On a class of a coupled nonlinear viscoelastic Kirchhoff equations variable-exponents: global existence, blow up, growth and decay of solutions

Abstract

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term.

1 Introduction

Numerous previous research works in the field of problems with variable exponents acknowledged the role of dampers which appear in many areas of applied sciences. The objective of this work is to provide further insight into the complex interactions that exist between dampers and variable exponents.

In the absence of a variable exponent, the reader can view the following papers related to the topic being studied: the general decay, blow-up, and growth of solutions [1, 3–5, 7–9, 11–15].

We present our current problem of a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents, where we combine several damping terms into one, of course in the general case ($\eta \geq 0$).

In this work, we try to collect many studies in one paper, where the reader has in his hands four different proofs with the methods used under appropriate conditions, while comparing the differences.

First, we list some previous works similar to ours. Whereas in the absence of the source term with ($\eta =0$), the study is found in [16], the authors studied the global existence and general decay of solutions for a quasilinear system with degenerate damping terms.

As for the presence of the source and degenerate damping terms in the absence of dispersion term, we mention the work done by the authors in [26], where they obtained global existence of solutions. Then, they proved the general decay result. Finally, they proved the finite time blow-up result of solutions with negative initial energy. For more information in this context, you can see the papers [6, 17, 18, 20, 22, 23].

On the other hand, there are many works that deal with the variable exponent. We mention, for example, our work [24]. In the presence of delay, the authors have demonstrated a nonlinear Kirchhoff-type equation with distributed delay and variable exponents. Under a suitable hypothesis they proved the blow-up of solutions, and by using an integral inequality due to Komornik, they obtained the general decay result. See also [2, 10, 21, 29], and [25], each of which examines a different problem with appropriate conditions.

In this work, we are examining the following problem:

$$\begin{aligned} \textstyle\begin{cases} \vert \Phi _{t} \vert ^{\eta }\Phi _{tt}-\mathcal{T}( \Vert \nabla \Phi \Vert _{2}^{2})\Delta \Phi +\int _{0}^{t}h_{1}(t- \varsigma )\Delta \Phi (\varsigma )\,d\varsigma -\Delta \Phi _{tt}+g_{1}( \Phi _{t}) =f_{1} ( \Phi ,\Psi ), \\ \vert \Psi _{t} \vert ^{\eta }\Psi _{tt}-\mathcal{T}( \Vert \nabla \Psi \Vert _{2}^{2})\Delta \Psi +\int _{0}^{t}h_{2}(t- \varsigma )\Delta \Psi (\varsigma )\,d\varsigma -\Delta \Psi _{tt}+g_{2}( \Psi _{t}) =f_{2} ( \Phi ,\Psi ), \\ \Phi ( x,t ) =\Psi ( x,t ) =0, \quad ( x,t ) \in \partial \Omega \times ( 0,T ) , \\ \Phi ( x,0 ) =\Phi _{0} ( x ) ,\qquad \Phi _{t} ( x,0 ) =\Phi _{1} ( x ) , \quad x\in \Omega , \\ \Psi ( x,0 ) =\Psi _{0} ( x ) ,\qquad \Psi _{t} ( x,0 ) =\Psi _{1} ( x ) , \quad x\in \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.1)

where

$$ g_{1}(\Phi _{t}):=\zeta _{1} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)-2} \Phi _{t}(t), \qquad g_{2}(\Psi _{t}):=\zeta _{3} \bigl\vert \Psi _{t}(t) \bigr\vert ^{s(x)-2} \Psi _{t}(t), $$

in which $\eta \geq 0$ for $N=1,2$ and $0<\eta \leq \frac{2}{N-2}$ for $N\geq 3$, and $h_{i}(\cdot):R^{+}\rightarrow R^{+}$ $(i=1,2)$ symbolizes the relaxation function, $-\Delta _{tt} ( \cdot )$ symbolizes the dispersion term, and $\mathcal{T}(\sigma )$ is a positive locally Lipschitz function for $\gamma ,\sigma \geq 0$ such that $\mathcal{T}(\sigma )=\alpha _{1}+\alpha _{2}\sigma ^{\gamma}$, and

$$ \textstyle\begin{cases} f_{1}(\Phi ,\Psi )=a_{1} \vert \Phi +\Psi \vert ^{2(q(x)+1)}(\Phi + \Psi )+b_{1} \vert \Phi \vert ^{q(x)}.\Phi . \vert \Psi \vert ^{q(x)+2}, \\ f_{2}(\Phi ,\Psi )=a_{1} \vert \Phi +\Psi \vert ^{2(q(x)+1)}(\Phi + \Psi )+b_{1} \vert \Psi \vert ^{q(x)}.\Psi . \vert v \vert ^{q(x)+2}. \end{cases} $$

(1.2)

In this context, we consider $q(\cdot)$, $m(\cdot)$, and $s(\cdot)$ are variable exponents defined as measurable functions on Ω̅ in the following manner:

$$\begin{aligned} &1\leq q^{-}\leq q(x)\leq q^{+}\leq q^{*}, \\ &2\leq m^{-}\leq m(x)\leq m^{+}\leq m^{*}, \\ &2\leq s^{-}\leq s(x)\leq s^{+}\leq s^{*}, \end{aligned}$$

(1.3)

where

$$\begin{aligned} &q^{-}= \inf_{x\in \overline{\Omega}} q(x),\qquad m^{-}= \inf_{x\in \overline{\Omega}} m(x),\qquad s^{-}= \inf _{x\in \overline{\Omega}} s(x), \\ &q^{+}= \sup_{x\in \overline{\Omega}} q(x), \qquad m^{+}= \sup_{x\in \overline{\Omega}} m(x),\qquad s^{+}= \sup_{x\in \overline{\Omega}} s(x), \end{aligned}$$

(1.4)

with

$$ \max \bigl\{ m^{+},s^{+}\bigr\} \leq 2q^{-}+1 $$

(1.5)

and

$$ m^{*},s^{*}=\frac{2(n-1)}{n-2} \quad \text{if } n\geq 3. $$

(1.6)

As for the division of the paper, it is as follows. In the following section, we present the hypotheses, concepts, and lemmas essential for our study. In Sect. 3, we obtain global existence of the solution of (1.1). Next, Sects. 4 and 5 are dedicated to proving the blow-up result, followed by the exponential growth of solutions. In Sect. 6, we establish the general decay when $f_{1}=f_{2}=0$. Finally, we present the general conclusion in the last section.

2 Preliminaries

In this section, we give some related theory and put suitable hypotheses for the proof of our result.

(H1) Put a nonincreasing and differentiable function $h_{i}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ where

$$\begin{aligned} &h_{i}(t)\geq 0 , \quad 1- \int _{0}^{\infty }h_{i} ( \varsigma ) \,d\varsigma =l_{i}>0, \quad i=1,2. \end{aligned}$$

(2.1)

(H2) One can find $\xi _{1},\xi _{2}>0$ in a way that

$$\begin{aligned} &h_{i}^{\prime } ( t ) \leq -\xi _{i} h_{i} ( t ) , \quad t\geq 0, i=1,2. \end{aligned}$$

(2.2)

Lemma 2.1

There exists $F(\Phi , \Psi )$ defined by

$$\begin{aligned} F(\Phi ,\Psi ) =&\frac{1}{2(q(x)+2)} \bigl[\Phi f_{1}(\Phi ,\Psi )+ \Phi f_{2}(\Phi ,\Psi ) \bigr] \\ =&\frac{1}{2(q(x)+2)} \bigl[a_{1} \vert \Phi +\Psi \vert ^{2(q(x)+2)}+2 b_{1} \vert \Phi .\Psi \vert ^{q(x)+2} \bigr] \geq 0, \end{aligned}$$

where

$$\begin{aligned} \frac{\partial F}{\partial \Phi}=f_{1}(\Phi ,\Psi ), \qquad \frac{\partial F}{\partial \Psi}=f_{2}(\Phi ,\Psi ). \end{aligned}$$

For simplification, we put $\alpha _{1}=\alpha _{2}=1$ and $a_{1}=b_{1} = 1$ for convenience.

Lemma 2.2

[6] One can find $c_{0}>0$ and $c_{1}>0$ in a way that

$$\begin{aligned} \frac{c_{0}}{2(q(x)+2)} \bigl( \vert \Phi \vert ^{2(q(x)+2)}+ \vert \Psi \vert ^{2(q(x)+2)} \bigr) \leq& F(\Phi ,\Psi ) \\ \leq& \frac{c_{1}}{2(q(x)+2)} \bigl( \vert \Phi \vert ^{2(q(x)+2)}+ \vert \Psi \vert ^{2(q(x)+2)} \bigr). \end{aligned}$$

(2.3)

Now, we consider $q:\Omega \rightarrow [1,\infty )$ is a measurable function.

After, introducing the Lebesgue space with a variable exponent $q(\cdot)$ as follows:

$$ L^{q(\cdot)}(\Omega )= \biggl\{ \Phi :\Omega \rightarrow \mathbb{R}; \text{ measurable in } \Omega : \int _{\Omega} \vert \Phi \vert ^{q(\cdot)}\,dx< \infty \biggr\} , $$

we give the norm as follows:

$$ \Vert \Phi \Vert _{q(\cdot)}=\inf \biggl\{ \lambda >0: \int _{\Omega} \biggl\vert \frac{\Phi}{\lambda} \biggr\vert ^{q(x)}\,dx\leq 1 \biggr\} . $$

This space is fitted with the standard norm, $L^{q(\cdot)}(\Omega )$ is a Banach space. After that, we introduce the variable exponent Sobolev space $W^{1,q(\cdot)}(\Omega )$ as follows:

$$ W^{1,q(\cdot)}(\Omega )= \bigl\{ \Phi \in L^{q(\cdot)}(\Omega ); \nabla \Phi \text{ exists and } \vert \nabla \Phi \vert \in L^{q(\cdot)}(\Omega ) \bigr\} , $$

with the norm given by

$$ \Vert \Phi \Vert _{1,q(\cdot)}= \Vert \Phi \Vert _{q(\cdot)}+ \Vert \nabla \Phi \Vert _{q(\cdot)}, $$

$W^{1,q(\cdot)}(\Omega )$ is a Banach space, and the closure of $C^{\infty}_{0}(\Omega )$ is given by $W^{1,q(\cdot)}_{0}(\Omega )$.

For $\Phi \in W^{1,q(\cdot)}_{0}(\Omega )$, we give the equivalent norm

$$ \Vert \Phi \Vert _{1,q(\cdot)}= \Vert \nabla \Phi \Vert _{q(\cdot)}. $$

$W^{-1,q'(\cdot)}_{0}(\Omega )$ denotes the dual of $W^{1,q(\cdot)}_{0}(\Omega )$ in which $\frac{1}{q(\cdot)}+\frac{1}{q'(\cdot)}=1$.

Next, we offer the continuity condition of Log-Hölder:

$$\begin{aligned} \bigl\vert p(x)-p(y) \bigr\vert \leq -\frac{M_{1}}{\log \vert x-y \vert } \quad \text{and} \quad \bigl\vert m(x)-m(y) \bigr\vert \leq - \frac{M_{2}}{\log \vert x-y \vert } \end{aligned}$$

(2.4)

for all $x,y\in \Omega$, where $M_{1},M_{2}>0$ and $0<\varrho <1$ with $\vert x-y\vert <\varrho$.

Theorem 2.3

Assume that (2.1)–(2.2) hold. Then, for any $(\Phi _{0},\Phi _{1},\Psi _{0},\Psi _{1})\in \mathcal{H}$, (1.1) has a unique solution for some $T>0$:

$$\begin{aligned} &\Phi ,\Psi \in C\bigl([0,T]; H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\bigr), \\ &\Phi _{t}\in C\bigl([0,T]; H^{1}_{0}( \Omega )\bigr)\cap L^{m(x)}\bigl(\Omega \times (0,T)\bigr), \\ &\Psi _{t}\in C\bigl([0,T]; H^{1}_{0}( \Omega )\bigr)\cap L^{s(x)}\bigl(\Omega \times (0,T)\bigr), \end{aligned}$$

where

$$ \mathcal{H}= H^{1}_{0}(\Omega )\times L^{2}(\Omega )\times H^{1}_{0}( \Omega ) \times L^{2}(\Omega ). $$

Now, we define the functional of energy.

Lemma 2.4

Let (2.1)–(2.2) be satisfied and $(\Phi ,\Psi )$ be a solution of (1.1). Then the functional $\mathcal{E}(t)$ is nonincreasing and defined as follows:

$$\begin{aligned} \mathcal{E}(t) =&\frac{1}{\eta +2} \bigl[ \Vert \Phi _{t} \Vert _{\eta +2}^{ \eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr]+\frac{1}{2} \bigl[ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr] \\ &{}+\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2( \gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr] \\ &{}+\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}+\frac{1}{2} \bigl[(h_{1}o\nabla \Phi ) (t)+(h_{2}o\nabla \Psi ) (t) \bigr]- \int _{\Omega}F(\Phi ,\Psi )\,dx \end{aligned}$$

(2.5)

fulfills

$$\begin{aligned} \mathcal{E}'(t) =& \frac{1}{2} \bigl[ \bigl(h'_{1}o\nabla \Phi \bigr) (t)+ \bigl(h'_{2}o \nabla \Psi \bigr) (t) \bigr]- \frac{1}{2} \bigl[h_{1}(t) \Vert \nabla \Phi \Vert _{2}^{2}+h_{2}(t) \Vert \nabla \Psi \Vert _{2}^{2} \bigr] \\ &{}-\zeta _{1} \int _{\Omega} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)}\,dx-\zeta _{3} \int _{\Omega} \bigl\vert \Psi _{t}(t) \bigr\vert ^{s(x)}\,dx \\ \leq & 0. \end{aligned}$$

(2.6)

Proof

By multiplying (1.1)₁, (1.1)₂ by $\Phi _{t}$, $\Psi _{t}$ and integrating over Ω, we have

$$\begin{aligned} & \frac {d}{dt} \biggl\{ \frac{1}{\eta +2} \Vert \Phi _{t} \Vert _{\eta +2}^{ \eta +2}+\frac{1}{\eta +2} \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \frac{1}{2} \Vert \nabla \Phi _{t} \Vert _{2}^{2}+\frac{1}{2} \Vert \nabla \Psi _{t} \Vert _{2}^{2} \\ &\qquad +\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2( \gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr] \\ &\qquad +\frac{1}{2} \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \frac{1}{2} \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \\ &\qquad +\frac{1}{2}(h_{1}o\nabla \Phi ) (t)+ \frac{1}{2}(h_{2}o\nabla \Psi ) (t)- \int _{\Omega}F(\Phi ,\Psi )\,dx \biggr\} \\ &\quad =-\zeta _{1} \int _{\Omega} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx-\zeta _{3} \int _{\Omega} \bigl\vert \Psi _{t}(t) \bigr\vert ^{s(x)} \,dx \\ &\qquad +\frac{1}{2}\bigl(h_{1}'o\nabla \Phi \bigr)-\frac{1}{2}h_{1}(t) \Vert \nabla \Phi \Vert _{2}^{2}+\frac{1}{2}\bigl(h_{2}'o \nabla \Psi \bigr)-\frac{1}{2}h_{2}(t) \Vert \nabla \Psi \Vert _{2}^{2}. \end{aligned}$$

(2.7)

Hence, we find (2.5) and (2.6). Then, we have $\mathcal{E}$ is a nonincreasing function. This ends the proof. □

3 Global existence

Now, we show that the solution of (1.1) is uniformly bounded and global in time. For this purpose, we set

$$\begin{aligned} &I(t)= \biggl[ \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &\hphantom{I(t)}\quad +\frac{1}{(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr] \\ &\hphantom{I(t)}\quad + \bigl[(h_{1}o\nabla \Phi ) (t)+(h_{2}o\nabla \Psi ) (t) \bigr]-2\bigl(q^{-}+2\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx, \end{aligned}$$

(3.1)

$$\begin{aligned} &J(t)=\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &\hphantom{J(t)}\quad +\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2( \gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr] \\ &\hphantom{J(t)}\quad +\frac{1}{2} \bigl[(h_{1}o\nabla \Phi ) (t)+(h_{2}o\nabla \Psi ) (t) \bigr]- \int _{\Omega}F(\Phi ,\Psi )\,dx. \end{aligned}$$

(3.2)

Hence,

$$ \mathcal{E}(t)=J(t)+\frac{1}{\eta +2} \bigl[ \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr]+ \frac{1}{2} \bigl[ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr]. $$

(3.3)

Lemma 3.1

Suppose that the initial data $(\Phi _{0},\Phi _{1}),(\Psi _{0},\Psi _{1}) \in (H^{1}(\Omega ) \times L^{2}(\Omega ))^{2}$ satisfy $I(0)>0$ and

$$ \widehat{\xi}:=\frac{c_{1}C_{*}(p^{+})}{l} \biggl( \frac{(2p^{-}+4)\mathcal{E}(0)}{l(p^{-}+1)} \biggr)^{p^{+}+1}< 1. $$

(3.4)

Then $I(t)>0$ for any $t\in [0, T ]$.

Proof

Since $I(0)>0$, we deduce by continuity that there exists $0< T^{*}\leq T$ such that $I(t)\geq 0$ for all $t\in [0, T^{*}]$.

This implies that $\forall t\in [0, T^{*}]$,

$$\begin{aligned} J(t) \geq &\frac{q^{-}+1}{2(q^{-}+2)} \biggl[ \biggl(1- \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}+\frac{q^{-}+1}{2(q^{-}+2)} \biggl\{ \frac{1}{(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)} \bigr] \biggr\} \\ &{}+\frac{q^{-}+1}{2(q^{-}+2)} \bigl[(h_{1}o\nabla \Phi ) (t)+(h_{2}o \nabla \Psi ) (t) \bigr]+\frac{1}{2(q^{-}+2)}I(t) \\ \geq &\frac{q^{-}+1}{2(q^{-}+2)} \biggl\{ l_{1} \Vert \nabla \Phi \Vert _{2}^{2}+l_{2} \Vert \nabla \Psi \Vert _{2}^{2}+\frac{1}{(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)} \bigr] \\ & {} +(h_{1}o\nabla \Phi ) (t)+(h_{2}o\nabla \Psi ) (t) \biggr\} \\ \geq &\frac{q^{-}+1}{2(q^{-}+2)} \bigl(l_{1} \Vert \nabla \Phi \Vert _{2}^{2}+l_{2} \Vert \nabla \Psi \Vert _{2}^{2} \bigr). \end{aligned}$$

(3.5)

Hence, by (2.6) and (3.3), we get

$$\begin{aligned} \bigl(l_{1} \Vert \nabla \Phi \Vert _{2}^{2}+l_{2} \Vert \nabla \Psi \Vert _{2}^{2} \bigr) \leq & \frac{2(q^{-}+2)}{q^{-}+1}J(t)\leq \frac{2(q^{-}+2)}{q^{-}+1}\mathcal{E}(t) \\ \leq &\frac{2(q^{-}+2)}{q^{-}+1}\mathcal{E}(0), \quad \forall t\in \bigl[0, T^{*}\bigr] . \end{aligned}$$

(3.6)

On the other hand, by using (2.3), we get

$$\begin{aligned} 2\bigl(q^{-}+2\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx < &c_{1} \int _{\Omega} \bigl( \vert \Phi \vert ^{2p^{+}+4)}+ \vert \Psi \vert ^{2p^{+}+4)} \bigr)\,dx. \end{aligned}$$

(3.7)

Then the embedding $H^{1}_{0}(\Omega ))\hookrightarrow L^{2p^{+}+4}(\Omega )$ yields

$$\begin{aligned} \int _{\Omega} \bigl( \vert \Phi \vert ^{2p^{+}+4}+ \vert \Psi \vert ^{2p^{+}+4)} \bigr) \,dx \leq &C_{*} \bigl(p^{+}\bigr) \bigl( \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2p^{+}+4}+ \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2p^{+}+4} \bigr) \\ =&C_{*}\bigl(p^{+}\bigr) \bigl\{ \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2} \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2p^{+}+2} \\ & {} + \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2} \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2p^{+}+2} \bigr\} . \end{aligned}$$

(3.8)

By (3.6), we find

$$\begin{aligned} 2\bigl(q^{-}+2\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx&< c_{1}C_{*} \bigl(p^{+}\bigr) \biggl( \frac{2(p^{-}+2)\mathcal{E}(0)}{l_{1}( p^{-}+1)} \biggr)^{p^{+}+1} \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2} \\ &\quad +c_{1}C_{*}\bigl(p^{+}\bigr) \biggl( \frac{2(p^{-}+2)\mathcal{E}(0)}{l_{2}(p^{-}+1)} \biggr)^{p^{+}+1} \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2} \\ & < \widehat{\xi} \bigl(l_{1} \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2}+l_{2} \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2} \bigr), \end{aligned}$$

(3.9)

where

$$ \widehat{\xi}=c_{1}C_{*}\bigl(p^{+} \bigr)l^{-(p^{+}+2)} \biggl( \frac{(2p^{-}+4)\mathcal{E}(0)}{(p^{-}+1)} \biggr)^{p^{+}+1}, $$

the embedding constant $C_{*}(p^{+})$ and $l=\min (l_{1},l_{2})$.

By (3.4) and (2.1), we obtain

$$\begin{aligned} 2\bigl(q^{-}+2\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx < &\xi \biggl(1- \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggr) \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2} \\ &{}+ \widehat{\xi} \biggl(1- \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2}. \end{aligned}$$

(3.10)

According to (3.1),(3.6), and (3.10), we get

$$\begin{aligned} I(t) >& (1-\widehat{\xi}) \biggl\{ \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \bigl\Vert \nabla \Phi (t) \bigr\Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \bigl\Vert \nabla \Psi (t) \bigr\Vert _{2}^{2} \biggr\} \\ >&0, \quad \forall t\in \bigl[0,T^{*}\bigr]. \end{aligned}$$

(3.11)

By repeating this procedure, $T^{*}$ can be extended to T. This completes the proof. □

Remark 3.2

Under the conditions of Lemma 3.1, we have $J(t)\geq 0$, and consequently $\mathcal{E}(t)\geq 0$, $\forall t\in [0,T]$. Hence, by (3.3) and (3.5), we find

$$\begin{aligned} &\bigl\Vert \Phi _{t}(t) \bigr\Vert _{\eta +2}^{\eta +2}+ \bigl\Vert \Psi _{t}(t) \bigr\Vert _{ \eta +2}^{\eta +2} \leq (\eta +2)\mathcal{E}(0), \\ &\bigl\Vert \nabla \Phi _{t}(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla \Psi _{t}(t) \bigr\Vert _{2}^{2} \leq 2 \mathcal{E}(0), \\ &\Vert \nabla \Phi \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)}\leq \frac{2q^{-}+4}{q^{-}+1}(\gamma +1)\mathcal{E}(0). \end{aligned}$$

(3.12)

Theorem 3.3

Suppose that the hypotheses of Lemma 3.1hold, then the solution of (1.1) is global and bounded.

Proof

It suffices to show that

$$\begin{aligned} \bigl\Vert (\Phi ,\Psi ) \bigr\Vert _{H} :=& \Vert \Phi _{t} \Vert ^{\eta +2}_{ \eta +2}+ \Vert \Psi _{t} \Vert ^{\eta +2}_{\eta +2}+ \Vert \nabla \Phi \Vert ^{2}_{2}+ \Vert \nabla \Psi \Vert ^{2}_{2} \\ &{}+ \Vert \nabla \Phi _{t} \Vert ^{2}_{2}+ \Vert \nabla \Psi _{t} \Vert ^{2}_{2} \end{aligned}$$

is bounded independently of t. To achieve this, we use (3.12) to get

$$\begin{aligned} \mathcal{E}(0) >&\mathcal{E}(t)=J(t)+\frac{1}{\eta +2} \bigl[ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{ \eta +2} \bigr]+ \frac{1}{2} \bigl[ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr] \\ \geq &\frac{q^{-}+1}{2(q^{-}+2)} \bigl(l_{1} \Vert \nabla \Phi \Vert _{2}^{2}+l_{2} \Vert \nabla \Psi \Vert _{2}^{2} \bigr)+\frac{1}{\eta +2} \bigl[ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{ \eta +2} \bigr] \\ &{}+\frac{1}{2} \bigl[ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr]. \end{aligned}$$

(3.13)

Therefore,

$$ \bigl\Vert (\Phi ,\Psi ) \bigr\Vert _{H}\leq C\mathcal{E}(0), $$

where $C(q^{-},\eta ,l_{1},l_{2})$ is a positive constant. □

4 Blow-up

Here, we establish the blow-up result for the solution of (1.1) with negative initial energy. Initially, we introduce the following functional:

$$\begin{aligned} \mathbb{H}(t)=-\mathcal{E}(t) =&-\frac{1}{\eta +2} \bigl[ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr]- \frac{1}{2} \bigl[ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr] \\ &{}-\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2( \gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr] \\ &{}-\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}-\frac{1}{2} \bigl[(h_{1}o\nabla \Phi ) (t)+(h_{2}o\nabla \Psi ) (t) \bigr]+ \int _{\Omega}F(\Phi ,\Psi )\,dx. \end{aligned}$$

(4.1)

Theorem 4.1

Assume that (2.1)–(2.2) and $\mathcal{E}(0)<0$ hold. Then the solution of (1.1) blows up in finite time.

Proof

From (2.5), the following can be written:

$$ \mathcal{E}(t)\leq \mathcal{E}(0)\leq 0. $$

(4.2)

Therefore

$$\begin{aligned} \mathbb{H}'(t)=-\mathcal{E}'(t) \geq & \zeta _{1} \int _{\Omega} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)}\,dx-\zeta _{3} \int _{\Omega} \bigl\vert \Psi _{t}(t) \bigr\vert ^{s(x)}\,dx, \end{aligned}$$

(4.3)

hence

$$\begin{aligned} \mathbb{H}'(t) \geq &\zeta _{1} \int _{\Omega} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)}\,dx \geq 0 \\ \mathbb{H}'(t) \geq &\zeta _{3} \int _{\Omega} \bigl\vert \Psi _{t}(t) \bigr\vert ^{s(x)}\,dx \geq 0. \end{aligned}$$

(4.4)

By (4.1) and (2.3), we have

$$\begin{aligned} 0\leq \mathbb{H}(0)\leq \mathbb{H}(t) \leq & \int _{\Omega}F(\Phi , \Psi )\,dx \\ \leq & \int _{\Omega}\frac{c_{1}}{2(q(x)+2)} \bigl( \vert \Phi \vert ^{2(q(x)+2)}+ \vert \Psi \vert ^{2(q(x)+2)} \bigr)\,dx \\ \leq &\frac{c_{1}}{2(q^{-}+2)}\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr), \end{aligned}$$

(4.5)

where

$$ \varrho (\varphi )=\varrho _{q(\cdot)}(\varphi )= \int _{\Omega} \vert \varphi \vert ^{2(q(x)+2)}\,dx. $$

Lemma 4.2

Let $\exists c>0$ in a way that any solution of (1.1) satisfies

$$ \Vert \Phi \Vert ^{2(q^{-}+2)}_{2(q^{-}+2)}+ \Vert \Psi \Vert ^{2(q^{-}+2)}_{2(q^{-}+2)} \leq c\bigl(\varrho (\Phi )+\varrho (\Psi ) \bigr). $$

(4.6)

Proof

Let

$$ \Omega _{1}=\bigl\{ x\in \Omega : \bigl\vert \Phi (x,t) \bigr\vert \geq 1\bigr\} , \qquad \Omega _{2}=\bigl\{ x\in \Omega : \bigl\vert \Phi (x,t) \bigr\vert < 1\bigr\} , $$

(4.7)

we have

$$\begin{aligned} \varrho (\Phi ) =& \int _{\Omega _{1}} \vert \Phi \vert ^{2(q(x)+2)}\,dx+ \int _{\Omega _{2}} \vert \Phi \vert ^{2(q(x)+2)}\,dx \\ \geq & \int _{\Omega _{1}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx+ \int _{ \Omega _{2}} \vert \Phi \vert ^{2(q^{+}+2)}\,dx \\ \geq & \int _{\Omega _{1}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx+c \biggl( \int _{ \Omega _{2}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx \biggr)^{ \frac{2(q^{+}+2)}{2(q^{-}+2)}}, \end{aligned}$$

(4.8)

then

$$\begin{aligned} &\varrho (\Phi )\geq \int _{\Omega _{1}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx \\ &\biggl(\frac{\varrho (\Phi )}{c}\biggr)^{\frac{2(q^{-}+2)}{2(q^{+}+2)}}\geq \int _{\Omega _{1}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx. \end{aligned}$$

(4.9)

Hence, we get

$$\begin{aligned} \Vert \Phi \Vert ^{2(q^{-}+2)}_{2(q^{-}+2)} \leq &\varrho (\Phi )+c \bigl( \varrho (\Phi )\bigr)^{\frac{2(q^{-}+2)}{2(q^{+}+2)}} \\ \leq &\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+c\bigl(\varrho (\Phi )+ \varrho ( \Psi )\bigr)^{\frac{2(q^{-}+2)}{2(q^{+}+2)}} \\ \leq &\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)\bigl[1+c\bigl(\varrho (\Phi )+\varrho ( \Psi )\bigr)^{\frac{2(q^{-}+2)}{2(q^{+}+2)}-1}\bigr]. \end{aligned}$$

(4.10)

According to (4.5), we have

$$\begin{aligned} \frac{\mathbb{H}(0)}{c}\leq \bigl(\varrho (\Phi )+\varrho (\Psi )\bigr). \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \Phi \Vert ^{2(q^{-}+2)}_{2(q^{-}+2)} \leq &\bigl(\varrho ( \Phi )+ \varrho (\Psi )\bigr)\bigl[1+c\bigl(\mathbb{H}(0)\bigr)^{\frac{2(q^{-}+2)}{2(q^{+}+2)}-1} \bigr]. \end{aligned}$$

Hence

$$ \Vert \Phi \Vert ^{2(q^{-}+2)}_{2(q^{-}+2)} \leq c\bigl(\varrho ( \Phi )+ \varrho (\Psi )\bigr). $$

(4.11)

Using the same method, we find

$$ \Vert \Psi \Vert ^{2(q^{-}+2)}_{2(q^{-}+2)} \leq c\bigl(\varrho ( \Phi )+ \varrho (\Psi )\bigr). $$

(4.12)

By combining the previous two inequalities (4.11) and (4.12), we get the result we want (4.6). □

Corollary 4.3

$$\begin{aligned} & \int _{\Omega} \vert \Phi \vert ^{m(x)}\,dx\leq c \bigl(\bigl(\varrho (\Phi )+ \varrho (\Psi )\bigr)^{m^{-}/2(q^{-}+2)}+\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{m^{+}/2(q^{-}+2)} \bigr), \\ & \int _{\Omega} \vert \Psi \vert ^{s(y)}\,dy\leq c \bigl(\bigl(\varrho (\Phi )+ \varrho (\Psi )\bigr)^{s^{-}/2(q^{-}+2)} +\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{s^{+}/2(q^{-}+2)} \bigr). \end{aligned}$$

(4.13)

Proof

By (1.5), we get

$$\begin{aligned} \int _{\Omega} \vert \Phi \vert ^{m(x)}\,dx \leq & \int _{\Omega _{1}} \vert \Phi \vert ^{m^{+}}\,dx+ \int _{\Omega _{2}} \vert \Phi \vert ^{m^{-}}\,dx \\ \leq &c \biggl( \int _{\Omega _{1}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx \biggr)^{\frac{m^{+}}{2(q^{-}+2)}}+c \biggl( \int _{\Omega _{2}} \vert \Phi \vert ^{2(q^{-}+2)}\,dx \biggr)^{\frac{m^{-}}{2(q^{-}+2)}} \\ \leq &c \bigl( \Vert \Phi \Vert ^{m^{+}}_{2(q^{-}+2)}+ \Vert \Phi \Vert ^{m^{-}}_{2(q^{-}+2)} \bigr). \end{aligned}$$

(4.14)

Then, Lemma 4.2 gives (4.13)₁. Similarly, we get (4.13)₂. □

Here, we introduce the following new functional:

$$\begin{aligned} \mathfrak{D}(t) =&\mathbb{H}^{1-\alpha}(t)+ \frac{\varepsilon}{\eta +1} \int _{\Omega} \bigl[\Phi \vert \Phi _{t} \vert ^{\eta}\Phi _{t}+\Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t} \bigr]\,dx \\ &{}+\varepsilon \int _{\Omega} [\nabla \Phi _{t}\nabla \Phi + \nabla \Psi _{t}\nabla \Psi ]\,dx, \end{aligned}$$

(4.15)

where $0<\varepsilon$ will be considered later, and take

$$\begin{aligned} 0< \alpha < &\min \biggl\{ \biggl(1-\frac {1}{2(q^{-}+2)}- \frac{1}{\eta +2} \biggr),\frac{1+2\gamma}{4(\gamma +1)}, \frac{2q^{-}+4-m^{-}}{(2q^{-}+4)(m^{+}-1)}, \\ &{} \frac{2q^{-}+4-m^{+}}{(2q^{-}+4)(m^{+}-1)}, \frac{2q^{-}+4-r^{+}}{(2q^{-}+4)(s^{+}-1)}, \frac{2q^{-}+4-s^{-}}{(2q^{-}+4)(s^{+}-1)} \biggr\} < 1. \end{aligned}$$

(4.16)

By multiplying (1.1)₁, (1.1)₂ by Φ, Ψ and with the help of (4.15), we obtain

$$\begin{aligned} \mathfrak{D}'(t) =&(1-\alpha )\mathbb{H}^{-\alpha} \mathbb{H}'(t)+ \frac{\varepsilon}{\eta +1}\bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr) \\ &{}+ \underbrace{\varepsilon \int _{\Omega}\nabla \Phi \int ^{t}_{0}g(t-\varsigma ) \nabla \Phi ( \varsigma )\,d\varsigma \,dx}_{J_{1}} + \underbrace{\varepsilon \int _{\Omega}\nabla \Psi \int ^{t}_{0}h(t-\varsigma ) \nabla \Psi ( \varsigma )\,d\varsigma \,dx}_{J_{2}} \\ &{}- \underbrace{\varepsilon \zeta _{1} \int _{\Omega} \Phi \Phi _{t} \vert \Phi _{t} \vert ^{m(x)-2} \,dx}_{J_{3}}- \underbrace{ \varepsilon \zeta _{3} \int _{\Omega} \Psi \Psi _{t} \vert \Psi _{t} \vert ^{s(x)-2} \,dx}_{J_{4}} \\ &{}-\varepsilon \bigl( \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2}\bigr)-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+ \underbrace{\varepsilon \int _{\Omega}\bigl(\Phi f_{1}(\Phi ,\Psi )+\Psi f_{2}(\Phi ,\Psi )\bigr)\,dx}_{J_{5}}. \end{aligned}$$

By (2.1), we obtain

$$\begin{aligned} \mathfrak{D}'(t) \geq &(1-\alpha )\mathbb{H}^{-\alpha} \mathbb{H}'(t)+ \frac{\varepsilon}{\eta +1}\bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr) \\ &{}+ \underbrace{\varepsilon \int _{\Omega}\nabla \Phi \int ^{t}_{0}h_{1}(t-\varsigma ) \nabla \Phi (\varsigma )\,d\varsigma \,dx}_{J_{1}} + \underbrace{ \varepsilon \int _{\Omega}\nabla \Psi \int ^{t}_{0}h_{2}(t-\varsigma ) \nabla \Psi (\varsigma )\,d\varsigma \,dx}_{J_{2}} \\ &{}- \underbrace{\varepsilon \zeta _{1} \int _{\Omega} \Phi \Phi _{t} \vert \Phi _{t} \vert ^{m(x)-2} \,dx}_{J_{3}}- \underbrace{ \varepsilon \zeta _{3} \int _{\Omega} \Psi \Psi _{t} \vert \Psi _{t} \vert ^{s(x)-2} \,dx}_{J_{4}} \\ &{}-\varepsilon \bigl( \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2}\bigr)-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\underbrace{\varepsilon \bigl(2q^{-}+4\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx}_{J_{5}}. \end{aligned}$$

(4.17)

We have

$$\begin{aligned} &J_{1}=\varepsilon \int _{0}^{t}h_{1}(t-\varsigma ) \,d\varsigma \int _{ \Omega}\nabla \Phi .\bigl(\nabla \Phi (\varsigma )-\nabla \Phi (t)\bigr)\,dx\,d \varsigma +\varepsilon \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \Vert \nabla \Phi \Vert _{2}^{2} \\ &\hphantom{J_{1}}\geq \frac{\varepsilon}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \Vert \nabla \Phi \Vert _{2}^{2}- \frac{\varepsilon}{2}(h_{1}o \nabla \Phi ). \end{aligned}$$

(4.18)

$$\begin{aligned} &J_{2}=\varepsilon \int _{0}^{t}h_{2}(t-\varsigma )\,d \varsigma \int _{ \Omega}\nabla \Psi .\bigl(\nabla \Psi (\varsigma )-\nabla \Psi (t)\bigr)\,dx\,d \varsigma +\varepsilon \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \Vert \nabla \Psi \Vert _{2}^{2} \\ &\hphantom{J_{2}}\geq \frac{\varepsilon}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \Vert \nabla \Psi \Vert _{2}^{2}- \frac{\varepsilon}{2}(h_{2}o \nabla \Psi ). \end{aligned}$$

(4.19)

From (4.17), we find

$$\begin{aligned} \mathfrak{D}'(t) \geq &(1-\alpha )\mathbb{H}^{-\alpha} \mathbb{H}'(t)+ \frac{\varepsilon}{\eta +1}\bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}-\frac{\varepsilon}{2}(h_{1}o\nabla \Phi )-\frac{\varepsilon}{2}(h_{2}o \nabla \Psi )-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+J_{3}+J_{4}+J_{5}. \end{aligned}$$

(4.20)

At this stage we apply Young’s inequality, which gives us for $\delta _{1},\delta _{2}>0$ the following:

$$\begin{aligned} &J_{3} \leq \varepsilon \zeta _{1} \biggl\{ \frac{1}{m^{-}} \int _{ \Omega}\delta _{1}^{m(x)} \vert \Phi \vert ^{m(x)}\,dx+ \frac{m^{+}-1}{m^{+}} \int _{\Omega}\delta _{1}^{-\frac{m(x)}{m(x)-1}} \vert \Phi _{t} \vert ^{m(x)} \,dx \biggr\} , \end{aligned}$$

(4.21)

$$\begin{aligned} &J_{4} \leq \varepsilon \zeta _{3} \biggl\{ \frac{1}{s^{-}} \int _{ \Omega}\delta _{2}^{s(x)} \vert \Psi \vert ^{s(x)}\,dx+ \frac{s^{+}-1}{s^{+}} \int _{\Omega}\delta _{2}^{-\frac{s(x)}{s(x)-1}} \vert \Psi _{t} \vert ^{s(x)} \,dx \biggr\} . \end{aligned}$$

(4.22)

Therefore, by setting $\delta _{1}$, $\delta _{2}$ so that

$$\begin{aligned} \delta _{1}^{-\frac{m(x)}{m(x)-1}}=\zeta _{1}\kappa \mathbb{H}^{- \alpha}(t), \qquad \delta _{2}^{-\frac{s(x)}{s(x)-1}}= \zeta _{3}\kappa \mathbb{H}^{- \alpha}(t), \end{aligned}$$

(4.23)

substituting the previous two equalities into (4.20) gives us the following inequality:

$$\begin{aligned} \mathfrak{D}'(t) \geq &\bigl[(1-\alpha )-\varepsilon \kappa ( \widehat{m}+ \widehat{s})\bigr]\mathbb{H}^{-\alpha}\mathbb{H}'(t)+ \frac{\varepsilon}{\eta +1} \bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{ \eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr)-\frac{\varepsilon}{2}(h_{1}o \nabla \Phi )-\frac{\varepsilon}{2}(h_{2}o\nabla \Psi ) \\ &{}-\varepsilon \frac{\zeta _{1}}{ m^{-}} \int _{\Omega}(\zeta _{1} \kappa )^{1-m(x)} \mathbb{H}^{\alpha (m(x)-1)}(t) \vert \Phi \vert ^{m(x)}\,dx \\ &{}-\varepsilon \frac{\zeta _{3}}{ s^{-}} \int _{\Omega}(\zeta _{3} \kappa )^{1-s(x)}\mathbb{H}^{\alpha (s(x)-1)}(t) \vert \Psi \vert ^{s(x)}\,dx \\ &{}-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr)+J_{5}, \end{aligned}$$

(4.24)

where $\widehat{m}=\frac{m^{+}-1}{m^{-}}$, $\widehat{s}=\frac{s^{+}-1}{s^{-}}$.

Now, by using (4.5) and (4.13)₁, we have

$$\begin{aligned} & \frac{\zeta _{1}}{ m^{-}} \int _{\Omega}(\zeta _{1}\kappa )^{1-m(x)} \mathbb{H}^{\alpha (m(x)-1)}(t) \vert \Phi \vert ^{m(x)}\,dx \\ &\quad \leq \frac{\zeta _{1}}{ m^{-}} \int _{\Omega}(\zeta _{1}\kappa )^{1-m^{-}} \mathbb{H}^{\alpha (m^{+}-1)}(t) \vert \Phi \vert ^{m(x)}\,dx \\ &\quad =C_{1}\mathbb{H}^{\alpha (m^{+}-1)}(t) \int _{\Omega} \vert \Phi \vert ^{m(x)}\,dx \\ &\quad \leq C_{2} \bigl\{ \bigl(\varrho (\Phi )+\varrho (\Psi ) \bigr)^{ \frac{m^{-}}{2(q^{-}+2)}+\alpha (m^{+}-1)} \\ &\qquad \times\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{\frac{m^{+}}{2(q^{-}+2)}+\alpha (m^{+}-1)} \bigr\} . \end{aligned}$$

(4.25)

By (4.16), we find

$$\begin{aligned} &r=m^{-}+\alpha \bigl(2 q^{-}+4\bigr) \bigl(m^{+}-1\bigr)\leq \bigl(2q^{-}+4\bigr), \\ &r=m^{+}+\alpha \bigl(2 q^{-}+4\bigr) \bigl(m^{+}-1\bigr)\leq \bigl(2 q^{-}+4\bigr). \end{aligned}$$

We use the following inequality to advance the proof:

$$ Z^{\gamma}\leq Z+1\leq \biggl(1+\frac{1}{v}\biggr) (Z+v), \quad \forall Z\geq 0, 0< \gamma \leq 1, v>0, $$

(4.26)

with $v=\frac{1}{\mathbb{H}(0)}$. Then we have

$$\begin{aligned} \bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{\frac{m^{-}}{2(q^{-}+2)}+\alpha (m^{+}-1)} \leq &\biggl(1+ \frac{1}{\mathbb{H}(0)}\biggr) \bigl(\bigl(\varrho (\Phi )+\varrho ( \Psi )\bigr)+ \mathbb{H}(0) \bigr) \\ \leq &C_{3} \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{H}(t) \bigr) \end{aligned}$$

(4.27)

and

$$\begin{aligned} \bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{\frac{m^{+}}{2(q^{-}+2)}+\alpha (m^{+}-1)} \leq C_{3} \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{H}(t) \bigr), \end{aligned}$$

(4.28)

where $C_{3}=1+\frac{1}{\mathbb{H}(0)}$. Substituting (4.27) and (4.28) into (4.25), we get

$$\begin{aligned} & \frac{\zeta _{1}}{ m^{-}} \int _{\Omega}(\zeta _{1}\kappa )^{1-m(x)} \mathbb{H}^{\alpha (m(x)-1)}(t) \vert \Phi \vert ^{m(x)}\,dx \\ &\quad \leq C_{4} \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{H}(t) \bigr). \end{aligned}$$

(4.29)

Similarly, we get

$$\begin{aligned} & \frac{\zeta _{3}}{ s^{-}} \int _{\Omega}(\zeta _{3}\kappa )^{1-s(x)} \mathbb{H}^{\alpha (s(x)-1)}(t) \vert \Psi \vert ^{s(x)}\,dx \\ &\quad \leq C_{5} \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{H}(t) \bigr), \end{aligned}$$

(4.30)

where $C_{4}=C_{4}(\kappa )=C_{3}\frac{\zeta _{1}}{m^{-}}(\zeta _{1}\kappa )^{1-m^{-}}$, $C_{5}=C_{5}(\kappa )=C_{3}\frac{\zeta _{3})}{ s^{-}}(\zeta _{3} \kappa )^{1-s^{-}}$.

At this stage, combining (4.29), (4.30), and (4.24), and by (2.3) we find

$$\begin{aligned} \mathfrak{D}'(t) \geq &\bigl[(1-\alpha )-\varepsilon \kappa ( \widehat{m}+ \widehat{s})\bigr]\mathbb{H}^{-\alpha}\mathbb{H}'(t)+ \frac{\varepsilon}{\eta +1} \bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{ \eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr)-\frac{\varepsilon}{2}(h_{1}o \nabla \Phi )-\frac{\varepsilon}{2}(h_{2}o\nabla \Psi )+ J_{5} \\ &{}-\varepsilon ( C_{4}+C_{5}) \bigl(\bigl(\varrho ( \Phi )+\varrho (\Psi )\bigr)+ \mathbb{H}(t) \bigr)-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr). \end{aligned}$$

(4.31)

Now, for $0< a<1$, from (4.1) and (2.3) we have

$$\begin{aligned} J_{5} =&\varepsilon \bigl(2q^{-}+4\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx \\ =&\varepsilon a\bigl(2q^{-}+4\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx+\varepsilon (1-a) \bigl(2q^{-}+4\bigr) \mathbb{H}(t) \\ &{}+\frac{\varepsilon (1-a)(2q^{-}+4)}{\eta +2}\bigl( \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2} \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \bigl((h_{1}o \nabla \Phi )+(h_{2}o\nabla \Psi )\bigr) \\ &{}+\frac{\varepsilon (1-a)(q^{-}+2)}{(\gamma +1)}\bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr). \end{aligned}$$

(4.32)

Substituting (4.32) in (4.31) and applying (2.3) gives

$$\begin{aligned} \mathfrak{D}'(t) \geq & \bigl\{ (1-\alpha )-\varepsilon \kappa ( \widehat{m}+\widehat{s}) \bigr\} \mathbb{H}^{-\alpha}\mathbb{H}'(t) \\ &{}+\varepsilon \bigl\{ (1-a) \bigl(q^{-}+2\bigr)+1 \bigr\} \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr) \\ &{}+\varepsilon \biggl\{ \frac{\varepsilon (1-a)(2q^{-}+4)}{\eta +2}+ \frac{1}{\eta +1} \biggr\} \bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon \biggl\{ (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggr) \biggr\} \Vert \nabla \Phi \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \biggr\} \Vert \nabla \Psi \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (1-a) \bigl(q^{-}+2\bigr)- \frac{1}{2} \biggr\} (h_{1}o\nabla \Phi +h_{2}o \nabla \Psi ) \\ &{}+\varepsilon \biggl\{ \frac{(1-a)(q^{-}+2)}{\gamma +1}-1 \biggr\} \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)}\bigr) \\ &{}+\varepsilon \bigl\{ c_{0}a- \bigl(C_{4}(\kappa )+C_{5}(\kappa ) \bigr) \bigr\} \bigl(\varrho (\Phi )+\varrho (\Psi ) \bigr) \\ &{}+\varepsilon \bigl\{ (1-a) \bigl(2q^{-}+4\bigr)- \bigl(C_{4}(\kappa )+C_{5}( \kappa ) \bigr) \bigr\} \mathbb{H}(t). \end{aligned}$$

(4.33)

By choosing $0< a$ so small that

$$ \bigl(q^{-}+2\bigr) (1-a)>1+\gamma , $$

we have

$$\begin{aligned} &\lambda _{1}:=\bigl(q^{-}+2\bigr) (1-a)-1>0, \\ &\lambda _{2}:=\bigl(q^{-}+2\bigr) (1-a)- \frac{1}{2}>0, \\ &\lambda _{3}:=\frac{(q^{-}+2)(1-a)}{\gamma +1}-1>0. \end{aligned}$$

At this moment we present this supposition

$$ \max \biggl\{ \int _{0}^{\infty}h_{1}(\varsigma )\,d \varsigma , \int _{0}^{ \infty}h_{2}(\varsigma )\,d \varsigma \biggr\} < \frac {(q^{-}+2)(1-a)-1}{((q^{-}+2)(1-a)-\frac {1}{2})}= \frac {2\lambda _{1}}{2\lambda _{1}+1} $$

(4.34)

gives

$$\begin{aligned} &\lambda _{4}=\biggl\{ \bigl(\bigl(q^{-}+2\bigr) (1-a)-1 \bigr)- \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggl(\bigl(q^{-}+2\bigr) (1-a)-\frac{1}{2} \biggr) \biggr\} >0, \\ &\lambda _{5}= \biggl\{ \bigl(\bigl(q^{-}+2\bigr) (1-a)-1 \bigr)- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggl(\bigl(q^{-}+2\bigr) (1-a)-\frac{1}{2} \biggr) \biggr\} >0. \end{aligned}$$

Next, we choose κ large enough such that

$$\begin{aligned} &\lambda _{6}=ac_{0}- \bigl(C_{4}( \kappa )+C_{5}(\kappa ) \bigr)>0, \\ &\lambda _{7}=2\bigl(q^{-}+2\bigr) (1-a)- \bigl(C_{4}(\kappa )+C_{5}(\kappa ) \bigr)>0. \end{aligned}$$

At this point, take κ, a, and we pick ε in a way that

$$ \lambda _{8}=(1-\alpha )-\varepsilon \kappa (\widehat{m}+ \widehat{s})>0 $$

and

$$\begin{aligned} \mathfrak{D}(0) =&\mathbb{H}^{1-\alpha}(0)+ \frac{\varepsilon}{\eta +1} \int _{\Omega} \bigl[\Phi _{0} \vert \Phi _{1} \vert ^{\eta}\Phi _{1}+\Psi _{0} \vert \Psi _{1} \vert ^{\eta}\Psi _{1} \bigr]\,dx \\ &{}+\varepsilon \int _{\Omega} [\nabla \Phi _{1}\nabla \Phi _{0}+ \nabla \Psi _{1}\nabla \Psi _{0} ] \,dx>0. \end{aligned}$$

(4.35)

Hence, from (4.33) we deduce for some $\mu >0$

$$\begin{aligned} \mathfrak{D}'(t) \geq &\mu \bigl\{ \mathbb{H}(t)+ \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)} \\ &{}+ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} + \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2} +(h_{1}o \nabla \Phi )+(h_{2}o\nabla \Psi ) \\ &{}+\varrho (\Phi )+\varrho (\Psi ) \bigr\} \end{aligned}$$

(4.36)

and

$$ \mathfrak{D}(t)\geq \mathfrak{D}(0)>0, \quad t>0. $$

(4.37)

Next, by Hölder’s and Young’s inequalities, we find

$$\begin{aligned} \biggl\vert \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta} \Phi _{t}+ \Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr)\,dx \biggr\vert ^{ \frac{1}{1-\alpha}} \leq &C \bigl[ \Vert \Phi \Vert _{2(q^{-}+2)}^{ \frac{\theta}{1-\alpha}}+ \Vert \Phi _{t} \Vert _{\eta +2}^{ \frac{\mu}{1-\alpha}} \\ & {} + \Vert \Psi \Vert _{2(q^{-}+2)}^{\frac{\theta}{1-\alpha}}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\frac{\mu}{1-\alpha}} \bigr], \end{aligned}$$

(4.38)

where $\frac{1}{\mu}+\frac{1}{\theta}=1$.

Pick $\mu =(\eta +2)(1-\alpha )$ to get the below

$$ \frac{\theta}{1-\alpha}=\frac{\eta +2}{(1-\alpha )(\eta +2)-1}\leq 2\bigl(q^{-}+2 \bigr). $$

Consequently, by the application of (4.5), (4.16), and (4.26), we have

$$\begin{aligned} &\Vert \Phi \Vert _{2(q^{-}+2)}^{ \frac{\eta +2}{(1-\alpha )(\eta +2)-1}}\leq d\bigl( \Vert \Phi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)}+ \mathbb{H}(t)\bigr) \\ &\Vert \Psi \Vert _{2(q^{-}+2)}^{ \frac{\eta +2}{(1-\alpha )(\eta +2)-1}}\leq d\bigl( \Vert \Psi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)}+ \mathbb{H}(t)\bigr), \quad \forall t\geq 0. \end{aligned}$$

Then we have

$$\begin{aligned} & \biggl\vert \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta} \Phi _{t}+ \Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr)\,dx \biggr\vert ^{ \frac{1}{1-\alpha}} \\ &\quad \leq c \bigl\{ \varrho (\Phi )+\varrho (\Psi )+ \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \mathbb{H}(t) \bigr\} . \end{aligned}$$

(4.39)

In the same way, we have

$$\begin{aligned} \biggl\vert \int _{\Omega}(\nabla \Phi \nabla \Phi _{t}+\nabla \Psi \nabla \Psi _{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha}} \leq &C \bigl[ \Vert \nabla \Phi \Vert _{2}^{\frac{\theta}{1-\alpha}}+ \Vert \nabla \Phi _{t} \Vert _{2}^{\frac{\mu}{1-\alpha}} \\ & {} + \Vert \nabla \Psi \Vert _{2}^{\frac{\theta}{1-\alpha}}+ \Vert \nabla \Psi _{t} \Vert _{2}^{\frac{\mu}{1-\alpha}} \bigr], \end{aligned}$$

where $\frac{1}{\mu}+\frac{1}{\theta}=1$.

In this, by assuming $\theta =2(\gamma +1)(1-\alpha )$, we get

$$\begin{aligned} &\frac{\mu}{1-\alpha}=\frac{2(\gamma +1)}{2(1-\alpha )(\gamma +1)-1} \leq 2 \\ &\biggl\vert \int _{\Omega}(\nabla \Phi \nabla \Phi _{t}+\nabla \Psi \nabla \Psi _{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha}}\leq c \bigl\{ \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert ^{2( \gamma +1)}_{2} \\ &\hphantom{\biggl\vert \int _{\Omega}(\nabla \Phi \nabla \Phi _{t}+\nabla \Psi \nabla \Psi _{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha}}}\quad + \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr\} . \end{aligned}$$

(4.40)

Thus, by (4.39) and (4.40), we have

$$\begin{aligned} \mathfrak{D}^{\frac{1}{1-\alpha}}(t) =& \biggl(\mathbb{H}^{1-\alpha}(t)+ \frac{\varepsilon}{\eta +1} \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{ \eta}\Phi _{t}+\Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr)\,dx \\ &{}+\varepsilon \int _{\Omega}(\nabla \Phi _{t}\nabla \Phi +\nabla \Psi _{t}\nabla \Psi )\,dx \biggr)^{\frac{1}{1-\alpha}} \\ \leq &c \biggl(\mathbb{H}(t)+ \biggl\vert \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta} \Phi _{t}+\Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr) \,dx \biggr\vert ^{\frac{1}{1-\alpha}}+ \Vert \nabla \Phi \Vert _{2}^{ \frac{2}{1-\alpha}}+ \Vert \nabla \Psi \Vert _{2}^{\frac{2}{1-\alpha}} \\ & {} + \Vert \nabla \Phi _{t} \Vert _{2}^{\frac{2}{1-\alpha}}+ \Vert \nabla \Psi _{t} \Vert _{2}^{\frac{2}{1-\alpha}} \biggr) \\ \leq &c \bigl(\mathbb{H}(t)+ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla \Phi \Vert ^{2( \gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla \Phi _{t} \Vert _{2}^{2} \\ &{}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}+(h_{1}o \nabla \Phi )+(h_{2}o \nabla \Psi )+\varrho (\Phi )+\varrho (\Psi ) \bigr) \\ \leq &c \bigl\{ \mathbb{H}(t)+ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla \Phi \Vert ^{2( \gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \\ &{}+ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}+ \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2}+(h_{1}o \nabla \Phi )+(h_{2}o\nabla \Psi ) \\ &{}+\varrho (\Phi )+\varrho (\Psi ) \bigr\} . \end{aligned}$$

(4.41)

Now, (4.36) and (4.41) imply

$$ \mathfrak{D}'(t)\geq \lambda \mathfrak{D}^{\frac{1}{1-\alpha}}(t), $$

(4.42)

where $0< \lambda$, this relies only on β and c.

Further simplification of (4.42) leads us to

$$ \mathfrak{D}^{\frac{\alpha}{1-\alpha}}(t)\geq \frac{1}{\mathfrak{D}^{\frac{-\alpha}{1-\alpha}}(0)-\lambda \frac{\alpha}{(1-\alpha )} t}. $$

Hence, $\mathfrak{D}(t)$ blows up in time

$$ T\leq T^{*}= \frac{1-\alpha}{\lambda \alpha \mathfrak{D}^{\alpha /(1-\alpha )}(0)}. $$

This ends the proof. □

5 Growth of solution

Here, the exponential growth of solution of problem (1.1) is established with positive initial energy.

First, based on Theorem 3.3, we have a solution that is global in time. To achieve the objectives of our results in this section, we first introduce the following function:

$$\begin{aligned} \begin{aligned} \Upsilon (t):={}&\biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2}+(h_{1} \circ \nabla \Phi )\\ &+(h_{2} \circ \nabla \Psi ). \end{aligned} \end{aligned}$$

(5.1)

Then, we present the following lemma, which is similar to the one presented first by Vitillaro [28] to study a class of single wave equations, also see [27].

Lemma 5.1

Suppose that (2.1) and (2.2) hold. Let $(u, v,z,y)$ be a solution of (1.1). Assume further that

$$ \Vert \nabla \Phi _{0} \Vert _{2}^{2}+ \Vert \nabla \Psi _{0} \Vert _{2}^{2}> \alpha _{1}^{2}, \qquad \mathcal{E}(0) < d_{1}. $$

(5.2)

Then there exists a constant $\alpha _{2} > \alpha _{1}$ such that

$$ \Upsilon (t)>\alpha _{2}^{2} $$

(5.3)

and

$$ 2\bigl(p^{-}+2\bigr) \int _{0}^{L}F(\Phi ,\Psi )\,dx\geq (B\alpha _{2})^{2(p^{+}+2)} $$

(5.4)

with

$$\begin{aligned} &B= \biggl(\frac{c_{1}C_{*}(p^{+})}{l^{(p^{+}+2)}} \biggr)^{ \frac{1}{2(p^{+}+2)}}, \quad \alpha _{1}=B^{-\frac{p^{+}+2}{p^{+}+1}}, \\ & d_{1}= \biggl[\frac{1}{2(p^{-}+2)(1-a)}-\frac{1}{2(p^{-}+2)} \biggr] \alpha _{1}^{2}. \end{aligned}$$

(5.5)

We also know which number $d= [\frac{1}{2}-\frac{1}{2(p^{-}+2)} ]\alpha _{1}^{2}>d_{1}> \mathcal{E}(0)$.

Moreover, one can easily see that, from (5.5), the condition $\mathcal{E}(0) < d_{1}$ is equivalent to inequality (3.4).

Since $0< a<1$, we will appoint it later, we have $2<2(p^{-}+2)(1-a)<2(p^{-}+2)$.

For this purpose, we set the functional

$$\begin{aligned} \mathbb{T}(t)=d_{1}-\mathcal{E}(t) =&d_{1}- \frac{1}{\eta +2} \bigl[ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{ \eta +2}^{\eta +2} \bigr]-\frac{1}{2} \bigl[ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr] \\ &{}-\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla \Phi \Vert _{2}^{2( \gamma +1)}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr] \\ &{}-\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}-\frac{1}{2} \bigl[(h_{1}o\nabla \Phi ) (t)+(h_{2}o\nabla \Psi ) (t) \bigr]+ \int _{\Omega}F(\Phi ,\Psi )\,dx. \end{aligned}$$

(5.6)

Theorem 5.2

Assume that (2.1)–(2.2) are satisfied and $\mathcal{E}(0)< d_{1}$, then

$$ 2\bigl(q^{-}+2\bigr)>\frac{\eta +2}{\eta +1}. $$

(5.7)

Then the solution of problem (1.1) grows exponentially.

Proof

To achieve our goal, by (2.5) we first deduce

$$ \mathcal{E}(t)\leq \mathcal{E}(0)< d_{1}, $$

(5.8)

with the help of (4.3) and (4.4) and (5.2), (2.3), we have

$$\begin{aligned} 0\leq \mathbb{T}(0)\leq \mathbb{T}(t) \leq &d_{1}- \frac{1}{2}\alpha _{1}^{2}+ \frac{1}{2(p^{-}+2)} \bigl[ \Vert \Phi +\Psi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)}+2 \Vert \Phi \Psi \Vert _{(p^{-}+2)}^{p^{-}+2} \bigr] \\ \leq &d-\frac{1}{2}\alpha _{1}^{2}+ \frac{c_{1}}{2(p^{-}+2)} \bigl[ \Vert \Phi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)}+ \Vert \Psi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)} \bigr] \\ \leq &-\frac{1}{2(p^{-}+2)}\alpha _{1}^{2}+ \frac{c_{1}}{2(p^{-}+2)} \bigl[ \Vert \Phi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)}+ \Vert \Psi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)} \bigr] \\ \leq &\frac{c_{1}}{2(p^{-}+2)} \bigl[ \Vert \Phi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)}+ \Vert \Psi \Vert _{2(p^{-}+2)}^{2(p^{-}+2)} \bigr] . \end{aligned}$$

(5.9)

In this, we introduce the functional

$$\begin{aligned} \mathfrak{R}(t) =&\mathbb{T}(t)+\frac{\varepsilon}{\eta +1} \int _{ \Omega} \bigl[\Phi \vert \Phi _{t} \vert ^{\eta}\Phi _{t}+\Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t} \bigr]\,dx \\ &{}+\varepsilon \int _{\Omega} [\nabla \Phi _{t}\nabla v+\nabla \Psi _{t}\nabla \Psi ]\,dx, \end{aligned}$$

(5.10)

where $\varepsilon >0$.

From (1.1)₁, (1.1)₂, and (5.10), we get

$$\begin{aligned} \mathfrak{R}'(t) =&\mathbb{T}'(t)+ \frac{\varepsilon}{\eta +1}\bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{ \eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon \int _{\Omega}\nabla \Phi \int ^{t}_{0}h_{1}(t- \varsigma ) \nabla \Phi (\varsigma )\,d\varsigma \,dx +\varepsilon \int _{ \Omega}\nabla \Psi \int ^{t}_{0}h_{2}(t-\varsigma ) \nabla \Psi ( \varsigma )\,d\varsigma \,dx \\ &{}-\varepsilon \zeta _{1} \int _{\Omega} \Phi \Phi _{t} \vert \Phi _{t} \vert ^{m(x)-2} \,dx-\varepsilon \zeta _{3} \int _{\Omega} \Psi \Psi _{t} \vert \Psi _{t} \vert ^{s(x)-2} \,dx \\ &{}-\varepsilon \bigl( \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2}\bigr)-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\varepsilon \int _{\Omega}\bigl(\Phi f_{1}(\Phi ,\Psi )+\Psi f_{2}( \Phi ,\Psi )\bigr)\,dx. \end{aligned}$$

By (2.1), we find

$$\begin{aligned} \mathfrak{R}'(t) \geq &\mathbb{T}'(t)+ \frac{\varepsilon}{\eta +1}\bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{ \eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}\bigr) \\ &{}+ \underbrace{\varepsilon \int _{\Omega}\nabla \Phi \int ^{t}_{0}h_{1}(t-\varsigma ) \nabla \Phi (\varsigma )\,d\varsigma \,dx}_{I_{1}} + \underbrace{ \varepsilon \int _{\Omega}\nabla \Psi \int ^{t}_{0}h_{2}(t-\varsigma ) \nabla \Psi (\varsigma )\,d\varsigma \,dx}_{I_{2}} \\ &{}- \underbrace{\varepsilon \zeta _{1} \int _{\Omega} \Phi \Phi _{t} \vert \Phi _{t} \vert ^{m(x)-2} \,dx}_{I_{3}}- \underbrace{ \varepsilon \zeta _{3} \int _{\Omega} \Psi \Psi _{t} \vert \Psi _{t} \vert ^{s(x)-2} \,dx}_{I_{4}} \\ &{}-\varepsilon \bigl( \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2}\bigr)-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\underbrace{\varepsilon \bigl(2q^{-}+4\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx}_{I_{5}}. \end{aligned}$$

(5.11)

Similarly to $J_{1}$, $J_{2}$ in (4.18) and (4.19), we estimate $I_{1}$, $I_{2}$:

$$\begin{aligned} &I_{1}=J_{1}\geq \frac{\varepsilon}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \Vert \nabla \Phi \Vert _{2}^{2}- \frac{\varepsilon}{2}(h_{1}o \nabla \Phi ), \end{aligned}$$

(5.12)

$$\begin{aligned} &I_{2}=J_{2}\geq \frac{\varepsilon}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \Vert \nabla \Psi \Vert _{2}^{2}- \frac{\varepsilon}{2}(h_{2}o \nabla \Psi ). \end{aligned}$$

(5.13)

From (5.11), we find

$$\begin{aligned} \mathfrak{R}'(t) \geq &\mathbb{T}'(t)+ \frac{\varepsilon}{\eta +1}\bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{ \eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}\bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}-\frac{\varepsilon}{2}(h_{1}o\nabla \Phi )-\frac{\varepsilon}{2}(h_{2}o \nabla \Psi )-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+I_{3}+I_{4}+I_{5} . \end{aligned}$$

(5.14)

Similar to $J_{3}$ and $J_{4}$ in (4.21)–(4.22), we estimate $I_{3}$ and $I_{4}$. By Young’s inequality, we find for $\delta _{1},\delta _{2}>0$

$$\begin{aligned} I_{3} \leq &\varepsilon \zeta _{1} \biggl\{ \frac{1}{m^{-}} \int _{ \Omega}\delta _{1}^{m(x)} \vert \Phi \vert ^{m(x)}\,dx+ \frac{m^{+}-1}{m^{+}} \int _{\Omega}\delta _{1}^{-\frac{m(x)}{m(x)-1}} \vert \Phi _{t} \vert ^{m(x)} \,dx \biggr\} \end{aligned}$$

(5.15)

and

$$\begin{aligned} I_{4} \leq &\varepsilon \zeta _{3} \biggl\{ \frac{1}{s^{-}} \int _{ \Omega}\delta _{2}^{s(x)} \vert \Psi \vert ^{s(x)}\,dx+ \frac{s^{+}-1}{s^{+}} \int _{\Omega}\delta _{2}^{-\frac{s(x)}{s(x)-1}} \vert \Psi _{t} \vert ^{s(x)} \,dx \biggr\} . \end{aligned}$$

(5.16)

Therefore, by setting $\delta _{1}$, $\delta _{2}$ so that

$$\begin{aligned} \delta _{1}^{-\frac{m(x)}{m(x)-1}}=\frac{\zeta _{1}}{2}\kappa , \qquad \delta _{2}^{-\frac{s(x)}{s(x)-1}}=\frac{\zeta _{3}}{2}\kappa , \end{aligned}$$

(5.17)

substituting in (5.14), we obtain

$$\begin{aligned} \mathfrak{R}'(t) \geq &\bigl[1-\varepsilon \kappa (\widehat{m}+ \widehat{s})\bigr] \mathbb{T}'(t)+\frac{\varepsilon}{\eta +1} \bigl( \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr)-\frac{\varepsilon}{2}(h_{1}o \nabla \Phi )-\frac{\varepsilon}{2}(h_{2}o\nabla \Psi ) \\ &{}-\varepsilon \frac{\zeta _{1}}{m^{-}} \int _{\Omega}\biggl( \frac{\zeta _{1}\kappa}{2}\biggr)^{1-m(x)} \vert \Phi \vert ^{m(x)}\,dx- \varepsilon \frac{\zeta _{3}}{ s^{-}} \int _{\Omega}\biggl( \frac{\zeta _{3}\kappa}{2}\biggr)^{1-s(x)} \vert \Psi \vert ^{s(x)}\,dx \\ &{}-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)} \bigr)+I_{5}, \end{aligned}$$

(5.18)

where $\widehat{m}=\frac{m^{+}-1}{m^{-}}$, $\widehat{s}=\frac{s^{+}-1}{s^{-}}$. By using (4.5) and (4.13), we have

$$\begin{aligned} \frac{\zeta _{1}}{ m^{-}} \int _{\Omega}\biggl(\frac{\zeta _{1}\kappa}{2}\biggr)^{1-m(x)} \vert \Phi \vert ^{m(x)}\,dx \leq &\frac{\zeta _{1}}{ m^{-}} \int _{ \Omega}\biggl(\frac{\zeta _{1}\kappa}{2}\biggr)^{1-m^{-}} \vert \Phi \vert ^{m(x)}\,dx \\ =&C_{8} \int _{\Omega} \vert \Phi \vert ^{m(x)}\,dx \\ \leq &C_{9} \bigl\{ \bigl(\varrho (\Phi )+\varrho (\Psi ) \bigr)^{ \frac{m^{-}}{2(q^{-}+2)}} \\ &{}+\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{\frac{m^{+}}{2(q^{-}+2)}} \bigr\} . \end{aligned}$$

(5.19)

By (1.5), we get

$$\begin{aligned} r=m^{-}\leq \bigl(2q^{-}+4\bigr), \qquad r=m^{+}\leq \bigl(2 q^{-}+4\bigr) \end{aligned}$$

and by (4.26) with $v=\frac{1}{\mathbb{T}(0)}$. Then we have

$$\begin{aligned} \bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{\frac{m^{-}}{2(q^{-}+2)}} \leq &\biggl(1+ \frac{1}{\mathbb{T}(0)}\biggr) \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{T}(0) \bigr) \\ \leq &C_{10} \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{T}(t) \bigr) \end{aligned}$$

(5.20)

and

$$\begin{aligned} \bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)^{\frac{m^{+}}{2(q^{-}+2)}}\leq C_{10} \bigl(\bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \mathbb{T}(t) \bigr), \end{aligned}$$

(5.21)

where $C_{10}=1+\frac{1}{\mathbb{T}(0)}$. Substituting (5.20) and (5.21) into (5.19), we get

$$\begin{aligned} \frac{\zeta _{1}}{ m^{-}} \int _{\Omega}\biggl(\frac{\zeta _{1}\kappa}{2}\biggr)^{1-m(x)} \vert \Phi \vert ^{m(x)}\,dx\leq C_{11} \bigl(\bigl( \varrho (\Phi )+\varrho ( \Psi )\bigr)+\mathbb{T}(t) \bigr). \end{aligned}$$

(5.22)

Similarly, we find

$$\begin{aligned} \frac{\zeta _{3}}{ s^{-}} \int _{\Omega}\biggl(\frac{\zeta _{3}\kappa}{2}\biggr)^{1-s(x)} \vert \Psi \vert ^{s(x)}\,dx\leq C_{12} \bigl(\bigl( \varrho (\Phi )+\varrho ( \Psi )\bigr)+\mathbb{T}(t) \bigr), \end{aligned}$$

(5.23)

where $C_{11}=C_{11}(\kappa )=C_{9}\frac{\zeta _{1}}{m^{-}}( \frac{\zeta _{1}\kappa}{2})^{1-m^{-}}$, $C_{12}=C_{12}(\kappa )=C_{9}\frac{\zeta _{3}}{ s^{-}}( \frac{\zeta _{3}\kappa}{2})^{1-s^{-}}$.

Combining (5.22), (5.23), and (5.18), we have

$$\begin{aligned} \mathfrak{R}'(t) \geq &\bigl[1-\varepsilon \kappa (\widehat{m}+ \widehat{s})\bigr] \mathbb{T}'(t)+\frac{\varepsilon}{\eta +1} \bigl( \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \biggr] \\ &{}+\varepsilon \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr)-\frac{\varepsilon}{2}(h_{1}o \nabla \Phi )-\frac{\varepsilon}{2}(h_{2}o\nabla \Psi )+ I_{5} \\ &{}-\varepsilon ( C_{11}+C_{12}) \bigl(\bigl(\varrho ( \Phi )+\varrho (\Psi )\bigr)+ \mathbb{T}(t) \bigr)-\varepsilon \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr). \end{aligned}$$

(5.24)

Here, for $0< a<1$, from (5.6) and (2.3) we have

$$\begin{aligned} J_{7} =&\varepsilon \bigl(2q^{-}+4\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx \\ =&\varepsilon a\bigl(2q^{-}+4\bigr) \int _{\Omega}F(\Phi ,\Psi )\,dx+\varepsilon (1-a) \bigl(2q^{-}+4\bigr) \bigl( \mathbb{T}(t)-d_{1}\bigr) \\ &{}+\frac{\varepsilon (1-a)(2q^{-}+4)}{\eta +2}\bigl( \Vert \Phi _{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2} \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{2}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Psi \Vert _{2}^{2} \\ &{}+\varepsilon (1-a) \bigl(q^{-}+2\bigr) \bigl((h_{1}o \nabla \Phi )+(h_{2}o\nabla \Psi )\bigr) \\ &{}+\frac{\varepsilon (1-a)(q^{-}+2)}{(\gamma +1)}\bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2(\gamma +1)}\bigr). \end{aligned}$$

(5.25)

Substituting (5.25) in (5.24) and applying (2.3) and (5.4), we get

$$\begin{aligned} \mathfrak{R}'(t) \geq & \bigl\{ 1-\varepsilon \kappa (\widehat{m}+ \widehat{s}) \bigr\} \mathbb{T}'(t) \\ &{}+\varepsilon \bigl\{ (1-a) \bigl(q^{-}+2\bigr)+1 \bigr\} \bigl( \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} \bigr) \\ &{}+\varepsilon \biggl\{ \frac{\varepsilon (1-a)(2q^{-}+4)}{\eta +2}+ \frac{1}{\eta +1} \biggr\} \bigl( \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon \biggl\{ (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggr) \biggr\} \Vert \nabla \Phi \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (1-a) \bigl(q^{-}+2\bigr) \biggl(1- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggr) \biggr\} \Vert \nabla \Psi \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (1-a) \bigl(q^{-}+2\bigr)- \frac{1}{2} \biggr\} (h_{1}o\nabla \Phi +h_{2}o \nabla \Psi ) \\ &{}+\varepsilon \biggl\{ \frac{(1-a)(q^{-}+2)}{\gamma +1}-1 \biggr\} \bigl( \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)}\bigr) \\ &{}+\varepsilon \bigl\{ c_{0}\bigl( \underbrace{a-2 \bigl(p^{-}+2\bigr) (1-a)\,d_{1}(B\alpha _{2})^{-2(p^{+}+2)}}_{ \widehat{c}}\bigr)-C_{13}( \kappa ) \bigr\} \bigl(\varrho (\Phi )+\varrho ( \Psi ) \bigr) \\ &{}+\varepsilon \bigl\{ (1-a) \bigl(2q^{-}+4\bigr)-C_{13}( \kappa ) \bigr\} \mathbb{T}(t), \end{aligned}$$

(5.26)

where $C_{13}(\kappa )=C_{11}(\kappa )+C_{12}(\kappa )$, by (5.5),(2.3), and (5.4), one can check that $\widehat{c}>0$.

Here, assume $0< a$ so small that

$$ \bigl(q^{-}+2\bigr) (1-a)>1+\gamma , $$

we have

$$\begin{aligned} &\lambda _{1}:=\bigl(q^{-}+2\bigr) (1-a)-1>0, \\ &\lambda _{2}:=\bigl(q^{-}+2\bigr) (1-a)- \frac{1}{2}>0, \\ &\lambda _{3}:=\frac{(q^{-}+2)(1-a)}{\gamma +1}-1>0, \end{aligned}$$

and we assume

$$ \max \biggl\{ \int _{0}^{\infty}h_{1}(\varsigma )\,d \varsigma , \int _{0}^{ \infty}h_{2}(\varsigma )\,d \varsigma \biggr\} < \frac {(q^{-}+2)(1-a)-1}{((q^{-}+2)(1-a)-\frac {1}{2})}= \frac {2\lambda _{1}}{2\lambda _{1}+1}, $$

(5.27)

which gives

$$\begin{aligned} &\lambda _{4}= \biggl\{ \bigl(\bigl(q^{-}+2\bigr) (1-a)-1 \bigr)- \int _{0}^{t}h_{1}( \varsigma )\,d \varsigma \biggl(\bigl(q^{-}+2\bigr) (1-a)-\frac{1}{2} \biggr) \biggr\} >0, \\ &\lambda _{5}= \biggl\{ \bigl(\bigl(q^{-}+2\bigr) (1-a)-1 \bigr)- \int _{0}^{t}h_{2}( \varsigma )\,d \varsigma \biggl(\bigl(q^{-}+2\bigr) (1-a)-\frac{1}{2} \biggr) \biggr\} >0. \end{aligned}$$

Next, we pick κ large enough such that

$$\begin{aligned} \lambda _{6} =&c_{0}\widehat{c}-C_{13}( \kappa )>0, \\ \lambda _{7} =&2\bigl(q^{-}+2\bigr) (1-a)-C_{13}(\kappa )>0. \end{aligned}$$

At this point, we fix κ, a and select ε so small that

$$ \lambda _{8}=1-\varepsilon \kappa (\widehat{m}+\widehat{s})>0 $$

and

$$\begin{aligned} \mathfrak{R}(0) =&\mathbb{T}(0)+\frac{\varepsilon}{\eta +1} \int _{ \Omega} \bigl[\Phi _{0} \vert \Phi _{1} \vert ^{\eta}\Phi _{1}+\Psi _{0} \vert \Psi _{1} \vert ^{\eta}\Psi _{1} \bigr]\,dx \\ &{}+\varepsilon \int _{\Omega} [\nabla \Phi _{1}\nabla \Phi _{0}+ \nabla \Psi _{1}\nabla \Psi _{0} ] \,dx>0, \end{aligned}$$

(5.28)

and from (5.9) and (5.10)

$$\begin{aligned} \mathfrak{R}(t) \leq & c \bigl[ \Vert \Phi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)}+ \Vert \Psi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)} \bigr]. \end{aligned}$$

(5.29)

Thus, for some $\mu _{1}>0$, (5.26) implies

$$\begin{aligned} \mathfrak{R}'(t) \geq &\mu _{1} \bigl\{ \mathbb{T}(t)+ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert w_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla v \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)} \\ &{}+ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2} + \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2} +(h_{1}o \nabla \Phi )+(h_{2}o\nabla \Psi ) \\ &{}+\varrho (\Phi )+\varrho (\Psi ) \bigr\} \end{aligned}$$

(5.30)

and

$$ \mathfrak{R}(t)\geq \mathfrak{R}(0)>0, \quad t>0. $$

(5.31)

After, by Hölder’s and Young’s inequalities, we find

$$\begin{aligned} \biggl\vert \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta} \Phi _{t}+ \Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr)\,dx \biggr\vert \leq &C \bigl[ \Vert \Phi \Vert _{2(q^{-}+2)}^{\theta}+ \Vert \Phi _{t} \Vert _{ \eta +2}^{\mu} \\ & {} + \Vert \Psi \Vert _{2(q^{-}+2)}^{\theta}+ \Vert \Psi _{t} \Vert _{\eta +2}^{ \mu} \bigr] , \end{aligned}$$

(5.32)

where $\frac{1}{\mu}+\frac{1}{\theta}=1$. Next, assume $\mu =(\eta +2)$ to reach

$$ \theta =\frac{(\eta +2)}{(\eta +1)}\leq 2\bigl(q^{-}+2\bigr). $$

By using (5.7) and (4.26), we find

$$\begin{aligned} &\Vert \Phi \Vert _{2(q^{-}+2)}^{\frac{\eta +2}{(\eta +1)}}\leq K\bigl( \Vert \Phi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)}+\mathbb{T}(t)\bigr) \\ &\Vert \Psi \Vert _{2(q^{-}+2)}^{\frac{\eta +2}{(\eta +1)}}\leq K\bigl( \Vert \Psi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)}+\mathbb{T}(t)\bigr), \quad \forall t\geq 0. \end{aligned}$$

Then

$$\begin{aligned} & \biggl\vert \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta} \Phi _{t}+ \Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr)\,dx \biggr\vert \\ &\quad \leq c \bigl\{ \bigl(\varrho (\Phi )+\varrho (\Psi )\bigr)+ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \mathbb{T}(t) \bigr\} . \end{aligned}$$

(5.33)

Hence

$$\begin{aligned} \mathfrak{R}(t) =& \biggl(\mathbb{T}(t)+\frac{\varepsilon}{\eta +1} \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta}\Phi _{t}+\Psi \vert \Psi _{t} \vert ^{\eta}\Psi _{t}\bigr)\,dx \\ &{}+\varepsilon \int _{\Omega}(\nabla \Phi _{t}\nabla \Phi +\nabla \Psi _{t}\nabla \Psi )\,dx \biggr) \\ \leq &c \bigl(\mathbb{T}(t)+ \Vert \Phi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \Psi _{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla \Phi \Vert _{2}^{2}+ \Vert \nabla \Psi \Vert _{2}^{2} \\ &{}+ \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \Vert \nabla \Psi _{t} \Vert _{2}^{2}+ \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla \Psi \Vert _{2}^{2( \gamma +1)} \\ &{}+(h_{1}o\nabla \Phi )+(h_{2}o\nabla \Psi )+\bigl( \varrho (\Phi )+ \varrho (\Psi )\bigr) \bigr). \end{aligned}$$

(5.34)

From (5.30) and (5.34), we have

$$ \mathfrak{R}'(t)\geq \lambda _{1} \mathfrak{R}(t), $$

(5.35)

where $\lambda _{1}> 0$, this relies on $\mu _{1}$ and c. Hence, (5.35) gives

$$ \mathfrak{R}(t)\geq \mathfrak{R}(0)e^{(\lambda _{1} t)}\quad \forall t>0. $$

(5.36)

Then (5.29) and (5.36) imply

$$ \Vert \Phi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)}+ \Vert \Psi \Vert _{2(q^{-}+2)}^{2(q^{-}+2)} \geq C e^{(\lambda _{1} t)},\quad \forall t>0. $$

This implies that the solution grows exponentially with $L^{2(p^{-}+2)}$-norm. This ends the proof. □

6 General decay

In this section, we state and prove the general decay of system (1.1) in the case $f_{1}=f_{2}=0$. For this goal, problem (1.1) can be written as

$$ \textstyle\begin{cases} \vert \Phi _{t} \vert ^{\eta }\Phi _{tt}-\mathcal{T}( \Vert \nabla \Phi \Vert _{2}^{2})\Delta \Phi +\int _{0}^{t}h_{1}(t- \varsigma )\Delta \Phi (\varsigma )\,d\varsigma -\Delta \Phi _{tt}+g_{1}( \Phi _{t})=0, \\ \Phi ( x,0) =\Phi _{0}(x), \qquad \Phi _{t}( x,0) =\Phi _{1}( x), \quad \text{in } \Omega \\ \Phi ( x,t) =0, \quad \text{in } \partial \Omega \times (0, T), \end{cases} $$

(6.1)

where

$$ g_{1}(\Phi _{t})=\zeta _{1} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)-2} \Phi _{t}(t). $$

We introduce the modified functional of energy $\mathfrak{E}$ of (6.1) as follows:

$$\begin{aligned} \mathfrak{E}(t) =&\frac{1}{\eta +2} \Vert \Phi _{t} \Vert _{\eta +2}^{ \eta +2}+\frac{1}{2} \Vert \nabla \Phi _{t} \Vert _{2}^{2}+ \frac{1}{2(\gamma +1)} \Vert \nabla \Phi \Vert _{2}^{2(\gamma +1)} \\ &{}+\frac{1}{2} \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \Vert \nabla \Phi \Vert _{2}^{2}+ \frac{1}{2}(h_{1}o\nabla \Phi ) (t). \end{aligned}$$

(6.2)

From Lemma 2.4, the functional of energy satisfies

$$\begin{aligned} \mathfrak{E}'(t) \leq &-\zeta _{1} \int _{\Omega} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)}\,dx+ \frac{1}{2}\bigl(h'_{1}o \nabla \Phi \bigr) (t)-\frac{1}{2}h_{1}(t) \Vert \nabla \Phi \Vert _{2}^{2}\leq 0. \end{aligned}$$

(6.3)

Lemma 6.1

(Komornik, [19]) Assume a nonincreasing function $E:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ and suppose that $\exists \sigma ,\omega >0$ in a manner that

$$ \int _{\Im}^{\infty}E^{1+\aleph}(t)\,dt\leq \frac{1}{\omega}E^{\aleph}(0)E( \Im )=cE(\Im ), \quad \forall \Im >0. $$

(6.4)

Then we have $\forall t\geq 0$

$$ \textstyle\begin{cases} E(t)\leq cE(0)/(1+t)^{\frac{1}{\aleph}}, \quad \textit{if } \aleph >0, \\ E(t)\leq cE(0)e^{-\omega t}, \quad \textit{if } \aleph =0. \end{cases} $$

(6.5)

Theorem 6.2

Suppose that (1.3), (2.1)–(2.2), and (2.4) hold. Then there exist $c,\lambda >0$ so that the solution of (6.1) satisfies

$$ \textstyle\begin{cases} \mathfrak{E}(t)\leq c\mathfrak{E}(0)/(1+t)^{\frac{2}{m^{+}-2}}, \quad \textit{if } m^{+}>2, \\ \mathfrak{E}(t)\leq c\mathfrak{E}(0)e^{-\lambda t}, \quad \textit{if } m(x)=2. \end{cases} $$

(6.6)

Proof

Multiplying (6.1)₁ by $\Phi \mathfrak{E}^{p}(t)$ for $p>0$,

then integrating over $\Omega \times (\Im ,T)$, where $\Im < T$, gives

$$\begin{aligned} & \int _{\Im}^{T}\mathfrak{E}^{p}(t) \int _{\Omega} \biggl\{ \Phi \vert \Phi _{t} \vert ^{\eta }\Phi _{tt}-\mathcal{T}\bigl( \Vert \nabla \Phi \Vert _{2}^{2}\bigr)\Phi \Delta \Phi + \int _{0}^{t}h_{1}(t- \varsigma ) \Phi \Delta \Phi (\varsigma )\,d\varsigma \\ & \quad -\Phi \Delta \Phi _{tt}+\zeta _{1}\Phi \Phi _{t} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)-2} \biggr\} \,dx\,dt=0, \end{aligned}$$

(6.7)

we deduce that

$$\begin{aligned} & \int _{\Im}^{T}\mathfrak{E}^{p}(t) \int _{\Omega} \biggl\{ \frac{d}{dt} \frac{1}{\eta +1}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}\bigr)-\frac{1}{\eta +1} \vert \Phi _{t} \vert ^{\eta +2}+ \frac{d}{dt}(\nabla \Phi \nabla \Phi _{t})- \vert \nabla \Phi _{t} \vert ^{2} \\ &\qquad +\mathcal{T} \bigl( \Vert \nabla \Phi \Vert ^{2}_{2} \bigr) \vert \nabla \Phi \vert ^{2}- \int _{0}^{t}h_{1}(t-\varsigma ) \nabla \Phi \nabla \Phi (\varsigma )\,d\varsigma +\zeta _{1}\Phi \Phi _{t} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)-2} \biggr\} \,dx\,dt \\ &\quad =0. \end{aligned}$$

(6.8)

By (6.2) and the relation

$$\begin{aligned} &\frac{d}{dt} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}\bigl(\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}+\nabla \Phi \nabla \Phi _{t} \bigr) \,dx \biggr) \\ &\quad =p \mathfrak{E}^{p-1}(t)\mathfrak{E}'(t) \biggl( \int _{\Omega}\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}\,dx+ \int _{\Omega} \nabla \Phi \nabla \Phi _{t} \,dx \biggr) \\ &\qquad +\mathfrak{E}^{p}(t)\frac{d}{dt} \biggl( \int _{\Omega}\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}\,dx+ \int _{\Omega} \nabla \Phi \nabla \Phi _{t} \,dx \biggr), \end{aligned}$$

we deduce

$$\begin{aligned} &(\eta +2) \int _{\Im}^{T}\mathfrak{E}^{p+1}(t) \,dt \\ &\quad = \underbrace{ \int _{\Im}^{T}\frac{d}{dt} \biggl( \mathfrak{E}^{p}(t) \int _{\Omega}\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}\,dx \biggr)\,dt}_{I_{1}}- \underbrace{p \int _{\Im}^{T} \biggl(\mathfrak{E}^{p-1}(t) \mathfrak{E}'(t) \int _{\Omega}\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}\,dx \biggr)\,dt}_{I_{2}} \\ &\qquad +(\eta +1) \underbrace{ \int _{\Im}^{T}\frac{d}{dt} \biggl( \mathfrak{E}^{p}(t) \int _{\Omega}\nabla \Phi \nabla \Phi _{t}\,dx \biggr)\,dt}_{I_{3}} \\ &\qquad - \underbrace{(\eta +1)p \int _{\Im}^{T} \biggl(\mathfrak{E}^{p-1}(t) \mathfrak{E}'(t) \int _{\Omega}\nabla \Phi \nabla \Phi _{t}\,dx \biggr)\,dt}_{I_{4}}- \underbrace{\frac{\eta}{2} \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega} \vert \nabla \Phi _{t} \vert ^{2}\,dx \biggr)\,dt}_{I_{5}} \\ &\qquad + \underbrace{\frac{\eta +2}{2} \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \biggl(1- \int _{0}^{t}h_{1}(\varsigma )\,d \varsigma \biggr) \int _{\Omega} \vert \nabla \Phi \vert ^{2}\,dx \biggr)\,dt}_{I_{6}} \\ &\qquad + \underbrace{ \biggl((\eta +1)+\frac{\eta +2}{2(\gamma +1)} \biggr) \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega} \Vert \nabla \Phi \Vert ^{2\gamma}_{2} \vert \nabla \Phi \vert ^{2}\,dx \biggr)\,dt}_{I_{7}} \\ &\qquad + \underbrace{(\eta +1)\zeta _{1} \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}\Phi \Phi _{t} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)-2} \,dx \biggr) \,dt}_{I_{8}} \\ &\qquad + \underbrace{\frac{\eta +2}{2} \int _{\Im}^{T} \bigl(\mathfrak{E}^{p}(t) (h_{1}\circ \nabla \Phi ) (t) \bigr)\,dt}_{I_{9}} \\ &\qquad - \underbrace{(\eta +1) \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{0}^{t}h_{1}(t-\varsigma ) \int _{\Omega}\nabla \Phi \nabla \Phi (\varsigma )\,dx\,d\varsigma \biggr)\,dt}_{I_{10}}. \end{aligned}$$

(6.9)

Now, we estimate $I_{j},j=1,\ldots,10$, of the RHS in (6.9), we have

$$\begin{aligned} I_{1} =&\mathfrak{E}^{p}(T) \int _{\Omega}\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}(x,T)\,dx-\mathfrak{E}^{p}(\Im ) \int _{ \Omega}\Phi \vert \Phi _{t} \vert ^{\eta } \Phi _{t}(x, \Im )\,dx \\ \leq &c\mathfrak{E}^{p}(T) \bigl\{ \bigl\Vert \Phi (x,T) \bigr\Vert _{2}^{2}+ \bigl\Vert \Phi _{t}(x,T) \bigr\Vert _{\eta +2}^{\eta +2} \bigr\} \\ &{}+c\mathfrak{E}^{p}(\Im ) \bigl\{ \bigl\Vert \Phi (x,\Im ) \bigr\Vert _{2}^{2}+ \bigl\Vert \Phi _{t}(x,\Im ) \bigr\Vert _{\eta +2}^{\eta +2} \bigr\} \\ \leq &c\mathfrak{E}^{p}(T) \bigl\{ c_{*} \bigl\Vert \nabla \Phi (T) \bigr\Vert ^{2}_{2}+ \mathfrak{E}(T) \bigr\} \\ &{}+c\mathfrak{E}^{p}(\Im ) \bigl\{ c_{*} \bigl\Vert \nabla \Phi (\Im ) \bigr\Vert ^{2}_{2}+ \mathfrak{E}(\Im ) \bigr\} \\ \leq &c_{1} \bigl(\mathfrak{E}^{p+1}(T)+ \mathfrak{E}^{p+1}(\Im ) \bigr). \end{aligned}$$

(6.10)

Because $\mathfrak{E}$ is a nonincreasing function, we find

$$\begin{aligned} I_{1}\leq c\mathfrak{E}^{p+1}(\Im )\leq \mathfrak{E}^{p}(0) \mathfrak{E}(\Im )\leq c\mathfrak{E}(\Im ). \end{aligned}$$

(6.11)

Similarly, we find

$$\begin{aligned} &I_{2}\leq -p \int _{\Im}^{T}\mathfrak{E}^{p-1}(t) \mathfrak{E}'(t) \bigl(c_{*}\mathfrak{E}(t)+ \mathfrak{E}(t) \bigr)\,dt \\ &\hphantom{I_{2}}\leq -c \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}'(t)\,dt\leq c \mathfrak{E}^{p+1}(\Im )\leq c\mathcal{E}(\Im ), \end{aligned}$$

(6.12)

$$\begin{aligned} &I_{3}\leq c \int _{\Im}^{T}\mathfrak{E}^{p}(t) \bigl( \Vert \nabla \Phi \Vert ^{2}_{2}+ \Vert \nabla \Phi _{t} \Vert ^{2}_{2}\bigr)\,dt \\ &\hphantom{I_{3}}\leq c\mathfrak{E}^{p+1}(\Im )\leq \mathfrak{E}^{p}(0) \mathfrak{E}( \Im )\leq c\mathfrak{E}(\Im ), \end{aligned}$$

(6.13)

and

$$\begin{aligned} I_{4} \leq &-(\eta +1)p \int _{\Im}^{T}\mathfrak{E}^{p-1}(t) \mathfrak{E}'(t) \bigl(c\mathfrak{E}(t) \bigr)\,dt \\ \leq &-c \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}'(t)\,dt\leq c \mathfrak{E}^{p+1}(\Im )\leq c\mathfrak{E}(\Im ). \end{aligned}$$

(6.14)

Next, we get

$$\begin{aligned} I_{5} =&-\frac{\eta}{2}c \int _{\Im}^{T} \bigl(\mathfrak{E}^{p}(t) \Vert \nabla \Phi _{t} \Vert ^{2}_{2} \bigr)\,dt \\ \leq &c \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}(t)\,dt\leq c \mathfrak{E}^{p+1}(\Im )\leq c\mathfrak{E}( \Im ). \end{aligned}$$

(6.15)

After that, we get

$$\begin{aligned} I_{6} \leq &(\eta +2) \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}(t)\,dt \leq c\mathfrak{E}^{p+1}(\Im )\leq c\mathfrak{E}( \Im ). \end{aligned}$$

(6.16)

For the next term, we have

$$\begin{aligned} I_{7} =& \bigl(2(\gamma +1) (\eta +1)+(\eta +2) \bigr) \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \frac{ \Vert \nabla \Phi \Vert ^{2(\gamma +1)}_{2}}{2(\gamma +1)} \biggr)\,dt \\ \leq &c \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}(t)\,dt\leq c \mathfrak{E}^{p+1}(\Im )\leq c\mathfrak{E}( \Im ), \end{aligned}$$

(6.17)

by Young’s inequality, we find

$$\begin{aligned} I_{8} =&(\eta +1)\zeta _{1} \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}\Phi \Phi _{t} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)-2} \,dx \biggr)\,dt \\ \leq &\varepsilon \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{ \Omega} \bigl\vert \Phi (t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt \\ &{}+c \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{ \varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt \\ \leq &\varepsilon \int _{\Im}^{T}\mathfrak{E}^{p}(t) \biggl[ \int _{ \Omega _{+}} \bigl\vert \Phi (t) \bigr\vert ^{m^{+}} \,dx+ \int _{\Omega _{-}} \bigl\vert \Phi (t) \bigr\vert ^{m^{-}} \,dx \biggr]\,dt \\ &{}+c \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{ \varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt. \end{aligned}$$

Here, utilizing $H_{0}^{1}(\Omega )\hookrightarrow L^{m^{-}}(\Omega )$ and $H_{0}^{1}(\Omega )\hookrightarrow L^{m^{+}}(\Omega )$, we get

$$\begin{aligned} I_{8} \leq &\varepsilon \int _{\Im}^{T}\mathfrak{E}^{p}(t) \bigl[c \bigl\Vert \nabla \Phi (t) \bigr\Vert ^{m^{+}}_{2} +c \bigl\Vert \nabla \Phi (t) \bigr\Vert ^{m^{-}}_{2} \bigr]\,dt \\ &{}+c \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{ \varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt \\ \leq &\varepsilon \int _{\Im}^{T}\mathfrak{E}^{p}(t) \bigl[c \mathfrak{E}^{\frac{m^{+}-2}{2}}(0)\mathfrak{E}(t) +c\mathfrak{E}^{ \frac{m^{-}-2}{2}}(0) \mathfrak{E}(t) \bigr]\,dt \\ &{}+c \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{ \varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt \\ \leq &c\varepsilon \int _{\Im}^{T}\mathfrak{E}^{p+1}(t) \,dt+c \int _{ \Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{\varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt \end{aligned}$$

(6.18)

and

$$\begin{aligned} I_{9} \leq &(\eta +2) \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}(t)\,dt \leq c\mathfrak{E}^{p+1}(\Im )\leq c\mathfrak{E}( \Im ). \end{aligned}$$

(6.19)

By Young’s inequality, we get

$$\begin{aligned} I_{10} \leq &(\eta +1) \int _{\Im}^{T} (\mathfrak{E}^{p}(t) \bigl(c \Vert \nabla \Phi \Vert ^{2}_{2}+c(h_{1} \circ \nabla \Phi ) (t) \bigr)\,dt \\ \leq &c \int _{\Im}^{T}\mathfrak{E}^{p}(t) \mathfrak{E}(t)\,dt\leq c \mathfrak{E}^{p+1}(\Im )\leq c\mathfrak{E}( \Im ). \end{aligned}$$

(6.20)

By substituting (6.11)–(6.20) into (6.9), we find

$$\begin{aligned} \int _{\Im}^{T}\mathfrak{E}^{p+1}(t) \,dt \leq &c\varepsilon \int _{\Im}^{T} \mathfrak{E}^{p+1}(t) \,dt+c\mathfrak{E}(\Im ) \\ &{}+c \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{ \varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt. \end{aligned}$$

(6.21)

Now, we choose ε so small that

$$\begin{aligned} \int _{\Im}^{T}\mathfrak{E}^{p+1}(t) \,dt \leq &c\mathfrak{E}(\Im )+c \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega}c_{\varepsilon}(x) \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt. \end{aligned}$$

(6.22)

After, we fix ε, $c_{\varepsilon}(x)\leq M$ since $m(x)$ is bounded.

Then, by (6.3), we have

$$\begin{aligned} \int _{\Im}^{T}\mathfrak{E}^{p+1}(t) \,dt \leq &c\mathfrak{E}(\Im )+cM \int _{\Im}^{T} \biggl(\mathfrak{E}^{p}(t) \int _{\Omega} \bigl\vert \Phi _{t}(t) \bigr\vert ^{m(x)} \,dx \biggr)\,dt \\ \leq &c\mathfrak{E}(\Im )-\frac{cM}{\zeta _{1}} \int _{\Im}^{T} \mathfrak{E}^{p}(t) \mathfrak{E}'(t)\,dt \\ \leq &c\mathfrak{E}(\Im )+\frac{cM}{\zeta _{1}(p+1)} \bigl[ \mathfrak{E}^{p+1}( \Im )-\mathfrak{E}^{p+1}(T) \bigr]\leq c \mathfrak{E}(\Im ). \end{aligned}$$

(6.23)

Taking $T\rightarrow \infty$, we get

$$\begin{aligned} \int _{\Im}^{\infty}\mathfrak{E}^{p+1}(t) \,dt\leq c\mathfrak{E}(\Im ). \end{aligned}$$

(6.24)

Hence, Komornik’s Lemma 6.1 (with $\aleph =p=\frac{m^{+}-2}{2}$) gives (6.6). This ends the proof. □

7 Conclusion

In this paper, we investigated a coupled nonlinear viscoelastic Kirchhoff-type system with sources and variable exponents. Firstly, we showed the global existence of the solution. Next, we proved the blow-up result with negative initial energy. After that, we established the exponential growth of solution but with positive initial energy. At the end of this study we obtained the general decay by Komornik’s lemma in the case of absence of the source terms.

As for the future vision, we will apply the same method to study other systems, but with the addition of some damping terms.