Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral delay differential equations using the r-Lambert W function

1.   Introduction

The Navier-Stokes equations are a typical nonlinear system, which model the mechanics law for fluid flow and have been applied in many fields. There are many research findings on the Navier-Stokes system, involving well-posedness, long-time behavior, etc., ^{[1,2,3,4,5,6,7,8,9,10]}. Furthermore, to simulate the fluid movement modeled by the Navier-Stokes equations, some regularized systems are proposed, such as the Navier-Stokes-Voigt equations. The Navier-Stokes-Voigt equations were introduced by Oskolkov in 1973, which describe the motion of Kelvin-Voigt viscoelastic incompressible fluid. Based on the global well-posedness of 3D Navier-Stokes-Voigt equations in ^[11], many interesting results on long-time behavior of solutions have been obtained, such as the existence of global attractor and pullback attractors, determining modes and estimate on fractal dimension of attractor ^[12,13,14] and references therein for details.

The influence of past history term on dynamical system is well known, we refer to ^{[15,16,17,18]} for interesting conclusions, such as the global well-posedness, the existence of attractors and so on. In 2013, Gal and Tachim-Medjo ^[17] studied the Navier-Stokes-Voigt system with instantaneous viscous term and memory-type viscous term, and obtained the well-posedness of solution and exponential attractors of finite dimension. In 2018, Plinio et al. ^[18] considered the Navier-Stokes-Voigt system in ^[18], in which the instantaneous viscous term was completely replaced by the memory-type viscous term and the Ekman damping $\beta u$ was presented. The authors showed the existence of regular global and exponential attractors with finite dimension. The presence of Ekman damping was to eliminate the difficulties brought by the memory term in deriving the dissipation of system.

Some convergence results of solutions or attractors as perturbation vanishes for the non-autonomous dynamical systems without memory can be seen in ^{[19,20,21,22]}. However, there are few convergence results on the system with memory. Therefore, our purpose is to study the tempered pullback dynamics and robustness of the following 3D incompressible Navier-Stokes-Voigt equations on the bounded domain $\Omega$ with memory and the Ekman damping:

where $\Omega_{\tau} = (\tau, +\infty)\times\Omega$, $\partial\Omega_{\tau} = (\tau, +\infty)\times\partial\Omega$, $\Omega_{0} = (0, \infty)\times\Omega$, $\tau\in \mathbb{R}^+$ is the initial time, $\alpha > 0$ is a length scale parameter characterizing the elasticity of fluid, $\beta > 0$ is the Ekman dissipation constant, $u = (u_1(t, x), u_2(t, x), u_3(t, x))$ is the unknown velocity field of fluid, and $p$ is the unknown pressure. The non-autonomous external force is $f_{\varepsilon}(t, x) = f_1(x)+\varepsilon f_2(t, x)\ (0\leq\varepsilon < \varepsilon_0)$, where $\varepsilon_0$ is a fixed constant small enough. In addition, $u(\tau)$ is the initial velocity, and $\varphi(s, x)$ denotes the past history of velocity. The memory kernel $g:\ [0, \infty)\rightarrow [0, \infty)$ is supposed to be convex, smooth on $(0, \infty)$ and satisfies that

In general, we give the past history variable

which satisfies

Also, $\eta$ has the explicit representation

and

Next, we give the main features of this paper as follows.

1) Inspired by ^[18,23], we provide a detailed representation and Gronwall type estimates for the energy of (1.1) dependent on $\varepsilon$ in Lemma 2.1, with a focus on the parameters $\omega, \ \Lambda$ and the increasing function $J(\ast)$. Using these parameters, we construct the universe $\mathcal{D}$ and derive the existence of $\mathcal{D}-$pullback absorbing sets, see Lemma 4.9.

2) Via the decomposition method, we show that the process of the system has the property of $\mathcal{D}-\kappa-$pullback contraction in the space $N_V$, and the $\mathcal{D}-$pullback asymptotic compactness is obtained naturally. Based on the theory of attractor in ^[1,24], the $\mathcal{D}$-pullback attractors for the process $\{S^{\varepsilon}(t, \tau)\}$ in $N_V$ are derived, see Theorem 3.3.

3) When the perturbation parameter $\varepsilon\rightarrow 0$ with the non-autonomous external force, the robustness is obtained via the upper semi-continuity of pullback attractors of (1.1) by using the technique in ^[18,21,22], see Theorems 3.2 in Section 3.

This paper is organized as follows. Some preliminaries are given in Section 2, and the main results are stated in Section 3, which contains the global well-posedness of solution, the existence of pullback attractors and robustness. Finally, the detailed proofs are provided in Sections 4 and 5.

2.   Preliminaries

2.1.   Some functional spaces

● The Sobolev spaces

Let $E = \{u|u\in (C^{\infty}_0(\Omega))^3, \mbox{div} u = 0\}$, $H$ is the closure of $E$ in ${(L^2(\Omega))^3}$ topology with the norm and inner product as

$V$ is the closure of $E$ in ${(H^1(\Omega))^3}$ topology with the norm and inner product as

Also, we denote

$H$ and $V$ are Hilbert spaces with their dual spaces $H$ and $V'$ respectively, $\|\cdot\|_{*}$ and $\langle\cdot, \cdot\rangle$ denote the norm in $V'$ and the dual product between $V$ and $V'$ respectively, and also $H$ to itself.

● The fractional power functional spaces

Let $P_L$ be the Helmholz-Leray orthogonal projection in $(L^2(\Omega))^3$ onto $H$ ^[3,7], and

where

$A = -P_{L}\Delta$ is the Stokes operator with eigenvalues $\{\lambda_j\}^{\infty}_{j = 1}$ and orthonormal eigenfunctions $\{\omega_j\}^{\infty}_{j = 1}$.

Define the fractional operator $A^{s}$ by

for $u = {\sum_{j}}(u, \omega_j)\omega_{j}$ with the domain $D(A^{s}) = \{u|A^{s}u\in H\}$, and we use the norm of $D(A^{s})$ as

Especially, denote $W = D(A)$, and $V = D(A^{1/2})$ with norm $\|u\|_{1} = |A^{1/2}u| = \|u\|$ for any $u\in V$.

● The memory spaces

For any $s\in (0, \infty)$, we define $\mu(s) = -g'(s)$, which is nonnegative, absolutely continuous, decreasing ($\mu'\leq 0$ almost everywhere) and

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

Let

which is a Hilbert space on $\mathbb{R}^+$ with inner product and norm

Moreover, the extended memory space can be defined as

equipped with the norm

2.2.   Some inequalities and conclusions

● The bilinear and trilinear operators

The bilinear and trilinear operators are defined as follows ^[8]

Denote $B(u) = B(u, u)$, $B(u, v)$ is a continuous operator from $V\times V$ to $V'$, and there hold

● Some useful lemmas

Lemma 2.1. (^[23]) Assume that

1) A nonnegative function $h$ is locally summable on $\mathbb{R}^{+}$, and for any $\varepsilon\in (0, \varepsilon_0]$ and any $t\geq\tau\geq 0$ there holds

2) The nonnegative function $y_{\varepsilon}(t)$ is absolutely continuous on $[\tau, \infty)$, and satisfies for some constants $R, C_0\geq 0$ that

where $p, q, r\geq 0$, and $p-1 > (q-1)(1+r)\geq 0$.

3) Let $z(t)\geq 0$ be a continuous function on $(0, \infty)$ equivalent to $y_{\varepsilon}(t)$, which means there exist some constants $M\geq 1, \ L\geq 0$ such that

Then, there exist $\omega, \ \Lambda > 0$ and an increasing function $J(\ast):\ \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that

Remark 2.1. Under the assumptions in Lemma 2.1, there exists a constant $\theta\in (0, 1)$ satisfying

Denote

then

Lemma 2.2. (^[15]) Let $\eta$ be the past history variable and (1.2) holds. Then

3.   Main results

3.1.   Some assumptions

We assume that $f_1(x)$ and $f_2(t, x)$ satisfy the following hypotheses:

(C1) The function $f_1\in H$.

(C2) $f_2(t, x)$ is translation bounded in $L^{2}_{loc}(\mathbb{R}, H)$, which means there exists a constant $K > 0$ such that

and for any $t\in\mathbb{R}$, there also holds

where $\kappa, \delta$ are the same as parameters in (2.1) and (2.2) respectively.

3.2.   Equivalent problem

Construct the infinitesimal generator of right-translation semigroup on $M_X$

whose domain is

Given initial datum $U(\tau) = (u(\tau), \eta^\tau)\in N_V$, then (1.1) can be transformed into the following abstract form

3.3.   Main results for the system (3.2)

● Global well-posedness of solution

Definition 3.1 A function $U(t) = (u(t), \eta^t):\ [\tau, +\infty)\rightarrow N_V$ is called the weak solution to (3.2), if for any fixed $T > \tau$ there hold

(i) $U(t)\in C([\tau, T]; N_V), \ \frac{\partial u}{\partial t}\in L^2(\tau, T; V)$.

(ii) $U(\tau) = (u(\tau), \eta^\tau)$.

(iii) for any $w\in C^{1}([\tau, T]; V)$ with $w(T, x) = 0$, there holds

Theorem 3.2. Let $U(\tau)\in N_{V}$, and the hypotheses (C1)–(C2) hold. Then the global weak solution $U(t, x)$ to system (3.2) uniquely exists on $(\tau, T)$, which generates a strongly continuous process

and $S^{\varepsilon}(t, \tau)U(\tau) = U(t)$.

Proof. The global well-posedness of solution can be obtained by the Galerkin approximation method, energy estimates and compact scheme. The detailed proof can be found in ^[15,18] and is omitted here.

● Existence of $\mathcal{D}$-pullback attractors

Theorem 3.3. Assume $U(\tau)\in N_{V}$ and the hypotheses (C1)–(C2) hold. Then the process $S^{\varepsilon}(t, \tau):\ N_V\rightarrow N_V$ generated by the system (3.2) possesses a minimal family of $\mathcal{D}$-pullback attractors $\mathcal {A}^{\varepsilon} = \{A^{\varepsilon}(t)\}_{t\in \mathbb{R}}$ in $N_V$.

Proof. See Section 4.2.

When $\varepsilon = 0$, the system (3.2) can be reduced to the following autonomous system

Remark 3.1. The existence of global attractor $\mathcal{A}^0$ in $N_V$ can be achieved for the semigroup $S^0(t-\tau)$ generated by (3.4).

● Robustness: upper semi-continuity of $\mathcal{D}$-pullback attractors

Let $\bigvee$ be a metric space, and $\{\mathcal{A}^{\lambda}\}_{\lambda\in \bigvee}$ is a family of subsets in $X$. Then it is said that $\{\mathcal{A}^{\lambda}\}$ has the property of upper semi-continuity as $\lambda\rightarrow \lambda_{0}$ in $X$ if

The upper semi-continuity of attractors and related conclusions can be referred to ^[1,19,20,22] for more details.

In the following way, we intend to establish some results on the convergence between $\mathcal{D}$-pullback attractors $\mathcal{A}^\varepsilon$ to system (3.2) and global attractor $\mathcal{A}^0$ to system (3.4) as $\varepsilon\rightarrow 0$.

Theorem 3.4. Let $U(\tau)\in N_{V}$, $\mathcal{A}^{\varepsilon}$ is the family of $\mathcal{D}$-pullback attractors of $S^{\varepsilon}(t, \tau)$ in $N_V$ to system (3.2), and $\mathcal{A}^{0}$ is the global attractor of $S^{0}(t-\tau)$ in $N_V$ to system (3.4). Then the robustness of system is obtained by the following upper semi-continuity

Proof. See Section 5.

4.   Pullback dynamics

4.1.   Theory of dynamics

In this section, we first give the fundamental theory of attractors for dissipative systems, and the related conclusions can be seen in ^[1,2,3,7].

● Some relevant definitions

Definition 4.1. Assume that $\mathcal{P}(X)$ is the family of all nonempty subsets in a metric space $X$. If $\mathcal{D}$ is some nonempty class of families in the form $\widehat{D} = \{D(t):t\in\mathbb{R}\}\subset\mathcal{P}(X)$, where $D(t)\subset X$ is nonempty and bounded, then $\mathcal{D}$ is said to be a universe in $\mathcal{P}(X)$.

Definition 4.2. The family $\widehat{D}_{0} = \{D_{0}(t):t\in\mathbb{R}\}\subset\mathcal{P}(X)$ is $\mathcal{D}$-pullback absorbing for the process $S(\cdot, \cdot)$ on $X$ if for any $t\in\mathbb{R}$ and any $\widehat{D}\in\mathcal{D}$, there exists a $\tau_{0}(t, \widehat{D})\leq t$ such that

Definition 4.3. A process $S(\cdot, \cdot)$ on $X$ is said to be $\mathcal{D}$-pullback asymptotically compact if for any $t\in\mathbb{R}$, any $\widehat{D}\in\mathcal{D}$, and any sequences $\{\tau_{n}\}\subset(-\infty, t]$ and $\{x_{n}\}\subset X$ satisfying $\tau_{n}\rightarrow -\infty$ and $x_{n}\in D(\tau_{n})$, the sequence $\{S(t, \tau_{n})x_{n}\}$ is relatively compact in $X$.

The $\mathcal{D}$-pullback asymptotic compactness can be characterized by the Kuratowski measure of noncompactness $\kappa(B)\ (B\subset X)$, relating definition and properties can be referred to ^[25,26], and the definition of $\mathcal{D}-\kappa$-pullback contraction will be given as follows.

Definition 4.4. For any $t\in \mathbb{R}$ and $\varepsilon > 0$, a process $S(t, \tau)$ on $X$ is said to be $\mathcal{D}-\kappa$-pullback contracting if there exists a constant $T_{\mathcal{D}}(t, \varepsilon) > 0$ such that

Definition 4.5. A family $\mathcal{A}(t) = \{A(t)\}_{t \in \mathbb{R}}$ is called the $\mathcal{D}$-pullback attractors of process $S(t, \tau)$, if for any $t \in \mathbb{R}$ and any $\{D(t)\}\in \mathcal{D}$, the following properties hold.

(i) $A(t)$ is compact in $X$.

(ii) $S(t, \tau)A(\tau) = A(t)$, $t \ge \tau$.

(iii) $\lim_{\tau \to -\infty}{\rm dist}_{X}(S(t, \tau)D(\tau), A(t)) = 0$.

In addition, $\mathcal{D}$-pullback attractor $\mathcal{A}$ is said to be minimal if whenever $\hat{C}$ is another $\mathcal{D}$-attracting family of closed sets, then $A(t) \subset C(t)$ for all $t \in \mathbb{R}$.

● Some conclusions

Theorem 4.6. (^[1,27]) Let $S(\cdot, \cdot):\mathbb{R}^{2}_{d}\times X\rightarrow X$ be a continuous process, where $\mathbb{R}^{2}_{d} = \{(t, \tau)\in\mathbb{R}^{2} |\, t\geq\tau\}$, $\mathcal{D}$ is a universe in $\mathcal{P}(X)$, and a family $\widehat{D}_{0} = \{D_{0}(t):t\in\mathbb{R}\}\subset\mathcal{P}(X)$ is $\mathcal{D}$-pullback absorbing for $S(\cdot, \cdot)$, which is $\mathcal{D}$-pullback asymptotically compact. Then, the family of $\mathcal{D}$-pullback attractors $\mathcal{A_{D}} = \{A_{\mathcal{D}}(t):t\in\mathbb{R}\}$ exists and

Remark 4.1. If $\widehat{D}_{0}\in \mathcal{D}$, then $\mathcal{A_{D}}$ is minimal family of closed subsets attracting pullback to $\mathcal{D}$. It is said to be unique provided that $\widehat{D}_{0}\in \mathcal{D}$, $D_0(t)$ is closed for any $t\in \mathbb{R}$, and $\mathcal{D}$ is inclusion closed.

Theorem 4.7. (^[21]) Assume that $\mathcal{\tilde{D}} = \{\widehat{\tilde{D}}(t)\}$ is a family of sets in $X$, $S(\cdot, \cdot)$ is continuous, and, for any $t\in \mathbb{R}$, there exists a constant $T(t, \mathcal{D}, \mathcal{\tilde{D}})$ such that

If $S(\cdot, \cdot)$ is $\mathcal{D}$-pullback absorbing and $\mathcal{\hat{D}}-\kappa$-pullback contracting, then the $\mathcal{D}$-pullback attractors $\mathcal{A_{D}} = \{A_{\mathcal{D}}(t):t\in\mathbb{R}\}$ exist for $S(\cdot, \cdot)$.

Lemma 4.8. (^[28]) Assume that $S(\cdot, \cdot) = S_1(\cdot, \cdot)+S_2(\cdot, \cdot)$, $\mathcal{\tilde{D}} = \{\widehat{\tilde{D}}(t)\}$ is a family of subsets in $X$, and for any $t\in \mathbb{R}$ and any $\tau\in \mathbb{R}^+$ there hold

(i) For any $u(t-\tau)\in \tilde{D}(t-\tau)$,

(ii) For any $T\geq \tau$, $\cup_{0\leq \tau\leq T} S_2(t, t-\tau)\tilde{D}(t-\tau)$ is bounded, and $S_2(t, t-\tau)\tilde{D}(t-\tau)$ is relatively compact in $X$.

Then $S(\cdot, \cdot)$ is $\mathcal{\hat{D}}-\kappa$-pullback contracting in $X$.

4.2.   Proof of Theorem 3.3

From Theorems 3.2, we know that the system (3.2) generates a continuous process $S^{\varepsilon}(t, \tau)$ in $N_V$. To obtain the $\mathcal{D}$-pullback attractors, we need to establish the existence of $\mathcal{D}$-pullback absorbing set and the $\mathcal{D}$-pullback asymptotic compactness of $S^{\varepsilon}(t, \tau)$.

● Existence of $\mathcal{D}$-pullback absorbing set in $N_V$

Let $\mathcal{D}$ denote a family of all $\{D(t)\}_{t\in \mathbb{R}}\subset \mathcal{P}(N_V)$ satisfying

where $\omega = \omega_{3/4, 1, 4, 3, f_\varepsilon} > 0$ and $J(\cdot) = J_{3/4, 1, 4, 3, f_\varepsilon}(\cdot)$. Next, we establish the existence of $\mathcal{D}$-pullback absorbing set.

Lemma 4.9. Let $(u(\tau), \eta^{\tau})\in N_V$, then the process $\{S^{\varepsilon}(t, \tau)\}$ to system (3.2) possesses a $\mathcal{D}$-pullback absorbing set $\widehat{D}_{0}^{\varepsilon}(t) = \{D_{0}^{\varepsilon}(t)\}_{t\in \mathbb{R}}$ in $N_{V}$, where

with radius

Proof. Multiplying (3.2) by $u$, we have

that is

Multiplying (3.2) by $u_t$, we have

Then, the interpolation inequality and Young inequality lead to

To estimate the term $\int_{0}^{\infty}\mu(s)\frac{d}{ds}\|\eta\|^2ds$ in (4.3) and avoid the possible singularity of $\mu$ at zero, we refer to ^[18] and construct the following new function

where $\tilde{s}$ is fixed such that $\int_{0}^{\tilde{s}}\mu(s)ds\leq \kappa/2$. Also, if we set

then differentiating in $t$ leads to

We use the technique in ^[18] and set

where

For sufficient small $\varepsilon$, it leads to

where we choose $\varepsilon_0$ satisfying

and $0\leq \varepsilon < \varepsilon_0$. Then, there holds

and

Then by Lemma 2.1, there exist

and an increasing function

such that

which implies the conclusion holds.

Remark 4.2. For the semigroup $S^{0}(t-\tau)$, it has the global absorbing set $D^0_{0}$ in $N_V$, where

and

● $\mathcal{D}-\kappa$-pullback contraction of $S^{\varepsilon}(t, \tau)$ in $N_V$

To verify the pullback contraction of $S^{\varepsilon}(t, \tau)$, we decompose $S^{\varepsilon}(t, \tau)$ as follows

which solve the following two problems respectively

and

Lemma 4.10. Let $U(\tau)\in D_{0}^{\varepsilon}(\tau)$, then the solution $S_1^{\varepsilon}(t-\tau)U(\tau)$ to the system (4.11) satisfies

Proof. Multiplying (4.11) by $u_1$ and $\frac{\partial}{\partial t}u_1$ respectively, and repeating the reasonings as shown as in Lemma 4.9, in which $\beta = 0$ and $f_\varepsilon = 0$, we can derive the conclusion finally. The parameter $\omega$ is dependent on $\varepsilon$ and the increasing function $J(\ast)$ is different from the one in Lemma 4.9. Despite all this, these parameters can be unified in same representation, and the concrete details are omitted here.

Lemma 4.11. Let $U(\tau)\in D_{0}^{\varepsilon}(\tau)$, then for any $t\in \mathbb{R}$, there exists $C^\varepsilon(t) > 0$ such that the solution $S_2^{\varepsilon}(t, \tau)U(\tau)$ to the system (4.12) satisfies

Proof. Multiplying (4.12) by $Au_2$, we have

from the existence of pullback absorbing set, Lemma 4.10, the interpolation inequality and Young inequality, we have

Multiplying (4.12) by $A\partial_t u_2$, we have

By the existence of pullback absorbing set, Lemma 4.10 and Young inequality, one has

To estimate the term $\int_{0}^{\infty}\mu(s)\frac{d}{ds}|A\eta_2|^2ds$ in (4.14), we set

and differentiating in $t$ leads to

where

and

and

Also, from the fact that $\mu'(s)+\delta \mu(s)\leq 0$, we have

Thus

We use the technique in ^[18] and set

where

For sufficient small enough $\varepsilon$, it leads to

and there holds

it follows from the Gronwall lemma that

which means the conclusion holds.

Above all, Lemmas 4.10, 4.11 and 4.8 lead to

Lemma 4.12. Let $U(\tau)\in N_V$, then the process $S^{\varepsilon}(t, \tau):\ N_V\rightarrow N_V$ generated by the system (3.2) is $\mathcal{D}-\kappa$-pullback contracting in $N_V$.

Consequently, from Theorem 4.7, we can finish the proof of Theorem 3.3.

5.   Robustness

5.1.   Theories on robustness

By the definition of upper semi-continuity, the following lemmas can be used to obtain the robustness of pullback attractors for evolutionary systems.

Lemma 5.1. (^[20]) Let $\varepsilon\in (0, \varepsilon_0]$, $\{S^{\varepsilon}(t, \tau)\}$ is the process of evolutionary system with non-autonomous term (depending on $\varepsilon$), which is obtained by perturbing the semigroup $S^0(\tau)$ of system without $\varepsilon$, and, for any $t\in \mathbb{R}$, there also hold that

(i) $S^{\varepsilon}(t, \tau)$ has the pullback attractors $\mathcal{A}^{\varepsilon}(t)$, and $\mathcal{A}^{0}$ is the global attractor for $S^0(\tau)$.

(ii) For any $\tau\in \mathbb{R}^+$ and any $u\in X$, there holds uniformly that

(iii) There exists a compact subset $G\subset X$ such that

Then, for any $t\in \mathbb{R}$, there holds

Lemma 5.2. (^[21]) For any $t\in \mathbb{R}$, $\tau\in \mathbb{R}^+$, and $\varepsilon\in (0, \varepsilon_0]$, $\widehat{D}^\varepsilon_{0}(t) = \{D^\varepsilon_{0}(t):t\in\mathbb{R}\}$ is the pullback absorbing set for $S^{\varepsilon}(t, \tau)$, and $\widehat{C}_{0}^\varepsilon(t) = \{C_0^\varepsilon(t):t\in\mathbb{R}\}$ is a family of compact subsets in $X$. Assume that $S^\varepsilon(\cdot, \cdot) = S^\varepsilon_1(\cdot, \cdot)+S^\varepsilon_2(\cdot, \cdot)$, and there hold

(i) For any $u_{t-\tau}\in D^\varepsilon_{0}(t-\tau)$,

(ii) For any $T\geq \tau$, $\cup_{0\leq \tau\leq T} S^\varepsilon_2(t, t-\tau)D^\varepsilon_{0}(t-\tau)$ is bounded, and there exists a constant $T_{\mathcal{D}^\varepsilon_{0}}(t)$, independent of $\varepsilon$, such that

(iii) There is a compact subset $G\subset X$ such that

Then, the process $S^{\varepsilon}(t, \tau)$ has the pullback attractors $\mathcal{A}^{\varepsilon}(t)$, and

5.2.   Proof of Theorem 3.4

We give the following procedure to verify Theorem 3.4.

Lemma 5.3. Let $(u^\varepsilon, \eta^\varepsilon) = S^{\varepsilon}(t, \tau)U(\tau)$ be the solution to system (3.2), and $(u, \eta) = S^{0}(t-\tau)U(\tau)$ is the solution to system (3.4), then, for any bounded subset $B\subset N_V$, there holds

Proof. We know

and

Let $w^\varepsilon = u^\varepsilon-u$ and $\xi^\varepsilon = \eta^\varepsilon-\eta$, we can derive

and multiplying it by $w^\varepsilon$ leads to

it follows that

Integrating (5.5) over $[\tau, t]$, from Lemma 2.2 we derive that

that is

and the Gronwall inequality leads to

which means that the conclusion is finished.

Proof of Theorem 3.4. From (4.24) and the fact that $W\hookrightarrow V$ is compact, we know that there exists a compact subset $G\subset N_V$ such that

Combining Lemma 4.10, Lemma 5.2 and (5.9), we have

In addition, the confirmation of condition $(ii)$ in Lemma 5.1 is finished from Lemma 5.3, and we have

Acknowledgments

Rong Yang was partially supported by the Science and Technology Project of Beijing Municipal Education Commission (No. KM202210005011).

Conflict of interest

The authors declare there is no conflict of interest.