Longtime dynamics of solutions for higher-order $(m_{1},m_{2})$-coupled Kirchhoff models with higher-order rotational inertia and nonlocal damping

Abstract

The Kirchhoff model is derived from the vibration problem of stretchable strings. This paper focuses on the longtime dynamics of a higher-order $(m_{1},m_{2})$-coupled Kirchhoff system with higher-order rotational inertia and nonlocal damping. We first obtain the state of the model’s solutions in different spaces through prior estimation. After that, we immediately prove the existence and uniqueness of their solutions in different spaces through the Faedo-Galerkin method. Subsequently, we prove their family of global attractors using the compactness theorem. Finally, we reflect on the subsequent research of the model and point out relevant directions for further research on the model. In this way, we systematically study the longtime dynamics of the higher-order $(m_{1},m_{2})$-coupled Kirchhoff model with higher-order rotational inertia, thus enriching the relevant findings of higher-order coupled Kirchhoff models and laying a theoretical foundation for future practical applications.

1 Introduction

This study considers the longtime dynamics of the following higher-order $(m_{1},m_{2})$-coupled Kirchhoff model in a bounded smooth domain $\Omega \subset \mathbb{R}^{n}$:

$$ \textstyle\begin{cases} (1+\alpha (-\Delta )^{m_{1}})u_{tt} +N_{1}( \Vert \nabla ^{m_{1}}u \Vert ^{2})(- \Delta )^{m_{1}}u_{t}+M( \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2})(-\Delta )^{m_{1}}u \\ \quad {}+ g_{1}(u,v) =f_{1}(x),\\ (1+\beta (-\Delta )^{m_{2}})v_{tt}+N_{2}( \Vert \nabla ^{m_{2}}v \Vert ^{2})(- \Delta )^{m_{2}}v_{t}+M( \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2})(-\Delta )^{m_{2}}v \\ \quad {}+ g_{2}(u,v) =f_{2}(x),\end{cases} $$

(1)

under the following boundary conditions:

$$ \begin{aligned} &u(x)=0,\qquad \frac{\partial ^{i}u}{\partial \mathbf{n}^{i}}=0, \quad i=1,\dots ,m_{1}-1,m_{1}>1, \\ &v(x)=0,\qquad \frac{\partial ^{j}v}{\partial \mathbf{n}^{j}}=0,\quad j=1, \dots ,m_{2}-1,m_{2}>1, \end{aligned} $$

(2)

and the following initial conditions:

$$ \begin{aligned} &u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \qquad v(x,0)=v_{0}(x), \\ &v_{t}(x,0)=v_{1}(x),\quad x\in \Omega , \end{aligned} $$

(3)

where Δ is the Laplace operator, $\alpha \in (0,1]$ and $\beta \in (0,1]$ are rotational coefficients, $N_{1}$, $N_{2}$, $M_{1}$, and $M_{2}$ are scalar functions specified later, $g_{1}$ and $g_{2}$ are the given source terms, and $f_{1}$ and $f_{2}$ are the given functions.

Equation (1) is a set of generalized higher-order quasilinear wave equations. The proposed equation in this paper originated from the stretchable string vibration problem established by Kirchhoff in 1883:

$$ \rho h\frac{\partial ^{2}u}{\partial t^{2}}= \biggl\{ p_{0}+ \frac{Eh}{2L} \int _{0}^{L}\biggl(\frac{\partial u}{\partial x} \biggr)^{2}\,\mathrm{d}x \biggr\} \frac{\partial ^{2}u}{\partial t^{2}}, $$

(4)

where $u=u(x,t)$ is the lateral displacement at space coordinate x and time coordinate t, $0< x< L$, $t\ge 0$, E is the Young’s modulus, ρ is the mass density, h is the cross-sectional area, L is the length, and $p_{0}$ is the initial axial tension. In recent decades, the long-term behaviors of Kirchhoff equations in various forms have attracted much academic attention, and for abundant research results for some related system, we can refer to [1–12].

For instance, Chueshov [1] studied the well-posedness and long-term dynamic behaviors of the following Kirchhoff equation with a nonlinear strong damping term:

$$ u_{tt}+\sigma \bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u_{t}-\phi \bigl( \Vert \nabla u \Vert ^{2}\bigr) \Delta u+f(u)=h(x). $$

(5)

Moreover, Lin, Lv, and Lou [2] studied the global dynamics of the following generalized nonlinear Kirchhoff–Boussinesq equation with strong damping:

$$ u_{tt}+\alpha u_{t}-\beta \Delta u_{t}+\Delta ^{2}u=\operatorname{div}\bigl(g\bigl( \vert \nabla u \vert ^{2}\bigr)\nabla u\bigr)+\Delta h(u)+f(x). $$

(6)

This paper proved that the semigroup conformed to the squeezing property of the system, while demonstrating the existence of an exponential attractor. Then, the spectral interval theory verified that the system had an inertial manifold.

Nakao [3] investigated the initial-boundary value problem of a quasilinear Kirchhoff-type wave equation with standard dissipation $u_{t}$:

$$ u_{tt}-\bigl(1+ \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}\bigr)\Delta u+u_{t}+g(x,u)=f(x). $$

(7)

Under an external force, the stretchable string undergoes elastic deformation. Over time, elastic mechanics methods may not fully reflect the actual long-term characteristics of the string, and increasing attention is directed to the long-term properties of the strings with the rotational inertia effect. Wave equations with rotational inertia have become a research hotspot in mathematics and physics.

Chueshov and Lasiecka [13] proposed a plate model with rotational inertia,

$$\begin{aligned} &(1-\alpha \Delta )u_{tt}+\Delta ^{2}u- \beta \Delta u_{t}+\bigl(Q- \Vert \nabla u \Vert ^{2} \bigr)\Delta u=P(u,u_{t}), \end{aligned}$$

(8)

where $\alpha \ge 0$ represents the rotational inertia parameter, $\beta >0$ is the damping coefficient, Q is a parameter describing the internal stress acting on the plate, and P is a function representing the lateral load that may depend on u and $u_{t}$. When $\alpha >0$, this model of transverse inertia becomes the Rayleigh plate equation, and (8) is a pure hyperbolic problem. When $\alpha =0$, (8) becomes a Berger plate model with structural damping. Chueshov and Lasiecka [13] studied the well-posedness and longtime dynamic behavior of (8).

Niimura [14] studied the long-term dynamic behavior of autonomous beam equations with nonlocal structural damping and rotational inertia under initial boundary value conditions:

$$ (1-\alpha \Delta )u_{tt}+\Delta ^{2}u-N \bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u_{t}-M \bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u+f(u)=h(x). $$

(9)

The well-posedness of the global solution was established, and the existence of a global attractor was proved for the autonomous infinite dynamical system corresponding to $\alpha \in [0,1]$, while the existence of an exponential attractor was demonstrated.

With the advance of research, scholars have shifted their focus on the dynamics of higher-order Kirchhoff equations. Ye and Tao [15] studied the initial-boundary value problem of the following higher-order Kirchhoff-type equation with a nonlinear dissipation term:

$$ u_{tt}+\Phi \bigl( \bigl\Vert D^{m}u \bigr\Vert ^{2}\bigr) (-\Delta )^{m}u+a \vert u_{t} \vert ^{q-2}u_{t}=b \vert u \vert ^{r-2}u. $$

(10)

Lin and Zhu [16] studied the initial-boundary value problem of the following nonlinear nonlocal higher-order Kirchhoff-type equations:

$$ u_{tt}+M\bigl( \bigl\Vert D^{m}u \bigr\Vert ^{2}\bigr) (-\Delta )^{m}u+\beta (-\Delta )^{m}u_{t}+g(x,u_{t})=f(x). $$

(11)

The existence and uniqueness of the solutions were demonstrated, and the existence of a global attractor family was confirmed using the compactness method, thus obtaining the finite Hausdorff and fractal dimensions.

Ding and Yang [17] investigated the well-posedness, regularity, and longtime behavior of solutions for an extensible beam equation with fractional rotational inertia and structural nonlinear damping: $(1+(-\Delta )^{\theta})u_{tt}+\Delta ^{2}u-M(\|\nabla u\|^{2}) \Delta u+N(\|\nabla u\|^{2})(-\Delta )^{\omega}u_{t}+f(u)=g$, where the dissipative index is $\omega \in (0,1]$, and the rotational index is $\theta \in [0,\omega )$. To the best of our knowledge, a comprehensive study on the long-term dynamics of coupled Kirchhoff models incorporating rotational inertia is yet to be reported.

Originating from physics, a system coupling measures the dependence of two entities on each other. With suitable conditions or parameters, a connected system can be coupled, and its potential energy can enable the generation of new functions by combining the structural functions of different systems. As mathematical equations derived from physics, the Kirchhoff model is naturally considered a coupled system, and scholars gradually considered the dynamics of coupled Kirchhoff equations. For example, Wang and Zhang [18] studied the long-term dynamics of coupled beam equations with strong damping under nonlinear boundary conditions. Lin and Zhang [19] studied the initial boundary value problem of the following Kirchhoff coupling group with a source term and strong damping:

$$ \textstyle\begin{cases} u_{tt}-\beta \Delta u_{t}-M( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2})\Delta u+g_{1}(u,v)=f_{1}(x), \\ v_{tt}-\beta \Delta v_{t}-M( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2})\Delta v+g_{2}(u,v)=f_{2}(x). \end{cases} $$

(12)

The finite Hausdorff dimension of the global attractor was obtained in a previous work [19].

In recent years, Lin et al. [20–23] explored the dynamics of higher-order coupled Kirchhoff equations and obtained a series of excellent results.

Few existing studies have focused on higher-order coupled Kirchhoff problems, and higher-order $(m_{1},m_{2})$-coupled Kirchhoff models with a nonlinear strong damping have not been studied. The main difficulties lie in the estimation and processing of the harmonic term and the nonlinear damping term. In addition, the nonlinear damping also brings challenges when proving the uniqueness. Therefore, we propose a higher-order coupled Kirchhoff model with higher-order rotational inertia. Under reasonable assumptions, this paper overcame these difficulties by using Hölder, Young, Poincaré, and Gagliardo–Nirenberg inequalities, thus obtaining the global solution and the global attractor family. The conclusions could fill the gap of the global attractor family for higher-order coupled models with higher-order rotational inertia (regardless of whether $m_{1}$ equals $m_{2}$) and lay the foundation for subsequent engineering applications.

The rest of this paper is organized as follows. Section 2 provides the fundamentals for this work and states the main results. Section 3 proves the main results. Finally, the summary and prospects are presented in Sect. 4.

2 Preparatory knowledge and statement of main results

This section introduces the assumptions for this work and presents the main results.

In this paper, $\|\cdot \|$ and $(\cdot ,\cdot )$ denote the norm and inner product in $H=L^{2}(\Omega )$. Let $V_{k}=D((-\Delta )^{\frac {k}{2}})$ be the scale of the Hilbert space generated by the Laplacian with Dirichlet boundary condition on H and endowed with standard inner product and norm, respectively, $(\cdot ,\cdot )_{V_{k}}=((-\Delta )^{\frac {k}{2}}\cdot ,(-\Delta )^{ \frac {k}{2}}\cdot )$, and $\|\cdot \|_{V_{k}}=\|(-\Delta )^{\frac {k}{2}}\cdot \|$. The main goal here is to study the well-posedness and long-term dynamics of problem (1)–(3) under the following set of assumptions:

• (A1) Function $M(s)$ is continuous on the interval $[0,+\infty )$, $M(s)\in C^{1}(\mathbb{R}^{+})$, and

  • 1) $M'(s)\ge 0$,

  • 2) $M(0)\equiv M_{0}>0$.

• (A2) For any $u,v\in H$, if $J(u,v)=\int _{\Omega}[G_{1}(u,v)+G_{2}(u,v)]\,\mathrm{d}x$, where $G_{1}(u,v)=\int _{0}^{u}g_{1}(s,v)\,\mathrm{d}s$, $G_{2}(u,v)=\int _{0}^{v}g_{2}(u,s)\,\mathrm{d}s$, then for any $\mu \ge 0$, there exist $C_{1}\ge 0$, $C_{\mu}\ge 0$, $C'_{\mu}\ge 0$ such that

  $$\begin{aligned}& G_{1}(u,v)+G_{2}(u,v)-C_{1}J(u,v)+\mu \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)\ge -C_{\mu}, \\& J(u,v)+\mu \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)\ge -C'_{ \mu}. \end{aligned}$$

• (A3) Function $g_{j}(u,v)\in C^{1}(\mathbb{R})$ ($j=1,2$), and

  $$\begin{aligned}& \bigl\vert g_{j}(u,v) \bigr\vert \le C_{2}\bigl(1+ \vert u \vert ^{p_{j}}+ \vert v \vert ^{q_{j}}\bigr); \\& \bigl\vert g_{ju}(u,v) \bigr\vert \le C_{3} \bigl(1+ \vert u \vert ^{p_{j}-1}+ \vert v \vert ^{q_{j}} \bigr); \\& \bigl\vert g_{jv}(u,v) \bigr\vert \le C_{4} \bigl(1+ \vert u \vert ^{p_{j}}+ \vert v \vert ^{q_{j}-1} \bigr). \end{aligned}$$

  Specifically, when $n=1,2$, $1\le p_{j}(q_{j})$; when $3\le n\le 2m$, $1\le p_{j}(q_{j})\le \frac{n}{n-2}$; when $2m< n$, $1\le p_{j}(q_{j})\le \frac{n}{n-2m}$, where $m=\min \{m_{1},m_{2}\}$.

• (A4) One has $N_{j}(s_{j})\ge N_{j0}$, where $N_{j0}$ ($j=1,2$) are positive constants and $\rho _{1},\rho _{2}>0$. Thus, $M(s_{1}+s_{2})-\rho _{1} N_{1}(s_{1})-\rho _{2} N_{2}(s_{2})>0$.

Then, the research phase space of this study is obtained as follows:

$$\begin{aligned}& V_{0} =H,\qquad V_{1}=H_{0}^{1}( \Omega ),\qquad V_{k}=H^{k}(\Omega ) \cap H_{0}^{1}(\Omega ), \\& X_{\alpha 0\times \beta 0} =V_{m_{1}}\times V_{m_{1}}\times V_{m_{2}} \times V_{m_{2}}, \\& X_{\alpha k_{1}\times \beta k_{2}} =V_{m_{1}+k_{1}}\times V_{m_{1}+k_{1}} \times V_{m_{2}+k_{2}}\times V_{m_{2}+k_{2}}, \\& k_{1} =0,1,2,\dots ,m_{1},\qquad k_{2}=0,1,2, \dots ,m_{2}, \end{aligned}$$

and the norms of the corresponding spaces are as follows:

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha k_{1}\times \beta k_{2}}}&= \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2}. \end{aligned} $$

Meanwhile, the general form of the Poincaré inequality is $\lambda _{1}\|\nabla ^{r}u\|^{2}\le \|\nabla ^{r+1}u\|^{2}$, where $\lambda _{1}$ is the first eigenvalue of −Δ with a homogeneous Dirichlet boundary on Ω. In this paper, $C_{i}$ is a constant, and $C(\cdot )$ is a constant depending on the parameters in the parentheses.

The main results of this paper are as follows.

Theorem 1

Suppose that assumptions (A1)–(A4) hold. If $f_{1}\in V_{k_{1}}$, $f_{2}\in V_{k_{2}}$ and initial data $(u_{0},u_{1},v_{0},v_{1})\in X_{\alpha k_{1}\times \beta k_{2}}$, $k_{1}=0,1,2, \dots ,m_{1}$, $k_{2}=0,1,2,\dots ,m_{2}$, then for $\forall \alpha , \beta \in (0,1]$, problem (1)–(3) admits a unique solution $(u, v)$ satisfying

$$\begin{aligned}& u \in L^{\infty}(0,\infty ;V_{m_{1}+k_{1}}); \\& u_{t} \in L^{\infty}(0,\infty ;V_{m_{1}+k_{1}})\cap L^{2}(0,T;V_{m_{1}+k_{1}}); \\& v \in L^{\infty}(0,\infty ;V_{m_{2}+k_{2}}); \\& v_{t} \in L^{\infty}(0,\infty ;V_{m_{2}+k_{2}})\cap L^{2}(0,T;V_{m_{2}+k_{2}}). \end{aligned}$$

Theorem 2

Suppose that assumptions (A1)–(A4) hold. If $f_{1}\in V_{m_{1}}$, $f_{2}\in V_{m_{2}}$ and initial data $(u_{0},u_{1},v_{0},v_{1})\in X_{\alpha m_{1}\times \beta m_{2}}$, then for $\forall \alpha , \beta \in (0,1]$, problem (1)–(3) has a global attractor family $\mathcal{A}$ in $X_{\alpha 0\times \beta 0}$:

$$\begin{aligned}& \mathcal{A}=\{A_{\alpha k_{1}\times \beta k_{2}}\},\qquad A_{\alpha k_{1} \times \beta k_{2}}=\omega (B_{{\alpha k_{1}\times \beta k_{2}},0})= \bigcap_{\tau \ge 0} \overline{ \bigcup_{t\ge \tau}S(t)B_{{\alpha k_{1}\times \beta k_{2}},0}}, \\& k_{1}=1,2,\dots ,m_{1},\qquad k_{2}=1,2, \dots ,m_{2}, \end{aligned}$$

where $B_{{\alpha k_{1}\times \beta k_{2}},0}=\{(u,u_{t},v,v_{t})\in X_{ \alpha k_{1}\times \beta k_{2}}:\|(u,u_{t},v,v_{t})\|^{2}_{X_{\alpha k_{1} \times \beta k_{2}}}=\|\nabla ^{m_{1}+k_{1}}u\|^{2}+\|\nabla ^{k_{1}}u_{t} \|^{2} +\alpha \|\nabla ^{m_{1}+k_{1}}u_{t}\|^{2}+\|\nabla ^{m_{2}+k_{2}}v \|^{2}+\|\nabla ^{k_{2}}v_{t}\|^{2}+\beta \|\nabla ^{m_{2}+k_{2}}v_{t} \|^{2}\le C(R_{\alpha 0\times \beta 0})+C(R_{\alpha k_{1}\times \beta k_{2}})\}$ are bounded absorbing sets in $X_{\alpha 0\times \beta 0}$, $B_{{\alpha k_{1}\times \beta k_{2}},0}$ are compact in $X_{\alpha 0\times \beta 0}$, $A_{\alpha k_{1}\times \beta k_{2}}\subset X_{\alpha 0\times \beta 0}$. Moreover,

• 1) $S(t)A_{\alpha k_{1}\times \beta k_{2}}=A_{\alpha k_{1}\times \beta k_{2}}$, for all $t\ge 0$,

• 2) Sets $A_{\alpha k_{1}\times \beta k_{2}}$ attract all bounded sets in $X_{\alpha 0\times \beta 0}$, i.e., for all $B_{\alpha k_{1}\times \beta k_{2}}\subset X_{\alpha 0 \times \beta 0}$, they are mapped to bounded sets in $X_{\alpha 0\times \beta 0}$, and

  $$ \begin{aligned} &\operatorname{dist}\bigl(S(t)B_{\alpha k_{1}\times \beta k_{2}},A_{\alpha k_{1} \times \beta k_{2}}\bigr) \\ &\quad =\sup _{x\in B_{\alpha k_{1}\times \beta k_{2}}} \inf_{y\in A_{\alpha k_{1}\times \beta k_{2}}} \bigl\Vert S(t)x-y \bigr\Vert _{X_{ \alpha 0\times \beta 0}}\to 0\quad (t\to \infty ), \end{aligned} $$

  where $\{S(t)\}_{t\ge 0}$ is the solution semigroup generated by problem (1)–(3).

3 Proof of the main results

This section presents the proof of the existence and uniqueness of the solutions and the family of global attractors for problem (1)–(3).

Let $\varepsilon >0$ be small enough, and $\frac{\lambda ^{m_{1}}_{1}}{2}N_{10}-2-4\varepsilon -2\varepsilon ^{2} \ge 0$, $\frac{N_{10}}{4\alpha}-2\varepsilon -\varepsilon ^{2}\ge 0$, $\frac{\lambda ^{m_{2}}_{1}}{2}N_{20}-2-4\varepsilon -2\varepsilon ^{2} \ge 0$, and $\frac{N_{20}}{4\beta}-2\varepsilon -\varepsilon ^{2}\ge 0$.

Lemma 1

([24])

Let $y:\mathbb{R}^{+}\to \mathbb{R}^{+}$ be an absolutely continuous positive function, which satisfies the following differential inequality for some $\delta >0$:

$$ \frac{\mathrm{d}}{\mathrm{d}t}y(t)+2\delta y(t)\le g(t)y(t)+K,\quad t>0, $$

where $K\ge 0$, and $a\ge 0$ if $t\ge s\ge 0$ so that $\int ^{t}_{s}g(\tau )\,\mathrm{d}\tau \le \delta (t-s)+a$. Then,

$$ y(t)\le \mathrm{e}^{a} y(0)\mathrm{e}^{-\delta t}+ \frac{K\mathrm{e}^{a}}{\delta},\quad t\ge 0. $$

Lemma 2

([16])

Let X be a Banach space, then the continuous operator semigroup $\{S(t)\}_{t\ge 0}$ satisfies the following:

• (1) Semigroup $\{S(t)\}_{t\ge 0}$ is uniformly bounded in X, i.e., for all $R_{0}>0$, there exists a positive constant $C_{0}(R_{0})$ that when $\| u\|_{X}\le R_{0}$,

  $$ \bigl\Vert S(t)u \bigr\Vert _{X}\le C_{0}(R_{0}), \quad \textit{for all } t\in [0,+ \infty ); $$

• (2) There exists a bounded absorbing set $B_{0}$ in X, and for any bounded set $B\subset X$, there exists a moment $t_{0}$ such that

  $$\begin{aligned} S(t)B\subset B_{0},\quad t\ge t_{0}; \end{aligned}$$

• (3) If $t>0$, and $S(t)$ is a fully continuous operator,

then the semigroup $\{S(t)\}_{t\ge 0}$ has a global attractor A in X, and

$$\begin{aligned} A=\omega (B_{0})=\bigcap_{\tau \ge 0} \overline{\bigcup_{t\ge \tau}S(t)B_{0}}. \end{aligned}$$

Lemma 3

Suppose that assumptions (A1)–(A4) hold. If $f_{j}\in H$ ($j=1,2$) and initial data $(u_{0},u_{1},v_{0},v_{1})\in X_{\alpha 0\times \beta 0}$, then for $R_{\alpha 0\times \beta 0}>0$, there exist positive constants $C(R_{\alpha 0\times \beta 0})$ and $t_{\alpha 0\times \beta 0}$ so that when $t\ge t_{\alpha 0\times \beta 0}$, $(u, y_{1}, v, y_{2})$ determined by problem (1)–(3) satisfies

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha 0\times \beta 0}}&= \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2} \\ &\quad{}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2}\le C(R_{ \alpha 0\times \beta 0}),\quad \textit{for }\forall \alpha ,\beta \in (0,1], \end{aligned} $$

(13)

where $y_{1}=u_{t}+\varepsilon u$, $y_{2}=v_{t}+\varepsilon v$.

Proof

Multiplying the first equation of (1) by $y_{1}$ in H and the second by $y_{2}$ in H, we have

$$ \begin{aligned} &\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \biggl[ \Vert y_{1} \Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}+ \int _{0}^{ \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2}}M( \tau )\,\mathrm{d}\tau +2J(u,v) \biggr] \\ &\quad \quad{} + \varepsilon M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \cdot \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \quad{}- \varepsilon \bigl( \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}\bigr) - \varepsilon \bigl( \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\ &\quad \quad {}+ \varepsilon ^{2} \bigl((u,y_{1})+\alpha \bigl((-\Delta )^{m_{1}}u,y_{1} \bigr)+(v,y_{2})+ \beta \bigl((-\Delta )^{m_{2}}v,y_{2} \bigr)\bigr) \\ &\quad \quad{} + N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad{}-\varepsilon N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl(\nabla ^{m_{1}}y_{1},\nabla ^{m_{1}}u\bigr) - \varepsilon N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl(\nabla ^{m_{2}}y_{2}, \nabla ^{m_{2}}v\bigr) \\ &\quad \quad {}+\varepsilon \bigl(g_{1}(u,v),u\bigr)+\varepsilon \bigl(g_{2}(u,v),v \bigr)=(f_{1},y_{1})+(f_{2},y_{2}). \end{aligned} $$

(14)

Using Hölder, Young, and Poincaré inequalities, we have

$$\begin{aligned}& -\varepsilon \bigl( \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+ \beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\& \quad \quad{}+\varepsilon ^{2}\bigl((u,y_{1})+\alpha \bigl((- \Delta )^{m_{1}}u,y_{1}\bigr)\bigr)+ \varepsilon ^{2}\bigl((v,y_{2})+\beta \bigl((-\Delta )^{m_{2}}v,y_{2}\bigr)\bigr) \\& \quad \ge \biggl(-\varepsilon -\frac{\varepsilon ^{2}}{2}\biggr) \bigl( \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\& \quad \quad{}-\frac{\varepsilon ^{2}}{2}\bigl( \Vert u \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \Vert v \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\& \quad \ge \biggl(-\varepsilon -\frac{\varepsilon ^{2}}{2}\biggr) \bigl( \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}\bigr)+\biggl(- \varepsilon - \frac{\varepsilon ^{2}}{2}\biggr) \bigl( \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\& \quad \quad{}- \frac{\varepsilon ^{2}}{2}\bigl(\lambda ^{-m_{1}}_{1}+ \alpha \bigr) \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}- \frac{\varepsilon ^{2}}{2}\bigl(\lambda ^{-m_{2}}_{1}+ \beta \bigr) \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}, \end{aligned}$$

(15)

$$\begin{aligned}& N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+N_{2} \bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\& \quad \quad{}-\varepsilon N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl(\nabla ^{m_{1}}y_{1}, \nabla ^{m_{1}}u\bigr)-\varepsilon N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl(\nabla ^{m_{2}}y_{2}, \nabla ^{m_{2}}v\bigr) \\& \quad \ge \frac{1}{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \frac{1}{2}N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\& \quad \quad{}-\frac{\varepsilon ^{2}}{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}-\frac{\varepsilon ^{2}}{2}N_{2} \bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}, \end{aligned}$$

(16)

$$\begin{aligned}& (f_{1},y_{1})+(f_{2},y_{2}) \le \Vert f_{1} \Vert \Vert y_{1} \Vert + \Vert f_{2} \Vert \Vert y_{2} \Vert \le \frac{1}{2} \Vert y_{1} \Vert ^{2}+ \frac{1}{2} \Vert y_{2} \Vert ^{2}+ \frac{1}{2} \Vert f_{1} \Vert ^{2}+ \frac{1}{2} \Vert f_{2} \Vert ^{2}. \end{aligned}$$

(17)

Inserting the above estimates into (14), we have

$$ \begin{aligned} &\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \biggl[ \Vert y_{1} \Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}+ \int _{0}^{ \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2}}M( \tau )\,\mathrm{d}\tau +2J(u,v) \biggr] \\ &\quad \quad{}+\frac{1}{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}-\biggl( \frac{1}{2}+\varepsilon +\frac{\varepsilon ^{2}}{2}\biggr) \Vert y_{1} \Vert ^{2} \\ &\quad \quad {}-\biggl( \varepsilon + \frac{\varepsilon ^{2}}{2}\biggr)\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+\frac{1}{2}N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}-\biggl(\frac{1}{2}+\varepsilon + \frac{\varepsilon ^{2}}{2} \biggr) \Vert y_{2} \Vert ^{2}-\biggl(\varepsilon + \frac{\varepsilon ^{2}}{2}\biggr)\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}+ \varepsilon M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \quad {}- \frac{\varepsilon ^{2}}{2}\bigl(N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr)+\lambda ^{-m_{1}}_{1}+ \alpha \bigr) \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2} \\ &\quad \quad{}-\frac{\varepsilon ^{2}}{2}\bigl(N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)+ \lambda ^{-m_{2}}_{1}+\beta \bigr) \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2} \\ &\quad \le -\varepsilon \bigl(g_{1}(u,v),u\bigr)-\varepsilon \bigl(g_{2}(u,v),v\bigr)+ \frac{1}{2} \Vert f_{1} \Vert ^{2}+\frac{1}{2} \Vert f_{2} \Vert ^{2}. \end{aligned} $$

(18)

According to (A1),

$$ \begin{aligned} &\varepsilon M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \ge \frac{\varepsilon}{4} \int _{0}^{ \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2}}M(\tau )\,\mathrm{d}\tau \\ &\quad \quad{}+\frac{3\varepsilon}{4}M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)\cdot \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr), \end{aligned} $$

(19)

and according to (A2),

$$ -\varepsilon \bigl(g_{1}(u,v),u\bigr)-\varepsilon \bigl(g_{2}(u,v),v\bigr)\le - \varepsilon C_{1}J(u,v)+ \varepsilon \mu \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)+\varepsilon C_{\mu}. $$

(20)

Inserting (18) and (19) into (20), we have

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \biggl[ \Vert y_{1} \Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}+ \int _{0}^{ \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2}}M( \tau )\,\mathrm{d}\tau +2J(u,v) \biggr] \\ &\quad \quad{}+\biggl(\frac{\lambda ^{m_{1}}_{1}}{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr)-1-2 \varepsilon - \varepsilon ^{2}\biggr) \Vert y_{1} \Vert ^{2} \\ &\quad \quad {}+\biggl(\frac{1}{2\alpha}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr)-2\varepsilon -\varepsilon ^{2}\biggr)\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2} \\ &\quad \quad{}+\biggl(\frac{\lambda ^{m_{2}}_{1}}{2}N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-1-2 \varepsilon - \varepsilon ^{2}\biggr) \Vert y_{2} \Vert ^{2} \\ &\quad \quad {}+\biggl(\frac{1}{2\beta}N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-2\varepsilon - \varepsilon ^{2}\biggr)\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad{}+\frac{\varepsilon}{2} \int _{0}^{ \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2}}M(\tau )\,\mathrm{d}\tau +2 \varepsilon C_{1}J(u,v) \\ &\quad \quad{}+\biggl(\frac{3\varepsilon}{2}M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-2\varepsilon \mu - \varepsilon ^{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr)- \varepsilon ^{2}\lambda ^{-m_{1}}_{1}\biggr) \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2} \\ &\quad \quad{}+\biggl(\frac{3\varepsilon}{2}M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-2\varepsilon \mu - \varepsilon ^{2}N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)- \varepsilon ^{2}\lambda ^{-m_{2}}_{1}\biggr) \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2} \\ &\quad \le 2\varepsilon C_{\mu}+ \Vert f_{1} \Vert ^{2}+ \Vert f_{2} \Vert ^{2}. \end{aligned}$$

(21)

Let $H_{1}(t)=\|y_{1}\|^{2}+\alpha \|\nabla ^{m_{1}}y_{1}\|^{2}+\|y_{2}\|^{2}+ \beta \|\nabla ^{m_{2}}y_{2}\|^{2}+\int _{0}^{\|\nabla ^{m_{1}}u\|^{2}+ \|\nabla ^{m_{2}}v\|^{2}}M(\tau )\,\mathrm{d}\tau +2J(u,v)$ and $\sigma _{1}=\min \{\frac{\lambda ^{m_{1}}_{1}}{2}N_{10}-2-4 \varepsilon -2\varepsilon ^{2},\frac{1}{2\alpha}N_{10}-2\varepsilon - \varepsilon ^{2},\frac{\lambda ^{m_{2}}_{1}}{2}N_{20}-2-4\varepsilon -2 \varepsilon ^{2},\frac{1}{2\beta}N_{10}-2\varepsilon -\varepsilon ^{2}, \frac{\varepsilon}{2},\varepsilon C_{1}\}$. Then we can infer from (21) that

$$ \frac{\mathrm{d}}{\mathrm{d}t}H_{1}(t)+\sigma _{1}H_{1}(t)\le 2 \varepsilon C_{\mu}+ \Vert f_{1} \Vert ^{2}+ \Vert f_{2} \Vert ^{2}. $$

(22)

According to Gronwall’s inequality, we have

$$ H_{1}(t)\le H_{1}(0) \mathrm{e}^{-\sigma _{1}t}+ \frac{2\varepsilon C_{\mu}+ \Vert f_{1} \Vert ^{2}+ \Vert f_{2} \Vert ^{2}}{\sigma _{1}}, $$

(23)

and

$$ \begin{aligned} H_{1}(t)&\ge \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad {}+M_{0} \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) +2J(u,v) \\ &\ge \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+ \beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad {}+ \frac{M_{0}}{2}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) -2C'_{\mu} \\ &\ge C_{5}\bigl( \Vert y_{1} \Vert ^{2}+ \Vert y_{2} \Vert ^{2}+ \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-2C'_{\mu}, \end{aligned} $$

(24)

according to (A1) and (A2), where $\mu =\frac{M_{0}}{4}$ and $C_{5}=\min \{1,\frac{M_{0}}{2}\}$. Thus,

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha 0\times \beta 0}}&= \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2} \\ &\quad {}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\le \frac{(H_{1}(t)+2C'_{\mu})}{C_{5}} \\ &\le \frac{H_{1}(0)\mathrm{e}^{-\sigma _{1}t}+2C'_{\mu}}{C_{5}}+ \frac{2\varepsilon C_{\mu}+ \Vert f_{1} \Vert ^{2}+ \Vert f_{2} \Vert ^{2}}{\sigma _{1}C_{5}}, \end{aligned} $$

(25)

i.e.,

$$ \overline{\lim_{t\to \infty}} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha 0 \times \beta 0}} \le \frac{2C'_{\mu}}{C_{5}}+ \frac{2\varepsilon C_{\mu}+ \Vert f_{1} \Vert ^{2}+ \Vert f_{2} \Vert ^{2}}{\sigma _{1}C_{5}}=R_{ \alpha 0\times \beta 0}. $$

(26)

Therefore, there exist positive constants $C(R_{\alpha 0\times \beta 0})$ and $t_{\alpha 0\times \beta 0}$ such that, whenever $t\ge t_{\alpha 0\times \beta 0}$,

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha 0\times \beta 0}}&= \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \Vert y_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}y_{1} \bigr\Vert ^{2} \\ &\quad {}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}+ \Vert y_{2} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}y_{2} \bigr\Vert ^{2} \\ &\le C(R_{\alpha 0\times \beta 0}). \end{aligned} $$

(27)

Thus, Lemma 3 is proved. □

Lemma 4

Suppose that assumptions (A1)–(A4) hold. If $f_{1}\in V_{k_{1}}$, $f_{2}\in V_{k_{2}}$, $k_{1}=1,2,\dots ,m_{1}$, $k_{2}=1,2,\dots ,m_{2}$, and initial data $(u_{0},u_{1},v_{0},v_{1})\in X_{\alpha k_{1}\times \beta k_{2}}$, then, for $R_{\alpha k_{1}\times \beta k_{2}}>0$, there exist positive constants $C(R_{\alpha k_{1}\times \beta k_{2}})$ and $t_{\alpha k_{1}\times \beta k_{2}}$ such that, whenever $t\ge t_{\alpha k_{1}\times \beta k_{2}}$, $(u,y_{1},v,y_{2})$ determined by problem (1)–(3) satisfies

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha k_{1}\times \beta k_{2}}}&= \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+ \beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\le C(R_{\alpha k_{1}\times \beta k_{2}}),\quad \textit{for } \forall \alpha ,\beta \in (0,1], \end{aligned} $$

(28)

where $y_{1}=u_{t}+\varepsilon u$, $y_{2}=v_{t}+\varepsilon v$.

Proof

Multiplying the first equation of (1) by $(-\Delta )^{k_{1}}y_{1}$, $k_{1}=1,2,\dots ,m_{1}$ in H and the second by $(-\Delta )^{k_{2}}y_{2}$, $k_{2}=1,2,\dots ,m_{2}$ in H and then integrating over Ω, we have

$$ \begin{aligned} &\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \bigl[ \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}+M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr) \bigr] \\ &\quad \quad {}+\varepsilon M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)\cdot \bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \quad{}-\varepsilon \bigl( \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}\bigr)-\varepsilon \bigl( \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \bigr) \\ &\quad \quad{}+\varepsilon ^{2}\bigl(\bigl(\nabla ^{k_{1}}u,\nabla ^{k_{1}}y_{1}\bigr)+ \alpha \bigl(\nabla ^{m_{1}+k_{1}}u,\nabla ^{m_{1}+k_{1}}y_{1}\bigr)\bigr) \\ &\quad \quad {}+ \varepsilon ^{2}\bigl(\bigl(\nabla ^{k_{2}}v,\nabla ^{k_{2}}y_{2}\bigr)+\beta \bigl( \nabla ^{m_{2}+k_{2}}v, \nabla ^{m_{2}+k_{2}}y_{2}\bigr)\bigr) \\ &\quad \quad{}+N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad{}-\varepsilon N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl(\nabla ^{m_{1}+k_{1}}y_{1}, \nabla ^{m_{1}+k_{1}}u\bigr) \\ &\quad\quad {}-\varepsilon N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl( \nabla ^{m_{2}+k_{2}}y_{2},\nabla ^{m_{2}+k_{2}}v\bigr) \\ &\quad \quad{}+\bigl(g_{1}(u,v),(-\Delta )^{k_{1}}y_{1} \bigr)+\bigl(g_{2}(u,v),(-\Delta )^{k_{2}}y_{2} \bigr) \\ &\quad = \frac{ \Vert \nabla ^{m_{1}+k_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}+k_{2}}v \Vert ^{2}}{2} \frac{\mathrm{d}}{\mathrm{d}t}M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \quad {}+\bigl(f_{1},(- \Delta )^{k_{1}}y_{1}\bigr)+\bigl(f_{2},(-\Delta )^{k_{2}}y_{2}\bigr). \end{aligned} $$

(29)

Using Hölder, Young, and Poincaré inequalities, we then have

$$\begin{aligned}& -\varepsilon \bigl( \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2} \bigr)-\varepsilon \bigl( \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\& \quad \quad{}+\varepsilon ^{2}\bigl(\bigl(\nabla ^{k_{1}}u,\nabla ^{k_{1}}y_{1}\bigr)+ \alpha \bigl(\nabla ^{m_{1}+k_{1}}u,\nabla ^{m_{1}+k_{1}}y_{1}\bigr)\bigr) \\& \quad \quad{}+\varepsilon ^{2}\bigl(\bigl(\nabla ^{k_{2}}v,\nabla ^{k_{2}}y_{2}\bigr)+ \beta \bigl(\nabla ^{m_{2}+k_{2}}v, \nabla ^{m_{2}+k_{2}}y_{2}\bigr)\bigr) \\& \quad \ge \biggl(-\varepsilon -\frac{\varepsilon ^{2}}{2}\biggr) \bigl( \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}\bigr) \\& \quad \quad {}+\biggl(-\varepsilon - \frac{\varepsilon ^{2}}{2}\biggr) \bigl( \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\& \quad \quad{}-\frac{\varepsilon ^{2}}{2}\bigl( \bigl\Vert \nabla ^{k_{1}}u \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}\bigr)-\frac{\varepsilon ^{2}}{2}\bigl( \bigl\Vert \nabla ^{k_{2}}v \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr) \\& \quad \ge -\biggl(\varepsilon +\frac{\varepsilon ^{2}}{2}\biggr) \bigl( \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}\bigr) \\& \quad \quad {}-\biggl(\varepsilon + \frac{\varepsilon ^{2}}{2}\biggr) \bigl( \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2}\bigr) \\& \quad \quad{}-\frac{\varepsilon ^{2}}{2}\bigl(\lambda ^{-m_{1}}_{1}+ \alpha \bigr) \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}-\frac{\varepsilon ^{2}}{2}\bigl(\lambda ^{-m_{2}}_{1}+ \beta \bigr) \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}, \end{aligned}$$

(30)

$$\begin{aligned}& N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\& \quad \quad{}-\varepsilon N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl(\nabla ^{m_{1}+k_{1}}y_{1}, \nabla ^{m_{1}+k_{1}}u\bigr)-\varepsilon N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl( \nabla ^{m_{2}+k_{2}}y_{2},\nabla ^{m_{2}+k_{2}}v\bigr) \\& \quad \ge \frac{1}{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \frac{1}{2}N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\& \quad \quad{}-\frac{\varepsilon ^{2}}{2}N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}-\frac{\varepsilon ^{2}}{2}N_{2} \bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}, \end{aligned}$$

(31)

$$\begin{aligned}& \bigl(g_{1}(u,v),(-\Delta )^{k_{1}}y_{1} \bigr)+\bigl(g_{2}(u,v),(-\Delta )^{k_{2}}y_{2} \bigr) \\& \quad \le \bigl\Vert g_{1}(u,v) \bigr\Vert \bigl\Vert \nabla ^{2k_{1}}y_{1} \bigr\Vert + \bigl\Vert g_{2}(u,v) \bigr\Vert \bigl\Vert \nabla ^{2k_{2}}y_{2} \bigr\Vert \\& \quad \le \frac{N_{10}}{8} \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \frac{2\lambda ^{k_{1}-m_{1}}_{1}}{N_{10}} \bigl\Vert g_{1}(u,v) \bigr\Vert ^{2} \\& \quad \quad {}+ \frac{N_{20}}{8} \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2}+ \frac{2\lambda ^{k_{2}-m_{2}}_{1}}{N_{20}} \bigl\Vert g_{2}(u,v) \bigr\Vert ^{2}, \end{aligned}$$

(32)

$$\begin{aligned}& \bigl(f_{1},(-\Delta )^{k_{1}}y_{1} \bigr)+\bigl(f_{2},(-\Delta )^{k_{2}}y_{2}\bigr) \\& \quad \le \bigl\Vert \nabla ^{k_{1}}f_{1} \bigr\Vert \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert + \bigl\Vert \nabla ^{k_{2}}f_{2} \bigr\Vert \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert \\& \quad \le \frac{1}{2} \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\frac{1}{2} \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+ \frac{1}{2} \bigl\Vert \nabla ^{k_{1}}f_{1} \bigr\Vert ^{2}+\frac{1}{2} \bigl\Vert \nabla ^{k_{2}}f_{2} \bigr\Vert ^{2}, \end{aligned}$$

(33)

and

$$\begin{aligned}& \begin{aligned}[b] \bigl\Vert g_{1}(u,v) \bigr\Vert ^{2}&= \int _{\Omega} \bigl\vert g_{1}(u,v) \bigr\vert ^{2}\,\mathrm{d}x\le \int _{\Omega} \bigl\vert C_{2}\bigl(1+ \vert u \vert ^{p_{1}}+ \vert v \vert ^{q_{1}}\bigr) \bigr\vert ^{2}\,\mathrm{d}x \\ &\le C_{6} \int _{\Omega}\bigl(1+ \vert u \vert ^{2p_{1}}+ \vert v \vert ^{2q_{1}}\bigr)\,\mathrm{d}x\le C_{7}\bigl(1+ \Vert u \Vert ^{2p_{1}}_{2p_{1}}+ \Vert v \Vert ^{2q_{1}}_{2q_{1}}\bigr), \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned}& \bigl\Vert g_{2}(u,v) \bigr\Vert ^{2}\le C_{8}\bigl(1+ \Vert u \Vert ^{2p_{2}}_{2p_{2}}+ \Vert v \Vert ^{2q_{2}}_{2q_{2}} \bigr), \end{aligned}$$

(35)

according to (A3). Furthermore, based on the Gagliardo–Nirenberg inequality, we can conclude that

$$ \textstyle\begin{cases} \Vert u \Vert ^{2p_{j}}_{2p_{j}}\le C_{9j} \Vert \nabla ^{m_{1}}u \Vert ^{ \frac{n(p_{j}-1)}{m_{1}}} \Vert u \Vert ^{\frac{2m_{1}p_{j}-n(p_{j}-1)}{m_{1}}}, \\ \Vert v \Vert ^{2q_{j}}_{2q_{j}}\le C_{10j} \Vert \nabla ^{m_{2}}v \Vert ^{ \frac{n(q_{j}-1)}{m_{2}}} \Vert v \Vert ^{\frac{2m_{2}q_{j}-n(q_{j}-1)}{m_{2}}}. \end{cases} $$

Thus, we have

$$ \bigl\Vert g_{1}(u,v) \bigr\Vert ^{2}+ \bigl\Vert g_{2}(u,v) \bigr\Vert ^{2}\le C(R_{\alpha 0\times \beta 0}). $$

(36)

Inserting (31)–(33) and (36) into (29), we have

$$ \begin{aligned} &\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \bigl[ \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad {}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2}+M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)\bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr) \bigr] \\ &\quad \quad{}+ \frac{N_{1}( \Vert \nabla ^{m_{1}}u \Vert ^{2})\lambda ^{m_{1}}_{1}-2-4\varepsilon -2\varepsilon ^{2}}{4} \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2} \\ &\quad \quad{}+\biggl(\frac{2N_{1}( \Vert \nabla ^{m_{1}}u \Vert ^{2})-N_{10}}{8\alpha}- \varepsilon -\frac{\varepsilon ^{2}}{2}\biggr) \alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2} \\ &\quad \quad{}+ \frac{N_{2}( \Vert \nabla ^{m_{2}}v \Vert ^{2})\lambda ^{m_{2}}_{1}-2-4\varepsilon -2\varepsilon ^{2}}{4} \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad{}+\biggl(\frac{2N_{2}( \Vert \nabla ^{m_{2}}v \Vert ^{2})-N_{20}}{8\beta}- \varepsilon -\frac{\varepsilon ^{2}}{2}\biggr) \beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad \quad{}+\frac{\varepsilon}{2} M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2} \bigr) \\ &\quad \quad{}+ \biggl(\frac{\varepsilon}{2} M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-\frac{\varepsilon ^{2}}{2}N_{1} \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \\ &\quad \quad {}- \frac{\varepsilon ^{2}}{2}\bigl(\lambda ^{-m_{1}}_{1}+\alpha \bigr) \biggr) \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2} \\ &\quad \quad{}+ \biggl(\frac{\varepsilon}{2} M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)-\frac{\varepsilon ^{2}}{2}N_{2} \bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \quad {}- \frac{\varepsilon ^{2}}{2}\bigl(\lambda ^{-m_{2}}_{1}+\beta \bigr) \biggr) \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2} \\ &\quad \le \frac{ \Vert \nabla ^{m_{1}+k_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}+k_{2}}v \Vert ^{2}}{2} \frac{\mathrm{d}}{\mathrm{d}t}M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)+\frac{1}{2} \bigl\Vert \nabla ^{k_{1}}f_{1} \bigr\Vert ^{2} \\ &\quad \quad{}+\frac{1}{2} \bigl\Vert \nabla ^{k_{2}}f_{2} \bigr\Vert ^{2}+C(R_{0},\lambda _{1}) \\ &\quad \le \bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr) \cdot M'\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \\ &\quad \quad{}\times \bigl(\bigl(\nabla ^{m_{1}}u,\nabla ^{m_{1}}u_{t} \bigr)+\bigl(\nabla ^{m_{2}}v, \nabla ^{m_{2}}v_{t} \bigr)\bigr) \\ &\quad \quad {}+\frac{1}{2} \bigl\Vert \nabla ^{k_{1}}f_{1} \bigr\Vert ^{2}+ \frac{1}{2} \bigl\Vert \nabla ^{k_{2}}f_{2} \bigr\Vert ^{2}+C(R_{0}, \lambda _{1}) \\ &\quad \le C_{9}\bigl( \bigl\Vert \nabla ^{m_{1}}u_{t} \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}v_{t} \bigr\Vert \bigr)\cdot \bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr)+ \frac{1}{2} \bigl\Vert \nabla ^{k_{1}}f_{1} \bigr\Vert ^{2} \\ &\quad \quad{}+\frac{1}{2} \bigl\Vert \nabla ^{k_{2}}f_{2} \bigr\Vert ^{2}+C(R_{\alpha 0\times \beta 0},\lambda _{1}). \end{aligned} $$

(37)

Letting $H_{2}(t)=\|\nabla ^{k_{1}}y_{1}\|^{2}+\alpha \|\nabla ^{m_{1}+k_{1}}y_{1} \|^{2}+\|\nabla ^{k_{2}}y_{2}\|^{2}+\beta \|\nabla ^{m_{2}+k_{2}}y_{2} \|^{2}+M(\|\nabla ^{m_{1}}u\|^{2}+\|\nabla ^{m_{2}}v\|^{2})\cdot (\| \nabla ^{m_{1}+k_{1}}u\|^{2}+\|\nabla ^{m_{2}+k_{2}}v\|^{2})$ and $\sigma _{2}=\min \{ \frac{\lambda ^{m_{1}}_{1}N_{10}-2-4\varepsilon -2\varepsilon ^{2}}{4}, \frac{\lambda ^{m_{2}}_{1}N_{20}-2-4\varepsilon -2\varepsilon ^{2}}{4}, \frac{N_{10}}{8\alpha}-\varepsilon -\frac{\varepsilon ^{2}}{2}, \frac{N_{20}}{8\beta}-\varepsilon -\frac{\varepsilon ^{2}}{2}, \frac{\varepsilon}{2}\}$, we have

$$ \begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}H_{2}(t)+\sigma _{2}H_{2}(t) \\ &\quad \le C_{11}\bigl( \bigl\Vert \nabla ^{m_{1}}u_{t} \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}v_{t} \bigr\Vert \bigr)H_{2}(t)+ \bigl\Vert \nabla ^{k_{1}}f_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}f_{2} \bigr\Vert ^{2}+C(R_{\alpha 0\times \beta 0}, \lambda _{1}). \end{aligned} $$

(38)

Taking the scalar product of (1) in H with $u_{t}$, $v_{t}$, we have

$$ \begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl[ \Vert u_{t} \Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}}u_{t} \bigr\Vert ^{2}+ \Vert v_{t} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}v_{t} \bigr\Vert ^{2} \\ &\quad{}+ \int _{0}^{ \Vert \nabla ^{m_{1}}u \Vert ^{2}+ \Vert \nabla ^{m_{2}}v \Vert ^{2}}M( \tau )\,\mathrm{d}\tau +2J(u,v)- 2(f_{1},u)-2(f_{2},v) \biggr] \\ &\quad{}+N_{1} \bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}u_{t} \bigr\Vert ^{2}+N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}v_{t} \bigr\Vert ^{2}=0, \end{aligned} $$

(39)

and integrating (39) on $(0,t)$ yields

$$ \begin{aligned} & \int _{0}^{t}\bigl( \bigl\Vert \nabla ^{m_{1}}u_{t}(\tau ) \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v_{t}(\tau ) \bigr\Vert ^{2}\bigr)\,\mathrm{d}\tau \\ &\quad \le \frac{1}{\min \{N_{10},N_{20}\}} \int _{0}^{t}\bigl(N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u( \tau ) \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}u_{t}(\tau ) \bigr\Vert ^{2} \\ &\quad \quad {}+N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v( \tau ) \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}v_{t}(\tau ) \bigr\Vert ^{2}\bigr) \,\mathrm{d}\tau \\ &\quad \le \frac{1}{\min \{N_{10},N_{20}\}}\biggl( \Vert u_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}u_{1} \bigr\Vert ^{2}+ \Vert v_{1} \Vert ^{2} +\beta \bigl\Vert \nabla ^{m_{2}}v_{1} \bigr\Vert ^{2} \\ &\quad \quad{}+ \int _{0}^{ \Vert \nabla ^{m_{1}}u_{0} \Vert ^{2}+ \Vert \nabla ^{m_{2}}v_{0} \Vert ^{2}}M(\tau )\,\mathrm{d}\tau +2J(u_{0},v_{0})-2(f_{1},u_{0})-2(f_{2},v_{0}) \biggr) \le C_{12}. \end{aligned} $$

(40)

Then,

$$ C_{11} \int _{s}^{t}\bigl(\bigl( \bigl\Vert \nabla ^{m_{1}}u_{t}(\tau ) \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}v_{t}( \tau ) \bigr\Vert \bigr)\bigr)\,\mathrm{d} \tau \le \frac{\sigma _{2}}{2}(t-s)+a $$

(41)

for $t>s\ge 0$ and some $a>0$. Based on (38), (41), and Lemma 1, we obtain

$$ H_{2}(t)\le C_{13}H_{2}(0) \mathrm{e}^{-\frac{\sigma _{2}}{2}t}+C_{14}. $$

(42)

According to (A1), we have

$$ \begin{aligned} H_{2}(t)&\ge \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad{}+M\bigl( \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2}\bigr)\cdot \bigl( \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr) \\ &\ge C_{15}\bigl( \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}\bigr), \end{aligned} $$

(43)

and then,

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha k_{1}\times \beta k_{2}}}&= \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2} \le \frac{C_{13}H_{2}(0)\mathrm{e}^{-\frac{\sigma _{2}}{2}t}+C_{14}}{C_{15}}, \end{aligned} $$

(44)

i.e.,

$$ \overline{\lim_{t\to \infty}} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha k_{1} \times \beta k_{2}}} \le R_{\alpha k_{1}\times \beta k_{2}}. $$

(45)

Therefore, there exist positive constants $C(R_{\alpha k_{1}\times \beta k_{2}})$ and $t_{\alpha k_{1}\times \beta k_{2}}$ such that whenever $t\ge t_{\alpha k_{1}\times \beta k_{2}}$, the obtained $(u,y_{1},v,y_{2})$ satisfies

$$ \begin{aligned} \bigl\Vert (u,y_{1},v,y_{2}) \bigr\Vert ^{2}_{X_{\alpha k_{1}\times \beta k_{2}}}&= \bigl\Vert \nabla ^{k_{1}}y_{1} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}y_{1} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}y_{2} \bigr\Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}+k_{2}}y_{2} \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2} \\ &\le C(R_{\alpha k_{1}\times \beta k_{2}}),\quad k_{1}=1,2,\dots ,m_{1}, k_{2}=1,2,\dots ,m_{2}. \end{aligned} $$

(46)

Thus, Lemma 4 is proved. □

Proof of Theorem 1

According to previous findings [16] and the Faedo–Galerkin method, problem (1)–(3) has global solutions, which follows by combining with Lemmas 3 and 4.

Let $(u^{1},v^{1})$ and $(u^{2},v^{2})$ be two solutions of problem (1)–(3) corresponding to the same initial data, respectively, $w=u^{1}-u^{2}$, $z=v^{1}-v^{2}$. Then, $(w,z)$ solves

$$ \textstyle\begin{cases} (1+\alpha (-\Delta )^{m_{1}})w_{tt}+\frac{1}{2}\sigma _{12}(t)(- \Delta )^{m_{1}}w_{t}+\frac{1}{2}\Phi _{12}(t)(-\Delta )^{m_{1}}w \\ \quad {}+ G_{1}(u^{1},u^{2},v^{1},v^{2};t)=0, \\ (1+\beta (-\Delta )^{m_{2}})z_{tt}+\frac{1}{2}\sigma _{34}(t)(- \Delta )^{m_{2}}z_{t}+\frac{1}{2}\Phi _{12}(t)(-\Delta )^{m_{2}}z \\ \quad {}+ G_{2}(u^{1},u^{2},v^{1},v^{2};t)=0, \end{cases} $$

(47)

where

$$\begin{aligned}& \sigma _{12} =\sigma _{1}(t)+\sigma _{2}(t), \qquad \Phi _{12}(t)= \Phi _{1}(t)+ \Phi _{2}(t), \\& \sigma _{i}(t) =N_{1}\bigl( \bigl\Vert \nabla ^{m_{1}}u^{i} \bigr\Vert ^{2}\bigr),\qquad \Phi _{i}(t)=M\bigl( \bigl\Vert \nabla ^{m_{1}}u^{i} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v^{i} \bigr\Vert ^{2}\bigr),\quad i=1,2, \\& \sigma _{34} =\sigma _{3}(t)+\sigma _{4}(t),\qquad \sigma _{j}(t)=N_{2}\bigl( \bigl\Vert \nabla ^{m_{2}}v^{j} \bigr\Vert ^{2}\bigr),\quad j=3,4, \\& G_{1}\bigl(u^{1},u^{2},v^{1},v^{2};t \bigr) \\& \quad =\frac{1}{2}\bigl\{ \bigl[\sigma _{1}(t)- \sigma _{2}(t)\bigr](-\Delta )^{m_{1}}\bigl(u^{1}_{t}+u^{2}_{t} \bigr)+\bigl[\Phi _{1}(t)- \Phi _{2}(t)\bigr](-\Delta )^{m_{1}}\bigl(u^{1}+u^{2}\bigr)\bigr\} \\& \quad \quad{}+g_{1}(u_{1},v_{1})-g_{1}(u_{2},v_{2}), \\& G_{2}\bigl(u^{1},u^{2},v^{1},v^{2};t \bigr) \\& \quad =\frac{1}{2}\bigl\{ \bigl[\sigma _{3}(t)- \sigma _{4}(t)\bigr](-\Delta )^{m_{2}}\bigl(v^{1}_{t}+v^{2}_{t} \bigr)+\bigl[\Phi _{1}(t)- \Phi _{2}(t)\bigr](-\Delta )^{m_{2}}\bigl(v^{1}+v^{2}\bigr)\bigr\} \\& \quad \quad{}+g_{2}(u_{1},v_{1})-g_{2}(u_{2},v_{2}). \end{aligned}$$

According to Lemma 3,

$$ \sigma '_{12}\le C(R_{\alpha 0\times \beta 0}) \bigl( \bigl\Vert \nabla ^{m_{1}}u^{1}_{t} \bigr\Vert + \bigl\Vert \nabla ^{m_{1}}u^{2}_{t} \bigr\Vert \bigr),\sigma '_{34}\le C(R_{\alpha 0 \times \beta 0}) \bigl( \bigl\Vert \nabla ^{m_{2}}v^{1}_{t} \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}v^{2}_{t} \bigr\Vert \bigr). $$

Taking the scalar product of (47) in H with $w_{t}$, $z_{t}$, we obtain

$$ \begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\biggl[ \Vert w_{t} \Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \Vert z_{t} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2} \\ &\quad {}+ \frac{1}{4}\Phi _{0}\cdot \bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr)\biggr] \\ &\quad{} + \frac{1}{2}\sigma _{12}(t) \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \frac{1}{2} \sigma _{34}(t) \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2} \\ &\quad{} + \bigl(G_{1}\bigl(u^{1},u^{2},v^{1},v^{2};t \bigr),w_{t}\bigr)+\bigl(G_{2}\bigl(u^{1},u^{2},v^{1},v^{2};t \bigr),z_{t}\bigr)=0. \end{aligned} $$

(48)

According to Lemma 3 and (A1), $M_{0}\le M\le C(R_{0},H_{1}(0))\equiv M_{1}$. When $\frac{\mathrm{d}}{\mathrm{d}t}(\|\nabla ^{m_{1}}w\|^{2}+\|\nabla ^{m_{2}}z \|^{2})\ge 0$, we have $\Phi _{0}=2M_{0}$; otherwise $\Phi _{0}=2M_{1}$.

Letting $(G_{1}(u^{1},u^{2},v^{1},v^{2};t),w_{t})=G_{11}+G_{12}+G_{13}$ and $(G_{2}(u^{1},u^{2},v^{1},v^{2};t),z_{t})=G_{21}+G_{22}+G_{23}$, we have

$$\begin{aligned} G_{11}&=\frac{1}{2}\bigl(\sigma _{1}(t)-\sigma _{2}(t)\bigr) \bigl(\nabla ^{m_{1}}\bigl(u^{1}_{t}+u^{2}_{t} \bigr), \nabla ^{m_{1}}w_{t}\bigr) \\ &\le C(R_{\alpha 0\times \beta 0}) \bigl( \bigl\Vert \nabla ^{m_{1}}u^{1}_{t} \bigr\Vert + \bigl\Vert \nabla ^{m_{1}}u^{2}_{t} \bigr\Vert \bigr) \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert \\ &\le \frac{\sigma _{120}}{8} \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \frac{2C(R_{\alpha 0\times \beta 0})}{\sigma _{120}}\bigl( \bigl\Vert \nabla ^{m_{1}}u^{1}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{1}}u^{2}_{t} \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}, \end{aligned}$$

(49)

$$\begin{aligned} G_{12}&=\frac{1}{2}\bigl(\Phi _{1}(t)-\Phi _{2}(t)\bigr) \bigl(\nabla ^{m_{1}} \bigl(u^{1}+u^{2}\bigr), \nabla ^{m_{1}}w_{t} \bigr) \\ &\le C(R_{\alpha 0\times \beta 0}) \bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert \bigr) \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert \\ &\le \frac{\sigma _{120}}{8} \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \frac{2C(R_{\alpha 0\times \beta 0})}{\sigma _{120}}\bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr), \end{aligned}$$

(50)

$$\begin{aligned} G_{13}&=\bigl(g_{1}(u_{1},v_{1})-g_{1}(u_{2},v_{2}),w_{t} \bigr)\le C(R_{\alpha 0 \times \beta 0}) \bigl( \Vert w_{t} \Vert ^{2}+ \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr), \end{aligned}$$

(51)

$$\begin{aligned} G_{21}&=\frac{1}{2}\bigl(\sigma _{3}(t)-\sigma _{4}(t)\bigr) \bigl(\nabla ^{m_{2}}\bigl(v^{1}_{t}+v^{2}_{t} \bigr), \nabla ^{m_{2}}z_{t}\bigr) \\ &\le C(R_{\alpha 0\times \beta 0}) \bigl( \bigl\Vert \nabla ^{m_{2}}v^{1}_{t} \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}v^{2}_{t} \bigr\Vert \bigr) \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert \\ &\le \frac{\sigma _{340}}{8} \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2}+ \frac{2C(R_{\alpha 0\times \beta 0})}{\sigma _{340}}\bigl( \bigl\Vert \nabla ^{m_{2}}v^{1}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v^{2}_{t} \bigr\Vert ^{2}\bigr) \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}, \end{aligned}$$

(52)

$$\begin{aligned} G_{22}&=\frac{1}{2}\bigl(\Phi _{1}(t)-\Phi _{2}(t)\bigr) \bigl(\nabla ^{m_{2}} \bigl(v^{1}+v^{2}\bigr), \nabla ^{m_{2}}z_{t} \bigr) \\ &\le C(R_{\alpha 0\times \beta 0}) \bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert + \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert \bigr) \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert \\ &\le \frac{\sigma _{340}}{8} \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2}+ \frac{2C(R_{\alpha 0\times \beta 0})}{\sigma _{340}}\bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr), \end{aligned}$$

(53)

$$\begin{aligned} G_{23}&=\bigl(g_{2}(u_{1},v_{1})-g_{2}(u_{2},v_{2}),z_{t} \bigr)\le C(R_{\alpha 0 \times \beta 0}) \bigl( \Vert z_{t} \Vert ^{2}+ \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr), \end{aligned}$$

(54)

where $\sigma _{120}=2N_{10}$ and $\sigma _{340}=2N_{20}$.

Inserting (49)–(54) into (48), we have

$$ \begin{aligned} &\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}\biggl[ \Vert w_{t} \Vert ^{2}+ \alpha \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \Vert z_{t} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2} \\ &\quad \quad {}+ \frac{1}{4}\Phi _{0}\cdot \bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr)\biggr] \\ &\quad \le C_{16}\bigl(1+ \bigl\Vert \nabla ^{m_{1}}u^{1}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{1}}u^{2}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v^{1}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v^{2}_{t} \bigr\Vert ^{2}\bigr) \\ &\quad \quad{}\times \biggl[ \Vert w_{t} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \Vert z_{t} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2} \\ &\quad \quad {}+ \frac{1}{4}\Phi _{0}\cdot \bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr)\biggr]. \end{aligned} $$

(55)

Solving this differential inequality yields

$$ \begin{aligned} &\biggl[ \Vert w_{t} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}w_{t} \bigr\Vert ^{2}+ \Vert z_{t} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}z_{t} \bigr\Vert ^{2} \\ &\quad \quad {}+\frac{1}{4}\Phi _{0}\cdot \bigl( \bigl\Vert \nabla ^{m_{1}}w \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z \bigr\Vert ^{2}\bigr)\biggr] \\ &\quad \le \biggl[ \Vert w_{1} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}w_{1} \bigr\Vert ^{2}+ \Vert z_{1} \Vert ^{2}+ \beta \bigl\Vert \nabla ^{m_{2}}z_{1} \bigr\Vert ^{2} \\ &\quad \quad {}+ \frac{1}{4}\Phi _{0}\cdot \bigl( \bigl\Vert \nabla ^{m_{1}}w_{0} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}z_{0} \bigr\Vert ^{2}\bigr) \biggr] \\ &\quad \quad{}\times \exp \biggl( \int _{0}^{t}C_{17}\bigl(1+ \bigl\Vert \nabla ^{m_{1}}u^{1}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{1}}u^{2}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v^{1}_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v^{2}_{t} \bigr\Vert ^{2}\bigr)\,\mathrm{d}s \biggr). \end{aligned} $$

(56)

Thus, the uniqueness of the solutions is proved.

Therefore, problem (1)–(3) possess a unique solution $(u,v)$. Theorem 1 is proved. □

According to Theorem 1, we can define a nonlinear operator $\{S(t)\}_{t\ge 0}$ the on space $X_{\alpha 0\times \beta 0}: S(t)(u_{0},u_{1},v_{0},v_{1})=(u,u_{t},v,v_{t})$, for all $t\ge 0$. Theorem 1 shows that $\{S(t)\}_{t\ge 0}$ forms a continuous semigroup in $X_{\alpha 0\times \beta 0}$. Before proving the existence of a family of global attractors, we first present their definition.

Definition 1

Let $X_{0}$ be a Banach space and $\{S(t)\}_{t\ge 0}$ a continuous operator semigroup. If there exists a compact set $A_{k_{1}\times k_{2}}$ satisfying the following conditions:

• (i) (Invariance) All $A_{k_{1}\times k_{2}}$ are invariant sets under the action of semigroup $\{S(t)\}_{t\ge 0}$,

  $$ S(t)A_{k_{1}\times k_{2}}=A_{k_{1}\times k_{2}},\quad \forall t\ge 0; $$

• (ii) (Attractiveness) All $A_{k_{1}\times k_{2}}$ attract all bounded sets in $X_{0}$, i.e., for any bounded $B\subset X_{0}$,

  $$ \operatorname{dist}\bigl(S(t)B,A_{k_{1}\times k_{2}}\bigr)=\sup_{x\in B} \inf_{y\in A_{k_{1} \times k_{2}}} \bigl\Vert S(t)x-y \bigr\Vert _{X_{0}}\to 0,\quad \text{as } t\to \infty . $$

  In particular, when $t\to \infty$, all trajectories $S(t)u_{0}$ from $u_{0}$ converge to $A_{k_{1}\times k_{2}}$, i.e.,

  $$ \operatorname{dist}\bigl(S(t)u_{0},A_{k_{1}\times k_{2}}\bigr)\to 0,\quad \text{as } t \to \infty . $$

then, a compact set $A_{k}$ is a global attractor of the semigroup $\{S(t)\}_{t\ge 0}$. Let $\mathcal{A}=\{A_{k_{1}\times k_{2}}\subset X_{0}: k_{1}=1,2,\dots ,m_{1},k_{2}=1,2, \dots ,m_{2}\}$ be a family of subsets in $X_{0}$. Then $\mathcal{A}$ is called the global attractor family in $X_{0}$.

Proof of Theorem 2

By Lemma 3, for all $R_{\alpha 0\times \beta 0}>0$, we have $\|(u_{0},u_{1},v_{0},v_{1})\|_{X_{\alpha 0\times \beta 0}}\le R_{ \alpha 0\times \beta 0}$. Thus,

$$ \begin{aligned} \bigl\Vert S(t) (u_{0},u_{1},v_{0},v_{1}) \bigr\Vert ^{2}_{X_{\alpha 0 \times \beta 0}}&= \bigl\Vert \nabla ^{m_{1}}u \bigr\Vert ^{2}+ \Vert u_{t} \Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}}u_{t} \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{m_{2}}v \bigr\Vert ^{2} \\ &\quad{}+ \Vert v_{t} \Vert ^{2}+\beta \bigl\Vert \nabla ^{m_{2}}v_{t} \bigr\Vert ^{2} \\ &\le C(R_{\alpha 0\times \beta 0}), \end{aligned} $$

indicating that $\{S(t)\}_{t\ge 0}$ are uniformly bounded in $X_{\alpha 0\times \beta 0}$.

Further,

$$ \begin{aligned}B_{{\alpha k_{1}\times \beta k_{2}},0}&=\bigl\{ (u,u_{t},v,v_{t}) \in X_{\alpha k_{1}\times \beta k_{2}}: \\ &\quad \bigl\Vert (u,u_{t},v,v_{t}) \bigr\Vert ^{2}_{X_{\alpha k_{1}\times \beta k_{2}}}= \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{1}}u_{t} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}u_{t} \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}v_{t} \bigr\Vert ^{2}+ \beta \bigl\Vert \nabla ^{m_{2}+k_{2}}v_{t} \bigr\Vert ^{2}\le C(R_{\alpha 0\times \beta 0})+C(R_{\alpha k_{1}\times \beta k_{2}})\bigr\} \end{aligned} $$

are bounded absorbing sets of the semigroup $\{S(t)\}_{t\ge 0}$ in $X_{\alpha 0\times \beta 0}$.

Because $X_{\alpha k_{1}\times \beta k_{2}}\hookrightarrow \hookrightarrow X_{ \alpha 0\times \beta 0}$ are compactly embedded, i.e., bounded sets in $X_{\alpha k_{1}\times \beta k_{2}}$ are compact sets in $X_{\alpha 0\times \beta 0}$, the solution semigroup $\{S(t)\}_{t\ge 0}$ is a fully continuous operator.

To sum up, we obtained the global attractor family $\mathcal{A}=\{A_{\alpha k_{1}\times \beta k_{2}}\}$ of the solution semigroup $\{S(t)\}_{t\ge 0}$ in $X_{\alpha 0\times \beta 0}$, and

$$\begin{aligned} &A_{\alpha k_{1}\times \beta k_{2}}=\omega (B_{{\alpha k_{1}\times \beta k_{2}},0})=\bigcap _{\tau \ge 0} \overline{\bigcup_{t\ge \tau}S(t)B_{{\alpha k_{1}\times \beta k_{2}},0}}, \\ &A_{\alpha k_{1}\times \beta k_{2}}\subset X_{\alpha 0\times \beta 0}, \quad k_{1}=1,2, \dots ,m_{1},k_{2}=1,2,\dots ,m_{2}, \text{ for } \forall \alpha ,\beta \in (0,1]. \end{aligned}$$

Theorem 2 is proved. □

Note 1

Lemma 4 and Theorem 2 show that bounded absorbing sets

$$ \begin{aligned} B_{{\alpha k_{1}\times \beta k_{2}},0}&=\bigl\{ (u,u_{t},v,v_{t}) \in X_{\alpha k_{1}\times \beta k_{2}}: \\ &\quad \bigl\Vert (u,u_{t},v,v_{t}) \bigr\Vert ^{2}_{X_{\alpha k_{1}\times \beta k_{2}}}= \bigl\Vert \nabla ^{m_{1}+k_{1}}u \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{1}}u_{t} \bigr\Vert ^{2}+\alpha \bigl\Vert \nabla ^{m_{1}+k_{1}}u_{t} \bigr\Vert ^{2} \\ &\quad{}+ \bigl\Vert \nabla ^{m_{2}+k_{2}}v \bigr\Vert ^{2}+ \bigl\Vert \nabla ^{k_{2}}v_{t} \bigr\Vert ^{2}+ \beta \bigl\Vert \nabla ^{m_{2}+k_{2}}v_{t} \bigr\Vert ^{2}\le C(R_{\alpha 0\times \beta 0})+C(R_{\alpha k_{1}\times \beta k_{2}})\bigr\} \end{aligned} $$

are compact sets in $X_{\alpha 0\times \beta 0}$. Therefore, based on condition 3 in Lemma 2, the operator semigroup $\{S(t)\}_{t\ge 0}$ only needs to be a continuous operator. According to Theorem 1, the semigroup $\{S(t)\}_{t\ge 0}$ is already continuous. Thus, the global attractor family $\mathcal{A}=\{A_{\alpha k_{1}\times \beta k_{2}}\}$ of problem (1)–(3) in $X_{\alpha 0\times \beta 0}$ can also be obtained.

4 Summary and prospects

This paper investigated higher-order $(m_{1},m_{2})$-coupled Kirchhoff systems with higher-order rotational inertia and nonlocal damping. For the first time, we systematically defined the family of global attractors of problem (1)–(3) and proved its existence. The findings enriched the relevant findings on higher-order coupled Kirchhoff models and laid a theoretical foundation for future practical applications.

Despite defining and proving the existence of the global attractor family of the higher-order $(m_{1},m_{2})$-coupled Kirchhoff system, many questions concerning such models still require further investigation:

• 1. The higher-order $(m_{1},m_{2})$-coupled Kirchhoff system in this paper is autonomous, while the relatively complex nonautonomous higher-order $(m_{1},m_{2})$-coupled Kirchhoff systems and higher-order $(m_{1},m_{2})$-coupled Kirchhoff systems with delays have not been studied. Thus, it is very meaningful to study the asymptotic behaviors of such systems;

• 2. This paper focused mainly on the global attractor family of dynamic systems, while many other properties were not explored, such as the dimension estimate, the exponential attractor family, and the inertial manifold family. The scarce relevant theoretical results warrant further research efforts.