Optimal decay of an abstract nonlinear viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping

Abstract

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping and a nonlinear source term. Under some suitable assumptions on the weight of the delayed feedback, the weight of the non-delayed feedback and the behavior of the relaxation function, we establish two explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some properties of the convex functions. Moreover, we give some applications and examples. This work generalizes the previous results without time delay term to those with delay in the nonlinear damping.

Keywords

Abstract evolution equation general decay nonlinear damping nonlinear source term time delay

1. Introduction

Let H be a real Hilbert space with inner product and related norm denoted by $⟨ \cdot, \cdot ⟩$ and $‖ \cdot ‖$ , respectively. Let $A : D (A) ⟶ H$ be a self-adjoint linear positive operator with domain $D (A) \subset H$ such that the embedding is dense and compact. $h : R_{+} ⟶ R_{+}$ is the kernel of the memory term and $τ > 0$ represents a time delay.

In this work, we consider the following second-order abstract evolution equation $\begin{matrix} (1.1) & \{\begin{matrix} u_{t t} (t) + A u (t) - \int_{0}^{t} h (t - s) A u (s) d s \\ + μ_{1} Q (u_{t} (t)) + μ_{2} P (u_{t} (t - τ)) = F (u (t)), & t \in (0, + \infty), \\ u_{t} (t - τ) = f_{0} (t - τ), & t \in (0, τ), \\ u (0) = u_{0}, u_{t} (0) = u_{1}, \end{matrix} \end{matrix}$ where the initial datum $(u_{0}, u_{1}, f_{0})$ belongs to a suitable space. $μ_{1}$ is a positive constant and $μ_{2}$ is a real number. The functions $Q, P : H \to H$ and $F : D (A^{\frac{1}{2}}) \to H$ satisfy some assumptions.

The delay effects often arise in many practical problems, even small delay may destabilize a system which is asymptotically stable in the absence of delay, in this sense, see [7,17,38]. A large part in the literature is available addressing the stability, instability and the connection between the memory term, the frictional damping and the delay terms. In particular, for wave equation with constant or variable delay, we refer to read [5,38,39]. They showed that the frictional damping term is strong enough to stabilize the system when the weight of the delay be sufficiently small. In the absence of delay ( $μ_{2} = 0$ ), Messaoudi in [32] gave a general decay rate where the exponential and the polynomial decay rates are special cases. Precisely, he considered a relaxation function satisfies $\begin{matrix} (1.2) & h^{'} (t) ⩽ - ζ (t) h (t), \forall t \in R_{+}, \end{matrix}$ where ζ is a nonincreasing positive differentiable function. Under the above condition on h, Benaissa et al. in [9] studied the energy decay of solutions by introducing a suitable Lyapunov functional, under a relation between the weight of the delay term in the feedback and the weight of the term without delay. They considered the following equation $\begin{matrix} u_{t t} (t) - Δ u (t) + \int_{0}^{t} h (t - s) Δ u (s) d s + μ_{1} g_{1} (u_{t} (t)) + μ_{2} g_{2} (u_{t} (t - τ)) = 0, \end{matrix}$ where $μ_{1}$ and $μ_{2}$ are positive real numbers. The function $g_{1} : R \to R$ is a non-decreasing continuous function and $g_{2} : R \to R$ is an odd non-decreasing function of the class $C^{1} (R)$ . When the functions $g_{1}$ and $g_{2}$ are linear, Kirane and Said-Houari [25] proved the global well-posedness of the system under some restrictions on the parameters $μ_{1}$ and $μ_{2}$ , and established a general decay result of energy if $0 < μ_{2} ⩽ μ_{1}$ , but they assumed the constants $μ_{1}$ and $μ_{2}$ are positive constants. Subsequently, the results were extended and improved by Liu [27,28] and Dai and Yang [16]. We can also find some interesting results on delayed system by using (1.2) in [15,18,20,21,29] and so on. After that, motivated by the work of Lasiecka and Tataru [26], Alabau-Boussouira et al. [3] introduced the following condition $\begin{matrix} (1.3) & h^{'} (t) ⩽ - G (h (t)), \forall t \in R_{+}, \end{matrix}$ where G is a convex function, which appeared in many papers, see [13,33,40]. Mustafa in [34] considered the following plate equation $\begin{matrix} u_{t t} (t) + Δ^{2} u (t) - \int_{0}^{+ \infty} h (s) Δ^{2} u (t - s) d s + μ_{1} u_{t} | u_{t} |^{k - 2} + μ_{2} u_{t} (t - τ) {| u_{t} (t - τ) |}^{k - 2} = 0, \end{matrix}$ where $k ⩾ 2$ . He established an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Recently, Mustafa in [36] established an explicit energy decay result where the exponential and the polynomial decay rates are recovered, under a general condition on the relaxation function, $h^{'} (t) ⩽ - ζ (t) h^{p} (t)$ , with $1 ⩽ p < 2$ . In [37], the same author established an optimal explicit and general decay results when the relaxation function h satisfies $\begin{matrix} (1.4) & h^{'} (t) ⩽ - ζ (t) G (h (t)), \forall t \in R_{+}, \end{matrix}$ where G is a positive function of class $C^{1} (R^{+})$ , $G (0) = G^{'} (0) = 0$ and G is linear or strictly increasing and strictly convex $C^{2}$ function on $(0, r]$ , $r < h (0)$ . Al-Mahdi et al. in [2] established the same stability result for the following system: $\begin{matrix} u_{t t} (t) + A u (t) - \int_{0}^{t} h (t - s) A u (s) d s + η (t) g (u_{t}) = \nabla F (u (t)), t \in (0, + \infty), \end{matrix}$ where $\nabla F$ denotes the gradient of a Gateaux differentiable functional $F : D (A^{\frac{1}{2}}) \to R$ and $η \in C^{1} (0, + \infty)$ is a positive nonincreasing function. For some works with the condition (1.4), we refer to [8,14,22,31,35].

There are different results according to the general decay for a several problems with internal or boundary feedback and for constant or variable delay. For instance, Boukhatem and Benabderrahmane in [11] considered a variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term. They established a general decay result where the kernel memory satisfies the equation (1.2) and $\begin{matrix} \int_{0}^{+ \infty} \frac{h (s)}{G^{- 1} (- h^{'} (s))} d s + sup_{s \in R_{+}} \frac{h (s)}{G^{- 1} (- h^{'} (s))} < + \infty, \end{matrix}$ where G is an increasing positive strictly convex function satisfying some additional properties. In the case of distributed delay, Aili and Khemmoudj in [1] considered the following viscoelastic wave equation of Kirchhoff type with a nonlinear internal damping $\begin{array}{l} | u_{t} |^{l} u_{t t} - M (‖ \nabla u ‖_{2}^{2}) Δ u - Δ u_{t t} + \int_{0}^{t} h (t - s) Δ u (s) d s \\ + μ_{0} g_{1} (u_{t}) + \int_{τ_{1}}^{τ_{2}} μ (s) g_{2} (u_{t} (x, t - s)) d s + f (u) = 0, in Ω \times R^{+}, \end{array}$ where Ω is a bounded domain in $R^{n}$ , $n \in N^{*}$ , l and $μ_{0}$ are two positive constants. $M (s) = m_{0} + m_{1} s^{β}$ for $m_{0} > 0$ , $m_{1} ⩾ 0$ , $β ⩾ 1$ and $s ⩾ 0$ , $μ : [τ_{1}, τ_{2}] \to R$ is a bounded function, where $τ_{1}$ and $τ_{2}$ are two real numbers satisfying $0 ⩽ τ_{1} < τ_{2}$ . They established the general decay result under some general assumptions on $μ_{0}$ and μ where the viscoelastic kernels h satisfy the condition (1.3). For more general decay results in the presence of delay, see [9,12,24,30] and references therein.

In this work, we are interested in giving an optimal explicit and a general decay rate of the solution of the problem (1.1) under more general assumptions on the function h, $μ_{1}$ and the weight of delay $μ_{2}$ . More precisely, we are intending to extend the results of Benaissa et al. in [9]and Messaoudi [31] to the semilinear abstract viscoelastic equation with time delay in the nonlinear damping term and nonlinear source term; the system (1.1). To the best of our knowledge, there is no decay result for problems with delay where the relaxation functions satisfy (1.4) in the abstract form. Moreover, our problem generalizes the earlier problems without time delay term to those with delay in the nonlinear internal damping term.

The paper is organized as follows. In Section 2, we state and prove some preliminary results under suitable hypothesis. In Section 3, we present some technical lemmas needed for our work. Then, we establish the decay results of the energy by using the energy method to produce a suitable Lyapunov functional and some properties of the convex functions in the Section 4. Finally, Section 5 is devoted to give some concrete applications to illustrate our abstract result.

2. Preliminaries

In this section, we present some assumptions in order to state and prove our main result. Then, we establish several preliminary results needed in this work.

We consider the following assumptions:

There exist a positive constant a satisfying $\begin{matrix} (2.1) & {‖ A^{\frac{1}{2}} u ‖}^{2} ⩾ a ‖ u ‖^{2}, \forall u \in D (A^{\frac{1}{2}}) . \end{matrix}$

The kernel memory function $h : R_{+} ⟶ R_{+}$ is of class $C^{1}$ nonincreasing function satisfying $\begin{matrix} (2.2) & h (0) > 0, h_{0} = \int_{0}^{+ \infty} h (s) d s < 1 . \end{matrix}$ Moreover, there exists a $C^{1}$ function $G : (0, + \infty) \to (0, + \infty)$ which is linear or strictly increasing and strictly convex $C^{2}$ function on $(0, r]$ , $r ⩽ h (0)$ , with $G (0) = G^{'} (0) = 0$ , such that $\begin{matrix} (2.3) & h^{'} (t) ⩽ - ζ (t) G (h (t)), \forall t ⩾ 0, \end{matrix}$ where $ζ : R_{+} \to R_{+}$ is a nonincreasing differentiable function.

The functions $Q, P : H \to H$ , $F : D (A^{\frac{1}{2}}) \to H$ are locally Lipschitz mappings such that there exist continuous and differentiable mappings $P : H \to R_{+}$ and $F : D (A^{\frac{1}{2}}) \to R_{+}$ satisfying $\begin{matrix} D F = F, D P = P . \end{matrix}$ and, for $α_{1}$ , $α_{3}$ and $α_{2} > 1$ be a positive constants, we have $\begin{array}{l} (2.4) & ⟨ F (u), u ⟩ ⩾ F (u), \forall u \in D (A^{\frac{1}{2}}), \\ (2.5) & ⟨ Q (u), u ⟩ ⩾ α_{1} P (u), \forall u \in H, \\ (2.6) & | ⟨ P (u), v ⟩ | ⩽ (α_{2} - 1) P (u) + P (v) and P (u) ⩽ α_{3} {‖ A^{\frac{1}{2}} u ‖}^{2} . \end{array}$

Moreover, for $i = 1, 2$ , there exists an increasing continuous function $ψ_{i} : R_{+} \to R_{+}$ , with $ψ_{i} (0) = 0$ , such that $\begin{array}{l} (2.7) & ⟨ Q (u), v ⟩ ⩽ ψ_{1} (⟨ u, v ⟩), \forall u, v \in H, \\ (2.8) & | ⟨ F (u), v ⟩ | ⩽ ψ_{2} (‖ A^{\frac{1}{2}} u ‖) ‖ A^{\frac{1}{2}} u ‖ ‖ A^{\frac{1}{2}} v ‖, \forall u, v \in D (A^{\frac{1}{2}}) . \end{array}$

The coefficients of delay and dissipation satisfy $\begin{matrix} (2.9) & | μ_{2} | < \frac{μ_{1} α_{1}}{α_{2}} . \end{matrix}$

Remark 2.1.

Similarly to [35], from (A2), we clearly deduce that ${lim}_{t \to + \infty} h (s) = 0$ . Therefore, there exists $t_{1} ⩾ 0$ , such that $\begin{matrix} (2.10) & h (t_{1}) = r \Rightarrow h (t) ⩽ r, \forall t ⩾ t_{1} . \end{matrix}$ By using the fact that h and ζ are positive nonincreasing continuous functions and G is positive continuous function, we have, for all $t \in [0, t_{1}]$ , $\begin{matrix} δ_{1} ⩽ ζ (t) G (h (t)) ⩽ δ_{2} . \end{matrix}$ for $δ_{1}$ and $δ_{2}$ are positive constants, which gives, for all $t \in [0, t_{1}]$ , $\begin{matrix} (2.11) & h^{'} (t) ⩽ - ζ (t) G (h (t)) ⩽ - \frac{δ_{1}}{h (0)} h (0) ⩽ - \frac{δ_{1}}{h (0)} h (t) . \end{matrix}$

If G is a strictly increasing and strictly convex $C^{2}$ function on $(0, r]$ , with $G (0) = G^{'} (0) = 0$ , then G has an extension $\overline{G}$ which is a strictly increasing and strictly convex $C^{2}$ function on $(0, + \infty)$ . Moreover, we can define $\overline{G}$ by $\begin{matrix} (2.12) & \overline{G} (t) = \frac{c}{2} t^{2} + (b - c r) t + (a + \frac{c}{2} r^{2} - b r), for t > r, \end{matrix}$ where $a = G (r)$ , $b = G^{'} (r)$ and $c = G^{″} (r)$ .

If F is a convex function on $[a, b]$ , $f : Ω \to [a, b]$ and h are integrable functions on $Ω, h (x) ⩾ 0$ , and $\int_{Ω} h (x) d x = k > 0$ , then Jensen’s inequality states that $\begin{matrix} F [\frac{1}{k} \int_{Ω} f (x) h (x) d x] ⩽ \frac{1}{k} \int_{Ω} F [f (x)] h (x) d x . \end{matrix}$

In order to state and prove the desired results, as in [38], we introduce the variable z by $\begin{matrix} z (ρ, t) = u_{t} (t - ρ τ), ρ \in (0, 1), t > 0, \end{matrix}$ therefore, the problem (1.1) can be recast as the following problem $\begin{matrix} (2.13) & \{\begin{matrix} u_{t t} (t) + A u (t) - \int_{0}^{t} h (s) A u (s) d s \\ + μ_{1} Q (u_{t} (t)) + μ_{2} P (z (1, t)) = F (u (t)), & t \in (0, + \infty), \\ τ z_{t} (ρ, t) + z_{ρ} (ρ, t) = 0, & ρ \in (0, 1), t > 0, \\ z (1, t) = f_{0} (t - τ), & t \in (0, τ), \\ z (0, t) = u_{t} (t), & t ⩾ 0, \\ u (0) = u_{0}, u_{t} (0) = u_{1} . \end{matrix} \end{matrix}$

Now, let us define the modified energy functional E associated with problem (2.13) by $\begin{matrix} (2.14) & \begin{matrix} E (t) & = \frac{1}{2} (‖ u_{t} ‖^{2} + (1 - \int_{0}^{t} h (s) d s) {‖ A^{\frac{1}{2}} u ‖}^{2} \\ + (h ◇ A^{\frac{1}{2}} u) (t) - 2 F (u) + ξ τ \int_{0}^{1} P (z (ρ, t)) d ρ), \forall t \in R_{+}, \end{matrix} \end{matrix}$ with the initial energy $\begin{matrix} (2.15) & E (0) = \frac{1}{2} (‖ u_{1} ‖^{2} + {‖ A^{\frac{1}{2}} u_{0} ‖}^{2} - 2 F (u_{0}) + ξ τ \int_{0}^{1} P (z (ρ, 0)) d ρ), \end{matrix}$ where $\begin{matrix} (2.16) & (h ◇ A^{\frac{1}{2}} u) (t) = \int_{0}^{t} h (t - s) {‖ A^{\frac{1}{2}} u (t) - A^{\frac{1}{2}} u (s) ‖}^{2} d s \end{matrix}$ and ξ a positive constant such that $\begin{matrix} (2.17) & | μ_{2} | (α_{2} - 1) < ξ < μ_{1} α_{1} - | μ_{2} |, \end{matrix}$ note that ξ exists according to (2.9).
Lemma 2.2.
Assume that (A1)–(A4) hold. Then, the energy functional defined by ( 2.14 ) satisfies $\begin{matrix} (2.18) & E^{'} (t) ⩽ \frac{1}{2} (h^{'} ◇ A^{\frac{1}{2}} u) (t) - c_{1} ⟨ Q (u_{t}), u_{t} ⟩ ⩽ 0, \forall t \in R_{+} . \end{matrix}$
Proof.
Using the first equation of (2.13), we obtain $\begin{matrix} ⟨ u_{t t}, u_{t} ⟩ + ⟨ A u, u_{t} ⟩ - ⟨ \int_{0}^{t} h (t - s) A u (s) d s, u_{t} ⟩ + μ_{1} ⟨ Q (u_{t}), u_{t} ⟩ + μ_{2} ⟨ P (z (1, t)), u_{t} ⟩ = ⟨ F (u), u_{t} ⟩ . \end{matrix}$ Then, we have $\begin{matrix} (2.19) & \begin{matrix} \frac{1}{2} \frac{d}{d t} (‖ u_{t} ‖^{2} + {‖ A^{\frac{1}{2}} u ‖}^{2} - 2 F (u)) + μ_{1} ⟨ Q (u_{t}), u_{t} ⟩ + μ_{2} ⟨ P (z (1, t)), u_{t} ⟩ \\ = \int_{0}^{t} h (t - s) ⟨ A u (s), u_{t} ⟩ d s . \end{matrix} \end{matrix}$ On the other hand, we can easily check that $\begin{matrix} (2.20) & \begin{matrix} 2 & \int_{0}^{t} h (t - s) ⟨ A u (s), u_{t} ⟩ d s \\ = \frac{d}{d t} [\int_{0}^{t} h (s) d s {‖ A^{\frac{1}{2}} u ‖}^{2} - (h ◇ A^{\frac{1}{2}} u) (t)] + (h^{'} ◇ A^{\frac{1}{2}} u) (t) - h (t) {‖ A^{\frac{1}{2}} u ‖}^{2} . \end{matrix} \end{matrix}$ Consequently, we have $\begin{array}{l} \frac{d}{d t} {\frac{1}{2} (‖ u_{t} ‖^{2} + (1 - \int_{0}^{t} h (s) d s) {‖ A^{\frac{1}{2}} u ‖}^{2} + (h ◇ A^{\frac{1}{2}} u) (t)) - F (u)} \\ - \frac{1}{2} (h^{'} ◇ A^{\frac{1}{2}} u) (t) + \frac{1}{2} h (t) {‖ A^{\frac{1}{2}} u ‖}^{2} + μ_{1} ⟨ Q (u_{t}), u_{t} ⟩ + μ_{2} ⟨ P (z (1, t)), u_{t} ⟩ = 0, \end{array}$ using (2.6), we arrive at $\begin{array}{l} \frac{d}{d t} {\frac{1}{2} (‖ u_{t} ‖^{2} + (1 - \int_{0}^{t} h (s) d s) {‖ A^{\frac{1}{2}} u ‖}^{2} + (h ◇ A^{\frac{1}{2}} u) (t)) - F (u)} \\ ⩽ | μ_{2} | [(α_{2} - 1) P (z (1, t)) + P (u_{t})] - μ_{1} ⟨ Q (u_{t}), u_{t} ⟩ \\ (2.21) & + \frac{1}{2} (h^{'} ◇ A^{\frac{1}{2}} u) (t) - \frac{1}{2} h (t) {‖ A^{\frac{1}{2}} u ‖}^{2} . \end{array}$ Similarly, by the second equation of (2.13), we have $\begin{array}{rcl} \frac{d}{d t} (ξ τ \int_{0}^{1} P (z (ρ, t)) d ρ) & = & ξ τ \int_{0}^{1} ⟨ z_{t}, P (z (ρ, t)) ⟩ d ρ = - ξ \int_{0}^{1} \frac{\partial}{\partial ρ} P (z (ρ, t)) d ρ \\ (2.22) & = & ξ [P (u_{t}) - P (z (1, t))] . \end{array}$ By replacing (2.22) in (2.21) and using (2.5), we obtain $\begin{array}{rcl} E^{'} (t) & ⩽ & \frac{1}{2} (h^{'} ◇ A^{\frac{1}{2}} u) (t) - \frac{1}{2} h (t) {‖ A^{\frac{1}{2}} u ‖}^{2} + (\frac{| μ_{2} |}{α_{1}} - μ_{1} + \frac{ξ}{α_{1}}) ⟨ Q (u_{t}), u_{t} ⟩ \\ + (| μ_{2} | (α_{2} - 1) - ξ) P (z (1, t)) . \end{array}$ From (2.17), we obtain $\begin{matrix} E^{'} (t) ⩽ \frac{1}{2} (h^{'} ◇ A^{\frac{1}{2}} u) (t) - c_{1} ⟨ Q (u_{t}), u_{t} ⟩, \end{matrix}$ where $\begin{matrix} c_{1} = min {\frac{| μ_{2} |}{α_{1}} - μ_{1} + \frac{ξ}{α_{1}}, | μ_{2} | (α_{2} - 1) - ξ} . \end{matrix}$ which is positive by (2.17). This completes the proof of the Lemma. □

In the following result, we state, without proof, the local existence of (1.1), see [9,10].
Proposition 2.3.
Assume that the assumptions (A1)–(A4) hold and for an initial datum $(u_{0}, u_{1}, f_{0}) \in D (A^{\frac{1}{2}}) \times H \times L^{2} (0, 1; H)$ . Then, the system ( 1.1 ) has a unique local mild solution.

By using Lemma 2.2, we prove the global existence of solution of problem (2.13) under small initial conditions. Theorem 2.4.
Assume that (A1)–(A4) hold and there exist a positive constant $ρ_{0}$ such that for any $(u_{0}, u_{1}, f_{0}) \in D (A^{\frac{1}{2}}) \times H \times L^{2} (0, 1; H)$ satisfying $\begin{matrix} {(‖ A^{\frac{1}{2}} u_{0} ‖ + ‖ u_{1} ‖^{2} + \int_{0}^{τ} {\tilde{f}}_{0} (s) d s)}^{\frac{1}{2}} < ρ_{0}, \end{matrix}$ where ${\tilde{f}}_{0} (s) = P (f_{0} (s))$ . Then, the problem ( 2.13 ) admits a unique mild solution u on $[0, + \infty)$ .
Proof.
From the proposition 2.3, the problem admits a unique local solution u in a maximal time interval $[0, T)$ .

Now, similarly to [4] and by using (2.4), (2.8) and (2.15), we have $\begin{matrix} E (0) ⩾ \frac{1}{2} ‖ u_{1} ‖^{2} + \frac{1}{2} {‖ A^{\frac{1}{2}} u_{0} ‖}^{2} - F (u_{0}) ⩾ \frac{1}{2} ‖ u_{1} ‖^{2} + \frac{l}{4} {‖ A^{\frac{1}{2}} u_{0} ‖}^{2} ⩾ 0, \end{matrix}$ if $ψ_{2} (‖ A^{\frac{1}{2}} u ‖) < \frac{l}{4}$ where $l = (1 - h_{0})$ . Furthermore, we show that if $\begin{matrix} (2.23) & ψ_{2} (‖ A^{\frac{1}{2}} u ‖) < \frac{l}{4} and ψ_{2} (2 {(\frac{E (0)}{l})}^{\frac{1}{2}}) < \frac{l}{4}, \end{matrix}$ then $\begin{array}{rcl} E (t) & ⩾ & \frac{1}{2} {‖ u_{t} (t) ‖}^{2} + \frac{1}{2} (1 - \int_{0}^{t} h (s) d s) {‖ A^{\frac{1}{2}} u (t) ‖}^{2} - F (u (t)) \\ ⩾ & \frac{1}{2} {‖ u_{t} (t) ‖}^{2} + \frac{1 - h_{0}}{2} {‖ A^{\frac{1}{2}} u (t) ‖}^{2} - F (u (t)) \\ (2.24) & ⩾ & \frac{1}{2} {‖ u_{t} (t) ‖}^{2} + \frac{l}{4} {‖ A^{\frac{1}{2}} u (t) ‖}^{2}, \forall t \in [0, T) . \end{array}$ Now, let ν be the supremum of all $s \in [0, T)$ such that (2.24) holds true for any $t \in [0, s]$ . Suppose $ν < T$ . By continuity of the function E, we obtain $\begin{matrix} (2.25) & E (ν) ⩾ \frac{1}{2} {‖ u_{t} (ν) ‖}^{2} + \frac{l}{4} {‖ A^{\frac{1}{2}} u (ν) ‖}^{2} ⩾ 0 . \end{matrix}$ Hence, from (2.25), we have $\begin{matrix} ψ_{2} (‖ A^{\frac{1}{2}} u (ν) ‖) ⩽ ψ_{2} (2 {(\frac{E (ν)}{l})}^{\frac{1}{2}}) ⩽ ψ_{2} (2 {(\frac{E (0)}{l})}^{\frac{1}{2}}) < \frac{l}{4}, \end{matrix}$ which gives $\begin{array}{rcl} E (ν) & ⩾ & \frac{1}{2} {‖ u_{t} (ν) ‖}^{2} + \frac{1 - \int_{0}^{ν} h (s) d s}{2} {‖ A^{\frac{1}{2}} u (ν) ‖}^{2} - F (u (ν)) \\ ⩾ & \frac{1}{2} {‖ u_{t} (ν) ‖}^{2} + (\frac{l}{2} - \frac{l}{4}) {‖ A^{\frac{1}{2}} u (ν) ‖}^{2} = \frac{1}{2} {‖ u_{t} (ν) ‖}^{2} + \frac{l}{4} {‖ A^{\frac{1}{2}} u (ν) ‖}^{2} . \end{array}$ This contradicts the maximality of ν. Let $\begin{matrix} ρ_{0} = \frac{\sqrt{l}}{2} ψ_{2}^{- 1} (\frac{l}{4}) > 0 . \end{matrix}$ Then $ψ_{2} (‖ A^{\frac{1}{2}} u_{0} ‖) < \frac{l}{4}$ for any $u_{0} \in D (A^{\frac{1}{2}})$ and any $u_{1} \in H$ such that $\begin{matrix} (2.26) & {({‖ A^{\frac{1}{2}} u_{0} ‖}^{2} + ‖ u_{1} ‖^{2} + \int_{0}^{τ} {\tilde{f}}_{0} (s) d s)}^{\frac{1}{2}} < ρ_{0}, \end{matrix}$ where ${\tilde{f}}_{0} (s) = P (f_{0} (s))$ . This assumption implies that $‖ A^{\frac{1}{2}} u_{0} ‖ < ρ_{0}$ , so, we have $\begin{matrix} ψ_{2} (‖ A^{\frac{1}{2}} u_{0} ‖) < ψ_{2} (ρ_{0}) = ψ_{2} (\frac{\sqrt{l}}{2} ψ_{2}^{- 1} (\frac{l}{4})) . \end{matrix}$ Moreover, by using (2.4) and (2.8), we obtain $\begin{array}{rcl} E (0) & ⩽ & \frac{1}{2} ‖ u_{1} ‖^{2} + \frac{1}{2} ‖ A^{\frac{1}{2}} u_{0} ‖ + \frac{1}{2} \int_{0}^{τ} {\tilde{f}}_{0} (s) d s \\ ⩽ & (‖ A^{\frac{1}{2}} u_{0} ‖ + ‖ u_{1} ‖^{2} + \int_{0}^{τ} {\tilde{f}}_{0} (s) d s) < ρ_{0}^{2}, \end{array}$ and, by the definition of $ρ_{0}$ , we deduce that $\begin{matrix} ψ_{2} (2 {(\frac{E (0)}{l})}^{\frac{1}{2}}) < ψ_{2} (ψ_{2}^{- 1} (\frac{l}{4})) = \frac{l}{4} . \end{matrix}$ In addition, under the assumption (2.26) and (2.23), we get $\begin{matrix} (2.27) & 0 ⩽ \frac{1}{2} {‖ u_{t} (t) ‖}^{2} + \frac{l}{4} {‖ A^{\frac{1}{2}} u (t) ‖}^{2} ⩽ E (t) ⩽ E (0) ⩽ ρ_{0}^{2} . \end{matrix}$ Thus, the energy function is nonnegative on $[0, T)$ and bounded which means that the solution exists on $[0, + \infty)$ and from (2.27), we have $\begin{matrix} (2.28) & ψ_{2} (‖ A^{\frac{1}{2}} u (t) ‖) ⩽ ψ_{2} (2 {(\frac{E (t)}{l})}^{\frac{1}{2}}) ⩽ ψ_{2} (2 {(\frac{E (0)}{l})}^{\frac{1}{2}}) < \frac{l}{4}, \forall t ⩾ 0 . \end{matrix}$ This completes the proof of Theorem 2.4. □

3. Technical lemmas

In this section, we present some technical lemmas needed to prove our main results.

Lemma 3.1.
Let u be the solution of ( 2.13 ). Then the functional $\begin{matrix} (3.1) & I_{1} (t) = ⟨ u_{t} (t), u (t) ⟩, \end{matrix}$ satisfies, for all $t ⩾ 0$ $\begin{array}{rcl} (3.2) & I_{1}^{'} (t) & ⩽ & ‖ u_{t} ‖^{2} - [\frac{l}{2} - α_{3} (| μ_{2} | + \frac{μ_{1}}{a})] {‖ A^{\frac{1}{2}} u ‖}^{2} + | μ_{2} | (α_{2} - 1) P (z (1, t)) \\ (3.3) & + \frac{μ_{1}}{4 α_{3}} {‖ Q (u_{t}) ‖}^{2} + \frac{C_{α}}{l} (k ◇ A^{\frac{1}{2}} u) (t), \end{array}$ for any $0 < α < 1$ , where $\begin{matrix} (3.4) & C_{α} = \int_{0}^{+ \infty} \frac{h^{2} (s)}{α h (s) - h^{'} (s)} d s and k (t) = α h (t) - h^{'} (t), \end{matrix}$ were introduced in Jin et al. [ 23 ].
Proof.
Differentiating (3.1) with respect to t, we find $\begin{matrix} I_{1}^{'} (t) = ‖ u_{t} ‖^{2} + ⟨ u_{t t} (t), u (t) ⟩ . \end{matrix}$ On the other hand, multiplying the first equation of (2.13) by $u (t)$ , we have $\begin{array}{l} ⟨ u_{t t} (t), u (t) ⟩ + ⟨ A u (t), u (t) ⟩ - ⟨ \int_{0}^{t} h (t - s) A u (s) d s, u (t) ⟩ + μ_{1} ⟨ Q (u_{t} (t)), u (t) ⟩ \\ + μ_{2} ⟨ P (z (1, t)), u (t) ⟩ = ⟨ F (u (t)), u (t) ⟩, \end{array}$ By the definition of $A^{\frac{1}{2}}$ , we have $\begin{array}{rcl} I_{1}^{'} (t) & = & ‖ u_{t} ‖^{2} - (1 - \int_{0}^{t} h (s) d s) {‖ A^{\frac{1}{2}} u ‖}^{2} + ⟨ \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (s) - u (t)) d s, A^{\frac{1}{2}} u (t) ⟩ \\ (3.5) & - μ_{1} ⟨ Q (u_{t} (t)), u (t) ⟩ - μ_{2} ⟨ P (z (1, t)), u (t) ⟩ + ⟨ F (u (t)), u (t) ⟩ . \end{array}$ Then, by using Cauchy–Schwarz’s and Young’s inequalities, we obtain $\begin{matrix} (3.6) & \begin{matrix} ⟨ \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (s) - u (t)) d s, A^{\frac{1}{2}} u (t) ⟩ \\ ⩽ \frac{l}{4} {‖ A^{\frac{1}{2}} u ‖}^{2} + \frac{1}{l} {‖ \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (s) - u (t)) d s ‖}^{2} . \end{matrix} \end{matrix}$ Moreover, we have $\begin{array}{l} {‖ \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (s) - u (t)) d s ‖}^{2} \\ ⩽ {(\int_{0}^{t} h (t - s) ‖ A^{\frac{1}{2}} (u (s) - u (t)) ‖ d s)}^{2} \\ ⩽ {(\int_{0}^{t} \frac{h (t - s)}{\sqrt{α h (t - s) - h^{'} (t - s)}} \sqrt{α h (t - s) - h^{'} (t - s)} ‖ A^{\frac{1}{2}} (u (s) - u (t)) ‖ d s)}^{2} \\ ⩽ (\int_{0}^{t} \frac{h^{2} (s)}{α h (s) - h^{'} (s)} d s) \int_{0}^{t} (α h (t - s) - h^{'} (t - s)) {‖ A^{\frac{1}{2}} (u (s) - u (t)) ‖}^{2} d s \\ (3.7) & ⩽ C_{α} (k ◇ A^{\frac{1}{2}} u) (t) . \end{array}$ Using (2.1), Young’s and Cauchy Schwarz’ inequalities, we obtain, for any $α_{3} > 0$ , $\begin{matrix} (3.8) & - μ_{1} ⟨ Q (u_{t} (t)), u (t) ⟩ ⩽ \frac{μ_{1}}{4 α_{3}} {‖ Q (u_{t}) ‖}^{2} + \frac{α_{3} μ_{1}}{a} {‖ A^{\frac{1}{2}} u ‖}^{2} . \end{matrix}$ Next, using (2.6) we get $\begin{matrix} (3.9) & - μ_{2} ⟨ P (z (1, t)), u (t) ⟩ ⩽ | μ_{2} | ((α_{2} - 1) P (z (1, t)) + α_{3} {‖ A^{\frac{1}{2}} u ‖}^{2}) . \end{matrix}$ By using (2.8) and (2.28), we have, for $u \in D (A^{\frac{1}{2}})$ $\begin{matrix} (3.10) & ⟨ F (u (t)), u (t) ⟩ ⩽ ψ_{2} (‖ A^{\frac{1}{2}} u ‖) {‖ A^{\frac{1}{2}} u ‖}^{2} ⩽ \frac{l}{4} {‖ A^{\frac{1}{2}} u ‖}^{2}, \end{matrix}$ Substituting the inequalities (3.6), (3.7), (3.8), (3.9) and (3.10) in (3.5), we get (3.2). □
Lemma 3.2.
Let u be the solution of ( 2.13 ). Then the functional $\begin{matrix} (3.11) & I_{2} (t) = - ⟨ u_{t} (t), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩, \end{matrix}$ satisfies, for $ε > 0$ and for all $t ⩾ 0$ $\begin{array}{rcl} I_{2}^{'} (t) & ⩽ & - (\int_{0}^{t} h (s) d s - ε) ‖ u_{t} ‖^{2} + ε {‖ A^{\frac{1}{2}} u ‖}^{2} + \frac{c_{3} (1 + C_{α})}{ε} (k ◇ A^{\frac{1}{2}} u) (t) \\ (3.12) & + | μ_{2} | (α_{2} - 1) P (z (1, t)) + ε {‖ Q (u_{t}) ‖}^{2}, \end{array}$ where $c = max {\frac{c_{2}}{2 a}, ε (| μ_{2} | α_{3} + 1) + \frac{μ_{1}^{2}}{4 a} + \frac{17 l^{2}}{32} + \frac{α^{2}}{2 a}}$
Proof.
Differentiating (3.11) with respect to t, we find $\begin{matrix} I_{2}^{'} (t) & = - ⟨ u_{t t} (t), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ - ⟨ u_{t} (t), \int_{0}^{t} h^{'} (t - s) (u (t) - u (s)) d s ⟩ \\ - \int_{0}^{t} h (s) d s ‖ u_{t} ‖^{2} . \end{matrix}$ Then, using the first equation of (2.13) and using the definition of $A^{\frac{1}{2}}$ , we get $\begin{array}{rcl} I_{2}^{'} (t) & = & - \int_{0}^{t} h (s) d s ‖ u_{t} ‖^{2} + (1 - \int_{0}^{t} h (s) d s) ⟨ A^{\frac{1}{2}} u (t), \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (t) - u (s)) d s ⟩ \\ + μ_{2} ⟨ P (z (1, t)), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ \\ - ⟨ F (u (t)), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ \\ - ⟨ u_{t} (t), \int_{0}^{t} h^{'} (t - s) (u (t) - u (s)) d s ⟩ + μ_{1} ⟨ Q (u_{t} (t)), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ \\ (3.13) & + {‖ \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (s) - u (t)) d s ‖}^{2} . \end{array}$ By using Cauchy–Schwarz’s and Young’s inequalities and (3.7), we get, for $ε > 0$ , $\begin{matrix} (1 - \int_{0}^{t} h (s) d s) ⟨ A^{\frac{1}{2}} u (t), \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (t) - u (s)) d s ⟩ ⩽ \frac{ε}{2} {‖ A^{\frac{1}{2}} u ‖}^{2} + \frac{l^{2} C_{α}}{2 ε} (k ◇ A^{\frac{1}{2}} u) (t) \end{matrix}$ and $\begin{matrix} μ_{1} ⟨ Q (u_{t} (t)), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ ⩽ ε {‖ Q (u_{t}) ‖}^{2} + \frac{μ_{1}^{2} C_{α}}{4 a ε} (k ◇ A^{\frac{1}{2}} u) (t) . \end{matrix}$ Furthermore, by (2.6) and (3.7) we obtain $\begin{matrix} μ_{2} ⟨ P (z (1, t)), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ ⩽ | μ_{2} | ((α_{2} - 1) P (z (1, t)) + α_{3} C_{α} (k ◇ A^{\frac{1}{2}} u) (t)) . \end{matrix}$ By using (2.8), (2.28), (3.7) and Young’s inequality, we have $\begin{matrix} (3.14) & ⟨ F (u (t)), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ ⩽ \frac{ε}{2} {‖ A^{\frac{1}{2}} u ‖}^{2} + \frac{l^{2} C_{α}}{32 ε} (k ◇ A^{\frac{1}{2}} u) (t) . \end{matrix}$ On the other hand and by using (3.4), we get $\begin{array}{l} - ⟨ u_{t} (t), \int_{0}^{t} h^{'} (t - s) (u (t) - u (s)) d s ⟩ \\ = ⟨ u_{t} (t), \int_{0}^{t} k (t - s) (u (t) - u (s)) d s ⟩ - ⟨ u_{t} (t), \int_{0}^{t} α h (t - s) (u (t) - u (s)) d s ⟩ \\ ⩽ ε {‖ u_{t} (t) ‖}^{2} + \frac{1}{2 ε} {(\int_{0}^{t} \sqrt{k (t - s)} \sqrt{k (t - s)} ‖ u (t) - u (s) ‖ d s)}^{2} \\ + \frac{α^{2}}{2 ε} {(\int_{0}^{t} h (t - s) ‖ u (t) - u (s) ‖ d s)}^{2} \\ ⩽ ε {‖ u_{t} (t) ‖}^{2} + (\frac{\int_{0}^{t} k (s) d s}{2 ε a} + \frac{α^{2} C_{α}}{2 ε a}) (k ◇ A^{\frac{1}{2}} u) (t) \\ ⩽ ε {‖ u_{t} (t) ‖}^{2} + (\frac{c_{2}}{2 ε a} + \frac{α^{2} C_{α}}{2 ε a}) (k ◇ A^{\frac{1}{2}} u) (t), \end{array}$ where $c_{2} = α h_{0} + h (0)$ . Then, inserting these five inequalities and the inequality (3.7) in (3.13), we get (3.12). □
Lemma 3.3.
Let u be the solution of ( 2.13 ). Then the functional $\begin{matrix} (3.15) & I_{3} (t) = τ e^{2 τ} \int_{0}^{1} e^{- 2 τ ρ} P (z (ρ, t)) d ρ, \end{matrix}$ satisfies, for all $t ⩾ 0$ $\begin{matrix} (3.16) & I_{3}^{'} (t) ⩽ - 2 τ \int_{0}^{1} P (z (ρ, t)) d ρ + e^{2 τ} ‖ u_{t} ‖^{2} + \frac{e^{2 τ}}{4 α_{1}^{2}} {‖ Q (u_{t}) ‖}^{2} - P (z (1, t)) . \end{matrix}$
Proof.
By using the second equation of (2.13), we get $\begin{array}{rcl} I_{3}^{'} (t) & = & - e^{2 τ} \int_{0}^{1} e^{- 2 τ ρ} ⟨ z_{ρ} (ρ, t), P (z (ρ, t)) ⟩ d ρ \\ = & e^{2 τ} (- 2 τ \int_{0}^{1} e^{- 2 τ ρ} P (z (ρ, t)) d ρ - \int_{0}^{1} \frac{\partial}{\partial ρ} e^{- 2 τ ρ} P (z (ρ, t)) d ρ) . \end{array}$ Then, by integrating by parts and $z (0, t) = u_{t} (t)$ , we get $\begin{matrix} I_{3}^{'} (t) = - 2 τ e^{2 τ} \int_{0}^{1} e^{- 2 τ ρ} P (z (ρ, t)) d ρ + e^{2 τ} P (u_{t}) - P (z (1, t)) . \end{matrix}$ By using (2.5), Cauchy–Schwarz’s and Young’s inequalities, we get $\begin{matrix} P (u_{t}) ⩽ ‖ u_{t} ‖^{2} + \frac{1}{4 α_{1}^{2}} {‖ Q (u_{t}) ‖}^{2}, \end{matrix}$ then, using the fact that $e^{- 2 τ ρ} ⩾ e^{- 2 τ}$ , for any $ρ \in (0, 1)$ , we obtain (3.16). □
Lemma 3.4.
Let u be the solution of ( 2.13 ). Then the functional $\begin{matrix} (3.17) & I_{4} (t) = \int_{0}^{t} f (t - s) {‖ A^{\frac{1}{2}} u (s) ‖}^{2} d s, \end{matrix}$ where $f (t) = \int_{t}^{+ \infty} h (s) d s$ , satisfies, $\begin{matrix} (3.18) & I_{4}^{'} (t) ⩽ - \frac{1}{2} (h ◇ A^{\frac{1}{2}} u) (t) + 3 (1 - l) {‖ A^{\frac{1}{2}} u ‖}^{2}, \forall t ⩾ 0 . \end{matrix}$
Proof.
By differentiating (3.17), we get $\begin{matrix} I_{4}^{'} (t) = f (0) {‖ A^{\frac{1}{2}} u ‖}^{2} + \int_{0}^{t} f^{'} (t - s) {‖ A^{\frac{1}{2}} u (s) ‖}^{2} d s . \end{matrix}$ Then, by using Young’s inequality and the fact $f^{'} (t) = - h (t)$ , we obtain $\begin{array}{rcl} I_{4}^{'} (t) & = & f (0) {‖ A^{\frac{1}{2}} u ‖}^{2} - \int_{0}^{t} h (t - s) {‖ A^{\frac{1}{2}} u (s) ‖}^{2} d s \\ ⩽ & h_{0} {‖ A^{\frac{1}{2}} u ‖}^{2} - \int_{0}^{t} h (t - s) {‖ A^{\frac{1}{2}} (u (t) - u (s)) ‖}^{2} d s \\ - 2 ⟨ A^{\frac{1}{2}} u, \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (t) - u (s)) d s ⟩ . \end{array}$ But $\begin{array}{rcl} - 2 ⟨ A^{\frac{1}{2}} u, \int_{0}^{t} h (t - s) A^{\frac{1}{2}} (u (t) - u (s)) d s ⟩ & ⩽ & \frac{\int_{0}^{t} h (s) d s}{2 h_{0}} \int_{0}^{t} h (t - s) {‖ A^{\frac{1}{2}} (u (t) - u (s)) ‖}^{2} d s \\ + 2 h_{0} {‖ A^{\frac{1}{2}} u ‖}^{2} . \end{array}$ Moreover, as $\int_{0}^{t} h (s) d s ⩽ f (0) = h_{0}$ and by using (2.1), we get (3.18) where $h_{0} = 1 - l$ . □

4. General decay

In this section, we shall state and prove an explicit and general decay rate result of the energy function E. In the first subsection, we give the general decay theorem under the assumption (2.7) in the case of linear function $ψ_{1}$ and in the second subsection we derive the asymptotic decay in the case of nonlinear function $ψ_{1}$ by using some properties of the convex functions. For this, we construct a Lyapunov functional L equivalent to E, with which we can show the desired result. Throughout this section, c is used to denote generic positive constants. Let $\begin{matrix} (4.1) & L (t) = M E (t) + \sum_{i = 1}^{3} N_{i} I_{i} (t), \end{matrix}$ where M, $N_{1}$ , $N_{2}$ and $N_{3}$ are positive constants.

Lemma 4.1.
Assume that (A1)–(A4) hold, there exist two positive constants $β_{1}$ and $β_{2}$ such that $\begin{matrix} (4.2) & β_{1} E (t) ⩽ L (t) ⩽ β_{2} E (t) . \end{matrix}$
Proof.
Using Cauchy–Schwarz’s and Young’s inequalities, we have $\begin{array}{l} | L (t) - M E (t) | \\ ⩽ N_{1} | ⟨ u_{t}, u ⟩ | + N_{2} | ⟨ u_{t} (t), \int_{0}^{t} h (t - s) (u (t) - u (s)) d s ⟩ | \\ + N_{3} τ e^{2 τ} \int_{0}^{1} e^{- 2 τ ρ} P (z (ρ, t)) d ρ \\ ⩽ \frac{N_{1} + N_{2}}{2} ‖ u_{t} ‖^{2} + \frac{N_{1}}{2 a} {‖ A^{\frac{1}{2}} u ‖}^{2} + \frac{h_{0} N_{2}}{2 a} (h ◇ A^{\frac{1}{2}} u) (t) + N_{3} τ e^{2 τ} \int_{0}^{1} P (z (ρ, t)) d ρ \\ ⩽ C E (t) . \end{array}$ Then, by choosing M so large, we have $L \sim E$ . □
Lemma 4.2.
Under a suitable choice of M, $N_{1}$ , $N_{2}$ and $N_{3}$ , the Lyapunov functional L defined in ( 4.1 ) satisfies $\begin{matrix} (4.3) & L^{'} (t) ⩽ - 3 (1 - l) {‖ A^{\frac{1}{2}} u ‖}^{2} - ‖ u_{t} ‖^{2} + \frac{1}{4} (h ◇ A^{\frac{1}{2}} u) (t) + c_{4} {‖ Q (u_{t}) ‖}^{2}, \forall t ⩾ t_{1}, \end{matrix}$ where $c_{4} = (ε N_{2} + \frac{μ_{1}}{4 α_{3}} N_{1} + \frac{e^{2 τ}}{4 α_{1}^{2}} N_{3})$ .
Proof.
Combining (4.1), (2.18), (3.2), (3.12) and (3.16). Then, by using (3.4) and for $h_{1} = \int_{0}^{t_{1}} h (s) d s > 0$ , where $t_{1}$ was introduced in (2.10), we have, for all $t ⩾ t_{1}$ , $\begin{array}{rcl} L^{'} (t) & ⩽ & - [(h_{1} - ε) N_{2} - N_{1} - e^{2 τ} N_{3}] ‖ u_{t} ‖^{2} - [\frac{l}{2} N_{1} - α_{3} (| μ_{2} | + \frac{μ_{1}}{a}) N_{1} - ε N_{2}] {‖ A^{\frac{1}{2}} u ‖}^{2} \\ + \frac{α M}{2} (h ◇ A^{\frac{1}{2}} u) (t) + (ε N_{2} + \frac{μ_{1}}{4 α_{3}} N_{1} + \frac{e^{2 τ}}{4 α_{1}^{2}} N_{3}) {‖ Q (u_{t}) ‖}^{2} \\ - [\frac{M}{2} - \frac{C_{α}}{l} N_{1} - \frac{c (C_{α} + 1)}{ε} N_{2}] (k ◇ A^{\frac{1}{2}} u) (t) - 2 N_{3} τ \int_{0}^{1} P (z (ρ, t)) d ρ \\ - (N_{3} - | μ_{2} | (α_{2} - 1) (N_{1} + N_{2})) P (z (1, t)) . \end{array}$ Consequently, by taking $ε = \frac{l}{4 N_{2}}$ , we obtain $\begin{array}{rcl} L^{'} (t) & ⩽ & - [h_{1} N_{2} - \frac{l}{4} - N_{1} - e^{2 τ} N_{3}] ‖ u_{t} ‖^{2} - [\frac{l}{2} N_{1} - α_{3} (| μ_{2} | + \frac{μ_{1}}{a}) N_{1} - \frac{l}{4}] {‖ A^{\frac{1}{2}} u ‖}^{2} \\ - [\frac{M}{2} - \frac{4 c}{l} N_{2}^{2} - C_{α} (\frac{1}{l} N_{1} + \frac{4 c}{l} N_{2}^{2})] (k ◇ A^{\frac{1}{2}} u) (t) + \frac{α M}{2} (h ◇ A^{\frac{1}{2}} u) (t) \\ + (ε N_{2} + \frac{μ_{1}}{4 α_{3}} N_{1} + \frac{e^{2 τ}}{4 α_{1}^{2}} N_{3}) {‖ Q (u_{t}) ‖}^{2} - 2 N_{3} τ \int_{0}^{1} P (z (ρ, t)) d ρ \\ - [N_{3} - | μ_{2} | (α_{2} - 1) (N_{1} + N_{2})] P (z (1, t)) . \end{array}$

At this point, let take $α_{3} = \frac{a l}{4 (μ_{1} + a | μ_{2} |)}$ and choose $N_{1}$ large enough so that $\begin{matrix} \frac{l}{4} N_{1} - \frac{l}{4} > 4 (1 - l) . \end{matrix}$ Then, let pick $N_{3}$ and $N_{2}$ big enough so that $\begin{array}{l} N_{3} - | μ_{2} | (α_{2} - 1) (N_{1} + N_{2}) > 0, \\ h_{1} N_{2} - \frac{l}{4} - N_{1} - e^{2 τ} N_{3} > 1 . \end{array}$ Now, as $\frac{α h^{2} (s)}{α h (s) - h^{'} (s)} < h (s)$ and by using the Lebesgue dominated convergence theorem ( $h (\cdot)$ being a control function), we have $\begin{matrix} α C_{α} = \int_{0}^{+ \infty} \frac{α h^{2} (s)}{α h (s) - h^{'} (s)} d s \to 0 as α \to 0 . \end{matrix}$ Consequently, there is $0 < α_{0} < 1$ such that if $α < α_{0}$ , then $\begin{matrix} α C_{α} < \frac{1}{8 [\frac{1}{l} N_{1} + \frac{4 c}{l} N_{2}^{2}]} . \end{matrix}$ Then, we choose M large enough such that (4.2) is satisfied $\begin{matrix} \frac{M}{2} - \frac{4 c}{l} N_{2}^{2} > 0, \end{matrix}$ then, for M fixed, we choose α so that $\begin{matrix} α = \frac{1}{2 M} < α_{0}, \end{matrix}$ which gives $\begin{matrix} \frac{M}{2} - \frac{4 c}{l} N_{2}^{2} - C_{α} (\frac{1}{l} N_{1} + \frac{4 c}{l} N_{2}^{2}) > 0 . \end{matrix}$ Therefore, we arrive at $\begin{array}{rcl} L^{'} (t) & ⩽ & - 4 (1 - l) {‖ A^{\frac{1}{2}} u ‖}^{2} - ‖ u_{t} ‖^{2} + \frac{1}{4} (h ◇ A^{\frac{1}{2}} u) (t) - 2 N_{3} τ \int_{0}^{1} P (z (ρ, t)) d ρ \\ + (ε N_{2} + \frac{μ_{1}}{4 α_{3}} N_{1} + \frac{e^{2 τ}}{4 α_{1}^{2}} N_{3}) {‖ Q (u_{t}) ‖}^{2}, \end{array}$ which yields (4.3). □

4.1. First general decay

Theorem 4.3.
Assume that (A1)–(A4) hold and $ψ_{1}$ is linear. Then there exist positive constants $k_{1}$ , $k_{2}$ , $k_{3}$ and $k_{4}$ such that the solution of ( 1.1 ) satisfies, for all $t ⩾ t_{1}$ , $\begin{array}{l} (4.4) & E (t) ⩽ k_{1} e^{- k_{2} \int_{t_{1}}^{t} ζ (s) d s}, if G is linear \\ (4.5) & E (t) ⩽ k_{4} G_{1}^{- 1} (k_{3} \int_{t_{1}}^{t} ζ (s) d s), if G is nonlinear, \end{array}$ where $G_{1} (t) = \int_{t}^{r} \frac{d s}{s G^{'} (s)}$ , which is strictly decreasing and convex on $(0, r]$ , with ${lim}_{t \to 0} G_{1} (t) = + \infty$ .
Proof.
By using (2.11) and (2.18), we conclude that, for any $t_{1} ⩾ 0$ , $\begin{matrix} (4.6) & \begin{matrix} \int_{0}^{t_{1}} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s \\ ⩽ \frac{- h (0)}{δ_{1}} \int_{0}^{t_{1}} h^{'} (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s ⩽ - c E^{'} (t) . \end{matrix} \end{matrix}$ Inserting this estimate in (4.3) and introducing the following function F which is equivalent to E by $\begin{matrix} F (t) = L (t) + c E (t), \end{matrix}$ where c is a positive constant. In the case of $ψ_{1}$ is linear and by using (2.18) and (2.7), we have $\begin{matrix} (4.7) & {‖ Q (u_{t}) ‖}^{2} ⩽ c^{'} ⟨ Q (u_{t}), u_{t} ⟩ ⩽ - c E^{'} (t), \end{matrix}$ On the other hand, we have for some constant $m > 0$ and for all $t_{1} ⩾ 0$ , $\begin{array}{rcl} L^{'} (t) & ⩽ & - 3 (1 - l) {‖ A^{\frac{1}{2}} u ‖}^{2} - ‖ u_{t} ‖^{2} + \frac{1}{4} (h ◇ A^{\frac{1}{2}} u) (t) + c_{4} {‖ Q (u_{t}) ‖}^{2} \\ ⩽ & - m E (t) + c (h ◇ A^{\frac{1}{2}} u) (t) + c_{4} {‖ Q (u_{t}) ‖}^{2} \\ ⩽ & - m E (t) - c E^{'} (t) + c \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s + c_{4} {‖ Q (u_{t}) ‖}^{2}, \end{array}$ and by using (4.7), we obtain $\begin{matrix} (4.8) & F^{'} (t) ⩽ - m E (t) + c \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s - c E^{'} (t) . \end{matrix}$ Let consider the following two cases.
Case 1: G is linear.
By multiplying (4.8) by ζ and using (A2) and (2.18), we get $\begin{array}{rcl} ζ (t) F^{'} (t) & ⩽ & - m ζ (t) E (t) + c ζ (t) \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s - c ζ (t) E^{'} (t) \\ ⩽ & - m ζ (t) E (t) + c \int_{t_{1}}^{t} ζ (t) h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s - c ζ (t) E^{'} (t) \\ ⩽ & - m ζ (t) E (t) - c \int_{t_{1}}^{t} h^{'} (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s - c ζ (t) E^{'} (t), \end{array}$ by using the fact that ζ is nonincreasing, we deduce $\begin{matrix} ζ (t) F^{'} (t) ⩽ - m ζ (t) E (t) - 2 c E^{'} (t), \end{matrix}$ which gives $\begin{matrix} {(ζ F + 2 c E)}^{'} (t) ⩽ - m ζ (t) E (t) . \end{matrix}$ Consequently, by integrating this last over $(t_{1}, t)$ and using the fact that $ζ F + 2 c E \sim E$ , we obtain $\begin{matrix} E (t) ⩽ k_{1} e^{- k_{2} \int_{t_{1}}^{t} ζ (s) d s} \forall t ⩾ t_{1}, \end{matrix}$ where $k_{1}$ and $k_{2}$ are positive constants.
Case 2: G is nonlinear.
Firstly, we introduce the following function $\begin{matrix} L_{1} (t) = L (t) + I_{4} (t), \end{matrix}$ which is nonnegative by using Lemmas 3.4 and 4.2. Moreover, it satisfies $\begin{array}{rcl} L_{1}^{'} (t) & ⩽ & - (1 - l) {‖ A^{\frac{1}{2}} u ‖}^{2} - ‖ u_{t} ‖^{2} - \frac{1}{4} (h ◇ A^{\frac{1}{2}} u) (t) + c_{4} {‖ Q (u_{t}) ‖}^{2} \\ ⩽ & - β E (t) + c ⟨ Q (u_{t}), u_{t} ⟩, \end{array}$ where β is a positive constant. Then, using (2.18) and (2.7), we get $\begin{array}{rcl} L_{1}^{'} (t) & ⩽ & - β E (t) + c ⟨ Q (u_{t}), u_{t} ⟩ \\ ⩽ & - β E (t) - c E^{'} (t), \end{array}$ Hence, $\begin{matrix} β \int_{t_{1}}^{t} E (s) d s ⩽ L_{1} (t_{1}) - L_{1} (t) ⩽ L_{1} (t_{1}), \end{matrix}$ this gives $\begin{matrix} (4.9) & \int_{0}^{+ \infty} E (s) d s < + \infty . \end{matrix}$ Let now define the function I by $\begin{matrix} I (t) = p \int_{t_{1}}^{t} {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s, \forall t ⩾ t_{1}, \end{matrix}$ where p be a positive constant, so $I (t) > 0$ , for all $t ⩾ t_{1}$ , otherwise (4.8) leads to an exponential decay. Furthermore, we choose p so that $\begin{matrix} (4.10) & I (t) < 1 . \end{matrix}$ We also define the function λ by $\begin{matrix} λ (t) = - \int_{t_{1}}^{t} h^{'} (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s, \forall t ⩾ t_{1}, \end{matrix}$ for $t_{1}$ small enough and by (2.18), we observe that $\begin{matrix} (4.11) & λ (t) ⩽ - c E^{'} (t) . \end{matrix}$ Since G is strictly convex on $(0, r]$ and $G (0) = 0$ , then $\begin{matrix} (4.12) & G (θ x) ⩽ θ G (x), for some θ \in [0, 1] and x \in (0, r] . \end{matrix}$ By using the assumption (A2), (4.10) and Jensen’s inequality, we get $\begin{array}{rcl} λ (t) & = & \frac{1}{p I (t)} \int_{t_{1}}^{t} I (t) (- h^{'} (s)) p {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s \\ ⩾ & \frac{1}{p I (t)} \int_{t_{1}}^{t} I (t) ζ (s) G (h (s)) p {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s \\ ⩾ & \frac{ζ (t)}{p I (t)} \int_{t_{1}}^{t} G (I (t) h (s)) p {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s \\ ⩾ & \frac{ζ (t)}{p} G (\frac{1}{I (t)} \int_{t_{1}}^{t} I (t) h (s) p {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s) \\ = & \frac{ζ (t)}{p} G (p \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s) \\ = & \frac{ζ (t)}{p} \overline{G} (p \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s) \end{array}$ where $\overline{G}$ is an extension of G, which is strictly increasing and strictly convex $C^{2}$ on $(0, + \infty)$ , see Remark 2.1. The use of this fact and since ζ is a positive nonincreasing function, we obtain $\begin{matrix} (4.13) & \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s ⩽ \frac{1}{p} {(\overline{G})}^{- 1} (\frac{p λ (t)}{ζ (t)}), \end{matrix}$ and (4.8) becomes $\begin{matrix} (4.14) & F^{'} (t) ⩽ - m E (t) + c {(\overline{G})}^{- 1} (\frac{p λ (t)}{ζ (t)}), \forall t ⩾ t_{1} . \end{matrix}$ Let $0 < r_{1} < r$ , then we define the functional $F_{1}$ by $\begin{matrix} F_{1} (t) = {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) F (t) + E (t), \end{matrix}$ by using (4.14) and the fact that $E^{'} ⩽ 0$ , $G^{'} > 0$ and $G^{″} > 0$ , we conclude that $F_{1}$ is equivalent to E and $\begin{array}{rcl} F_{1}^{'} (t) & = & r_{1} \frac{E^{'} (t)}{E (0)} {\overline{G}}^{″} (r_{1} \frac{E (t)}{E (0)}) F (t) + {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) F^{'} (t) + E^{'} (t) \\ (4.15) & ⩽ & - m E (t) {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) + c {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) {(\overline{G})}^{- 1} (\frac{p λ (t)}{ζ (t)}) + E^{'} (t) . \end{array}$ Let ${\overline{G}}^{}$ be the convex conjugate of G in the sense of Young (see [6] pp. 61–64), which is given by $\begin{matrix} (4.16) & {\overline{G}}^{} (s) = s {({\overline{G}}^{'})}^{- 1} (s) - \overline{G} [{({\overline{G}}^{'})}^{- 1} (s)] \end{matrix}$ and it satisfies the following Young’s inequality $\begin{matrix} (4.17) & A B ⩽ {\overline{G}}^{} (A) + \overline{G} (B) . \end{matrix}$ By taking $\begin{matrix} A = {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) and B = {\overline{G}}^{- 1} (\frac{p λ (t)}{ζ (t)}), \end{matrix}$ and using (4.17), we obtain $\begin{array}{rcl} {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) {\overline{G}}^{- 1} (\frac{p λ (t)}{ζ (t)}) & ⩽ & {\overline{G}}^{} ({\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)})) + \frac{p λ (t)}{ζ (t)} \\ ⩽ & r_{1} \frac{E (t)}{E (0)} {\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) - \overline{G} (r_{1} \frac{E (t)}{E (0)}) + \frac{p λ (t)}{ζ (t)}, \end{array}$ then, multiplying by $ζ (t)$ and using (4.11) and the fact that, as $r_{1} \frac{E (t)}{E (0)} < r$ , ${\overline{G}}^{'} (r_{1} \frac{E (t)}{E (0)}) = G^{'} (r_{1} \frac{E (t)}{E (0)})$ to obtain $\begin{matrix} ζ (t) F_{1}^{'} (t) ⩽ - m ζ (t) E (t) G^{'} (r_{1} \frac{E (t)}{E (0)}) + c r_{1} ζ (t) \frac{E (t)}{E (0)} G^{'} (r_{1} \frac{E (t)}{E (0)}) - c E^{'} (t) . \end{matrix}$ On the other hand, the functional $F_{2} = ζ F_{1} + c E$ is equivalent to E which means for some $γ_{1}$ and $γ_{2}$ , we have $\begin{matrix} (4.18) & γ_{1} F_{2} (t) ⩽ E (t) ⩽ γ_{2} F_{2} (t), \end{matrix}$ and under a suitable choice of $r_{1}$ and for a positive constant k, we find $\begin{matrix} (4.19) & F_{2}^{'} (t) ⩽ - k ζ (t) \frac{E (t)}{E (0)} G^{'} (r_{1} \frac{E (t)}{E (0)}) = - k ζ (t) G_{2} (\frac{E (t)}{E (0)}), \end{matrix}$ where $G_{2} (t) = t G^{'} (r_{1} t)$ . Since $G_{2}^{'} (t) = G^{'} (r_{1} t) + r_{1} t G^{″} (r_{1} t)$ , and using the strict convexity of G on $(0, r]$ , we find that $G_{2}, G_{2}^{'} > 0$ on $(0, 1]$ . Finally, with $\begin{matrix} R (t) = γ_{1} \frac{F_{2} (t)}{E (0)}, \end{matrix}$ then, by using (4.18) and (4.19), we have $R \sim E$ and for some positive constant $k_{3}$ , (4.19) gives $\begin{matrix} R^{'} (t) ⩽ - k_{3} ζ (t) G_{2} (R (t)), \forall t ⩾ t_{1} . \end{matrix}$ A simple integration over $(t_{1}, t)$ , we find $\begin{matrix} \int_{t_{1}}^{t} \frac{- R^{'} (s)}{G_{2} (R (s))} d s ⩾ k_{3} \int_{t_{1}}^{t} ζ (s) d s . \end{matrix}$ Since $r_{1} R (t_{1}) < r$ , we obtain $\begin{matrix} G_{1} (r_{1} R (t)) = \int_{r_{1} R (t)}^{r_{1} R (t_{1})} \frac{d s}{s G^{'} (s)} ⩾ k_{3} \int_{t_{1}}^{t} ζ (s) d s . \end{matrix}$ Using the fact that $G_{1}$ is strictly decreasing function on $(0, r]$ and ${lim}_{t \to 0} G_{1} (t) = + \infty$ . Then $\begin{matrix} R (t) ⩽ \frac{1}{r_{1}} G_{1}^{- 1} (k_{3} \int_{t_{1}}^{t} ζ (s) d s), \end{matrix}$ consequently, by using the fact that R is equivalent to E, the stability estimate (4.5) is established. This completes the proof. □
Example 4.4.
Assume that (A2) holds with $G (s) = s^{p}$ , where $1 ⩽ p < 2$ . Then, the decay rate of E is given by $\begin{matrix} (4.20) & E (t) ⩽ \{\begin{matrix} k_{1} e^{- k_{2} \int_{0}^{t} ζ (s) d s}, & if p = 1 \\ k_{3} {(1 + \int_{0}^{t} ζ (s) d s)}^{\frac{- 1}{p - 1}}, & if 1 < p < 2, \end{matrix} \end{matrix}$ where $k_{1}$ , $k_{2}$ and $k_{3}$ are positive constants.

In this example, we can show that h not be necessarily of exponential or polynomial decay but under general assumption on the relaxation function h which gives a much larger class of functions h, the uniform stability of the system (1.1) is established with an explicit formula of the decay rates of the energy.
Remark 4.5.
The decay rate of E given by (4.3) is optimal in the sense that it’s consistent with the decay rate of h given by (2.3) where (4.4) and (4.5) provide the best decay rates expected under the very general assumption on h.
4.2. Second general decay

Theorem 4.6.
Assume that (A1)–(A4) hold and $ψ_{1}$ is nonlinear. Then there exist positive constants $k_{5}$ , $k_{6}$ , $k_{7}$ , $k_{8}$ and ${\tilde{r}}_{2}$ such that the solution of ( 1.1 ) satisfies, for all $t ⩾ t_{1}$ , $\begin{matrix} (4.21) & E (t) ⩽ K_{1}^{- 1} (k_{5} \int_{t_{1}}^{t} ζ (s) d s + k_{6}), if G is linear \end{matrix}$ where $K_{1} (t) = \int_{t}^{1} \frac{d s}{K_{2} (s)}$ and, for all $t ⩾ t_{2}$ $\begin{matrix} (4.22) & E (t) ⩽ k_{8} (t - t_{1}) W_{1}^{- 1} (\frac{k_{7}}{(t - t_{1}) \int_{t_{2}}^{t} ζ (s) d s}), if G is nonlinear, \end{matrix}$ where $W_{1} (t) = t W^{'} ({\tilde{r}}_{2} t)$ , $W = {({\overline{G}}^{- 1} + {\overline{K}}^{- 1})}^{- 1}$ .
Proof.
In this subsection, we assume that the nonlinear function $ψ_{1}$ is given by, for $s ⩾ 0$ , $\begin{matrix} ψ_{1} (s) = K^{- 1} (s), \end{matrix}$ where $K : R_{+} \to R_{+}$ be a strict convex and increasing function of class $C^{1} (R_{+}) \cap C^{2} (R_{+})$ satisfying $K (0) = 0$ , and K is linear on $[0, \tilde{r}]$ (or $K^{'} (0) = 0$ and $K^{″} > 0$ on $[0, \tilde{r}]$ ).

Consequently, the condition (2.7) can be return to $\begin{matrix} (4.23) & {‖ Q (u_{t}) ‖}^{2} ⩽ K^{- 1} (⟨ Q (u_{t}), u_{t} ⟩) ⩽ K^{- 1} (J (t)), \end{matrix}$ where J is given by $\begin{matrix} (4.24) & J (t) = ⟨ Q (u_{t}), u_{t} ⟩, \forall u \in H, \end{matrix}$

Let consider the following two cases.
Case 1: G is linear.
Multiplying (4.3) by ζ and by using (4.23), we obtain, for all $t ⩾ t_{1}$ , $\begin{array}{rcl} ζ (t) L^{'} (t) & ⩽ & - m ζ (t) E (t) + c \int_{t_{1}}^{t} ζ (t) h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s + C_{1} ζ (t) {‖ Q (u_{t}) ‖}^{2} \\ ⩽ & - m ζ (t) E (t) - c E^{'} (t) + C_{1} ζ (t) K^{- 1} (J (t)), \end{array}$ which gives, by using the fact that ζ is nonincreasing, $\begin{matrix} (4.25) & {\tilde{F}}^{'} (t) ⩽ - m ζ (t) E (t) + c ζ (t) K^{- 1} (J (t)), \forall t ⩾ t_{1} \end{matrix}$ where $\tilde{F} : = ζ L + c E$ , which is equivalent to E. Now, for $0 < {\tilde{r}}_{1} < \tilde{r}$ and $\tilde{c} > 0$ , then we define the functional ${\tilde{F}}_{1}$ by $\begin{matrix} {\tilde{F}}_{1} (t) : = K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) \tilde{F} (t) + \tilde{c} E (t), \end{matrix}$ by using $\tilde{F} \sim E$ , we conclude that ${\tilde{F}}_{1}$ is equivalent to E which means for some $γ_{3}$ and $γ_{4}$ , we have $\begin{matrix} (4.26) & γ_{3} {\tilde{F}}_{1} (t) ⩽ E (t) ⩽ γ_{4} {\tilde{F}}_{1} (t) . \end{matrix}$ Noting that $E^{'} ⩽ 0$ , $K^{'} > 0$ and $K^{″} > 0$ , and using (4.25), we infer $\begin{array}{rcl} {\tilde{F}}_{1}^{'} (t) & = & {\tilde{r}}_{1} \frac{E^{'} (t)}{E (0)} K^{″} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) \tilde{F} (t) + K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) {\tilde{F}}^{'} (t) + \tilde{c} E^{'} (t) \\ (4.27) & ⩽ & - m ζ (t) E (t) K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) + c ζ (t) K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) K^{- 1} (J (t)) + \tilde{c} E^{'} (t) . \end{array}$ Let $K^{}$ be the convex conjugate of K in the sense of Young (see [6] pp. 61–64), similarly to (4.16) and (4.17), with $A = K^{'} (r_{1} \frac{E (t)}{E (0)})$ and $B = K^{- 1} (J (t))$ , using (2.18) and (4.24), we arrive at $\begin{array}{rcl} {\tilde{F}}_{1}^{'} (t) & ⩽ & - m ζ (t) E (t) K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) + c {\tilde{r}}_{1} ζ (t) \frac{E (t)}{E (0)} K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) + c ζ (t) J (t) + \tilde{c} E^{'} (t) \\ ⩽ & - m ζ (t) E (t) K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) + c {\tilde{r}}_{1} ζ (t) \frac{E (t)}{E (0)} K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) - c E^{'} (t) + \tilde{c} E^{'} (t) . \end{array}$ Therefore, under a suitable choice of ${\tilde{r}}_{1}$ , $\tilde{c}$ and for a positive constant k, we find $\begin{matrix} (4.28) & {\tilde{F}}_{1}^{'} (t) ⩽ - k ζ (t) \frac{E (t)}{E (0)} K^{'} ({\tilde{r}}_{1} \frac{E (t)}{E (0)}) = - k ζ (t) K_{2} (\frac{E (t)}{E (0)}), \end{matrix}$ where $K_{2} (t) = t K^{'} ({\tilde{r}}_{1} t)$ . Since $K_{2}^{'} (t) = K^{'} ({\tilde{r}}_{1} t) + {\tilde{r}}_{1} t K^{″} ({\tilde{r}}_{1} t)$ , and using the strict convexity of K on $(0, \tilde{r}]$ , we find that $K_{2}, K_{2}^{'} > 0$ on $(0, 1]$ . Finally, with $\begin{matrix} \tilde{R} (t) = γ_{3} \frac{{\tilde{F}}_{1} (t)}{E (0)}, \end{matrix}$ then, by using (4.26) and (4.28), we have $\tilde{R} \sim E$ and for some positive constant $k_{5}$ , (4.28) gives $\begin{matrix} {\tilde{R}}^{'} (t) ⩽ - k_{5} ζ (t) K_{2} (\tilde{R} (t)) . \end{matrix}$ A simple integration over $(t_{1}, t)$ , we find $\begin{matrix} \tilde{R} (t) ⩽ K_{1}^{- 1} (k_{5} \int_{t_{1}}^{t} ζ (s) d s + k_{6}), \end{matrix}$ for some $k_{6} > 0$ with $K_{1} (t) = \int_{t}^{1} \frac{d s}{K_{2} (s)}$ . Consequently, by using the fact that $\tilde{R}$ is equivalent to E, the stability estimate (4.21) is established.
Case 2: G is nonlinear.
Let define the function $\tilde{I}$ by $\begin{matrix} \tilde{I} (t) = \frac{p}{(t - t_{1})} \int_{t_{1}}^{t} {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s, \end{matrix}$ where p be a positive constant, so $\tilde{I} (t) > 0$ , for all $t_{1} ⩾ 0$ . Furthermore, by a particular choice of p so that $\tilde{I} (t) < 1$ and by repeating exactly the same arguments to get (4.13), we have $\begin{matrix} (4.29) & \int_{t_{1}}^{t} h (s) {‖ A^{\frac{1}{2}} (u (t) - u (t - s)) ‖}^{2} d s ⩽ \frac{(t - t_{1})}{p} {\overline{G}}^{- 1} (\frac{p λ (t)}{(t - t_{1}) ζ (t)}) . \end{matrix}$ Consequently, by using (4.3), (4.6), (4.23) and (4.29), we obtain, for all $t ⩾ t_{1}$ , $\begin{matrix} (4.30) & L^{'} (t) ⩽ - m E (t) - c E^{'} (t) + c (t - t_{1}) {\overline{G}}^{- 1} (\frac{p λ (t)}{(t - t_{1}) ζ (t)}) + c K^{- 1} (J (t)) . \end{matrix}$ Let $\overline{K}$ is an extension of K, which is strictly increasing and strictly convex. Since ${lim}_{t \to + \infty} \frac{1}{t - t_{1}} = 0$ , so, there exists $t_{2} > t_{1}$ such that $\frac{1}{t - t_{1}} < 1$ , for $t > t_{2}$ . Then, by using (4.12) with $θ = \frac{1}{t - t_{1}} < 1$ , we obtain $\begin{matrix} {\overline{K}}^{- 1} (J (t)) ⩽ (t - t_{1}) {\overline{K}}^{- 1} (\frac{J (t)}{(t - t_{1})}), \forall t ⩾ t_{2} . \end{matrix}$ Hence, $\begin{matrix} (4.31) & {\tilde{L}}^{'} (t) ⩽ - m E (t) + c (t - t_{1}) {\overline{G}}^{- 1} (\frac{p λ (t)}{(t - t_{1}) ζ (t)}) + (t - t_{1}) {\overline{K}}^{- 1} (\frac{J (t)}{(t - t_{1})}), \end{matrix}$ where $\tilde{L} : = L + c E$ , which is equivalent to E. Let $\begin{matrix} r_{2} = min {r, \tilde{r}}, χ (t) = max {\frac{p λ (t)}{(t - t_{1}) ζ (t)}, \frac{J (t)}{(t - t_{1})}} and W = {({\overline{G}}^{- 1} + {\overline{K}}^{- 1})}^{- 1} . \end{matrix}$ Consequently, (4.31) reduce to $\begin{matrix} (4.32) & {\tilde{L}}^{'} (t) ⩽ - m E (t) + c (t - t_{1}) W^{- 1} (χ (t)), \forall t ⩾ t_{2} . \end{matrix}$ Now, for $0 < {\tilde{r}}_{2} < r_{2}$ , then we define the functional $F_{3}$ by $\begin{matrix} F_{3} (t) : = W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) \tilde{L} (t), \forall t ⩾ t_{2}, \end{matrix}$ by using (4.30) and the fact that $E^{'} ⩽ 0$ , $W^{'} > 0$ and $W^{″} > 0$ on $(0, r_{2}]$ , we conclude that $F_{3}$ is equivalent to E which means for some $γ_{5}$ and $γ_{6}$ , we have $\begin{matrix} (4.33) & γ_{5} F_{3} (t) ⩽ E (t) ⩽ γ_{6} F_{3} (t), \end{matrix}$ and $\begin{array}{rcl} F_{3}^{'} (t) & = & (\frac{- {\tilde{r}}_{2}}{{(t - t_{1})}^{2}} \cdot \frac{E (t)}{E (0)} + \frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E^{'} (t)}{E (0)}) W^{″} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) \tilde{L} (t) \\ + W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) {\tilde{L}}^{'} (t), \\ (4.34) & ⩽ & - m E (t) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) + c (t - t_{1}) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) W^{- 1} (χ (t)) . \end{array}$ Let $W^{}$ be the convex conjugate of W in the sense of Young (see [6] pp. 61–64), similarly to (4.16) and (4.17), with $\begin{matrix} A = W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) and B = W^{- 1} (χ (t)), \end{matrix}$ then, by using (2.18), we arrive at $\begin{matrix} (4.35) & F_{3}^{'} (t) ⩽ - m E (t) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) + c {\tilde{r}}_{2} \frac{E (t)}{E (0)} W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) - c (t - t_{1}) χ (t) . \end{matrix}$ On other hand, by using (2.18), (4.11) and (4.24), we have $\begin{array}{rcl} (t - t_{1}) ζ (t) χ (t) & ⩽ & p λ (t) + ζ (t) J (t) ⩽ p λ (t) + ζ (0) J (t) \\ ⩽ & - c E^{'} (t) . \end{array}$ Furthermore, multiplying (4.35) by $ζ (t)$ and using the fact that ${\tilde{r}}_{2} \frac{E (t)}{E (0)} < r_{2}$ , we get $\begin{matrix} ζ (t) F_{3}^{'} (t) ⩽ - m E (t) ζ (t) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) + c {\tilde{r}}_{2} ζ (t) \cdot \frac{E (t)}{E (0)} W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) - c E^{'} (t) . \end{matrix}$ Then, using the fact that ζ is nonincreasing function, we arrive at, for all $t ⩽ t_{2}$ , $\begin{matrix} F_{4}^{'} (t) ⩽ - m E (t) ζ (t) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) + c {\tilde{r}}_{2} ζ (t) \cdot \frac{E (t)}{E (0)} W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) - c E^{'} (t) . \end{matrix}$ where $F_{4} : = ζ F_{3} + c E$ which is equivalent to E. Therefore, for a suitable choice of ${\tilde{r}}_{2}$ and for a positive constant k, we get $\begin{matrix} k ζ (t) (\frac{E (t)}{E (0)}) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) ⩽ - F_{4}^{'} (t), \forall t ⩾ t_{2} . \end{matrix}$ An integration of this last yields $\begin{matrix} \int_{t_{2}}^{t} k ζ (s) (\frac{E (s)}{E (0)}) W^{'} (\frac{{\tilde{r}}_{2}}{(s - t_{1})} \cdot \frac{E (s)}{E (0)}) d s ⩽ - \int_{t_{2}}^{t} F_{4}^{'} (s) d s ⩽ F_{4} (t_{2}) . \end{matrix}$ By using the facts that $E^{'} ⩽ 0$ , $W^{'}, W^{″} > 0$ , we deduce that the map $t \mapsto E (t) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)})$ is nonincreasing. Consequently, we obtain $\begin{matrix} (4.36) & k (\frac{E (t)}{E (0)}) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) \int_{t_{2}}^{t} ζ (s) d s ⩽ F_{4} (t_{2}) . \end{matrix}$ Multiplying each side of (4.36) by $\frac{1}{t - t_{1}}$ , we arrive at $\begin{matrix} (4.37) & k (\frac{E (t)}{E (0)}) W^{'} (\frac{{\tilde{r}}_{2}}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) \int_{t_{2}}^{t} ζ (s) d s ⩽ \frac{k_{7}}{(t - t_{1})} . \end{matrix}$ Next, we set $W_{1} (t) = t W^{'} ({\tilde{r}}_{2} t)$ which is strictly increasing, then, we get $\begin{matrix} (4.38) & k W_{1} (\frac{1}{(t - t_{1})} \cdot \frac{E (t)}{E (0)}) \int_{t_{2}}^{t} ζ (s) d s ⩽ \frac{k_{7}}{(t - t_{1})} . \end{matrix}$ Finally, for two positive constants $k_{7}$ and $k_{8}$ , we obtain $\begin{matrix} E (t) ⩽ k_{8} (t - t_{1}) W_{1}^{- 1} (\frac{k_{7}}{(t - t_{1}) \int_{t_{2}}^{t} ζ (s) d s}) . \end{matrix}$ This completes the proof. □
Example 4.7.
Let consider the following examples:
G is linear.

Let $h (t) = a e^{- b (1 + t)}$ , where $a, b > 0$ is small enough so that (2.3) is satisfied, then $\begin{matrix} h^{'} (t) = - b G (h (t)), with G (t) = t, and ζ (t) = b . \end{matrix}$ Furthermore, we assume that $ψ_{1} (t) = c t^{\frac{1}{q}}$ , with $q > 1$ . Then, $K (t) = c t^{q}$ , applying (4.21) with $\begin{matrix} K_{1}^{- 1} (t) = {(c_{1} t + c_{2})}^{\frac{1}{1 - q}}, \end{matrix}$ consequently, we get $\begin{matrix} E (t) ⩽ \frac{1}{{(c_{3} t + c_{4})}^{\frac{1}{q - 1}}} . \end{matrix}$

G is nonlinear.

Let $h (t) = \frac{a}{{(1 + t)}^{2}}$ , where a is chosen so that (2.3) be satisfied. Then, for b is a fixed constant, we have $\begin{matrix} h^{'} (t) = - b G (h (t)), with G (t) = t^{\frac{3}{2}} . \end{matrix}$ Now, let $K (t) = c t^{3}$ , then $\begin{matrix} W (t) = {(G^{- 1} + H^{- 1})}^{- 1} (t) = {(\frac{\sqrt{1 + 4 t} - 1}{2})}^{3} \end{matrix}$ and $\begin{matrix} W_{2} (t) = \frac{3 t}{\sqrt{1 + 4 t}} {(\frac{\sqrt{1 + 4 t} - 1}{2})}^{3} ⩽ c t^{\frac{3}{2}}, \end{matrix}$ consequently, we get $\begin{matrix} E (t) ⩽ \frac{c}{{(t - t_{1})}^{\frac{1}{3}}} . \end{matrix}$

5. Applications

We can seek our result in many problems. Let Ω be a bounded and regular domain of $R^{n}$ , with $n ⩾ 1$ . In this section, we present only two illustrative problems.

5.1. Wave equations

We consider the following viscoelastic wave equation with a delay term in the non-linear internal feedback: $\begin{matrix} (5.1) & \{\begin{matrix} u_{t t} (t) - Δ u + \int_{0}^{t} h (t - s) Δ u (s) d s + μ_{1} | u_{t} (t) |^{m - 2} u_{t} (t) \\ + μ_{2} | u_{t} (t - τ) |^{m - 2} u_{t} (t - τ) = b | u (t) |^{p - 2} u (t), & x \in Ω, t \in (0, + \infty), \\ u (x, t) = 0, & x \in \partial Ω, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ u_{t} (t - τ) = f_{0} (t - τ), & t \in (0, τ), \end{matrix} \end{matrix}$ with initial data $(u_{0}, u_{1}, f_{0}) \in [H^{2} (Ω) \cap H_{0}^{1} (Ω)] \times H_{0}^{1} (Ω) \times H^{1} (- τ, 0; L^{2} (Ω))$ . m, b and P are positive constants with $2 ⩽ m < p ⩽ \frac{2 n}{n - 2}$ , if $n ⩾ 3$ and $2 ⩽ m < p ⩽ + \infty$ , if $n = 1, 2$ .

This problem is a particular case of (1.1) with $\begin{matrix} H = L^{2} (Ω), and A u = - Δ u, \forall u \in D (A), \end{matrix}$ where $D (A) = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ . It is well known that A is a positive self-adjoint operator with $D (A^{\frac{1}{2}}) = H_{0}^{1} (Ω)$ and $‖ u ‖_{D (A^{\frac{1}{2}})} = ‖ \nabla u ‖_{2}$ .

Furthermore, The assumption (A3) is verified, for $\begin{array}{l} Q (u) = P (u) = | u |^{m - 2} u, F (u) = b | u |^{p - 2} u, \\ P (u) = \frac{1}{m} \int_{Ω} | u |^{m} d x and F (u) = \frac{b}{p} \int_{Ω} | u |^{P} d x . \end{array}$

5.2. Plate equations

We can also consider the following system $\begin{matrix} \{\begin{matrix} u_{t t} (t) + Δ^{2} u (t) - \int_{0}^{t} h (t - s) Δ^{2} u (s) d s + μ_{1} Q (u_{t} (t)) \\ + μ_{2} P (u_{t} (t - τ)) = (\int_{Ω} K (x, y) u^{2} (y, t) d y) u (x, t), & t \in (0, + \infty), \\ u (x, t) = Δ u (x, t) = 0, & x \in \partial Ω, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ u_{t} (t - τ) = f_{0} (t - τ) & t \in (0, τ), \end{matrix} \end{matrix}$ with initial data $(u_{0}, u_{1}, f_{0}) \in [H^{4} (Ω) \cap H_{0}^{2} (Ω)] \times H_{0}^{2} (Ω) \times H^{1} (- τ, 0; L^{2} (Ω))$ , the function K is a bounded function from $Ω \times Ω \to (0, + \infty)$ such that $K (x, y) = K (y, x)$ and $\begin{matrix} \exists α, β > 0, α ⩽ K (x, y) ⩽ β, \forall x, y \in Ω . \end{matrix}$ The function $Q : R \to R$ is a non-decreasing continuous function and there exists a strictly increasing function $q_{0} \in C^{1} (R^{+})$ with $q_{0} (0) = 0$ such that $\begin{matrix} (5.2) & \{\begin{matrix} q_{0} (| s |) ⩽ | Q (s) | ⩽ q_{0}^{- 1} (| s |), & \forall | s | ⩽ ε, \\ c_{1} | s | ⩽ | Q (s) | ⩽ c_{2} | s |, & \forall | s | ⩾ ε, \end{matrix} \end{matrix}$ where $c_{1}$ , $c_{2}$ are positive constants. The function $P : R \to R$ is an odd non-decreasing function of the class $C^{1}$ such that there exist $a_{1}, a_{2}, a_{3} > 0$ $\begin{matrix} a_{1} s P (s) ⩽ P (s) ⩽ a_{2} s Q (s), | P^{'} (s) | ⩽ a_{3} . \end{matrix}$ where $P (s) = \int_{0}^{s} P (r) d r$ . Moreover, the assumption (A3) is verified for $\begin{matrix} F (u) = (\int_{Ω} K (x, y) u^{2} (y) d y) u (x), F (u) = \frac{1}{4} \int_{Ω \times Ω} K (x, y) u^{2} (x) u^{2} (y) d x d y . \end{matrix}$

This two applications generalize the earlier problems from [2,8,9,19,22,34] that treated the particular case of the linear/without delay wave equation, while in this work, we consider an abstract setting and a very general class of memory kernel functions.

Footnotes

Acknowledgements

The authors are highly grateful to the anonymous referee for his/her valuable comments and suggestions for the improvement of the paper. This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria (for the first and second authors). Baowei Feng is supported by the National Natural Science Foundation of China, grant #11701465.

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