General decay results for a viscoelastic wave equation with a variable exponent nonlinearity

Abstract

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type.

Keywords

Viscoelastic general decay relaxation function variable exponent

1. Introduction

Let $Ω \subset R^{N}$ ( $N ⩾ 1$ ) be a bounded domain with a smooth boundary $\partial Ω$ . This work is concerned with the following initial-boundary value problem: $\begin{matrix} (P) & \{\begin{matrix} u_{t t} - Δ u + \int_{0}^{t} g (t - s) Δ u (\cdot, s) d s + a | u_{t} |^{m (\cdot) - 2} u_{t} = 0, & in Ω \times (0, T), \\ u = 0, & on \partial Ω \times (0, T), \\ u (\cdot, 0) = u_{0}, u_{t} (\cdot, 0) = u_{1}, & in Ω, \end{matrix} \end{matrix}$ where a is a positive constant, $u_{0}, u_{1}$ are given initial data, g and m are positive functions to be specified later. This equation can be considered as a model of the dynamics of a viscoelastic body with nonlinear frictional damping of variable-exponent type.

Inspired by the work of Dafermos [7] in 1970, the well-posedness and stability of viscoelastic equations had received a considerable amount of attention. Many stability results had been established with different types of relaxation functions, see [13,17] and the references therein.

Concerning the decay rates of viscoelastic equations with linear and nonlinear (constant exponent) frictional damping, we mention the work of Cavalcanti et al. [6] in which they studied the problem $\begin{matrix} (1) & \{\begin{matrix} u_{t t} - Δ u + \int_{0}^{t} g (t - s) Δ u (\cdot, s) d s + a (x) u_{t} + f (x, t, u) = 0, & in Ω \times (0, \infty) \\ u = 0, & on \partial Ω \times (0, \infty) \\ u (\cdot, 0) = u_{0}, u_{t} (\cdot, 0) = u_{1}, & in Ω, \end{matrix} \end{matrix}$ where $a : Ω \subset R^{N} ⟶ R$ is a bounded function that satisfies, for some constant $a_{0}$ , $\begin{matrix} a (x) ⩾ a_{0} > 0 a.e. on ω \subset Ω, \end{matrix}$ $f \in C^{1} (\bar{Ω} \times [0, \infty) \times R)$ satisfies for some constants $C_{0}, p > 0$ , $\begin{matrix} | f (x, t, s) | ⩽ C_{0} (1 + | s |^{p + 1}), \forall (x, t, s) \in \bar{Ω} \times [0, \infty) \times R, \end{matrix}$ and the positive relaxation function g satisfies, for two constants $ξ_{1}, ξ_{2} > 0$ , $\begin{matrix} - ξ_{1} g (t) ⩽ g^{'} (t) ⩽ - ξ_{2} g (t), \forall t ⩾ 0 . \end{matrix}$ Under some additional conditions on f and a geometric condition on ω, they proved an exponential decay result of the energy associated to (1). Berrimi and Messaoudi in [5] obtained the same result by weakening some of the conditions on g and dropping the geometric on ω. Fabrizio and Polidoro [9] considered (1) with $a ⩾ 0$ being a real constant and proved that the exponential decay of g is a necessary condition for the exponential stability of the system. Messaoudi [12] investigated the following system: $\begin{matrix} (2) & \{\begin{matrix} u_{t t} - Δ u + \int_{0}^{t} g (t - s) Δ u (\cdot, s) d s + a | u_{t} |^{m - 2} u_{t} = 0, & in Ω \times (0, \infty) \\ u = 0, & on \partial Ω \times (0, \infty) \\ u (\cdot, 0) = u_{0}, u_{t} (\cdot, 0) = u_{1}, & in Ω, \end{matrix} \end{matrix}$ where $a > 0$ and $\begin{matrix} 1 < m ⩽ \frac{2 N}{N - 2}, if N ⩾ 3 and m > 1, if N = 1, 2 \end{matrix}$ and g satisfies, for some non-increasing differentiable positive function ξ, $\begin{matrix} g^{'} (t) ⩽ - ξ (t) g (t), \forall t ⩾ 0 . \end{matrix}$ He proved a general decay rate for the solution of (2) and his result gives exponential and polynomial decay rates as special cases. Very recently, Belhannache et al. [4] used the ideas of Jin et al. [11] and Mustafa [17] and extended the result of Messaoudi [12] to the case where the relaxation function satisfies $\begin{matrix} g^{'} (t) ⩽ - ξ (t) G (g (t)), \forall t ⩾ 0 . \end{matrix}$ Their result generalizes all the existing results related to general decay rates of viscoelastic wave equations with linear or nonlinear frictional damping.

Various physical phenomena such as flows of electro-rheological fluids, fluids with temperature dependent viscocity, nonlinear viscoelasticity, filteration processes through a porous media, or image processing are modeled using partial differential equations with variable exponents [20]. Recently, these equations had been investigated by many researchers. Most of the results treating hyperbolic problems with variable exponents dealt with blow up and non global existence [1,2,10,14–16,18]. For the stability of nonlinear damped wave equations with variable exponent nonlinearities, we have only few results. We mention the work of Messaoudi et al. [14] in which they considered the equation of the form $\begin{matrix} (3) & \{\begin{matrix} u_{t t} - div (| \nabla u |^{r (\cdot) - 2} \nabla u) + | u_{t} |^{m (\cdot) - 2} u_{t} = 0, & in Ω \times (0, T) \\ u (x, t) = 0, & on \partial Ω \times (0, T) \\ u (\cdot, 0) = u_{0}, u_{t} (\cdot, 0) = u_{1}, & in Ω, \end{matrix} \end{matrix}$ where the variable exponents $m (\cdot)$ and $r (\cdot)$ are measurable functions satisfying $\begin{matrix} 2 ⩽ r_{1} ⩽ r (x) ⩽ r_{2} ⩽ m_{1} ⩽ m (x) ⩽ m_{2} ⩽ r_{1}^{*} \end{matrix}$ with $\begin{matrix} r_{1} : = ess \inf_{x \in Ω} r (x), r_{2} : = ess \sup_{x \in Ω} r (x), m_{1} : = ess \inf_{x \in Ω} m (x), m_{2} : = ess \sup_{x \in Ω} m (x), \end{matrix}$ and $\begin{matrix} r_{1}^{*} : = \{\begin{matrix} \frac{N r_{1}}{N - r_{1}} & if N > r_{1} \\ \infty & if N = r_{1} . \end{matrix} \end{matrix}$ Under some additional conditions on m and r, they showed that the solution of (3) decays either exponentially or polynomially depending on the exponent $m (\cdot)$ . They also presented some numerical simulations to demonstrate their results.

To the best of our knowledge, the stability of ( P ) has not been discussed by any researcher and no result is available. It is our purpose in this paper to investigate this issue and to extend the result of Belhannache et al. [4] to the case of nonlinear frictional damping of variable-exponent type. The paper is divided into two sections, in addition to the introduction. In Section 2, we present some preliminary results, state and prove technical lemmas needed for the proof of our main results. In Section 3, we state and prove our main results.

2. Preliminaries

In this section, we introduce the notion of Lebesgue and Sobolev spaces with variable exponents (see [8,19]), state our assumptions and also present some preliminary results needed in the sequel. The constant exponent Lebesgue space with its usual norm is denoted by $(L^{p} (Ω), ‖ \cdot ‖_{p})$ , where $1 ⩽ p ⩽ \infty$ . Let $p (\cdot) : Ω \subset R^{N} ⟶ [1, \infty]$ be a measurable function, the Lebesgue space with variable exponent $p (\cdot)$ is defined by $\begin{matrix} L^{p (\cdot)} (Ω) : = {v : Ω ⟶ R measurable | \exists λ > 0 for which ϱ_{p (\cdot)} (λ v) < \infty}, \end{matrix}$ where $\begin{matrix} ϱ_{p (\cdot)} (v) : = \int_{Ω} {| v (x) |}^{p (x)} d x . \end{matrix}$ The space $L^{p (\cdot)} (Ω)$ endowed with Luxemburg–Nakano norm, given by $\begin{matrix} ‖ v ‖_{p (\cdot)} : = inf {λ > 0 | ϱ_{p (\cdot)} (\frac{v}{λ}) < \infty}, \forall v \in L^{p (\cdot)} (Ω), \end{matrix}$ is Banach. It is separable if $p (\cdot)$ is bounded and reflexive if $1 < p_{1} ⩽ p_{2} < \infty$ , where $\begin{matrix} p_{1} : = ess \inf_{x \in Ω} p (x) and p_{2} : = ess \sup_{x \in Ω} p (x) . \end{matrix}$ The variable-exponent Sobolev space is defined as $\begin{matrix} W^{1, p (\cdot)} (Ω) : = {v \in L^{p (\cdot)} (Ω) : | \nabla v | \in L^{p (\cdot)} (Ω)} . \end{matrix}$ The space $W^{1, p (\cdot)} (Ω)$ endowed with the norm $\begin{matrix} ‖ v ‖_{W^{1, p (\cdot)}} : = ‖ v ‖_{p (\cdot)} + ‖ \nabla v ‖_{p (\cdot)} \end{matrix}$ is Banach, it is separable if $p (\cdot)$ is bounded and reflexive if $1 < p_{1} ⩽ p_{2} < \infty$ .

Definition 2.1.
The exponent $p (\cdot) : Ω ⟶ [1, \infty)$ is said to satisfy the log-Hölder continuity condition, if there exists $A > 0$ such that for any $0 < δ < 1$ , we have $\begin{matrix} | p (x) - p (y) | ⩽ - \frac{A}{log | x - y |}, for a.e. x, y \in Ω with | x - y | < δ . \end{matrix}$

We will use the Sobolev embedding $H_{0}^{1} (Ω) ↪ L^{q} (Ω)$ , for any $q ⩾ 1$ if $N = 1, 2$ and $1 ⩽ q ⩽ \frac{2 N}{N - 2}$ if $N ⩾ 3$ , this implies that there exists $c_{e} > 0$ such that $\begin{matrix} ‖ v ‖_{q} ⩽ c_{e} ‖ \nabla v ‖_{2}, \forall v \in H_{0}^{1} (Ω) . \end{matrix}$

We assume that the relaxation function g and the variable exponent function $m (\cdot)$ satisfy the following hypotheses:
$g : [0, + \infty) ⟶ (0, + \infty)$ is a non-increasing differentiable function such that $\begin{matrix} 1 - \int_{0}^{\infty} g (s) d s = l > 0 . \end{matrix}$

There exist a non-increasing differentiable function $ξ : [0, + \infty) ⟶ (0, + \infty)$ and a $C^{1}$ -function $G : [0, \infty) ⟶ [0, \infty)$ which is either linear or strictly increasing and strictly convex $C^{2}$ -function on $(0, r]$ , $r < g (0)$ , with $G (0) = G^{'} (0) = 0$ such that $\begin{matrix} g^{'} (t) ⩽ - ξ (t) G (g (t)), \forall t ⩾ 0 . \end{matrix}$

$m : \bar{Ω} ⟶ [1, \infty)$ is a log-Hölder continuous function such that $\begin{matrix} m_{1} : = ess \inf_{x \in Ω} m (x), m_{2} : = ess \sup_{x \in Ω} m (x) and 1 < m_{1} ⩽ m (x) ⩽ m_{2}, \end{matrix}$ with $m_{2} < \infty$ if $n = 1, 2$ and $m_{2} ⩽ \frac{2 n}{n - 2}$ if $n ⩾ 3$ .
Remark 2.1 ([17]).

There is a strictly convex and strictly increasing $C^{2}$ function $\bar{G} : [0, + \infty) ⟶ [0, + \infty)$ which is an extension of G. For instance, we can define $\bar{G}$ , for any $t > r$ , by $\begin{matrix} \bar{G} (t) : = \frac{G^{″} (r)}{2} t^{2} + (G^{'} (r) - G^{″} (r) r) t + (G (r) + \frac{G^{″} (r)}{2} r^{2} - G^{'} (r) r) . \end{matrix}$

Lemma 2.1 ([17]).

Under assumptions (A.1)–(A.2), there exists $β_{0} > 0$ such that $\begin{matrix} (4) & ξ (t) g (t) ⩽ - β g^{'} (t), \forall t \in [0, t_{0}], \end{matrix}$ where $t_{0} : = g^{- 1} (r)$ .

Definition 2.2.
A function $u : Ω \times [0, T] ⟶ R$ is said to be a weak solution of problem ( P ) if $\begin{array}{l} u \in L^{\infty} ((0, T); H_{0}^{1} (Ω)) \cap W^{1, \infty} ((0, T); L^{2} (Ω)) \cap W^{2, \infty} ((0, T); H^{- 1} (Ω)), \\ u_{t} \in L^{m (\cdot)} (Ω \times (0, T)) . \end{array}$ and it satisfies, for almost every $t \in [0, T]$ , $\begin{matrix} (5) & \{\begin{matrix} \frac{d}{d t} \int_{Ω} u_{t} (x, t) v (x) d x + \int_{Ω} \nabla u (x, t) \cdot \nabla v (x) d x + \int_{Ω} \nabla v (x) \cdot \int_{0}^{t} g (t - s) \nabla u (x, s) d s d x \\ + a \int_{Ω} | u_{t} (x, t) |^{m (x) - 2} u_{t} (x, t) v (x) d x = 0, \forall v \in H_{0}^{1} (Ω), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), for a.e. x \in Ω . \end{matrix} \end{matrix}$
We state the existence theorem whose proof can be established by combining the arguments of [4,16].
Theorem 2.1.
Suppose conditions (A.1), (A.3) hold and $(u_{0}, u_{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ . Then, for any $T > 0$ , Problem ( P ) has a weak solution $\begin{array}{l} u \in L^{\infty} ((0, T); H_{0}^{1} (Ω)) \cap W^{1, \infty} ((0, T); L^{2} (Ω)) \cap W^{2, \infty} ((0, T); H^{- 1} (Ω)), \\ u_{t} \in L^{m (\cdot)} (Ω \times (0, T)) . \end{array}$

The energy functional associated to problem ( P ) is given by $\begin{matrix} E (t) : = \frac{1}{2} [{‖ u_{t} (t) ‖}_{2}^{2} + (1 - \int_{0}^{t} g (s) d s) {‖ \nabla u (t) ‖}_{2}^{2} + (g \circ \nabla u) (t)], \end{matrix}$ for any $t ⩾ 0$ , where for $v \in L_{loc}^{2} (R_{+}; L^{2} (Ω))$ , $\begin{matrix} (g \circ v) (t) : = \int_{0}^{t} g (t - s) {‖ v (t) - v (s) ‖}^{2} d s . \end{matrix}$ Multiplying the equation in ( P ) by $u_{t}$ , integrating over Ω and using the boundary condition, we obtain $\begin{matrix} (6) & E^{'} (t) = - \frac{1}{2} g (t) {‖ \nabla u (t) ‖}_{2}^{2} + \frac{1}{2} (g^{'} \circ \nabla u) (t) - a \int_{Ω} | u_{t} |^{m (x)} d x ⩽ 0 . \end{matrix}$ As in Jin et al. [11], we set, for any $0 < α < 1$ , $\begin{matrix} C_{α} : = \int_{0}^{\infty} \frac{g^{2} (s)}{α g (s) - g^{'} (s)} d s and h_{α} (t) : = α g (t) - g^{'} (t) . \end{matrix}$
Lemma 2.2 ([11]).

Assume that condition (A.1) holds. Then for any $v \in L_{loc}^{2} ([0, + \infty); L^{2} (Ω))$ , we have $\begin{matrix} (7) & \int_{Ω} {(\int_{0}^{t} g (t - s) (v (t) - v (s)) d s)}^{2} d x ⩽ C_{α} (h_{α} \circ v) (t), \forall t ⩾ 0 . \end{matrix}$

Lemma 2.3 ([17]).

Suppose that (A.1)–(A.2) hold. Then, for any $t ⩾ t_{0}$ , we have $\begin{matrix} \int_{0}^{t_{0}} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s ⩽ - c E^{'} (t) . \end{matrix}$

Lemma 2.4 (Jensen’s inequality).

Let $F : [a, b] ⟶ R$ be a convex function. Assume that the functions $f : Ω ⟶ [a, b]$ and $h : Ω ⟶ R$ are integrable such that $h (x) ⩾ 0$ , for any $x \in Ω$ and $\int_{Ω} h (x) d x = k > 0$ . Then, $\begin{matrix} F (\frac{1}{k} \int_{Ω} f (x) h (x) d x) ⩽ \frac{1}{k} \int_{Ω} F (f (x)) h (x) d x . \end{matrix}$

Lemma 2.5.
Suppose that assumption (A.1) holds and $m_{1} ⩾ 2$ . Then, the functionals $I_{1}$ and $I_{2}$ defined by $\begin{matrix} I_{1} (t) : = \int_{Ω} u u_{t} d x and I_{2} (t) : = - \int_{Ω} u_{t} \int_{0}^{t} g (t - s) (u (t) - u (s)) d s d x \end{matrix}$ satisfy, for any $δ > 0$ and $t > 0$ , $\begin{matrix} (8) & I_{1}^{'} (t) ⩽ ‖ u_{t} ‖_{2}^{2} - \frac{l}{4} ‖ \nabla u ‖_{2}^{2} + c C_{α} (h_{α} \circ \nabla u) (t) + c \int_{Ω} | u_{t} |^{m (x)} d x \end{matrix}$ and $\begin{array}{l} I_{2}^{'} (t) ⩽ & - (\int_{0}^{t} g (s) d s - δ) ‖ u_{t} ‖_{2}^{2} + δ ‖ \nabla u ‖_{2}^{2} + \frac{c}{δ} (C_{α} + 1) (h_{α} \circ \nabla u) (t) \\ (9) & + c_{0} δ {(1 - l)}^{m_{1} - 1} (g \circ \nabla u) (t) + \int_{Ω} c_{δ} (x) | u_{t} |^{m (x)} d x, \end{array}$ where $\begin{matrix} c_{0} : = c_{e}^{m_{1}} {(\frac{2}{l} E (0))}^{(m_{1} - 2) / 2} + c_{e}^{m_{2}} {(\frac{2}{l} E (0))}^{(m_{2} - 2) / 2} \end{matrix}$ and $\begin{matrix} c_{δ} (x) : = a^{\frac{m (x)}{1 - m (x)}} δ^{\frac{1}{1 - m (x)}} (m (x) - 1) {(m (x))}^{\frac{m (x)}{1 - m (x)}} . \end{matrix}$
Proof.
Differentiate $I_{1}$ , use the differential equation in ( P ), exploit Young’s inequality and Lemma 2.2 to get, for any $ε > 0$ , $\begin{array}{l} I_{1}^{'} (t) = & ‖ u_{t} ‖_{2}^{2} - (1 - \int_{0}^{t} g (s) d s) ‖ \nabla u ‖_{2}^{2} \\ + \int_{Ω} \nabla u (t) \cdot \int_{0}^{t} g (t - s) (\nabla u (s) - \nabla u (t)) d s d x \\ - a \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} u d x \\ ⩽ & ‖ u_{t} ‖_{2}^{2} - l ‖ \nabla u ‖_{2}^{2} - \frac{l}{2} ‖ \nabla u ‖_{2}^{2} \\ + \frac{1}{2 l} \int_{Ω} {(\int_{0}^{t} g (t - s) | \nabla u (s) - \nabla u (t) | d s)}^{2} d x \\ + ε \int_{Ω} | u |^{m (x)} d x + \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x \\ (10) & ⩽ & ‖ u_{t} ‖_{2}^{2} - \frac{l}{2} ‖ \nabla u ‖_{2}^{2} + c C_{α} (h_{α} \circ \nabla u) (t) + ε \int_{Ω} | u |^{m (x)} d x + \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x . \end{array}$ Next, we set $\begin{matrix} Ω_{+} : = {x \in Ω : | u (x, t) | ⩾ 1} and Ω_{-} : = {x \in Ω : | u (x, t) | ⩽ 1} . \end{matrix}$ Then, $\begin{array}{l} \int_{Ω} | u |^{m (x)} d x & = \int_{Ω +} | u |^{m (x)} d x + \int_{Ω_{-}} | u |^{m (x)} d x ⩽ \int_{Ω +} | u |^{m_{2}} d x + \int_{Ω_{-}} | u |^{m_{1}} d x \\ ⩽ \int_{Ω} | u |^{m_{2}} d x + \int_{Ω} | u |^{m_{1}} d x ⩽ (c_{e}^{m_{1}} ‖ \nabla u ‖_{2}^{m_{1}} + c_{e}^{m_{2}} ‖ \nabla u ‖_{2}^{m_{2}}) \\ = (c_{e}^{m_{1}} ‖ \nabla u ‖_{2}^{m_{1} - 2} + c_{e}^{m_{2}} ‖ \nabla u ‖_{2}^{m_{2} - 2}) ‖ \nabla u ‖_{2}^{2} \\ = [c_{e}^{m_{1}} {(\frac{2}{l} E (t))}^{(m_{1} - 2) / 2} + c_{e}^{m_{2}} {(\frac{2}{l} E (t))}^{(m_{2} - 2) / 2}] ‖ \nabla u ‖_{2}^{2} \\ ⩽ [c_{e}^{m_{1}} {(\frac{2}{l} E (0))}^{(m_{1} - 2) / 2} + c_{e}^{m_{2}} {(\frac{2}{l} E (0))}^{(m_{2} - 2) / 2}] ‖ \nabla u ‖_{2}^{2} \\ = c_{0} ‖ \nabla u ‖_{2}^{2} . \end{array}$ Therefore, $\begin{matrix} I_{1}^{'} (t) ⩽ ‖ u_{t} ‖_{2}^{2} - (\frac{l}{2} - c_{0} ε) ‖ \nabla u ‖_{2}^{2} + c C_{α} (h \circ \nabla u) (t) + \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x . \end{matrix}$ By fixing $ε = \frac{l}{4 c_{0}}$ , $c_{ε} (x)$ remains bounded, therefore we obtain the required estimate (8).

Differentiating $I_{2}$ and using the differential equation in ( P ), we obtain $\begin{array}{l} I_{2}^{'} (t) = & (1 - \int_{0}^{t} g (s) d s) \int_{Ω} \nabla u \cdot \int_{0}^{t} g (t - s) (\nabla u (t) - \nabla u (s)) d s d x \\ + \int_{Ω} {(\int_{0}^{t} g (t - s) | \nabla u (t) - \nabla u (s) | d s)}^{2} d x \\ + a \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} \int_{0}^{t} g (t - s) (u (t) - u (s)) d s d x \\ (11) & - \int_{Ω} u_{t} \int_{0}^{t} g^{'} (t - s) (u (t) - u (s)) d s d x - (\int_{0}^{t} g (s) d s) ‖ u_{t} ‖_{2}^{2} . \end{array}$ Now we estimate the terms in the right-hand side of the above equality. First, for any $δ > 0$ , we have $\begin{array}{l} (1 - \int_{0}^{t} g (s) d s) \int_{Ω} \nabla u \cdot \int_{0}^{t} g (t - s) (\nabla u (t) - \nabla u (s)) d s d x \\ + \int_{Ω} {(\int_{0}^{t} g (t - s) | \nabla u (t) - \nabla u (s) | d s)}^{2} d x ⩽ δ ‖ \nabla u ‖_{2}^{2} + \frac{c}{δ} C_{α} (h_{α} \circ \nabla u) (t) . \end{array}$ Next, for almost every $x \in Ω$ fixed, we have $\begin{array}{l} \int_{0}^{t} g (t - s) | u (t) - u (s) | d s ⩽ & {(\int_{0}^{t} g (s) d s)}^{\frac{m (x) - 1}{m (x)}} {(\int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m (x)} d s)}^{\frac{1}{m (x)}} \\ ⩽ & {(1 - l)}^{\frac{m (x) - 1}{m (x)}} {(\int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m (x)} d s)}^{\frac{1}{m (x)}} . \end{array}$ Therefore, for almost every $x \in Ω$ , we have $\begin{matrix} {| \int_{0}^{t} g (t - s) (u (t) - u (s)) d s |}^{m (x)} ⩽ {(1 - l)}^{m_{1} - 1} \int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m (x)} d s . \end{matrix}$ This entails that $\begin{array}{l} a \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} \int_{0}^{t} g (t - s) (u (t) - u (s)) d s d x \\ ⩽ δ \int_{Ω} {| \int_{0}^{t} g (t - s) (u (t) - u (s)) d s |}^{m (x)} d x + \int_{Ω} c_{δ} (x) | u_{t} |^{m (x)} d x \\ ⩽ {(1 - l)}^{m_{1} - 1} δ \int_{Ω} \int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m (x)} d s + \int_{Ω} c_{δ} (x) | u_{t} |^{m (x)} d x . \end{array}$ Similarly to above, we arrive at $\begin{array}{l} \int_{Ω} \int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m (x)} d s d x \\ ⩽ \int_{Ω_{+}} \int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m_{2}} d s d x + \int_{Ω_{-}} \int_{0}^{t} g (t - s) {| u (t) - u (s) |}^{m_{1}} d s d x \\ ⩽ \int_{0}^{t} g (t - s) {‖ u (t) - u (s) ‖}_{m_{2}}^{m_{2}} d s + \int_{0}^{t} g (t - s) {‖ u (t) - u (s) ‖}_{m_{1}}^{m_{1}} d s \\ ⩽ [c_{e}^{m_{2}} {(\frac{2}{l} E (0))}^{\frac{m_{2} - 2}{2}} + c_{e}^{m_{1}} {(\frac{2}{l} E (0))}^{\frac{m_{1} - 2}{2}}] \int_{0}^{t} g (t - s) {‖ \nabla u (t) - \nabla u (s) ‖}_{2}^{2} d s . \end{array}$ Therefore, $\begin{matrix} a \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} \int_{0}^{t} g (t - s) (u (t) - u (s)) d s d x ⩽ c_{0} δ {(1 - l)}^{m_{1} - 1} (g \circ \nabla u) (t) + \int_{Ω} c_{δ} (x) | u_{t} |^{m (x)} d x . \end{matrix}$ Finally, $\begin{array}{l} - \int_{Ω} u_{t} \int_{0}^{t} g^{'} (t - s) (u (t) - u (s)) d s d x \\ = \int_{Ω} u_{t} \int_{0}^{t} h_{α} (t - s) (u (t) - u (s)) d s d x - \int_{Ω} u_{t} \int_{0}^{t} α g (t - s) (u (t) - u (s)) d s d x \\ ⩽ δ ‖ u_{t} ‖_{2}^{2} + \frac{c}{δ} (C_{α} + 1) (h_{α} \circ \nabla u) (t) . \end{array}$ Combining the above estimates with (11), we get the required result. □
Lemma 2.6.
Suppose that assumption (A.1) holds and m satisfies $1 < m_{1} < 2$ . Then, the functionals $I_{1}$ and $I_{2}$ satisfy, for any $δ > 0$ and $t > 0$ , $\begin{array}{l} (12) & I_{1}^{'} (t) ⩽ ‖ u_{t} ‖_{2}^{2} - \frac{l}{4} ‖ \nabla u ‖_{2}^{2} + c C_{α} (h_{α} \circ \nabla u) (t) + c [\int_{Ω} | u_{t} |^{m (x)} d x + {(\int_{Ω} | u_{t} |^{m (x)} d x)}^{m_{1} - 1}] \end{array}$ and $\begin{array}{l} I_{2}^{'} (t) ⩽ & - (\int_{0}^{t} g (s) d s - δ) ‖ u_{t} ‖_{2}^{2} + c δ ‖ \nabla u ‖_{2}^{2} + \frac{c}{δ} (C_{α} + 1) (h_{α} \circ \nabla u) (t) \\ (13) & + δ (g \circ \nabla u) (t) + \frac{c}{δ} [\int_{Ω} | u_{t} |^{m (x)} d x + {(\int_{Ω} | u_{t} |^{m (x)} d x)}^{m_{1} - 1}] . \end{array}$
Proof.
We re-estimate the last term of the right-hand side of (10) as follows: let $\begin{matrix} Ω_{1} = {x \in Ω : m (x) < 2} and Ω_{2} = Ω ∖ Ω_{1}, \end{matrix}$ then $\begin{matrix} (14) & - a \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} u d x = - a \int_{Ω_{1}} | u_{t} |^{m (x) - 2} u_{t} u d x - a \int_{Ω_{2}} | u_{t} |^{m (x) - 2} u_{t} u d x . \end{matrix}$ Since $m (x) ⩾ 2 m (x) - 2 ⩾ 2 m_{1} - 2$ on $Ω_{1}$ , it follows that $\begin{array}{l} - a \int_{Ω_{1}} | u_{t} |^{m (x) - 2} u_{t} u d x ⩽ & \frac{l}{8 c_{e}^{2}} \int_{Ω_{1}} | u |^{2} d x + \frac{2 a^{2} c_{e}^{2}}{l} \int_{Ω_{1}} | u_{t} |^{2 m (x) - 2} d x \\ ⩽ & \frac{l}{8} ‖ \nabla u ‖_{2}^{2} + \frac{2 a^{2} c_{e}^{2}}{l} [\int_{Ω_{1}^{+}} | u_{t} |^{2 m (x) - 2} d x + \int_{Ω_{1}^{-}} | u_{t} |^{2 m (x) - 2} d x] \\ ⩽ & \frac{l}{8} ‖ \nabla u ‖_{2}^{2} + \frac{2 a^{2} c_{e}^{2}}{l} [\int_{Ω_{1}^{+}} | u_{t} |^{m (x)} d x + \int_{Ω_{1}^{-}} | u_{t} |^{2 m_{1} - 2} d x] \\ ⩽ & \frac{l}{8} ‖ \nabla u ‖_{2}^{2} + c [\int_{Ω} | u_{t} |^{m (x)} d x + {(\int_{Ω_{1}^{-}} | u_{t} |^{2} d x)}^{m_{1} - 1}] \\ (15) & ⩽ & \frac{l}{8} ‖ \nabla u ‖_{2}^{2} + c [\int_{Ω} | u_{t} |^{m (x)} d x + {(\int_{Ω} | u_{t} |^{m (x)} d x)}^{m_{1} - 1}], \end{array}$ where $Ω_{1}^{-} : = {x \in Ω_{1} : | u_{t} (x, t) | < 1}$ and $Ω_{1}^{+} : = Ω_{1} ∖ Ω_{1}^{-}$ .

Next, for any $ε > 0$ we have, by the case $m (x) ⩾ 2$ , $\begin{matrix} (16) & - a \int_{Ω_{2}} | u_{t} |^{2 m (x) - 2} u_{t} u d x ⩽ c_{0} ε ‖ \nabla u ‖_{2}^{2} + \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x, \end{matrix}$ where $\begin{matrix} c_{ε} (x) = a^{\frac{m (x)}{m (x) - 1}} ε^{\frac{1}{1 - m (x)}} (m (x) - 1) {(m (x))}^{\frac{m (x)}{1 - m (x)}} . \end{matrix}$ Therefore, by combining (14)–(16), we arrive at $\begin{array}{l} I_{1}^{'} (t) ⩽ & ‖ u_{t} ‖_{2}^{2} - (\frac{3 l}{8} - c_{0} ε) ‖ \nabla u ‖_{2}^{2} + c C_{α} (h_{α} \circ \nabla u) (t) \\ + c [\int_{Ω} (1 + c_{ε} (x)) | u_{t} |^{m (x)} d x + {(\int_{Ω} | u_{t} |^{m (x)} d x)}^{m_{1} - 1}] . \end{array}$ By fixing $ε = \frac{l}{8 c_{0}}$ , $c_{ε} (x)$ remains bounded and, consequently, we obtain (12).

Similarly, we re-estimate the third term in the right-hand side of (11) to get, for any $δ > 0$ , $\begin{array}{l} a \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} \int_{0}^{t} g (t - s) (u (t) - u (s)) d s d x \\ ⩽ c δ ‖ \nabla u ‖_{2}^{2} + c δ (g \circ \nabla u) (t) + \frac{c}{δ} [\int_{Ω} | u_{t} |^{m (x)} d x + {(\int_{Ω} | u_{t} |^{m (x)} d x)}^{m_{1} - 1}] . \end{array}$ Hence, estimate (13) is established. □

We have the following lemma from Jin et al. [11]. Lemma 2.7.
Assume that (A.1) holds, then the functional $I_{3}$ , defined by $\begin{matrix} I_{3} (t) : = \int_{0}^{t} f (t - s) {‖ \nabla u (s) ‖}_{2}^{2} d s, \end{matrix}$ where $f (t) = \int_{t}^{\infty} g (s) d s$ , satisfies $\begin{matrix} (17) & I_{3}^{'} (t) ⩽ - \frac{1}{2} (g \circ \nabla u) (t) + 3 (1 - l) ‖ \nabla u ‖_{2}^{2}, \forall t ⩾ 0 . \end{matrix}$
Lemma 2.8.
Suppose that $m_{1} ⩾ 2$ . Then, the functional $L$ , defined by $\begin{matrix} L (t) : = N E (t) + ε_{1} I_{1} (t) + ε_{2} I_{2} (t), \end{matrix}$ satisfies, for a suitable choice of $N, ε_{1}, ε_{2} > 0$ , $\begin{matrix} (18) & L \sim E \end{matrix}$ and $\begin{matrix} (19) & L^{'} (t) ⩽ - c_{1} {‖ u_{t} (t) ‖}_{2}^{2} - 4 (1 - l) {‖ \nabla u (t) ‖}_{2}^{2} + \frac{1}{4} (g \circ \nabla u) (t), \forall t ⩾ t_{0} . \end{matrix}$
Proof.
It is not difficult to establish the equivalence $L \sim E$ . To prove (19), we begin by exploiting estimates (8) and (9) to get $\begin{array}{l} L^{'} (t) ⩽ & - [(\int_{0}^{t} g (s) d s - δ) ε_{2} - ε_{1}] {‖ u_{t} (t) ‖}_{2}^{2} - (\frac{l}{4} ε_{1} - δ ε_{2}) {‖ \nabla u (t) ‖}_{2}^{2} \\ - [\frac{N}{2} - \frac{c}{δ} ε_{2} - c C_{α} (ε_{1} + \frac{ε_{2}}{δ})] (h_{α} \circ \nabla u) (t) \\ + (c_{0} δ {(1 - l)}^{m_{1} - 1} ε_{2} + \frac{α N}{2}) (g \circ \nabla u) (t) \\ (20) & - \int_{Ω} (a N - c ε_{1} - ε_{2} c_{δ} (x)) | u_{t} |^{m (x)} d x . \end{array}$ Now, set $g_{0} = \int_{0}^{t_{0}} g (s) d s$ and choose δ small enough so that $\begin{matrix} δ < min {\frac{1}{2} g_{0}, \frac{1}{16} l g_{0}, \frac{l g_{0}}{1024 c_{0} {(1 - l)}^{m_{1}}}} . \end{matrix}$ Whence δ is fixed, $c_{δ} (x)$ is bounded and a choice of $ε_{1} = \frac{3}{8} g_{0} ε_{2}$ satisfies $\begin{matrix} \frac{1}{4} g_{0} ε_{2} < ε_{1} < \frac{1}{2} g_{0} ε_{2} \end{matrix}$ and makes $\begin{array}{l} c_{1} : = (g_{0} - δ) ε_{2} - ε_{1} > \frac{1}{2} g_{0} ε_{2} - \frac{3}{8} g_{0} ε_{2} = \frac{1}{8} g_{0} ε_{2} > 0, \\ c_{2} : = \frac{1}{4} l ε_{1} - δ ε_{2} > \frac{3}{32} l g_{0} ε_{2} - \frac{1}{16} l g_{0} ε_{2} = \frac{1}{32} l g_{0} ε_{2} > 0 . \end{array}$ By setting $\begin{matrix} ε_{2} = \frac{1}{8 c_{0} δ {(1 - l)}^{m_{1} - 1}}, \end{matrix}$ we obtain $\begin{matrix} c_{0} δ {(1 - l)}^{m_{1} - 1} ε_{2} = \frac{1}{8} and c_{2} > \frac{1}{32} l g_{0} ε_{2} = \frac{l g_{0}}{256 c_{0} δ {(1 - l)}^{m_{1} - 1}} > 4 (1 - l) . \end{matrix}$ Therefore, inequality (20) becomes, for any $t ⩾ t_{0}$ , $\begin{array}{l} L^{'} (t) ⩽ & - c_{1} {‖ u_{t} (t) ‖}^{2} - 4 (1 - l) {‖ \nabla u (t) ‖}^{2} \\ - [\frac{N}{2} - \frac{c}{δ} ε_{2} - c C_{α} (ε_{1} + \frac{ε_{2}}{δ})] (h_{α} \circ \nabla u) (t) \\ (21) & + (\frac{1}{8} + \frac{α N}{2}) (g \circ \nabla u) (t) - (a N - c (ε_{1} + ε_{2})) \int_{Ω} | u_{t} |^{m (x)} d x . \end{array}$ Using (A.1) and the fact that $\frac{α g^{2} (s)}{α g (s) - g^{'} (s)} < g (s)$ , we infer, from the Lebesgue Dominated Convergence Theorem, that $\begin{matrix} lim_{α \to 0^{+}} α C_{α} = lim_{α \to 0^{+}} \int_{0}^{\infty} \frac{α g^{2} (s)}{α g (s) - g^{'} (s)} d s = 0 . \end{matrix}$ So there exists $0 < α_{0} < 1$ such that $\begin{matrix} α C_{α} < \frac{1}{16 c (ε_{1} + \frac{1}{δ} ε_{2})}, whenever 0 < α < α_{0} . \end{matrix}$ Now, we choose N large enough so that $L \sim E$ and $\begin{matrix} N > max {\frac{4 c}{δ} ε_{2}, \frac{1}{4 α_{0}}, \frac{c}{a} (ε_{1} + ε_{2})}, \end{matrix}$ then for $α = \frac{1}{4 N}$ , we have $\begin{matrix} \frac{N}{4} - \frac{c}{δ} ε_{2} > 0 and α < α_{0} . \end{matrix}$ This gives $\begin{matrix} \frac{N}{2} - \frac{c}{δ} ε_{2} - c C_{α} (ε_{1} + \frac{ε_{2}}{δ}) > \frac{N}{2} - \frac{c}{δ} ε_{2} - \frac{1}{16 α} = \frac{N}{4} - \frac{c}{δ} ε_{2} > 0 . \end{matrix}$ Hence, we arrived at the desired estimate. □
Lemma 2.9.
Suppose that $1 < m_{1} < 2$ . Then, the functional $L$ , defined by $\begin{matrix} L (t) : = N E (t) + ε_{1} I_{1} (t) + ε_{2} I_{2} (t) \end{matrix}$ satisfies, for a suitable choice of $N, ε_{1}, ε_{2} > 0$ , $\begin{matrix} L \sim E, \end{matrix}$ and the estimate $\begin{array}{l} (22) & L^{'} (t) ⩽ - c_{1} {‖ u^{'} (t) ‖}_{2}^{2} - 4 (1 - l) {‖ \nabla u (t) ‖}_{2}^{2} + \frac{1}{4} (g \circ \nabla u) (t) + c {(- E^{'} (t))}^{m_{1} - 1}, \forall t ⩾ t_{0} . \end{array}$
Proof.
The proof goes similar to that of Lemma 2.8 with some changes in the following terms: $\begin{matrix} δ < min {\frac{1}{2} g_{0}, \frac{1}{16 c} l g_{0}, \frac{l g_{0}}{1024 (1 - l)}} and ε_{2} = \frac{1}{8 δ} . \end{matrix}$ □

3. General decay

In this section, we establish our main decay results. We, first start with some lemmas.

Lemma 3.1.

Assume that (A.1) and (A.3) hold and $m_{1} ⩾ 2$ . Then, $\begin{matrix} (23) & \int_{0}^{\infty} E (s) d s < \infty . \end{matrix}$

Proof.

Set $L : = L + I_{3}$ , then a combination of estimates (17) and (19) yields $\begin{matrix} L^{'} (t) ⩽ - c_{1} {‖ u^{'} (t) ‖}_{2}^{2} - (1 - l) {‖ \nabla u (t) ‖}_{2}^{2} - \frac{1}{4} (g \circ \nabla u) (t) ⩽ - c E (t), \forall t ⩾ t_{0} . \end{matrix}$ An integration over $(t_{0}, t)$ , gives $\begin{matrix} c \int_{t_{0}}^{t} E (s) d s ⩽ - L (t) + L (t_{0}) ⩽ L (t_{0}), \forall t > t_{0} . \end{matrix}$ Exploiting the continuity of E, we obtain $\begin{matrix} \int_{0}^{\infty} E (s) d s < \infty . \end{matrix}$ □

Lemma 3.2.

Assume that (A.1) and (A.3) hold and $1 < m_{1} < 2$ . Then, $\begin{matrix} (24) & \int_{0}^{\infty} E^{\frac{1}{m_{1} - 1}} (s) d s < \infty . \end{matrix}$ Furthermore, $\begin{matrix} (25) & \int_{t_{0}}^{t} E (s) d s ⩽ c {(t - t_{0})}^{2 - m_{1}}, \forall t ⩾ t_{0} . \end{matrix}$

Proof.

Set $L : = L + I_{3}$ , then a combination of estimates (17) and (22) from Lemmas 2.7 and 2.9 gives $\begin{array}{l} L^{'} (t) ⩽ & - c_{1} {‖ u^{'} (t) ‖}_{2}^{2} - (1 - l) {‖ \nabla u (t) ‖}_{2}^{2} - \frac{1}{4} (g \circ \nabla u) (t) + c {(- E^{'} (t))}^{m_{1} - 1} \\ ⩽ & - c E (t) + c {(- E^{'} (t))}^{m_{1} - 1}, \forall t ⩾ t_{0} . \end{array}$ Multiplying both sides of the above estimate by $E^{q}$ , with $q = \frac{2 - m_{1}}{m_{1} - 1}$ , then taking advantage of Young’s inequality on the second term in the right-hand side of the resultant, we get, for a fixed $0 < ε < 1$ , $\begin{matrix} E^{q} (t) L^{'} (t) ⩽ - c (1 - ε) E^{q + 1} (t) - \frac{c}{ε} E^{'} (t), \forall t ⩾ t_{0} . \end{matrix}$ Thus, using the non-increasing property of E, we obtain, for some $β_{1} > 0$ , $\begin{matrix} {(E^{q} L + c E)}^{'} (t) ⩽ - β_{1} E^{q + 1} (t), \forall t ⩾ t_{0}, \end{matrix}$ and integration over $(t_{0}, t)$ , gives $\begin{matrix} β_{1} \int_{t_{0}}^{t} E^{q + 1} (s) d s ⩽ (E^{q} L + c E) (t_{0}), \forall t > t_{0} . \end{matrix}$ Hence, we obtain, owing to the continuity of E, $\begin{matrix} \int_{0}^{\infty} E^{\frac{1}{m_{1} - 1}} (s) d s < \infty . \end{matrix}$ Moreover, from Hölder’s inequality, we have $\begin{matrix} \int_{t_{0}}^{t} E (s) d s ⩽ {(t - t_{0})}^{\frac{q}{q + 1}} {(\int_{t_{0}}^{t} E^{q + 1} (s) d s)}^{\frac{1}{q + 1}} ⩽ c {(t - t_{0})}^{2 - m_{1}}, \forall t ⩾ t_{0} . \end{matrix}$ This completes the proof. □

Recalling (6), we easily see that $\begin{matrix} (26) & θ (t) : = - \int_{t_{0}}^{t} g^{'} (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s ⩽ - c E^{'} (t), \forall t ⩾ t_{0} . \end{matrix}$

Lemma 3.3.

Assume that conditions (A.1)–(A.3) hold. Then, there exists $γ \in (0, 1)$ such that for any $t ⩾ t_{0}$ , we have the following estimates $\begin{array}{l} (27) & \int_{t_{0}}^{t} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s ⩽ \frac{1}{γ} {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t)}), if m_{1} ⩾ 2; \\ (28) & \int_{t_{0}}^{t} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s ⩽ \frac{1}{γ} {(t - t_{0})}^{2 - m_{1}} {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t) {(t - t_{0})}^{2 - m_{1}}}), \end{array}$ if $1 < m_{1} < 2$ .

Proof.

For $m_{1} ⩾ 2$ , it follows from (23) that $\begin{array}{l} \int_{t_{0}}^{t} {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s & ⩽ 2 \int_{t_{0}}^{t} ({‖ \nabla u (t) ‖}_{2}^{2} + {‖ \nabla u (t - s) ‖}_{2}^{2}) d s \\ ⩽ \frac{4}{l} \int_{t_{0}}^{t} (E (t) + E (t - s)) d s \\ ⩽ \frac{8}{l} \int_{t_{0}}^{t} E (s) d s < \infty, \forall t ⩾ t_{0}, \end{array}$ therefore, we can choose $γ \in (0, 1)$ such that $\begin{matrix} (29) & η (t) : = γ \int_{t_{0}}^{t} {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s < 1, \forall t ⩾ t_{0} . \end{matrix}$ From the strict convexity of $\bar{G}$ and the fact that $\bar{G} (0) = 0$ , we have $\begin{matrix} \bar{G} (s τ) ⩽ s \bar{G} (τ), for any τ \in [0, \infty) and 0 ⩽ s ⩽ 1 . \end{matrix}$ Taking advantage of the above and owing to (A.2), (29) and Jensen’s inequality, we have $\begin{array}{l} θ (t) & = - \frac{1}{η (t)} \int_{t_{0}}^{t} η (t) g^{'} (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩾ \frac{1}{η (t)} \int_{t_{0}}^{t} η (t) ξ (s) G (g (s)) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩾ \frac{ξ (t)}{η (t)} \int_{t_{0}}^{t} \bar{G} (η (t) g (s)) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩾ \frac{ξ (t)}{γ} \bar{G} (γ \int_{t_{0}}^{t} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s), \forall t > t_{0}, \end{array}$ which implies (27).

In the case where $1 < m_{1} < 2$ , we define η by $\begin{matrix} (30) & η (t) = \frac{γ}{{(t - t_{0})}^{2 - m_{1}}} \int_{t_{0}}^{t} {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s, \forall t > t_{0} . \end{matrix}$ Then, estimate (25) allows us to pick $γ \in (0, 1)$ so that $η (t) < 1$ for any $t > t_{0}$ . Therefore, we have, for any $t > t_{0}$ , $\begin{array}{l} θ (t) & = - \frac{1}{η (t)} \int_{t_{0}}^{t} η (t) g^{'} (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩾ \frac{1}{η (t)} \int_{t_{0}}^{t} η (t) ξ (s) G (g (s)) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩾ \frac{ξ (t)}{η (t)} \int_{t_{0}}^{t} \bar{G} (η (t) g (s)) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩾ \frac{ξ (t) {(t - t_{0})}^{2 - m_{1}}}{γ} \bar{G} (\frac{γ}{{(t - t_{0})}^{2 - m_{1}}} \int_{t_{0}}^{t} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s), \end{array}$ from which we obtain (28). □

Theorem 3.1.

Let $(u_{0}, u_{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ . Assume that hypotheses (A.1)–(A.3) hold and $m_{1} ⩾ 2$ . Then, there exist positive constants $C, λ, λ_{1}, λ_{2}$ such that the energy functional associated to problem ( P ) satisfies $\begin{matrix} E (t) ⩽ C exp (- λ \int_{t_{0}}^{t} ξ (s) d s), \forall t ⩾ t_{0}, if G is linear; \end{matrix}$ and $\begin{matrix} E (t) ⩽ λ_{2} G_{0}^{- 1} (λ_{1} \int_{t_{0}}^{t} ξ (s) d s), \forall t ⩾ t_{0}, if G is nonlinear, \end{matrix}$ where $G_{0} (t) : = \int_{t}^{r} \frac{1}{s G^{'} (s)} d s$ and $g (t_{0}) = r$ .

Proof.

Case $G$ is linear. It follows from (A.2), (6) and Lemma 2.8 that, for some $β_{2} > 0$ , $\begin{array}{l} ξ (t) L^{'} (t) & ⩽ - β_{2} ξ (t) E (t) + c ξ (t) (g \circ \nabla u) (t) ⩽ - β_{2} ξ (t) E (t) - c (g^{'} \circ \nabla u) (t) \\ ⩽ - β_{2} ξ (t) E (t) - c E^{'} (t), \forall t ⩾ t_{0} . \end{array}$ Therefore, $ξ L + c E \sim E$ and a simple integration over $(t_{0}, t)$ yields, for some $C, λ > 0$ , $\begin{matrix} E (t) ⩽ C exp (- λ \int_{t_{0}}^{t} ξ (s) d s), \forall t ⩾ t_{0} . \end{matrix}$ Case $G$ is nonlinear. From lemmas 2.3, 2.8 and estimate (27) we have, for some $β_{3} > 0$ , $\begin{array}{l} L^{'} (t) & ⩽ - β_{3} E (t) + c \int_{0}^{t_{0}} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s + c \int_{t_{0}}^{t} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ ⩽ - β_{3} E (t) - c E^{'} (t) + \frac{c}{γ} {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t)}), \forall t ⩾ t_{0} . \end{array}$ Let $F : = L + c E \sim E$ , then $\begin{matrix} (31) & F^{'} (t) ⩽ - β_{3} E (t) + \frac{c}{γ} {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t)}), \forall t ⩾ t_{0} . \end{matrix}$ Let $0 < ε_{0} < r$ and define a functional $F_{1}$ by $\begin{matrix} F_{1} (t) : = {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) F (t) + c E (t), \forall t ⩾ t_{0} . \end{matrix}$ Then using (6), (A.2) and estimate (31), we have $F_{1} \sim E$ and $\begin{array}{l} F_{1}^{'} (t) & = ε_{0} \frac{E^{'} (t)}{E (0)} {\bar{G}}^{″} (ε_{0} \frac{E (t)}{E (0)}) F (t) + {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) F^{'} (t) + E^{'} (t) \\ (32) & ⩽ - β_{3} E (t) {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) + c {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t)}), \forall t ⩾ t_{0} . \end{array}$ Let ${\bar{G}}^{*}$ be the convex conjugate of $\bar{G}$ (see [3, pp. 61–64]), which is given by $\begin{matrix} (33) & {\bar{G}}^{*} (s) = s {({\bar{G}}^{'})}^{- 1} (s) - \bar{G} [{({\bar{G}}^{'})}^{- 1} (s)] \end{matrix}$ and satisfies the following generalized Young inequality $\begin{matrix} (34) & A B ⩽ {\bar{G}}^{*} (A) + \bar{G} (B) . \end{matrix}$ Set $A = {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)})$ , $B = {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t)})$ , then a combination of (32)–(34) gives $\begin{array}{l} F_{1}^{'} (t) & ⩽ - β_{3} E (t) {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) + c {\bar{G}}^{*} [{\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)})] + c \frac{γ θ (t)}{ξ (t)} \\ = - (β_{3} E (0) - c ε_{0}) \frac{E (t)}{E (0)} {\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) + c \frac{θ (t)}{ξ (t)}, \forall t ⩾ t_{0} . \end{array}$ Multiply this latter estimate by $ξ (t)$ and use (26) to get $\begin{array}{l} ξ (t) F_{1}^{'} (t) & ⩽ - (β_{3} E (0) - c ε_{0}) ξ (t) \frac{E (t)}{E (0)} G^{'} (ε_{0} \frac{E (t)}{E (0)}) + c θ (t) \\ ⩽ - (β_{3} E (0) - c ε_{0}) ξ (t) \frac{E (t)}{E (0)} G^{'} (ε_{0} \frac{E (t)}{E (0)}) - c E^{'} (t), \forall t ⩾ t_{0}, \end{array}$ since $ε_{0} \frac{E (t)}{E (0)} < r$ , hence ${\bar{G}}^{'} (ε_{0} \frac{E (t)}{E (0)}) = G^{'} (ε_{0} \frac{E (t)}{E (0)})$ .

Take $ε_{0}$ smaller, if needed, to obtain, for some positive constant $β_{4}$ , $\begin{matrix} ξ (t) F_{1}^{'} (t) ⩽ - β_{4} ξ (t) \frac{E (t)}{E (0)} G^{'} (ε_{0} \frac{E (t)}{E (0)}) - c E^{'} (t), \forall t ⩾ t_{0} . \end{matrix}$ Hence by setting $F_{2} = ξ F_{1} + c E$ , we have, for two constants $α_{1}, α_{2} > 0$ , $\begin{matrix} (35) & α_{1} F_{2} (t) ⩽ E (t) ⩽ α_{2} F_{2} (t), \forall t ⩾ t_{0} \end{matrix}$ and $\begin{matrix} (36) & F_{2}^{'} (t) ⩽ - β_{4} ξ (t) \frac{E (t)}{E (0)} G^{'} (ε_{0} \frac{E (t)}{E (0)}), \forall t ⩾ t_{0} . \end{matrix}$ Now, let $\begin{matrix} R (t) : = \frac{α_{1} F_{2} (t)}{E (0)} and G_{2} (τ) = τ G^{'} (ε_{0} τ), \end{matrix}$ then we deduce from (A.2) that $G_{2}, G_{2}^{'} > 0$ on $(0, 1]$ , and from (35) and (36) that $R \sim E$ and $\begin{matrix} (37) & - \frac{R^{'} (t)}{G_{2} (R (t))} ⩾ λ_{1} ξ (t), \forall t > t_{0}, \end{matrix}$ where $λ_{1} = \frac{α_{1} β_{3}}{E (0)} > 0$ . Integrating over $(t_{0}, t)$ and making an appropriate change of variable, we obtain $\begin{matrix} \int_{ε_{0} R (t)}^{ε_{0} R (t_{0})} \frac{1}{s H^{'} (s)} d s ⩾ λ_{1} \int_{t_{0}}^{t} ξ (s) d s, \forall t > t_{0} . \end{matrix}$ Hence, $\begin{matrix} E (t) ⩽ λ_{2} G_{0}^{- 1} (λ_{1} \int_{t_{0}}^{t} ξ (s) d s), \forall t > t_{0}, \end{matrix}$ where $G_{0} = \int_{t}^{r} \frac{1}{s G (s)} d s$ and $λ_{2} > 0$ . □

Example 3.1.

Consider the relaxation function $g (t) = a exp (- α t)$ , where a, α are positive constants and a is chosen so that hypothesis (A.1) is satisfied, then $\begin{matrix} g^{'} (t) = - α G (g (t)) with G (s) = s . \end{matrix}$ Therefore, it follows from Theorem 3.1 that $\begin{matrix} E (t) ⩽ C exp (- β t), for t large, \end{matrix}$ where $β = α λ$ .

Consider $g (t) = a e^{- {(1 + t)}^{ν}}$ , for $0 < ν < 1$ and a is chosen so that condition (A.1) is satisfied, then $\begin{matrix} g^{'} (t) = ξ (t) G (g (t)) with ξ (t) = ν {(1 + t)}^{ν - 1} and G (s) = s . \end{matrix}$ Theorem (3.1) entails that $\begin{matrix} E (t) ⩽ C e^{- λ {(1 + t)}^{ν}}, for t large . \end{matrix}$

Consider the following relaxation function, for $ν > 1$ , $\begin{matrix} g (t) = \frac{a}{{(1 + t)}^{ν}} \end{matrix}$ and a is chosen so that hypothesis (A.1) remains valid. Then $\begin{matrix} g^{'} (t) = - b G (g (t)) with G (s) = s^{p}, \end{matrix}$ where b is a fixed constant, $p = \frac{1 + ν}{ν}$ and it satisfies $1 < p < 2$ . Then, we deduce from Theorem (3.1) that $\begin{matrix} E (t) ⩽ \frac{C}{{(1 + t)}^{ν}}, for t large . \end{matrix}$

Theorem 3.2.

Let $(u_{0}, u_{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ . Assume that hypotheses (A.1)–(A.3) hold and $1 < m_{1} < 2$ . Then, there exist three positive constants $C, λ_{1}, λ_{2}$ and $t_{1} > t_{0}$ such that the energy functional associated to problem ( P ) satisfies $\begin{matrix} E (t) ⩽ C {(\int_{t_{0}}^{t} ξ (s) d s)}^{1 - m_{1}}, \forall t > t_{0}, if G is linear; \end{matrix}$ and, if G is nonlinear, we have $\begin{matrix} E (t) ⩽ λ_{2} {(t - t_{0})}^{2 - m_{1}} G_{1}^{- 1} [λ_{1} {({(t - t_{0})}^{\frac{m_{1} - 2}{m_{1} - 1}} \int_{t_{1}}^{t} ξ (s) d s)}^{- 1}], \forall t > t_{1}, \end{matrix}$ where $G_{1} (τ) : = τ^{\frac{1}{m_{1} - 1}} G^{'} (τ)$ .

Proof.

Case $G$ is linear. It follows from (A.2), (6) and lemma 2.9, that, for some $γ_{1} > 0$ , $\begin{array}{l} ξ (t) L^{'} (t) & ⩽ - γ_{1} ξ (t) E (t) + c ξ (t) (g \circ \nabla u) (t) + c ξ (t) {(- E^{'} (t))}^{m_{1} - 1} \\ ⩽ - γ_{1} ξ (t) E (t) - c E^{'} (t) + c ξ (t) {(- E^{'} (t))}^{m_{1} - 1}, \forall t ⩾ t_{0} . \end{array}$ Set $L_{1} : = ξ L + c E \sim E$ , then repeating the same computations as in the proof of the estimate (25) in lemma 3.2, we get, for $q = \frac{2 - m_{1}}{m_{1} - 1}$ and for any $ε > 0$ , $\begin{matrix} E^{q} (t) L_{1}^{'} (t) ⩽ - (γ_{1} - ε) ξ (t) E^{q + 1} (t) - c E^{'} (t), \forall t ⩾ t_{0} . \end{matrix}$ Let $L_{2} : = E^{q} L_{1} + c E \sim E$ and fix $ε < γ_{1}$ , then from the above estimate and the non-increasing property of E we conclude that, for some $γ_{2}, γ_{3} > 0$ , $\begin{matrix} L_{2}^{'} (t) ⩽ - γ_{2} ξ (t) E^{q + 1} (t) ⩽ - γ_{3} ξ (t) L_{2}^{q + 1} (t), \forall t ⩾ t_{0} . \end{matrix}$ A simple integration over $(t_{0}, t)$ and recalling that $L_{2} \sim E$ , we get for some $C > 0$ , $\begin{matrix} E (t) ⩽ C {(\int_{t_{0}}^{t} ξ (s) d s)}^{1 - m_{1}}, \forall t > t_{0} . \end{matrix}$ Case $G$ is nonlinear. It follows, from lemmas 2.3, 2.9 and estimate (28), that $\begin{array}{l} L^{'} (t) ⩽ & - γ_{4} E (t) + c \int_{0}^{t_{0}} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s \\ + c \int_{t_{0}}^{t} g (s) {‖ \nabla u (t) - \nabla u (t - s) ‖}_{2}^{2} d s + c {(- E^{'} (t))}^{m_{1} - 1} \\ ⩽ & - γ_{4} E (t) - c E^{'} (t) + \frac{c}{γ} {(t - t_{0})}^{2 - m_{1}} {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t) {(t - t_{0})}^{2 - m_{1}}}) \\ + c {(- E^{'} (t))}^{m_{1} - 1}, \forall t ⩾ t_{0}, \end{array}$ where $γ_{4} > 0$ . The functional $F : = L + c E \sim E$ and satisfies, for any $t ⩾ t_{0}$ , $\begin{matrix} (38) & F^{'} (t) ⩽ - γ_{4} E (t) + \frac{c}{γ} {(t - t_{0})}^{2 - m_{1}} {\bar{G}}^{- 1} (\frac{γ θ (t)}{ξ (t) {(t - t_{0})}^{2 - m_{1}}}) + c {(- E^{'} (t))}^{m_{1} - 1} . \end{matrix}$ For $0 < ε_{1} < 1$ fixed, we define a functional $F_{1}$ by $\begin{matrix} F_{1} (t) : = {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) F (t), \forall t > t_{0} . \end{matrix}$ According to (A.2), (6), (38) and the generalized Young inequality we have, for any $t > t_{0}$ , $\begin{array}{l} F_{1}^{'} (t) = & (\frac{ε_{1} E^{'} (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}} - \frac{ε_{1} (2 - m_{1}) E (t)}{E (0) {(t - t_{0})}^{3 - m_{1}}}) {\bar{G}}^{″} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) F (t) \\ + {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) F^{'} (t) \\ ⩽ & - γ_{4} E (t) {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) + c {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{σ}}) {(- E^{'} (t))}^{m_{1} - 1} \\ + \frac{c}{γ} {(t - t_{0})}^{2 - m_{1}} {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) {\bar{G}}^{- 1} (\frac{γ θ (t)}{{(t - t_{0})}^{2 - m_{1}} ξ (t)}) \\ ⩽ & - γ_{4} E (t) {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) + \frac{c}{γ} {(t - t_{0})}^{2 - m_{1}} {\bar{G}}^{*} [{\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}})] \\ + \frac{c}{γ} {(t - t_{0})}^{2 - m_{1}} \bar{G} [{\bar{G}}^{- 1} (\frac{γ θ (t)}{{(t - t_{0})}^{2 - m_{1}} ξ (t)})] \\ + c {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) {(- E^{'} (t))}^{m_{1} - 1} \\ ⩽ & - (γ_{4} E (0) - c ε_{1}) \frac{E (t)}{E (0)} {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) + \frac{c θ (t)}{ξ (t)} \\ + c {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) {(- E^{'} (t))}^{m_{1} - 1} \\ ⩽ & - γ_{5} E (t) {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) + \frac{c θ (t)}{ξ (t)} \\ + c {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) {(- E^{'} (t))}^{m_{1} - 1}, \end{array}$ where $γ_{5} > 0$ . Multiplying the last estimate by $ξ (t)$ and exploiting (26) we obtain, for any $t > t_{0}$ , $\begin{array}{l} ξ (t) F_{1}^{'} (t) ⩽ & - γ_{5} ξ (t) E (t) {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) - c E^{'} (t) \\ + c ξ (t) {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) {(- E^{'} (t))}^{m_{1} - 1} . \end{array}$ By setting $F_{2} : = ξ F_{1} + c E$ , we obtain, for any $t > t_{0}$ , $\begin{array}{l} F_{2}^{'} (t) ⩽ & - γ_{5} ξ (t) E (t) {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) \\ + c {\bar{G}}^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) {(- E^{'} (t))}^{m_{1} - 1} . \end{array}$ Similarly to the computations of the proof of (25) in Lemma 3.2, we multiply the above estimate with $E^{q}, (q = \frac{2 - m_{1}}{m_{1} - 1})$ , and apply Hölder’s inequality, to obtain, for some $γ_{6} > 0$ , $\begin{matrix} F_{3}^{'} (t) ⩽ - γ_{6} ξ (t) {(\frac{E (t)}{E (0)})}^{q + 1} \bar{G^{'}} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}), \forall t > t_{0}, \end{matrix}$ where $F_{3} : = E^{q} F_{2} + c E$ .

We deduce form ${lim}_{t \to \infty} \frac{1}{{(t - t_{0})}^{2 - m}} = 0$ , that, there exists $t_{1} > t_{0}$ such that $\frac{1}{{(t - t_{0})}^{2 - m_{1}}} < 1$ for any $t ⩾ t_{1}$ , which implies $\begin{matrix} ξ (t) {(\frac{E (t)}{E (0)})}^{q + 1} G^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) ⩽ - c F_{3}^{'} (t), \forall t > t_{1} . \end{matrix}$ The non-increasing property of E entails that the map $\begin{matrix} t \mapsto {(\frac{E (t)}{E (0)})}^{q + 1} G^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) \end{matrix}$ is non-increasing and, therefore, upon integration over $(t_{1}, t)$ , we arrive at $\begin{matrix} {(\frac{E (t)}{E (0)})}^{q + 1} G^{'} (\frac{ε_{1} E (t)}{E (0) {(t - t_{0})}^{2 - m_{1}}}) \int_{t_{1}}^{t} ξ (s) d s ⩽ c F_{3} (t_{1}), \forall t > t_{1} . \end{matrix}$ By multiplying both sides by ${(\frac{ε_{1}}{{(t - t_{0})}^{2 - m_{1}}})}^{q + 1}$ and setting $G_{1} (τ) : = τ^{q + 1} G^{'} (τ)$ , which is strictly increasing, we obtain, for some positive constants $λ_{1}, λ_{2}$ , $\begin{matrix} E (t) ⩽ λ_{2} {(t - t_{0})}^{2 - m_{1}} G_{0}^{- 1} [λ_{1} {({(t - t_{0})}^{\frac{2 - m_{1}}{m_{1} - 1}} \int_{t_{1}}^{t} ξ (s) d s)}^{- 1}], \forall t > t_{1} . \end{matrix}$ This completes the proof. □

Example 3.2.

Let $g (t) = a exp (- α t)$ , where a, α are positive constants and a is chosen so that hypothesis (A.1) is satisfied, then $\begin{matrix} g^{'} (t) = - α G (g (t)) with G (s) = s . \end{matrix}$ Then, Theorem 3.2 implies that $\begin{matrix} E (t) ⩽ \frac{c}{{(t - t_{0})}^{m_{1} - 1}}, \forall t > t_{1} . \end{matrix}$

For $g (t) = a e^{- α {(1 + t)}^{ν}}$ , where $α > 0$ , $0 < ν < 1$ and a is chosen so that condition (A.1) is satisfied, then $\begin{matrix} g^{'} (t) = - ξ (t) G (g (t)) with ξ (t) = ν {(1 + t)}^{ν - 1} and G (s) = s . \end{matrix}$ We deduce from Theorem 3.2 that $\begin{matrix} E (t) ⩽ \frac{c}{{(1 + t)}^{ν (m_{1} - 1)}}, for t large enough . \end{matrix}$

Consider $\begin{matrix} g (t) = \frac{a}{{(1 + t)}^{ν}}, \end{matrix}$ where $ν > 1$ and a is chosen so that hypothesis (A.1) remains valid. Then $\begin{matrix} g^{'} (t) = - b G (g (t)) with G (s) = s^{p}, \end{matrix}$ where b is a fixed constant, $p = \frac{1 + ν}{ν}$ and it satisfies $1 < p < 2$ . Then, $G_{1} (s) = p t^{\frac{ν + m_{1} - 1}{ν (m_{1} - 1)}}$ and 3.2 we obtain, for t large, $\begin{matrix} E (t) ⩽ \frac{c}{{(1 + t)}^{\frac{(m_{1} - 1) (m_{1} + ν - 2)}{m_{1} + ν - 1}}} . \end{matrix}$

Footnotes

Acknowledgements

The authors thank King Fahd University of Petroleum and Minerals for its past and present support. The second author is sponsored by the University of Sharjah under grant no. 2002144089, 2019–2020.

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