Stability Results for a System of Nonlinear Viscoelastic Plate Equations with Nonlinear Frictional Damping and Logarithmic Source Terms

Abstract

In this paper, we consider a system of nonlinear viscoelastic plate equations with nonlinear frictional damping and logarithmic source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system. Our results substantially improve some earlier related results in the literature.

1 Introduction

In this paper, we investigate the decay of solutions of the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}+\Delta ^2 u+ u+f_1(u,z) -\int _0^t g(t-s)\Delta ^2 u(s)ds =\kappa u \ln \vert u \vert ,&{} \text {in }\Omega ,\; t>0,\\ z_{tt}+\Delta ^2 z+ z+ f_2(u,z) + h (z_t) =\kappa z \ln \vert z \vert ,&{}\text {in }\Omega ,\; t>0, \qquad \text {(P)} \\ u(\cdot ,t)=z(\cdot ,t)= \frac{\partial u}{\partial \eta }=\frac{\partial z}{\partial \eta }=0,&{} \text {on}\;\partial \Omega , t\ge 0,\\ (u(0),z(0))=(u_{0},z_{0}), (u_{t}(0),z_{t}(0))=(u_{1},z_{1}),&{} \text {in }\Omega , \end{array}\right. } \end{aligned}$$

with

$$\begin{aligned} {\left\{ \begin{array}{ll} f_{1}(u,z)=a{\vert {u+z}\vert }^{2\left( \rho +1\right) }\left( u+z \right) +b{\vert {u}\vert }^{\rho } u {\vert {z}\vert }^{\rho +2},\\ f_{2}(u,z)=a{\vert {u+z}\vert }^{2\left( \rho +1\right) }\left( u+z \right) +b{\vert {z}\vert }^{\rho } z {\vert {u}\vert }^{\rho +2}, \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega$ is a bounded and regular domain of $\mathbb {R}^2$, with smooth boundary $\partial \Omega$. The vector $\eta$ is the unit outer normal to $\partial \Omega$. The constants $a,b>0$, $\rho >-1$, and $\kappa$ is a small positive number satisfying some specific conditions.

The logarithmic nonlinearity is of much interest in physics, since it appears naturally in inflation cosmology and supersymmetric field theories, quantum mechanics, nuclear physics, optics, and geophysics [1,2,3,4,5]. This type of problem has attracted the attention of many authors, and several decays and blow-up results have been established. We start with the pioneering work of Birula and Mycielski [4] in which they investigated the following problem:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} u_{tt}- u_{xx} +u-\varepsilon u \ln {\vert u\vert }^2=0,&{}\text {in }[a,b]\times (0,T),\\ u(a,t)=u(b,t)=0,&{}\text {in } (0,T),\\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),&{}\text {in } [a,b], \end{array}\right. } \end{aligned} \end{aligned}$$

(1.2)

and proved that wave equations with the logarithmic nonlinearity have stable, localized, soliton-like solutions in any number of dimensions. In [6], Cazenave and Haraux established the existence and uniqueness of the solution for the following Cauchy problem:

$$\begin{aligned} u_{tt} -\Delta u= u \ln {\vert u\vert }^k, \text { in }\mathbb {R}^3. \end{aligned}$$

(1.3)

Gorka [5] considered the corresponding one-dimensional Cauchy problem for Eq. 1.3 and established the global existence of weak solutions for all $(u_{0},u_1) \in H^{1}_{0} \times L^2$ by using some compactness arguments. For more results dealing with logarithmic nonlinearity, we refer the reader to see [7,8,9,10].

Regarding the plate equations, we start with the result obtained by Lagnese [11] when he considered a viscoelastic plate equation and proved a decay result by introducing a dissipative mechanism on the boundary. In [12], Rivera et al. considered a viscoelastic plate equation and established an exponential decay result for relaxation functions decaying exponentially. For more results in this direction, we refer the reader to see [13,14,15,16,17].

For viscoelastic problems, Dafermos [18] considered a one-dimensional viscoelastic problem of the form

$$\rho u_{tt}=c u_{xx}-\int _{-\infty }^{t}g(t-s)u_{xx}(s)ds,$$

and established various existing results and then proved, for smooth monotone decreasing relaxation functions, that the solutions go to zero as t goes to infinity. However, no rate of decay has been specified. In [19], Cavalcanti et al. considered the equation

$$\begin{aligned} u_{tt}-\Delta u+\int _{0}^{t}g(t-s)\Delta u(x,s)ds+a(x)u_{t}+{\vert u \vert }^{p-1} u=0, \quad \text {in}~ \Omega \times (0,\infty ), \end{aligned}$$

(1.4)

and established an exponential decay result under some geometrical restrictions and for relaxation functions decaying exponentially. Messaoudi [20, 21] generalized the decay rates allowing a wide class of kernels, among which those of exponential and polynomial decay types are only special cases. After that, several steps were done by generalizing the conditions imposed on the relaxation functions; we mention among them the work of [22,23,24,25,26,27,28].

Motivated by the above works, we intend to investigate the stability of the coupled system (P) with a general form of nonlinear coupling terms $f_1,f_2$ and subject to nonlinear weak frictional damping and viscoelastic damping acting simultaneously and complementarily in the domain. In other words, we study the competition between the nonlinear feedback and viscoelasticity and establish general decay rates for the energy without imposing any growth assumption near the origin on h and strongly weakening the usual assumptions on g. The obtained energy decay rates are not necessarily of exponential or polynomial types.

Let us note here that though the logarithmic nonlinearity is somehow weaker than the polynomial nonlinearity, both the existence and stability results are not obtained by straightforward application of the method used for polynomial nonlinearity.

This paper is organized as follows. In Sect. 2, we present some notation and material needed for our work. In Sect. 3, we present the global existence of the solutions of the problem. Some technical lemmas and the decay results are presented in Sects. 4 and 5, respectively.

2 Preliminaries

In this section, we present some material needed for the proof of our results. We use the standard Lebesgue space $L^{2}(\Omega )$ and Sobolev space $H^{2}_{0}(\Omega )$ with their usual scalar products and norms. Throughout this paper, c is used to denote a generic positive constant. We consider the following hypotheses:

(A1):

$g: \mathbb {R}^+\rightarrow \mathbb {R}^+$ is a $C^{1}$ nonincreasing function satisfying

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

(2.1)

and there exists a $C^1$ function $G:(0,\infty )\rightarrow (0,\infty )$ which is linear or it is strictly increasing and strictly convex $C^2$ function on (0, r], $r\le g(0)$, with $G(0)=G^{\prime }(0)=0$, such that

$$\begin{aligned} g^{\prime }(t)\le -\xi (t) G(g(t)),\quad \forall t\ge 0, \end{aligned}$$

(2.2)

where $\xi (t)$ is a positive nonincreasing differentiable function.

(A2):

$h:\mathrm {I\hspace{-1.0pt}R}\rightarrow \mathrm {I\hspace{-1.0pt}R}$ is nondecreasing $C^{1}$ function such that there exists a $C^2$ convex and increasing function $H:\mathbb {R}^+\rightarrow \mathbb {R}^+$ satisfying $H(0)=0$ and $H^{\prime }(0)=0$ or H is linear on $[0,\varepsilon ]$ such that

$$\begin{aligned} \begin{aligned}&c_1\vert s\vert \le \vert h(s)\vert \le c_2 \vert s\vert , \text { if }\vert s\vert \ge \varepsilon ,\\&\vert s\vert ^2+h^2(s)\le H^{-1}\left( sh(s)\right) ,\text { if }\vert s\vert \le \varepsilon , \end{aligned} \end{aligned}$$

(2.3)

where $\varepsilon ,c_{1},c_{2}$ are positive constants.

(A3):

The constant $\kappa$ in (P) satisfies $0<\kappa <\kappa _0$, where $\kappa _0$ is the positive real number satisfying

$$\begin{aligned} {\sqrt{\frac{2\pi \ell }{\kappa _0 c_p}}}=e^{-\frac{3}{2}-\frac{1}{\kappa _0}} \end{aligned}$$

(2.4)

and $c_{p}$ is the smallest positive number satisfying

$$\begin{aligned} \Vert \nabla u\Vert ^2_2 \le c_{p} \Vert \Delta u\Vert ^2_{2} ,\quad \forall u \in H_{0}^{2}(\Omega ), \end{aligned}$$

where $\Vert . \Vert _2=\Vert .\Vert _{L^{2}(\Omega )}$.

Remark 2.1

If h satisfies

$$\begin{aligned} \begin{aligned}&h_0(\vert s\vert )\le \vert h(s)\vert \le h_0^{-1}(\vert s\vert ), \quad \vert s\vert \le \varepsilon \\&c_1\vert s\vert \le \vert h(s)\vert \le c_2 \vert s\vert , \quad \vert s\vert \ge \varepsilon \end{aligned} \end{aligned}$$

(2.5)

for some strictly increasing function $h_0 \in C^1([0,+\infty ))$, with $h_0(0)=0$, and positive constants $c_1,c_2,\varepsilon$ and the function $H(s)=\sqrt{\frac{s}{2}}h_0\left( \frac{s}{2}\right)$, is strictly convex $C^2$ function on $(0,\varepsilon ]$ when $h_0$ is nonlinear, then (A2) is staisfied. This kind of hypothesis, where (A2) is weaker, was considered by Liu and Zuazua [29] and Alabau-Boussouira [30].

Remark 2.2

If G is a strictly increasing and strictly convex $C^2$ function on (0, r], with $G(0) = G'(0) = 0$, then it has an extension $\overline{G}$, which is strictly increasing and strictly convex $C^2$ function on $(0,+\infty )$.

Remark 2.3

Concerning the functions $f_1$ and $f_2$, we note that

$$\begin{aligned} uf_1(u,z)+zf_2(u,z)=2(\rho +2)F(u,z),\quad \forall (u,z)\in \mathbb {R}^2, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} F(u,z)=\frac{1}{2(\rho +2)}\left[ a {\vert u+z \vert }^{2(\rho +2)}+2b {\vert uz\vert }^{\rho +2}\right] . \end{aligned} \end{aligned}$$

Lemma 2.1

[31, 32] (Logarithmic Sobolev inequality) Let v be any function in $H^{1}_{0}(\Omega )$ and a be any positive real number. Then,

$$\begin{aligned} \int _{\Omega }v^2 \ln {\vert v\vert }dx \le \frac{1}{2}{\Vert v\Vert }^{2}_{2}\ln {\Vert v\Vert }^2_{2}+\frac{a^2}{2\pi }{\Vert \nabla v\Vert }^{2}_{2} -(1+\ln {a}){\Vert v\Vert }^{2}_{2}. \end{aligned}$$

(2.6)

Corollary 2.2

Let v be any function in $H^{2}_{0}(\Omega )$ and a be any positive real number. Then,

$$\begin{aligned} \int _{\Omega }v^2 \ln {\vert v\vert }dx \le \frac{1}{2}{\Vert v\Vert }^{2}_{2}\ln {\Vert v\Vert }^2_{2}+\frac{ c_{p} a^2}{2\pi }{\Vert \Delta v\Vert }^{2}_{2} -(1+\ln {a}){\Vert v\Vert }^{2}_{2} . \end{aligned}$$

(2.7)

For completeness we state, without proof, the local existence result of system (P) which can be proved by the same way established in [33,34,35].

Proposition 2.3

Assume $(A1)-(A3)$ hold and $(u_{0},u_{1}),(z_{0},z_{1})\in H_{0}^{2}(\Omega )\times L^2(\Omega )$. Then, system (P) has a local weak solution

$$u\in C([0,T],H^2_0(\Omega ))\cap C^1([0,T],L^2(\Omega )).$$

We define the energy functional E(t) associated with system (P) as follows:

$$\begin{aligned} \begin{aligned} E(t):=&\frac{1}{2}\Big [\Vert u_t\Vert _{2}^{2}+\Vert z_t\Vert _{2}^{2}\Big ]+\frac{1}{2}\left( 1-\int _0^t g(s)ds\right) \Vert \Delta u\Vert _{2}^{2}+\frac{1}{2}\Vert \Delta z\Vert _{2}^{2}+\dfrac{1}{2}(g \circ \Delta u)(t)\\&+ \frac{\kappa +2}{4}\Big [\Vert u\Vert _{2}^{2}+\Vert z\Vert _{2}^{2}\Big ] +2(\rho +2)\int _{\Omega } F(u,z) dx-\frac{\kappa }{2}\int _{\Omega } u^2 \ln \vert u\vert dx-\frac{\kappa }{2} \int _{\Omega }z^2 \ln \vert z \vert dx.\end{aligned} \end{aligned}$$

(2.8)

By multiplying the two equations in (P) by $u_t$ and $z_t$, respectively, integrating over $\Omega ,$ using integration by parts and adding the results, we get

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}E(t)=\dfrac{1}{2}(g'\circ \Delta u)(t)-\dfrac{1}{2}g(t)\Vert \Delta u \Vert ^2- \int _\Omega z_t h(z_t)dx \le 0. \end{aligned} \end{aligned}$$

(2.9)

3 Global Existence

In this section, we state the global existence result which can be proved using the potential wells corresponding to the Logarithmic nonlinearity as in [33,34,35,36]. For this purpose, we define the following functions:

$$\begin{aligned} J(u,z)= & {} \frac{1}{2}\Big [\ell \Vert \Delta u\Vert _{2}^{2}+\Vert \Delta z\Vert _{2}^{2}\Big ]+\frac{\kappa +2}{4}\Big [\Vert u\Vert _{2}^{2}+\Vert z\Vert _{2}^{2}\Big ]\nonumber \\{} & {} -\frac{\kappa }{2}\int _{\Omega } u^2 \ln \vert u\vert dx-\frac{\kappa }{2} \int _{\Omega }z^2 \ln \vert z \vert dx.\end{aligned}$$

(3.1)

$$\begin{aligned} I(u,z)= & {} \ell \Vert \Delta u\Vert _{2}^{2}+\Vert \Delta z\Vert _{2}^{2}+ \Vert u\Vert _{2}^{2}+\Vert z\Vert _{2}^{2}\nonumber \\{} & {} -\kappa \int _{\Omega } u^2 \ln \vert u\vert dx- \kappa \int _{\Omega }z^2 \ln \vert z \vert dx. \end{aligned}$$

(3.2)

It is clear that

$$\begin{aligned} E(t)\ge \frac{1}{2}\left( {\Vert u_t\Vert }_2^2+{\Vert z_t\Vert }_2^2\right) +J(u,z)+2(\rho +2)\int _{\Omega } F(u,z) dx. \end{aligned}$$

(3.3)

We define the potential well (stable set)

$$\begin{aligned} \mathcal {W}=\{(u,z)\in H^{2}_{0}(\Omega )\times H^{2}_{0}(\Omega ), I(u,z)>0\}\cup \{(0,0)\}. \end{aligned}$$

The potential well depth is defined by

$$\begin{aligned} 0<d=\inf _{(u,z)}{\{\sup _{\lambda \ge 0}{J(\lambda u,\lambda z):(u,z)\in H^{2}_{0}(\Omega )\times H^{2}_{0}(\Omega ), \Vert \Delta u\Vert _2 \ne 0\text { and } \Vert \Delta z\Vert _2 \ne 0\}}}, \end{aligned}$$

(3.4)

and the well-known Nehari-manifold

$$\begin{aligned} \mathcal {N}=\{(u,z):(u,z)\in H^{2}_{0}(\Omega )\times H^{2}_{0}(\Omega ) : I(u,z)=0, \Vert \Delta u\Vert _2 \ne 0\text { and } \Vert \Delta z\Vert _2 \ne 0\}. \end{aligned}$$

(3.5)

As in [33, 37, 38], the potential well depth d satisfies

$$\begin{aligned} 0<d=\inf _{(u,z)\in \mathcal {N}}{J(u,z)}, \end{aligned}$$

(3.6)

and

$$\begin{aligned} d \ge \frac{\ell \pi }{\kappa } e^{2+\frac{2}{\kappa }}. \end{aligned}$$

(3.7)

Lemma 3.1

Let $(u_{0},u_{1}),(z_{0},z_{1})\in H_{0}^{2}(\Omega )\times L^2(\Omega )$ such that

$$\begin{aligned} 0<E(0)<d \text { and }I(u_0,z_0)>0. \end{aligned}$$

(3.8)

Then, any solution of $\text {(P)}$, $(u, z)\in \mathcal {W}$.

Proof

Let T be the maximal existence time of a weak solution of (u, z). From Eqs. 2.9 and 3.3, we have for any $t\in [0,T)$,

$$\begin{aligned} \begin{aligned} \frac{1}{2}\left( \Vert u_t\Vert ^2+\Vert z_t\Vert ^2\right)&+ 2(\rho +2)\int _\Omega F(u,z) dx+ J(u,z)\\&\le \frac{1}{2}\left( \Vert u_1\Vert ^2 +\Vert z_1\Vert ^2\right) + 2(\rho +2)\int _\Omega F(u_0,z_0)dx+ J(u_0,z_0)\\&<d. \end{aligned} \end{aligned}$$

(3.9)

Then, we claim that $(u(t),z(t))\in \mathcal {W}$ for all $t\in [0,T)$. If not, then there is a $t_0 \in (0,T)$ such that $I(u(t_0),z(t_0))<0$. Using the continuity of I(u(t), z(t)) in t, we deduce that there exists a $t_*\in (0,T)$ such that $I(u(t_*),z(t_*))=0$. Then, using the definition of d in Eq. 3.4 gives

$$d\le J(u(t_*),z(t_*))\le E(u(t_*),z(t_*))\le E(0)<d,$$

which is a contradiction. $\square$

4 Technical Lemmas

In this section, we state and prove some essential lemmas needed in the proof of our decay results.

Lemma 4.1

[35] Assume that g satisfies (A1). Then, for $u\in H^{2}_{0}(\Omega ),$ we have

$$\begin{aligned} \int _{\Omega }\left( \int _{0}^{t}g(t-s)(u(t)-u(s))ds\right) ^{2}dx\le c(go \Delta u)(t) \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega }\left( \int _{0}^{t} g^{\prime }(t-s)(u(t)-u(s))ds\right) ^{2}dx\le -c(g^{\prime } o \Delta u)(t). \end{aligned}$$

Proof

$$\begin{aligned} \int _{\Omega }\left( \int _{0}^{t}g(t\!-\!s)(u(t)\!-\!u(s))ds\right) ^{2}dx\!=\!\int _{\Omega }\left( \int _{0}^{t}\sqrt{g(t\!-\!s)}\sqrt{g(t\!-\!s)}(u(t)\!-\!u(s))ds\right) ^{2}dx. \end{aligned}$$

By applying Cauchy-Schwarz’ and Poincaré’s inequalities, we can show that

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left( \int _{0}^{t}g(t-s)(u(t)-u(s))ds\right) ^{2}dx\\&\quad \le \int _{\Omega }\left( \int _{0}^{t}g(t-s)ds\right) \left( \int _{0}^{t}g(t-s)(u(t)-u(s))^{2}ds\right) dx\\&\quad \le (1-\ell )c(go \Delta u)(t)\\&\quad \le c(go \Delta u)(t). \end{aligned} \end{aligned}$$

(4.1)

Similarly, the second inequality in Lemma 4.1 can be proved. $\square$

Lemma 4.2

[39] There exist two positive constants $\Lambda _1$ and $\Lambda _2$ such that

$$\begin{aligned} \begin{aligned} \int _{\Omega }{\vert {f_i(u,z)}\vert }^{2}dx \le {\Lambda _i{\left( \ell {\vert \vert \ {\Delta u} \vert \vert }_2^{2}+{\vert \vert \ {\Delta z}\vert \vert }_2^{2}\right) }^{2\rho +3}} ,\quad i=1,2. \end{aligned} \end{aligned}$$

(4.2)

Lemma 4.3

Assume that $(A1-A3)$ and Eq. 3.8 hold. Then, the functional

$$\begin{aligned} \begin{aligned} I_1(t)= \int _{\Omega }u u_{t}dx + \int _{\Omega } z z_{t}dx \end{aligned} \end{aligned}$$

(4.3)

satisfies, along the solutions of System $\text {(P)}$,

$$\begin{aligned} I_1^{\prime }(t)\le & {} \int _{\Omega }\left( u_{t} ^{2}+\! z_{t}^{2}\right) dx-\! \frac{\ell }{2}\int _{\Omega } \vert \Delta u \vert ^2 dx-\!\frac{\ell }{2}\int _{\Omega } \vert \Delta z \vert ^2 dx\!+c(g\circ \Delta u)(t)\!+ c \int _\Omega h^2 (z_t) dx\nonumber \\{} & {} -\! \int _{\Omega }u^2 dx \!-\! \int _{\Omega }z^2 dx\!-\!2(\rho \!+2)\int _{\Omega }F(u,z)dx\!+\! \kappa \int _{\Omega } u^2 \ln \vert u \vert dx\! +\! \kappa \int _{\Omega } z^2 \ln \vert z \vert dx.\nonumber \\ \end{aligned}$$

(4.4)

Proof

We differentiate $I_1(t)$ and use integration by parts, to get

$$\begin{aligned} I_1^{\prime }(t)= & {} \int _{\Omega } u_{t}^2 dx + \int _{\Omega } u \Big [-\Delta ^2 u - u-f_1(u,z) \ln \vert u \vert \Big ] dx\nonumber \\{} & {} +\int _{\Omega } z_{t}^2 dx+ \int _{\Omega } z \Big [-\Delta ^2 z- z- f_2(u,z)-h(z_t)+\kappa z \ln \vert z \vert \Big ]dx\nonumber \\= & {} \int _{\Omega }\left( \vert u_{t} \vert ^{2}+\vert z_{t} \vert ^{2}\right) dx- \int _{\Omega }\left( \vert \Delta u \vert ^2 + \vert \Delta z \vert ^2 \right) dx\nonumber \\{} & {} - \int _{\Omega }u^2 dx - \int _{\Omega }z^2 dx-2(\rho +2)\int _{\Omega }F(u,z)dx+\int _{\Omega }\Delta u (t) \int _0^t g(t-s)\Delta u(s)ds\nonumber \\{} & {} + \kappa \int _{\Omega } u^2 \ln \vert u \vert dx + \kappa \int _{\Omega } z^2 \ln \vert z \vert dx-\int _{\Omega } z h(z_t) dx. \end{aligned}$$

(4.5)

Applying Young’s inequality, we obtain, for $\varepsilon _1 >0$,

$$\begin{aligned} \begin{aligned} \int _{\Omega }\Delta u (t) \int _0^t g(t-s)\Delta u(s)dsdx\le c\varepsilon _1\int _\Omega \vert \Delta u \vert ^2 dx+\dfrac{c }{\varepsilon _1} (g\circ \Delta u)(t). \end{aligned} \end{aligned}$$

(4.6)

Similarly, we find for $\varepsilon _2 >0$,

$$\begin{aligned} \begin{aligned} \int _\Omega z h(z_t)dx \le \frac{\varepsilon _2 c_{p}}{2} \int _\Omega \vert \Delta z \vert ^2 dx + c \varepsilon _2 \int _\Omega h^2 (z_t) dx. \end{aligned} \end{aligned}$$

(4.7)

Combining the above estimates, we arrive at

$$\begin{aligned} I_1^{\prime }(t)\le & {} \int _{\Omega }\left( \vert u_{t} \vert ^{2}+\vert z_{t} \vert ^{2}\right) dx- \left[ 1-c\varepsilon _1\right] \int _{\Omega } \vert \Delta u \vert ^2 dx- \left[ 1-\frac{\varepsilon _2 c_p}{2}\right] \int _{\Omega } \vert \Delta z \vert ^2 dx\nonumber \\{} & {} -\! \int _{\Omega }u^2 dx \!-\! \int _{\Omega }z^2 dx\!-\!2(\rho \!+\!2)\int _{\Omega }F(u,z)dx\!+\! \kappa \int _{\Omega } u^2 \ln \vert u \vert dx \!+\! \kappa \int _{\Omega } z^2 \ln \vert z \vert dx\nonumber \\{} & {} +\dfrac{c }{\varepsilon _1} (g\circ \Delta u)(t)+ c\varepsilon _2 \int _\Omega h^2 (z_t) dx. \end{aligned}$$

(4.8)

By choosing $\varepsilon _1, \varepsilon _2$ small enough, we obtain the desired result. $\square$

Lemma 4.4

Assume that $(A1-A3)$ and Eq. 3.8 hold. Then, the functional

$$I_2(t):=-\int _\Omega u_t\int _{0}^{t}g(t-s)( u(t)- u(s))ds dx$$

satisfies along the solutions of System $\text {(P)}$, for any $\varepsilon _2>0$ and $0< \varepsilon _0 < 1$,

$$\begin{aligned} I_2'(t)\le & {} -\left( \int _0^t g(s) ds-\frac{\varepsilon _2}{4}\right) \int _\Omega u_t^2 dx + \frac{c }{\varepsilon _2} (-g' \circ \Delta u)(t)+\frac{c }{\varepsilon _2} (g \circ \Delta u)(t)\nonumber \\{} & {} + c \varepsilon _2 \int _\Omega \vert \Delta u \vert ^2 dx+c \varepsilon _2 \int _\Omega \vert \Delta z \vert ^2 dx+{c_{\varepsilon _0,\varepsilon _2}} (g \circ \Delta u)^{\frac{1}{1+\varepsilon _0}}(t). \end{aligned}$$

(4.9)

Proof

Differentiating $I_2$, using the equations of (P), we obtain

$$\begin{aligned} I_2' (t)= & {} - \left( \int _0^t g(s) ds\right) \int _\Omega u_t^2 dx - \int _\Omega u_t\int _{0}^{t}g'(t-s)(u(t)- u(s))dsdx\nonumber \\{} & {} - \int _\Omega u_{tt}\int _{0}^{t}g(t-s)(u(t)- u(s))dsdx. \end{aligned}$$

(4.10)

Taking into account (P), and using integration by parts, we obtain

$$\begin{aligned} I_2' (t)= & {} - \left( \int _0^t g(s) ds\right) \int _\Omega u_t^2 dx \underset{\mathcal {I}_{1}}{\underbrace{-\int _\Omega u_t\int _{0}^{t}g'(t-s)(u(t)- u(s))dsdx}}\nonumber \\{} & {} +\underset{\mathcal {I}_{2}}{\underbrace{\int _\Omega \Delta u \int _{0}^{t}g(t-s)(\Delta u(t)- \Delta u(s))dsdx}}\nonumber \\{} & {} +\underset{\mathcal {I}_{3}}{\underbrace{\int _\Omega u \int _{0}^{t}g(t-s)( u(t)- u(s))dsdx}}\nonumber \\{} & {} +\underset{\mathcal {I}_{4}}{\underbrace{\int _\Omega f_1(u,z) \int _{0}^{t}g(t-s)( u(t)- u(s))dsdx}}\nonumber \\{} & {} +\underset{\mathcal {I}_{5}}{\underbrace{\int _\Omega \int _{0}^{t} g(t-s)(\Delta u(t)- \Delta u(s))ds \int _{0}^{t} g(t-s)\Delta u(s)dsdx}}\nonumber \\{} & {} - \underset{\mathcal {I}_{6}}{\underbrace{\int _\Omega u \ln \vert u \vert ^\kappa \int _{0}^{t}g(t-s)(u(t)- u(s))dsdx}}. \end{aligned}$$

(4.11)

Now, using Young’s and Poincaré’s inequalities and Lemma 4.1, we get for any $\varepsilon _2>0$,

$$\begin{aligned} \mathcal {I}_1\le & {} \frac{\varepsilon _2}{4} \int _\Omega u_t^2 dx -\frac{c}{\varepsilon _2} (g' \circ \Delta u)(t)\end{aligned}$$

(4.12)

$$\begin{aligned} \mathcal {I}_2 , \mathcal {I}_3\le & {} c\varepsilon _2 \int _\Omega \vert \Delta u \vert ^2(t) dx + \frac{c }{\varepsilon _2} (g \circ \Delta u)(t). \end{aligned}$$

(4.13)

Similarly, using Lemmas 4.1 and 4.2, we obtain

$$\begin{aligned} \begin{aligned} \mathcal {I}_4&\le \varepsilon _2 \int _\Omega f^2_1(u,z) dx + \frac{c }{\varepsilon _2} (g \circ \Delta u)(t)\\&\le c\varepsilon _2\left( \int _\Omega \vert \Delta u \vert ^2dx + \int _\Omega \vert \Delta z \vert ^2 dx\right) + \frac{c }{\varepsilon _2} (g \circ \Delta u)(t) \end{aligned} \end{aligned}$$

(4.14)

and

$$\begin{aligned} \begin{aligned} \mathcal {I}_5 \le c\varepsilon _2 \int _\Omega \vert \Delta u \vert ^2 dx + c {\varepsilon _2}(g \circ \Delta u)(t). \end{aligned} \end{aligned}$$

(4.15)

Using the embedding of $H^{2}_{0}(\Omega )$ in $L^{\infty }(\Omega )$ and performing the same calulactions as before, we get, for any $\varepsilon _2 >0$ and any $\epsilon _0 \in (0,1)$,

$$\begin{aligned}{} & {} \int _{\Omega }u\ln {\vert u\vert ^k}\int _{0}^{t}g(t-s)(u(t)-u(s))dsdx \\\le & {} k\int _{\Omega }\left( u^2 +d_{\epsilon _0}\vert u\vert ^{1- {\epsilon _0}}\right) \left| \int _{0}^{t}g(t-s)(u(t)-u(s))dsdx\right| \\\le & {} c\int _{\Omega }\vert u\vert ^2\left| \int _{0}^{t}g(t-s)(u(t)-u(s))ds\right| dx+\varepsilon _2 \int _{\Omega } u^2 dx\\{} & {} +c_{\epsilon _0,\varepsilon _2}\int _{\Omega }\left| \int _{0}^{t}g(t-s)(u(t)-u(s))ds\right| ^{\frac{2}{1+\epsilon _0}} dx\\\le & {} c\varepsilon _2\int _\Omega \vert \Delta u \vert ^2 dx +\frac{c}{\varepsilon _2} \int _{\Omega }\left| \int _{0}^{t}g(t-s)(u(t)-u(s))ds \right| ^{2} dx\\{} & {} +c_{\epsilon _0,\varepsilon _2} \int _{\Omega }\left| \int _{0}^{t}g(t-s)(u(t)-u(s))ds\right| ^{\frac{2}{1+\epsilon _0}} dx, \end{aligned}$$

then, using Holder’s inequality and Lemma 4.1, we find

$$\begin{aligned} \begin{aligned}&\mathcal {I}_6\le c\varepsilon _2\int _\Omega \vert \Delta u \vert ^2 dx +\frac{c}{\varepsilon _2}(go\Delta u)(t)+c_{\epsilon _0 ,\varepsilon _2}(go\Delta u)^{\frac{1}{1+\epsilon _0}}(t). \end{aligned} \end{aligned}$$

(4.16)

Combining Eqs. 4.11–4.16, we obtain the desired result. $\square$

Lemma 4.5

Assume $(A1)-(A3)$ hold and $(u_{0},u_{1}),(z_{0},z_{1})\in \mathcal {W}\times L^2(\Omega )$ be given. Assume further   $0<E(0)<\alpha \beta _0<d$, where

$$\beta _0= \left( {\frac{\ell \pi }{k}}\right) e^{2+\frac{2}{k}}\quad \text { and }\quad 0<\sqrt{\frac{2}{k}}e^{\frac{1}{k}} \alpha ^{\frac{1}{2}} <1,$$

then the functional $\mathcal {L}$ defined by

$$\begin{aligned} \mathcal {L}(t)= N E(t)+ N_1 I_1(t)+ N_2 I_2 (t) \end{aligned}$$

(4.17)

satisfies, for any $t_0>0$

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

(4.18)

and

$$\begin{aligned} \mathcal {L}'(t) \!\le \! -\!c E(t)\!+\!c (g \circ \Delta u)(t)\!+\!{c_{\varepsilon _0}} (g \circ \! \Delta u)^{\frac{1}{1+\varepsilon _0}}(t)\!+\!c \int _\Omega \left( z_t^2\!+\! h^2 (z_t) \right) dx,\quad \text {for all } t\!\ge \! t_0. \end{aligned}$$

(4.19)

Proof

The proof of Eq. 4.18 is similar to the one in [33]. For the proof of Eq. 4.19, since g is positive and $g(0)>0$, then, for any $t_0>0$, we have

$$\int _{0}^{t}g(s)ds\ge \int _{0}^{t_0}g(s)ds=g_0>0, \quad \forall t\ge t_0.$$

By using Eqs. 2.9, 4.4, 4.9, and the definition of E, we obtain for any $\gamma >0$ and $t\ge t_0$,

$$\begin{aligned} \mathcal {L}^{\prime }(t)\le & {} -\gamma E(t)-\left[ N_2 (g_0-\varepsilon _2)-N_1-\frac{\gamma }{2}\right] \int _\Omega \vert u_t \vert ^2 dx\nonumber \\{} & {} -\left[ \frac{N_1 \ell }{2} -c \varepsilon _2 N_2-\frac{\gamma }{2}\right] \int _\Omega \vert \Delta u \vert ^2 dx -\left[ \frac{N_1 \ell }{2}-c \varepsilon _2 N_2-\frac{\gamma }{2}\right] \int _\Omega \vert \Delta z \vert ^2 dx\nonumber \\{} & {} +\left[ cN_1 +\frac{c N_2}{\varepsilon _2}+\frac{\gamma }{2}\right] (g \circ \Delta u)(t)+\left[ \frac{N}{2} -\frac{c N_2}{\varepsilon _2}\right] (g' \circ \Delta u)(t)\nonumber \\{} & {} -\!\left( N_1\!-\!\frac{\gamma }{2}\right) \Big [\Vert u\Vert _{2}^{2}\!+\!\Vert z\Vert _{2}^{2}\Big ]\!-\!2(\rho +2)\left( N_1\!-\!\frac{\gamma }{2}\right) \int _{\Omega }F(u,z)dx\!+\! {c_{\varepsilon _0,\varepsilon _2}} (g \circ \! \Delta u)^{\frac{1}{1+\varepsilon _0}}(t)\nonumber \\{} & {} + \left( N _1+ \frac{\gamma }{2}\right) \int _\Omega z_t^2 dx + c N_1 \int _\Omega h^2(z_t) dx\nonumber \\{} & {} +\kappa \left( N_1 -\frac{\gamma }{2}\right) \int _{\Omega } u^2 \ln \vert u \vert dx + \kappa \left( N_1-\frac{\gamma }{2}\right) \int _{\Omega } z^2 \ln \vert z \vert dx. \end{aligned}$$

(4.20)

Using the Logarithmic Sobolev inequality Eq. 2.7, we obtain

$$\begin{aligned} \mathcal {L}^{\prime }(t)\le & {} -\gamma E(t)-\left[ N_2 (g_0-\varepsilon _2)-N_1-\frac{\gamma }{2}\right] \int _\Omega \vert u_t \vert ^2 dx\nonumber \\{} & {} -\left[ \frac{N_1 \ell }{2} -c \varepsilon _2 N_2-\frac{\gamma }{2}-\kappa \left( N_1-\frac{\gamma }{2}\right) \frac{c_p a^2}{ 2\pi }\right] \left( \int _\Omega \vert \Delta u \vert ^2 dx+ \int _\Omega \vert \Delta z \vert ^2 dx\right) \nonumber \\{} & {} +\left[ cN_1 +\frac{c N_2}{\varepsilon _2}+\frac{\gamma }{2}\right] (g \circ \Delta u)(t)+\left[ \frac{N}{2} -\frac{c N_2}{\varepsilon _2}\right] (g' \circ \Delta u)(t)\nonumber \\{} & {} -2(\rho +2)\left( N_1-\frac{\gamma }{2}\right) \int _{\Omega }F(u,z)dx+ {c_{\varepsilon _0,\varepsilon _2}} (g \circ \Delta u)^{\frac{1}{1+\varepsilon _0}}(t)\\{} & {} + \left( N_1 + \frac{\gamma }{2}\right) \int _\Omega z_t^2 dx + c N_1 \int _\Omega h^2(z_t) dx\nonumber \\{} & {} -\left[ N_1-\frac{\gamma (\kappa +2)}{4}+\kappa \left( N_1-\frac{\gamma }{2}\right) \left( (1+\ln a)-\frac{1}{2} \ln \Vert u\Vert _2^2\right) \right] \Vert u\Vert _{2}^{2}\nonumber \\{} & {} -\left[ N_1-\frac{\gamma (\kappa +2)}{4}+\kappa \left( N_1-\frac{\gamma }{2}\right) \left( (1+\ln a)-\frac{1}{2} \ln \Vert z\Vert _2^2\right) \right] \Vert z\Vert _{2}^{2}.\nonumber \end{aligned}$$

(4.21)

We choose $\varepsilon _2$ small enough such that

$$\begin{aligned} g_0-\varepsilon _2 > \frac{g_0}{2},~~ \text {and}~~ \varepsilon _2 < \frac{\ell g_0}{16c}. \end{aligned}$$

Whence $\varepsilon _2$ is fixed, the choice of any two positive constants $N_1$ and $N_2$ satisfying

$$\begin{aligned} \frac{g_0N_2}{4}< N_1< \frac{g_0N_2}{2} \end{aligned}$$

will make

$$\begin{aligned} \alpha _1:=N_2 (g_0-\varepsilon _2)-N_1>0, ~ \text {and}~\alpha _2:=\frac{N_1 \ell }{2} -c \varepsilon _2 N_2>0. \end{aligned}$$

(4.22)

So, Eq. 4.21 becomes

$$\begin{aligned} \mathcal {L}^{\prime }(t)\le & {} -\gamma E(t)-\left[ \alpha _1-\frac{\gamma }{2}\right] \int _\Omega \vert u_t \vert ^2 dx\nonumber \\{} & {} -\left[ \alpha _2-\frac{\gamma }{2}-\kappa \left( N_1-\frac{\gamma }{2}\right) \frac{c_p a^2}{ 2\pi }\right] \left( \int _\Omega \vert \Delta u \vert ^2 dx+ \int _\Omega \vert \Delta z \vert ^2 dx\right) \nonumber \\{} & {} +c(g \circ \Delta u)(t)+\left[ \frac{N}{2} -\frac{c N_2}{\varepsilon _2}\right] (g' \circ \Delta u)(t)\nonumber \\{} & {} -2(\rho +2)\left( N_1-\frac{\gamma }{2}\right) \int _{\Omega } F(u,z) dx + {c_{\varepsilon _0}} (g \circ \Delta u)^{\frac{1}{1+\varepsilon _0}}(t) \\{} & {} + \left( N_1 + \frac{\gamma }{2}\right) \int _\Omega z_t^2 dx + c N_1 \int _\Omega h^2(z_t) dx\nonumber \\{} & {} -\left[ N_1-\frac{\gamma (\kappa +2)}{4}+\kappa \left( N_1-\frac{\gamma }{2}\right) \left( (1+\ln a)-\frac{1}{2} \ln \Vert u\Vert _2^2\right) \right] \Vert u\Vert _{2}^{2}\nonumber \\{} & {} -\left[ N_1-\frac{\gamma (\kappa +2)}{4}+\kappa \left( N_1-\frac{\gamma }{2}\right) \left( (1+\ln a)-\frac{1}{2} \ln \Vert z\Vert _2^2\right) \right] \Vert z\Vert _{2}^{2}.\nonumber \end{aligned}$$

(4.23)

Then, using (3.8) and selecting $\gamma$ and $\kappa$ small enough such that

$$\begin{aligned} \alpha _1-\frac{\gamma }{2}>0, ~\text {and}~\alpha _2-\frac{\gamma }{2}-\kappa \left( N_1-\frac{\gamma }{2}\right) \frac{C_\rho a^2}{ 2\pi }>0, \end{aligned}$$

(4.24)

Using Eqs. 2.8, 2.9, and the fact that $u\in \mathcal {W}$ (Lemma 3.1), we have

$$\begin{aligned} \ln {\Vert u\Vert _2^2}<\ln {\left( \frac{4}{\kappa }E(t)\right) }< & {} \ln {\left( \frac{4}{k}E(0)\right) }\nonumber \\< & {} \ln {\left( \frac{4}{\kappa }\alpha \beta _0\right) }\\= & {} \ln {\left( \frac{4\pi }{\kappa ^2}\alpha e^{2+\frac{2}{\kappa }}\right) }.\nonumber \end{aligned}$$

(4.25)

By picking $0<\gamma <\frac{4}{k+2}$ and taking a satisfying

$$2\frac{\sqrt{\ell \pi }}{\kappa }e^{\frac{1}{\kappa }} \alpha ^{\frac{1}{2}}<a<\sqrt{\frac{2\ell \pi }{\kappa }},$$

we guarantee the following:

$$\begin{aligned} N_1-\frac{\gamma (\kappa +2)}{4}> & {} 0,\\ \left( N_1-\frac{\gamma }{2}\right) \left[ (1+\ln {a})-\frac{1}{2}\ln {\Vert u\Vert _2^2}\right]> & {} 0, \\ \left( N_1-\frac{\gamma }{2}\right) \left[ (1+\ln {a})-\frac{1}{2}\ln {\Vert z\Vert _2^2}\right]> & {} 0. \end{aligned}$$

Finally, we choose N large enough so that $\frac{N}{2} -\frac{cN_2}{\varepsilon _2}>0$ and Eq. 4.18 remains true. Therefore, we arrive at the desired result Eq. 4.19. $\square$

5 Stability

In this section, we state and prove the stability result of system (P). For this purpose, we have the following lemmas and remarks.

Lemma 5.1

Under the assumption (A2), we have the following estimates:

$$\begin{aligned} \begin{aligned} \int _{\Omega }z_t+h^2(z_t)dx \le c \int _{\Omega }z_t h(z_t)dx,\quad \text { if { H} is linear} \end{aligned} \end{aligned}$$

(5.1)

$$\begin{aligned} \begin{aligned} \int _{\Omega }z_t^2+h^2(z_t)dx \le cH^{-1}(\chi _0(t))-cE^{\prime }(t),\quad \text { if { H} is nonlinear} \end{aligned} \end{aligned}$$

(5.2)

where

$$\begin{aligned} \chi _0(t):=\frac{1}{\vert \Omega _2\vert }\int _{\Omega _2}z_{t}(t)h(z_{t}(t))dx \le -cE^{\prime }(t) \end{aligned}$$

(5.3)

and

$$\Omega _{2}=\{x\in \Omega :\vert z_{t}(t) \vert \le \varepsilon _{1} \},\quad \text {where}\;\; \varepsilon _{1}=\min \{\varepsilon , h_0(\varepsilon )\}. $$

Proof

Case 1: H is linear. Then, using (A2), we have

$$c_{1} \vert z_t \vert \le \vert h(z_t) \vert \le c_{2} \vert z_t \vert ,$$

and hence

$$\begin{aligned} z_t^2+h^{2}(z_t) \le c z_th(z_t). \end{aligned}$$

(5.4)

So, Eq. 5.1 is established.

Case 2. H is nonlinear on $[0,\varepsilon ]$. Let $0< \varepsilon _1\le \varepsilon$ such that

$$sh(s)\le \min {\{ \varepsilon ,h( \varepsilon )\}},\quad \text {for all }\vert s\vert \le \varepsilon _1.$$

Using (A2) and Remark 2.1, we have, for $\varepsilon _{1} \le \vert s \vert \le \varepsilon ,$

$$\vert h(s) \vert \le \frac{h_{0}^{-1}(\vert s \vert )}{\vert s \vert }\vert s \vert \le \frac{h_{0}^{-1}(\vert \varepsilon \vert )}{\vert \varepsilon _{1} \vert }\vert s \vert $$

and

$$\vert h(s) \vert \ge \frac{h_{0}(\vert s \vert )}{\vert s \vert }\vert s \vert \ge \frac{h_{0}(\vert \varepsilon _{1} \vert )}{\vert \varepsilon \vert }\vert s \vert .$$

Therefore, we deduce that

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} s^2+h^2(s)\le H^{-1}(sh(s))\quad \text {for all}\quad \vert s \vert < \varepsilon _{1} \\ \quad c^{\prime }_{1} \vert s \vert \le \vert h(s) \vert \le c^{\prime }_{2} \vert s \vert \quad \text {for all}\quad \vert s \vert \ge \varepsilon _{1}. \end{aligned} \end{array}\right. } \end{aligned}$$

(5.5)

We define the following partition:

$$\Omega _{1}=\{x\in \Omega :\vert z_{t} \vert > \varepsilon _{1} \},\quad \Omega _{2}=\{x\in \Omega :\vert z_{t} \vert \le \varepsilon _{1} \}.$$

Using Eq. 5.5 and recalling the definition of $\varepsilon _{1}$, we get on $\Omega _{2}$

$$\begin{aligned} z_{t}h(z_{t}) \le \ \varepsilon _{1} h_{0}^{-1}(\ \varepsilon _{1}) \le h_{0}(\varepsilon )\varepsilon =h(\varepsilon ^{2}). \end{aligned}$$

(5.6)

Let

$$\chi _0(t):=\frac{1}{\vert \Omega _{2}\vert }\int _{\Omega _{2}}z_{t}h(z_{t})dx,$$

then using Jensen’s inequality, with the fact that $H^{-1}$ is concave, we get

$$\begin{aligned} H^{-1}\left( \chi _0(t)\right) \ge c \int _{\Omega _{2}}H^{-1}(z_{t}h(z_{t}))dx. \end{aligned}$$

(5.7)

Thus, combining Eqs. 2.9, 5.5, and 5.7, we arrive at

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left( z_t^2+h^{2}(z_t)\right) dx=\int _{\Omega _{2}}\left( z_t^2+h^{2}(z_{t})\right) dx+\int _{\Omega _{1}}\left( z_t^2+h^{2}(z_{t})\right) dx\\&\quad \le \int _{\Omega _{2}}H^{-1}\left( z_{t}h(z_{t})\right) dx+c\int _{\Omega _{1}}z_{t}h(z_{t})dx\\&\quad \le cH^{-1}(\chi _0(t))-cE^{\prime }(t). \end{aligned} \end{aligned}$$

(5.8)

This completes the proof of this lemma. $\square$

Lemma 5.2

Assume that (A1) and Eq. 3.8 hold, we have the following estimate:

$$\begin{aligned} (go\Delta u)(t) \le \frac{t}{q} \overline{G}^{-1}\left( \frac{qI(t)}{t\xi (t)}\right) , \end{aligned}$$

(5.9)

where $I(t):=(-g^{\prime }o \Delta u)(t)$, $q \in (0,1)$ and $\overline{G}$ is an extension of G such that $\overline{G}$ is strictly increasing and strictly convex $C^2$ function on $(0,\infty )$.

Proof

We define the following quantity:

$$\lambda (t):=\frac{q}{t}\int _{0}^{t}\int _{\Omega }{\vert \Delta u(t)-\Delta u(t-s)\vert }^{2}dxds,\quad t>0.$$

By using Eqs. 2.8, 2.9, and the fact that $u\in \mathcal {W}$, we easily see that

$$\lambda (t)\le \frac{8cq E(0)}{\ell },$$

then choosing $q\in (0,1)$ small enough so that, for all $t>0$,

$$\begin{aligned} \lambda (t)<1. \end{aligned}$$

(5.10)

Since G is strictly convex on (0, r] and $G(0)=0,$ then

$$\begin{aligned} G(\theta v)\le \theta G(v),\text { }0\le \theta \le 1\text { and }v\in (0,r]. \end{aligned}$$

(5.11)

Define

$$\begin{aligned} I(t):=(-g^{\prime }o \Delta u)(t)\le -cE^{\prime }(t). \end{aligned}$$

(5.12)

The use of Eqs. 2.2, 5.10, 5.11 and Jensen’s inequality leads to

$$\begin{aligned} I(t)= & {} \frac{1}{q\lambda (t)}\int _{0}^{t}\lambda (t)(-g^{\prime }(s))\int _{\Omega }{q\vert \Delta u(t)- \Delta u(t-s)\vert }^{2}dxds\nonumber \\\ge & {} \frac{1}{q\lambda (t)}\int _{0}^{t}\lambda (t)\xi (s) G(g(s))\int _{\Omega }{q\vert \Delta u(t)-\Delta u(t-s)\vert }^{2}dxds\nonumber \\\ge & {} \frac{\xi (t)}{q\lambda (t)}\int _{0}^{t}G(\lambda (t)g(s))\int _{\Omega }{q\vert \Delta u(t)-\Delta u(t-s)\vert }^{2}dxds\\\ge & {} \frac{t\xi (t)}{q} G\biggl (\frac{q}{t}\int _{0}^{t}g(s)\int _{\Omega }{\vert \Delta u(t)-\Delta u(t-s)\vert }^{2}dxds\biggr )\nonumber \\= & {} \frac{t\xi (t)}{q}\overline{G}\biggl (\frac{q}{t} \int _{0}^{t}g(s)\int _{\Omega }{\vert \Delta u(t)-\Delta u(t-s)\vert }^{2}dxds\biggr ).\nonumber \end{aligned}$$

(5.13)

This gives Eq. 5.9. $\square$

Remark 5.1

Using the fact that $(go\Delta u)(t)\le cE(t)\le cE(0),$ we obtain

$$\begin{aligned} \begin{aligned}&(go\Delta u)(t)=(go\Delta u)^{\frac{\epsilon _0}{1+\epsilon _0}}(t) (go\Delta u)^{\frac{1}{1+\epsilon _0}}(t)\\&\quad \le c (go\Delta u)^{\frac{1}{1+\epsilon _0}}(t) \end{aligned} \end{aligned}$$

(5.14)

Remark 5.2

In the case of G is linear and since $\xi$ is nonincreasing, we have

$$\begin{aligned} \begin{aligned}&\xi (t)(g\circ \Delta u)^{\frac{1}{1+\epsilon _0}}(t)=\left( \xi ^{\epsilon _0} (t)\xi (t)(g\circ \Delta u)(t)\right) ^{\frac{1}{1+\epsilon _0}}\\&\qquad \le \left( \xi ^{\epsilon _0} (0)\xi (t)(g\circ \Delta u)(t)\right) ^{\frac{1}{1+\epsilon _0}}\\&\qquad \le c\left( \xi (t)(g\circ \Delta u)(t)\right) ^{\frac{1}{1+\epsilon _0}}\\&\qquad \le c(-E^{\prime }(t))^{\frac{1}{(1+\epsilon _0 )}} . \end{aligned} \end{aligned}$$

(5.15)

Theorem 5.3

Let $(u_{0},u_{1}),(z_{0},z_{1})\in \mathcal {W}\times L^2(\Omega )$ be given. Assume that $(A1)-(A3)$ and Eq. 3.8 hold and H is linear. Then, there exist strictly positive constants c, $k_2$ and $\bar{\varepsilon }$ such that the solution of $\text {(P)}$ satisfies, for all $t>t_1=\max \{t_0, 1\}$,

$$\begin{aligned} E(t)\le & {} c\left( 1+\int _{t_0}^{t}\xi ^{1+\epsilon _0}(s)ds\right) ^{\frac{-1}{\epsilon _0}}, \qquad \text {if { G} is linear.}\end{aligned}$$

(5.16)

$$\begin{aligned} E(t)\le & {} c t^{\frac{1}{1+\epsilon _0}} {K_{2}}^{-1} \left( \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}\displaystyle \int _{t_1}^{t}\xi (s) ds } \right) , \qquad \text {if { G} is nonlinear} \end{aligned}$$

(5.17)

where ${K_2}(s)=s K^{\prime }(\bar{\varepsilon }s)$ and $K=\left( \left[ \left( \overline{G}\right) ^{-1}\right] ^{\frac{1}{1+\epsilon _0}}\right) ^{-1}$.

Proof

Case 1: G is linear

We multiply Eq. 4.19 by $\xi (t)$ and use Eqs. 2.2, 2.9, 5.1, 5.9, and 5.14 to get

$$\begin{aligned} \begin{aligned}&\xi (t) L^{\prime }(t) \le -m \xi (t)E(t)+c \left( -E^{\prime }(t) \right) ^{\frac{1}{(1+\epsilon _0)}}-c E^{\prime }(t),\quad \forall t\ge t_0. \end{aligned} \end{aligned}$$

(5.18)

Multiply Eq. 5.18 by $\xi ^{\epsilon _0}(t) E^{\epsilon _0}(t)$ and use Young’s inequality to obtain

$$\begin{aligned} \begin{aligned} \xi ^{\epsilon _0 +1}(t)E^{\epsilon _0}(t)L^{\prime }(t)&\le -m \xi ^{\epsilon _0 +1}(t)E^{\epsilon _0+1}(t)+c\left( \xi E\right) ^{\epsilon _0}(t)\left( -E^{\prime }(t) \right) ^{\frac{1}{\epsilon _0 +1}}-cE^{\prime }(t)\\&\le -m \xi ^{\epsilon _0 +1}(t)E^{\epsilon _0+1}(t) +c\left( \varepsilon ^{\prime } \xi ^{\epsilon _0+1}(t)E^{\epsilon _0+1}-c_{\varepsilon ^{\prime }}E^{\prime }(t)\right) \\&=-(m-\varepsilon ^{\prime } c)\xi ^{\epsilon _0 +1}(t)E^{\epsilon _0+1}-cE^{\prime }(t),\quad \forall t\ge t_0. \end{aligned} \end{aligned}$$

We then choose $0<\varepsilon ^{\prime } <\frac{m}{c}$ and use that $\xi ^{\prime } \le 0$ and $E^{\prime } \le 0$, to get, for $c_1 =m-\varepsilon ^{\prime } c$,

$$\begin{aligned} \left( \xi ^{\epsilon _0+1}E^{\epsilon _0} L+cE\right) ^{\prime }(t) \le -c_1 \xi ^{\epsilon _0+1}(t) E^{\epsilon _0+1}(t),\quad \forall t\ge t_0 . \end{aligned}$$

Let $L_1=\xi ^{\gamma +1}E^{\gamma }L +cE$. Then $L_1\sim E$ (thanks to Eq. 4.18) and

$$\begin{aligned} L_1^{\prime }(t) \le -c \xi ^{\epsilon _0+1}(t) L_1^{\epsilon _0+1}(t), \text { }\forall t \ge t_0. \end{aligned}$$

Integrating over $(t_0,t)$ and using the fact that $L_1 \sim E$, we obtain Eq. 5.16.

Case 2: G is nonlinear.

Using Eqs. 2.9, 4.19, 5.1, 5.9, and 5.14, we obtain, $\forall \; t\ge t_0$,

$$\begin{aligned} L^{\prime }(t)\le -m E(t)+ct^{\frac{1}{1+\epsilon _0}}\left[ \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t\xi (t)}\right) \right] ^{\frac{1}{1+\epsilon _0}}-cE^{\prime }(t). \end{aligned}$$

(5.19)

Combining the strictly increasing property of $\overline{G}$ and the fact that $\frac{1}{t} < 1$ whenever $t > 1$, we obtain

$$\begin{aligned} \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t\xi (t)}\right) \le \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t^{\frac{1}{1+\epsilon _0}}\xi (t)}\right) \end{aligned}$$

(5.20)

and, then, Eq. 5.19 becomes

$$\begin{aligned} \begin{aligned}&L^{\prime }(t)\le -m E(t)+ct^{\frac{1}{1+\epsilon _0}}\left[ \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t^{\frac{1}{1+\epsilon _0}}\xi (t)}\right) \right] ^{\frac{1}{1+\epsilon _0}}-cE^{\prime }(t),\quad \forall t>t_1, \end{aligned} \end{aligned}$$

(5.21)

where $t_1=\max \{t_0,1\}$.

Let $F_1(t)=L(t)+cE(t)\sim E$, $K=\left( \left[ \left( \overline{G}\right) ^{-1}\right] ^{\frac{1}{1+\epsilon _0}}\right) ^{-1},$ and $\chi (t)=\frac{qI(t)}{t^{\frac{1}{1+\epsilon _0}}\xi (t)}$, then Eq. 5.21 reduces to

$$\begin{aligned} F_{1}^{\prime }(t)\le -m E(t)+ct^{\frac{1}{1+\epsilon _0}} K^{-1}(\chi (t)),\quad \forall t>t_1. \end{aligned}$$

(5.22)

Now, for $\bar{\varepsilon }<r$ and using Eq. 5.22 and the fact that $E^{\prime }\le 0$, $K^{\prime }>0, K^{\prime \prime }>0$ on (0, r],  we find that the functional $F_{2},$ defined by

$$F_{2}(t):=K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) F_{1}(t), \quad \forall t>t_1,$$

satisfies, for some $\alpha _{1},\alpha _{2}>0.$

$$\begin{aligned} \alpha _{1} F_{2}(t)\le E(t)\le \alpha _{2}F_{2}(t) \end{aligned}$$

(5.23)

and, for all $t>t_1$,

$$\begin{aligned} \begin{aligned}&F_{2}^{\prime }(t)\le -m E(t)K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +ct^{\frac{1}{1+\epsilon _0}}K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) K^{-1}(\chi (t)). \end{aligned} \end{aligned}$$

(5.24)

Let $K^{*}$ be the convex conjugate of K in the sense of Young, and then apply the generalized Young inequality with $A=K^{\prime }\left( \frac{\varepsilon _{1}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right)$ and $B=K^{-1}(\chi (t)),$ we arrive at

$$\begin{aligned} F_{2}^{\prime }(t)\le & {} -m E(t)K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c t^{\frac{1}{1+\epsilon _0}}K^{*}\left( K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \right) \nonumber \\{} & {} +c t^{\frac{1}{1+\epsilon _0}}\chi (t)\nonumber \\\le & {} -m E(t)K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c\frac{E(t)}{E(0)}K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \nonumber \\{} & {} +ct^{\frac{1}{1+\epsilon _0}}\chi (t),\;\;\forall \;t>t_1. \end{aligned}$$

(5.25)

Then, multiplying Eq. 5.25 by $\xi (t)$, using Eq. 5.12, and setting $F_{3}:=\xi F_{2}+cE \sim E$, we obtain, for all $t>t_1,$

$$\begin{aligned} \begin{aligned}&F_{3}^{\prime }(t)\le -m \xi (t) E(t)K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c \bar{\varepsilon } \xi (t) \cdot \frac{E(t)}{E(0)}K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) . \end{aligned} \end{aligned}$$

This gives, for a suitable choice of $\bar{\varepsilon }$,

$$\begin{aligned} \int _{t_1}^{t} k \left( \frac{E(s)}{E(0)} \right) K^{\prime }\left( \frac{\bar{\varepsilon }}{s^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(s)}{E(0)}\right) \xi (s) ds \le - \int _{t_1}^{t} F_{3}^{\prime }(s)ds\le F_3(t_1). \end{aligned}$$

(5.26)

Using the fact that the map $t \mapsto E(t)K^{\prime }\left( \frac{\bar{\varepsilon }}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right)$ is nonincreasing and setting ${K_2}(s)=s K^{\prime }(\bar{\varepsilon }s)$, we obtain,

$$\begin{aligned} k K_{2} \left( \frac{1}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \int _{t_1}^{t} \xi (s) ds \le \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}},\quad \forall t>t_1. \end{aligned}$$

(5.27)

Finally, for two positive constants $k_2$ and $k_3$, we infer

$$\begin{aligned} E(t) \le k_3 t^{\frac{1}{1+\epsilon _0}} {K_{2}}^{-1} \left( \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}\displaystyle \int _{t_1}^{t}\xi (s) ds } \right) ,\;\;\forall \;t>t_1. \end{aligned}$$

(5.28)

This finishes the proof. $\square$

Remark 5.3

Using the strictly increasing and strictly convex properties of $\overline{H}$ and $\overline{G}$, setting $\theta ={\left( \frac{1}{t}\right) }^{\frac{1}{1+\epsilon _0}} < 1, \quad \forall t >1$ and using

$$\begin{aligned} \overline{H}(\theta z)\le \theta \overline{H}(z),\text { }0\le \theta \le 1\text { and }z\in (0,\varepsilon ], \end{aligned}$$

(5.29)

we obtain

$$\overline{H}^{-1}(\chi _0(t)) \le t^{\frac{1}{1+\epsilon _0}}\overline{H}^{-1}\left( \frac{\chi _0(t)}{t^{\frac{1}{1+\epsilon _0}}}\right) , \quad \forall t>1$$

and

$$\left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t\xi (t)}\right) \le \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t^{\frac{1}{1+\epsilon _0}}\xi (t)}\right) ,\quad \forall t>1.$$

Notation 5.1

$$\begin{aligned} \text {Let } r_0=\min {\{r_1,\varepsilon \}},\qquad \chi (t)=\max {\Big \{\frac{qI(t)}{t^{\frac{1}{1+\epsilon _0}}\xi (t)},\frac{\chi _0(t)}{t^{\frac{1}{1+\epsilon _0}}}}\Big \}, \end{aligned}$$

(5.30)

$F(t)=t^{1+\epsilon _0}$, $\mathcal {L}:=\xi L \sim E$, $J=\left( F^{-1} +H^{-1}\right) ^{-1},\chi _1(t)=\max {\{-E^{\prime }(t),\chi _0(t)\}}$ and

$$W=\left( \left[ \left( \overline{G}\right) ^{-1}\right] ^{\frac{1}{1+\epsilon _0}}+\overline{H}^{-1}\right) ^{-1}.$$

Theorem 5.4

Let $(u_{0},u_{1}),(z_{0},z_{1})\in \mathcal {W}\times L^2(\Omega )$ be given. Assume that $(A1)-(A3)$ and Eq. 3.8 hold and H is nonlinear. Then, there exist strictly positive constants $c_3, c_4, k_2, k_3$ and $\varepsilon _2$ such that the solution of $\text {(P)}$ satisfies, for all $t>t_1$,

$$\begin{aligned} E(t)\le J_1^{-1}\left( c_{3}\int _{t_0}^{t}\xi (s)ds+c_{4}\right) ,\text { if { G} is linear,} \end{aligned}$$

(5.31)

and

$$\begin{aligned} E(t) \le k_3 t^{\frac{1}{1+\epsilon _0}} {W_{2}}^{-1} \left( \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}\displaystyle \int _{t_1}^{t}\xi (s) ds } \right) , \text { if { G} is nonlinear,} \end{aligned}$$

(5.32)

where $J_{1}(t)=\int _{t}^{1}\frac{1}{J_{2}(s)}ds,\;\;\;\; J_{2}(t)=t J^{\prime }(\varepsilon _{1}t),$ and $W_{2}(t)=tW'(\varepsilon _2 t).$

Proof

Case 1: G is linear

We multiply Eq. 4.19 by $\xi (t)$ and use Eqs. 5.2, 5.14, and 5.15, and we get, $\forall t\ge t_0$,

$$\begin{aligned} \begin{aligned} {\mathcal {L}}'(t)&\le -m \xi (t) E(t)+c \left( -E^{\prime }(t) \right) ^{\frac{1}{(1+\epsilon _0)}} + c \xi (t) H^{-1}(\chi _0(t))\\&\le -m \xi (t) E(t)+cF^{-1}(-E^{\prime }(t)) + c H^{-1}(\chi _0(t)) \\&\le -m \xi (t) E(t)+ c \xi (t) J^{-1}(\chi _1(t)). \end{aligned} \end{aligned}$$

(5.33)

Now, for $\varepsilon _{2}<\varepsilon$ and $c_{0}>0$, using Eq. 5.33 and the fact that $E^{\prime }\le 0$, $J^{\prime }>0, J^{\prime \prime }>0$ on $(0,\varepsilon ],$ we obtain, for all $t\ge t_0$

$$\begin{aligned} \mathcal {L}_{1}^{\prime }(t)= & {} \varepsilon _{2}\frac{E^{\prime }(t)}{E(0)}J^{\prime \prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) \mathcal {L}(t)+ J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) {\mathcal {L}}^{\prime }(t)\nonumber \\\le & {} -m E(t)J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) +c \xi (t) J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) J^{-1}(\chi _1(t)), \end{aligned}$$

(5.34)

where $\mathcal {L}_{1}(t):=J' \left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) \mathcal {L}(t)\sim E$.

Using the generalized Young inequality with $A=J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right)$ and $B=J^{-1}(\chi _1(t)),$ Eq. 5.3, we obtain, for a suitable choice of $\varepsilon _{2}$ and $c_{0},$

$$\begin{aligned}{} & {} \mathcal {L}_{1}^{\prime }(t)\le -m E(t)J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) +c \xi (t) J^{*}\left( H^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) \right) +c \xi (t) \chi _0(t)\\\le & {} -m E(t)J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) +c\varepsilon _{2} \xi (t) \frac{E(t)}{E(0)}J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) -c E^{\prime }(t)\\{} & {} -c \xi (t) \frac{E^{\prime }(t)}{E(0)}J^{\prime }\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) \\= & {} -c \xi (t) J_{2}\left( \varepsilon _{2}\frac{E(t)}{E(0)}\right) , \forall t\ge t_0, \end{aligned}$$

where $J^{*}$ is the convex conjugate of J and $J_{2}(t)=t J^{\prime }(\varepsilon _{2}t).$

Let $R_1(t)=\varepsilon \frac{\alpha _{3}{\mathcal {L}_1}(t)}{E(0)}\sim E,\qquad 0<\varepsilon <1,$ then using the last estimate, we get, for some positive constant $c_3$,

$$R_1^{\prime }(t)\le -c_3 \xi (t) J_{2}(R_1(t)),\quad \forall t\ge t_0.$$

Then, a simple integration gives, for some $c_{4}>0,$

$$\begin{aligned} R_1(t)\le J_1^{-1}\left( c_{3}\int _{t_0}^{t} \xi (s) ds +c_{4}\right) ,\quad \forall t\ge t_0, \end{aligned}$$

(5.35)

where $J_{1}(t)=\int _{t}^{1}\frac{1}{J_{2}(s)}ds.$

Case 2. G is nonlinear.

Using Eqs. 4.19, 5.2, 5.9, and 5.14, we obtain, $\forall t\ge t_0$,

$$\begin{aligned} L^{\prime }(t)\le -m E(t)+ct^{\frac{1}{1+\epsilon _0}}\left[ \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t\xi (t)}\right) \right] ^{\frac{1}{1+\epsilon _0}}+c H^{-1}(\chi _0(t))-cE^{\prime }(t). \end{aligned}$$

(5.36)

Using Remark 5.3 and Eqs. 5.30, 5.36 becomes

$$\begin{aligned} \begin{aligned} \mathcal {F}^{\prime }(t)&\le -m E(t)+ct^{\frac{1}{1+\epsilon _0}}\left[ \left( \overline{G}\right) ^{-1}\left( \frac{qI(t)}{t^{\frac{1}{1+\epsilon _0}}\xi (t)}\right) \right] ^{\frac{1}{1+\epsilon _0}} +ct^{\frac{1}{1+\epsilon _0}}\overline{H}^{-1}\left( \frac{\chi _0(t)}{t^{\frac{1}{1+\epsilon _0}}}\right) \\&\le -m E(t)+ct^{\frac{1}{1+\epsilon _0}}W^{-1}(\chi (t)), \end{aligned} \end{aligned}$$

(5.37)

where $\mathcal {F}(t)=L(t)+cE(t)\sim E$.

Now, for $\varepsilon _{2}<r_0$ and using Eq. 5.36 and the fact that $E^{\prime }\le 0$, $W^{\prime }>0, W^{\prime \prime }>0$ on $(0,r_0],$ we deduce that

$$\begin{aligned} \begin{aligned}&\mathcal {F}_{1}^{\prime }(t)\le -m E(t)W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +ct^{\frac{1}{1+\epsilon _0}}W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) W^{-1}(\chi (t)), \end{aligned} \end{aligned}$$

(5.38)

where $\mathcal {F}_{1}(t):=W^{\prime }\left( \frac{\varepsilon _{2}}{(t-t_{1})^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \mathcal {F}(t)\sim E, \quad \forall t>t_1$.

Applying the generalized Young inequality with $A=W^{\prime }\left( \frac{\varepsilon _{2}}{t-t_{1}} \cdot \frac{E(t)}{E(0)}\right)$ and $B=W^{-1}(\chi (t)),$ we arrive at

$$\begin{aligned} \mathcal {F}_{1}^{\prime }(t)\le & {} -m E(t)W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c t^{\frac{1}{1+\epsilon _0}}W^{*}\left( W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \right) \nonumber \\{} & {} +c t^{\frac{1}{1+\epsilon _0}}\chi (t)\nonumber \\\le & {} -m E(t)W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c\epsilon _2\frac{E(t)}{E(0)}W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \nonumber \\{} & {} +ct^{\frac{1}{1+\epsilon _0}}\chi (t). \end{aligned}$$

(5.39)

So, multiplying Eq. 5.39 by $\xi (t)$ and using Eqs. 5.3, 5.12, and 5.30 give, for a suitable choice of $\varepsilon _2$,

$$\begin{aligned} \begin{aligned} \xi (t)\mathcal {F}_{1}^{\prime }(t)&\le -m \xi (t) E(t)W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c \varepsilon _{2} \xi (t) \cdot \frac{E(t)}{E(0)}W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \\&\qquad -cE^{\prime }(t)\\&\le -m \xi (t) E(t)W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) +c \varepsilon _2 \xi (t) \cdot \frac{E(t)}{E(0)}W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \\&\le -k \xi (t) \left( \frac{E(t)}{E(0)}\right) W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) , \qquad \forall t>t_1, \end{aligned} \end{aligned}$$

where $\mathcal {F}_{2}:=\xi \mathcal {F}_{1}+cE \sim E$. Hence, we deduce that

$$\begin{aligned} \int _{t_1}^{t} k \left( \frac{E(s)}{E(0)} \right) W^{\prime }\left( \frac{\varepsilon _{2}}{s^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(s)}{E(0)}\right) \xi (s) ds \le - \int _{t_1}^{t} \mathcal {F}_{2}^{\prime }(s)ds\le \mathcal {F}_{2}(t_1). \end{aligned}$$

(5.40)

Using the fact that the map $t \mapsto E(t)W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right)$ is nonincreasing, we have

$$\begin{aligned} \begin{aligned}&k \left( \frac{E(t)}{E(0)} \right) W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \int _{t_1}^{t} \xi (s) ds \\&\qquad \le \int _{t_1}^{t} k \left( \frac{E(s)}{E(0)} \right) W^{\prime }\left( \frac{\varepsilon _{2}}{s^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(s)}{E(0)}\right) \xi (s) ds\le \mathcal {F}_{2}(t_1),\qquad \forall t>t_1. \end{aligned} \end{aligned}$$

(5.41)

Multiplying each side of Eq. 5.41 by $\frac{1}{t^{\frac{1}{1+\epsilon _0}}}$, we have

$$\begin{aligned} k \left( \frac{1}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) W^{\prime }\left( \frac{\varepsilon _{2}}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \int _{t_1}^{t} \xi (s) ds \le \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}},\qquad \forall t>t_1. \end{aligned}$$

(5.42)

Next, we set $W_{2}(s)=s W^{\prime }(\varepsilon _{2}s)$ which is strictly increasing, then we obtain,

$$\begin{aligned} k W_{2} \left( \frac{1}{t^{\frac{1}{1+\epsilon _0}}} \cdot \frac{E(t)}{E(0)}\right) \int _{t_1}^{t} \xi (s) ds \le \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}},\qquad \forall t>t_1. \end{aligned}$$

(5.43)

Finally, for two positive constants $k_2$ and $k_3$, we obtain

$$\begin{aligned} E(t) \le k_3 t^{\frac{1}{1+\epsilon _0}} {W_{2}}^{-1} \left( \frac{k_2}{t^{\frac{1}{1+\epsilon _0}}\int _{t_1}^{t}\xi (s) ds } \right) . \end{aligned}$$

(5.44)

This finishes the proof. $\square$

Example 5.1

The following examples illustrate our results:

• 1. $h_0$ and G are linear. Let $g(t)=a e^{-b(1+t)},$ where $b>0$ and $a>0$ is small enough so that Eq. 2.1 is satisfied, then $g^{\prime }(t)=-\xi (t) G(g(t))$ where $G(t)=t$ and $\xi (t)=b.$ For the frictional nonlinearity, assume that $h_0(t)=ct$, so $H(t)=\sqrt{t} h_0(\sqrt{t})=ct$. Therefore, we can use Eq. 5.16 to obtain

  $$\begin{aligned} E(t) \le \frac{c}{(1+t)^\frac{1}{\epsilon _0}}. \end{aligned}$$

  (5.45)

• 2. $h_0$ is linear, and G is nonlinear. Let $g(t)=\frac{a}{(1+t)^q}$, where $q >1+\epsilon _0$ and a is chosen so that hypothesis Eq. 2.1 remains valid. Then,

  $$g^{\prime }(t)=-b G(g(t)),\qquad \text {with}\qquad G(s)=s^{\frac{q+1}{q}},$$

  where b is a fixed constant. For the frictional nonlinearity, we assume that $h_0(t)=ct,$ and $H(t)=\sqrt{t} h_0(\sqrt{t})=ct$. Since $K(s)=s^{\frac{(\epsilon _0+1)(q+1)}{q}}$. Then, Eq. 5.17 gives, $\forall t\ge t_1$

  $$\begin{aligned} E(t)\le \frac{c}{t^{\frac{q-1-\epsilon _0}{(1+\epsilon _0)^2 (q+1)}}} \end{aligned}$$

  (5.46)

• 3. $h_0$ is nonlinear, and G is linear. Let $g(t)=a e^{-b(1+t)},$ where $b>0$ and $a>0$ is small enough so that Eq. 2.1 is satisfied, then $g^{\prime }(t)=-\xi (t) G(g(t))$ where $G(t)=t$ and $\xi (t)=b.$ Also, assume that $h_0(t)=ct^2,$ where $H(t)=\sqrt{t} h_0(\sqrt{t})=ct^{\frac{3}{2}}$. Then, after taking $\epsilon _0=\frac{1}{2}$, we have

  $$F(t)=t^\frac{3}{2}$$

  and

  $$J(t)=c t^\frac{3}{2}.$$

  Therefore, applying Eq. 5.31, we obtain

  $$\begin{aligned} E(t)\le \frac{c}{(1+t)^2} \end{aligned}$$

  (5.47)

• 4. $h_0$ and G are nonlinear. Let $g(t)=\frac{a}{(1+t)^4}$, where a is chosen so that hypothesis Eq. 2.1 remains valid. Then

  $$g^{\prime }(t)=-b G(g(t)),\qquad \text {with}\qquad G(s)=s^{\frac{5}{4}},$$

  where b is a fixed constant. For the frictional nonlinearity, let $h_0(t)=c t^2$ and $H(t)=c t^\frac{3}{2}$. Then, with $\epsilon _0=\frac{1}{5},$ we obtain

  $$W=\left( \left[ \left( \overline{G}\right) ^{-1}\right] ^{\frac{1}{1+\epsilon _0}}+\overline{H}^{-1}\right) ^{-1}=cs^\frac{3}{2}$$

  and

  $$\begin{aligned} \begin{aligned} W_2(s)=cs^\frac{3}{2}\ \end{aligned} \end{aligned}$$

  Therefore, applying Eq. 5.32, we obtain, $\forall t\ge t_1$

  $$E(t)\le \frac{c}{t^{\frac{7}{18}}}.$$