Asymptotic behavior for a class of viscoelastic equations with memory lacking instantaneous damping

1.   Introduction

In this paper, we mainly study the following initial-boundary value problem for viscoelastic equation with hereditary memory:

where $\Omega\subset\mathbb{R}^3$ is a bounded smooth domain, $\nu > 0$, and the forcing term $g = g(x)\in L^{2}(\Omega)$ is given.

Next, we establish the following hypotheses for the kernel function $k(s)$

$(H_{1})$ Let $\mu(s) = -k'(s)$, and assume

and there exists $\delta > 0$, such that

and let

$(H_{2})$ The nonlinearity $f\in C^1$ satisfies $f(0) = 0$ and also fulfills the following conditions

and

where $c, \lambda_1$ are positive constants and $\lambda_1$ is the first eigenvalue of $-\Delta$ in $H_0^1(\Omega)$ with Dirichlet boundary condition. From (1.5), it's easy to get that there exist $\lambda(0 < \lambda < \lambda_1)$ and $c_f\geq 0$; such that

Let $F(s) = \int_0^s f(\sigma)d\sigma$, then

The equation associated with Eq (1.1) is as follows

which mainly describes a pure dispersion wave process, such as the motion equation of strain-arc wave of linear elastic rod considering transverse inertia and ion-acoustic wave propagation equation in space transformation with weak nonlinear effects (see e.g., ^[1,2,3,4]).

In recent years, the following types of equations have been studied by many scholars (see e.g., ^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18]} and the references therein)

Many researchers considered different kinds of cases, respectively, when the parameters $\rho, \gamma, \alpha, \nu, \mu = 0$ or $\rho, \gamma, \alpha, \nu, \mu\neq 0$ under different situations. However, they only obtained global existence of solutions and the energy decay results^{[6,7,8,15,16]}. In particular, in ^[10,11,17], the scholars only proved blow-up result, decay result and global existence result of solutions under various kinds of conditions and when the dispersion term and dissipative term don't be contained. Next, we analyze several key results in detail. Araújo et al.^[18] established well-posedness result when $\gamma\geq0, \nu, \mu = 0$ and proved the existence of global attractor when $\nu, \mu = 0$ and $-\Delta u_t$ was included.

Qin et al.^[19] proved the existence of uniform attractors in non-autonomous case by improving the result of ^[18] when $-\Delta u_t$ was still included. Recently, Conti et al.^[20] obtained the existence of global attractors and optimal regularity of global attractors for the following equation when the nonlinearity $f$ meets critical growth

with $\rho \in [0, 4]$ and $\rho < 4$, respectively. Therefore, based on the above existing research, we devote to obtain the existence of global attractors in higher regular space for the problem (1.1) which doesn't contain the strong damping $-\Delta u_t$ in this paper.

Firstly, because the Eq (1.1) doesn't contain strong dissipative term $-\Delta u_t$, which makes that the Eq (1.1) is different from usual viscoelastic equations. Next, for the Eq (1.1), its dissipation is only generated by memory term with weaker dissipation rather than the strong dissipative term $-\Delta u_t$, which leads to the need of higher regularity to ensure compactness, so the multiplier $A^{\kappa}u_t$ will be used to obtain our result. We use new analytical techniques to obtain the upper semicontinuity of global attractors. Thus, our results complement the existing conclusions because we only use the memory dissipation to prove the existence and the semicontinuity of global attractors.

In addition, to the best of our knowledge, the key point for proving the existence of global attractors is to verify the existence of bounded absorbing set and the compactness of the semigroup in some sense. However, the absence of term $-\Delta u_t$ causes that energy dissipation of Eq (1.1) is lower than usual viscoelastic equation, and its dissipation only is presented by the memory term. Hence, this will lead to two main difficulties. On the one hand, the absence of term $-\Delta u_t$ makes the equation lacks strong structural damping. On the other hand, to ensure strong convergence of the solution in $L^2(0, T;H_0^1(\Omega))$, how to obtain higher regularity of solutions. Thereby, for obtaining dissipative and compactness of semigroup, we will use analysis techniques and the ideas in ^[21,22] to overcome these difficulties.

The plan of this paper is as follows. In Section 2, we recall some basic concepts and useful results that will be used later. In Section 3, firstly, the bounded absorbing set is obtained. Secondly, we verify asymptotic compact of semigroup by contractive function method ^[23,24]. Finally, the existence of global attractors $\mathcal{A}$ is proved in $H_0^1(\Omega) \times H_0^1(\Omega) \times L_{\mu}^2(\mathbb{R}^+; H_0^1(\Omega))$. In section 4, we obtain the upper semicontinuity of global attractors.

2.   Preliminaries

Following the Dafermos' idea of introducing an additional variable $\eta^t$, the past history of $u$, whose evolution is ruled by a first-order hyperbolic equation (see e.g., ^[25] and references therein). Thus the original problem (1.1) can be translated into a dynamical system on a phase space with two components (see ^[26]). In particular, in the following, we introduce the past history of $u$ in the, i.e.

Provided that let $\eta^t_t = \frac{\partial}{\partial t}\eta^t, \quad \eta^t_s = \frac{\partial}{\partial s}\eta^t$, then we have

Historical variable $u_0(\cdot, -s)$ of $u$ satisfies the following condition

where $\Re > 0$ and $\sigma \leq \delta$ ($\delta$ is from (1.3)).

By $(H_1)$ and (2.1), (2.2), we get

Thus, if we assume $\nu - m_0 = 1$, then the system (1.1) can be rewrite as

with initial-boundary condition

In the whole paper, unless otherwise stated, $z(t) = (u(t), u_t(t), \eta^t)$ is the solution of systems (2.6), (2.7) with initial value $z_0 = (u_0, u_1, \eta^0)$.

For conveniences, hereafter let $|\cdot |_p$ be the norm of $L^p(\Omega)(p\geq 1)$. Let $\langle\cdot, \cdot \rangle$ be the inner product of $L^2(\Omega)$, $\|\cdot\|_0^2$ be the equivalent norm $H_0^1(\Omega)$. Denote $A = -\Delta$ with domain $D(A) = H^2(\Omega)\cap H_0^1(\Omega)$.

Denoting the weight space $\mathcal{V}_1 = L_\mu ^2(\mathbb{R^+}; H_0^1(\Omega))$, $\mathcal{V}_2 = L_\mu ^2(\mathbb{R}^+; D(A))$ and its inner product and norm are

Then phase spaces of the Eq (2.6) are

and their corresponding norms are

In addition, denote $\mathcal{V}_{\kappa} = D(A^{\frac{\kappa+1}{2}})(\kappa\in(0, \frac{1}{2}))$ and let $\|\cdot\|_{\kappa}$ be the norm of $\mathcal{V}_{\kappa}$. Then we can also define phase space of the Eq (1.1) is

and the corresponding norm is ${\left\| \cdot \right\|^2_{\mathcal{M}_{\kappa}}} = \left\| \cdot \right\|^2_{\kappa} + \left\| \cdot \right\|^2_{\kappa} + \left\|\cdot \right\|_{\mu, {\kappa}}^2.$

And there exists the following compact embedding

Definition 2.1. Let $X$ be a Banach spaces and $\mathcal{X}$ be a family of operators defined on it. We say that $\{S(t)\}_{t\geq0}$ is a continuous semigroup on $X$ if $\{S(t)\}_{t\geq0}$ fulfills

and satisfies

(i)$S(0) = Id$(Identity operator);

(ii)$S(t+s) = S(t)S(s), \forall t, s\geq 0$.

The main results of this paper (the existence of global attractors) can be obtained by the following definitions and theorem. Next, let's talk about it (it's similar to ^[14,23,26]).

Definition 2.2. Let $X, Y$ be two Banach spaces and $B$ be a bounded subset of $X\times Y$. We call a function $\phi(\cdot, \cdot; \cdot, \cdot)$, defined on $(X\times X) \times (Y\times Y)$, to be a contractive function if for any sequence $\{(x_n, y_n)\}_{n = 1}^\infty \subset B$, there is a subsequence $\{(x_{n_k}, y_{n_k})\}_{k = 1}^\infty\subset\{(x_n, y_n)\}_{n = 1}^\infty$ satisfies

We denote the set of all contractive functions on $B\times B$ by $\mathfrak{E}(B)$.

Lemma 2.3. Let $X, Y$ be two Banach spaces and $B$ be a bounded subset of $X\times Y$, $\{S(t)\}_{t\geq 0}$ is semigroup with a bounded absorbing set $B_0$ on $X\times Y$. Moreover, assume that for any $\varepsilon > 0$ there exist $T = T(B; \varepsilon)$ and $\phi_T(\cdot, \cdot; \cdot, \cdot)\in \mathfrak{E}(B)$ such that

where $\phi_T$ depends on $T$. Then the semigroup $\{S(t)\}_{t\geq 0}$ is asymptotically compact in $X\times Y$.

In the following theorem, we give a method to verify the asymptotically compactness of a semigroup generated by the Eq (1.1), which will be used in our later discussion.

Theorem 2.4. Let $X, Y$ be two Banach spaces and $\{S(t)\}_{t\geq 0}$ be a continuous semigroup on $X \times Y$. Then $\{S(t)\}_{t\geq 0}$ has a global attractor in $X \times Y$. Provided that the following conditions hold:

(i) $\{S(t)\}_{t\geq 0}$ has a bounded absorbing set $B_0$ on $X \times Y$;

(ii) $\{S(t)\}_{t\geq 0}$ is a contractive semigroup on $X \times Y$.

Lemma 2.5. Let $X\subset\subset H\subset Y$ be Banach spaces, with $X$ reflexive. Suppose that $u_n$ is a sequence that is uniformly bounded in $L^2(0, T; X)$ and $du_n/ dt$ is uniformly bounded in $L^p(0, T; Y)$, for some $p > 1$. Then there is a subsequence of $u_n$ that converges strongly in $L^2(0, T; H)$.

3.   Global attractors in $\mathcal{M}_1$

Throughout the paper, we assume that $\Omega\subset \mathbb{R}^n(n = 3)$ be bounded smooth domain, the kernel function $\mu$ and the nonlinearity satisfy $(H_1)$ and $(H_2)$ respectively, and $g\in L^2(\Omega)$.

Firstly, the well-posedness result for the Eq (1.1) can be obtained by the $Faedo$-$Galerkin$ method (see e.g., ^[18]). Thereout, we only give the final result.

Lemma 3.1. For any $T > 0$ and $z_{0} = (u_0, u_1, \eta^{0})\in \mathcal {M}_{1}$, the problem $(1.1)$ has unique weak solution $z = (u(x, t), u_t(x, t), \eta^{t})$ satisfying

and

By Lemma 3.1, the semigroup $\{S(t)\}_{t\geq 0}$ in $\mathcal{M}_1$ will be defined as the following:

and it is a strongly continuous semigroup on $\mathcal{M}_1$.

Lemma 3.2. For some $\mathcal{R} > 0$ and

then there exists a constant $\mathcal{R}_1 = \mathcal{R}_1(\mathcal{R})$, such that for any $t \geq 0$, the following estimate holds:

Proof. Multiplying the first equation of (2.6) by $u_t$, and integrating over $\Omega$, we obtain that

Next, let $E(t) = |u_t|^2_2+\|u_t\|^2_0 + \|u\|^2_0+\|\eta^t\|^2_{\mu, 0}-2\langle F(u), 1\rangle- 2\langle g, u\rangle$, then by $(H_2)$, $H\ddot{o}lder$ inequality and Young inequality, we can get that

and

hold for any $t\geq0$.

In addition, it's easy to obtain that

Integrating (3.4) about $t$ from 0 to $t$, and combining with (3.2), (3.3), we have

where $\mathcal{R}_1 = \mathcal{R}_1(\|z(0)\|_{\mathcal{M}_1})$ depends on $\|z(0)\|_{\mathcal{M}_1}$.

Lemma 3.3. For any $T > 0$, $z_0\in \mathcal{M}_1$ and $\|z_0\|_{\mathcal{M}_1}\leq \mathcal{R}$, then there exists a constant $\mathcal{K}_1 = \mathcal{K}_1(\mathcal{R}, T)$, it follows that

holds for any $t \in [0, T]$.

Proof. Multiplying the first equation of (2.6) by $u_{tt}$, and integrating over $\Omega$, we obtain that

Using Lemma3.2, $H\ddot{o}lder$ inequality and Young inequality, then

By Lemma 3.2, we have

Combining with (3.8) and $t \in [0, T]$, we get

Just let $\mathcal{K}_1 = C\big[1+\mathcal{R}_1^6+|g|_2^2\big](1+T)$, then (3.6) holds.

Lemma 3.4. Provided that $(u(t), u_t(t), \eta^t)$ is a sufficiently regular solution of $(2.6)$, $(2.7)$. Then, for the functional

it satisfies the following estimate

And we can also obtain

where $H(t) = \frac{1}{2}|u_t|_2^2+\frac{1}{2}\|u\|_0^2+\frac{1}{2}\|u_t\|_0^2+\frac{1}{2}\|\eta^t\|_{\mu, 0}^2$, and $k_0 = k_0(m_0)$ is a positive constant.

Proof. First of all, by Hölder inequality and Young inequality, it's easy to get that

Next, taking the derivative about $t$ for $\Lambda_0(t)$, we have

Now, we sequentially deal with the two terms on the right of (3.13).

The estimate for the first term is as follows

The estimate for the second term, by concerning the second equation of (2.6), we obtain

and

Thus, combining with (3.13)–(3.15), then (3.11) holds.

Lemma 3.5. Assuming that $(u(t), u_t(t), \eta^t)$ is a sufficiently regular solution of $(2.6)$, $(2.7)$. Then the functional

fulfills the following control

And we can obtain differential inequality

where $k$ is a positive constant and $\varepsilon \in (0, 1)$.

Proof. Using $H\ddot{o}lder$ inequality, Young inequality and $Poincar\acute{e}$ inequality, it's easy to get

Furthermore, taking the time-derivative for $\mathcal{N}(t)$ and combining with the first equation (2.6), it follows that

where $k_1\geq \frac{1}{\lambda_1}+1$. Next, we dispose each term on the right side of (3.20), it follows that

and by (1.6)

By (3.20)–(3.22) and the definition of $H(t)$, it yields

Theorem 3.6. There exists a constant $\mathcal{R}_0$, such that, for any $T_0 = T_0(\|z_0\|_{\mathcal{M}_1}) > 0$, whenever

then for all $t\geq T_0$, we have

Proof. According to the definition of $H(t)$, we can obtain

In addition, the following functional can be defined

By Lemma 3.4 and Lemma 3.5, we get

Let its perturbation $\epsilon$ be small enough and $C$ be sufficiently large, and combining with $(H_2)$, then it yields

However, combining with $(H_2)$, (3.11), (3.18) and (3.23), we have

i.e.

Thus, when $\delta$ is fixed, then we can choose appropriate $l, \epsilon, C$, such that

and

Furthermore, let $\gamma = \epsilon(1-\varepsilon), \gamma_0 = \max\{\frac{C^2}{m_0}+\frac{1+2\epsilon}{2\lambda_1^2}, \epsilon c_f|\Omega|\}$, then by (3.27), we obtain

Using (3.25), we have

From Gronwall Lemma, it's easy to obtain

Using (3.25) again, we have

Hence, for any $t\geq T_0 = \frac{3C}{2\gamma}\ln \frac{2\gamma\mathcal{L}(0)}{3\gamma_0(|g|_2^2+1)+\frac{2\epsilon c_f}{C}|\Omega|}$, we obtain

where $\mathcal{R}_0 = \frac{12\gamma_0 }{\gamma}(|g|_2^2+1)+\frac{8\epsilon c_f}{C}|\Omega|$.

Therefore, we can know that the set

is a bounded absorbing set for semigroup $\{S(t)\}_{t\geq 0}$ on $\mathcal{M}_1$.

Corollary 3.7. There exists a constant $C_{\mathcal{R}_0}$, such that, for all $t\geq T_0$, we have

Proof. Integrating (3.29) about $t$ from $t$ to $t+1$, and combining with (3.25) and Lemma 3.7, the above estimate is easily obtained.

Lemma 3.8. For any $T > 0$, there exists a constant $\mathcal{R}_3 > 0$, such that, whenever

it follows that

Proof. Multiplying the first equation of (2.6) by $A^{\kappa}u_t$, and integrating over $\Omega$, we obtain that

where $E_1(t) = \frac{1}{2}\big[|A^{\frac{\kappa}{2}}u_t|^2_{2}+\|u_t\|^2_{\kappa} + \|u\|^2_{\kappa} +\|\eta^t\|^2_{\mu, {\kappa}}\big]$.

Due to

and by $(H_2)$ and Lemma 3.2, we obtain

Then by (3.33)–(3.35), we have

where $h, h_1$ are positive constant.

Hence, using Gronwall lemma, we can obtain that

holds for any $t\in [0, T]$. This proof is finished.

Lemma 3.9. For any $t \in [0, T]$, there exists a constant $\mathcal{R}_5 > 0$, such that, whenever

it follows that

Proof. Firstly, the first equation of the system (2.6), it can be rewritten

Next, let $v = u_t+\delta_1 u$, and multiplying (3.38) by $A^{\kappa}v$, and integrating over $\Omega$, we obtain that

In addition, we deal with each term on the right of (3.39), it yields by using $Minkowski$ inequality

and

next, we deal with the nonlinear term by using Hölder inequality and Sobolev embedding theorem, it yields

which, together with (3.39)–(3.42), obtains

Where $h_0, h_2$ are positive constant.

Let $\delta_1$ be small enough, such that

Then combining Lemma 3.1, Lemma 3.2, Lemma 3.8 and Gronwall lemma, we get

Moreover, integrating (3.41) about t from t to t+1, then we have

where $\mathcal{R}_5 = \mathcal{R}_5(Q(\|z_0\|_{\kappa}), \beta_0, {\beta_1}, \delta_1, \mathcal{R}_1, \mathcal{R}_3, \mathcal{K}_1, |g|_2)$.

In order to prove the existence of global attractor for $\{S(t)\}_{t\geq 0}$ on $\mathcal{M}_1$, we have to verify some compactness for the semigroup $\{S(t)\}_{t\ge 0}$. For further purpose, we will give asymptotically compact theorem of the semigroup on $\mathcal{M}_1$.

Theorem 3.10. The semigroup $\{S(t)\}_{t\geq 0}$ associated with problem $(2.6)$, $(2.7)$ is asymptotically compact on $\mathcal{M}_1$.

Proof. Firstly, Let $z^1(t) = (u^1(t), u^1_t(t), \eta^t_1), z^2(t) = (u^2(t), u^2_t(t), \eta^t_2)$ are two solutions of (2.6) corresponding with the initial data $z_0^1 = (u_0^1, u_1^1, \eta^0_1), z^2_0 = (u_0^2, u_1^2, \eta^0_2)$ respectively. Setting $z(t) = (\omega(t), \omega_t(t), \theta^t) = (u^1(t)-u^2(t), u^1_t(t)-u^2_t(t), \eta^t_1-\eta^t_2)$, then $z(t)$ satisfies the following equation

with initial-boundary conditions

Similar to the definition of $H(t)$, we let $\mathcal{H}_{\omega}(t) = \frac{1}{2}|\omega_t(t)|_2^2+\frac{1}{2}\|\omega(t)\|_0^2+\frac{1}{2}\|\omega_t(t)\|_0^2 + \frac{1}{2}\|\theta^t\|_{\mu, 0}^2$, then according to Lemma 3.4 and Lemma 3.5, we obtain

$(i)$ Let $\Lambda_{\omega}(t) = -\langle\theta^t, \omega_t\rangle_{\mathcal{M}_1}-\int_{0}^{\infty}\mu(s)\langle\theta^t, \omega_t\rangle ds$, we have

and

$(ii)$ Assuming that $\mathcal{N}_{\omega}(t) = \int_{\Omega}{\omega_t}\omega dx+\int_{\Omega}\nabla \omega_t \nabla \omega dx$, we can also obtain that

and

$(iii)$ Obviously, we can get it easily

By $(H_2)$, it's easy to obtain

where $Q(\cdot)$ denotes monotonically increasing function.

Combining with (3.51) and (3.52), we have

Secondly, we can define the following functional

By (3.48) and (3.50), we get

Next, let its perturbation $\epsilon$ be small enough and $C_1$ be sufficiently large, then yields

Therefore, we can also deduce easily that

In the same way, let $\varepsilon > 0$ be small enough and $C_1$ is sufficiently large, such that

and

and let $\alpha_0 = \epsilon(1-\varepsilon)$, then (3.56) becomes

Using Gronwall lemma, we can deduce that

holds for any $T > \frac{3C}{2\beta}\ln{\frac{2\mathcal{L}_\omega(0)}{C\varepsilon}}$.

Combining with (3.55) and (3.58), then we have

where

Next, for all $T\geq T_1 > 0$, we prove $\phi^T(z^1, z^2) \in \mathfrak{E}(B)$.

Indeed, let $z_n(t) = (u_n(t), u_{nt}(t), \eta_n^{t}) \in B_0$ be the solution with initial value $z_n(0) = z^n_0 = (u^n_0, u^n_1, \eta_n^0)\in B_0$. According to Corollary 3.11 and Lemma 3.8, we know that the sequence $\{(u_n(t), u_{nt}(t), \eta_n^{t})\}$ is uniformly bounded in $\mathcal{M}_1$. That is

Because we know that the embedding $D(A^{\frac{1+\kappa}{2}})\hookrightarrow D(A^{\frac{1}{2}})$ is compact by (2.8) and $D(A^{\frac{1}{2}})\hookrightarrow L^{2}(\Omega)$ is continuous, then combining with Lemma 2.5, it's easy to get that there exists subsequence (still note $\{(u_n(t), u_{nt}(t), \eta_n^{t})\}$) of $\{(u_n(t), u_{nt}(t), \eta_n^{t})\}$ such that

Thanks to Theorem 3.7 and Theorem 3.10, we can deduce the main result of this paper as the following theorem:

Theorem 3.11. The semigroup $\{S(t)\}_{t\geq 0}$ for the problem $(2.6)$, $(2.7)$ possesses a global attractor $\mathcal{A}$ in $\mathcal{M}_1$; and $\mathcal{A}$ is non-empty, compact, invariant in $\mathcal{M}_1$ and attracts any bounded set of $\mathcal{M}_1$ with respect to $\mathcal{M}_1$-norm.

4.   Upper semicontinuity of attractors in $\mathcal{M}_1$

Let's consider the following equations

with initial-boundary condition

where $\omega\in [0, 1]$ is a disturbance parameter.

Let $\omega = 0$, then the above equation is transformed into Eq (2.6) with (2.7). Using the proof method of above section word by word, we have the following lemma:

Lemma 4.1. The semigroup $\{S_{\omega}(t)\}_{t\ge 0}$ associated with Eq $\rm (4.1)$ with $\rm(4.2)$ possesses a compact global attractor $\mathcal{A}_{\omega}$ for any $\omega \ge 0$.

Then $\mathcal{A}_{0} = \mathcal{A}$($\mathcal{A}$ from Theorem 3.11).

Lemma 4.2. For any $T\ge 0$ and $z_0\in \mathcal{A}_{\omega}$, then we have

for $0\leq t\leq T$ holds, where C is independently of $\omega$.

Proof. Let $z = (u, u_t, \eta^t)$ and $z^{\omega} = (v, v_t, \xi^t)$ are unique solutions of Eqs (2.6) and (4.1) with initial value $z_0\in \mathcal{A}_{\omega}$ respectively. Setting $w = u-v, \zeta^t = \eta^t-\xi^t$, then $(w, w_t, \zeta^t)$ is a unique solution of the following equations

with initial-boundary condition

Multiplying the first equation of (4.3) by $w_{t}$ in $L^2(\Omega)$, we obtain

According to Hölder inequality, $z^{\omega}\in \mathcal{A}_{\omega}$ and Lemma 3.2, there exist a constant $\alpha > 0$ such that

where $Q_{\mathcal{R}_0} = Q_{\mathcal{R}_0}(\mathcal{R}_0)$ is independently $\omega$.

Let $C = \frac{Q_{\mathcal{R}_0}}{\alpha}$ and applying Gronwall Lemma, we have

Theorem 4.3. Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth boundary, and we assume that $f$ satisfies $\rm(1.4)$–$\rm(1.7)$ and $\rm(1.2)$, $\rm(1.3)$ holds, given $g\in L^2(\Omega)$, then

Proof. For any $\varepsilon > 0$, since $\mathcal{A}_{\omega}$ is an universal bounded subset of $\mathcal{M}_1$ for any $\omega\in [0, 1]$, and $\mathcal{A}$ is a compact attracting set for $\{S(t)\}_{t\geq 0}$ on $\mathcal{M}_1$. So there exists $T > 0$ such that $S(t)(T)\mathcal{A}_{\omega} \subset N(\mathcal{A}, \frac{\varepsilon}{2})$. On the other hand, associating with the invariance of $\mathcal{A}_{\omega}$ and Lemma (4.2), for any $t \geq T$, we have

as $\omega$ small enough. Setting $\varepsilon = \frac{\omega}{C}$, so we have

here $\nu \geq C$ is a constant, which completes the proof of the desired results.

Acknowledgments

The authors would like to thank the referees for their many helpful comments and suggestions. The research is financially supported by Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science and Technology), National Natural Science Foundation of China (Nos. 11101053, 71471020).

Conflict of interest

The authors declare that they have no competing interests.