In this section, we first state the result about the well-posedness of problem (1). Under assumptions, we can derive an existence result by the standard Faedo–Galerkin method (see [15, 18]).
Theorem 5
Suppose that \(k>0\), \(\alpha , \beta , \delta _{1}, \delta _{2}>0\) and \((F1)\)–\((F2)\) hold. If \(h_{B}, h_{S}\in L^{2}(0,L)\), \((u_{0}, v_{0}, u_{1}, v_{1})\in E_{1}\), then for any given \(T>0\), there exists a unique solution \((u,v)\) of (1)–(3) such that
$$\begin{aligned}& u\in C\bigl([0,T], Y_{2}\bigr),\qquad u_{t}\in C \bigl([0,T], Y_{0}\bigr),\\& v\in C\bigl([0,T], Y_{1}\bigr),\qquad v_{t}\in C \bigl([0,T], Y_{0}\bigr). \end{aligned}$$
Furthermore, \((u_{0}, v_{0}, u_{1}, v_{1})\rightarrow (u(t), v(t), u_{t}(t), v_{t}(t))\) is continuous in \(E_{1}\).
Consequently, it admits to define a \(C^{0}\) semigroup
$$ S(t): (u_{0}, v_{0}, u_{1}, v_{1})\rightarrow \bigl(u(t), v(t), u_{t}(t), v_{t}(t)\bigr),\quad t \in \mathbb{R}^{+}, $$
and it maps \(E_{1}\) into itself.
To obtain the existence of exponential attractors, we need to prove some lemmas as follows.
Lemma 6
Given any \(R>0\) and any two initial data \(z_{1}=(u_{11}, v_{11}, u_{12}, v_{12})\), \(z_{2}=(u_{21}, v_{21}, u_{22}, v_{22}) \in E_{0}\) such that \(\| z_{i}\| _{0}\leq R\), there holds
$$ \bigl\Vert S(t)z_{1}-S(t)z_{2} \bigr\Vert _{0}\leq e^{Kt} \Vert z_{1}-z_{2} \Vert _{0},\quad \forall t\in \mathbb{R}^{+}, $$
(8)
for some \(K=K(R)>0\).
Proof
Given two solutions \(z^{1}=(u^{1},v^{1},u_{t}^{1},v_{t}^{1})\) and \(z^{2}=(u^{2},v^{2},u_{t}^{2},v_{t}^{2})\), corresponding to different initial data \(z_{1}\) and \(z_{2}\), the difference \(z^{1}-z^{2}=(\omega ^{1},\omega ^{2},\omega _{t}^{1},\omega _{t}^{2})\) fulfills
$$ \begin{aligned}[b]&\frac{d}{dt}\bigl(\alpha \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+ \beta \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} + \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}\bigr)+2 \delta _{1} \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}+ 2\delta _{2} \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2} \\ &\quad {}+2\bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+},\omega _{t}^{1}\bigr)+2\bigl(f_{B} \bigl(u^{1}\bigr)-f_{B}\bigl(u^{2}\bigr), \omega _{t}^{1}\bigr) \\ &\quad {}-2\bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+},\omega _{t}^{2}\bigr)+2\bigl(f_{S} \bigl(v^{1}\bigr)-f_{S}\bigl(v^{2}\bigr), \omega _{t}^{2}\bigr)=0. \end{aligned} $$
(9)
Using (4) and Hölder’s inequality, we have
$$\begin{aligned}& \begin{aligned}[b] -2\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\omega _{t}^{1}\bigr)& \leq 2k \bigl\Vert \bigl(u^{1}-v^{1}\bigr)^{+}- \bigl(u^{2}-v^{2}\bigr)^{+} \bigr\Vert \bigl\Vert \omega _{t}^{1} \bigr\Vert \\ &\leq 2k \bigl\Vert \omega ^{1}-\omega ^{2} \bigr\Vert \bigl\Vert \omega _{t}^{1} \bigr\Vert \\ & \leq \frac{2k}{\lambda ^{2}} \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+ \frac{2k}{\lambda ^{2}} \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} +k \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}, \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned}& \begin{aligned}[b]2\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\omega _{t}^{2}\bigr)& \leq 2k \bigl\Vert \bigl(u^{1}-v^{1}\bigr)^{+}- \bigl(u^{2}-v^{2}\bigr)^{+} \bigr\Vert \bigl\Vert \omega _{t}^{2} \bigr\Vert \\ &\leq 2k \bigl\Vert \omega ^{1}-\omega ^{2} \bigr\Vert \bigl\Vert \omega _{t}^{2} \bigr\Vert \\ & \leq \frac{2k}{\lambda ^{2}} \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+ \frac{2k}{\lambda ^{2}} \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} +k \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}. \end{aligned} \end{aligned}$$
(11)
By \((F2)\) and Lemma 3, as well as the Sobolev embedding theorems, we know that \(f_{B}(u)\), \(f_{B}'(u)\), \(f_{B}''(u)\), \(f_{S}(u)\), \(~f_{S}'(u)\), \(f_{S}''(u)\) are uniformly bounded in \(L^{\infty}\). That is, there exists a constant \(M>0\) such that
$$ \bigl\vert f_{B}(u) \bigr\vert _{L^{\infty}}, \bigl\vert f_{B}'(u) \bigr\vert _{L^{\infty}}, \bigl\vert f_{B}''(u) \bigr\vert _{L^{ \infty}}, \bigl\vert f_{S}(u) \bigr\vert _{L^{\infty}}, \bigl\vert f_{S}'(u) \bigr\vert _{L^{\infty}}, \bigl\vert f_{S}''(u) \bigr\vert _{L^{ \infty}}\leq M. $$
(12)
Therefore
$$\begin{aligned}& \begin{aligned}[b]-2\bigl(f_{B}\bigl(u^{1} \bigr)-f_{B}\bigl(u^{2}\bigr),\omega _{t}^{1}\bigr)&=-2\bigl(f_{B}' \bigl( \theta u^{1}+(1-\theta )u^{2}\bigr)\omega ^{1},\omega _{t}^{1}\bigr) \\ & \leq 2\| f_{B}'(\theta u^{1}+(1-\theta )u^{2}\| _{ \infty} \Vert \omega ^{1}\| \bigl\Vert \omega _{t}^{1} \bigr\Vert \leq 2M \bigl\Vert \omega ^{1} \bigr\Vert \bigl\Vert \omega _{t}^{1} \bigr\Vert \\ &\leq \frac{M}{\lambda ^{2}} \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2} +M \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}, \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned}& \begin{aligned}[b]-2\bigl(f_{S}\bigl(v^{1} \bigr)-f_{S}\bigl(v^{2}\bigr),\omega _{t}^{2}\bigr)&=-2\bigl(f_{S}' \bigl( \theta v^{1}+(1-\theta )v^{2}\bigr)\omega ^{2},\omega _{t}^{2}\bigr) \\ & \leq 2\| f_{S}'(\theta v^{1}+(1-\theta )v^{2}\| _{ \infty} \Vert \omega ^{2}\| \bigl\Vert \omega _{t}^{2} \bigr\Vert \leq 2M \bigl\Vert \omega ^{2} \bigr\Vert \bigl\Vert \omega _{t}^{2} \bigr\Vert \\ &\leq \frac{M}{\lambda ^{2}} \bigl\Vert \Delta \omega ^{2} \bigr\Vert ^{2} +M \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}. \end{aligned} \end{aligned}$$
(14)
Combining with the above estimates, we have
$$ \begin{aligned}[b]&\frac{d}{dt}\bigl(\alpha \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+ \beta \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} + \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}\bigr) \\ &\quad \leq \frac{4k+M}{\lambda ^{2}} \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+ \frac{4k+M}{\lambda ^{2}} \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} +(k+M) \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}+(k+M) \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}. \end{aligned} $$
(15)
Thus, we can find a positive constant \(K=\max\{\frac{4k+M}{\alpha \lambda ^{2}}, \frac{4k+M}{\beta \lambda ^{2}}, k+M\}\) such that
$$ \begin{aligned}[b] &\frac{d}{dt}\bigl(\alpha \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} + \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}\bigr)\\ &\quad \leq K\bigl(\alpha \bigl\Vert \Delta \omega ^{1} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla \omega ^{2} \bigr\Vert ^{2} + \bigl\Vert \omega _{t}^{1} \bigr\Vert ^{2}+ \bigl\Vert \omega _{t}^{2} \bigr\Vert ^{2}\bigr). \end{aligned} $$
(16)
The assertion follows from the Gronwall lemma. □
Lemma 7
There exists \(C \geq 0\) such that
$$ \sup_{z_{0}\in \mathcal{B}_{1}} \bigl\Vert z_{t}(t) \bigr\Vert _{0}\leq C. $$
Proof
From (1) we have
$$ u_{tt}=-\alpha \Delta ^{2}u-\delta _{1}u_{t}-k(u-v)^{+}-f_{B}(u)+h_{B} $$
and
$$ v_{tt}=\beta \Delta v-\delta _{2}v_{t}+k(u-v)^{+}-f_{S}(u)+h_{S}. $$
By exploiting Lemma 3, Lemma 4, and (12), we get
$$ \Vert u_{tt} \Vert \leq \alpha \bigl\Vert \Delta ^{2}u \bigr\Vert +\delta _{1} \Vert u_{t} \Vert +k \Vert u-v \Vert + \bigl\Vert f_{B}(u) \bigr\Vert + \Vert h_{B} \Vert \leq C $$
(17)
and
$$ \Vert v_{tt} \Vert \leq \beta \Vert \Delta v \Vert + \delta _{2} \Vert v_{t} \Vert +k \Vert u-v \Vert + \bigl\Vert f_{S}(u) \bigr\Vert + \Vert h_{S} \Vert \leq C. $$
(18)
Further, by virtue of Lemma 4, we can get \(\| \Delta u_{t}\| \leq C\), \(\| \nabla v_{t}\| \leq C\), thus
$$ \bigl\Vert z_{t}(t) \bigr\Vert _{0}^{2}= \alpha \Vert \Delta u_{t} \Vert ^{2}+\beta \Vert \nabla v_{t} \Vert ^{2}+ \Vert u_{tt} \Vert ^{2}+ \Vert v_{tt} \Vert ^{2}\leq C. $$
(19)
Namely,
$$ \sup_{z_{0}\in \mathcal{B}_{1}} \bigl\Vert z_{t}(t) \bigr\Vert _{0}\leq C. $$
 □
Like the method in [21], we define
$$ \mathcal {X}= \overline{\bigcup_{\tau \geq t_{1}}S(\tau ) \mathcal {B}_{1}}^{E_{0}}. $$
(20)
Lemma 8
For every \(T>0\), the mapping \((t,z_{0})\mapsto S(t)z_{0}\) is Lipschitz continuous on \([0,T]\times \mathcal {X}\).
Proof
For \(z_{1}, z_{2}\in \mathcal {X}\) and \(t_{1}, t_{2}\in [0,T]\), we have
$$ \bigl\Vert S(t_{1})z_{1}-S(t_{2})z_{2} \bigr\Vert _{0}\leq \bigl\Vert S(t_{1})z_{1}-S(t_{1})z_{2} \bigr\Vert _{0}+ \bigl\Vert S(t_{1})z_{2}-S(t_{2})z_{2} \bigr\Vert _{0}. $$
(21)
The first term of the above inequality is handled by estimate (8). Concerning the second one,
$$ \bigl\Vert S(t_{1})z_{2}-S(t_{2})z_{2} \bigr\Vert _{0}= \bigl\Vert z(t_{1})-z(t_{2}) \bigr\Vert _{0}\leq \biggl\vert \int _{t_{1}}^{t_{2}} \bigl\Vert z_{t}( \tau ) \bigr\Vert _{0}\,d\tau \biggr\vert \leq C \vert t_{1}-t_{2} \vert . $$
(22)
Hence
$$ \bigl\Vert S(t_{1})z_{1}-S(t_{2})z_{2} \bigr\Vert _{0}\leq L\bigl[ \vert t_{1}-t_{2} \vert + \Vert z_{1}-z_{2} \Vert _{0} \bigr] $$
(23)
for some \(L=L(T)\geq 0\). □
Lemma 9
Let \(\mathcal {X}\subset E_{0}\) be a compact invariant subset. Assume that there exists a time \(t_{\ast}>0\) such that the map \(S(t_{\ast}):\mathcal {X}\rightarrow \mathcal {X}\) admits a decomposition of the form
$$ S(t_{\ast})=S_{0}+S_{1},\quad S_{0}:\mathcal {X}\rightarrow E_{0},\quad S_{1}: \mathcal {X}\rightarrow E_{1}, $$
where \(S_{0}\) satisfies
$$ \bigl\Vert S_{0}(z_{1})-S_{0}(z_{2}) \bigr\Vert _{0}\leq \frac{1}{8} \Vert z_{1}-z_{2} \Vert _{0},\quad \forall z_{1}, z_{2}\in \mathcal {X}, $$
and \(S_{1}\) satisfies
$$ \bigl\Vert S_{1}(z_{1})-S_{1}(z_{2}) \bigr\Vert _{1}\leq C_{\ast} \Vert z_{1}-z_{2} \Vert _{0},\quad \forall z_{1}, z_{2}\in \mathcal {X}, $$
for some \(C_{\ast}>0\).
Proof
For \(z_{0}\in \mathcal {X}\), we denote by \(S_{0}(t)z_{0}\) the solution at time t of the linear homogeneous problem associated with (1)–(3), and let \(S_{1}(t)z_{0}=S(t)z_{0}-S_{0}(t)z_{0}\).
Given two solutions
$$ z^{1}(t)=\bigl(u^{1},v^{1};u_{t}^{1},v_{t}^{1} \bigr)\quad \text{and}\quad z^{2}(t)=\bigl(u^{2},v^{2};u_{t}^{2},v_{t}^{2} \bigr) $$
originating from \(z_{1}, z_{2}\in \mathcal {X}\), respectively.
Set \(\overline{z}=z^{1}-z^{2}=(\bar{u},\bar{v};\bar{u_{t}},\bar{v_{t}})\) and decompose zÌ… into the sum
$$ \bar{z}=\bar{z}_{d}+\bar{z}_{c}=(\omega _{1},\omega _{3};\omega _{1t}, \omega _{3t})+ (\omega _{2},\omega _{4};\omega _{2t},\omega _{4t}), $$
where \(\bar{z}_{d}\) satisfies
$$ \textstyle\begin{cases} \omega _{1tt}+\alpha \Delta ^{2}\omega _{1}+\delta _{1}\omega _{1t}=0, \\ \omega _{3tt}-\beta \Delta \omega _{3}+\delta _{2}\omega _{3t}=0, \\ \overline{z}_{d}(0)=z_{1}-z_{2}, \end{cases} $$
(24)
and \(\bar{z}_{c}\) satisfies
$$ \textstyle\begin{cases} \omega _{2tt}+\alpha \Delta ^{2}\omega _{2}+\delta _{1}\omega _{2t}+k(u^{1}-v^{1})^{+}-k(u^{2}-v^{2})^{+}+f_{B}(u^{1})-f_{B}(u^{2})=0, \\ \omega _{4tt}-\beta \Delta \omega _{4}+\delta _{2}\omega _{4t}-k(u^{1}-v^{1})^{+}+k(u^{2}-v^{2})^{+}+f_{S}(v^{1})-f_{S}(v^{2})=0, \\ \overline{z}_{c}(0)=0. \end{cases} $$
(25)
It is apparent that \(\overline{z}_{d}(t)=S_{0}(t)z_{1}-S_{0}(t)z_{2}\) and \(\overline{z}_{c}(t)=S_{1}(t)z_{1}-S_{1}(t)z_{2}\).
For (24), taking the scalar product of the first and second equations of (24) with \(2\omega _{1t}+\delta _{1}\omega _{1}\) and \(2\omega _{3t}+\delta _{2}\omega _{3}\) in \(L^{2}(0,L)\), respectively, we infer that
$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert \omega _{1t} \Vert ^{2}+\alpha \Vert \Delta \omega _{1} \Vert ^{2}+\delta _{1}(\omega _{1t}, \omega _{1}) +\frac{1}{2}\delta _{1}^{2} \Vert \omega _{1} \Vert ^{2}+ \Vert \omega _{3t} \Vert ^{2}+ \beta \Vert \nabla \omega _{3} \Vert ^{2}+\delta _{2}(\omega _{3t},\omega _{3}) \\ &\quad {}+\frac{1}{2}\delta _{2}^{2} \Vert \omega _{3} \Vert ^{2}\biggr) + \delta _{1}\bigl( \Vert \omega _{1t} \Vert ^{2}+\alpha \Vert \Delta \omega _{1} \Vert ^{2}\bigr)+ \delta _{2} \bigl( \Vert \omega _{3t} \Vert ^{2}+ \beta \Vert \nabla \omega _{3} \Vert ^{2}\bigr)=0. \end{aligned}$$
(26)
Denote
$$ \begin{aligned}[b]E(t)={}& \Vert \omega _{1t} \Vert ^{2}+ \alpha \Vert \Delta \omega _{1} \Vert ^{2}+\delta _{1}(\omega _{1t},\omega _{1}) + \frac{1}{2}\delta _{1}^{2} \Vert \omega _{1} \Vert ^{2} \\ &{}+ \Vert \omega _{3t} \Vert ^{2}+ \beta \Vert \nabla \omega _{3} \Vert ^{2}+\delta _{2}( \omega _{3t},\omega _{3}) + \frac{1}{2}\delta _{2}^{2} \Vert \omega _{3} \Vert ^{2}. \end{aligned} $$
Due to the inequalities \(\delta _{1}(\omega _{1t},\omega _{1})\leq \frac{1}{2}\| \omega _{1t}\| ^{2} +\frac{1}{2}\delta _{1}^{2}\| \omega _{1}\| ^{2}\), \(\delta _{2}(\omega _{3t},\omega _{3}) \leq \frac{1}{2}\| \omega _{3t}\| ^{2} +\frac{1}{2} \delta _{2}^{2}\| \omega _{3}\| ^{2}\), we have
$$ \begin{aligned}[b]E(t)&\leq \frac{3}{2} \Vert \omega _{1t} \Vert ^{2}+ \alpha \Vert \Delta \omega _{1} \Vert ^{2}+\delta _{1}^{2} \Vert \omega _{1} \Vert ^{2} +\frac{3}{2} \Vert \omega _{3t} \Vert ^{2}+ \beta \Vert \nabla \omega _{3} \Vert ^{2}+ \delta _{2}^{2} \Vert \omega _{3} \Vert ^{2} \\ & \leq \frac{3}{2} \Vert \omega _{1t} \Vert ^{2}+\biggl(\alpha + \frac{\delta _{1}^{2}}{\lambda ^{2}}\biggr) \Vert \Delta \omega _{1} \Vert ^{2}+\frac{3}{2} \Vert \omega _{3t} \Vert ^{2}+ \biggl( \beta + \frac{\delta _{2}^{2}}{\lambda ^{2}}\biggr) \Vert \nabla \omega _{3} \Vert ^{2}. \end{aligned} $$
(27)
Let \(\kappa =\max\{\frac{3}{2}, 1+ \frac{\delta _{1}^{2}}{\lambda ^{2}\alpha}, 1+ \frac{\delta _{2}^{2}}{\lambda ^{2}\beta}\}>0\), we get
$$ E(t)\leq \kappa \bigl\Vert \overline{z}_{d}(t) \bigr\Vert _{0}^{2}. $$
(28)
Meanwhile
$$ E(t)\geq \frac{1}{2} \Vert \omega _{1t} \Vert ^{2}+ \alpha \Vert \Delta \omega _{1} \Vert ^{2}+\frac{1}{2} \Vert \omega _{3t} \Vert ^{2}+ \beta \Vert \nabla \omega _{3} \Vert ^{2}\geq \frac{1}{2} \bigl\Vert \overline{z}_{d}(t) \bigr\Vert _{0}^{2}. $$
(29)
Therefore, \(E(t)\) satisfies
$$ \frac{1}{2} \bigl\Vert \overline{z}_{d}(t) \bigr\Vert _{0}^{2}\leq E(t) \leq \kappa \bigl\Vert \overline{z}_{d}(t) \bigr\Vert _{0}^{2}. $$
(30)
Let \(\delta =\min\{\delta _{1}, \delta _{2}\}>0\), by (26), we deduce that
$$ \frac{d}{dt}E(t)\leq -\delta \bigl\Vert \overline{z}_{d}(t) \bigr\Vert _{0}^{2}. $$
(31)
Combining (30) with (31), we get
$$ \frac{d}{dt}E(t)\leq -\frac{\delta}{\kappa}E(t). $$
(32)
Using (30) and the Gronwall lemma, we end up with
$$ \bigl\Vert \overline{z}_{d}(t) \bigr\Vert _{0}^{2} \leq 2\kappa e^{- \frac{\delta}{\kappa}t} \bigl\Vert \overline{z}_{d}(0) \bigr\Vert _{0}^{2}, $$
(33)
namely,
$$ \bigl\Vert S_{0}(t)z_{1}-S_{0}(t)z_{2} \bigr\Vert _{0}\leq \sqrt{2\kappa}e^{- \frac{\delta}{2\kappa}t} \Vert z_{1}-z_{2} \Vert _{0}. $$
(34)
Choose \(t_{\ast}=\frac{2\kappa}{\delta}\ln8\sqrt{2\kappa}\), we get
$$ \bigl\Vert S_{0}(t_{\ast})z_{1}-S_{0}(t_{\ast})z_{2} \bigr\Vert _{0} \leq \frac{1}{8} \Vert z_{1}-z_{2} \Vert _{0}. $$
(35)
For system (25), choose \(0<\varepsilon <1\). Taking the scalar product of the first and second equations of (25) with \(\Delta ^{2}\phi =\Delta ^{2}\omega _{2t}+\varepsilon \Delta ^{2} \omega _{2}\) and \(-\Delta \psi =-\Delta \omega _{4t}-\varepsilon \Delta \omega _{4}\) in \(L^{2}(0,L)\), respectively, we find
$$ \begin{aligned}[b]&\frac{1}{2}\frac{d}{dt}\bigl( \Vert \Delta \phi \Vert ^{2}+ \alpha \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}\bigr) \\ &\quad {}+\alpha \varepsilon \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+( \delta _{1}-\varepsilon ) \Vert \Delta \phi \Vert ^{2} - \varepsilon (\delta _{1}- \varepsilon ) \bigl(\omega _{2},\Delta ^{2}\phi \bigr) \\ &\quad {}+\beta \varepsilon \Vert \Delta \omega _{4} \Vert ^{2}+( \delta _{2}-\varepsilon ) \Vert \nabla \psi \Vert ^{2} - \varepsilon (\delta _{2}-\varepsilon ) ( \omega _{4},-\Delta \psi ) \\ &\quad {}+\bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+},\Delta ^{2}\phi \bigr)+\bigl(-k\bigl(u^{1}-v^{1} \bigr)^{+}+k\bigl(u^{2}-v^{2} \bigr)^{+},- \Delta \psi \bigr) \\ &\quad {}+\bigl(f_{B}\bigl(u^{1}\bigr)-f_{B} \bigl(u^{2}\bigr),\Delta ^{2}\phi \bigr)+ \bigl(f_{S}\bigl(v^{1}\bigr)-f_{S} \bigl(v^{2}\bigr),- \Delta \psi \bigr)=0. \end{aligned} $$
(36)
Thanks to Young’s inequality and Hölder’s inequality, we have
$$\begin{aligned} & \bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2}\phi \bigr) \\ &\quad=\frac{d}{dt}\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2} \omega _{2}\bigr)+ \varepsilon \bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}-\bigl(k\bigl(u^{1}-v^{1}\bigr)_{t}^{+}-k \bigl(u^{2}-v^{2}\bigr)_{t}^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\quad\geq \frac{d}{dt}\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2} \omega _{2}\bigr)+ \varepsilon \bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}-k \bigl\Vert \bigl(u^{1}-v^{1}\bigr)_{t}^{+}- \bigl(u^{2}-v^{2}\bigr)_{t}^{+} \bigr\Vert \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert \\ &\quad\geq \frac{d}{dt}\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2} \omega _{2}\bigr)+ \varepsilon \bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}-k \Vert \bar{u}_{t}-\bar{v}_{t} \Vert \bigl\Vert \Delta ^{2} \omega _{2} \bigr\Vert \\ &\quad\geq \frac{d}{dt}\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2} \omega _{2}\bigr)+ \varepsilon \bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}-k \Vert \bar{u}_{t} \Vert \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert - \Vert \bar{v}_{t} \Vert \bigl\Vert \Delta ^{2} \omega _{2} \bigr\Vert \\ &\quad\geq \frac{d}{dt}\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2} \omega _{2}\bigr)+ \varepsilon \bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}-\frac{\varepsilon \alpha}{4} \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}-\frac{8k^{2}}{\varepsilon \alpha} \Vert \bar{u}_{t} \Vert ^{2} -\frac{8k^{2}}{\varepsilon \alpha} \Vert \bar{v}_{t} \Vert ^{2} \end{aligned}$$
(37)
and
$$ \begin{aligned}[b]& \bigl(-k\bigl(u^{1}-v^{1} \bigr)^{+}+k\bigl(u^{2}-v^{2} \bigr)^{+},-\Delta \psi \bigr)\\ &\quad =-k\bigl(\bigl(u^{1}-v^{1} \bigr)_{x}^{+}-\bigl(u^{2}-v^{2} \bigr)_{x}^{+}, \nabla \psi \bigr) \\ & \quad\geq -k \bigl\Vert \bigl(u^{1}-v^{1} \bigr)_{x}^{+}-\bigl(u^{2}-v^{2} \bigr)_{x}^{+} \bigr\Vert \Vert \nabla \psi \Vert \geq -k \Vert \nabla \bar{u}-\nabla \bar{v} \Vert \Vert \nabla \psi \Vert \\ & \quad\geq -k \Vert \nabla \bar{u} \Vert \Vert \nabla \psi \Vert -k \Vert \nabla \bar{v} \Vert \Vert \nabla \psi \Vert \geq -\frac{\delta _{2}}{4} \Vert \nabla \psi \Vert ^{2}- \frac{8k^{2}}{\delta _{2}} \Vert \nabla \bar{u} \Vert ^{2}- \frac{8k^{2}}{\delta _{2}} \Vert \nabla \bar{v} \Vert ^{2}. \end{aligned} $$
(38)
Denote \(\varphi (t)=\theta u^{1}(t)+(1-\theta )u^{2}(t)\), \(\sigma (t)=\theta v^{1}(t)+(1- \theta )v^{2}(t)\), applying Lemma 3, we have
$$ \begin{gathered} \bigl\Vert \varphi _{t}(t) \bigr\Vert \leq \theta \bigl\Vert u_{t}^{1}(t) \bigr\Vert +(1-\theta ) \bigl\Vert u_{t}^{2}(t) \bigr\Vert \leq R_{0}, \\ \bigl\Vert \nabla \sigma (t) \bigr\Vert \leq \theta \bigl\Vert \nabla v^{1}(t) \bigr\Vert +(1-\theta ) \bigl\Vert \nabla v^{2}(t) \bigr\Vert \leq R_{0}. \end{gathered} $$
(39)
By (12) and (39), we achieve
$$ \begin{aligned}[b]\bigl(f_{B}\bigl(u^{1} \bigr)-f_{B}\bigl(u^{2}\bigr),\Delta ^{2}\phi \bigr)={}&\bigl(f'_{B}\bigl( \varphi (t)\bigr)\bar{u}, \Delta ^{2}\omega _{2t}\bigr)+\varepsilon \bigl(f'_{B}\bigl( \varphi (t)\bigr)\bar{u},\Delta ^{2}\omega _{2}\bigr) \\ ={}&\frac{d}{dt}\bigl(f'_{B}\bigl(\varphi (t)\bigr)\bar{u},\Delta ^{2}\omega _{2}\bigr)+ \varepsilon \bigl(f'_{B}\bigl(\varphi (t)\bigr) \bar{u},\Delta ^{2}\omega _{2}\bigr)\\ &{}- \bigl(f''_{B}\bigl( \varphi (t)\bigr) \varphi _{t}(t)\bar{u},\Delta ^{2}\omega _{2} \bigr)-\bigl(f'_{B}\bigl( \varphi (t)\bigr) \bar{u}_{t},\Delta ^{2}\omega _{2}\bigr) \\ \geq{}& \frac{d}{dt}\bigl(f'_{B}\bigl( \varphi (t)\bigr)\bar{u},\Delta ^{2}\omega _{2}\bigr)+ \varepsilon \bigl(f'_{B}\bigl(\varphi (t)\bigr) \bar{u},\Delta ^{2}\omega _{2}\bigr)\\ &{} -MR_{0} \Vert \bar{u} \Vert \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert -M \Vert \bar{u}_{t} \Vert \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert \\ \geq{}& \frac{d}{dt}\bigl(f'_{B}\bigl( \varphi (t)\bigr)\bar{u},\Delta ^{2}\omega _{2}\bigr)+ \varepsilon \bigl(f'_{B}\bigl(\varphi (t)\bigr) \bar{u},\Delta ^{2}\omega _{2}\bigr)\\ &{} - \frac{\varepsilon \alpha}{4} \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}-\frac{8M^{2}R_{0}^{2}}{\varepsilon \alpha} \Vert \bar{u} \Vert ^{2} -\frac{8M^{2}}{\varepsilon \alpha} \Vert \bar{u}_{t} \Vert ^{2} \end{aligned} $$
(40)
and
$$ \begin{aligned}[b]\bigl(f_{S}\bigl(v^{1} \bigr)-f_{S}\bigl(v^{2}\bigr),-\Delta \psi \bigr)&= \bigl(f'_{S}\bigl(\sigma (t)\bigr) \bar{v},-\Delta \psi \bigr)\\ &=\bigl(f''_{S}\bigl(\sigma (t)\bigr)\nabla \sigma (t)\bar{v}, \nabla \psi \bigr) +\bigl(f'_{S} \bigl(\sigma (t)\bigr)\nabla \bar{v},\nabla \psi \bigr) \\ & \geq -MR_{0} \Vert \bar{v} \Vert \Vert \nabla \psi \Vert -M \Vert \nabla \bar{v} \Vert \Vert \nabla \psi \Vert \\ &\geq -\frac{\delta _{2}}{4} \Vert \nabla \psi \Vert ^{2}- \frac{8M^{2}R_{0}^{2}}{\delta _{2}} \Vert \bar{v} \Vert ^{2}- \frac{8M^{2}}{\delta _{2}} \Vert \nabla \bar{v} \Vert ^{2}. \end{aligned} $$
(41)
Therefore, together with (36)–(41), it leads to
$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \Delta \phi \Vert ^{2}+\alpha \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+2\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}+2 \bigl(f'_{B}\bigl(\varphi (t)\bigr)\bar{u}, \Delta ^{2}\omega _{2}\bigr)\bigr)+\varepsilon \alpha \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+2(\delta _{1}- \varepsilon ) \Vert \Delta \phi \Vert ^{2} -2\varepsilon ( \delta _{1}-\varepsilon ) \bigl(\Delta ^{2}\omega _{2},\phi \bigr) \\ &\qquad{}+2\varepsilon \beta \Vert \Delta \omega _{4} \Vert ^{2}+2\biggl( \frac{\delta _{2}}{2}-\varepsilon \biggr) \Vert \nabla \psi \Vert ^{2} -2\varepsilon (\delta _{2}- \varepsilon ) (-\Delta \omega _{4},\psi ) \\ &\qquad{}+2\varepsilon \bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+},\Delta ^{2} \omega _{2}\bigr)+2 \varepsilon \bigl(f'_{B}\bigl(\varphi (t)\bigr) \bar{u},\Delta ^{2} \omega _{2}\bigr) \\ & \quad\leq 2\biggl(\frac{8k^{2}}{\varepsilon \alpha} \Vert \bar{u}_{t} \Vert ^{2} +\frac{8k^{2}}{\varepsilon \alpha} \Vert \bar{v}_{t} \Vert ^{2}+\frac{8k^{2}}{\delta _{2}} \Vert \nabla \bar{u} \Vert ^{2}+ \frac{8k^{2}}{\delta _{2}} \Vert \nabla \bar{v} \Vert ^{2} \\ &\qquad{}+\frac{8M^{2}R_{0}^{2}}{\varepsilon \alpha} \Vert \bar{u} \Vert ^{2} + \frac{8M^{2}}{\varepsilon \alpha} \Vert \bar{u}_{t} \Vert ^{2}+ \frac{8M^{2}R_{0}^{2}}{\delta _{2}} \Vert \bar{v} \Vert ^{2}+ \frac{8M^{2}}{\delta _{2}} \Vert \nabla \bar{v} \Vert ^{2}\biggr). \end{aligned}$$
(42)
Furthermore, by Young’s inequality and Hölder’s inequality, we have
$$ \begin{aligned}[b]& \varepsilon \alpha \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+2(\delta _{1}- \varepsilon ) \Vert \Delta \phi \Vert ^{2} -2\varepsilon (\delta _{1}-\varepsilon ) \bigl(\Delta ^{2} \omega _{2},\phi \bigr) \\ &\quad\geq \varepsilon \alpha \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2} +2(\delta _{1}- \varepsilon ) \Vert \Delta \phi \Vert ^{2}- \frac{2\varepsilon \delta _{1}}{\lambda} \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert \Vert \Delta \phi \Vert \\ &\quad\geq \frac{\varepsilon \alpha}{2} \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+2\biggl(\delta _{1}- \varepsilon - \frac{4\varepsilon \delta _{1}^{2}}{\lambda ^{2}\alpha}\biggr) \Vert \Delta \phi \Vert ^{2} \end{aligned} $$
(43)
and
$$ \begin{aligned}[b] & 2\varepsilon \beta \Vert \Delta \omega _{4} \Vert ^{2}+2\biggl(\frac{\delta _{2}}{2}-\varepsilon \biggr) \Vert \nabla \psi \Vert ^{2} -2\varepsilon (\delta _{2}- \varepsilon ) (- \Delta \omega _{4},\psi ) \\ &\quad\geq 2\varepsilon \beta \Vert \Delta \omega _{4} \Vert ^{2}+( \delta _{2}-2\varepsilon ) \Vert \nabla \psi \Vert ^{2} - \frac{2\varepsilon \delta _{2}}{\lambda} \Vert \Delta \omega _{4} \Vert \Vert \nabla \psi \Vert \\ & \quad\geq \varepsilon \beta \Vert \Delta \omega _{4} \Vert ^{2}+\biggl( \delta _{2}-2\varepsilon - \frac{4\varepsilon \delta _{2}^{2}}{\lambda ^{2}\beta}\biggr) \Vert \nabla \psi \Vert ^{2}. \end{aligned} $$
(44)
Thus, we can choose ε small enough such that
$$ \delta _{1}-\varepsilon - \frac{4\varepsilon \delta _{1}^{2}}{\lambda ^{2}\alpha}\geq \frac{\delta _{1}}{2},\qquad \delta _{2}-2\varepsilon - \frac{4\varepsilon \delta _{2}^{2}}{\lambda ^{2}\beta}\geq \frac{\delta _{2}}{2}. $$
And let \(\varepsilon _{0}=\min\{\frac{\varepsilon}{2}, \delta _{1}, \frac{\delta _{2}}{2}\}\), we conclude from (42) that
$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \Delta \phi \Vert ^{2}+\alpha \bigl\Vert \Delta ^{2}\omega _{2} \bigr\Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+2\bigl(k\bigl(u^{1}-v^{1} \bigr)^{+}-k\bigl(u^{2}-v^{2} \bigr)^{+}, \Delta ^{2}\omega _{2}\bigr) \\ &\qquad{}+ \bigl(f'_{B}\bigl(\varphi (t)\bigr)\bar{u},\Delta ^{2}\omega _{2}\bigr)\bigr)+\varepsilon _{0} \bigl( \Vert \Delta \phi \Vert ^{2}+\alpha \bigl\Vert \Delta ^{2} \omega _{2} \bigr\Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}+ \beta \Vert \Delta \omega _{4} \Vert ^{2} \\ &\qquad{}+2\bigl(k\bigl(u^{1}-v^{1}\bigr)^{+}-k \bigl(u^{2}-v^{2}\bigr)^{+},\Delta ^{2}\omega _{2}\bigr)+ \bigl(f'_{B} \bigl( \varphi (t)\bigr)\bar{u},\Delta ^{2}\omega _{2} \bigr)\bigr) \\ &\quad\leq 2\biggl(\frac{8k^{2}}{\varepsilon \alpha} \Vert \bar{u}_{t} \Vert ^{2} +\frac{8k^{2}}{\varepsilon \alpha} \Vert \bar{v}_{t} \Vert ^{2}+\frac{8k^{2}}{\delta _{2}} \Vert \nabla \bar{u} \Vert ^{2}+ \frac{8k^{2}}{\delta _{2}} \Vert \nabla \bar{v} \Vert ^{2} \\ &\qquad{}+\frac{8M^{2}R_{0}^{2}}{\varepsilon \alpha} \Vert \bar{u} \Vert ^{2} + \frac{8M^{2}}{\varepsilon \alpha} \Vert \bar{u}_{t} \Vert ^{2}+ \frac{8M^{2}R_{0}^{2}}{\delta _{2}} \Vert \bar{v} \Vert ^{2}+ \frac{8M^{2}}{\delta _{2}} \Vert \nabla \bar{v} \Vert ^{2}\biggr). \end{aligned}$$
(45)
Therefore, we arrive at
$$\begin{aligned} & \frac{d}{dt}\biggl( \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2} \omega _{2}+ \frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \biggr\Vert ^{2}+ \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B}\bigl(\varphi (t)\bigr) \bar{u} \biggr\Vert ^{2} \\ &\qquad{}+ \Vert \Delta \phi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}\biggr) \\ &\qquad{}+\varepsilon _{0}\biggl( \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \biggr\Vert ^{2} \\ &\qquad{}+ \biggl\Vert \sqrt{\frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B}\bigl( \varphi (t)\bigr)\bar{u} \biggr\Vert ^{2}+ \Vert \Delta \phi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}\biggr) \\ &\quad\leq \biggl(\frac{16k^{2}}{\varepsilon \alpha}+ \frac{16M^{2}}{\varepsilon \alpha}\biggr) \Vert \bar{u}_{t} \Vert ^{2} +\frac{16k^{2}}{\varepsilon \alpha} \Vert \bar{v}_{t} \Vert ^{2}+ \frac{16k^{2}}{\delta _{2}} \Vert \nabla \bar{u} \Vert ^{2}+ \biggl( \frac{16k^{2}}{\delta _{2}}+ \frac{16M^{2}}{\delta _{2}}\biggr) \Vert \nabla \bar{v} \Vert ^{2} \\ &\qquad{}+\frac{16M^{2}R_{0}^{2}}{\varepsilon \alpha} \Vert \bar{u} \Vert ^{2}+ \frac{16M^{2}R_{0}^{2}}{\delta _{2}} \Vert \bar{v} \Vert ^{2} \\ &\qquad{}+ \frac{4k^{2}}{\alpha} \int _{\Omega} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \bigl(\bigl(u^{1}-v^{1} \bigr)_{t}^{+}-\bigl(u^{2}-v^{2} \bigr)_{t}^{+}\bigr)\,dx \\ &\qquad{}+\frac{4}{\alpha}\bigl(f''_{B}( \varphi )\varphi _{t}(t)\bar{u},f'_{B}( \varphi )\bar{u}\bigr) +\frac{4}{\alpha}\bigl(f'_{B}( \varphi )\bar{u}_{t},f'_{B}( \varphi ) \bar{u}\bigr) \\ &\qquad{}+\frac{2\varepsilon _{0}k^{2}}{\alpha} \bigl\Vert \bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+} \bigr\Vert ^{2} +\frac{2\varepsilon _{0}}{\alpha} \bigl\Vert f'_{B}(\varphi )\bar{u} \bigr\Vert ^{2}. \end{aligned}$$
(46)
Moreover, by exploiting conditions (12), (39) and Young’s inequality, as well as Hölder’s inequality, we have
$$ \begin{aligned}[b]& \frac{4k^{2}}{\alpha} \int _{\Omega} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \bigl(\bigl(u^{1}-v^{1} \bigr)_{t}^{+}-\bigl(u^{2}-v^{2} \bigr)_{t}^{+}\bigr)\,dx \\ &\quad\leq \frac{4k^{2}}{\alpha} \bigl\Vert \bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+} \bigr\Vert \bigl\Vert \bigl(u^{1}-v^{1} \bigr)_{t}^{+}-\bigl(u^{2}-v^{2} \bigr)_{t}^{+} \bigr\Vert \\ & \quad\leq \frac{4k^{2}}{\alpha} \Vert \bar{u}-\bar{v} \Vert \Vert \bar{u}_{t}-\bar{v}_{t} \Vert \\ & \quad\leq \frac{2k^{2}}{\alpha} \Vert \bar{u}-\bar{v} \Vert ^{2}+ \frac{2k^{2}}{\alpha} \Vert \bar{u}_{t}-\bar{v}_{t} \Vert ^{2} \\ & \quad\leq \frac{4k^{2}}{\alpha} \Vert \bar{u} \Vert ^{2}+ \frac{4k^{2}}{\alpha} \Vert \bar{v} \Vert ^{2}+ \frac{4k^{2}}{\alpha} \Vert \bar{u}_{t} \Vert ^{2}+ \frac{4k^{2}}{\alpha} \Vert \bar{v}_{t} \Vert ^{2} \end{aligned} $$
(47)
and
$$ \frac{4}{\alpha}\bigl(f''_{B}( \varphi )\varphi _{t}(t)\bar{u},f'_{B}( \varphi )\bar{u}\bigr)\leq \frac{4}{\alpha}M^{2}R_{0} \Vert \bar{u} \Vert ^{2}, $$
(48)
and
$$ \frac{4}{\alpha}\bigl(f'_{B}(\varphi ) \bar{u}_{t},f'_{B}(\varphi )\bar{u}\bigr) \leq \frac{4}{\alpha}M^{2} \Vert \bar{u}_{t} \Vert \Vert \bar{u} \Vert \leq \frac{2M^{2}}{\alpha} \Vert \bar{u}_{t} \Vert ^{2}+\frac{2M^{2}}{\alpha} \Vert \bar{u} \Vert ^{2}, $$
(49)
and
$$\begin{aligned}& \frac{2\varepsilon _{0}k^{2}}{\alpha} \bigl\Vert \bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+} \bigr\Vert ^{2}\leq \frac{2\varepsilon _{0}k^{2}}{\alpha} \Vert \bar{u}-\bar{v} \Vert ^{2}\leq \frac{4\varepsilon _{0}k^{2}}{\alpha} \Vert \bar{u} \Vert ^{2}+ \frac{4\varepsilon _{0}k^{2}}{\alpha} \Vert \bar{v} \Vert ^{2}, \end{aligned}$$
(50)
$$\begin{aligned}& \frac{2\varepsilon _{0}}{\alpha} \bigl\Vert f'_{B}(\varphi ) \bar{u} \bigr\Vert ^{2}\leq \frac{2\varepsilon _{0}M^{2}}{\alpha} \Vert \bar{u} \Vert ^{2}. \end{aligned}$$
(51)
Combining with (46)–(51), it leads to
$$\begin{aligned} & \frac{d}{dt}\biggl( \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2} \omega _{2}+ \frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \biggr\Vert ^{2}+ \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B}\bigl(\varphi (t)\bigr) \bar{u} \biggr\Vert ^{2} \\ &\qquad{}+ \Vert \Delta \phi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}\biggr) \\ &\qquad{}+\varepsilon _{0}\biggl( \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \biggr\Vert ^{2} \\ &\qquad{}+ \biggl\Vert \sqrt{\frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B}\bigl( \varphi (t)\bigr)\bar{u} \biggr\Vert ^{2}+ \Vert \Delta \phi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}\biggr) \\ &\quad\leq \biggl(\frac{16k^{2}}{\varepsilon \alpha}+ \frac{16M^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha} + \frac{2M^{2}}{\alpha}\biggr) \Vert \bar{u}_{t} \Vert ^{2}+\biggl( \frac{16k^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha}\biggr) \Vert \bar{v}_{t} \Vert ^{2} \\ &\qquad{}+\biggl( \frac{16k^{2}}{\delta _{2}}+ \frac{16M^{2}}{\delta _{2}}\biggr) \Vert \nabla \bar{v} \Vert ^{2} \\ &\qquad{}+ \frac{16k^{2}}{\delta _{2}} \Vert \nabla \bar{u} \Vert ^{2}+\biggl( \frac{16M^{2}R_{0}^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha} + \frac{4M^{2}R_{0}}{\alpha}+ \frac{2M^{2}}{\alpha}+ \frac{4\varepsilon _{0}k^{2}}{\alpha}+ \frac{4\varepsilon _{0}M^{2}}{\alpha}\biggr) \Vert \bar{u} \Vert ^{2} \\ &\qquad{}+\biggl(\frac{16M^{2}R_{0}^{2}}{\delta _{2}}+\frac{4k^{2}}{\alpha}+ \frac{4\varepsilon _{0}k^{2}}{\alpha} \biggr) \Vert \bar{v} \Vert ^{2} \\ & \quad\leq \biggl(\frac{16k^{2}}{\varepsilon \alpha}+ \frac{16M^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha} + \frac{2M^{2}}{\alpha}\biggr) \Vert \bar{u}_{t} \Vert ^{2}+\biggl( \frac{16k^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha}\biggr) \Vert \bar{v}_{t} \Vert ^{2} \\ &\qquad{}+\biggl(\frac{16k^{2}}{\delta _{2}}+ \frac{16M^{2}R_{0}^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha} + \frac{4M^{2}R_{0}}{\alpha}+\frac{2M^{2}}{\alpha}+ \frac{4\varepsilon _{0}k^{2}}{\alpha}+ \frac{4\varepsilon _{0}M^{2}}{\alpha}\biggr)/ \lambda ^{2} \Vert \Delta \bar{u} \Vert ^{2} \\ &\qquad{}+\biggl(\frac{16k^{2}}{\delta _{2}}+\frac{16M^{2}}{\delta _{2}}+ \biggl( \frac{16M^{2}R_{0}^{2}}{\delta _{2}}+\frac{4k^{2}}{\alpha}+ \frac{4\varepsilon _{0}k^{2}}{\alpha}\biggr)/\lambda ^{2}\biggr) \Vert \nabla \bar{v} \Vert ^{2}. \end{aligned}$$
(52)
Let
$$ \begin{aligned}\Lambda ={}&\max\biggl\{ \frac{16k^{2}}{\varepsilon \alpha}+ \frac{16M^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha} + \frac{2M^{2}}{\alpha}, \biggl( \frac{16k^{2}}{\delta _{2}}+ \frac{16M^{2}R_{0}^{2}}{\varepsilon \alpha}+\frac{4k^{2}}{\alpha} + \frac{4M^{2}R_{0}}{\alpha} \\ &{}+\frac{2M^{2}}{\alpha}+\frac{4\varepsilon _{0}k^{2}}{\alpha}+ \frac{4\varepsilon _{0}M^{2}}{\alpha}\biggr)\biggm/ \lambda ^{2}\alpha , \biggl( \frac{16k^{2}}{\delta _{2}}+ \frac{16M^{2}}{\delta _{2}}\biggr)\biggm/\beta\\ &{} + \biggl( \frac{16M^{2}R_{0}^{2}}{\delta _{2}}+ \frac{4k^{2}}{\alpha}+ \frac{4\varepsilon _{0}k^{2}}{\alpha}\biggr)\biggm/\lambda ^{2}\beta )\biggr\} . \end{aligned} $$
We can deduce from (52) that
$$\begin{aligned} &\frac{d}{dt}\biggl( \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2} \omega _{2}+ \frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}-v^{1} \bigr)^{+}-\bigl(u^{2}-v^{2} \bigr)^{+}\bigr) \biggr\Vert ^{2}+ \biggl\Vert \sqrt{ \frac{\alpha}{2}}\Delta ^{2}\omega _{2}+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B}\bigl(\varphi (t)\bigr) \bar{u} \biggr\Vert ^{2} \\ &\qquad{}+ \Vert \Delta \phi \Vert ^{2}+\beta \Vert \Delta \omega _{4} \Vert ^{2}+ \Vert \nabla \psi \Vert ^{2}\biggr) \\ &\quad\leq \Lambda \bigl(\alpha \Vert \Delta \bar{u} \Vert ^{2}+\beta \Vert \nabla \bar{v} \Vert ^{2} + \Vert \bar{u}_{t} \Vert ^{2}+ \Vert \bar{v}_{t} \Vert ^{2}\bigr)=\Lambda \bigl\Vert \bar{z}(t) \bigr\Vert _{0}^{2}\leq \Lambda e^{Kt} \Vert z_{1}-z_{2} \Vert _{0}^{2}. \end{aligned}$$
(53)
Integrating (53) over \((0,t_{\ast})\), we have that
$$\begin{aligned} &\biggl\Vert \sqrt{\frac{\alpha}{2}}\Delta ^{2}\omega _{2}(t_{ \ast})+\frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}(t_{\ast})-v^{1}(t_{ \ast}) \bigr)^{+}-\bigl(u^{2}(t_{\ast})-v^{2}(t_{\ast}) \bigr)^{+}\bigr) \biggr\Vert ^{2} \\ &\qquad{}+ \biggl\Vert \sqrt{\frac{\alpha}{2}}\Delta ^{2}\omega _{2}(t_{\ast})+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B} \bigl(\varphi (t_{\ast})\bigr)\bar{u}(t_{ \ast}) \biggr\Vert ^{2} \\ &\qquad{}+ \bigl\Vert \Delta \phi (t_{\ast}) \bigr\Vert ^{2}+ \beta \bigl\Vert \Delta \omega _{4}(t_{\ast}) \bigr\Vert ^{2}+ \bigl\Vert \nabla \psi (t_{\ast}) \bigr\Vert ^{2} \\ &\quad\leq \biggl\Vert \sqrt{\frac{\alpha}{2}}\Delta ^{2}\omega _{2}(0)+ \frac{\sqrt{2}k}{\sqrt{\alpha}} \bigl(\bigl(u^{1}(0)-v^{1}(0) \bigr)^{+}-\bigl(u^{2}(0)-v^{2}(0) \bigr)^{+}\bigr) \biggr\Vert ^{2} \\ &\qquad{}+ \biggl\Vert \sqrt{\frac{\alpha}{2}}\Delta ^{2}\omega _{2}(0)+ \frac{\sqrt{2}}{\sqrt{\alpha}} f'_{B} \bigl(\varphi (0)\bigr)\bar{u}(0) \biggr\Vert ^{2}+ \bigl\Vert \Delta \phi (0) \bigr\Vert ^{2}+\beta \bigl\Vert \Delta \omega _{4}(0) \bigr\Vert ^{2}+ \bigl\Vert \nabla \psi (0) \bigr\Vert ^{2} \\ &\qquad{}+ \int _{0}^{t_{\ast}}\Lambda e^{Kt} \Vert z_{1}-z_{2} \Vert _{0}^{2}\,dt \end{aligned}$$
(54)
$$\begin{aligned} &\quad\leq \frac{\Lambda}{K}\bigl(e^{Kt_{\ast}}-1\bigr) \Vert z_{1}-z_{2} \Vert _{0}^{2}+ \alpha \bigl\Vert \Delta ^{2}\omega _{2}(0) \bigr\Vert ^{2} \\ &\qquad{}+\frac{4k^{2}}{\alpha} \bigl\Vert \bigl(u^{1}(0)-v^{1}(0) \bigr)^{+}-\bigl(u^{2}(0)-v^{2}(0) \bigr)^{+} \bigr\Vert ^{2} \\ &\qquad{}+\alpha \bigl\Vert \Delta ^{2}\omega _{2}(0) \bigr\Vert ^{2}+ \frac{4}{\alpha} \bigl\Vert f'_{B} \bigl(\varphi (0)\bigr)\bar{u}(0) \bigr\Vert ^{2}+ \bigl\Vert \Delta \phi (0) \bigr\Vert ^{2}+\beta \bigl\Vert \Delta \omega _{4}(0) \bigr\Vert ^{2}+ \bigl\Vert \nabla \psi (0) \bigr\Vert ^{2} \\ &\quad\leq \frac{\Lambda}{K}\bigl(e^{Kt_{\ast}}-1\bigr) \Vert z_{1}-z_{2} \Vert _{0}^{2}+ \frac{4k^{2}}{\alpha} \bigl\Vert \bar{u}(0)-\bar{v}(0) \bigr\Vert ^{2}+\frac{4}{\alpha} \bigl\Vert f'_{B} \bigl(\varphi (0)\bigr)\bar{u}(0) \bigr\Vert ^{2} \\ &\quad\leq \frac{\Lambda}{K}\bigl(e^{Kt_{\ast}}-1\bigr) \Vert z_{1}-z_{2} \Vert _{0}^{2}+ \frac{8k^{2}}{\alpha}\bigl( \bigl\Vert \bar{u}(0) \bigr\Vert ^{2}+ \bigl\Vert \bar{v}(0) \bigr\Vert ^{2}\bigr)+ \frac{4M^{2}}{\alpha} \bigl\Vert \bar{u}(0) \bigr\Vert ^{2} \\ &\quad\leq \frac{\Lambda}{K}\bigl(e^{Kt_{\ast}}-1\bigr)+\biggl( \frac{8k^{2}}{\alpha}+ \frac{4M^{2}}{\alpha}\biggr)/\lambda ^{2} \bigl\Vert \Delta \bar{u}(0) \bigr\Vert ^{2}+\frac{8k^{2}}{\alpha \lambda ^{2}} \bigl\Vert \nabla \bar{v}(0) \bigr\Vert ^{2} \\ & \quad\leq \frac{\Lambda}{K}\bigl(e^{Kt_{\ast}}-1\bigr)+\biggl( \frac{8k^{2}}{\alpha}+ \frac{4M^{2}}{\alpha}\biggr)/\alpha \lambda ^{2} \Vert z_{1}-z_{2} \Vert _{0}^{2}+\frac{8k^{2}}{\alpha \beta \lambda ^{2}} \Vert z_{1}-z_{2} \Vert _{0}^{2} \\ &\quad\leq C_{\ast} \Vert z_{1}-z_{2} \Vert _{0}^{2}, \end{aligned}$$
(55)
where \(C_{\ast}=\frac{\Lambda}{K}(e^{Kt_{\ast}}-1)+ \frac{8k^{2}+4M^{2}}{\alpha ^{2}\lambda ^{2}}+ \frac{8k^{2}}{\alpha \beta \lambda ^{2}}\). Applying (12), Hölder’s inequality, and Cauchy’s inequality as well as Lemma 3, we conclude from (54) that
$$ \alpha \bigl\Vert \Delta ^{2}\omega _{2}(t_{\ast}) \bigr\Vert ^{2}+ \beta \bigl\Vert \Delta \omega _{4}(t_{\ast}) \bigr\Vert ^{2} + \bigl\Vert \Delta \phi (t_{\ast}) \bigr\Vert ^{2}+ \bigl\Vert \nabla \psi (t_{ \ast}) \bigr\Vert ^{2}\leq C_{\ast} \Vert z_{1}-z_{2} \Vert _{0}^{2}, $$
namely,
$$ \Vert \bar{z}_{\ast} \Vert _{1}^{2}\leq C_{\ast} \Vert z_{1}-z_{2} \Vert _{0}^{2}. $$
This completes the proof of Lemma 9. □
Our main result reads as follows.
Theorem 10
Under conditions \((F1)\)–\((F2)\), the semigroup \(S(t)\) acting on \(E_{0}\) possesses an exponential attractor \(\mathcal{E}\).
Proof
Lemma 8, Lemma 9, and Theorem 2 imply the existence of an exponential attractor. □