ω) x ψ_{x} d x + \frac{ρ_{2}}{2} \int_{0}^{1} x \frac{d ψ_{t}^{2}}{d x} d x - β \int_{0}^{1} x ψ_{x} ψ_{t} d x \\ = & \frac{k_{2}}{2} ψ_{x}^{2} (1, t) - \frac{k_{2}}{2} \int_{0}^{1} ψ_{x}^{2} d x - k_{1} \int_{0}^{1} (φ_{x} + ψ + l ω) x ψ_{x} d x \\ (4.11) & + \frac{ρ_{2}}{2} ψ_{t}^{2} (1, t) - \frac{ρ_{2}}{2} \int_{0}^{1} ψ_{t}^{2} d x - β \int_{0}^{1} x ψ_{x} ψ_{t} d x . \end{array}$ Young’s inequality and Poincare’s inequality give, for any $ε_{2} > 0$ , $\begin{array}{l} (4.12) & - β \int_{0}^{1} x ψ_{x} ψ_{t} d x ⩽ ε_{2} \int_{0}^{1} ψ_{x}^{2} d x + \frac{c}{ε_{2}} \int_{0}^{1} ψ_{t}^{2} d x, \end{array}$ and $\begin{array}{l} (4.13) & - k_{1} \int_{0}^{1} (φ_{x} + ψ + l ω) x ψ_{x} d x ⩽ 2 c k_{1} δ_{2} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + \frac{k_{1}}{8 δ_{2}} \int_{0}^{1} ψ_{x}^{2} d x . \end{array}$ Using the same estimate as (4.6), we arrive at $\begin{array}{l} (4.14) & ψ_{x}^{2} (1, t) ⩽ c [ψ_{t}^{2} (1, t) + {(- σ_{2}^{'} ⊙ ψ (1, t))}^{2} + σ_{2}^{2} (t) ψ^{2} (1, t)] . \end{array}$ Inserting (4.12)–(4.14) into (4.11), we obtain (4.10). □
Lemma 4.4.
The functional $I_{3} (t)$ , defined by $\begin{array}{l} I_{3} (t) : = - ρ_{1} k_{3} \int_{0}^{1} (ω_{x} - l φ) \int_{0}^{x} ω_{t} (y) d y d x - ρ_{1} k_{1} \int_{0}^{1} φ_{t} \int_{0}^{x} (φ_{y} + ψ + l ω) (y) d y d x, \end{array}$ satisfies for any $ε_{0}, ε_{3}, δ_{1} > 0$ , $\begin{array}{l} I_{3}^{'} (t) ⩽ & - (k_{3}^{2} - \frac{c}{ε_{3}} σ_{1}) \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + (k_{3}^{2} + ε_{3} + \frac{c}{ε_{3}} σ_{1}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x \\ + (ε_{0} - ρ_{1} k_{1} + \frac{ρ_{1} l | k_{3} - k_{1} |}{2} δ_{1}) \int_{0}^{1} φ_{t}^{2} d x + \frac{c}{ε_{3}} σ_{1} \int_{0}^{1} ψ_{x}^{2} d x + \frac{c}{ε_{0}} \int_{0}^{1} ψ_{t}^{2} d x \\ (4.15) & + (ρ_{1} k_{3} + \frac{c l | k_{3} - k_{1} |}{2 δ_{1}}) \int_{0}^{1} ω_{t}^{2} d x + c [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}] . \end{array}$
Proof.
By using (1.1)₁ and (1.1)₃, we have $\begin{array}{l} I_{3}^{'} (t) = & - ρ_{1} k_{3} \int_{0}^{1} ω_{x t} \int_{0}^{x} ω_{t} (y) d y d x + ρ_{1} k_{3} l \int_{0}^{1} φ_{t} \int_{0}^{x} ω_{t} (y) d y d x \\ - k_{3} \int_{0}^{1} (ω_{x} - l φ) \int_{0}^{x} [k_{3} {(ω_{y} - l φ)}_{y} - k_{1} l (φ_{y} + ψ + l ω)] d y d x \\ - k_{1} \int_{0}^{1} [k_{1} {(φ_{x} + ψ + l ω)}_{x} + k_{3} l (ω_{x} - l φ)] \int_{0}^{x} (φ_{y} + ψ + l ω) d y d x \\ - ρ_{1} k_{1} \int_{0}^{1} φ_{t} \int_{0}^{x} φ_{y t} d y d x - ρ_{1} k_{1} \int_{0}^{1} φ_{t} \int_{0}^{x} ψ_{t} d y d x - ρ_{1} k_{1} \int_{0}^{1} φ_{t} \int_{0}^{x} l ω_{t} d y d x . \end{array}$ By (2.2), we get $\begin{array}{l} I_{3}^{'} (t) = & - k_{3}^{2} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + ρ_{1} k_{3} \int_{0}^{1} ω_{t}^{2} d x + ρ_{1} k_{3} l \int_{0}^{1} φ_{t} \int_{0}^{x} ω_{t} d y d x \\ - k_{1}^{2} (φ_{x} + ψ + l ω) (1, t) \int_{0}^{1} (φ_{x} + ψ + l ω) d x - ρ_{1} k_{1} \int_{0}^{1} φ_{t}^{2} d x \\ (4.16) & - ρ_{1} k_{1} \int_{0}^{1} φ_{t} \int_{0}^{x} ψ_{t} d y d x - ρ_{1} k_{1} l \int_{0}^{1} φ_{t} \int_{0}^{x} ω_{t} d y d x . \end{array}$ Using Young’s inequality and (4.6)–(4.7), we can obtain (4.15). □
Lemma 4.5.
Assume that (A1)–(A2) hold and $ρ_{2} k_{1} = ρ_{1} k_{2}$ . Then the functional $I_{4} (t)$ , defined by $\begin{array}{l} I_{4} (t) : = ρ_{2} \int_{0}^{1} ψ_{t} (φ_{x} + ψ + l ω) d x + \frac{k_{2} ρ_{1}}{k_{1}} \int_{0}^{1} φ_{t} ψ_{x} d x . \end{array}$ satisfies, for any $ε_{0}, ε_{1} > 0$ , $\begin{array}{l} I_{4}^{'} (t) ⩽ & - (k_{1} - c σ_{1} - ε_{0}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + ε_{0} \int_{0}^{1} ω_{t}^{2} d x + \frac{c}{ε_{0}} \int_{0}^{1} ψ_{t}^{2} d x \\ + (\frac{l k_{2} k_{3}}{2 k_{1}} ε_{1} + c σ_{1}) \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + (\frac{l k_{2} k_{3}}{2 k_{1} ε_{1}} + c σ_{1} + c σ_{2}) \int_{0}^{1} ψ_{x}^{2} d x \\ (4.17) & + c [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}] + c [ψ_{t}^{2} (1, t) + {(- σ_{2}^{'} ⊙ ψ (1, t))}^{2}] . \end{array}$
Proof.
A straight forward differentiation, yields using (1.1), $\begin{array}{l} I_{4}^{'} (t) = & \int_{0}^{1} [k_{2} ψ_{x x} - k_{1} (φ_{x} + ψ + l ω) - β ψ_{t}] (φ_{x} + ψ + l ω) d x \\ + ρ_{2} \int_{0}^{1} ψ_{t} φ_{x t} d x + ρ_{2} \int_{0}^{1} ψ_{t}^{2} d x + ρ_{2} l \int_{0}^{1} ψ_{t} ω_{t} d x + \frac{k_{2}}{k_{1}} ρ_{1} \int_{0}^{1} φ_{t} ψ_{x t} d x \\ + \frac{k_{2}}{k_{1}} \int_{0}^{1} [k_{1} {(φ_{x} + ψ + l ω)}_{x} + l k_{3} (ω_{x} - l φ)] ψ_{x} d x . \end{array}$ Recalling the boundary conditions, we have $\begin{array}{l} I_{4}^{'} (t) = & k_{2} ψ_{x} (1, t) (φ_{x} + ψ + l ω) (1, t) - k_{1} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + ρ_{2} \int_{0}^{1} ψ_{t}^{2} d x \\ - \int_{0}^{1} β ψ_{t} (φ_{x} + ψ + l ω) d x + \frac{k_{2}}{k_{1}} ρ_{1} ψ_{t} (1, t) (φ_{t}) (1, t) \\ + ρ_{2} l \int_{0}^{1} ψ_{t} ω_{t} d x + \frac{l k_{2} k_{3}}{k_{1}} \int_{0}^{1} (ω_{x} - l φ) ψ_{x} d x + (ρ_{2} - \frac{k_{2}}{k_{1}} ρ_{1}) \int_{0}^{1} ψ_{t} φ_{x t} d x . \end{array}$ Then (4.17) follows from Young’s inequality, Poincaré’s inequality and $ρ_{1} k_{2} = ρ_{2} k_{1}$ . □
Lemma 4.6.
Assume that (A1)–(A2) hold and $k_{1} = k_{3}$ . Then the functional $I_{5} (t)$ , defined by $\begin{array}{l} I_{5} (t) : = I_{51} (t) + I_{52} (t), \end{array}$ where $\begin{array}{l} I_{51} (t) = - ρ_{1} \int_{0}^{1} ω_{t} (φ_{x} + ψ + l ω) d x - \frac{k_{3} ρ_{1}}{k_{1}} \int_{0}^{1} φ_{t} (ω_{x} - l ω) d x, \end{array}$ and $\begin{array}{l} I_{52} (t) = 2 ε_{5} ρ_{1} k_{3} \int_{0}^{1} (1 - 2 x) ω_{t} (ω_{t} - l φ) d x, \end{array}$ satisfies, for any $ε_{5} > 0$ , the following estimate $\begin{array}{l} I_{5}^{'} (t) ⩽ & - (\frac{l k_{3}^{2}}{k_{1}} - c σ_{1} - 4 ε_{5} k_{3}^{2}) \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + c \int_{0}^{1} ψ_{t}^{2} d x \\ + (l k_{1} + c σ_{1} + c ε_{5}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x - (\frac{ρ_{1} l}{2} - 4 ε_{5} ρ_{1} k_{3}) \int_{0}^{1} ω_{t}^{2} d x \\ + c σ_{1} \int_{0}^{1} ψ_{x}^{2} d x + (\frac{l k_{3} ρ_{1}}{k_{1}} + c ε_{5}) \int_{0}^{1} φ_{t}^{2} d x \\ (4.18) & - ε_{5} ρ_{1} k_{3} ω_{t}^{2} (0, t) + \frac{c}{ε_{5}} [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}] . \end{array}$
Proof.
From (1.1) we obtain $\begin{array}{l} I_{51}^{'} (t) = & - \int_{0}^{1} [k_{3} {(ω_{x} - l φ)}_{x} - l k_{1} (φ_{x} + ψ + l ω)] (φ_{x} + ψ + l ω) \\ - ρ_{1} \int_{0}^{1} ω_{t} φ_{x t} d x - ρ_{1} \int_{0}^{1} ω_{t} ψ_{t} d x - ρ_{1} l \int_{0}^{1} ω_{t}^{2} d x \\ - \frac{k_{3}}{k_{1}} \int_{0}^{1} [k_{1} {(φ_{x} + ψ + l ω)}_{x} + l k_{3} (ω_{x} - l φ)] (ω_{x} - l φ) d x \\ - \frac{k_{3}}{k_{1}} ρ_{1} \int_{0}^{1} φ_{t} ω_{x t} d x + \frac{l k_{3} ρ_{1}}{k_{1}} \int_{0}^{1} φ_{t}^{2} d x . \end{array}$ Using again (1.1), we arrive at $\begin{array}{l} I_{52}^{'} (t) = & l k_{1} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + ρ_{1} (1 - \frac{k_{3}}{k_{1}}) \int_{0}^{1} φ_{t} ω_{x t} d x \\ - ρ_{1} \int_{0}^{1} ω_{t} ψ_{t} d x - ρ_{1} l \int_{0}^{1} ω_{t}^{2} d x - k_{3} (φ_{x} + ψ + ω) (1, t) (ω_{x} - l φ) (1, t) \\ (4.19) & - \frac{l k_{3}^{2}}{k_{1}} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + \frac{l k_{3} ρ_{1}}{k_{1}} \int_{0}^{1} φ_{t}^{2} d x . \end{array}$

We use Young’s and Poincaré’s inequalities to derive, for any $ε_{5} > 0$ , $\begin{array}{l} (4.20) & - ρ_{1} \int_{0}^{1} ω_{t} ψ_{t} d x ⩽ \frac{ρ_{1} l}{2} \int_{0}^{1} ω_{t}^{2} d x + c \int_{0}^{1} ψ_{t}^{2} d x, \end{array}$ and $\begin{array}{l} (4.21) & - k_{3} (φ_{x} + ψ + ω) (1, t) (ω_{x} - l φ) (1, t) ⩽ ε_{5} k_{3}^{2} {(ω_{x} - l φ)}^{2} (1, t) + \frac{c}{ε_{5}} {(φ_{x} + ψ + l ω)}^{2} (1, t) . \end{array}$ Combining (4.20) and (4.21) with (4.19) and noting (4.6)–(4.7), we infer that,for any $ε_{5} > 0$ , $\begin{array}{l} I_{51}^{'} (t) ⩽ & (l k_{1} + c σ_{1}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x - (\frac{l k_{3}^{2}}{k_{1}} - c σ_{1}) \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x \\ + c σ_{1} \int_{0}^{1} ψ_{x}^{2} d x - \frac{ρ_{1} l}{2} \int_{0}^{1} ω_{t}^{2} d x + c \int_{0}^{1} ψ_{t}^{2} d x + ρ_{1} (1 - \frac{k_{3}}{k_{1}}) \int_{0}^{1} φ_{t} ω_{x t} d x \\ + \frac{l k_{3} ρ_{1}}{k_{1}} \int_{0}^{1} φ_{t}^{2} d x + ε_{5} k_{3}^{2} {(ω_{x} - l φ)}^{2} (1, t) \\ (4.22) & + \frac{c}{ε_{5}} [φ^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}] . \end{array}$ On the other hand, $\begin{array}{l} I_{52}^{'} (t) = & - ε_{5} k_{3}^{2} {(ω_{x} - l φ)}^{2} (1, t) + 2 ε_{5} k_{3}^{2} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x - ε_{5} ρ_{1} k_{3} ω_{t}^{2} (0, t) \\ - 2 ε_{5} k_{3} l k_{1} \int_{0}^{1} (1 - 2 x) (φ_{x} + ψ + l ω) (ω_{x} - l φ) d x \\ (4.23) & + 2 ε_{5} ρ_{1} k_{3} \int_{0}^{1} ω_{t}^{2} d x - 2 ε_{5} ρ_{1} k_{3} l \int_{0}^{1} (1 - 2 x) ω_{t} φ_{t} d x . \end{array}$ By Young’s inequality, we conclude that for any $ε_{5} > 0$ , $\begin{array}{l} - 2 ε_{5} k_{3} l k_{1} \int_{0}^{1} (1 - 2 x) (φ_{x} + ψ + l ω) (ω_{x} - l φ) d x \\ (4.24) & ⩽ 2 ε_{5} k_{3}^{2} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + c ε_{5} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x, \end{array}$ and $\begin{array}{l} - 2 ε_{5} ρ_{1} k_{3} l \int_{0}^{1} (1 - 2 x) ω_{t} φ_{t} d x ⩽ 2 ε_{5} ρ_{1} k_{3} \int_{0}^{1} ω_{t}^{2} d x + c ε_{5} \int_{0}^{1} φ_{t}^{2} d x, \end{array}$ which, together with (4.23) and (4.24), implies for any $ε_{5} > 0$ , $\begin{array}{l} I_{52}^{'} (t) ⩽ & - ε_{5} k_{3}^{2} {(ω_{x} - l φ)}^{2} (1, t) + 4 ε_{5} k_{3}^{2} \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x - ε_{5} ρ_{1} k_{3} ω_{t}^{2} (0, t) \\ + c ε_{5} \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x + 2 ε_{5} ρ_{1} k_{3} \int_{0}^{1} ω_{t}^{2} d x \\ (4.25) & + 4 ε_{5} ρ_{1} k_{3} \int_{0}^{1} ω_{t}^{2} d x + c ε_{5} \int_{0}^{1} φ_{t}^{2} d x . \end{array}$ Therefore, it follows from (4.22) and (4.25) that for any $ε_{5} > 0$ , $\begin{array}{l} I_{5}^{'} (t) ⩽ & - (\frac{l k_{3}^{2}}{k_{1}} - c σ_{1} - 4 ε_{5} k_{3}^{2}) \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x + c \int_{0}^{1} ψ_{t}^{2} d x \\ + (l k_{1} + c σ_{1} + c ε_{5}) \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x - (\frac{ρ_{1} l}{2} - 4 ε_{5} ρ_{1} k_{3}) \int_{0}^{1} ω_{t}^{2} d x \\ + c σ_{1} \int_{0}^{1} ψ_{x}^{2} d x + (\frac{l k_{3} ρ_{1}}{k_{1}} + c ε_{5}) \int_{0}^{1} φ_{t}^{2} d x + ρ_{1} (1 - \frac{k_{3}}{k_{1}}) \int_{0}^{1} φ_{t} ω_{x t} d x \\ - ε_{5} ρ_{1} k_{3} ω_{t}^{2} (0, t) + \frac{c}{ε_{5}} [φ_{t}^{2} (1, t) + {(- σ_{1}^{'} ⊙ φ (1, t))}^{2}], \end{array}$ Recalling that $k_{1} = k_{3}$ , (4.18) is etablished. □

The following lemma plays an important role to establish the optimal energy decay. Lemma 4.7.
The functional $I_{6} (t)$ , defined by $\begin{array}{l} I_{6} (t) : = \int_{0}^{t} σ_{1} (t - s) φ^{2} (1, s) d s + \int_{0}^{t} σ_{2} (t - s) ψ^{2} (1, s) d s, \end{array}$ satifies $\begin{array}{l} (4.26) & I_{6}^{'} (t) ⩽ \frac{1}{2} σ_{1}^{'} \circ φ (1, t) + \frac{1}{2} σ_{2}^{'} \circ ψ (1, t) + 3 σ_{1} (0) φ^{2} (1, t) + 3 σ_{2} (0) ψ^{2} (1, t) . \end{array}$
Proof.
A direct computation leads to $\begin{array}{l} I_{6}^{'} (t) = & \int_{0}^{t} σ_{1}^{'} (t - s) φ^{2} (1, s) d s + σ_{1} (0) φ^{2} (1, t) + σ_{2} (0) ψ^{2} (1, t) + \int_{0}^{t} σ_{2}^{'} (t - s) ψ^{2} (1, s) d s \\ = & \int_{0}^{t} σ_{1}^{'} (t - s) {[φ (1, s) - φ (1, t)]}^{2} d s + 2 φ (1, t) \int_{0}^{t} σ_{1}^{'} (t - s) [φ (1, s) - φ (1, t)] d s \\ + σ_{1} (t) φ^{2} (1, t) + \int_{0}^{t} σ_{2}^{'} (t - s) {[ψ (1, s) - ψ (1, t)]}^{2} d s \\ + 2 ψ (1, t) \int_{0}^{t} σ_{2}^{'} (t - s) [ψ (1, s) - ψ (1, t)] d s + σ_{2} (t) ψ^{2} (1, t) . \end{array}$ Applying Young’s and Hölder’s inequalities, we see that $\begin{array}{l} 2 u (1, t) \int_{0}^{t} σ^{'} (t - s) [u (s) - u (t)] d s \\ ⩽ 2 σ (0) u^{2} (1, t) + \frac{1}{2 σ (0)} {(\int_{0}^{t} \sqrt{- σ^{'} (t - s)} \sqrt{- σ^{'} (t - s)} [u (s) - u (t)] d s)}^{2} \\ ⩽ 2 σ (0) u^{2} (1, t) + \frac{\int_{0}^{t} σ^{'} (s) d s}{2 σ (0)} \int_{0}^{t} σ^{'} (t - s) {[u (s) - u (t)]}^{2} d s . \end{array}$ For $i = 1, 2$ , since $\begin{array}{l} σ_{i} (t) ⩽ σ_{1} (0) and \frac{\int_{0}^{t} σ_{i}^{'} (s) d s}{2 σ_{i} (0)} ⩾ - \frac{1}{2}, \end{array}$ we can obtain (4.26). □

4.2. Proof of Theorem 3.3

Proof.
For any $0 < ν_{i} < 1$ , motivated by Jin et al. [23] (see also Mustafa [35]), we define $\begin{array}{l} C_{ν_{i}} = \int_{0}^{\infty} \frac{{[σ_{i}^{'} (s)]}^{2}}{σ_{i}^{″} (s) - ν_{i} σ_{i}^{'} (s)} d s and h_{i} (t) = σ_{i}^{″} (s) - ν_{i} σ_{i}^{'} (s) . \end{array}$ It follows from Hölder’s inequality that $\begin{array}{l} {(- σ_{i}^{'} ⊙ u)}^{2} & = {(\int_{0}^{t} (- σ_{i}^{'} (t - s)) (u (t) - u (s)) d s)}^{2} \\ = {(\int_{0}^{t} \frac{- σ_{i}^{'} (t - s)}{\sqrt{σ_{i}^{″} (t - s) - ν_{i} σ_{i}^{'} (t - s)}} \sqrt{σ_{i}^{″} (t - s) - ν_{i} σ_{i}^{'} (t - s)} (u (t) - u (s)) d s)}^{2} \\ ⩽ (\int_{0}^{t} \frac{{[σ_{i}^{'} (s)]}^{2}}{σ_{i}^{″} (s) - ν_{i} σ_{i}^{'} (s)} d s) \int_{0}^{t} (σ_{i}^{″} (t - s) - ν_{i} σ_{i}^{'} (t - s)) {(u (t) - u (s))}^{2} d s \\ (4.27) & ⩽ C_{ν_{i}} (h_{i} \circ u) . \end{array}$ Now we define the functional $L (t)$ by $\begin{array}{l} L (t) : = N E (t) + N_{1} I_{1} (t) + N_{2} I_{2} (t) + N_{3} I_{3} (t) + N_{4} I_{4} (t) + N_{5} I_{5} (t), \end{array}$ where $N, N_{i} (i = 1, 2, 3, 4, 5)$ are positive constants. We know that there exist two constants $β_{1}, β_{2} > 0$ such that $\begin{array}{l} (4.28) & β_{1} E (t) ⩽ L (t) ⩽ β_{2} E (t) . \end{array}$ In fact, using Young’s inequality and Poincaré’s inequality, we have $\begin{array}{l} | L (t) - N E (t) | ⩽ c E (t) . \end{array}$ Taking N large (if needed), we can get (4.28) with $β_{1} = N - c > 0$ and $β_{2} = N + c > 0$ . Combining (4.1), (4.4), (4.10), (4.15), (4.17) and (4.18), taking $\begin{array}{l} δ_{1} = 1, δ_{2} = \frac{k_{1}}{k_{2}}, ε_{1} = \frac{k_{3}}{k_{2}}, N_{1} = k_{3} N_{3}, N_{2} = 4 k_{3} N_{3}, N_{4} = N_{5} = 1, \end{array}$ and noting $σ_{i}^{″} = ν_{i} σ_{i}^{'} + h_{i}$ , we see that $\begin{array}{l} L^{'} (t) ⩽ & - (α_{1} k_{1} N - \frac{c}{ε_{1}^{'}} k_{3} N_{3} - \frac{c}{ε_{3}} N_{3} - c - \frac{c}{ε_{5}}) φ_{t}^{2} (1, t) \\ - (α_{2} k_{2} N - \frac{c}{ε_{1}^{'}} k_{3} N_{3} - 4 c k_{3} N_{3} - c) ψ_{t}^{2} (1, t) \\ - (β N - \frac{c}{ε_{1}^{'}} k_{3} N_{3} - \frac{4 c}{ε_{2}} k_{3} N_{3} - \frac{c}{ε_{0}} N_{3} - \frac{c}{ε_{0}} - \frac{c}{ε_{5}}) \int_{0}^{1} ψ_{t}^{2} d x \\ - [ρ_{1} (k_{1} + k_{3}) N_{3} - ρ_{1} l (\frac{| k_{3} - k_{1} |}{2} N_{3} + \frac{k_{3}}{k_{1}}) - ε_{0} N_{3} - c ε_{5}] \int_{0}^{1} φ_{t}^{2} d x \\ - (\frac{ρ_{1} l}{2} - \frac{c l | k_{3} - k_{1} |}{2} N_{3} - ε_{0} - 4 ε_{5} ρ_{1} k_{3}) \int_{0}^{1} ω_{t}^{2} d x \\ - [k_{1} (1 - k_{1} N_{3} - l - k_{3} N_{3} - \frac{8 c k_{1} k_{3}}{k_{2}} N_{3}) - ε_{4} - c σ_{1} - c ε_{1}^{'} k_{3} N_{3} \\ - \frac{c}{ε_{1}^{'}} σ_{1} k_{3} N_{3} - ε_{3} N_{3} - \frac{c}{ε_{3}} σ_{1} N_{3} - c ε_{5}] \int_{0}^{1} {(φ_{x} + ψ + l ω)}^{2} d x \\ - [\frac{1}{2} k_{2} k_{3} N_{3} - \frac{l k_{2}^{2}}{2 k_{1}} - 4 k_{3} ε_{2} N_{3} - c ε_{1}^{'} k_{3} N_{3} \\ - \frac{c}{ε_{1}^{'}} k_{3} N_{3} σ_{1} - \frac{c}{ε_{1}^{'}} k_{3} N_{3} σ_{2} \frac{c}{ε_{3}} N_{3} σ_{2} - c_{1} σ_{1} - c σ_{2}] \int_{0}^{1} ψ_{x}^{2} d x \\ - [\frac{l k_{3}^{2}}{2 k_{1}} - c σ_{1} - 4 ε_{5} k_{3}^{2} - (c ε_{1}^{'} + \frac{c}{ε_{1}^{'}} σ_{1}) k_{3} N_{3} \\ + k_{3}^{2} N_{3} - \frac{c}{ε_{3}} σ_{1} N_{3} - c σ_{1}] \int_{0}^{1} {(ω_{x} - l φ)}^{2} d x \\ - \frac{α_{1} k_{1}}{2} N ν_{1} σ_{1}^{'} \circ φ (1, t) - \frac{α_{2} k_{2}}{2} N ν_{2} σ_{2}^{'} \circ ψ (1, t) \\ (4.29) & - (\frac{α_{1} k_{1}}{2} N - c C_{ν_{1}}) (h_{1} \circ φ (1, t)) - (\frac{α_{2} k_{2}}{2} N - c C_{ν_{2}}) (h_{2} \circ ψ (1, t)) . \end{array}$ Firstly we choose $N_{3}$ small so that $\begin{array}{l} ρ_{1} - c | k_{3} - k_{1} | N_{3} > 0, 1 - k_{3} N_{3} - k_{1} N_{3} - \frac{8 c k_{1} k_{3}}{k_{2}} N_{3} > 0 . \end{array}$ Then we take l so small that $\begin{array}{l} ρ_{1} (k_{1} + k_{3}) N_{3} - ρ_{1} l (\frac{| k_{3} - k_{1} |}{2} N_{3} + \frac{k_{3}}{k_{1}}) > 0, \frac{1}{2} k_{2} k_{3} N_{3} - \frac{l k_{2}^{2}}{2 k_{1}} > 0, \end{array}$ and $\begin{array}{l} 1 - k_{3} N_{3} - k_{1} N_{3} - \frac{8 c k_{1} k_{3}}{k_{2}} N_{3} - l > 0 . \end{array}$

For fixed $N_{3}$ and l, we take $ε_{0}, ε_{1}^{'}, ε_{2}, ε_{3}, ε_{4}$ and $ε_{5}$ so small that $\begin{array}{l} ρ_{1} (k_{1} + k_{3}) N_{3} - ρ_{1} l (\frac{| k_{3} - k_{1} |}{2} N_{3} + \frac{k_{3}}{k_{1}}) - ε_{0} N_{3} - c ε_{5} > 0, \\ \frac{ρ_{1} l}{2} - \frac{c l | k_{3} - k_{1} |}{2} N_{3} - ε_{0} - 4 ε_{5} ρ_{1} k_{3} > 0, \\ k_{1} (1 - k_{1} N_{3} - l - k_{3} N_{3} - \frac{8 c k_{1} k_{3}}{k_{2}} N_{3}) - ε_{4} - c ε_{1}^{'} k_{3} N_{3} - ε_{3} N_{3} - c ε_{5} > 0, \\ \frac{1}{2} k_{2} k_{3} N_{3} - \frac{l k_{2}^{2}}{2 k_{1}} - 4 k_{3} ε_{2} N_{3} - c ε_{1}^{'} k_{3} N_{3} > 0, \end{array}$ and $\begin{array}{l} \frac{l k_{3}^{2}}{2 k_{1}} - 4 ε_{5} k_{3}^{2} - c ε_{1}^{'} k_{3} N_{3} + k_{3}^{2} N_{3} > 0 . \end{array}$

For $i = 1, 2$ , noting $- σ_{i}^{'} > 0$ and $σ_{i}^{″} > 0$ , we obtain that for each $s \in [0, \infty)$ , $\begin{array}{l} lim_{ν_{i} \to 0} \frac{ν_{i} {[σ_{i}^{'} (s)]}^{2}}{σ_{i}^{″} (s) - ν_{i} σ_{i}^{'} (s)} d s = 0 and \frac{ν_{i} {[σ_{i}^{'} (s)]}^{2}}{σ_{i}^{″} (s) - ν_{i} σ_{i}^{'} (s)} < - σ_{i}^{'} (s) . \end{array}$ We infer from Lebesgue dominated convergence theorem that $\begin{array}{l} lim_{ν_{i} \to 0} ν_{i} C_{ν_{i}} = lim_{ν_{i} \to 0} \int_{0}^{\infty} \frac{ν_{i} {[σ_{i}^{'} (s)]}^{2}}{σ_{i}^{″} (s) - ν_{i} σ_{i}^{'} (s)} d s = 0 . \end{array}$ Thus, for a $ϵ_{0} > 0$ , there exist $0 < γ_{i} < 1$ such that if $ν_{i} < γ_{i}$ then $\begin{array}{l} (4.30) & ν_{i} C_{ν_{i}} < \frac{ϵ_{0}}{4 c} . \end{array}$ At this point we take N even larger such that $\begin{array}{l} α_{1} k_{1} N - \frac{c}{ε_{1}^{'}} k_{3} N_{3} - \frac{c}{ε_{3}} N_{3} - c - \frac{c}{ε_{5}} > 1, α_{2} k_{2} N - \frac{c}{ε_{1}^{'}} k_{3} N_{3} - 4 c k_{3} N_{3} - c > 1, \end{array}$ and $\begin{array}{l} β N - \frac{c}{ε_{1}^{'}} k_{3} N_{3} - \frac{4 c}{ε_{2}} k_{3} N_{3} - \frac{c}{ε_{0}} N_{3} - \frac{c}{ε_{0}} - \frac{c}{ε_{5}} > 0, \end{array}$ and choose $ν_{i}$ satisfying $\begin{array}{l} ν_{1} = \frac{1}{2 α_{1} k_{1} N} < γ_{1} and δ_{2} = \frac{1}{2 α_{2} k_{2} N} < γ_{2}, \end{array}$ which yields $\begin{array}{l} \frac{α_{1} k_{1}}{2} N - c C_{ν_{1}} > 0 and \frac{α_{2} k_{2}}{2} N - c C_{ν_{2}} > 0 . \end{array}$ As ${lim}_{t \to + \infty} k_{i} (t) = 0$ , then there exists $t_{1} > 0$ large such that for any $t ⩾ t_{1}$ , $\begin{array}{l} k_{1} (1 - k_{1} N_{3} - l - k_{3} N_{3} - \frac{8 c k_{1} k_{3}}{k_{2}} N_{3}) - ε_{4} - c σ_{1} \\ - c ε_{1}^{'} k_{3} N_{3} - \frac{c}{ε_{1}^{'}} σ_{1} k_{3} N_{3} - ε_{3} N_{3} - \frac{c}{ε_{3}} σ_{1} N_{3} - c ε_{5} > 0, \\ \frac{1}{2} k_{2} k_{3} N_{3} - \frac{l k_{2}^{2}}{2 k_{1}} - 4 k_{3} ε_{2} N_{3} - c ε_{1}^{'} k_{3} N_{3} \\ - \frac{c}{ε_{1}^{'}} k_{3} N_{3} σ_{1} - \frac{c}{ε_{1}^{'}} k_{3} N_{3} σ_{2} - \frac{c}{ε_{3}} N_{3} σ_{2} - c_{1} σ_{1} - c σ_{2} > 0, \end{array}$ and $\begin{array}{l} \frac{l k_{3}^{2}}{2 k_{1}} - c σ_{1} - 4 ε_{5} k_{3}^{2} - (c ε_{1}^{'} + \frac{c}{ε_{1}^{'}} σ_{1}) k_{3} N_{3} + k_{3}^{2} N_{3} - \frac{c}{ε_{3}} σ_{1} N_{3} - c σ_{1} > 0 . \end{array}$ Then from (4.29) we get there exist a positive constant $β_{3}$ such that for any $t ⩾ t_{1}$ , $\begin{array}{l} L^{'} (t) ⩽ & - β_{3} \int_{0}^{1} [ρ_{1} φ_{t}^{2} + ρ_{2} ψ_{t}^{2} + ρ_{1} ω_{t}^{2} + k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} + k_{3} {(ω_{x} - l φ)}^{2}] d x \\ (4.31) & - φ_{t}^{2} (1, t) - ψ_{t}^{2} (1, t) - \frac{ϵ_{0}}{4} σ_{1}^{'} \circ φ (1, t) - \frac{ϵ_{0}}{4} σ_{2}^{'} \circ ψ (1, t) . \end{array}$ It follows from (2.5) and (4.1) that for any $t ⩾ t_{1}$ , $\begin{array}{l} \int_{0}^{t_{1}} (- σ_{1}^{'} (s)) {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ ⩽ \frac{1}{d} \int_{0}^{t_{1}} σ_{1}^{″} (s) {[φ (1, t) - φ (1, t - s)]}^{2} d s ⩽ - c E^{'} (t), \end{array}$ and $\begin{array}{l} \int_{0}^{t_{1}} (- σ_{2}^{'} (s)) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s \\ ⩽ \frac{1}{d} \int_{0}^{t_{1}} σ_{2}^{″} (s) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s ⩽ - c E^{'} (t), \end{array}$ By (4.31), we derive that there exists a constant $m > 0$ such that $\begin{array}{l} L^{'} (t) ⩽ & - m E (t) - c \int_{0}^{t} σ_{1}^{'} (s) {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ - c \int_{0}^{t} σ_{2}^{'} (s) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s \\ ⩽ & - m E (t) - 2 c E^{'} (t) - c \int_{t_{1}}^{t} σ_{1}^{'} (s) {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ (4.32) & - c \int_{t_{1}}^{t} σ_{2}^{'} (s) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s . \end{array}$ Define the functional $F (t) : = L (t) + 2 c E (t) \sim E (t)$ , we infer from (4.32) that $\begin{array}{l} F^{'} (t) ⩽ & - m E (t) - c \int_{t_{1}}^{t} k_{1}^{'} (s) {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ (4.33) & - c \int_{t_{1}}^{t} k_{2}^{'} (s) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s . \end{array}$

We consider the following two cases.

Case 1. The particular case $M_{i} (t) = c t^{p_{i}}$ $(i = 1, 2)$ .

(I) $p_{1} = p_{2} = 1$ .

We multiply (4.33) by $η (t) = min {η_{1} (t), η_{2} (t)}$ , use (4.1) and (A1)–(A2) to conclude for any $t ⩾ t_{1}$ , $\begin{array}{l} η (t) F^{'} (t) ⩽ - q η (t) E (t) - c E^{'} (t) . \end{array}$ As $η (t)$ is a continuous nonincreasing function and $η^{'} (t) ⩽ 0$ for a.e. t, we get $\begin{array}{l} {(η F + c E)}^{'} (t) ⩽ η (t) F^{'} (t) + c E^{'} (t) ⩽ - m η (t) E (t), a.e. t ⩾ t_{0} . \end{array}$ By $η F + c E \sim E$ , we obtain that there exist two constants $c_{1}, c_{2} > 0$ , $\begin{array}{l} E (t) ⩽ c_{1} e^{- c_{2} \int_{t_{1}}^{t} η (s) d s} . \end{array}$ (II) $1 ⩽ p_{i} < 2$ and $p_{1} \neq p_{2}$ .

Define $J (t)$ by $\begin{array}{l} J (t) = L (t) + ϵ_{0} I_{5} (t), \end{array}$ where $ϵ_{0}$ is defined in (4.30). It follows from (4.26) and (4.31) that $J (t) ⩾ 0$ and for any $t ⩾ t_{1}$ , $\begin{array}{l} J^{'} (t) ⩽ & - β_{3} \int_{0}^{1} [ρ_{1} φ_{t}^{2} + ρ_{2} ψ_{t}^{2} + ρ_{1} ω_{t}^{2} + k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} + k_{3} {(ω_{x} - l φ)}^{2}] d x \\ - φ_{t}^{2} (1, t) - ψ_{t}^{2} (1, t) + 3 σ_{1} (0) ϵ_{0} φ^{2} (1, t) + 3 σ_{2} (0) ψ^{2} (1, t) \\ + \frac{ϵ_{0}}{4} σ_{1}^{'} \circ φ (1, t) + \frac{ϵ_{0}}{4} σ_{2}^{'} \circ (3 ω - ψ) (1, t) \\ ⩽ & - β_{3} \int_{0}^{1} [ρ_{1} φ_{t}^{2} + ρ_{2} ψ_{t}^{2} + ρ_{1} ω_{t}^{2} + k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} + k_{3} {(ω_{x} - l φ)}^{2}] d x \\ - φ_{t}^{2} (1, t) - ψ_{t}^{2} (1, t) + c ϵ_{0} \int_{0}^{1} [k_{1} {(φ_{x} + ψ + l ω)}^{2} + k_{2} ψ_{x}^{2} + k_{3} {(ω_{x} - l φ)}^{2}] d x \\ (4.34) & + \frac{ϵ_{0}}{4} σ_{1}^{'} \circ φ (1, t) + \frac{ϵ_{0}}{4} σ_{2}^{'} \circ (3 ω - ψ) (1, t) . \end{array}$ In (4.34), we pick $ϵ_{0} > 0$ small enough such that $\begin{array}{l} β_{3} - c ϵ_{0} > 0 . \end{array}$ Then there exists some constant $β_{4} > 0$ such that for any $t ⩾ t_{1}$ , $\begin{array}{l} J^{'} (t) ⩽ - β_{4} E (t) . \end{array}$ This implies $\begin{array}{l} β_{4} \int_{t_{1}}^{t} E (s) d s ⩽ J (t_{1}) - J (t) ⩽ J (t_{1}) . \end{array}$ Hence $\begin{array}{l} (4.35) & \int_{0}^{\infty} E (s) d s < \infty . \end{array}$ Denote $\begin{array}{l} J_{1} (t) = \int_{t_{1}}^{t} {[φ (1, s) - φ (1, t)]}^{2} d s and J_{2} (t) = \int_{t_{1}}^{t} {[ψ (1, s) - ψ (1, t)]}^{2} d s . \end{array}$ We have $\begin{array}{l} J_{1} (t) ⩽ c \int_{t_{1}}^{t} E (s) d s and J_{2} (t) ⩽ c \int_{t_{1}}^{t} E (s) d s . \end{array}$ Without loss of generality assuming $t_{1}$ so large that $J_{i} (t_{1}) > 0$ , then for any $t ⩾ t_{1}$ , $\begin{array}{l} 0 < J_{i} (t_{1}) ⩽ J_{i} (t) < \infty . \end{array}$ Using (4.1), (A1)–(A2) and Jensen’s inequality, we get from (4.31) that for some constant $q > 0$ , $\begin{array}{l} L^{'} (t) ⩽ & - q E (t) + \frac{c J_{1} (t)}{J_{1} (t)} [{(- σ_{1}^{'})}^{p_{1} \cdot \frac{1}{p_{1}}} \circ φ (1, t)] + \frac{c J_{2} (t)}{J_{2} (t)} [{(- σ_{2}^{'})}^{p_{2} \cdot \frac{1}{p_{2}}} \circ ψ (1, t)] \\ ⩽ & - q E (t) + c J_{1} (t) {[\frac{1}{J_{1} (t)} {(- σ_{1}^{'})}^{p_{1}} \circ φ (1, t)]}^{\frac{1}{p_{1}}} \\ + c J_{2} (t) {[\frac{1}{J_{2} (t)} {(- σ_{2}^{'})}^{p_{2}} \circ ψ (1, t)]}^{\frac{1}{p_{2}}} \\ ⩽ & - q E (t) + c J_{1}^{1 - \frac{1}{p_{1}}} (t) {[\frac{σ_{1}^{″}}{η_{1}} \circ φ (1, t)]}^{\frac{1}{p_{1}}} + c J_{2}^{1 - \frac{1}{p_{2}}} (t) {[\frac{σ_{2}^{″}}{η_{2}} \circ ψ (1, t)]}^{\frac{1}{p_{2}}} \\ ⩽ & - q E (t) + \frac{c}{{[η_{1} (t)]}^{\frac{1}{p_{1}}}} {[σ_{1}^{″} \circ φ (1, t)]}^{\frac{1}{p_{1}}} + \frac{c}{{[η_{2} (t)]}^{\frac{1}{p_{2}}}} {[σ_{2}^{″} \circ ψ (1, t)]}^{\frac{1}{p_{2}}} \\ (4.36) & ⩽ & - q E (t) + \frac{c}{{[η_{1} (t)]}^{\frac{1}{p_{1}}}} {[- E^{'} (t)]}^{\frac{1}{p_{1}}} + \frac{c}{{[η_{2} (t)]}^{\frac{1}{p_{2}}}} {[- E^{'} (t)]}^{\frac{1}{p_{2}}} . \end{array}$

Multiplying (4.31) by $E^{p - 1} (t)$ $(p = max {p_{1}, p_{2}})$ and using (4.1), we derive $\begin{array}{l} {(L E^{p - 1})}^{'} (t) & ⩽ L^{'} (t) E^{p - 1} (t) \\ ⩽ - q E^{p} (t) + c {[- \frac{E^{'} (t)}{η_{1} (t)}]}^{\frac{1}{p_{1}}} E^{p - 1} (t) + c {[- \frac{E^{'} (t)}{η_{2} (t)}]}^{\frac{1}{p_{2}}} E^{p - 1} (t) . \end{array}$ We apply Young’s inequality to obtain $\begin{array}{l} {(L E^{p - 1})}^{'} (t) ⩽ - q E^{p} (t) + ε E^{\frac{p_{1} (p - 1)}{p_{1} - 1}} (t) + \frac{c}{ε} [- \frac{E^{'} (t)}{η_{1} (t)}] + ε E^{\frac{p_{2} (p - 1)}{p_{2} - 1}} (t) + \frac{c}{ε} [- \frac{E^{'} (t)}{η_{2} (t)}] . \end{array}$ Noting $\frac{p_{i} (p - 1)}{p_{i} - 1} ⩾ p$ , taking $ε < \frac{1}{2} q$ , we see that $\begin{array}{l} (4.37) & {(L E^{p - 1})}^{'} (t) ⩽ - q^{'} E^{p} (t) - c \frac{E^{'} (t)}{η_{1} (t)} - c \frac{E^{'} (t)}{η_{2} (t)} . \end{array}$ Multiplying (4.37) by $η (t)$ and denoting $F (t) = η L E^{p - 1} + c E \sim E$ , we have $\begin{array}{l} F^{'} (t) ⩽ - q^{'} η (t) E^{p} (t), \end{array}$ where $q^{'} = \frac{q}{q - 1}$ . Then there exists a constant $q_{0} > 0$ such that $\begin{array}{l} F^{'} (t) ⩽ - q_{0} η (t) F^{p} (t), \end{array}$ which yields there exists a constant $c_{3} > 0$ such that $\begin{array}{l} E (t) ⩽ c_{3} {(1 + \int_{t_{1}}^{t} η (s) d s)}^{- \frac{1}{p - 1}} . \end{array}$ Noting the boundness of $η (t)$ and $E (t)$ and combining (I) and (II) and, we can get (3.4).

Case 2. The general case of $M_{i} (t)$ .

Taking into account (4.35), and defining $\begin{array}{l} J_{1} (t) : = q \int_{t_{1}}^{t} {[φ (1, t) - φ (1, t - s)]}^{2} d s, \end{array}$ and $\begin{array}{l} J_{2} (t) : = q \int_{t_{1}}^{t} {[ψ (1, t) - ψ (1, t - s)]}^{2} d s, \end{array}$ we can take $0 < q < 1$ such that for $i = 1, 2$ , $\begin{array}{l} (4.38) & J_{i} (t) < 1, \forall t ⩾ t_{1} . \end{array}$ Without loss of generality, we assume that $I_{i} (t) > 0$ for all $t ⩾ t_{1}$ . In the sequel we define $\begin{array}{l} π_{1} (t) : = \int_{t_{1}}^{t} σ_{1}^{″} (s) {[φ (1, t) - φ (1, t - s)]}^{2} d s, \end{array}$ and $\begin{array}{l} π_{2} (t) : = \int_{t_{1}}^{t} σ_{2}^{″} (s) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s . \end{array}$ It follows from (4.1) that $π_{i} (t) ⩽ - c E^{'} (t)$ $(i = 1, 2)$ . As $M_{i} (t)$ is strictly convex on $(0, r]$ and $M_{i} (0) = 0$ , we derive $\begin{array}{l} M_{i} (θ x) ⩽ θ M_{i} (x), i = 1, 2, 0 ⩽ θ ⩽ 1, x \in (0, r] . \end{array}$ Using Jensen’s inequality, (2.1) and (4.38), we obtain $\begin{array}{l} π_{1} (t) & = \frac{1}{q J_{1} (t)} \int_{t_{1}}^{t} J_{1} (t) (σ_{1}^{″} (s)) q {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ ⩾ \frac{1}{q J_{1} (t)} \int_{t_{1}}^{t} J_{1} (t) η_{1} (s) M_{1} (- σ_{1}^{'} (s)) q {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ ⩾ \frac{η_{1} (t)}{q J_{1} (t)} \int_{t_{1}}^{t} M_{1} (J_{1} (t) (- σ_{1}^{'} (s))) q {[φ (1, t) - φ (1, t - s)]}^{2} d s \\ ⩾ \frac{η_{1} (t)}{q} M_{1} (\frac{1}{J_{1} (t)} \int_{t_{1}}^{t} J_{1} (t) (- σ_{1}^{'} (s)) q {[φ (1, t) - φ (1, t - s)]}^{2} d s) \\ = \frac{η_{1} (t)}{q} M_{1} (q \int_{t_{1}}^{t} (- σ_{1}^{'}) (s) {[φ (1, t) - φ (1, t - s)]}^{2} d s) \\ (4.39) & = \frac{η_{1} (t)}{q} {\overline{M}}_{1} (q \int_{t_{1}}^{t} (- σ_{1}^{'} (s)) {[φ (1, t) - φ (1, t - s)]}^{2} d s), \end{array}$ where ${\overline{M}}_{1}$ , which is strictly convex and increasing function on $(0, \infty)$ of $C^{2}$ class, is the extension of $M_{1}$ . By (4.39) we have $\begin{array}{l} \int_{t_{1}}^{t} (- σ_{1}^{'} (s)) {[φ (1, t) - φ (1, t - s)]}^{2} d s ⩽ \frac{1}{q} {\overline{M}}_{1}^{- 1} (\frac{q π_{1} (t)}{η_{1} (t)}) . \end{array}$ Similarly, $\begin{array}{l} \int_{t_{1}}^{t} (- σ_{2}^{'} (s)) {[ψ (1, t) - ψ (1, t - s)]}^{2} d s ⩽ \frac{1}{q} {\overline{M}}_{2}^{- 1} (\frac{q π_{2} (t)}{η_{2} (t)}) . \end{array}$ We get from (4.33) that for any $t ⩾ t_{1}$ , $\begin{array}{l} (4.40) & F^{'} (t) ⩽ - m E (t) + c {\overline{M}}_{1}^{- 1} (\frac{q π_{1} (t)}{η_{1} (t)}) + c {\overline{M}}_{2}^{- 1} (\frac{q π_{2} (t)}{η_{2} (t)}) . \end{array}$ Let’s denote $\begin{array}{l} M_{0} (t) = min {{\overline{M}}_{1}^{'}, {\overline{M}}_{2}^{'}} . \end{array}$ For $r_{0} < r$ , we define the function $K_{1} (t)$ $\begin{array}{l} K_{1} (t) = {\overline{M}}_{0}^{'} (r_{0} \frac{E (t)}{E (0)}) F (t) + E (t) \sim E (t) . \end{array}$ Since $E^{'} (t) ⩽ 0$ , ${\overline{M}}_{0}^{'} > 0$ and ${\overline{M}}_{0}^{″} > 0$ , we infer from (4.40) that $\begin{array}{l} K_{1}^{'} (t) = & r_{0} \frac{E^{'} (t)}{E (0)} M_{0}^{'} (r_{0} \frac{E (t)}{E (0)}) F (t) + M_{0} (r_{0} \frac{E (t)}{E (0)}) F^{'} (t) + E^{'} (t) \\ ⩽ & - m E (t) M_{0} (r_{0} \frac{E (t)}{E (0)}) + c M_{0} (r_{0} \frac{E (t)}{E (0)}) {\overline{M}}_{1}^{- 1} (\frac{q π_{1} (t)}{η_{1} (t)}) \\ (4.41) & + c M_{0} (r_{0} \frac{E (t)}{E (0)}) {\overline{M}}_{2}^{- 1} (\frac{q π_{2} (t)}{η_{2} (t)}) . \end{array}$ We denote the conjugate function of the convex function ${\overline{M}}_{i}$ by ${\overline{M}}_{i}^{}$ (see Arnold [7]), then $\begin{array}{l} {\overline{M}}_{i}^{} (s) = s {({\overline{M}}_{i}^{'})}^{- 1} (s) - {\overline{M}}_{i} [{({\overline{M}}_{i}^{'})}^{- 1} (s)], i = 1, 2 . \end{array}$ satisfies Young’s inequality, $\begin{array}{l} A B_{i} ⩽ {\overline{M}}_{i}^{} (A) + {\overline{M}}_{i} (B_{i}), i = 1, 2 . \end{array}$ Taking $A = M_{0} (r_{0} \frac{E (t)}{E (0)})$ and $B = {\overline{M}}_{i}^{- 1} (\frac{q π_{i} (t)}{η_{i} (t)})$ , and using ${\overline{M}}_{i}^{} (s) ⩽ s {({\overline{M}}_{i}^{'})}^{- 1} (s)$ and (4.41), we have $\begin{array}{l} K_{1}^{'} (t) ⩽ & - m E (t) M_{0} (r_{0} \frac{E (t)}{E (0)}) + c {\overline{M}}_{1}^{} (M_{0} (r_{0} \frac{E (t)}{E (0)})) + c \frac{q π_{1} (t)}{η_{1} (t)} \\ + c {\overline{M}}_{2}^{} (M_{0} (r_{0} \frac{E (t)}{E (0)})) + c \frac{q π_{2} (t)}{η_{2} (t)} \\ ⩽ & - m E (t) M_{0} (r_{0} \frac{E (t)}{E (0)}) + c M_{0} (r_{0} \frac{E (t)}{E (0)}) {({\overline{M}}_{1}^{'})}^{- 1} (M_{0} (r_{0} \frac{E (t)}{E (0)})) + c \frac{q π_{1} (t)}{η_{1} (t)} \\ + c M_{0} (r_{0} \frac{E (t)}{E (0)}) {({\overline{M}}_{2}^{'})}^{- 1} (M_{0} (r_{0} \frac{E (t)}{E (0)})) + c \frac{q π_{2} (t)}{η_{2} (t)} \\ ⩽ & - m E (t) M_{0} (r_{0} \frac{E (t)}{E (0)}) + c M_{0} (r_{0} \frac{E (t)}{E (0)}) {({\overline{M}}_{1}^{'})}^{- 1} ({\overline{M}}_{1}^{'} (r_{0} \frac{E (t)}{E (0)})) + c \frac{q π_{1} (t)}{η_{1} (t)} \\ + c M_{0} (r_{0} \frac{E (t)}{E (0)}) {({\overline{M}}_{2}^{'})}^{- 1} ({\overline{M}}_{2}^{'} (r_{0} \frac{E (t)}{E (0)})) + c \frac{q π_{2} (t)}{η_{2} (t)} \\ (4.42) & ⩽ & - (m E (0) - c r_{0}) \frac{E (t)}{E (0)} M_{0} (r_{0} \frac{E (t)}{E (0)}) + c q (\frac{π_{1} (t)}{η_{1} (t)} + \frac{π_{2} (t)}{η_{2} (t)}) . \end{array}$

We multiply (4.42) by $η (t)$ to derive $\begin{array}{l} η (t) K_{1}^{'} (t) & ⩽ - (m E (0) - c r_{0}) η (t) \frac{E (t)}{E (0)} M_{0} (r_{0} \frac{E (t)}{E (0)}) + c q (π_{1} (t) + π_{2} (t)) \\ (4.43) & ⩽ - (m E (0) - c r_{0}) η (t) \frac{E (t)}{E (0)} M_{0} (r_{0} \frac{E (t)}{E (0)}) - c E^{'} (t) . \end{array}$ Define the functional $K_{2} (t)$ by $\begin{array}{l} K_{2} (t) = η (t) K_{1} (t) + c E (t) . \end{array}$ Then there exist constants $β_{5} > 0$ and $β_{6} > 0$ such that $\begin{array}{l} (4.44) & β_{5} K_{2} (t) ⩽ E (t) ⩽ β_{6} K_{2} (t) . \end{array}$ Choosing a suitable $r_{0} > 0$ , we conclude from (4.43) that for a constant $ζ > 0$ , $\begin{array}{l} (4.45) & K_{2}^{'} (t) ⩽ - ζ η (t) \frac{E (t)}{E (0)} M_{0} (r_{0} \frac{E (t)}{E (0)}) : = - ζ η (t) M_{3} (\frac{E (t)}{E (0)}), \end{array}$ where $M_{3} (t) = t M_{0} (r_{0} t)$ . From $0 ⩽ r_{0} \frac{E (t)}{E (0)} < r$ we infer that for any $t > 0$ $\begin{array}{l} M_{0} (r_{0} \frac{E (t)}{E (0)}) & = min {{\overline{M}}_{1}^{'} (r_{0} \frac{E (t)}{E (0)}), {\overline{M}}_{2}^{'} (r_{0} \frac{E (t)}{E (0)})} \\ = min {M_{1}^{'} (r_{0} \frac{E (t)}{E (0)}), M_{2}^{'} (r_{0} \frac{E (t)}{E (0)})} . \end{array}$

Denote $R (t) = \frac{β_{5} K_{2} (t)}{E (0)}$ . By (4.44), we obtain $\begin{array}{l} (4.46) & R (t) \sim E (t) . \end{array}$ Since $M_{3}^{'} (t) = M_{0} (r_{0} t) + r_{0} t M_{0}^{'} (r_{0} t)$ , noting the strict convexity of $M_{0}$ on $(0, r]$ , we know $M_{3}^{'} (t), M_{3} (t) > 0$ on $(0, 1]$ . In view of (4.45), we get for any $t ⩾ t_{1}$ , $\begin{array}{l} (4.47) & R^{'} (t) ⩽ - ζ_{1} η (t) M_{3} (R (t)), \end{array}$ where $ζ_{1}$ is a positive constant. Integrating (4.47) over $(t_{1}, t)$ , we have $\begin{array}{l} \int_{t_{1}}^{t} \frac{- R^{'} (s)}{M_{3} (R (s))} d s ⩾ ζ_{1} \int_{t_{1}}^{t} η (s) d s \Rightarrow \int_{r_{0} R (t)}^{r_{0} R (t_{1})} \frac{1}{s M_{0} (s)} d s ⩾ ζ_{1} \int_{t_{1}}^{t} η (s) d s . \end{array}$ Since $M_{4}$ , given by $\begin{array}{l} M_{4} (t) = \int_{t}^{r} \frac{1}{s M_{0} (s)} d s, \end{array}$ is strictly decreasing on $(0, r]$ and ${lim}_{t \to 0} M_{4} (t) = + \infty$ , we get $\begin{array}{l} (4.48) & R (t) ⩽ \frac{1}{r_{0}} M_{4}^{- 1} (ζ_{1} \int_{t_{1}}^{t} η (s) d s) . \end{array}$ Then (3.3) follows from (4.46) and (4.48). The proof is complete. □
5. Numerical tests

In the numerical part of this paper, we solve the system (1.1) using the finite volume method by discretizing the system on the space-time domain $[0, 1] \times [0, 1]$ using second order finite difference method. We implement Lax–Wendroff method. For a similar construction, we refer to [2,15,21]. According the our theoretical results, the choice of the functions $σ_{i}$ for $i = 1, 2$ as exponential and polynomial once conduct us to the study of four different Tests. For the following choices these Tests are:

Test 1. $σ_{1} (t) = e^{- 2 t}$ and $σ_{2} (t) = \frac{5}{(1 + t)}$ .

Test 2. $σ_{1} (t) = e^{- 2 t}$ and $σ_{2} (t) = 2 \sqrt{t + 1} e^{- \sqrt{t + 1}} + e^{- \sqrt{t + 1}}$ .

Test 3. $σ_{1} (t) = σ_{2} (t) = 2 \sqrt{t} e^{- \sqrt{t}} + e^{- \sqrt{t}}$ .

Test 4. $σ_{1} (t) = σ_{2} (t) = \frac{1}{(1 + t)}$ .

In order to ensure the scheme stability, we use a temporal and spatial steps satisfying the stability the Courant–Friedrichs–Lewy (CFL) inequality

Δ t = 0.002 < d x = 0.02

, where

Δ t

represents the time step and

Δ x

the spatial step. The spatial interval

[0, 1]

is subdivided into 200 subintervals, where the temporal interval

[0, 1]

is deduced from the stability condition above. We run our code for 10000 time steps using the following initial conditions:

\begin{array}{l} φ (x, 0) = φ_{t} (x, 0) = 5 \cdot 10^{- 2} x (1 - x); in [0, 1] . \\ (5.1) & ψ (x, 0) = ψ_{t} (x, 0) = 5 \cdot 10^{- 2} x^{2} (1 - x); in [0, 1] . \\ ω (x, 0) = ω_{t} (x, 0) = 5 \cdot 10^{- 2} x^{2} (1 - x); in [0, 1] . \end{array}

Fig. 1.

TEST 1.

Fig. 2.

TEST 2.

Fig. 3.

TEST 3.

Fig. 4.

TEST 4.

Fig. 5.

Energy functions for Tests 1, 2, 3 and 4.

In Fig. 1–4, we plot the damping behavior of the three waves φ, ψ and ω for the cross sections $x = 0.25$ , $x = 0.5$ and $x = 1$ . Moreover, we have to mention that the cutting solution at $x = 1$ , represent the non-Dirichlet memory boundary condition for φ and ψ. While $ω (1, t) = 0$ represents the Dirichlet boundary condition for ω. In Fig. 5, we plot and compare the decay behavior of resulting energies for the Test 1–4.

As a final conclusion, we obtained an exponential decay for the Tests 1 and 2 and a polynomial decay for the Tests 3 and 4. This is consequence of the choice of the functions $σ_{i}$ for $i = 1, 2$ and their values on the spacial interval $[0, 1]$ .

6. Concluding remarks and open problems

In this paper, Bresse system with one damping and with memory-type boundary conditions was considered, we proved the well-posedness, stability results were proved under the condition of equal wave speeds, and some numerical approximations based on finite volume were presented to validate the theoretical results obtained in this paper. The proofs were based on multipliers techniques and the construction of Lyapunov functional. We need to mention here that the stability results for the same model under the memory-type boundary conditions only is an open problem, also the condition of equal wave speeds is it a necessary condition? It is an open problem as well.

Footnotes

Acknowledgement

The authors are grateful to the editor and referees for their comments which helped to improve the paper. This work was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah. Grant No. 2002144089, 2019-2020.

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