New decay results for Timoshenko system in the light of the second spectrum of frequency with infinite memory and nonlinear damping of variable exponent type

Abstract

In this study, we consider a one-dimensional Timoshenko system with two damping terms in the context of the second frequency spectrum. One damping is viscoelastic with infinite memory, while the other is a non-linear frictional damping of variable exponent type. These damping terms are simultaneously and complementary acting on the shear force in the domain. We establish, for the first time to the best of our knowledge, explicit and general energy decay rates for this system with infinite memory. We use Sobolev embedding and the multiplier approach to get our decay results. These results generalize and improve some earlier related results in the literature.

Keywords

Timoshenko system second frequency spectrum multiplier method infinite memory exponential and polynomial decay variable exponents

1. Introduction

In 1921, Timoshenko [29] introduced the following system of hyperbolic partial differential equations as a model to describe the dynamics of a thick beam: $\begin{array}{c} (1.1) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.2) & ρ_{2} ψ_{t t} - b ψ_{x x} + κ (φ_{x} + ψ) = 0, \end{array}$ where $(x, t) \in [0, L] \times R^{+}$ , $ρ_{1}, ρ_{2}, b, κ > 0$ , and φ and ψ represent, respectively, the transverse displacement and the angular rotation. Viscoelastic-type Timoshenko system had received a considerable attention since the work of Ammar-Khodja et al. [10] in which the authors considered the following viscoelastic Timoshenko system: $\begin{array}{c} (1.3) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.4) & ρ_{2} ψ_{t t} - b ψ_{x x} + κ (φ_{x} + ψ) + \int_{0}^{t} θ (t - s) ψ_{x x} (x, s) d s = 0, \end{array}$ and the authors established the uniform stability of the system if and only if the system has equal speeds of wave propagation and the relaxation function θ is a positive nonincreasing differentiable $L^{1}$ function defined on $(0, \infty)$ . After that, several researchers have established stability results using different assumptions on the relaxation function θ. See for example [2,8,9,16,18,21,27,28].

The reader can notice that most of the studies in the literature deal with the Timoshenko model (1.1)–(1.2). This model is characterized by two natural frequencies which leads to a physical paradox, known as the second spectrum, which was not noticed in Timoshenko’s original work. In this regard, various types of dissipations (frictional, heat, viscoelastic) have been used to stabilize the system. This required (mathematically) some relation, called the equal-speed propagation. However, this is unrealistic requirement. Then, this paradox of second spectrum was later discovered in a series of analytical studies. Elishakoff [12] proposed the following truncated version of the classical Timoshenko system $\begin{array}{c} (1.5) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.6) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) = 0, \end{array}$ where $(x, t) \in [0, L] \times R^{+}$ . It is clear that the model (1.5)–(1.6) eliminates the anomaly of the second spectrum. The model (1.5)–(1.6) has been proposed and only few results regarding the stability have been established. For example, Almeida Júnior et al [4] considered the following two truncated Bresse–Timoshenko systems with constant delay terms: $\begin{array}{c} (1.7) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} + μ_{1} φ_{t} + μ_{2} φ_{t} (x, t - τ) = 0, \\ (1.8) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) = 0, \end{array}$ and $\begin{array}{c} (1.9) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.10) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) + μ_{1} ψ_{t} + μ_{2} ψ_{t} (x, t - τ) = 0, \end{array}$ and obtained exponential decay rates depend only on the assumption $μ_{1} > μ_{2}$ . On the other hand, Feng et al. [15] investigated the following system with time-varying delay $\begin{array}{c} (1.11) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.12) & - ρ_{2} φ_{x t t} - b ψ_{x x} + κ (φ_{x} + ψ) + ξ_{1} ψ_{t} + ξ_{2} ψ_{t} (x, t - τ (t)) = 0, \end{array}$ and under some specific assumptions on the delay terms and the constants $ξ_{1}$ and $ξ_{2}$ , the authors obtained exponential stability result regardless of the wave propagations velocities. For more results in this direction, we refer the reader to [6–8,13,14,25,26]. In this paper, we are concerned with a viscoelastic Timoshenko type system in the light of the second spectrum of frequency with nonlinear damping of variable exponent type; that is $\begin{array}{c} (1.13) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.14) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) + \int_{0}^{+ \infty} θ (s) ψ_{x x} (t - s) d s + β (t) | ψ_{t} |^{σ (\cdot) - 2} ψ_{t} = 0, \end{array}$ where $(x, t) \in [0, L] \times R^{+}$ . The function $β (t)$ is the time-dependent coefficient of the damping term and $σ (x)$ is the variable exponent function. Both $β (t)$ and $σ (x)$ satisfy some specific conditions to be mentioned in the next section. We consider (1.13)–(1.14) subject to the following initial and boundary conditions $\begin{matrix} (1.15) & \{\begin{array}{ll} φ (x, 0) = φ_{0} (x), φ_{t} (x, 0) = φ_{1} (x), \\ φ_{t t} (x, 0) = φ_{2} (x), ψ_{t} (x, 0) = ψ_{1} (x), & x \in [0, L], \\ ψ (x, - t) = ψ_{0} (x, t), & \forall t ⩾ 0, x \in [0, L], \\ φ_{x} (0, t) = φ_{x} (L, t) = ψ (0, t) = ψ (L, t) = 0, & \forall t > 0, \end{array} \end{matrix}$ with minimal conditions (to be specified later) on the $L^{1} (0, \infty)$ relaxation function θ, which is more general than the ones imposed in some related problems in the literature. Note that the system (1.13)–(1.14) correspond to the Timoshenko beam model which is free of second spectrum of frequency and its damaging consequences for wave propagation speed. We establish explicit and general energy decay results by using the Multiplier method and appropriate conditions on the variable exponents, relaxation function, the time-dependent coefficient, and the initial data. For the stability of equations with infinite memories, we refer the reader to [3,11,17,20,22] and the references therein.

From the boundary conditions in our system (1.15), it is clear that Poincaré’s inequality can not be applied over φ. So, integration (1.13) over $(0, L)$ and using the boundary conditions, we get $\begin{aligned} (1.16) & \frac{d^{2}}{d t^{2}} \int_{0}^{L} φ (x, t) d x = 0 . \end{aligned}$ By solving the differential equation (1.16) and using the initial condition, we get $\begin{aligned} (1.17) & \int_{0}^{L} φ (x, t) d x = \int_{0}^{L} φ_{0} (x) d x + t \int_{0}^{L} φ_{1} (x) d x \end{aligned}$ Now, if we introduce, new variable, $\begin{array}{r} \tilde{φ} (x, t) = φ (x, t) - \frac{1}{L} \int_{0}^{L} φ (x, t) d x, \end{array}$ we obtain $\begin{aligned} (1.18) & \int_{0}^{L} \tilde{φ} (x, t) d x = 0 . \end{aligned}$ Thus, we can use the application of Poincaré’s inequality on $\tilde{φ}$ . Thus, we work with $(\tilde{φ}, ψ)$ instead of $(φ, ψ)$ but, for simplicity, we write $(φ, ψ)$ .

1.1. Assumptions

Throughout this paper, we denote c to be a generic positive constant and we take into account the following hypotheses:

The relaxation function $θ : R^{+} \to R^{+}$ is a $C^{1}$ nonincreasing function satisfying $\begin{matrix} (1.19) & θ (0) > 0, b - \int_{0}^{\infty} θ (s) d s = ℓ > 0, \end{matrix}$ and $\begin{matrix} (1.20) & θ^{'} (t) ⩽ - ξ (t) θ (t), \forall t ⩾ 0, \end{matrix}$ where ξ is a positive nonincreasing differentiable function.

$σ : [0, L] \to [1, \infty)$ is a continuous function such that $\begin{matrix} σ_{1} : = \underset{x \in [0, L]}{essinf} σ (x), σ_{2} : = \underset{x \in [0, L]}{esssup} σ (x), \end{matrix}$ and $1 < σ_{1} ⩽ σ (x) ⩽ σ_{2} < + \infty$ . Moreover, the variable function σ satisfies the log-Hölder continuity condition; that is for any δ with $0 < δ < 1$ , there exists a constant $A > 0$ such that, $\begin{matrix} (1.21) & | σ (x) - σ (y) | ⩽ - \frac{A}{log | x - y |}, for all x, y \in Ω, with | x - y | < δ . \end{matrix}$

There exists a positive constant $m_{0}$ , such that $\begin{matrix} (1.22) & \int_{0}^{L} ψ_{0 x}^{2} (s) d x ⩽ m_{0}, \forall s ⩾ 0, \end{matrix}$ where $ψ_{0} (x, s)$ is defined in the boundary conditions (1.15). Further more, we define $\begin{matrix} L_{θ} (0, L) = {u : R^{+} \to H_{0}^{1} (0, L), \int_{0}^{+ \infty} θ (s) ‖ u_{x} ‖_{2}^{2} d s < + \infty} \end{matrix}$ which is a Hilbert space.

The time-dependent coefficient $β : [0, \infty) \to (0, \infty)$ is a nonincreasing $C^{1}$ function satisfying $\int_{0}^{\infty} β (s) d s = \infty$ .

For completeness, we state, without proof, the existence of the solutions of the system (1.13)–(1.14). First, we introduce the following spaces:

\begin{matrix} (1.23) & \begin{aligned} L_{*}^{2} (0, L) : = {u \in L^{2} (0, L) : \int_{0}^{L} u (x) d x = 0}, \\ H_{*}^{1} (0, L) : = H^{1} (0, L) \cap L_{*}^{2} (0, L), \end{aligned} \end{matrix}

and

\begin{matrix} H : = H_{*}^{1} (0, L) \times H_{*}^{1} (0, L) \times L_{*}^{2} (0, L) \times H_{0}^{1} (0, L) \times L^{2} (0, L) . \end{matrix}

Now, the existence of the solutions of the system (1.13)–(1.14) can be stated in the following proposition.

Proposition 1.1.
Assume that conditions (A1)–(A4) hold and $(φ_{0}, φ_{1}, φ_{2}, ψ_{0} (\cdot, 0), ψ_{1}) \in H$ , then the system ( 1.13 )–( 1.14 ) has a unique weak solution such that for any $T > 0$ , $\begin{array}{c} φ \in L^{\infty} ([0, T); H_{}^{1} (0, L)), \\ ψ \in L^{\infty} ([0, T); H_{0}^{1} (0, L)), \end{array}$ and* $\begin{array}{c} φ_{t} \in L^{\infty} ((0, T); H_{}^{1} (0, L)), \\ φ_{t t} \in L^{\infty} ((0, T); L_{}^{2} (0, L)), \\ ψ_{t} \in L^{\infty} ((0, T); L^{2} (0, L)) \cap L^{σ (\cdot)} ((0, L) \times (0, T)) . \end{array}$

The proof of the above existence result is similar to the proof of Theorem 2.1 in [5] by using the Galerkin approximation method. Also, the proof of the weak convergence of the nonlinear term of variable exponent type is similar to the proof in [23]. Now, we introduce the “modified” energy associated to the system (1.13)–(1.14) as follows: $\begin{array}{rcl} E (t) & = & \frac{ρ_{1}}{2} \int_{0}^{L} φ_{t}^{2} d x + \frac{ρ_{1} ρ_{2}}{2 κ} \int_{0}^{L} φ_{t t}^{2} d x \\ + \frac{ρ_{2}}{2} \int_{0}^{L} φ_{x t}^{2} d x + \frac{κ}{2} \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x \\ (1.24) & + \frac{ℓ}{2} \int_{0}^{L} ψ_{x}^{2} d x + \frac{1}{2} (θ \circ ψ_{x}) (t), \end{array}$ where, for $u : R \to L^{2} (0, L)$ and $θ : R^{+} \to R^{+}$ , $\begin{matrix} (θ \circ u) (t) = \int_{0}^{+ \infty} θ (s) \int_{0}^{L} {(u (t) - u (t - s))}^{2} d x d s . \end{matrix}$
2. Main results

In this section, we state our main results and the proofs would be in Section 4.

Theorem 2.1.
Assume that the conditions (A1)–(A4) hold and $σ_{1} = σ_{2} = 2$ . Then, there exist constants $γ_{0} \in (0, 1)$ and $δ_{1} > 0$ such that, for all $t \in R^{+}$ and for all $δ_{0} \in (0, γ_{0}]$ , then the energy functional of the system ( 1.13 )–( 1.14 ), defined in ( 1.24 ), satisfies the following decay estimate $\begin{matrix} (2.1) & E (t) ⩽ δ_{1} (1 + \int_{0}^{t} {(θ (s))}^{1 - δ_{0}} d s) e^{- δ_{0} \int_{0}^{t} (β ξ) (s) d s} + \frac{c m}{δ_{0}} \int_{t}^{+ \infty} θ (s) d s, \end{matrix}$ where $δ_{1} = max {E (0), \frac{c m}{δ_{0}} {(θ (0))}^{δ_{0}}}$ and the constant m is defined in ( 3.41 ).
Theorem 2.2.
Assume that conditions (A1)–(A4) hold, $1 < σ_{1} < 2$ and $σ_{2} \neq 2$ . Then, the energy functional defined in ( 1.24 ) satisfies for a positive constants C, $\begin{matrix} (2.2) & E (t) ⩽ C {(1 + t)}^{\frac{- 1}{ϑ}} {(ξ β)}^{- \frac{ϑ + 1}{ϑ}} [1 + \int_{0}^{t} {(ξ β)}^{\frac{ϑ + 1}{ϑ}} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s], \end{matrix}$ where $ϑ = max {\frac{2 - σ_{1}}{2 σ_{1} - 2}, \frac{σ_{2}}{2} - 1}$ .
Theorem 2.3.
Assume that conditions $(A_{1} - A_{4})$ hold, $σ_{1} ⩾ 2$ and $σ_{2} > 2$ . Then, the energy functional ( 1.24 ) satisfies for a positive constants C, $\begin{matrix} (2.3) & E (t) ⩽ C {(1 + t)}^{- \frac{2}{σ_{2} - 2}} {(ξ β)}^{- \frac{σ_{2}}{σ_{2} - 2}} (1 + \int_{0}^{t} {(1 + s)}^{\frac{2}{σ_{2} - 2}} {(ξ β)}^{\frac{σ_{2}}{σ_{2} - 2}} (s) h^{\frac{σ_{2}}{2}} d s) . \end{matrix}$

3. Technical lemmas

In this section, we state and establish several lemmas needed for the proofs of our main results.

Lemma 3.1.
The energy $E (t)$ defined in ( 1.24 ) satisfies $\begin{matrix} (3.1) & E^{'} (t) = \frac{1}{2} (θ^{'} \circ ψ_{x}) (t) - β (t) \int_{0}^{L} | ψ_{t} |^{σ (\cdot)} d x ⩽ 0 . \end{matrix}$
Proof.
The proof of (3.1) can be archived by multiplying (1.13) by $φ_{t}$ and (1.14) by $ψ_{t}$ and integrating on $(0, L)$ as follows: $\begin{matrix} (3.2) & \begin{array}{c} ρ_{1} \int_{0}^{L} φ_{t} φ_{t t} d x - κ \int_{0}^{L} φ_{t} {(φ_{x} + ψ)}_{x} d x = 0, \\ - ρ_{2} \int_{0}^{L} ψ_{t} φ_{t t x} d x - b \int_{0}^{L} ψ_{t} ψ_{x x} d x + κ \int_{0}^{L} ψ_{t} (φ_{x} + ψ) d x \\ + \int_{0}^{L} ψ_{t} \int_{0}^{+ \infty} θ (s) ψ_{x x} (t - s) d s d x + β (t) \int_{0}^{L} | ψ_{t} |^{σ (x) - 2} ψ_{t}^{2} d x = 0 . \end{array} \end{matrix}$ Using integration by parts and the boundary conditions (1.15) we get $\begin{matrix} (3.3) & \begin{array}{c} \frac{ρ_{1}}{2} \int_{0}^{L} \frac{d}{d t} {(φ_{t})}^{2} d x + κ \int_{0}^{L} φ_{x t} (φ_{x} + ψ) d x = 0, \\ + ρ_{2} \int_{0}^{L} ψ_{x t} φ_{t t} d x + \frac{b}{2} \int_{0}^{L} \frac{d}{d t} ψ_{x}^{2} d x + κ \int_{0}^{L} ψ_{t} (φ_{x} + ψ) d x \\ - \int_{0}^{L} ψ_{x t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x + β (t) \int_{0}^{L} | ψ_{t} |^{σ (x)} d x = 0 . \end{array} \end{matrix}$ From (1.13), we find that $ψ_{x} = \frac{ρ_{1}}{κ} φ_{t t} - φ_{x x}$ . This gives $ψ_{x t} = \frac{ρ_{1}}{κ} φ_{t t t} - φ_{x x t}$ . Imposing this fact in the above equation, we obtain $\begin{matrix} (3.4) & \begin{array}{c} \frac{ρ_{1}}{2} \int_{0}^{L} \frac{d}{d t} {(φ_{t})}^{2} d x + κ \int_{0}^{L} φ_{x t} (φ_{x} + ψ) d x = 0, \\ + ρ_{2} \int_{0}^{L} (\frac{ρ_{1}}{κ} φ_{t t t} - φ_{x x t}) φ_{t t} d x + \frac{b}{2} \int_{0}^{L} \frac{d}{d t} ψ_{x}^{2} d x + κ \int_{0}^{L} ψ_{t} (φ_{x} + ψ) d x \\ + \int_{0}^{L} ψ_{x t} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x - \int_{0}^{L} ψ_{x t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t) d s d x \\ + β (t) \int_{0}^{L} | ψ_{t} |^{σ (x)} d x = 0 . \end{array} \end{matrix}$ By differentiation yields $\begin{matrix} (3.5) & \begin{array}{c} κ \int_{0}^{L} φ_{x t} (φ_{x} + ψ) d x + κ \int_{0}^{L} ψ_{t} (φ_{x} + ψ) d x = \frac{κ}{2} \frac{d}{d t} \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x, \\ - \int_{0}^{L} ψ_{x t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t) d s d x = - \frac{1}{2} (\int_{0}^{+ \infty} θ (s) d s) \frac{d}{d t} \int_{0}^{L} ψ_{x}^{2} d x . \end{array} \end{matrix}$ Finally, recalling that $\begin{matrix} \frac{1}{2} \frac{d}{d t} (θ \circ ψ_{x}) (t) = \frac{1}{2} (θ^{'} \circ ψ_{x}) (t) + \int_{0}^{L} ψ_{x t} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x, \end{matrix}$ and summing up all the results with using $b - \int_{0}^{\infty} θ (s) d s = ℓ$ , we obtain (3.1). □
Lemma 3.2.
The second-order energy of the system ( 1.13 )–( 1.14 ) is defined by $\begin{aligned} E_{1} (t) = & \frac{ρ_{1}}{2} \int_{0}^{L} φ_{t t}^{2} d x + \frac{ρ_{1} ρ_{2}}{2 κ} \int_{0}^{L} φ_{t t t}^{2} d x + \frac{ρ_{2}}{2} \int_{0}^{L} φ_{x t t}^{2} d x + \frac{κ}{2} \int_{0}^{L} {(φ_{x} + ψ)}_{t}^{2} d x \\ + \frac{ℓ}{2} \int_{0}^{L} ψ_{x t}^{2} d x + \frac{1}{2} (θ \circ ψ_{x t}) (t) . \end{aligned}$ and satisfies the following uniform bound $\begin{matrix} (3.6) & \begin{array}{r} E_{1} (t) ⩽ C, \forall t ⩾ 0, \end{array} \end{matrix}$ where C is a positive constant.
Proof.
By taking the derivative of all the equations of the system (1.13)–(1.14) with respect to t, then multiplying (1.13) by $ϕ_{t t}$ and (1.14) by $ψ_{t t}$ , and integrating over $(0, L)$ , we get $\begin{matrix} (3.7) & \begin{aligned} E_{1}^{'} (t) = & \int_{0}^{L} \int_{0}^{\infty} θ^{'} (s) {| ψ_{x t} (t) - ψ_{x t} (t - s) |}^{2} d s d x \\ - β (t) \int_{0}^{L} (σ (x) - 1) | ψ_{t} |^{σ (x) - 2} ψ_{t t}^{2} d x - β^{'} (t) \int_{0}^{L} | ψ_{t} |^{σ (x) - 2} ψ_{t} ψ_{t t} d x . \end{aligned} \end{matrix}$

Using the fact $θ^{'} ⩽ 0$ , and applying Young’s inequality, we get $\begin{matrix} (3.8) & \begin{aligned} E_{1}^{'} (t) ⩽ & - β^{'} (t) \int_{0}^{L} [\frac{| ψ_{t} |^{σ (x) - 2} ψ_{t}}{\sqrt{β (t) (σ (x) - 1) | ψ_{t} |^{σ (x) - 2}}}] [ψ_{t t} \sqrt{β (t) (σ (x) - 1) | ψ_{t} |^{σ (x) - 2}}] d x \\ - β (t) \int_{0}^{L} (σ (x) - 1) | ψ_{t} |^{σ (x) - 2} ψ_{t t}^{2} d x \\ ⩽ & - \int_{0}^{L} {[\frac{| ψ_{t} |^{σ (x) - 2} ψ_{t}}{2 \sqrt{β (t) (σ (x) - 1) | ψ_{t} |^{σ (x) - 2}}}]}^{2} + {[ψ_{t t} \sqrt{β (t) (σ (x) - 1) | ψ_{t} |^{σ (x) - 2}}]}^{2} d x \\ - β (t) \int_{0}^{L} (σ (x) - 1) | ψ_{t} |^{σ (x) - 2} ψ_{t t}^{2} d x \\ ⩽ & \frac{1}{4} \int_{0}^{L} {(\frac{- β^{'} (t)}{β (t)})}^{2} \frac{β (t) | ψ_{t} |^{σ (x)}}{(σ (x) - 1)} d x . \end{aligned} \end{matrix}$ As in the argument in [24], since $\int_{0}^{\infty} β (t) d t = \infty$ , one can show that ${lim}_{t \to \infty} \frac{- β^{'} (t)}{β (t)} \neq \infty$ . In fact, if ${lim}_{t \to \infty} \frac{- β^{'} (t)}{β (t)} = \infty$ , then for a given $M > 0$ , there exists $ε > 0$ such that $\frac{- β^{'} (t)}{β (t)} ⩾ M$ for any $t > 0$ . Integrating this inequality, we find $M \int_{ε}^{t} β (s) d s ⩽ - \int_{ε}^{t} β^{'} (s) d s ⩽ β (ε)$ . This is contradiction with $\int_{M}^{\infty} β (t) d t = \infty$ . Hence, we conclude that $\frac{- β^{'} (t)}{β (t)}$ is bounded; that is for some positive constant $α_{0}$ , we have $\frac{- β^{'} (t)}{β (t)} ⩽ α_{0}$ . Thus, $\begin{matrix} (3.9) & \begin{array}{r} E_{1}^{'} (t) ⩽ \frac{{α_{0}}^{2}}{4 (σ_{1} - 1)} β (t) \int_{0}^{L} | ψ_{t} |^{σ (x)} d x = \frac{{α_{0}}^{2}}{4 (σ_{1} - 1)} (- E^{'} (t)) . \end{array} \end{matrix}$ Integrate this inequality over $(0, t)$ , we have $\begin{matrix} (3.10) & \begin{aligned} [E_{1} (t) - E_{1} (0)] & ⩽ (\frac{α_{0}^{2} β}{4 (σ_{1} - 1)}) [E (0) - E (t)] . \end{aligned} \end{matrix}$ Thus, we arrive at $\begin{matrix} (3.11) & \begin{aligned} E_{1} (t) & ⩽ (\frac{α_{0}^{2} β}{4 (σ_{1} - 1)}) E (0) + E_{1} (0) = C . \end{aligned} \end{matrix}$ This is the desired result. □
Lemma 3.3 (Gagliardo-Nirenberg interpolation inequality).

For some $c > 0$ and any $v > 2$ , we have $\begin{matrix} (3.12) & \begin{array}{r} ‖ ψ ‖_{v} ⩽ ‖ ψ_{x} ‖_{2}^{\frac{1}{2} - \frac{1}{v}} ‖ ψ ‖_{2}^{\frac{1}{2} + \frac{1}{v}}, \forall ψ \in W^{1, 2} (0, L) . \end{array} \end{matrix}$

As a consequence of the above interpolation inequality (3.12), we have for $1 < σ_{1} < 2$ , $\begin{matrix} (3.13) & \begin{aligned} ‖ ψ_{t} ‖_{\frac{σ_{1}}{σ_{1} - 1}} & ⩽ ‖ ψ_{x t} ‖_{2}^{\frac{2 - σ_{1}}{2 σ_{1}}} ‖ ψ_{t} ‖_{2}^{\frac{3 σ_{1} - 2}{2 σ_{1}}} ⩽ {(E_{1}^{'} (t))}^{\frac{2 - σ_{1}}{2 σ_{1}}} ‖ ψ_{t} ‖_{2}^{\frac{3 σ_{1} - 2}{2 σ_{1}}} \\ ⩽ {(E_{1}^{'} (t))}^{\frac{2 - σ_{1}}{2 σ_{1}}} ‖ ψ_{t} ‖_{2}^{\frac{3 σ_{1} - 2}{2 σ_{1}}} ⩽ {(C)}^{\frac{2 - σ_{1}}{4 σ_{1}}} {‖ ψ_{t} ‖_{2}^{2}}^{\frac{3 σ_{1} - 2}{4 σ_{1}}} \\ ⩽ c {(E (t))}^{\frac{3 σ_{1} - 2}{4 σ_{1}}} . \end{aligned} \end{matrix}$

Lemma 3.4.
Under the assumptions $(A_{2})$ and $(A_{4})$ , the following do hold: $\begin{matrix} (3.14) & \begin{aligned} c β (t) \int_{0}^{L} ψ_{t}^{2} d x ⩽ - c E^{'} (t), if σ_{1} = σ_{2} = 2 . \\ c β (t) \int_{0}^{L} ψ_{t}^{2} d x ⩽ c ε β E (t) - C_{ε} E^{- \hat{ϑ}} (E^{'} (t)), if σ_{1} ⩾ 2, σ_{2} > 2 . \\ c β (t) \int_{0}^{L} ψ_{t}^{2} d x ⩽ c ε_{1} β E + c ε_{2} β E - C_{ε_{1}} (E^{'} (t)) E^{- \hat{ϑ}} - C_{ε_{2}} (E^{'} (t)) E^{- \hat{ϑ}}, \\ if 1 < σ_{1} < 2, σ_{2} \neq 2, \end{aligned} \end{matrix}$ where $\hat{ϑ} = \frac{σ_{2}}{2} - 1 > 0$ .
Proof.
The proof of ${(3.14)}_{1}$ is direct by imposing $σ (x) = 2$ and combining with (3.1). To prove ${(3.14)}_{2}$ , we set the following partitions $\begin{array}{r} (3.15) & Ω_{1} = {x \in [0, L] : | ψ_{t} | ⩾ 1} and Ω_{2} = {x \in [0, L] : | ψ_{t} | < 1} . \end{array}$ Using of Hölder and Young inequalities and (3.1), we obtain for $Ω_{1}$ , $\begin{aligned} (3.16) & c β (t) \int_{Ω_{1}} ψ_{t}^{2} d x ⩽ c β (t) \int_{0}^{L} | ψ_{t} |^{σ (x)} d x ⩽ - c E^{'} (t), \end{aligned}$ and for $Ω_{2}$ , we get $\begin{aligned} c β (t) \int_{Ω_{2}} ψ_{t}^{2} d x & ⩽ c β (t) {(\int_{Ω_{2}} | ψ_{t} |^{σ_{2}} d x)}^{\frac{2}{σ_{2}}} \\ ⩽ c β (t) {(\int_{Ω_{2}} | ψ_{t} |^{σ (x)} d x)}^{\frac{2}{σ_{2}}} ⩽ β^{1 - \frac{2}{σ_{2}}} {(β \int_{0}^{L} | ψ_{t} |^{σ (x)} d x)}^{\frac{2}{σ_{2}}} \\ ⩽ c β^{1 - \frac{2}{σ_{2}}} {(- E^{'} (t))}^{\frac{2}{σ_{2}}} . \end{aligned}$ Note that $\begin{matrix} β^{1 - \frac{2}{σ_{2}}} {(- E^{'} (t))}^{\frac{2}{σ_{2}}} = \frac{β^{1 - \frac{2}{σ_{2}}} E^{\hat{ϑ}} {(- E^{'} (t))}^{\frac{2}{σ_{2}}}}{E^{\hat{ϑ}}}, \end{matrix}$ where $\hat{ϑ} = \frac{σ_{2}}{2} - 1 > 0$ . Young’s inequality gives for a positive constant ε, $\begin{matrix} \frac{β^{1 - \frac{2}{σ_{2}}} E^{\hat{ϑ}} {(- E^{'} (t))}^{\frac{2}{σ_{2}}}}{E^{\hat{ϑ}}} ⩽ ε β E^{\hat{ϑ} + 1} + C_{ε} (- E^{'} (t)) . \end{matrix}$ Therefore, we have $\begin{array}{rcl} (3.17) & c β (t) \int_{Ω_{2}} ψ_{t}^{2} d x & ⩽ & \frac{ε β E^{\hat{ϑ} + 1} + C_{ε} (- E^{'} (t))}{E^{\hat{ϑ}}} . \end{array}$ Combining (3.16) and (3.17), the proof of ${(3.14)}_{2}$ is completed. To prove ${(3.14)}_{3}$ , we discus two cases:

Case 1: If $σ_{2} ⩽ 2$ , then on $Ω_{2}$ , we have $\begin{array}{rcl} (3.18) & c β (t) \int_{Ω_{2}} ψ_{t}^{2} d x & ⩽ & c β (t) \int_{0}^{L} | ψ_{t} |^{σ (x)} d x ⩽ - c E^{'} (t) . \end{array}$ However, on $Ω_{1}$ , by using the estimate in (3.13), we have $\begin{aligned} c β (t) \int_{Ω_{1}} ψ_{t}^{2} d x & = c β (t) \int_{Ω_{1}} ψ_{t} ψ_{t} d x ⩽ c β {(\int_{Ω_{1}} | ψ_{t} |^{σ_{2}} d x)}^{\frac{1}{σ_{2}}} {(\int_{Ω_{1}} | ψ_{t} |^{\frac{σ_{2}}{σ_{1} - 1}} d x)}^{\frac{σ_{1} - 1}{σ_{2}}} \\ ⩽ c β^{1 - \frac{1}{σ_{1}}} {(β \int_{0}^{L} | ψ_{t} |^{σ (x)} d x)}^{\frac{1}{σ_{2}}} ‖ ψ_{t} ‖_{\frac{σ_{1}}{σ_{1} - 1}} \\ ⩽ c β {(- E^{'} (t))}^{\frac{1}{σ_{2}}} {(E (t))}^{\frac{3 σ_{1} - 2}{4 σ_{1}}} . \end{aligned}$ Using Young’s inequality for $γ = σ_{1}$ and $γ^{} = \frac{σ_{1}}{σ_{1} - 1}$ , we have for a positive constant ε, $\begin{aligned} (3.19) & c β (t) \int_{Ω_{1}} ψ_{t}^{2} d x ⩽ ε β {[E (t)]}^{\frac{3 σ_{1} - 2}{4 (σ_{1} - 1)}} + C_{ε} (- E^{'} (t)) . \end{aligned}$ Combining (3.18) and (3.19), we obtain $\begin{aligned} (3.20) & c β (t) \int_{0}^{L} ψ_{t}^{2} d x ⩽ - c E^{'} (t) + c ε β {[E (t)]}^{\frac{3 σ_{1} - 2}{4 (σ_{1} - 1)}} + C_{ε} (- E^{'} (t)) . \end{aligned}$ Since $\frac{3 σ_{1} - 2}{4 (σ_{1} - 1)} > 1$ , then we have $\begin{array}{c} (3.21) & c β (t) \int_{0}^{L} ψ_{t}^{2} d x ⩽ - c E^{'} (t) + c ε β E (t) + C_{ε} (- E^{'} (t)) . \end{array}$ Case 2: If $σ_{2} > 2$ , on $Ω_{2}$ , we have $\begin{matrix} (3.22) & \begin{aligned} c β (t) \int_{Ω_{2}} ψ_{t}^{2} d x & ⩽ c β (t) {(\int_{Ω_{2}} | ψ_{t} |^{σ_{2}} d x)}^{\frac{2}{σ_{2}}} ⩽ c β {(\int_{Ω_{2}} | ψ_{t} |^{σ (x)} d x)}^{\frac{2}{σ_{2}}} \\ ⩽ c β^{1 - \frac{2}{σ_{1}}} {(β \int_{0}^{L} | ψ_{t} |^{σ (x)} d x)}^{\frac{2}{σ_{2}}} ⩽ c β {(- E^{'} (t))}^{\frac{2}{σ_{2}}}, \end{aligned} \end{matrix}$ and on $Ω_{1}$ , we have $\begin{array}{c} (3.23) & c β (t) \int_{Ω_{1}} ψ_{t}^{2} d x ⩽ c β {(- E^{'} (t))}^{\frac{1}{σ_{2}}} {(E (t))}^{\frac{3 σ_{1} - 2}{4 σ_{1}}} . \end{array}$ Now, a combination of (3.22) and (3.23), we find $\begin{array}{rcl} (3.24) & c β (t) \int_{0}^{L} ψ_{t}^{2} d x & ⩽ & c β {(- E^{'} (t))}^{\frac{2}{σ_{2}}} + c β {(- E^{'} (t))}^{\frac{1}{σ_{2}}} {(E (t))}^{\frac{3 σ_{1} - 2}{4 σ_{1}}} . \end{array}$ Multiply the last inequality by $E^{\hat{ϑ}}$ where $\hat{ϑ} = \frac{σ_{2}}{2} - 1 > 0$ , we get $\begin{array}{rcl} (3.25) & c E^{\hat{ϑ}} β (t) \int_{0}^{L} ψ_{t}^{2} d x & ⩽ & c β E^{\hat{ϑ}} {(- E^{'} (t))}^{\frac{2}{σ_{2}}} + c β E^{\hat{ϑ}} {(- E^{'} (t))}^{\frac{1}{σ_{2}}} {(E (t))}^{\frac{3 σ_{1} - 2}{4 σ_{1}}} . \end{array}$ Using Young’s inequality twice, to get for a positive constant $ε_{1}, ε_{2} > 0$ $\begin{aligned} c E^{\hat{ϑ}} β (t) \int_{0}^{L} ψ_{t}^{2} d x & ⩽ c ε_{1} β E^{\hat{ϑ} + 1} + C_{ε_{1}} (- E^{'} (t)) + c C_{ε_{2}} (- E^{'} (t)) + c β ε_{2} E^{\frac{σ_{1}}{σ_{1} - 1} (\frac{3 σ_{1} - 2}{4 σ_{1}} + \hat{ϑ})} \\ ⩽ c β ε_{1} E^{\hat{ϑ} + 1} + C_{ε_{1}} (- E^{'} (t)) + c C_{ε_{2}} (- E^{'} (t)) + c β ε_{2} E^{\frac{σ_{2}}{2}} . \end{aligned}$ This is because $\frac{σ_{1}}{σ_{1} - 1} (\frac{3 σ_{1} - 2}{4 σ_{1}} + \hat{ϑ}) ⩾ \frac{σ_{2}}{2}$ and $E (t)$ is nonincreasing function. Now, we have $\begin{array}{rcl} (3.26) & c β (t) \int_{0}^{L} ψ_{t}^{2} d x & ⩽ & c ε_{1} β E + c ε_{2} β E - C_{ε_{1}} (E^{'} (t)) E^{- \hat{ϑ}} - C_{ε_{2}} (E^{'} (t)) E^{- \hat{ϑ}}, \end{array}$ where $\hat{ϑ} = \frac{σ_{2}}{2} - 1 > 0$ . This is the end of the proof. □
Lemma 3.5.
Assume that* (A2) and (A4) hold, then for any $λ, η > 0$ , we have $\begin{aligned} - β (t) \int_{0}^{L} ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x ⩽ & (c_{1} + c_{e}) λ \int_{0}^{L} | ψ_{x} |^{2} d x \\ + β (t) \int_{Ω } C_{λ} (x) | ψ_{t} |^{2 σ (x) - 2} d x \\ (3.27) & + β (t) \int_{Ω } C_{λ} (x) | ψ_{t} |^{σ (x)} d x, \end{aligned}$ and $\begin{aligned} - β (t) \int_{0}^{L} (φ_{x} + ψ) | ψ_{t} |^{σ (x) - 2} ψ_{t} d x \\ ⩽ c λ \int_{0}^{L} | ψ_{x} |^{2} d x + β (t) \int_{Ω } C_{η} (x) | ψ_{t} |^{2 σ (x) - 2} d x \\ (3.28) & + β (t) \int_{Ω } C_{η} (x) | ψ_{t} |^{σ (x)} d x . \end{aligned}$
Proof.
To prove the estimate (3.27), we consider the following partition $\begin{matrix} Ω_{} = {x \in [0, L] : σ (x) < 2}, Ω_{ } = {x \in [0, L] : σ (x) ⩾ 2} . \end{matrix}$ Then, $\begin{aligned} - β (t) \int_{0}^{L} ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x = & - β (t) \int_{Ω } ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x \\ (3.29) & - β (t) \int_{Ω } ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x . \end{aligned}$ Since β is nonincreasing function, we find $\begin{aligned} (3.30) & - β (t) \int_{Ω } ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x ⩽ λ c_{e} \int_{Ω } ψ_{x}^{2} d x + β (t) \int_{Ω } C_{λ} (x) | ψ_{t} |^{2 σ (x) - 2} d x, \end{aligned}$ where $c_{e}$ is the embedding constant. On the other hand, $\begin{aligned} (3.31) & - β (t) \int_{Ω } ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x ⩽ λ β (t) \int_{Ω } | ψ_{t} |^{σ (x)} d x + β (t) \int_{Ω } C_{λ} (x) | ψ_{t} |^{σ (x)} d x . \end{aligned}$ We estimate the first integral in (3.31) as follows: $\begin{matrix} (3.32) & \begin{aligned} λ β (t) \int_{Ω } | ψ |^{σ (x)} d x & ⩽ λ β (t) \int_{0}^{L} | ψ |^{σ (x)} d x \\ = λ β (t) \int_{Ω_{+}} | ψ |^{σ (x)} d x + λ β (t) \int_{Ω_{-}} | ψ |^{σ (x)} d x \\ ⩽ λ β (t) \int_{Ω_{+}} | ψ |^{σ_{2}} d x + λ β (t) \int_{Ω_{-}} | ψ |^{σ_{1}} d x \\ ⩽ λ β (t) \int_{0}^{L} | ψ |^{σ_{2}} d x + λ β (t) \int_{0}^{L} | ψ |^{σ_{1}} d x \\ ⩽ λ c_{e}^{σ_{1}} ‖ ψ_{x} ‖_{2}^{σ_{1}} + λ c_{e}^{σ_{2}} ‖ ψ_{x} ‖_{2}^{σ_{2}} \\ ⩽ λ (c_{e}^{σ_{1}} ‖ ψ_{x} ‖_{2}^{σ_{1} - 2} + c_{e}^{σ_{2}} ‖ ψ_{x} ‖_{2}^{σ_{2} - 2}) ‖ ψ_{x} ‖_{2}^{2} \\ ⩽ λ (c_{e}^{σ_{1}} {(\frac{2}{ℓ} E (0))}^{σ_{1} - 2} + c_{e}^{σ_{2}} {(\frac{2}{ℓ} E (0))}^{σ_{2} - 2}) ‖ ψ_{x} ‖_{2}^{2} \\ ⩽ c_{1} λ ‖ ψ_{x} ‖_{2}^{2}, \end{aligned} \end{matrix}$ where $\begin{matrix} Ω_{+} = {x \in [0, L] : | ψ (x, t) | ⩾ 1}, Ω_{-} = {x \in [0, L] : | ψ (x, t) | < 1} \end{matrix}$ and $\begin{matrix} (3.33) & \begin{array}{r} c_{1} = (c_{e}^{σ_{1}} {(\frac{2}{ℓ} E (0))}^{σ_{1} - 2} + c_{e}^{σ_{2}} {(\frac{2}{ℓ} E (0))}^{σ_{2} - 2}) . \end{array} \end{matrix}$ Therefore, (3.31) becomes $\begin{aligned} (3.34) & - β (t) \int_{Ω } ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x ⩽ c_{1} λ \int_{0}^{L} | ψ_{x} |^{2} d x + β (t) \int_{Ω } C_{λ} (x) | ψ_{t} |^{σ (x)} d x . \end{aligned}$ Combining (3.30) and (3.34), the proof of (3.27) is finished and the proof of the estimate (3.28) can be achieved by following the same steps in the proof of (3.27). □
Lemma 3.6.
For any* $λ > 0$ , we have the following estimate: $\begin{matrix} (3.35) & \begin{aligned} \int_{0}^{L} ψ_{t} | ψ_{t} |^{σ (x) - 2} \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ ⩽ C_{0} λ {(b - ℓ)}^{σ_{1} - 1} (θ \circ ψ_{x}) (t) + \int_{0}^{L} c_{λ} (x) | ψ_{t} |^{σ (x)} d x . \end{aligned} \end{matrix}$
Proof.
To prove (3.35), we have for almost every $x \in [0, L]$ fixed, $\begin{matrix} (3.36) & \begin{aligned} \int_{0}^{+ \infty} θ (s) | ψ (t) - ψ (t - s) | d s \\ ⩽ {(\int_{0}^{+ \infty} θ (s) d s)}^{\frac{σ (x) - 1}{σ (x)}} {(\int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ (x)} d s)}^{\frac{1}{σ (x)}} \\ \times {(b - ℓ)}^{\frac{σ (x) - 1}{σ (x)}} {(\int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ (x)} d s)}^{\frac{1}{σ (x)}} . \end{aligned} \end{matrix}$ Therefore, for almost every $x \in [0, L]$ , we have $\begin{matrix} (3.37) & {| \int_{0}^{+ \infty} θ (s) | ψ (t) - ψ (t - s) | d s |}^{σ (x)} ⩽ {(b - ℓ)}^{σ_{1} - 1} \int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ (x)} d s . \end{matrix}$ Recalling Young’s, Hölder’s and Poincaré’s inequalities, we get $\begin{matrix} (3.38) & \begin{aligned} \int_{0}^{L} | ψ_{t} |^{σ (x) - 2} ψ_{t} \int_{0}^{+ \infty} θ (s) (ψ (t) - ψ (t - s)) d s d x \\ ⩽ λ \int_{0}^{L} {| \int_{0}^{+ \infty} θ (s) (ψ (t) - ψ (t - s)) d s |}^{σ (x)} d x + \int_{0}^{L} c_{λ} (x) | ψ_{t} |^{σ (x)} d x \\ ⩽ λ {(b - ℓ)}^{σ_{1} - 1} \int_{0}^{L} \int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ (x)} d s d x + \int_{0}^{L} c_{λ} (x) | ψ_{t} |^{σ (x)} d x . \end{aligned} \end{matrix}$ Similarly, using the above partition $\begin{matrix} Ω_{+} = {x \in [0, L] : | ψ (x, t) | ⩾ 1}, Ω_{-} = {x \in [0, L] : | ψ (x, t) | < 1}, \end{matrix}$ we have $\begin{matrix} (3.39) & \begin{aligned} \int_{0}^{L} \int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ (x)} d s d x \\ ⩽ \int_{Ω_{+}} \int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ_{2}} d s d x \\ + \int_{Ω_{-}} \int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}^{σ_{1}} d s d x \\ ⩽ \int_{0}^{+ \infty} θ (s) {‖ ψ (t) - ψ (t - s) ‖}_{σ_{2}}^{σ_{2}} d s + \int_{0}^{+ \infty} θ (s) {‖ ψ (t) - ψ (t - s) ‖}_{σ_{1}}^{σ_{1}} d s \\ ⩽ [c_{e}^{σ_{2}} Υ_{0}^{\frac{σ_{2} - 2}{2}} + c_{e}^{σ_{1}} Υ_{0}^{\frac{σ_{1} - 2}{2}}] \int_{0}^{+ \infty} θ (s) {| ψ (t) - ψ (t - s) |}_{2}^{2} d s, \end{aligned} \end{matrix}$ where $Υ_{0} : = ‖ ψ (t) - ψ (t - s) ‖_{2}^{2}$ . Therefore, equation (3.38) becomes $\begin{matrix} (3.40) & \begin{aligned} \int_{0}^{L} | ψ_{t} |^{σ (x) - 2} ψ_{t} \int_{0}^{+ \infty} θ (s) (ψ (t) - ψ (t - s)) d s d x \\ ⩽ C_{0} λ {(b - ℓ)}^{σ_{1} - 1} (θ \circ ψ_{x}) (t) + \int_{0}^{L} c_{λ} (x) | ψ_{t} |^{σ (x)} d x, \end{aligned} \end{matrix}$ where $C_{0} = c_{e}^{σ_{2}} Υ_{0}^{\frac{σ_{2} - 2}{2}} + c_{e}^{σ_{1}} Υ_{0}^{\frac{σ_{1} - 2}{2}}$ and this completes the proof of the estimate (3.35). □
Lemma 3.7.
For $ψ \in L_{l o c}^{2} ([0, + \infty), H_{0}^{1} (0, L))$ , we have $\begin{matrix} \int_{0}^{L} {(\int_{0}^{+ \infty} θ (s) (ψ (t) - ψ (t - s)) d s)}^{2} d x ⩽ (b - ℓ) C_{p}^{2} (θ \circ ψ_{x}) (t), \end{matrix}$ where $C_{p}$ is the Poincaré constant.
Proof.
The result follows easily by applying Cauchy-Schwarz’ and Poincaré’s inequalities as follows: $\begin{aligned} \int_{0}^{L} {(\int_{0}^{+ \infty} θ (s) (ψ (t) - ψ (t - s)) d s)}^{2} d x \\ = \int_{0}^{L} {(\int_{0}^{+ \infty} \sqrt{θ (s)} \sqrt{θ (s)} (ψ (t) - ψ (t - s)) d s)}^{2} d x \\ ⩽ (\int_{0}^{L} θ (s) d s) \int_{0}^{L} \int_{0}^{+ \infty} θ (s) {(ψ (t) - ψ (t - s))}^{2} d s d x \\ ⩽ (b - ℓ) C_{p}^{2} (θ \circ ψ_{x}) (t) . \end{aligned}$ □
Lemma 3.8.
There exists a positive constant m such that $\begin{matrix} (3.41) & \int_{0}^{L} \int_{t}^{+ \infty} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s d x ⩽ m \int_{t}^{+ \infty} θ (s) d s . \end{matrix}$
Proof.
Using (1.22), (1.24), the definition $ψ (x, - t) = ψ_{0} (x, t)$ , $t ⩾ 0$ , $x \in [0, L]$ , and the fact that E is nonincreasing imply that $\begin{aligned} \int_{0}^{L} {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d x & ⩽ 4 sup_{τ \in R} \int_{0}^{L} ψ_{x}^{2} (τ) d x \\ ⩽ 4 sup_{τ < 0} \int_{0}^{L} ψ_{x}^{2} (τ) d x + 4 sup_{τ > 0} \int_{0}^{L} ψ_{x}^{2} (τ) d x \\ ⩽ 4 sup_{τ > 0} \int_{0}^{L} ψ_{0 x}^{2} (τ) d x + c E (0) . \end{aligned}$ Thanks to (A3), there exists a positive constant $m = c (m_{0}^{2} + E (0))$ where $m_{0}$ is defined in (1.22). Therefore, the proof of (3.41) is established. □
Lemma 3.9.
The functional $\begin{matrix} χ (t) = - ρ_{1} \int_{0}^{L} φ_{t} φ d x, \end{matrix}$ satisfies, along the solution of ( 1.13 )–( 1.14 ), for any $ε_{1} > 0$ , $\begin{array}{r} (3.42) & χ^{'} (t) ⩽ - ρ_{1} \int_{0}^{L} φ_{t}^{2} d x + c ε_{1} \int_{0}^{L} ψ_{x}^{2} d x + c \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x . \end{array}$
Proof.
Taking the derivative of $χ (t)$ and using (1.13), we have $\begin{matrix} χ^{'} (t) = - ρ_{1} \int_{0}^{L} φ_{t}^{2} d x + κ \int_{0}^{L} (φ_{x} + ψ) φ_{x} d x . \end{matrix}$ By using Young’s and Poincaré’s inequalities, we get for any $ε_{1} > 0$ , $\begin{array}{rcl} κ \int_{0}^{L} (φ_{x} + ψ) φ_{x} d x & = & κ \int_{0}^{L} (φ_{x} + ψ) (φ_{x} + ψ - ψ) d x \\ = & κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - κ \int_{0}^{L} (φ_{x} + ψ) ψ d x \\ ⩽ & κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + ε_{1} \int_{0}^{L} ψ^{2} d x + c_{ε_{1}} \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x \\ ⩽ & c \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + c ε_{1} \int_{0}^{L} ψ_{x}^{2} d x, \end{array}$ which implies (3.42). □
Lemma 3.10.
The functional $\begin{matrix} F (t) = ρ_{2} \int_{0}^{L} φ_{x t} φ_{x} d x, \end{matrix}$ satisfies along the solution of ( 1.13 )–( 1.14 ), for any $t > 0$ , $\begin{array}{rcl} F^{'} (t) & ⩽ & - \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} φ_{t t}^{2} d x - \frac{(b - ℓ)}{4} \int_{0}^{L} ψ_{x}^{2} d x + c \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x \\ (3.43) & + c (θ \circ ψ_{x}) + c β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x + c β (t) \int_{Ω } | ψ_{t} |^{σ (x)} d x \end{array}$
Proof.
To prove the estimate (3.43), we multiply (1.14) by ψ and use the fact that $ψ_{x} = \frac{ρ_{1}}{κ} φ_{t t} - φ_{x x}$ , to obtain $\begin{aligned} \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} φ_{t t}^{2} d x - ρ_{2} \int_{0}^{L} φ_{t t} φ_{x x} d x + b \int_{0}^{L} ψ_{x}^{2} d x \\ + k \int_{0}^{L} (φ_{x} + ψ) ψ d x - \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x \\ (3.44) & + β (t) \int_{0}^{L} ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x = 0 . \end{aligned}$ Clearly, $\begin{array}{rcl} \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) ψ_{x} (s) d s d x & = & \int_{0}^{+ \infty} θ (s) d s \int_{0}^{L} ψ_{x}^{2} d x \\ (3.45) & + \int_{0}^{L} ψ_{x} \int_{0}^{t} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x, \end{array}$ which, together with (3.44), gives $\begin{array}{rcl} F^{'} (t) & = & ρ_{2} \int_{0}^{L} φ_{t t x} φ_{x} d x + ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x \\ = & - \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} φ_{t t}^{2} d x - ℓ \int_{0}^{L} ψ_{x}^{2} d x - κ \int_{0}^{L} (φ_{x} + ψ) ψ d x \\ + ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x + \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x \\ (3.46) & - β (t) \int_{0}^{L} ψ | ψ_{t} |^{σ (x) - 2} ψ_{t} d x . \end{array}$ For any $ε_{2} > 0$ , we get from Young’s inequality and Poincaré’s inequality that $\begin{array}{r} (3.47) & - κ \int_{0}^{L} (φ_{x} + ψ) ψ d x ⩽ ε_{2} \int_{0}^{L} ψ_{x}^{2} d x + \frac{c}{ε_{2}} \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x . \end{array}$ Using Lemma 3.7, we find that $\begin{aligned} \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x \\ ⩽ ε_{2} \int_{0}^{L} ψ_{x}^{2} d x + \frac{c}{ε_{2}} \int_{0}^{L} {(\int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s)}^{2} d x \\ (3.48) & ⩽ ε_{2} \int_{0}^{L} ψ_{x}^{2} d x + \frac{c}{ε_{2}} (θ \circ ψ_{x}) . \end{aligned}$ Inserting (3.47), (3.27) and (3.48) into (3.46), we get $\begin{array}{rcl} F^{'} (t) & ⩽ & - \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} φ_{t t}^{2} d x - [(b - ℓ) - (c_{1} + c_{e}) λ - 2 ε_{2}] \int_{0}^{L} ψ_{x}^{2} d x + \frac{c}{ε_{2}} \int_{0}^{L} {(φ_{x} + ψ)}^{2} \\ + ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x + \frac{c}{ε_{2}} (θ \circ ψ_{x}) + c β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x + c β (t) \int_{Ω } | ψ_{t} |^{σ (x)} d x . \end{array}$ Choosing $λ = \frac{b - ℓ}{2 (c_{1} + c_{e})}$ and then selecting $ε_{2} = \frac{b - ℓ}{8}$ , we get (3.43). □
Lemma 3.11.
Define the functional $μ (t)$ by $\begin{matrix} μ (t) = μ_{1} (t) + \frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) μ_{2} (t), \end{matrix}$ where $\begin{matrix} μ_{1} (t) = - ρ_{2} \int_{0}^{L} φ_{x t} (φ_{x} + ψ) d x - \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{x t} ψ d x - \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x, \end{matrix}$ and $\begin{matrix} μ_{2} (t) = \frac{ρ_{2}}{b - ℓ} \int_{0}^{L} φ_{x t} \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x . \end{matrix}$

Then, for any $t ⩾ t_{1}$ , the functional $μ (t)$ satisfies, along the solution of ( 1.13 )–( 1.14 ) and for any $ε_{4} > 0$ and $ε_{6} > 0$ , $\begin{array}{rcl} μ^{'} (t) & ⩽ & - ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x - \frac{κ}{4} \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + ε_{4} \int_{0}^{L} ψ_{x}^{2} d x \\ (3.49) & + c (θ \circ ψ_{x}) + c β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x + c β (t) \int_{Ω } | ψ_{t} |^{σ (x)} d x . \end{array}$
Proof.
To prove the estimate (3.49), we multiply (1.14) by $(φ_{x} + ψ)$ and use integration by parts to get $\begin{array}{c} - ρ_{2} \int_{0}^{L} φ_{t t x} (φ_{x} + ψ) d x + b \int_{0}^{L} ψ_{x} {(φ_{x} + ψ)}_{x} d x + κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x \\ - \int_{0}^{L} {(φ_{x} + ψ)}_{x} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x + β (t) \int_{0}^{L} (φ_{x} + ψ) ψ_{t} | ψ_{t} |^{σ (x) - 2} d x = 0, \end{array}$ which, using ${(φ_{x} + ψ)}_{x} = \frac{ρ_{1}}{κ} φ_{t t}$ , implies that $\begin{array}{c} - ρ_{2} \int_{0}^{L} φ_{t t x} (φ_{x} + ψ) d x + \frac{b ρ_{1}}{κ} \int_{0}^{L} ψ_{x} φ_{t t} d x + κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x \\ (3.50) & - \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x + β (t) \int_{0}^{L} (φ_{x} + ψ) ψ_{t} | ψ_{t} |^{σ (x) - 2} d x = 0 . \end{array}$ It follows that $\begin{matrix} φ_{t t x} (φ_{x} + ψ) = \frac{d}{d t} [φ_{x t} (φ_{x} + ψ)] - φ_{x t} {(φ_{x} + ψ)}_{t}, \end{matrix}$ then we infer from (3.50) that $\begin{array}{c} \frac{d}{d t} [- ρ_{2} \int_{0}^{L} φ_{x t} (φ_{x} + ψ) d x] + ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x \\ + ρ_{2} \int_{0}^{L} φ_{x t} ψ_{t} d x + \frac{b ρ_{1}}{κ} \int_{0}^{L} ψ_{x} φ_{t t} d x \\ + κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x \\ (3.51) & + β (t) \int_{0}^{L} (φ_{x} + ψ) ψ_{t} | ψ_{t} |^{σ (x) - 2} d x = 0 . \end{array}$

In view of $\frac{d}{d t} (φ_{x t} ψ) = φ_{t t x} ψ + φ_{x t} ψ_{t}$ , we have $\begin{array}{r} (3.52) & - \frac{b ρ_{1}}{κ} \frac{d}{d t} \int_{0}^{L} φ_{x t} ψ d x = - \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{t t x} ψ d x - \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{x t} ψ_{t} d x . \end{array}$ Combining (3.51) and (3.52), we obtain $\begin{matrix} (3.53) & \begin{aligned} \frac{d}{d t} [- ρ_{2} \int_{0}^{L} φ_{x t} (φ_{x} + ψ) d x - \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{x t} ψ d x] \\ = - ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x - κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x \\ - b (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) \int_{0}^{L} φ_{x t} ψ_{t} d x + \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t t} \int_{0}^{\infty} θ (s) ψ_{x} (t - s) d s d x \\ - β (t) \int_{0}^{L} (φ_{x} + ψ) ψ_{t} | ψ_{t} |^{σ (x) - 2} d x . \end{aligned} \end{matrix}$ The direct computation yields $\begin{array}{c} \frac{d}{d t} [\frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x] \\ = \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t t} \int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s d x + \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t} \int_{0}^{+ \infty} θ^{'} (s) ψ_{x} (t - s) d s d x \\ (3.54) & + \frac{c ρ_{1}}{κ} \int_{0}^{L} φ_{t} ψ_{x} d x . \end{array}$ Then it follows from (3.53) and (3.54) that $\begin{array}{rcl} μ_{1}^{'} (t) & = & - ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x - κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - b (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) \int_{0}^{L} φ_{x t} ψ_{t} d x \\ - \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t} \int_{0}^{+ \infty} θ^{'} (s) (ψ_{x} (t - s) - ψ_{x} (t)) d s d x - \frac{c ρ_{1}}{κ} \int_{0}^{L} φ_{t} ψ_{x} d x \\ (3.55) & - β (t) \int_{0}^{L} (φ_{x} + ψ) ψ_{t} | ψ_{t} |^{σ (x) - 2} d x . \end{array}$ Obviously, we have $\begin{array}{rcl} (3.56) & μ_{2}^{'} (t) & = & \frac{ρ_{2}}{α} [\int_{0}^{L} φ_{t t x} \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x + α \int_{0}^{L} φ_{x t} ψ_{t} d x], \end{array}$ where $α : = b - ℓ$ . By using (1.14), we derive $\begin{array}{c} ρ_{2} \int_{0}^{L} φ_{t t x} \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ = \int_{0}^{L} [- b ψ_{x x} + κ (φ_{x} + ψ)] \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ + [\int_{0}^{L} \int_{0}^{+ \infty} θ (s) ψ_{x x} (t - s) d s + β (t) ψ_{t} | ψ_{t} |^{σ (x) - 2}] \\ \times \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ = b \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x \\ + κ \int_{0}^{L} (φ_{x} + ψ) \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ - \int_{0}^{L} (\int_{0}^{+ \infty} θ (s) ψ_{x} (t - s) d s) \\ \times (\int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s) d x \\ + β (t) \int_{0}^{L} ψ_{t} | ψ_{t} |^{σ (x) - 2} \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ = (b - α) \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x \\ + κ \int_{0}^{L} (φ_{x} + ψ) \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \\ + \int_{0}^{L} {(\int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s)}^{2} d x, \\ + β (t) \int_{0}^{L} ψ_{t} | ψ_{t} |^{σ (x) - 2} \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x \end{array}$ which, along with (3.56), implies that $\begin{array}{r} (3.57) & μ_{2}^{'} (t) = \sum_{i = 1}^{5} I_{i} (t) + ρ_{2} \int_{0}^{L} φ_{x t} ψ_{t} d x, \end{array}$ where $\begin{array}{rcl} I_{1} (t) & = & \frac{ρ_{2} (b - α)}{α} \int_{0}^{L} ψ_{x} \int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s d x, \\ I_{2} (t) & = & \frac{ρ_{2} κ}{α} \int_{0}^{L} (φ_{x} + ψ) \int_{0}^{+ \infty} θ (s) [ψ (t) - ψ (t - s)] d s d x, \\ I_{3} (t) & = & \frac{ρ_{2}}{α} \int_{0}^{L} {(\int_{0}^{+ \infty} θ (s) [ψ_{x} (t) - ψ_{x} (t - s)] d s)}^{2} d x, \\ I_{4} (t) & = & \frac{ρ_{2}}{α} β (t) \int_{0}^{L} ψ_{t} | ψ_{t} |^{σ (x) - 2} \int_{0}^{\infty} θ (s) [ψ (t) - ψ (t - s)] d s d x . \end{array}$ Multiplying (3.57) by $\frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b})$ , and adding the result to (3.55), we get $\begin{array}{rcl} μ^{'} (t) & = & - ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x - κ \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + \frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) \sum_{i = 1}^{5} I_{i} (t) \\ - \frac{c ρ_{1}}{κ} \int_{0}^{L} φ_{t} ψ_{x} d x - \frac{ρ_{1}}{κ} \int_{0}^{L} φ_{t} \int_{0}^{+ \infty} θ^{'} (s) [ψ_{x} (t - s) - ψ_{x} (t)] d s d x \\ (3.58) & - β (t) \int_{0}^{L} (φ_{x} + ψ) ψ_{t} | ψ_{t} |^{σ (x) - 2} d x . \end{array}$ By using Lemma 3.7, Young’s and Poincaré’s inequalities and the estimate (3.35), we get the following estimates for any $t ⩾ t_{1}$ , and for any $ε_{i} > 0$ ( $i = 4, 5$ ) $\begin{array}{c} (3.59) & \frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) I_{1} ⩽ ε_{4} \int_{0}^{L} ψ_{x}^{2} d x + \frac{c}{ε_{4}} (θ \circ ψ_{x}), \\ (3.60) & \frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) I_{2} ⩽ ε_{5} \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + \frac{c}{ε_{5}} (θ \circ ψ_{x}), \\ (3.61) & \frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) I_{3} ⩽ c (θ \circ ψ_{x}), \end{array}$ Finally, we have $\begin{array}{r} (3.62) & \frac{b}{ρ_{2}} (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) I_{4} ⩽ c β (t) \int_{0}^{L} | ψ_{t} |^{σ (x)} d x + c (θ \circ ψ_{x}), \end{array}$ Replacing (3.59)–(3.62) into (3.58) and using the estimate (3.28), we see that $\begin{array}{rcl} μ^{'} (t) & ⩽ & - ρ_{2} \int_{0}^{L} φ_{x t}^{2} d x - (κ - ε_{5} - c η) \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x + ε_{4} \int_{0}^{L} ψ_{x}^{2} d x \\ (3.63) & + c (θ \circ ψ_{x}) + c β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x + c β (t) \int_{Ω } | ψ_{t} |^{σ (x)} d x . \end{array}$ In (3.63), by taking $\begin{matrix} ε_{5} = \frac{κ}{2}, η = \frac{κ}{4 c}, \end{matrix}$ we obtain (3.49). □

Now, we introduce the following functional $\begin{array}{r} L (t) = N E (t) + N_{1} χ (t) + N_{2} F (t) + N_{3} μ (t), \end{array}$ where N, $N_{1}$ , $N_{2}$ and $N_{3}$ are positive constants will be chosen later. It is easy to get that, for N large, the functional $L (t)$ is equivalent to the energy functional E, i.e., there exist constants $β_{1} > 0$ and $β_{2} > 0$ such that $\begin{array}{r} β_{1} E (t) ⩽ L (t) ⩽ β_{2} E (t) . \end{array}$ Lemma 3.12.
The functional $L (t)$ satisfies, along the solution, for any $t ⩾ t_{1}$ , $\begin{array}{rcl} L^{'} (t) & ⩽ & - \int_{0}^{L} φ_{t}^{2} d x - \int_{0}^{L} φ_{t t}^{2} d x - \int_{0}^{L} ψ_{x}^{2} d x \\ (3.64) & - \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - \int_{0}^{L} φ_{x t}^{2} d x + c (θ \circ ψ_{x}) + c β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x . \end{array}$
Proof.
Combining (3.1), (3.42), (3.43) and (3.49), we have $\begin{array}{rcl} L^{'} (t) & ⩽ & - ρ_{1} N_{1} \int_{0}^{L} φ_{t}^{2} d x - \frac{ρ_{1} ρ_{2}}{κ} N_{2} \int_{0}^{L} φ_{t t}^{2} d x \\ - (\frac{b - ℓ}{4} N_{2} - c ε_{1} N_{1} - ε_{4} N_{3}) \int_{0}^{L} ψ_{x}^{2} d x \\ - (\frac{κ}{4} N_{3} - c N_{1} - c N_{2}) \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - (ρ_{2} N_{3} - ρ_{2} N_{2}) \int_{0}^{L} φ_{x t}^{2} d x \\ - (N - c N_{2} - c N_{3}) \int_{0}^{L} | ψ_{t} |^{σ (\cdot)} d x + (c N_{2} + c N_{3}) (θ \circ ψ_{x}) \\ (3.65) & - \frac{N}{2} (θ^{'} \circ ψ_{x}) + c (N_{2} + N_{3}) β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x . \end{array}$

Taking $\begin{matrix} ε_{1} = \frac{1}{c N_{1}}, ε_{4} = \frac{1}{N_{3}}, N_{1} = N_{2}, \end{matrix}$ we conclude that for any $t ⩾ t_{1}$ , $\begin{array}{rcl} L^{'} (t) & ⩽ & - ρ_{1} N_{2} \int_{0}^{L} φ_{t}^{2} d x - \frac{ρ_{1} ρ_{2}}{κ} N_{2} \int_{0}^{L} φ_{t t}^{2} d x - (\frac{b - ℓ}{4} N_{2} - 2) \int_{0}^{L} ψ_{x}^{2} d x \\ - (\frac{κ}{4} N_{3} - 2 c N_{2}) \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - (ρ_{2} N_{3} - ρ_{2} N_{2}) \int_{0}^{L} φ_{x t}^{2} d x \\ - (N - c N_{2} - c N_{3}) \int_{0}^{L} | ψ_{t} |^{σ (\cdot)} d x + (c N_{2} + c N_{3}) (θ \circ ψ_{x}) \\ (3.66) & - \frac{N}{2} (θ^{'} \circ ψ_{x}) + c (N_{2} + N_{3}) β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x . \end{array}$

Firstly, we take $\begin{matrix} N_{2} = \frac{12}{b - ℓ}, \end{matrix}$

After fixing $N_{1}$ and $N_{2}$ , we choose $N_{3}$ $\begin{matrix} N_{3} ⩾ max {N_{2}, \frac{8 c N_{2}}{κ}} \end{matrix}$ Finally, we choose N large enough such that satisfying $\begin{matrix} N - c N_{2} - c N_{3} > 0, \end{matrix}$ Then we get for any $t ⩾ t_{1}$ , $\begin{array}{rcl} L^{'} (t) & ⩽ & - \int_{0}^{L} φ_{t}^{2} d x - \int_{0}^{L} φ_{t t}^{2} d x - \int_{0}^{L} ψ_{x}^{2} d x \\ - \int_{0}^{L} {(φ_{x} + ψ)}^{2} d x - \int_{0}^{L} φ_{x t}^{2} d x + c (θ \circ ψ_{x}) + c β (t) \int_{Ω } | ψ_{t} |^{2 σ (x) - 2} d x . \end{array}$ Which completes the proof of estimate (3.64). □
Lemma 3.13.
If the assumptions (A1)–(A4) hold, then we have the following estimates: $\begin{matrix} (3.67) & \begin{array}{r} c β (t) \int_{Ω_{}} | ψ_{t} |^{2 σ (x) - 2} d x = \{\begin{array}{ll} 0, & σ_{1} ⩾ 2; \\ c ε E (t) + c β (t) E^{\frac{2 σ_{1} - 2}{2 - σ_{1}}} \int_{Ω_{}} C_{ε} | ψ_{t} |^{σ (x)}, & 1 < σ_{1} < 2 . \end{array} \end{array} \end{matrix}$
Proof.
The first estimate is clear because if $σ_{1} ⩾ 2$ , then $meas (Ω_{}) = 0$ . However, if $1 < σ_{1} < 2$ , then by using Young’s inequality, we have $\begin{array}{rcl} (3.68) & E^{\frac{2 - σ_{1}}{2 σ_{1} - 2}} | ψ_{t} |^{2 σ (x) - 2} & ⩽ & ε E^{\frac{σ (x)}{2 - σ (x)} \frac{2 - σ_{1}}{2 σ_{1} - 2}} + C_{ε} | ψ_{t} |^{σ (x)} . \end{array}$ Since, $\frac{σ (x)}{2 - σ (x)} \frac{2 - σ_{1}}{2 σ_{1} - 2} ⩾ \frac{2 - σ_{1}}{2 σ_{1} - 2} + 1$ , and E is nonincreasing, we get $\begin{array}{rcl} (3.69) & E^{\frac{2 - σ_{1}}{2 σ_{1} - 2}} | ψ_{t} |^{2 σ (x) - 2} & ⩽ & c ε E^{\frac{2 - σ_{1}}{2 σ_{1} - 2} + 1} + C_{ε} | ψ_{t} |^{σ (x)} . \end{array}$ Multiplying by $\frac{β (t)}{E^{\frac{2 - σ_{1}}{2 σ_{1} - 2}}}$ and integrating with respect to x, $\begin{array}{rcl} (3.70) & c β (t) \int_{Ω_{}} | ψ_{t} |^{2 σ (x) - 2} d x & ⩽ & c ε E (t) + c β (t) E^{\frac{2 σ_{1} - 2}{2 - σ_{1}}} \int_{Ω_{*}} C_{ε} | ψ_{t} |^{σ (x)} . \end{array}$ This finishes the proof. □

4. The proofs of the main theorems

In this section, we prove our main results stated in Section 2.

4.1. The proof of Theorem 2.1

To prove the energy decay in (2.1), we start by multiplying (3.64) by $β (t)$ . Then, we impose the estimate of $c β (t) \int_{0}^{L} ψ_{t}^{2} d x$ in ${(3.14)}_{1}$ . The modified (3.64) becomes $\begin{matrix} (4.1) & \begin{aligned} L_{1}^{'} (t) ⩽ & - c β E (t) + c β (t) \int_{0}^{t} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s \\ + c β (t) \int_{t}^{\infty} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s, \end{aligned} \end{matrix}$ where $L_{1} = β L + c β E \sim E$ . Using (1.24), (3.1) and the fact that ξ and θ are non increasing, we find that $\begin{matrix} (4.2) & \begin{aligned} c (β ξ) (t) \int_{0}^{t} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s & ⩽ - c β (t) \int_{0}^{t} ξ (s) θ^{'} (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s \\ ⩽ - c β (t) E^{'} (t), \forall t \in R^{+} . \end{aligned} \end{matrix}$ Now, multiplying (4.1) by $ξ (t)$ and combining with (4.2) and the estimate (3.41), we get $\begin{matrix} (4.3) & \begin{array}{r} K^{'} (t) ⩽ - c (β ξ) (t) E (t) + c m (β ξ) (t) \int_{t}^{\infty} θ (s) d s, \end{array} \end{matrix}$ where $K = ξ L_{1} + c β ξ \sim E$ . Let $h (t) = c m (ξ β) (t) \int_{t}^{+ \infty} θ (s) d s$ . Then, (4.3) becomes $\begin{matrix} (4.4) & K^{'} (t) ⩽ - γ_{0} (β ξ) K (t) + h (t) \end{matrix}$ for some $γ_{0} > 0$ . This last inequality remains true for any $δ_{0} \in (0, γ_{0}]$ ; that is $\begin{matrix} K^{'} (t) ⩽ - δ_{0} (β ξ) (t) K (t) + h (t), \forall t \in R^{+} . \end{matrix}$ Therefore, direct integration leads to $\begin{matrix} K (T) ⩽ e^{- δ_{0} \int_{0}^{T} (β ξ) (s) d s} (K (0) + \int_{0}^{T} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} h (t) d t) \end{matrix}$ and the fact that $K \sim E$ gives $\begin{matrix} (4.5) & E (T) ⩽ γ_{1} e^{- δ_{0} \int_{0}^{T} (β ξ) (s) d s} (K (0) + \int_{0}^{T} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} h (t) d t) . \end{matrix}$ It is obvious that $\begin{matrix} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} h (t) = \frac{c m}{δ_{0}} {(e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s})}^{'} \int_{t}^{+ \infty} g (s) d s, \forall t \in R^{+} . \end{matrix}$ Then, integration by parts leads to $\begin{aligned} \int_{0}^{T} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} h (t) d t \\ = \frac{c m}{δ_{0}} (e^{δ_{0} \int_{0}^{T} (β ξ) (s) d s} \int_{T}^{+ \infty} g (s) d s - \int_{0}^{+ \infty} g (s) d s + \int_{0}^{T} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} θ (t) d t) . \end{aligned}$ Consequently, combining with (4.5), we have $\begin{matrix} (4.6) & E (T) ⩽ γ_{1} (K (0) + \frac{c m}{δ_{0}} \int_{0}^{T} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} θ (t) d t) e^{- δ_{0} \int_{0}^{T} ξ (s) d s} + \frac{c m}{δ_{0}} \int_{T}^{+ \infty} θ (s) d s . \end{matrix}$ We observe that $\begin{matrix} {(e^{\int_{0}^{t} ξ (s) d s} θ (t))}^{'} ⩽ 0, \forall t \in R^{+} . \end{matrix}$ Consequently, we have $e^{\int_{0}^{t} (β ξ) (s) d s} θ (t) ⩽ θ (0)$ and $\begin{matrix} (4.7) & \int_{0}^{T} e^{δ_{0} \int_{0}^{t} (β ξ) (s) d s} θ (t) d t ⩽ {(θ (0))}^{δ_{0}} \int_{0}^{T} {(θ (t))}^{1 - δ_{0}} d t . \end{matrix}$ Finally, by combining (4.6) and (4.7), we obtain $\begin{matrix} (4.8) & E (t) ⩽ δ_{1} (1 + \int_{0}^{t} {(θ (s))}^{1 - δ_{0}} d s) e^{- δ_{0} \int_{0}^{t} (β ξ) (s) d s} + \frac{c m}{δ_{0}} \int_{t}^{+ \infty} θ (s) d s, \end{matrix}$ where $δ_{1} = max {γ_{1} K (0), \frac{c m}{δ_{0}} {(θ (0))}^{δ_{0}}}$ . Thus, the proof of (2.1) is completed.

4.2. The proof of Theorem 2.2

To prove (2.2), we first multiply (3.64) by $β (t)$ , to get $\begin{aligned} β L^{'} (t) ⩽ & - c β E (t) + c β \int_{0}^{L} ψ_{t}^{2} d x + c β \int_{0}^{L} \int_{0}^{\infty} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s d x \\ + c β (t) \int_{Ω_{*}} | ψ_{t} |^{2 σ (x) - 2} d x . \end{aligned}$ Imposing the estimate of $c β (t) \int_{Ω_{*}} | ψ_{t} |^{2 σ (x) - 2} d x$ given in (3.67) in the above equation and choosing ε small enough, we obtain $\begin{matrix} (4.9) & \begin{aligned} β L^{'} (t) ⩽ & - c β E (t) + c β \int_{0}^{L} ψ_{t}^{2} d x \\ + c β \int_{0}^{L} \int_{0}^{\infty} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}^{2} d s d x - c β E^{'} (t) E^{\frac{2 - σ_{1}}{2 σ_{1} - 2}} . \end{aligned} \end{matrix}$ Combining the estimate of $c β \int_{0}^{L} ψ_{t}^{2} d x$ in ${(3.14)}_{3}$ with (4.2) gives us $\begin{matrix} (4.10) & \begin{aligned} β L^{'} (t) ⩽ & - c β E (t)) + c ε_{1} β E + c ε_{2} β E - C_{ε_{1}} E^{'} (t) E^{- ϑ} - C_{ε_{2}} E^{'} (t) E^{- ϑ} - c β E^{'} (t) E^{\frac{2 - σ_{1}}{2 σ_{1} - 2}} \\ + c β \int_{0}^{t} θ (s) d s + c m β \int_{t}^{\infty} θ (s) d s . \end{aligned} \end{matrix}$ Multiplying (4.10) by $E^{ϑ} (t)$ , where $ϑ = max {\frac{2 - σ_{1}}{2 σ_{1} - 2}, \frac{σ_{2}}{2} - 1}$ , we get $\begin{matrix} (4.11) & \begin{aligned} E^{ϑ} (t) β H^{'} (t) ⩽ & - c β E^{ϑ + 1} (t) + c ε_{1} E^{ϑ + 1} (t) + ε_{2} β E^{ϑ + 1} (t) - C_{ε_{1}} E^{'} (t) - C_{ε_{2}} E^{'} (t) \\ + c β E^{ϑ} (t) \int_{0}^{t} θ (s) d s + c m β E^{ϑ} (t) \int_{t}^{\infty} θ (s) d s, \end{aligned} \end{matrix}$ and choosing $ε_{1}$ , $ε_{2}$ small enough, then we get $\begin{matrix} (4.12) & \begin{aligned} L_{2}^{'} (t) & ⩽ - c β E^{ϑ + 1} (t) + c β E^{ϑ} (t) \int_{0}^{t} θ (s) d s + c m β E^{ϑ} (t) \int_{t}^{\infty} θ (s) d s, \end{aligned} \end{matrix}$ where $L_{2} = E^{ϑ} β L + c β E \sim E$ . Multiplying (4.12) by $β^{ϑ} ξ^{ϑ + 1} (t)$ , using (3.1), and using the fact that $(ξ E)$ is non-increasing, we get $\begin{matrix} (4.13) & K^{'} (t) ⩽ - c {(ξ β)}^{ϑ + 1} (t) K^{ϑ + 1} (t) + {(ξ β E)}^{ϑ} (t) h (t), \end{matrix}$ where $h (t) = c m (ξ β) (t) \int_{t}^{\infty} θ (s) d s$ and $K = ξ L_{2} + c β E \sim E$ . Use of Young’s inequality, with $q = ϑ + 1$ and $q^{*} = \frac{ϑ + 1}{ϑ}$ , gives for some positive constants $c_{1}$ and $c_{2}$ $\begin{matrix} (4.14) & K^{'} (t) ⩽ - c_{1} {(ξ β)}^{ϑ + 1} (t) K^{ϑ + 1} (t) + c_{2} h^{ϑ + 1} (t) . \end{matrix}$ Multiply both sides of (4.14) by ${(ξ β)}^{η}$ , $η > 1$ , thus, we get $\begin{matrix} (4.15) & {(ξ β)}^{η} K^{'} (t) ⩽ - c_{1} {(ξ β)}^{ϑ + 1 + η} (t) K^{ϑ + 1} (t) + c_{2} {(ξ β)}^{η} h^{ϑ + 1} (t) . \end{matrix}$ As in [1], we set $χ : = ξ β > 0$ and it is nonincreasing, one can see that $\begin{matrix} (4.16) & {(χ^{η} K (t))}^{'} ⩽ - c_{1} χ^{ϑ + 1 + η} (t) K^{ϑ + 1} (t) + c_{2} χ^{η} h^{ϑ + 1} (t), \end{matrix}$ noting $Φ = χ^{η} K$ and taking $η = \frac{ϑ + 1}{ϑ}$ , we obtain $\begin{matrix} (4.17) & Φ^{'} (t) ⩽ - c_{1} Φ^{ϑ + 1} (t) + c_{2} χ^{η} (t) h^{ϑ + 1} (t) . \end{matrix}$ Let $\begin{matrix} (4.18) & H (t) : = Φ (t) - Λ (t); where Λ (t) = c_{2} {(1 + t)}^{\frac{- 1}{ϑ}} \int_{0}^{t} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s . \end{matrix}$ From the definition of Λ, we have $\begin{matrix} (4.19) & c_{2} χ^{η} (t) h^{ϑ + 1} (t) = Λ^{'} (t) + \frac{c_{2}}{ϑ} {(1 + t)}^{\frac{- 1}{ϑ} - 1} \int_{0}^{t} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s . \end{matrix}$ Since $χ^{η} (s) h^{ϑ + 1} {(1 + s)}^{\frac{1}{ϑ}} > 0$ , then we have for all $t ⩾ t_{0} > 0$ $\begin{matrix} ν : = \int_{0}^{t_{0}} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s ⩽ \int_{0}^{t} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s \end{matrix}$ and then $\begin{matrix} \frac{\int_{0}^{t} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s}{ν} ⩾ 1, \forall t ⩾ t_{0} \end{matrix}$ Thus, (4.19) yields, $\forall t ⩾ t_{0}$ , $\begin{matrix} (4.20) & c_{2} χ^{η} (t) h^{ϑ + 1} (t) ⩽ Λ^{'} (t) + \frac{1}{ϑ c_{2}^{ϑ} ν^{ϑ}} c_{2}^{ϑ + 1} {[{(1 + t)}^{\frac{- 1}{ϑ}}]}^{ϑ + 1} {[\int_{0}^{t} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s]}^{ϑ + 1} \end{matrix}$ we can choose $c_{2}$ large enough so that $\frac{1}{ϑ c_{2}^{ϑ} ν^{ϑ}} ⩽ c_{1}$ , and then we get $\begin{matrix} (4.21) & c_{2} χ^{η} (t) h^{ϑ + 1} (t) ⩽ Λ^{'} (t) + c_{1} Λ^{ϑ + 1}, \forall t ⩾ t_{0} . \end{matrix}$ Now using (4.21) and the definition of H, we get, $\forall t ⩾ t_{0}$ , $\begin{matrix} (4.22) & \begin{aligned} H^{'} (t) & = Φ^{'} (t) - Λ^{'} (t) ⩽ - c_{1} Φ^{ϑ + 1} (t) + c_{2} χ^{η} (t) h^{ϑ + 1} (t) - Λ^{'} (t) \\ ⩽ - c_{1} [{(H + Λ)}^{ϑ + 1} (t)] + c_{2} χ^{η} (t) h^{ϑ + 1} (t) - Λ^{'} (t) . \end{aligned} \end{matrix}$ Since $H (0) > 0$ . Then there exists $t_{1} > 0$ such that $H (t) > 0$ , $\forall t \in [0, t_{1})$ . Hence, $\begin{matrix} (4.23) & \begin{aligned} H^{'} (t) & ⩽ - c_{1} [H^{ϑ + 1} (t) + Λ^{ϑ + 1} (t)] + c_{2} χ^{η} (t) h^{ϑ + 1} (t) - Λ^{'} (t), \forall t \in [t_{0}, t_{1}) . \\ ⩽ - c_{1} [H^{ϑ + 1} (t) + Λ^{ϑ + 1} (t) - \frac{c_{2}}{c_{1}} χ^{η} (t) h^{ϑ + 1} (t) + \frac{1}{c_{1}} Λ^{'} (t)], \end{aligned} \end{matrix}$ Thus, $\begin{matrix} (4.24) & \begin{array}{r} H^{'} (t) ⩽ - c_{1} H^{ϑ + 1} (t), \forall t \in [t_{0}, t_{1}) . \end{array} \end{matrix}$ Integrate over $(t_{0}, t)$ , we have $\begin{matrix} (4.25) & H (t) ⩽ \frac{c}{{(t - t_{0})}^{\frac{1}{ϑ}}}, \forall t \in [t_{0}, t_{1}) . \end{matrix}$ If $t_{1} = + \infty$ , using again the definitions of H and Λ, we have, for t large enough, $\begin{matrix} (4.26) & Φ (t) ⩽ C {(1 + t)}^{\frac{- 1}{ϑ}} [1 + \int_{0}^{t} χ^{η} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s] . \end{matrix}$ If $t_{1} < + \infty$ , then there exists $t_{2} > t_{1}$ such that $H (t) ⩽ 0$ , $\forall t_{1} ⩽ t < t_{2}$ . Hence, (4.18) yields $Φ (t) ⩽ Λ (t)$ , $\forall t_{1} ⩽ t < t_{2}$ , consequently, we get (4.26). If $t_{2} = + \infty$ , we are done. Otherwise, there exists $t_{3} > t_{2}$ such that $H (t_{2}) = 0$ and $H (t) > 0$ , $\forall t_{2} < t < t_{3}$ , we then repeat the steps (4.23)–(4.25) on $[t_{2}, t_{3})$ to obtain (4.26). Therefore, (4.26) remains valid for all $t ⩾ t_{0}$ . Multiply (4.26) by $χ^{- η}$ and recall the definition of Φ, then for $η = \frac{ϑ + 1}{ϑ}$ we have $\begin{matrix} (4.27) & K (t) ⩽ C {(1 + t)}^{\frac{- 1}{ϑ}} χ^{- \frac{ϑ + 1}{ϑ}} [1 + \int_{0}^{t} χ^{\frac{ϑ + 1}{ϑ}} (s) h^{ϑ + 1} (s) {(1 + s)}^{\frac{1}{ϑ}} d s] \end{matrix}$ Using the fact $K \sim E$ , we have two cases:

If $ϑ = \frac{2 - σ_{1}}{2 σ_{1} - 2} > 0$ , then $ϑ + 1 = \frac{σ_{1}}{2 σ_{1} - 2}$ and $\frac{ϑ + 1}{ϑ} = \frac{σ_{1}}{2 - σ_{1}}$ , we get $\begin{matrix} (4.28) & \begin{array}{r} E (t) ⩽ C {(1 + t)}^{- \frac{2 σ_{1} - 2}{2 - σ_{1}}} χ^{- \frac{σ_{1}}{2 - σ_{1}}} (1 + \int_{0}^{t} {(1 + s)}^{\frac{2 σ_{1} - 2}{2 - σ_{1}}} χ^{\frac{σ_{1}}{2 - σ_{1}}} (s) h^{\frac{1}{σ_{1} - 1}} d s) . \end{array} \end{matrix}$ If $ϑ = \frac{σ_{2}}{2} - 1 > 0$ , we have $\begin{matrix} (4.29) & \begin{array}{r} E (t) ⩽ C {(1 + t)}^{- \frac{2}{σ_{2} - 2}} χ^{- \frac{σ_{2}}{σ_{2} - 2}} (1 + \int_{0}^{t} {(1 + s)}^{\frac{2}{σ_{2} - 2}} χ^{\frac{σ_{2}}{σ_{2} - 2}} (s) h^{\frac{σ_{2}}{2}} d s) . \end{array} \end{matrix}$ This establishes (2.2).

4.3. The proof of Theorem 2.3

To establish (2.3), we start multiplying (3.64) by β, recalling the estimate of $c β \int_{0}^{L} ψ_{t}^{2} d x$ in ${(3.14)}_{2}$ , to get $\begin{matrix} (4.30) & \begin{aligned} β L^{'} (t) ⩽ & - c β E (t) + c ε β E (t) - C_{ε} (E^{'} (t)) E^{- \hat{ϑ}} \\ + c β \int_{0}^{t} θ (s) {‖ ψ_{x} (t) - ψ_{x} (t - s) ‖}_{2}^{2} d s + c β \int_{t}^{\infty} θ (s) {| ψ_{x} (t) - ψ_{x} (t - s) |}_{2}^{2} d x . \end{aligned} \end{matrix}$ Multiplying (4.30) by $E^{\hat{ϑ}}$ where $\hat{ϑ} = \frac{σ_{2}}{2} - 1 > 0$ and imposing (4.2), we obtain $\begin{aligned} β E^{\hat{ϑ}} L^{'} (t) ⩽ & - c β E^{\hat{ϑ} + 1} (t) + c ε E^{\hat{ϑ} + 1} - C_{ε} (- E^{'} (t)) \\ + c β E^{\hat{ϑ}} \int_{0}^{t} θ (s) d s + c m β E^{\hat{ϑ}} \int_{t}^{\infty} θ (s) d x . \end{aligned}$

Choosing ε small enough leads to $\begin{matrix} (4.31) & L_{3} (t) ⩽ - c β E^{\hat{ϑ} + 1} (t) + c β E^{\hat{ϑ}} \int_{0}^{t} θ (s) d s + c m β E^{\hat{ϑ}} \int_{t}^{\infty} θ (s) d x, \end{matrix}$ where $L_{3} = β E^{\hat{ϑ}} L + c β E \sim E$ . Multiplying (4.31) by ${(ξ β)}^{\hat{ϑ}} ξ$ , combining with (3.1) and (4.2) and the nonincreasing of E, we get $\begin{matrix} (4.32) & \begin{array}{r} Y^{'} (t) ⩽ - c {(ξ β)}^{\hat{ϑ} + 1} E^{\hat{ϑ} + 1} (t) + c m β {(ξ β E)}^{\hat{ϑ}} ξ \int_{t}^{\infty} θ (s) d s, \end{array} \end{matrix}$ where $Y = ξ β L_{3} + 2 c β E \sim E$ . By letting $h (t) = c m (ξ β) (t) \int_{t}^{+ \infty} θ (s) d s$ .

Then, (4.32) becomes $\begin{matrix} (4.33) & Y^{'} (t) ⩽ - c {(ξ β)}^{\hat{ϑ} + 1} (t) E^{\hat{ϑ} + 1} (t) + c {(ξ β E)}^{\hat{ϑ}} (t) h (t) . \end{matrix}$ Let $χ : = ξ β > 0$ which is non-increasing, we get $\begin{matrix} (4.34) & Y^{'} (t) ⩽ - c χ^{\hat{ϑ} + 1} (t) Y^{\hat{ϑ} + 1} (t) + c {(χ E)}^{\hat{ϑ}} h (t) . \end{matrix}$ Use of Young’s inequality, with $ζ = \hat{ϑ} + 1$ and $ζ^{*} = \frac{\hat{ϑ} + 1}{\hat{ϑ}}$ , gives for some positive constant $c_{1}$ and $c_{2}$ $\begin{matrix} (4.35) & Y^{'} (t) ⩽ - c_{1} χ^{\hat{ϑ} + 1} (t) Y^{\hat{ϑ} + 1} (t) + c_{2} h^{\hat{ϑ} + 1} (t) . \end{matrix}$ Repeating the same steps in the last part of the proof Theorem 2.2 with replacing ϑ by $\hat{ϑ} = \frac{σ_{2} - 2}{2}$ , the proof of the decay (2.3) is completed.

5. Examples and remarks

In this section, we give some examples and remarks to illustrate and compare our stability results in Theorem 2.1, Theorem 2.2 and Theorem 2.3 with the some earlier results in the literature.

Example 5.1.

If $θ (s) = a e^{- {(s + 1)}^{b}}$ and $β \equiv 1$ . The constant $0 < b ⩽ 1$ and $a > 0$ small enough so that $θ (s)$ satisfies condition $(A_{1})$ . Therefore, $ξ (s) = b {(1 + s)}^{b - 1}$ and the energy decay (2.1) becomes for $δ_{0} = \frac{1}{2}$ , and some positive constants $c_{1}$ and $c_{2}$ $\begin{matrix} (5.1) & E (t) ⩽ c_{1} e^{- c_{2} (1 + t)} . \end{matrix}$ This is an exponential decay. If $θ (s) = \frac{a}{{(1 + s)}^{b}}$ and $β \equiv 1$ . The constant $b > 1$ and $a > 0$ small enough such that $θ (s)$ satisfies condition $(A_{1})$ . Therefore, $ξ (s) = b {(1 + s)}^{- 1}$ and the energy decay (2.1) becomes for $δ_{0} = \frac{1}{2}$ , and some positive constants $c_{1}$ and $c_{2}$ $\begin{matrix} (5.2) & E (t) ⩽ \frac{c_{1}}{{(1 + t)}^{c_{2}}} . \end{matrix}$ This is a polynomial decay. Then, we note that ${lim}_{t \to \infty} E (t) = 0$ .

Example 5.2.

If $θ (s) = a e^{- \sqrt{s}}$ . Therefore, $ξ (s) = \frac{1}{2 \sqrt{s}}$ , $\int_{t}^{\infty} θ (s) = 2 a (\sqrt{s} + 1) e^{- \sqrt{s}}$ , and $h (t) = ξ (t) \int_{t}^{\infty} θ (s) = a (\frac{\sqrt{t} + 1}{\sqrt{t}} e^{- \sqrt{t}})$ . Then, in case if $ϑ = \frac{2 - σ_{1}}{2 σ_{1} - 2} > 0$ , the energy decay (2.2) becomes for $β (t) = {(1 + t)}^{- λ}$ , $0 ⩽ λ ⩽ 1$ , $\begin{matrix} (5.3) & E (t) ⩽ C {(1 + t)}^{- \frac{2 σ_{1} - 2}{2 - σ_{1}}} {(1 + t)}^{\frac{σ_{1}}{4 - 2 σ_{1}} - λ} (1 + \int_{0}^{t} {(1 + s)}^{\frac{5 σ_{1} - 4}{2 (2 - σ_{1})} - λ} {(ξ β)}^{\frac{σ_{1}}{2 - σ_{1}}} (s) h^{\frac{σ_{1}}{2 - σ_{1}}} d s) . \end{matrix}$ Then, we obtain polynomial stability of the form $\begin{matrix} (5.4) & E (t) ⩽ C {(1 + t)}^{\frac{4 - 3 σ_{1}}{2 (2 - σ_{1})} - λ} . \end{matrix}$ It is clear that for $λ = 0$ and $σ_{1} > \frac{4}{3}$ , ${lim}_{t \to \infty} E (t) = 0$ . If $λ = 1$ , then we have for any $1 < σ_{1} < 2$ , ${lim}_{t \to \infty} E (t) = 0$ .

In the second case, if $ϑ = \frac{σ_{2}}{2} - 1 > 0$ , then we obtain for the same function polynomial stability of the form $\begin{matrix} (5.5) & E (t) ⩽ C {(1 + t)}^{\frac{σ_{2} - 4}{2 (σ_{2} - 2)} - λ} . \end{matrix}$ Then, we note that for $λ = 0$ and $σ_{1} < σ_{2} < 4$ , we have ${lim}_{t \to \infty} E (t) = 0$ . Also, If $λ = 1$ , then we have for any $1 < σ_{1} ⩽ σ_{2}$ , ${lim}_{t \to \infty} E (t) = 0$ .

Remark 5.3.

If there exists $ε_{0} \in (0, 1)$ , for which $\begin{matrix} (5.6) & \int_{0}^{+ \infty} {(θ (s))}^{1 - ε_{0}} d s < + \infty, \end{matrix}$ then we can choose $δ_{0} \in (0, γ_{1}]$ , $γ_{1} = min {ε_{0}, γ_{0}}$ such that $\begin{matrix} \int_{0}^{+ \infty} {(θ (s))}^{1 - δ_{0}} d s < + \infty, \end{matrix}$ and consequently, (2.1) takes the form $\begin{matrix} (5.7) & E (t) ⩽ δ_{2} (e^{- δ_{0} \int_{0}^{t} ξ (s) d s} + \int_{t}^{+ \infty} θ (s) d s) . \end{matrix}$

Remark 5.4.

In the absence of the viscoelastic term, the variable exponent frictional damping takes care of the stability of the system and we obtain similar results to the one in [24] $\begin{matrix} (5.8) & \{\begin{array}{ll} E (t) ⩽ c_{1} e^{- c_{2} \int_{0}^{t} β (s) d s}, & σ_{1} = σ_{2} = 2, \\ E (t) ⩽ c_{3} {(1 + \int_{0}^{t} β (s) d s)}^{\frac{- 2}{σ_{2} - 2}}, & σ_{1} ⩾ 2, and m_{2} > 2, \\ E (t) ⩽ c_{4} {(1 + \int_{0}^{t} β (s) d s)}^{\frac{- 1}{ϑ}}, & 1 < σ_{1} < 2, \end{array} \end{matrix}$ where $ϑ = max {\frac{2 - σ_{1}}{2 σ_{1} - 2}, \frac{σ_{2}}{2} - 1}$ and $c_{i}$ , $i = 1, \dots, 4$ are positive constants.

In the absence of the variable exponent frictional damping, the viscoelastic term takes care of the decay and we get similar result to the one in [19] $\begin{matrix} E (t) ⩽ δ_{1} (1 + \int_{0}^{t} {(θ (s))}^{1 - δ_{0}} d s) e^{- δ_{0} \int_{0}^{t} ξ (s) d s} + δ_{1} \int_{t}^{+ \infty} θ (s) d s . \end{matrix}$

Footnotes

Acknowledgement

The author would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The author also would like to thank the referee for his valuable comments which made significant improvements to our manuscript.

Funding

This work is funded by KFUPM, Grant No. INCB2311.

Availability of data and materials

Not applicable.

Declarations

Ethics approval and consent to participate

Ethics approval was not required for this research.

Competing interests

The author declares no competing interests.

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