New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

Abstract

This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function $k_{i}$, namely,

$$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$

We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when $k_{i}(s) = s^{p}$ and p covers the full admissible range $[1, 2)$.

Introduction

In this paper, we consider the following system:

$$\begin{aligned} \textstyle\begin{cases} u_{tt}(x,t)- \frac{1}{x}(xu_{x}(x,t))_{x}+\int _{0}^{t}k_{1}(t-s) \frac{1}{x}(xu_{x}(x,s))_{x}\,ds=f_{1}(u,v),\\ \quad x\in \Omega, t>0, \\ v_{tt}(x,t)- \frac{1}{x}(xv_{x}(x,t))_{x}+\int _{0}^{t}k_{2}(t-s) \frac{1}{x}(xv_{x}(x,s))_{x}\,ds=f_{1}(u,v),\\ \quad x\in \Omega, t>0, \\ u(x,t)=v(x,t)=0,\quad x \in \partial \Omega, t\geq 0, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x, 0)=u_{1}(x),\qquad v(x,0)=v_{0}(x),\\ v_{t}(x, 0)=v_{1}(x),\quad x\in \Omega, \\ u(L,t)=v(L,t)=0, \qquad\int _{0}^{L}xu(x,t)\,dx=\int _{0}^{L}xv(x,t)\,dx=0, \end{cases}\displaystyle \end{aligned}$$

(1)

where $\Omega =(0,L)$, $k_{i}:[0,+\infty )\longrightarrow (0,+\infty )$, ($i=1,2$), are non-increasing differentiable functions satisfying more general conditions to be mentioned later and

$$\begin{aligned} \textstyle\begin{cases} f_{1}(u,v)=a{ \vert {u+v} \vert }^{2 (r +1 )} (u+v )+b{ \vert {u} \vert }^{r} u { \vert {v} \vert }^{r+2}, \\ f_{2}(u,v)=a{ \vert {u+v} \vert }^{2 (r +1 )} (u+v )+b{ \vert {v} \vert }^{r} v { \vert {u} \vert }^{r+2}, \end{cases}\displaystyle \end{aligned}$$

(2)

where $r>-1$ and $a,b>0$.

Mixed nonlocal problems for parabolic and hyperbolic partial differential equations have received a great attention during the last few decades. These problems are especially inspired by modern physics and technology. They aim to describe many physical and biological phenomena. For instance, physical phenomena are modeled by initial boundary value problems with nonlocal constraints such as integral boundary conditions, when the data cannot be measured directly on the boundary, but the average value of the solution on the domain is known. Initial boundary value problems for second-order evolution partial differential equations and systems having nonlocal boundary conditions can be encountered in many scientific domains and many engineering models and are widely applied in heat transmission theory, underground water flow, medical science, biological processes, thermoelasticity, chemical reaction diffusion, plasma physics, chemical engineering, heat conduction processes, population dynamics, and control theory. See in this regard the work by Cannon [1], Shi [2], Capasso and Kunisch [3], Cahlon and Shi [4], Ionkin and Moiseev [5], Shi and Shilor [6], Choi and Chan [7], and Ewing and Lin [8]. In early work, most of the research on nonlocal mixed problems was devoted to the classical solutions. Later, mixed problems with integral conditions for both parabolic and hyperbolic equations were studied by Pulkina [9, 10], Yurchuk [11], Kartynnik [12], Mesloub and Bouziani [13], Mesloub and Messaoudi [14, 15], Mesloub [16], and Kamynin [17]. For instance, Said Mesloub and Fatiha Mesloub [18] obtained existence and uniqueness of the solution to the following problem:

$$\begin{aligned} u_{tt}- \frac{1}{x}(xu_{x})_{x}+ \int _{0}^{t}k(t-s)\frac{1}{x} (xu_{x} )_{x}\,ds+a u_{t}=f(t,x,u,u_{x}),\quad x\in (0,1), t>0, \end{aligned}$$

(3)

and proved that the solution blows up for large initial data and decays for sufficiently small initial data. Mesloub and Messaoud [14] considered the following nonlocal singular problem:

$$\begin{aligned} u_{tt}- \frac{1}{x}(xu_{x})_{x}+ \int _{0}^{t}g(t-s)\frac{1}{x} (xu_{x} )_{x}\,ds= \vert u \vert ^{p} u,\quad x\in (0,a), t>0, \end{aligned}$$

(4)

and proved blow-up result for large initial data and decay results of sufficiently small initial data enough for $p>2$. In [19], Draifia et al. proved a general decay result for the following singular one-dimensional viscoelastic system:

$$\begin{aligned} \textstyle\begin{cases} u_{tt}-\frac{1}{x}(xu_{x})_{x}+\int _{0}^{t}g_{1}(t-s) \frac{1}{x} (xu_{x}(x,s))_{x}\,ds= \vert v \vert ^{q+1} \vert u \vert ^{p-1}u,\quad\text{in }Q, \\ v_{tt}-\frac{1}{x}(xv_{x})_{x}+\int _{0}^{t}g_{2}(t-s) \frac{1}{x} (xv_{x}(x,s))_{x}\,ds= \vert u \vert ^{p+1} \vert v \vert ^{q-1}v,\quad\text{in }Q, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\quad x\in (0,\alpha ), \\ v(x,0)=v_{0}(x),\qquad v_{t}(x,0)=v_{1}(x),\quad x\in (0,\alpha ), \\ u(\alpha,t)=v(\alpha,t)=0,\qquad \int _{0}^{\alpha }xu(x,t)\,dx= \int _{0}^{\alpha }xv(x,t)\,dx=0, \end{cases}\displaystyle \end{aligned}$$

(5)

where $Q=(0,\alpha )\times (0,t)$ and $p,q >1$. Piskin and Ekinci [20] studied problem (1) when the Bessel operator has been replaced by a Kirchhoff operator with a degenerate damping terms. They proved the global existence and established a decay rate of solution and also a finite time blow up. Recently, Boulaaras et al. [21] treated problem (1) and proved the existence of a global solution to the problem using the potential-well theory. Moreover, they established a general decay result in which the relaxation functions $k_{1}$ and $k_{2}$ satisfy

$$\begin{aligned} k_{i}'(t)\leq - \xi (t) k_{i}^{p}(t),\quad 1\leq p < \frac{3}{2}. \end{aligned}$$

(6)

Motivated by the above work, we prove a general stability result of system (1) replacing the condition (6) used in [21] by a more general assumption of the form:

$$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$

Our decay result improves all the existing results in the literature related to this system.

This paper is divided into four sections. In Sect. 2, we state some assumptions needed in our work. Some technical lemmas will be given in Sect. 3. The statement with proof of the main result and some examples will be given in Sect. 4.

Preliminaries

In this section, we present some materials needed in the proof of our results. We also state, without proof, the global existence result for problem (1). Let $L^{p}_{x}=L^{p}_{x}(0,L)$ be the weighted Banach space equipped with the norm

$$\begin{aligned} \Vert u \Vert _{L^{p}_{x}}= \biggl( \int _{0}^{L} x u^{p} \,dx \biggr)^{\frac{1}{p}}. \end{aligned}$$

$L^{2}_{x}(0,L)$ is the Hilbert space of square integral functions having the finite norm

$$\begin{aligned} \Vert u \Vert _{L^{2}_{x}}= \biggl( \int _{0}^{L} x u^{2} \,dx \biggr)^{\frac{1}{2}}, \end{aligned}$$

$V=V^{1}_{x}(0,L)$ is the Hilbert space equipped with the norm

$$\begin{aligned} \Vert u \Vert _{V}= \bigl( \Vert u \Vert ^{2}_{L^{2}_{x}}+ \Vert u_{x} \Vert ^{2}_{L^{2}_{x}} \bigr)^{\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned} V_{0}=\bigl\{ u \in V \text{ such that } u(L)=0\bigr\} . \end{aligned}$$

Lemma 2.1

([14])

$\forall w\in V_{0}$, a Poincaré-type inequality is

$$\begin{aligned} \Vert w \Vert ^{2}_{L^{2}_{x}} \leq C_{p} \Vert w_{x} \Vert ^{2}_{L^{2}_{x}}. \end{aligned}$$

(7)

Remark 2.1

Notice that $\Vert u \Vert _{V_{0}}=\Vert u_{x} \Vert _{L^{2}_{x}}$ defines an equivalent norm on $V_{0}$.

Assumptions

$(A1)$ :

$k_{i}:\mathbb{R}_{+}\to \mathbb{R}_{+}$ (for $i=1,2$) are $C^{1}$ non-increasing functions satisfying

$$\begin{aligned} k_{i}(0)>0,\quad 1- \int _{0}^{+\infty }k_{i}(s)\,ds=:\ell _{i}>0. \end{aligned}$$

(8)

$(A2)$ :

There exist non-increasing differentiable functions $\xi _{i}:[0,+\infty )\longrightarrow (0,+\infty )$ and $\boldsymbol{C}^{1}$ functions $\Psi _{i}:[0,+\infty )\longrightarrow [0,+\infty )$ which are linear or strictly increasing and strictly convex $\boldsymbol{C}^{2}$ functions on $(0,\varepsilon ]$, $\varepsilon \leq k_{i}(0)$, with $\Psi _{i}(0)=\Psi _{i}'(0)=0$ such that

$$\begin{aligned} k_{i}'(t)\leq -\xi _{i}(t)\Psi _{i}\bigl(k_{i}(t)\bigr),\quad \forall t\geq 0 \text{ and for }i=1,2. \end{aligned}$$

(9)

Remark 2.2

The given functions $f_{1}$ and $f_{2}$ satisfy

$$\begin{aligned} uf_{1}(u,v)+vf_{2}(u,v)=2(r+2)F(u,v), \quad\forall (u,v)\in \mathbb{R}^{2}, \end{aligned}$$

where

$$\begin{aligned} 2(r+2) F(u,v)= \bigl[a { \vert u+v \vert }^{2(r+2)}+2b { \vert uv \vert }^{r+2} \bigr]. \end{aligned}$$

Lemma 2.2

(Jensen’s inequality)

Let $G:[a,b]\longrightarrow \mathbb{R}$ be a convex function. Assume that the functions $f:(0,L)\longrightarrow [a,b]$ and $h:(0,L)\longrightarrow \mathbb{R}$ are integrable such that $h(x)\geq 0$, for any $x\in (0,L)$ and $\int _{0}^{L}h(x)\,dx=k>0$. Then

$$\begin{aligned} G \biggl(\frac{1}{k} \int _{0}^{L} f(x)h(x)\,dx \biggr)\leq \frac{1}{k} \int _{0}^{L} G\bigl(f(x)\bigr)h(x)\,dx. \end{aligned}$$

Remark 2.3

If Ψ is a strictly increasing, strictly convex $C^{2}$ function over $(0, \varepsilon ]$ and satisfying $\Psi (0) = \Psi '(0) = 0$, then it has an extension, Ψ̅, that is also strictly increasing and strictly convex $C^{2}$ over $(0,\infty )$. For example, if $\Psi (\varepsilon ) = a, \Psi '(\varepsilon ) = b, \Psi ''( \varepsilon ) = c$, and for $t > \varepsilon$, Ψ̅ can be defined by

$$\begin{aligned} \overline{\Psi }(t)=\frac{c}{2} t^{2}+ (b-c \varepsilon )t+ \biggl(a+ \frac{c}{2} {\varepsilon }^{2} - b \varepsilon \biggr). \end{aligned}$$

(10)

Remark 2.4

Since $\Psi _{i}$ is strictly convex on $(0,\varepsilon ]$ and $\Psi _{i}(0)=0$,

$$\begin{aligned} \Psi _{i}(\theta z)\le \theta \Psi _{i}(z),\quad 0\le \theta \le 1, \forall z\in (0,\varepsilon ]\text{ and }i=1,2. \end{aligned}$$

(11)

The modified energy functional E associated to problem (1) is

$$\begin{aligned} E(t)={}&\frac{1}{2} \Vert u_{t} \Vert _{L_{x}^{2}}^{2}+ \Vert v_{t} \Vert _{L_{x}^{2}}^{2}+ \frac{1}{2} \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \Vert u_{x} \Vert _{L_{x}^{2}}^{2} \\ &{} +\frac{1}{2} \biggl(1- \int _{0}^{t} k_{2}(s)\,ds \biggr) \Vert v_{x} \Vert _{L_{x}^{2}}^{2} \\ &{} +\frac{1}{2}\bigl[(k_{1} o u_{x}) (t)+(k_{2} o v_{x}) (t)\bigr]- \int _{0}^{L}xF(u,v)\,dx, \end{aligned}$$

(12)

where, for any $w\in L^{2}_{loc} ([0,+\infty );L_{x}^{2}(0,L) )$ and $i=1,2$,

$$\begin{aligned} (k_{i}\circ w) (t):= \int _{0}^{t}k_{i}(t-s) \bigl\Vert w(t)-w(s) \bigr\Vert _{L_{x}^{2}}^{2}\,ds. \end{aligned}$$

Using (1) with direct differentiation gives

$$\begin{aligned} \frac{dE(t)}{dt}& = -\frac{1}{2} \bigl(k_{1}^{\prime } o u_{x}\bigr) (t)- \frac{1}{2}k_{1}(t) \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}- \frac{1}{2}\bigl(k_{2}^{\prime }o v_{x}\bigr) (t)-\frac{1}{2}k_{2}(t) \Vert v_{x} \Vert ^{2}_{L^{2}_{x}} \\ & \le \frac{1}{2}\bigl(k_{1}^{\prime } o u_{x}\bigr) (t)+\frac{1}{2}\bigl(k_{2}^{\prime }o v_{x}\bigr) (t) \le 0. \end{aligned}$$

(13)

Local and global existence

In this subsection, we state, without proof, the local and global existence results for system (1), which can be proved similarly to the ones in [14, 18] and [21].

Theorem 2.1

Assume that $(A1)$ and $(A2)$ hold. If $(u_{0},v_{0})\in V_{0}^{2}$ and $(u_{1},v_{1}) \in (L_{x}^{2})^{2}$. Then problem (1) has a unique local solution.

For the global existence, we introduce the following functionals:

$$\begin{aligned} J(t)={}&\frac{1}{2} \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \Vert u_{x} \Vert ^{2}_{L^{2}_{x}} +\frac{1}{2} \biggl(1- \int _{0}^{t} k_{2}(s)\,ds \biggr) \Vert v_{x} \Vert ^{2}_{L^{2}_{x}} \\ &{}+\frac{1}{2}\bigl[(k_{1}o u_{x}) (t)+(k_{2}o v_{x}) (t)\bigr]- \int _{0}^{L}x \bigl[a \vert u+v \vert ^{2(r+2)}+2b \vert uv \vert ^{(r+2)} \bigr]\,dx \end{aligned}$$

(14)

and

$$\begin{aligned} I(t)={}& \biggl(1- \int _{0}^{t} k_{1}(s)\,ds \biggr) \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+ \biggl(1- \int _{0}^{t}k_{2}(s)\,ds \biggr) \Vert v_{x} \Vert ^{2}_{L^{2}_{x}}+(k_{1}o u_{x}) (t)+(k_{2}o v_{x}) (t) \\ &{} -2(r+2) \int _{0}^{L}x \bigl[a \vert u+v \vert ^{2(r+2)}+2b \vert uv \vert ^{(r+2)} \bigr]\,dx. \end{aligned}$$

(15)

We notice that $E(t)=J(t)+\frac{1}{2}\Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+ \frac{1}{2}\Vert v_{t} \Vert ^{2}_{L^{2}_{x}}$.

Lemma 2.3

Suppose that $(A1)$ and $(A2)$ hold. Then, for any $(u_{0},v_{0})\in V_{0}^{2}$ and $(u_{1},v_{1}) \in (L^{2}_{x})^{2}$ satisfying

$$\begin{aligned} \textstyle\begin{cases} \beta =\eta { [\frac{2(r+2)}{r+1}E(0) ]}^{r+1} < 1, \\ I(0)=I(u_{0},v_{0})>0, \end{cases}\displaystyle \end{aligned}$$

(16)

there exists $t_{*} > 0$ such that

$$\begin{aligned} I(t)=I\bigl(u(t),v(t)\bigr) > 0, \quad\forall t \in [0,t_{*}). \end{aligned}$$

(17)

Remark 2.5

We can easily deduce from Lemma 2.3 that

$$\begin{aligned} \ell _{1}{ \Vert { u_{x}} \Vert }_{L_{x}^{2}}^{2}+\ell _{2}{ \Vert {v_{x}} \Vert }_{L_{x}^{2}}^{2} \le { \frac{2(\rho +2)}{\rho +1}E(0)},\quad \forall t \ge 0. \end{aligned}$$

(18)

Theorem 2.2

Assume that $(A1)$ and $(A2)$ hold. If $(u_{0},v_{0})\in V_{0}^{2}$ and $(u_{1},v_{1}) \in (L_{x}^{2})^{2}$ and satisfies (16), then the solution of (1) is global and bounded.

Technical lemmas

In this section, we establish several lemmas needed for the proof of our main result.

Lemma 3.1

There exist two positive constants $c_{1}$ and $c_{2}$ such that

$$\begin{aligned} \int _{0}^{L} x \bigl\vert f_{i}(u,v) \bigr\vert ^{2} \,dx \leq c_{i} \bigl(\ell _{1} \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+\ell _{2} \Vert v_{x} \Vert ^{2}_{L^{2}_{x}} \bigr)^{2r+3},\quad i=1,2. \end{aligned}$$

(19)

Proof

We prove inequality (19) for $f_{1}$ and the same result holds for $f_{2}$. It is clear that

$$\begin{aligned} \bigl\vert {f_{1}(u,v)} \bigr\vert &\le {C \bigl({ \vert {u+v} \vert }^{2r+3}+{ \vert {u} \vert }^{r+1} { \vert {v} \vert }^{r+2} \bigr)} \\ & \le C \bigl({ \vert u \vert }^{2r+3}+{ \vert v \vert }^{2r+3}+{ \vert {u} \vert }^{r+1} { \vert {v} \vert }^{r+2} \bigr). \end{aligned}$$

(20)

From (20) and Young’s inequality, with

$$\begin{aligned} q=\frac{2r+3}{r+1},\qquad q^{\prime }=\frac{2r+3}{r+2}, \end{aligned}$$

we get

$$\begin{aligned} { \vert {u} \vert }^{r+1}{ \vert {v} \vert }^{r+2}\le {c_{1} { \vert {u} \vert }^{2r+3} + c_{2}{ \vert {v} \vert }^{2r+3}}, \end{aligned}$$

hence

$$\begin{aligned} \bigl\vert f_{1}(u,v) \bigr\vert \le {C \bigl[{ \vert {u} \vert }^{2r+3} + { \vert {v} \vert }^{2r+3} \bigr]}. \end{aligned}$$

Consequently, by using (7), (12), (13) and the embedding $V_{0} \hookrightarrow L^{2(2r+3)}$, we obtain

$$\begin{aligned} \int _{0}^{L}x{ \bigl\vert {f_{1}(u,v)} \bigr\vert }^{2}\,dx &\le C \bigl( \Vert u \Vert ^{2(2r+3)}_{L_{x}^{2(2r+3)}}+ \Vert v \Vert ^{2(2r+3)}_{L_{x}^{2(2r+3)}} \bigr) \\ &\le {c_{1}{\bigl(\ell _{1} { \Vert {u_{x}} \Vert }_{L^{2}_{x}}^{2}+\ell _{2}{ \Vert { v_{x}} \Vert }_{L^{2}_{x}}^{2} \bigr)}^{2r+3}}. \end{aligned}$$

This completes the proof of Lemma 3.1. □

Lemma 3.2

([22])

There exist positive constants d and $t_{0}$ such that, for any $t\in [0,t_{0}]$, we have

$$\begin{aligned} k_{i}^{\prime }(t)\le -d k_{i}(t),\quad i=1,2. \end{aligned}$$

(21)

Lemma 3.3

If $(A1)$ holds. Then, for any $w\in V_{0}$, $0<\alpha <1$ and $i=1,2$, we have

$$\begin{aligned} \int _{0}^{L}x \biggl( \int _{0}^{t} k_{i}(t-s) \bigl(w(t)-w(s) \bigr)\,ds \biggr)^{2}\,dx \leq C_{\alpha,i}(h_{i}\circ w) (t), \end{aligned}$$

(22)

where $C_{\alpha,i}:=\int _{0}^{\infty }\frac{k_{i}^{2}(s)}{\alpha k_{i}(s)-k_{i}'(s)}\,ds$ and $h_{i}(t):=\alpha k_{i}(t)-k_{i}'(t)$.

Proof

The proof of this lemma goes similar to the one in [22]. □

Lemma 3.4

Under the assumptions $(A1)$ and $(A2)$, the functional

$$\begin{aligned} \Phi (t):= \int _{0}^{L}xuu_{t} \,dx+ \int _{0}^{L}xvv_{t} \,dx, \end{aligned}$$

satisfies, along with the solution of system (1), the estimate

$$\begin{aligned} \Phi ^{\prime }(t) \le{}& \Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+ \Vert v_{t} \Vert ^{2}_{L^{2}_{x}} - \frac{\ell _{1}}{2} \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}- \frac{\ell _{2}}{2} \Vert v_{x} \Vert ^{2}_{L^{2}_{x}} \\ &{} +C_{\alpha,1}(h_{1}\circ u_{x}) (t)+C_{\alpha,2}(h_{2}\circ v_{x}) (t)+ \int _{0}^{L}xF(u,v)\,dx. \end{aligned}$$

(23)

Proof

Direct differentiation, using (1), yields

$$\begin{aligned} \Phi ^{\prime }(t)={}& \int _{0}^{L}xu^{2}_{t}\,dx+ \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{x}^{2}\,dx \\ &{} + \int _{0}^{L}xu_{x} \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(s)-u_{x}(t)\bigr)\,ds\,dx \\ &{} + \int _{0}^{L}xv^{2}_{t}\,dx+ \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xv_{t}^{2}\,dx \\ &{} + \int _{0}^{L}xv_{x} \int _{0}^{t}k_{2}(t-s) \bigl(v_{x}(s)-v_{x}(t)\bigr)\,ds\,dx \\ &{} + \int _{0}^{L}x \bigl(uf_{1}(u,v)+vf_{2}(u,v) \bigr)\,dx. \end{aligned}$$

(24)

Using Young’s inequality, we obtain, for any $\delta _{1}, \delta _{2}\in (0,1)$,

$$\begin{aligned} \Phi ^{\prime }(t)\le{}& \int _{0}^{L}xu^{2}_{t}\,dx- \ell _{1} \int _{0}^{L}xu_{x}^{2}\,dx+ \frac{\delta _{1}}{2} \int _{0}^{L}xu_{x}^{2}\,dx \\ &{} +\frac{1}{2\delta _{1}} \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(s)-u_{x}(t)\bigr)\,ds \biggr)^{2}\,dx \\ &{} + \int _{0}^{L}xv^{2}_{t}\,dx- \ell _{2} \int _{0}^{L}xv_{x}^{2}\,dx+ \frac{\delta _{2}}{2} \int _{0}^{L}xv_{x}^{2}\,dx \\ &{} +\frac{1}{2\delta _{2}} \int _{0}^{L}x \biggl( \int _{0}^{t}k_{2}(t-s) \bigl(v_{x}(s)-v_{x}(t)\bigr)\,ds \biggr)^{2}\,dx \\ &{} + \int _{0}^{L}x F(u,v)\,dx. \end{aligned}$$

(25)

Taking $\delta _{1}=\ell _{1}$ and $\delta _{2}=\ell _{2}$ and using Lemma 3.3, we have

$$\begin{aligned} \Phi ^{\prime }(t)\le{}& \int _{0}^{L}xu^{2}_{t}\,dx- \frac{\ell _{1}}{2} \int _{0}^{L}xu_{x}^{2} \,dx+cC_{\alpha,1}(h_{1}\circ u_{x}) (t) \\ &{} + \int _{0}^{L}xv^{2}_{t}\,dx- \frac{\ell _{1}}{2} \int _{0}^{L}xv_{x}^{2} \,dx+cC_{ \alpha,2}(h_{2}\circ v_{x}) (t)+ \int _{0}^{L}x F(u,v)\,dx. \end{aligned}$$

(26)

 □

Let us introduce the functionals

$$\begin{aligned} \chi _{1}(t):=- \int _{0}^{L}x u_{t} \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \end{aligned}$$

and

$$\begin{aligned} \chi _{2}(t):=- \int _{0}^{L} xv_{t} \int _{0}^{t}k_{2}(t-s) \bigl(v(t)-v(s) \bigr)\,ds\,dx. \end{aligned}$$

Lemma 3.5

Assume that $(A1)$ and $(A2)$ hold. Then the functional

$$\begin{aligned} \chi (t):=\chi _{1}(t)+\chi _{2}(t) \end{aligned}$$

satisfies, along with the solution of (1), the following estimate:

$$\begin{aligned} \chi '(t)\le{}& {-} \biggl( \int _{0}^{t}k_{1}(s)\,ds-\delta \biggr) \Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+\frac{c}{\delta }(C_{\alpha,1}+1) (h_{1} \circ u_{x}) (t) \\ &{} - \biggl( \int _{0}^{t}k_{2}(s)\,ds-\delta \biggr) \Vert v_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert v_{x} \Vert ^{2}_{L^{2}_{x}}+ \frac{c}{\delta }(C_{\alpha,2}+1) (h_{2}\circ v_{x}) (t), \end{aligned}$$

(27)

where $0<\delta <1$.

Proof

Direct differentiation, using (1), gives

$$\begin{aligned} \chi _{1}'(t)={}&{-} \biggl( \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{t}^{2} \\ &{} + \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{x}(t) \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(t)-u_{x}(s)\bigr)\,ds\,dx \\ &{} + \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(t)-u_{x}(s)\bigr)\,ds \biggr)^{2}\,dx \\ &{} - \int _{0}^{L}x f_{1}(u,v) \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ &{} - \int _{0}^{L}x u_{t} \int _{0}^{t}k_{1}'(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx. \end{aligned}$$

(28)

Using Young’s inequality and Lemma 3.3, we get, for any $0<\delta <1$, the following:

$$\begin{aligned} &\biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{x}(t) \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(t)-u_{x}(s)\bigr)\,ds\,dx \\ &\qquad{} + \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(t)-u_{x}(s) \bigr\vert \,ds \biggr)^{2} \,dx \\ &\quad \leq \delta \int _{0}^{L}xu_{x}^{2}+ \frac{c}{\delta } \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(t)-u_{x}(s) \bigr\vert \,ds \biggr)^{2} \,dx \\ &\quad \leq \delta \int _{0}^{L}xu_{x}^{2}+ \frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u_{x}) (t). \end{aligned}$$

(29)

Using Young’s inequality, (18), (19) and (22), we have

$$\begin{aligned} & \int _{0}^{L} xf_{1}(u,v) \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ &\quad \le \delta \biggl( \int _{0}^{L} x{ \bigl\vert f_{1}(u,v) \bigr\vert }^{2} \,dx \biggr)+ \frac{1}{4\delta } \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds \biggr)^{2} \,dx \\ &\quad \le {c_{1} \delta {\bigl(\ell _{1} \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+\ell _{2} \Vert v_{x} \Vert _{L_{x}^{2}}^{2} \bigr)}^{2r+3}}+\frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u_{x}) (t) \\ &\quad \le c_{1}\delta { \biggl(\frac{2(r+2)}{r+1}E(0) \biggr)}^{2r+1}\bigl(\ell _{1} \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+\ell _{2} \Vert v_{x} \Vert _{L_{x}^{2}}^{2}\bigr)+ \frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u_{x}) (t) \\ &\quad \leq c\delta \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+c \delta \Vert v_{x} \Vert _{L_{x}^{2}}^{2}+ \frac{c}{\delta }C_{\alpha,1}(h_{1}\circ u_{x}) (t). \end{aligned}$$

(30)

Also, by applying Young’s inequality and Lemma 3.3, we obtain, for any $0<\delta <1$,

$$\begin{aligned} &- \int _{0}^{L} xu_{t} \int _{0}^{t}k_{1}'(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &\quad = \int _{0}^{L} x u_{t} \int _{0}^{t} h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx- \int _{0}^{L} x u_{t} \int _{0}^{t}\alpha k_{1}(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &\quad \leq \delta \Vert u_{t} \Vert _{L_{x}^{2}}^{2}+ \frac{1}{2\delta } \biggl( \int _{0}^{t}h_{1}(s)\,ds \biggr) (h_{1}\circ u) (t)+\frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u) (t) \\ &\quad \leq \delta \Vert u_{t} \Vert _{L_{x}^{2}}^{2}+ \frac{c}{\delta }(C_{\alpha,1}+1) (h_{1} \circ u_{x}) (t). \end{aligned}$$

(31)

Similarly, we have

$$\begin{aligned} - \int _{0}^{L} xv_{t} \int _{0}^{t}k_{2}'(t-s) \bigl(v(t)-v(s)\bigr)\,ds\,dx\leq \delta \Vert v_{t} \Vert _{L_{x}^{2}}^{2}+\frac{c}{\delta }(C_{\alpha,2}+1) (h_{2} \circ v_{x}) (t). \end{aligned}$$

(32)

A combination of all the above estimates gives

$$\begin{aligned} \chi _{1}'(t)\le - \biggl( \int _{0}^{t}k_{1}(s)\,ds-\delta \biggr) \Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+\frac{c}{\delta }(C_{\alpha,1}+1) (h_{1} \circ u_{x}) (t). \end{aligned}$$

(33)

Repeating the same calculations with $\chi _{2}$, we obtain

$$\begin{aligned} \chi _{2}'(t)\le - \biggl( \int _{0}^{t}k_{2}(s)\,ds-\delta \biggr) \Vert v_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert v_{x} \Vert ^{2}_{L^{2}_{x}}+\frac{c}{\delta }(C_{\alpha,2}+1) (h_{2} \circ v_{x}) (t). \end{aligned}$$

(34)

Therefore, (33) and (34) imply (27), which completes the proof of Lemma 3.5. □

Lemma 3.6

Assume that $(A1)$ and $(A2)$ hold. Then the functionals $J_{1}$ and $J_{2}$ defined by

$$\begin{aligned} J_{1}(t):= \int _{0}^{L}x \int _{0}^{t}K_{1}(t-s) \bigl\vert u_{x}(s) \bigr\vert ^{2}\,ds\,dx \end{aligned}$$

and

$$\begin{aligned} J_{2}(t):= \int _{0}^{L}x \int _{0}^{t}K_{2}(t-s) \bigl\vert v_{x}(s) \bigr\vert ^{2}\,ds\,dx \end{aligned}$$

satisfy, along with the solution of (1), the estimates

$$\begin{aligned} & J_{1}'(t)\leq 3(1-\ell ) \Vert u_{x} \Vert _{L_{x}^{2}}^{2}-\frac{1}{2}(k_{1} \circ u_{x}) (t), \end{aligned}$$

(35)

$$\begin{aligned} & J_{2}'(t)\leq 3(1-\ell ) \Vert v_{x} \Vert _{L_{x}^{2}}^{2}-\frac{1}{2}(k_{2} \circ v_{x}) (t), \end{aligned}$$

(36)

where $K_{i}(t):=\int _{t}^{\infty }k_{i}(s)\,ds$ (for $i=1,2$) and $\ell =\min \{\ell _{1},\ell _{2}\}$.

Proof

We will prove inequality (35) and the same proof also holds for (36). By Young’s inequality and the fact that $K_{1}^{\prime }(t)=-k_{1}(t)$, we see that

$$\begin{aligned} J_{1}^{\prime }(t)={}&K_{1}(0) \int _{0}^{L}x \bigl\vert u_{x}(t) \bigr\vert ^{2}\,dx- \int _{0}^{L}x \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(s) \bigr\vert ^{2} \,dx \\ ={}&{-} \int _{0}^{L}x \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(s)- u_{x}(t) \bigr\vert ^{2} \,ds \,dx \\ &{} -2 \int _{0}^{L} xu_{x}(t). \int _{0}^{t}k_{1}(t-s) \bigl( u_{x}(s)- u_{x}(t)\bigr)\,ds\,dx+K_{1}(t) \int _{0}^{L}x \bigl\vert u_{x}(t) \bigr\vert ^{2} \,dx. \end{aligned}$$

Now,

$$\begin{aligned} &-2 \int _{0}^{L}x u_{x}(t). \int _{0}^{t}k_{1}(t-s) \bigl( u_{x}(s)-u_{x}(t)\bigr)\,ds\,dx \\ &\quad \le 2(1-\ell _{1}) \int _{0}^{L}x \bigl\vert u_{x}(t) \bigr\vert ^{2} \,dx+ \frac{\int _{0}^{t}k_{1}(s)\,ds}{2(1-\ell _{1})} \int _{0}^{L}x \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(s)-u_{x}(t) \bigr\vert ^{2} \,ds\,dx. \end{aligned}$$

Using the facts that $K_{1}(0)=1-\ell _{1}$ and $\int _{0}^{t}k_{1}(s)\,ds \le 1-\ell _{1}$, (35) is established. □

Lemma 3.7

The functional L defined by

$$\begin{aligned} L(t):=NE(t)+N_{1} \phi (t)+N_{2} \chi (t) \end{aligned}$$

satisfies, for a suitable choice of $N,N_{1},N_{2}\ge 1$,

$$\begin{aligned} L(t)\sim E(t) \end{aligned}$$

(37)

and the estimate

$$\begin{aligned} L'(t)\leq {}&{-}4(1-\ell ) \bigl( \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+ \Vert v_{x} \Vert ^{2}_{L_{x}^{2}} \bigr)- \bigl( \Vert u_{t} \Vert ^{2}_{L_{x}^{2}}+ \Vert v_{t} \Vert ^{2}_{L_{x}^{2}} \bigr) \\ &{} +c \int _{0}^{L} xF(u,v)\,dx+\frac{1}{4} \bigl[(k_{1}\circ u_{x}) (t)+(k_{2} \circ v_{x}) (t) \bigr], \quad\forall t\geq t_{0}, \end{aligned}$$

(38)

where $t_{0}$ is introduced in Lemma 3.2and $\ell =\min \{\ell _{1},\ell _{2}\}$.

Proof

It is not difficult to prove that $L(t)\sim E(t)$. To establish (38), we choose $\delta =\frac{\ell }{4cN_{2}}$ where $\ell =\min \{\ell _{1},\ell _{2}\}$. We set $C_{\alpha }=\max \{C_{\alpha,1},C_{\alpha,2}\}$ and $k_{0}=\min \lbrace \int _{0}^{t_{0}}k_{1}(s)\,ds,\int _{0}^{t_{0}}k_{2}(s)\,ds \rbrace >0$. Now using (23) and (28) and recalling the fact that $k_{i}'=\alpha k_{i}-h_{i}$, we obtain, for any $t\geq t_{0}$,

$$\begin{aligned} L'(t)\leq{}&{ -}\frac{\ell }{4}(2N_{1}-1) \bigl( \Vert u_{x} \Vert ^{2}_{L_{x}^{2}}+ \Vert v_{x} \Vert ^{2}_{L_{x}^{2}} \bigr)- \biggl(k_{0}N_{2}-\frac{\ell }{4c}-N_{1} \biggr) \bigl( \Vert u_{t} \Vert ^{2}_{L_{x}^{2}}+ \Vert v_{t} \Vert ^{2}_{L_{x}^{2}} \bigr) \\ &{} -N_{1} \int _{0}^{L}x F(u,v)\,dx+\frac{\alpha }{2}N \bigl[(k_{1}\circ u_{x}) (t)+(k_{2} \circ v_{x}) (t) \bigr] \\ &{} - \biggl[\frac{1}{2}N-\frac{4c^{2}}{\ell }N^{2}_{2}-C_{\alpha } \biggl( \frac{4c^{2}}{\ell }N_{2}^{2}+cN_{1} \biggr) \biggr] \bigl[(h_{1}\circ u_{x}) (t)+(h_{2} \circ v_{x}) (t) \bigr]. \end{aligned}$$

First, we choose $N_{1}$ so large such that $\frac{\ell }{4}(2N_{1}-1)>4(1-\ell )$.

Then we select $N_{2}$ large enough so that $k_{0} N_{2}-\frac{\ell }{4c}-N_{1}>1$. Now, one can use the Lebesgue dominated convergence theorem with the fact that $\frac{\alpha k_{i}^{2}(s)}{\alpha k_{i}(s)-k_{i}'(s)}< k_{i}(s)$, for $i=1,2$, to prove that

$$\begin{aligned} \lim_{\alpha \rightarrow 0^{+}}\alpha C_{\alpha }=0. \end{aligned}$$

Therefore, there exists $\alpha _{0}\in (0,1)$ such that if $\alpha <\alpha _{0}$, then, we get $\alpha C_{\alpha }< \frac{1}{8 [\frac{4c^{2}}{\ell }N^{2}_{2}+cN_{1} ]}$. Then, by letting $\alpha =\frac{1}{2N}<\alpha _{0}$, we get $\frac{1}{4}N-\frac{4c^{2}}{\ell }N^{2}_{2}>0$. This leads to

$$\begin{aligned} \frac{1}{2}N-\frac{4c^{2}}{\ell }N^{2}_{2}-C_{\alpha } \biggl[ \frac{4c^{2}}{\ell }N_{2}^{2}+cN_{1} \biggr]>\frac{1}{4}N- \frac{4c^{2}}{\ell }N^{2}_{2}>0. \end{aligned}$$

Then, (38) is established. □

General decay result

In this section, we state and prove our main result.

Theorem 4.1

Let $(u_{0},v_{0}) \in V_{0}^{2}$ and $(u_{1},v_{1})\in (L_{x}^{2})^{2}$ be given and satisfying (16). Assume that $(A1)$ and $(A2)$ hold. If $\Psi _{1}$ and $\Psi _{2}$ are linear, then there exist two positive constants $\lambda _{1}$ and $\lambda _{2}$ such that the solution to problem (1) satisfies the estimate

$$\begin{aligned} E(t)\leq \lambda _{2}\exp \biggl(-\lambda _{1} \int _{t_{0}}^{t}\xi (s)\,ds \biggr),\quad \forall t\ge t_{0}, \end{aligned}$$

(39)

where $t_{0}$ is introduced in Lemma 3.2and $\xi (t)=\min \{\xi _{1}(t),\xi _{2}(t)\}$.

Proof

Using (21) and (13) we have, for any $t\geq t_{0}$,

$$\begin{aligned} \int _{0}^{t_{0}} k_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2} \,ds+ \int _{0}^{t_{0}} k_{2}(s) \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2} \,ds \leq -cE'(t). \end{aligned}$$

Using this inequality, the estimate (38) becomes, for some $m>0$ and for any $t\geq t_{0}$,

$$\begin{aligned} L'(t)\leq{} &{-}mE(t)-cE'(t)+c \int _{t_{0}}^{t} k_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2} \,ds \\ &{}+c \int _{t_{0}}^{t} k_{2}(s) \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2} \,ds. \end{aligned}$$

Let $\mathcal{L}:= L+cE\sim E$, we obtain

$$\begin{aligned} \mathcal{L}'(t)\leq {}& {-}mE(t)+c \int _{t_{0}}^{t} k_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2} \,ds \\ &{}+c \int _{t_{0}}^{t} k_{2}(s) \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2} \,ds, \quad\forall t\geq t_{0}. \end{aligned}$$

(40)

Multiply both sides of (40) by $\xi (t)=\min \{\xi _{1}(t),\xi _{2}(t)\}$ where ξ is non-increasing function and using $(A2)$ and (13) we get, for any $t\geq t_{0}$ and $m>0$, the following:

$$\begin{aligned} \xi (t)\mathcal{L}'(t)\leq{}& {-}m\xi (t)E(t)+c \int ^{t}_{0}\xi _{1}(s)k_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2}\,ds \\ &{}+c \int _{0}^{t}\xi _{2}(s)k_{2}(s) \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2}\,ds \\ \leq{} & {-}m\xi (t)E(t)-c \int _{0}^{t} k'_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2}\,ds \\ &{}\times c \int _{0}^{t} k'_{2}(s) \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert _{L_{x}^{2}}^{2}\,ds \\ \leq {}& {-}m\xi (t)E(t)-cE'(t). \end{aligned}$$

Since ξ is non-increasing, we have

$$\begin{aligned} (\xi \mathcal{L}+cE)'(t)\leq -m\xi (t)E(t), \quad\forall t\geq t_{0}. \end{aligned}$$

Integrating over $(t_{0},t)$ and using the fact that $\xi \mathcal{L}+cE\sim E$, then, for any $\lambda _{1},\lambda _{2} > 0$, we obtain

$$\begin{aligned} E(t)\leq \lambda _{2}\exp \biggl(-\lambda _{1} \int _{t_{0}}^{t}\xi (s)\,ds \biggr), \quad\forall t\geq t_{0}. \end{aligned}$$

 □

Theorem 4.2

Let $(u_{0},v_{0}) \in V_{0}^{2}$ and $(u_{1},v_{1})\in (L_{x}^{2})^{2}$ be given and satisfying (16). Assume that $(A1)$ and $(A2)$ hold. If $\Psi _{1}$ or $\Psi _{2}$ is nonlinear, then there exist two positive constants $\lambda _{1}$ and $\lambda _{2}$ such that the solution to problem (1) satisfies the estimate

$$\begin{aligned} E(t)\leq \lambda _{2} \Psi _{*}^{-1} \biggl( \lambda _{1} \int _{t_{0}}^{t}\xi (s)\,ds \biggr), \quad\forall t>t_{0}, \end{aligned}$$

(41)

where

$$\begin{aligned} \Psi _{*}(t)= \int _{t}^{r}\frac{1}{s H(s)}\,ds \quad\textit{with } H(t)= \min \bigl\{ \Psi _{1}'(t),\Psi _{2}'(t) \bigr\} . \end{aligned}$$

Proof

Using Lemmas 3.6 and 3.7, we easily see that

$$\begin{aligned} \mathcal{L}_{1}(t):=L(t)+J_{1}(t)+J_{2}(t) \end{aligned}$$

is nonnegative and, for any $t\geq t_{0}$, and, for some $C>0$,

$$\begin{aligned} \mathcal{L}_{1}'(t)\leq -c E(t). \end{aligned}$$

Therefore, we arrive at

$$\begin{aligned} \int _{0}^{\infty }E(s)\,ds< +\infty. \end{aligned}$$

(42)

Now, we define the following functionals:

$$\begin{aligned} I_{1}(t):=\gamma \int _{t_{0}}^{t} \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}} \,ds, \qquad I_{2}(t):=\gamma \int _{t_{0}}^{t} \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds. \end{aligned}$$

Thanks to (42), one can choose $0<\gamma <1$ so that

$$\begin{aligned} I_{i}(t)< 1, \quad \forall t \geq t_{0} \text{ and } i=1,2. \end{aligned}$$

(43)

Without loss of the generality, we assume that $I_{i}(t)>0$, for any $t> t_{0}$; otherwise, we get an exponential decay from (38). We also define the following functionals:

$$\begin{aligned} \eta _{1}(t):=- \int _{t_{0}}^{t} k_{1}'(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds, \text{ }\eta _{2}(t):=- \int _{t_{0}}^{t} k_{2}'(s) \bigl\Vert v_{x}(t)- v_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds \end{aligned}$$

and observe that

$$\begin{aligned} \eta _{1}(t)+\eta _{2}(t)\leq -cE'(t), \quad\forall t \geq t_{0}. \end{aligned}$$

(44)

Using (2.4), Assumption $(A2)$, inequality (43) and Jensen’s inequality, we obtain

$$\begin{aligned} \eta _{1}(t)&\leq \frac{1}{\gamma I_{1}(t)} \int _{t_{0}}^{t}\gamma I_{1}(t) \xi _{1}(s)\Psi _{1}\bigl(k_{1}(s)\bigr) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds \\ &\leq \frac{\xi _{1}(t)}{\gamma I_{1}(t)} \int _{t_{0}}^{t}\gamma \Psi _{1} \bigl(I_{1}(t)k_{1}(s)\bigr) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}} \,ds \\ &\leq \frac{\xi _{1}(t)}{\gamma }\Psi _{1} \biggl(\frac{1}{I_{1}(t)} \int _{t_{0}}^{t}\gamma I_{1}(t) k_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds \biggr) \\ &=\frac{\xi _{1}(t)}{\gamma }\bar{\Psi }_{1} \biggl(\gamma \int _{t_{0}}^{t} k_{1}(s) \bigl\Vert u_{x}(t)- u_{x}(t-s) \bigr\Vert ^{2}_{L_{x}^{2}} \,ds \biggr),\quad \forall t \geq t_{0}, \end{aligned}$$

where $\bar{\Psi }_{1}$ is defined in Remark (2.3). Then, we have

$$\begin{aligned} \int _{t_{0}}^{t} k_{1}(s) \bigl\Vert u(t)-u(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds \leq \frac{1}{\gamma } \bar{\Psi }^{-1}_{1} \biggl( \frac{\gamma \eta _{1}(t)}{\xi _{1}(t)} \biggr), \quad t\geq t_{0}. \end{aligned}$$

Similarly, we can have

$$\begin{aligned} \int _{t_{0}}^{t} k_{2}(s) \bigl\Vert v(t)- v(t-s) \bigr\Vert ^{2}_{L_{x}^{2}}\,ds \leq \frac{1}{\gamma } \bar{\Psi }^{-1}_{2} \biggl( \frac{\gamma \eta _{2}(t)}{\xi _{2}(t)} \biggr),\quad t \geq t_{0}. \end{aligned}$$

Thus, the estimate (40) becomes

$$\begin{aligned} F'(t)\leq -mE(t)+c \bar{\Psi }^{-1}_{1} \biggl( \frac{\gamma \eta _{1}(t)}{\xi _{1}(t)} \biggr)+c \bar{\Psi }^{-1}_{2} \biggl(\frac{\gamma \eta _{2}(t)}{\xi _{2}(t)} \biggr),\quad t\geq t_{0}. \end{aligned}$$

(45)

Set $H=\min \{\bar{\Psi }_{1}',\bar{\Psi }_{2}'\}$ and define the functional

$$\begin{aligned} F_{1}(t):= H \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr)F(t)+E(t), \quad \text{for } \varepsilon _{0} \in (0,\varepsilon ) \text{ and }t\geq t_{0}. \end{aligned}$$

Using the fact that $\bar{\Psi }_{i}'>0$, $\bar{\Psi }_{i}''>0$ and $E'\leq 0$, we also deduce that $F_{1}\sim E$. Further, we get

$$\begin{aligned} F_{1}'(t)=\varepsilon _{0}\frac{E'(t)}{E(0)} H' \biggl(\varepsilon _{0} \frac{E(t)}{E(0)} \biggr)F(t)+H \biggl(\varepsilon _{0} \frac{E(t)}{E(0)} \biggr)F'(t)+E'(t),\quad \text{for a.e }t \geq t_{0}. \end{aligned}$$

Recalling that $E'\leq 0$, then we drop the first and last terms of the above identity. Therefore, by using the estimate (45), we have

$$\begin{aligned} F_{1}'(t)\leq{} &{-}mE(t)H \biggl( \varepsilon _{0}\frac{E(t)}{E(0)} \biggr)+c H \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr) \bar{\Psi }^{-1}_{1} \biggl(\frac{\gamma \eta _{1}(t)}{\xi _{1}(t)} \biggr) \\ &{}+c H \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr)\bar{\Psi }^{-1}_{2} \biggl(\frac{\gamma \eta _{2}(t)}{\xi _{2}(t)} \biggr),\quad \text{for a.e }t\geq t_{0}. \end{aligned}$$

(46)

In the sense of Young [23], we let $\bar{\Psi }_{i}^{*}$ be the convex conjugate of $\bar{\Psi }_{i}$ such that

$$\begin{aligned} \bar{\Psi }_{i}^{*}(s)=s \bigl(\bar{\Psi }_{i}' \bigr)^{-1}(s)-\bar{\Psi }_{i} \bigl[ \bigl(\bar{\Psi }_{i}' \bigr)^{-1}(s) \bigr], \quad\text{for }i=1,2, \end{aligned}$$

(47)

and it satisfies the following generalized Young inequality:

$$\begin{aligned} AB_{i}\leq \bar{\Psi }_{i}^{*}(A)+ \bar{\Psi }_{i}(B_{i}),\quad \text{for }i=1,2. \end{aligned}$$

(48)

By letting $A=H (\varepsilon _{0}\frac{E(t)}{E(0)} )$, $B_{i}=\bar{\Psi }_{i}^{-1} ( \frac{\gamma \eta _{i}(t)}{\xi _{i}(t)} )$, for $i=1,2$, and combining (46)–(48), we have, for almost every $t\geq t_{0}$,

$$\begin{aligned} F_{1}'(t)\leq{} &{-}mE(t)H \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr)+c\bar{\Psi }_{1}^{*} \biggl[H \biggl(\varepsilon _{0} \frac{E(t)}{E(0)} \biggr) \biggr]+c \frac{\gamma \eta _{1}(t)}{\xi _{1}(t)} \\ &{}+c\bar{\Psi }_{2}^{*} \biggl[H \biggl(\varepsilon \frac{E(t)}{E(0)} \biggr) \biggr]+c\frac{\gamma \eta _{2}(t)}{\xi _{2}(t)} \\ \leq {}&{-}mE(t)H \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr)+cH \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr) \bigl(\bar{\Psi }_{1}' \bigr)^{-1} \biggl[\bar{\Psi }_{1}' \biggl(\varepsilon _{0} \frac{E(t)}{E(0)} \biggr) \biggr]+c \frac{\gamma \eta _{1}(t)}{\xi _{1}(t)} \\ &{}+cH \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr) \bigl(\bar{ \Psi }_{2}' \bigr)^{-1} \biggl[\bar{\Psi }_{2}' \biggl(\varepsilon _{0} \frac{E(t)}{E(0)} \biggr) \biggr]+c \frac{\gamma \eta _{2}(t)}{\xi _{2}(t)} \\ \leq{} &{-} \bigl(mE(0)-c\varepsilon _{0} \bigr)\frac{E(t)}{E(0)}H \biggl( \varepsilon _{0}\frac{E(t)}{E(0)} \biggr)+c \biggl( \frac{\gamma \eta _{1}(t)}{\xi _{1}(t)}+ \frac{\gamma \eta _{2}(t)}{\xi _{2}(t)} \biggr). \end{aligned}$$

Multiplying the above estimate by $\xi (t)=\min \{\xi _{1}(t),\xi _{2}(t)\}>0$ and using the fact in (44), we get

$$\begin{aligned} \xi (t)F_{1}'(t)\leq -\bigl(mE(0)-c\varepsilon _{0}\bigr)\xi (t) \frac{E(t)}{E(0)} H \biggl(\varepsilon _{0}\frac{E(t)}{E(0)} \biggr)-cE'(t), \quad\text{for a.e }t \geq t_{0}. \end{aligned}$$

Select $\varepsilon _{0}$ small enough so that $k_{0}:= mE(0)-c\varepsilon _{0} >0$, and we obtain

$$\begin{aligned} \xi (t)F_{1}'(t)\leq -k_{0}\xi (t) \frac{E(t)}{E(0)} H \biggl( \varepsilon _{0}\frac{E(t)}{E(0)} \biggr)-cE'(t),\quad \text{for a.e }t\geq t_{0}. \end{aligned}$$

Let $F_{2}=\xi F_{1}+cE\sim E$, we have, for some $\alpha _{1},\alpha _{2}>0$, the following equivalent inequality:

$$\begin{aligned} \alpha _{1} F_{2}(t)\leq E(t)\leq \alpha _{2} F_{2}(t), \quad \forall t\geq t_{0}. \end{aligned}$$

(49)

Hence, we have

$$\begin{aligned} F_{2}'(t)\leq -k_{0}\xi (t) \frac{E(t)}{E(0)}H \biggl(\varepsilon _{0} \frac{E(t)}{E(0)} \biggr), \quad\text{for a.e }t\geq t_{0}. \end{aligned}$$

(50)

Now, we set

$$\begin{aligned} H_{0}(t)=t H(\varepsilon _{0} t), \quad\forall t\in [0,1]. \end{aligned}$$

Using the fact that $\Psi _{i}'>0$ and $\Psi _{i}''>0$ on $(0,r]$ (for $i=1,2$), we deduce that $H_{0},H_{0}'>0$ $a.e$. on $(0,1]$. Now, we define the following functional:

$$\begin{aligned} R(t):=\frac{\alpha _{1} F_{2}(t)}{E(0)} \end{aligned}$$

and use (49) and (50) to show that $R\sim E$ and, for some $\beta _{1}>0$,

$$\begin{aligned} R'(t)\leq -\beta _{1}\xi (t)H_{0}\bigl(R(t) \bigr), \quad\text{for a.e }t\geq t_{0}. \end{aligned}$$

Integrating over the interval $(t_{0},t)$ and using a change of variables, we get

$$\begin{aligned} \int _{\varepsilon _{0} R(t)}^{\varepsilon _{0} R(0)}\frac{1}{s H(s)}\,ds \geq \beta _{1} \int _{t_{0}}^{t}\xi (s)\,ds; \end{aligned}$$

which gives

$$\begin{aligned} R(t)\leq \frac{1}{\varepsilon _{0}} \Psi _{*}^{-1} \biggl(\beta _{1} \int _{t_{0}}^{t}\xi (s)\,ds \biggr)\quad \forall t \geq t_{0}, \end{aligned}$$

where $\Psi _{*}(t):=\int _{t}^{r}\frac{1}{s H(s)}\,ds$. Since $R\sim E$, we have, for $\beta _{2}>0$,

$$\begin{aligned} E(t)\leq \beta _{2} \Psi _{*}^{-1} \biggl(\beta _{1} \int _{0}^{t}\xi (s)\,ds \biggr) \quad\forall t \geq t_{0}. \end{aligned}$$

This completes the proof. □

Example 4.3

• (1) Let $k_{1}(t)=ae^{-\alpha t}$ and $k_{2}(t)=\frac{b}{(1+t)^{q}}$, $q>1$. The constants a and b are chosen so that $(A1)$ is satisfied. Then there exists $C>0$ such that

  $$\begin{aligned} E(t)\leq \frac{C}{(1+t)^{q}},\quad \forall t>0. \end{aligned}$$

• (2) Let $k_{1}(t)=\frac{a}{(1+t)^{m}}$ and $k_{2}(t)=\frac{b}{(1+t)^{n}}$ with $m,n>1$. The constants a and b are chosen so that $(A1)$ is satisfied. Then there exists $C>0$ such that, for any $t>0$,

  $$\begin{aligned} E(t)\leq \frac{C}{(1+t)^{\nu }}, \quad\text{with }\nu =\min \{m,n \}. \end{aligned}$$

• (3) Let $k_{1}(t)=ae^{-\beta t}$ and $k_{2}(t)=be^{-(1+t)^{q}}$ with $0< q<1$. The constants a and b are chosen so that $(A1)$ is satisfied. Then there exist positive constants C and $\alpha _{1}$ such that

  $$\begin{aligned} E(t)\leq Ce^{-\alpha _{1}(1+t)^{\nu }}, \quad\text{for } t \text{ large}. \end{aligned}$$