Dynamics of a 3D Benjamin–Bona–Mahony equations with sublinear operator

Abstract

This paper is concerned with the tempered pullback dynamics for a three dimensional Benjamin–Bona–Mahony equations with sublinear operator on bounded domain, which describes the long time behavior for long waves model in shallow water with friction. By virtue of a new retarded Gronwall inequality, and using the energy equation method from J.M. Ball (Disc. Cont. Dyn. Syst. 10 (2004) 31–52) to achieve asymptotic compactness for solution process, the minimal family of pullback attractors has been obtained, which reduces a single trajectory under a sufficient condition.

Keywords

Benjamin–Bona–Mahony equation uniformly asymptotic stable pullback attractors

1. Introduction

The Benjamin–Bona–Mahony (BBM for short) equation is a well-known physical model discovered in 1972 (see Benjamin, Bona, and Mahony [4]), which incorporates dispersive effects for long waves in shallow water as $\begin{matrix} (1.1) & u_{t} + u_{x} + u u_{x} - u_{x x t} = 0 . \end{matrix}$ The BBM equation can be also seen as an improvement of the Korteweg–de Vries (KdV for short) equation $\begin{matrix} (1.2) & u_{t} + u_{x} + u u_{x} + u_{x x x} = 0 \end{matrix}$ for modeling long waves of small amplitude. By contrast with the KdV equation (1.2), the BBM model (1.1) is unstable in its high wave number components. For further exploration, while the KdV equation has an infinite number of integrals in motion, the BBM equation only possesses three. Both KdV and BBM systems cover the surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluid, hydromagnetic waves in cold plasma, acoustic waves in harmonic crystals. The research on well-posedness and dynamics of BBM equation endowed boundary condition and initial data, we can refer to [1,2,5–7,16–18,24,29,31–36,42,43] and references therein.

For the application of mathematical theory in engineer, time varying delay arises in fields of physical, chemical, biological, phase transition, thermal and economic phenomena, which means the motion is not only depend on the present state but also on some past occurrences, especially in control theory and its applications. The controller in engineer is always designed on boundary, which may generate memory or delay since the property of material, i.e., the control system is equivalent to a differential equation with boundary condition that contains memory or delay. By some transformations, the original control problem can be translated into differential equation with memory or delay subject to homogeneous boundary condition, which has been investigated by many mathematicians, see [8–15,20–23,37,38,41] and literatures therein. However, as far as we known, there are less results on the dynamics of control system for BBM equation, for instance the BBM model with memory (see [19]) or delay term. Our objective in this paper is to study the tempered pullback dynamics of the three dimensional BBM equation with sub-linear operator which is subjected to the periodic boundary condition and initial data $\begin{array}{rcl} (1.3) & \{\begin{matrix} u_{t} - △ u_{t} - ν △ u + \nabla \cdot \vec{F} (u) = g (t, u (t - ρ (t))), & (x, t) \in Ω_{τ}, \\ u (t, x_{i} + L e_{i}) = u (t, x_{i}), & (x, t) \in \partial Ω_{τ}, \\ u (τ, x) = u_{τ} (x), & x \in Ω, \\ u (t + τ, x) = ϕ (t, x), & (t, x) \in (- h, 0) \times Ω, \end{matrix} \end{array}$ where $Ω \subset R^{3}$ is a bounded domain with smooth boundary $\partial Ω$ , $Ω_{τ} = Ω \times (τ, + \infty)$ , $\partial Ω_{τ} = \partial Ω \times (τ, + \infty)$ , $Ω_{τ h} = Ω \times (τ - h, τ)$ , $τ \in R$ is the initial time, $u = u (t, x) = (u_{1} (t, x), u_{2} (t, x), u_{3} (t, x))$ is the unknown velocity vector field, ν is the kinematic viscosity of the fluid, $\vec{F}$ is a nonlinear vector function, $g (t, u (t - ρ (t)))$ is a sub-linear delay external force for the longtitude wave model which contains memory effects during a fixed interval of time of length $h > 0$ , $ρ (t)$ is an adequate delay function be given. $u_{0}$ is the initial data, ϕ denotes the state of delay in $[τ - h, τ]$ .

In system (1.3), if $ν = 0$ , then (1.3) reduces to a dispersive equation, which has the similar property with such as KdV model, wave equation and Schödinger system. If $F (u) = u + \frac{1}{2} u^{2}$ and $g (\cdot, \cdot) = g (x)$ for two dimension, then equation (1.3) reduces to $\begin{matrix} (1.4) & u_{t} + u_{x} + u u_{x} - ν u_{x x} - u_{x x t} = g (x) \end{matrix}$ which has been proposed as an essential law for propagation of long waves and incorporates nonlinear dispersive and dissipative effects (also seem as a generalized BBM equation). For the well-posedness of global solutions for this BBM equation and its related extension systems, we can refer to [1,2,5–7,17,24]. For the long-time behavior of global solution for the generalized BBM equation, such as the existence of finite dimensional global attractor and its structure, the stability and decay of solution, many mathematicians have obtained fruitful distinguished results, which can be found in [5,16,18,29,31–36,42,43] and the references therein. For the dynamical system of the high-dimensional BBM equation such as (1.3), we can refer to [26,27] and so on.

For the research on long-time asymptotic dynamics of BBM equation, the dissipation $- ν Δ u$ in (1.3) is necessary since dispersion is not enough to well-posedness and attracting. The control on BBM system also generates delay term, i.e., the sublinear operator, which possess less results on this interesting topic. The existence of global weak solution and minimal family of pullback attractors for the generalized BBM equation (1.3) will be proved at first, then we want to derived a sufficient condition when the pullback attractors reduces to a single trajectory, which coincides with the backward property of global solution.

The features of this present paper can be summarized as following.

(1) Let us recall the technique to deal with memory term, which originated from Dafermos. By a transformation, the PDEs with memory is equivalent to a new system, which can be solved by semigroup theory. One feature of this technique is to use a new produce space instead of original natural phase space.

For a PDE with delay, we also use the similar ides, i.e., the phase space should be extended to a new produce topology instead. The delay in (1.3) is sublinear operator which has finite domain, hence the original phase spaces $L^{p} (0, T; X)$ and X should be extended to $L^{p} ((- h, 0); X) \times L^{p} (0, T; X)$ and $C ([- h, 0]; X) \times X$ respectively, see Section 2.1.

The difference here is we only use the Galerkin method to deal with well-posedness since our sublinear operator is not the distributed delay, and the systems is not hyperbolic such as wave equation, which can be transformed to a coupled system and solved by semigroup technique.

(2) Since (1.3) contains the sublinear operator $\tilde{g} = g (t, u (t - ρ (t)))$ which is nonlinear, this is the main difficulty here, hence we should first estimate $\tilde{g}$ in the Fadeo–Galerkin approximated procedure to achieve the global existence of weak solution and continuous dependence on initial data, see Section 2.3 and Section 5.

(3) Based on the tempered universes which is no need to be compact or bounded, using the energy equation method (see [3,25,40]) to achieve $D$ -pullback asymptotic compactness for the processes $(U (t, τ); M_{V}^{2 q})$ and $(U (t, τ); C_{V})$ , we derive the minimal family of pullback attractors in $M_{V}^{2 q}$ and $C_{V}$ respectively. Here, we use a new definition of functional to overcome the compactness of nonlinear term $\tilde{g}$ to achieve the pullback asymptotically compact for the process.

(4) Comparing the 2D Navier–Stokes equation in [22,23], the phase space has more regularity. In addition, we need to balance the norm between H and $V$ in the energy equality, this is main difference from [22] and [23], which leads to some new estimate and definitions, such as $\int_{τ}^{t} ‖ u_{m} (s - ρ (s)) ‖_{V}^{2} d s$ . At last, we have derived a sufficient condition such that the pullback attractor reduces a single trajectory, which coincides to the backward property of global solution as in [30].

(5) For further research of our problem, since the memory $\int_{0}^{\infty} g (s) u_{x x} (t - s) d s$ of the BBM equation in [19] has some similar property with distributed delay $\int_{0}^{h} a (s) u (t - s) d s$ , using the technique in [19] and this presented paper, we can also derive the well-posedness of BBM equation with distributed delay and large time behavior as following remark, which is our objective for more detail in next paper.

Remark 1.1.
Considering the three dimensional BBM equation subjected to distributed delay $\begin{array}{rcl} (1.5) & \{\begin{matrix} u_{t} - △ u_{t} - ν △ u + \nabla \cdot \vec{F} (u) = \int_{0}^{h} a (s) g (u (t - s)) d s, & (x, t) \in Ω_{τ}, \\ u (t, x_{i} + L e_{i}) = u (t, x_{i}), & (x, t) \in \partial Ω_{τ}, \\ u (τ, x) = u_{τ} (x), & x \in Ω, \\ u (t + τ, x) = ϕ (t, x), & (t, x) \in (- h, 0) \times Ω, \end{matrix} \end{array}$ by constructing a new product space as $V = C_{η} ([- h, 0]; V) \times V$ , the global existence of mild solution in also can be derived in $V$ , here $C_{η} ([- h, 0]; V)$ is a weighted Banach space with delay which is different with $C ([- h, 0]; V)$ since the distributed delay.
Remark 1.2.
If the function $ρ (t)$ in the sublinear operator $g (t, u (t - ρ (t)))$ reduces to linear especially a constant, then the main theorems for BBM equation in this paper also hold, and the hypothesis on $g (\cdot, \cdot)$ can be weaken. Moreover, the finite fractal dimension of pullback attractor can also be proved for BBM equation with constant delay by using the technique in [39] except some extensions of basis in $L^{2} (Ω)$ .

This presented paper is organized as following. In Section 2, we shall state our main results. The proof will be presented in Sections 4 and 5. At last part, the further research will be shown.
2. Global well-posedness

2.1. Preliminaries

Throughout this paper, C denotes all generic positive constants, which is only dependent on Ω and some special parameters, but independent on the choice of time τ and t.

We set H is the closure of all functions in ${(L_{p}^{2} (Ω))}^{3}$ topology, V is the closure of all functions in $V = {(H_{p}^{1} (Ω))}^{3}$ topology with homogeneous case, denote the norm of Banach space $V$ as $‖ \cdot ‖$ , W is a homogeneous space with all functions in $W = {(H_{p}^{2} (Ω))}^{3}$ . The notations $| \cdot |_{2}$ and $(\cdot, \cdot)$ denote the norm and inner product of H, $‖ \cdot ‖_{V}$ and $((\cdot, \cdot))$ be the norm and inner product of V. It is easy to see the embeddings $V ↪ H \equiv H^{'} ↪ V^{'}$ and $W ↪ V ↪ H$ , here $H^{'}$ and $V^{'}$ are dual spaces of H and V respectively, where the injection is dense, continuous. The norm $‖ \cdot ‖_{*}$ denotes the norm in $V^{'}$ , $⟨ \cdot, \cdot ⟩$ be the dual product between V and $V^{'}$ (or H into itself).

The notation $A : = - Δ$ is a elliptic operator subject to the periodic boundary condition, and A is a self-adjoint positively defined operator on H. $A^{- 1}$ is a compact operator from H to H. Let the sequence ${ω_{j}}_{j = 1}^{\infty}$ be an orthonormal eigenfunctions of A and ${λ_{j}}_{j = 1}^{\infty}$ ( $0 < λ_{1} ⩽ λ_{2} ⩽ \dots$ ) be eigenvalues of the Stokes operator A for corresponding ${ω_{j}}_{j = 1}^{\infty}$ .

We denote $C_{H} = C ([- h, 0]; H)$ and $C_{V} = C ([- h, 0]; V)$ as two Banach spaces equipped with the norms $\begin{matrix} (2.1) & ‖ u ‖_{C_{H}} = sup_{θ \in [- h, 0]} {‖ u (t + θ) ‖}_{H} \end{matrix}$ and $\begin{matrix} (2.2) & ‖ u ‖_{C_{V}} = sup_{θ \in [- h, 0]} {‖ u (t + θ) ‖}_{V} \end{matrix}$ respectively, and also define space $C_{V} = C ([- h, 0]; V)$ similarly. Moreover, we introduce $\begin{array}{rcl} L_{H}^{p} = L^{p} (- h, 0; H), L_{V}^{p} = L^{p} (- h, 0; V), p \in [1, \infty] \end{array}$ as the time-delay Banach spaces with similar definition.

2.2. Hypotheses

For the global well-posedness of (1.3), we assume the continuous functions $ρ (\cdot)$ , $u_{τ}$ , $ϕ (\cdot)$ and $g (\cdot, t)$ satisfy

The delay function $ρ : [0, \infty) \to [0, h]$ satisfies $| \frac{d ρ}{d t} | ⩽ ρ^{*} < 1$ for some $ρ^{*}$ , i.e., $ρ \in C^{1} ([0, + \infty); [0, h])$ ;

The initial data $u_{τ} \in V \cap W$ and $ϕ : [- h, 0] \times V \to V$ satisfies $ϕ \in L^{2 q} (- h, 0; V)$ with $\frac{1}{p} + \frac{1}{q} = 1$ , $p > 1$ is a constant which is suitable for some functions in $L^{p} (\cdot)$ ;

The external force $g (\cdot, u) : [- h, \infty) \times H \to H$ is measurable with respect to $u \in H$ , $g (t, \cdot) : H \to H$ is continuous for all $t ⩾ τ$ , there exists functions $α, β : [τ, + \infty) \to [0, + \infty)$ such that $\begin{array}{rcl} (2.3) & {| g (t, u) |}^{2} ⩽ α (t) | u |^{2} + β (t) \end{array}$ for all $t ⩾ τ$ and $u \in H$ , where $α \in L^{p} (τ, T)$ and $β \in L^{1} (τ, T)$ , $1 ⩽ p ⩽ + \infty$ ;

The nonlinear vector function $\vec{F} (s) = (F_{1} (s), F_{2} (s), F_{3} (s))$ , $\forall s \in R$ has the following related notations as $\begin{matrix} (2.4) & f_{i} (s) = F_{i}^{'} (s), F_{i} (s) = \int_{0}^{s} F_{i} (r) d r, \end{matrix}$ where $\begin{matrix} (2.5) & \vec{f} (s) = (f_{1} (s), f_{2} (s), f_{3} (s)), \vec{F} (s) = (F_{1} (s), F_{2} (s), F_{3} (s)) . \end{matrix}$

Moreover, we assume that $F_{i}$ ( $i = 1, 2, 3$ ) are smooth functions satisfying $\begin{array}{l} (2.6) & F_{i} (0) = 0, | F_{i} (s) | ⩽ C_{1} | s | + C_{2} | s |^{2}, \\ (2.7) & | f_{i}^{'} (s) | ⩽ C_{1} + C_{2} | s |, \\ (2.8) & | F_{i} (s) | ⩽ C_{1} | s |^{2} + C_{2} | s |^{3}, \end{array}$ where $C_{1}$ , $C_{2}$ are positive constants.

2.3. Existence and uniqueness of global weak solution

In this subsection, for each $t \in (τ, T)$ ( $T > τ$ ), we denote $u^{t} = u (t + \cdot)$ . Based on the mathematical settings and assume $u_{τ} \in V \cap W$ , $ϕ \in L_{V}^{2 q}$ , then the problems (1.3) can be rewritten as the abstract equivalent form $\begin{array}{rcl} (2.9) & \{\begin{matrix} \frac{\partial u}{\partial t} + \frac{\partial}{\partial t} A u + ν A u + G (u) = P g (t, u (t - ρ (t))), \\ u (τ) = u_{τ}, u (t + τ) = ϕ (t, x), t \in (- h, 0), \end{matrix} \end{array}$ here $G (u) = P (\nabla \cdot \vec{F} (u))$ , $ϕ (0) = u_{τ}$ .

Definition 2.1 (The Hadamard weak solution).

The function $u (x, t) \in L^{\infty} ([τ, T]; V) \cap L^{2} (τ, T; V) \cap L^{p} (τ - h, T; V)$ is called the global weak Hadamard solution for system (2.9) if $\begin{array}{rcl} (2.10) & \frac{d}{d t} [(u, v) + ((u, v))] + ν ((u, v)) + (G (u), \nabla v) = (g (t, u (t - ρ (t))), v) \end{array}$ holds for $v \in V$ in the sense of $V^{'}$ which coincides with initial data $ϕ (t, x)$ and $ϕ (0, x) = u_{τ} (x)$ .

We shall give the global well-posedness of problem (1.3), which is consistent with Definition 2.1.

Theorem 2.1.
Suppose (1) $u_{τ} \in V \cap W$ , $ϕ \in L_{V}^{2 q}$ , (2) $g : R \times C_{H} \to H$ satisfy the hypotheses $(a) \sim (d)$ . Then there exist global weak solutions $u = u (\cdot, τ, u_{τ}, ϕ)$ of problem ( 1.3 ) that satisfies the Definition 2.1 and energy equality $\begin{array}{rcl} (2.11) & \frac{d}{d t} ({| u (t) |}^{2} + {‖ u (t) ‖}^{2}) + 2 ν ⟨ A u (t), u (t) ⟩ = 2 (g (t, u (t - ρ (t))), u (t)) . \end{array}$
Proof.
See Section 4. □
Remark 2.2.
In Theorem 2.1, the uniqueness of global weak solution is unknown, which is dependent on more assumption of nonlinear functions $F (\cdot)$ and $g (\cdot)$ .
Theorem 2.3.
If the hypotheses in Theorem 2.1 hold, and for all $r > 0$ , there exist constants $L (r) > 0$ and $\tilde{L} > 0$ , such that for $| u | ⩽ r$ , $| v | ⩽ r$ , the following inequalities $\begin{array}{rcl} (2.12) & | g (t, u) - g (t, v) | ⩽ L (r) γ^{1 / 2} (t) | u - v | \end{array}$ and $\begin{array}{rcl} (2.13) & | F_{i} (s_{1}) - F_{i} (s_{2}) | ⩽ \tilde{L} | s_{1} - s_{2} | \end{array}$ hold for all $t ⩾ τ$ , where $\begin{matrix} (2.14) & γ \in L^{\infty} (τ, T) : [τ, + \infty) \to [0, + \infty) \end{matrix}$ for all $τ < T$ . Then the global weak solution in Theorem 2.1 is unique.

Moreover, the weak solution is continuous dependent on initial data $u_{τ}$ and ϕ, which implies the weak solution generates a continuous process $U (t, τ)$ satisfies $\begin{array}{rcl} (2.15) & U (s, s) = Id, U (t, s) U (s, τ) = U (t, τ) \end{array}$ and hence $u \in C ([τ, T]; V)$ .
Proof.
The uniqueness follows from the estimate of resulting difference for arbitrary two global weak solutions, see Section 4. □
Remark 2.4.
In Theorem 2.3, if $γ \in L^{1} (τ, T)$ , the uniqueness also holds, which need some restriction on parameters.

3. Tempered pullback dynamics

3.1. Some extended assumptions for dynamics

For the global well-posedness of (1.3), we assume the continuous functions $ρ (\cdot)$ , $u_{τ}$ , $ϕ (\cdot)$ and $g (\cdot, t)$ satisfy

the delay function $ρ : R \to [0, h]$ , $| \frac{d ρ}{d t} | ⩽ ρ^{*} < 1$ for some $ρ^{*}$ ;

the function $g (\cdot, u) : [- h, + \infty) \times H \to H$ is measurable with respect to $u \in H$ . $g (t, \cdot) : H \to H$ is continuous for all $t ⩾ τ$ and there exists functions $α, β : R \to [0, + \infty)$ such that $\begin{array}{rcl} (3.1) & {| g (t, u) |}^{2} ⩽ α (t) | u |^{2} + β (t) \end{array}$ for all $t ⩾ τ$ and $u \in H$ with $α \in L_{loc}^{p} (R)$ and $β \in L_{loc}^{1} (R)$ , $1 ⩽ p ⩽ + \infty$ ;

$μ_{0} (t) = μ - \frac{\tilde{α} (t)}{ν λ_{1}^{2} (1 - ρ^{*})} > 0$ .

3.2. Phase spaces for pullback dynamics

Since the hypotheses has extended to $R$ in last subsection, then the global existence of unique weak solution can be extended also. We introduce some Banach spaces as phase space for pullback dynamics $\begin{array}{l} (3.2) & M_{V}^{2 q} = V \times L^{2 q} (- h, 0; V), \\ (3.3) & C_{V} = C ([- h, 0]; V) . \end{array}$

Considering the topology of $M_{V}^{2 q}$ , for $(u_{τ}, ϕ)$ in $M_{V}^{2 q}$ , the corresponding norm can be described as $\begin{array}{rcl} (3.4) & {‖ (u_{τ}, η) ‖}_{M_{V}^{2}} = ‖ u_{τ} ‖_{V} + {(\int_{- h}^{0} {‖ ϕ (s) ‖}_{V}^{2 q} d s)}^{1 / 2 q} . \end{array}$

3.3. Existence of pullback attractors

Based on the global well-posedness for problem (1.3), we denote $D (t, τ; u_{τ}, ϕ)$ as the set of all global solution in interval $[τ, T]$ for the initial datum $(u_{τ}, ϕ) \in M_{V}^{2 q}$ . Then our main result of pullback dynamics for this paper can be stated as the following theorems.

∙ Existence of pullback attractors in $M_{V}^{2 q}$ :

Theorem 3.1.
Assume that the hypotheses (a1), (b), (c1), (d) and (e) hold, then the global solution for problem ( 1.3 ) generates a process $(U (t, τ), M_{V}^{2 q})$ in $M_{V}^{2 q}$ $\begin{array}{rcl} (3.5) & U (t, τ) (u_{τ}, ϕ) = {u^{t} | u \in D (t, τ, u_{τ}, ϕ)} \end{array}$ for any $(u_{τ}, ϕ) \in M_{V}^{2 q}$ . Moreover, the process $(U (t, τ), M_{V}^{2 q})$ for problem ( 1.3 ) possesses a minimal family of pullback attractor $A_{M_{V}^{2 q}} (t)$ in the universes with fixed bounded sets, which is strictly invariant for the process.
Proof.
See Section 5 for more detail. □

∙ Existence of pullback attractors in $C_{V}$ : Theorem 3.2.
Assume that the hypotheses (a1), (b), (c1), (d) and (e) hold, then the global solution for problem ( 1.3 ) generates a process $(U (t, τ), C_{V})$ in $C_{V}$ $\begin{array}{rcl} (3.6) & U (t, τ) ϕ = {u^{t} | u \in D (t, τ, ϕ (0), ϕ)} \end{array}$ for arbitrary $ϕ \in C_{V}$ . Moreover, the process $(U (t, τ), C_{V})$ for problem ( 1.3 ) possesses a minimal family of pullback attractor $A_{C_{V}} (t)$ in the universes with fixed bounded sets, which is strictly invariant for the process.
Proof.
See Section 5 for more detail. □

3.4. When the pullback attractors reduce to a single trajectory

In this section, we shall present a sufficient condition that the pullback attractors above reduce to a single trajectory.

For our argument in next, we need more assumptions on F and g.

The nonlinear functions F, g satisfy that $\begin{array}{l} (3.7) & | F (s_{1}) - F (s_{2}) | ⩽ \tilde{L} | s_{1} - s_{2} |, \\ (3.8) & | g (t, u) - g (t, v) | ⩽ L (R) γ^{1 / 2} (t) | u - v |, \end{array}$ which guarantee that the uniqueness of global solution for problem (1.3).

Theorem 3.3.
Assume that $\begin{matrix} min {\frac{ν λ_{1}}{2} - \frac{\tilde{L}}{ν} - \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})}, \frac{ν}{2} - \frac{\tilde{L}}{ν λ_{1}} - \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1}^{2} (1 - ρ^{})}} > 0, \end{matrix}$ the initial data $(ϕ, φ) \in C_{V} \times V$ , if $g \in L_{loc}^{2} (R; H)$ , f satisfies hypotheses (a1), (b), (c1), (d), (e), and the assumption (H.6) holds, then the pullback attractors $A_{C_{V}} (t)$ reduces to a single trajectory which also implies the backward unique property of solution.
Proof.
See Section 5.5 for more detail. □
Remark 3.4.
For the 2D Navier–Stokes equation with Dirichlet or periodic boundary condition, we know that the global solution has the backward unique property, which means if the semigroup $S (t) : H \to H$ is injective for every $t > τ$ , then $S (t) u_{0} = S (t) v_{0}$ implies $u_{0} = v_{0}$ , see Section 2.5 in [28]. In the above theorem, we show that the pullback attractors reduce to a single trajectory for our problem (1.3), which coincides with backward property similar as in [28].

4. Proof of Theorem 2.1: Global well-posedness

In this section, we will use Faedo–Galerkin method to prove our Theorem 2.1.

Proof of Theorem 2.1.
Step 1: Existence of local solution for approximated equation

Denote ${w_{1}, w_{2}, \dots, w_{m}, \dots}$ as the orthogonal basis of Hilbert space V which is also formed by eigenfunctions for the Stokes operator A with the corresponding eigenvalues $\begin{matrix} (4.1) & 0 < λ_{1} ⩽ λ_{2} ⩽ \dots, lim_{j \to + \infty} λ_{j} = + \infty . \end{matrix}$

Let $V_{m} = span {w_{1}, w_{2}, \dots, w_{m}}$ , $P_{m} : V \to V_{m}$ be a projection, then we construct the approximated solution $u_{m} (t) = \sum_{j = 1}^{m} γ_{j m} (t) w_{j}$ ( $j = 1, 2, \dots, m$ ) of problem (2.9) in $V_{m}$ , where $γ_{j m} (t)$ is to be determined, then we derive that $u_{m} (t)$ satisfies the Cauchy problem $\begin{array}{l} (4.2) & \begin{matrix} \frac{d}{d t} [(u_{m} (t), w_{j}) + (A u_{m}, w_{j})] + ν (A u_{m} (t), w_{j}) + (P_{m} G (u_{m}), w_{j}) \\ = (P_{m} g (t, u_{m} (t - ρ (t))), w_{j}), \end{matrix} \\ (4.3) & u_{m} (τ) = P_{m} u_{τ}, \\ (4.4) & u_{m} (τ + s) = P_{m} ϕ (s), s \in [- h, 0] . \end{array}$

The problem (4.2)–(4.4) is a well-known ordinary functional differential equation with respect to the unknown variable $γ_{j m}$ . By the local existence of solution for ordinary differential equation, noting that $g (\cdot, \cdot)$ is Lipschitz continuous, we see that the problem (4.2)–(4.4) has a unique local solution.

Step 2: The Priori Estimates

The global solution will be derived by compact argument to the local approximated solution if it is uniformly bounded.

Multiplying (4.2) with $γ_{j m}$ of each term at both sides, summing from $j = 1$ to m, since $F (0) = 0$ and $\begin{array}{rcl} \int_{Ω} (\nabla \cdot \vec{F} (u_{m})) u_{m} d x & = & - \int_{Ω} \vec{F} (u_{m}) \cdot \nabla u_{m} d x \\ = & - \int_{\partial Ω} \vec{F} (u_{m}) \cdot \vec{n} d x \\ (4.5) & = & - \int_{\partial Ω} \vec{F} (0) \cdot \vec{n} d x = 0, \end{array}$ where $\vec{n}$ is the outer unit normal vector, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ u_{m} ‖_{H}^{2} + ‖ u_{m} ‖_{V}^{2}) + ν ‖ u_{m} ‖_{V}^{2} \\ = (g (t, u_{m} (t - ρ (t))), u_{m} (t)) \\ (4.6) & ⩽ \frac{ν}{2} {‖ u_{m} (t) ‖}_{V}^{2} + \frac{1}{2 ν λ_{1}} {‖ g (t, u_{m} (t - ρ (t))) ‖}_{H}^{2} . \end{array}$

Integrating (4.6) over $[τ, t]$ with the time variable, we deduce $\begin{array}{l} ‖ u_{m} ‖_{H}^{2} + ‖ u_{m} ‖_{V}^{2} + ν \int_{τ}^{t} {‖ u_{m} (s) ‖}_{V}^{2} d s \\ (4.7) & ⩽ | u_{τ} |_{H}^{2} + ‖ u_{τ} ‖_{V}^{2} + \frac{1}{ν λ_{1}} \int_{τ}^{t} {‖ g (u_{m} (s - ρ (s))) ‖}_{H}^{2} d s . \end{array}$

Using the Lemma 4.1 which will be proved in sequel, we have $\begin{array}{l} ‖ u_{m} ‖_{H}^{2} + ‖ u_{m} ‖_{V}^{2} + ν \int_{τ}^{t} {‖ u_{m} (s) ‖}_{V}^{2} d s \\ ⩽ ‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2} \\ + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} [\int_{- ρ (τ)}^{0} \tilde{α} (t + τ) {‖ ϕ (t) ‖}_{V}^{2} d t + \int_{τ}^{T} \tilde{α} (t) {‖ u (t) ‖}_{V}^{2} d t] + \int_{t}^{T} β (t) d t \\ ⩽ ‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2} \\ (4.8) & + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} [{‖ α (\cdot) ‖}_{L^{p} (τ, T)} {‖ ϕ (t) ‖}_{L^{2 q} (τ, T; V)} + \int_{τ}^{T} \tilde{α} (t) {‖ u (t) ‖}_{V}^{2} d t] + \int_{t}^{T} β (t) d t, \end{array}$ by the Gronwall Lemma, we conclude the uniform estimate of $u_{m}$ as $\begin{array}{rcl} (4.9) & {u_{m} (t)} is bounded in L^{\infty} (τ, T; V) \cap L^{2} (τ, T; V) . \end{array}$

Multiplying (4.2) with $A γ_{j m}$ of each term at both sides, summing from $j = 1$ to m, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ u_{m} ‖^{2} + | A u_{m} |_{2}^{2}) + ν | A u_{m} |_{2}^{2} \\ ⩽ | (g (t, u_{m} (t - ρ (t))), u_{m} (t)) | + | (\nabla \cdot \vec{F} (u_{m}), A u_{m} (t)) | \\ (4.10) & ⩽ \frac{ν}{4} {| A u_{m} (t) |}_{2}^{2} + \frac{1}{ν} {| g (t, u_{m} (t - ρ (t))) |}_{2}^{2} + {| (\nabla \cdot \vec{F} (u_{m}), A u_{m} (t)) |}_{2}^{2} . \end{array}$

By using the assumptions of nonlinear term $\vec{F}$ and $g (\cdot, \cdot)$ , the Agmon–Ladyzhenskaya inequality $‖ u_{m} ‖_{L^{4}} ⩽ C | \nabla u_{m} |_{2}^{3 / 4} | u_{m} |_{2}^{1 / 4}$ and $‖ \nabla u_{m} ‖_{L^{4}} ⩽ C | A u_{m} |_{2}^{3 / 4} | \nabla u_{m} |_{2}^{1 / 4}$ , we have $\begin{array}{rcl} | (\nabla \cdot \vec{F} (u_{m}), A u_{m} (t)) | & ⩽ & \int | F^{'} (u_{m}) | | \nabla u_{m} | | A u_{m} | d x \\ ⩽ & C \int_{Ω} (1 + | u_{m} |) | \nabla u_{m} | | A u_{m} | d x \\ ⩽ & C \int | \nabla u_{m} | | A u_{m} | d x + C \int_{Ω} | u_{m} | | \nabla u_{m} | | A u_{m} | d x \\ (4.11) & = & I_{1} + I_{2}, \end{array}$ where $\begin{array}{rcl} (4.12) & I_{1} ⩽ \frac{C}{ν} | \nabla u_{m} |_{2}^{2} + \frac{ν}{4} | A u_{m} |_{2}^{2} \end{array}$ and $\begin{array}{rcl} I_{2} & ⩽ & ‖ u_{m} ‖_{L^{4}} | A u_{m} |_{2} ‖ \nabla u_{m} ‖_{L^{4}} \\ ⩽ & C | u_{m} |^{1 / 4} | \nabla u_{m} |_{2}^{3 / 4} | A u_{m} |_{2} | A u_{m} |_{2}^{3 / 4} | \nabla u_{m} |_{2}^{1 / 4} \\ (4.13) & ⩽ & C | u_{m} |_{2}^{2} | \nabla u_{m} |_{2}^{8} + \frac{ν}{4} | A u_{m} |_{2}^{2} \end{array}$

Integrating (4.10) over $[τ, t]$ with the time variable, we deduce $\begin{array}{l} {| \nabla u_{m} (t) |}_{2}^{2} + {| A u_{m} (t) |}_{2}^{2} + \frac{ν}{2} \int_{τ}^{t} {| A u_{m} (r) |}_{2}^{2} d r \\ ⩽ | \nabla u_{τ} |_{2}^{2} + | A u_{τ} |_{2}^{2} + \frac{1}{ν} \int_{τ}^{t} {| g (u_{m} (r - ρ (r))) |}_{2}^{2} d r \\ (4.14) & + C \int_{τ}^{t} {| u_{m} (s) |}_{2}^{2} {| \nabla u_{m} (s) |}_{2}^{8} d s + \frac{C}{ν} \int_{τ}^{t} {| \nabla u_{m} (s) |}_{2}^{2} d s . \end{array}$

Using the Lemma 4.1 which will be proved in sequel, we have $\begin{array}{l} | \nabla u_{m} |_{2}^{2} + | A u_{m} |_{2}^{2} + \frac{ν}{2} \int_{τ}^{t} {| A u_{m} (s) |}_{2}^{2} d s \\ ⩽ ‖ u_{τ} ‖^{2} + | A u_{τ} |_{2}^{2} \\ + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} [\int_{- ρ (τ)}^{0} \tilde{α} (t + τ) {‖ ϕ (t) ‖}_{V}^{2} d t + \int_{τ}^{T} \tilde{α} (t) {‖ u (t) ‖}_{V}^{2} d t] + \int_{t}^{T} β (t) d t \\ + C \int_{τ}^{t} {| u_{m} (s) |}_{2}^{2} {| \nabla u_{m} (s) |}_{2}^{8} d s + \frac{C}{ν} \int_{τ}^{t} {| \nabla u_{m} (s) |}_{2}^{2} d s \\ ⩽ ‖ u_{τ} ‖^{2} + | A u_{τ} |_{2}^{2} \\ + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} [{‖ α (\cdot) ‖}_{L^{p} (τ, T)} {‖ ϕ (t) ‖}_{L^{2 q} (τ, T; V)} + \int_{τ}^{T} \tilde{α} (t) {‖ u (t) ‖}_{V}^{2} d t] + \int_{t}^{T} β (t) d t \\ (4.15) & + C \int_{τ}^{t} {| u_{m} (s) |}_{2}^{2} {| \nabla u_{m} (s) |}_{2}^{8} d s + \frac{C}{ν} \int_{τ}^{t} {| \nabla u_{m} (s) |}_{2}^{2} d s, \end{array}$ by the Gronwall Lemma, noting that the sequence $u_{m}$ is bounded in $L^{\infty} (τ, T; V) \cap L^{2} (τ, T; V)$ , we conclude $\begin{array}{rcl} (4.16) & {u_{m} (t)} is bounded in L^{\infty} (τ, T; W) \cap L^{2} (τ, T; W) . \end{array}$

Step 3: Compact argument and passing to limit for approximated equation

From (2.9), we see that $\begin{array}{rcl} (4.17) & \frac{d}{d t} [u_{m} + A u_{m}] = - ν A u_{m} - P_{m} G (u_{m}) + P_{m} g (t - ρ (t), u_{m} (t - ρ (t))) . \end{array}$

From (4.20), we obtain that $- ν A u_{m} \in L^{2} (τ, T; H)$ . By virtue of Lemma 4.1, it yields $P_{m} g (t - ρ (t), u_{m} (t - ρ (t))) \in L^{2} (τ, T; H)$ .

Moreover, from hypothesis (d), we derive $\begin{array}{rcl} {‖ P_{m} G (u_{m}) ‖}_{L^{2} (τ, T; H)}^{2} & ⩽ & \int_{τ}^{T} {| G (u_{m}) |}_{2}^{2} d s = \int_{τ}^{T} {| \nabla \cdot \vec{F} (u_{m}) |}_{2}^{2} d s \\ (4.18) & ⩽ & C [‖ u_{m} ‖_{L^{\infty} (τ, T; V)}^{2} + ‖ u_{m} ‖_{L^{2} (τ, T; V)}^{2}] . \end{array}$ i.e., $P_{m} G (u_{m})$ is uniformly bounded in $L^{2} (τ, T; H)$ .

Combining (4.17) and above assertions,from the hypothesis (d) and Lemma 4.1 which will be proved in sequel, we see that $\frac{d u_{m}}{d t} \in L^{2} (τ, T; H)$ . Using (4.8) and (4.9), we also derive that $u_{m} (\cdot + τ, x) \in L^{2 q} (- h, 0; V)$ . By Aubin–Lions compact argument and embeddings $W ↪ V ↪ H$ , we see that ${u_{m}}$ has a strong convergence subsequence as m tends to infinity (also denote ${u_{m} (x, t)}$ , the limit denote as $u (x, t)$ ) which satisfies that $\begin{array}{l} (4.19) & u_{m} ⇀ u weakly star in L^{\infty} (τ, T; V), \\ (4.20) & u_{m} ⇀ u weakly in L^{2} (τ, T; V), \\ (4.21) & u_{m} \to u in L^{2 q} (- h, 0; V) and L^{2} (τ, T; V) . \end{array}$ Then passing to limit in (4.2), we can conclude the existence of global weak solutions for (2.9) in the sense of Definition 2.1.

Step 4: The uniqueness and continuous dependence on initial data

Assume that $u (t; τ, ϕ)$ , $v (t; τ, ϕ)$ be two global weak solutions of (1.3) with corresponding initial data $(u_{τ}^{0}, ϕ_{0})$ and $(u_{τ}^{1}, ϕ_{1})$ , setting $w (t) = u (t) - v (t)$ , then $w (t)$ satisfies the Cauchy problem for equation $\begin{array}{rcl} (4.22) & w_{t} + A w_{t} + ν A w + G (u) - G (v) = g (t, u (t - ρ (t))) - g (t, v (t - ρ (t))) \end{array}$ with $w (τ) = w_{τ} = u_{τ}^{0} - u_{τ}^{1}$ and Taking inner product of (4.22) with w at both sides, using Young’s inequality, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ w ‖_{H}^{2} + ‖ w ‖_{V}^{2}) + ν ‖ w ‖_{V}^{2} \\ ⩽ | (G (u) - G (v), w) | + L (r) γ^{1 / 2} ‖ w ‖_{H} {‖ w (t - ρ (t)) ‖}_{H} \\ ⩽ | (G (u) - G (v), w) | + L (r) γ^{1 / 2} ‖ w ‖_{H} {‖ w (t - ρ (t)) ‖}_{H} \\ (4.23) & ⩽ | (G (u) - G (v), w) | + \frac{ν}{2} ‖ w ‖_{V}^{2} + \frac{L^{2} (r) γ (t)}{2 ν λ_{1}} {‖ w (t - ρ (t)) ‖}_{H}^{2} . \end{array}$

Moreover, by the hypothesis of $G (\cdot)$ in Theorem 2.3, we have $\begin{array}{rcl} | (G (u) - G (v), w) | & ⩽ & \int_{Ω} | \vec{F (u)} - \vec{F (v)} | | \nabla w | d x \\ ⩽ & C \int_{Ω} | w | | \nabla w | d x \\ (4.24) & ⩽ & \frac{ν}{2} ‖ w ‖_{V}^{2} + C ‖ w ‖_{H}^{2} . \end{array}$

Integrating (4.23) from τ to t, and noting that $\begin{array}{l} \int_{τ}^{t} {‖ w (s - ρ (s)) ‖}_{H}^{2} d s \\ (4.25) & ⩽ \frac{1 + \frac{1}{λ_{1}}}{2 (1 - ρ^{})} [\int_{τ - h}^{τ} {‖ ϕ (s) ‖}_{V}^{2} d s + \int_{τ}^{t} ({‖ w (s) ‖}_{H}^{2} + {‖ w (s) ‖}_{V}^{2}) d s], \end{array}$ from Lemma 4.1, then it yields $\begin{array}{l} {‖ w (t) ‖}_{H}^{2} + ‖ w ‖_{V}^{2} \\ ⩽ ‖ w_{τ} ‖_{H}^{2} + ‖ w_{τ} ‖_{V}^{2} + C \int_{τ}^{t} {‖ w (s) ‖}_{H}^{2} d s + \frac{L^{2} (r) (1 + λ_{1}) ‖ γ (t) ‖_{L^{\infty} (τ, T)}}{2 ν λ_{1}^{2} (1 - ρ^{})} \int_{τ - h}^{τ} {‖ ϕ (s) ‖}_{V}^{2} d s \\ + \frac{L^{2} (r) (1 + λ_{1}) ‖ γ (t) ‖_{L^{\infty} (τ, T)}}{2 ν λ_{1}^{2} (1 - ρ^{})} \int_{τ}^{t} ({‖ w (s) ‖}_{H}^{2} + {‖ w (s) ‖}_{V}^{2}) d s \\ ⩽ ‖ w_{τ} ‖_{H}^{2} + ‖ w_{τ} ‖_{V}^{2} + \frac{L^{2} (r) (1 + λ_{1}) ‖ γ (t) ‖_{L^{\infty} (τ, T)}}{2 ν λ_{1}^{2} (1 - ρ^{})} \int_{τ - h}^{τ} {‖ ϕ (s) ‖}_{V}^{2} d s \\ (4.26) & + C^{} \int_{τ}^{t} ({‖ w (s) ‖}_{H}^{2} + {‖ w (s) ‖}_{V}^{2}) d s, \end{array}$ by the Gronwall inequality, we conclude $\begin{array}{l} {‖ w (t) ‖}_{H}^{2} + {‖ w (t) ‖}_{V}^{2} \\ (4.27) & ⩽ (‖ w_{τ} ‖_{H}^{2} + ‖ w_{τ} ‖_{V}^{2} + \frac{L^{2} (r) (1 + λ_{1}) ‖ γ (t) ‖_{L^{\infty} (τ, T)}}{2 ν λ_{1}^{2} (1 - ρ^{})} \int_{τ - h}^{τ} {‖ ϕ (s) ‖}_{V}^{2} d s) e^{C^{} (T - τ)}, \end{array}$ which implies the uniqueness and continuous dependence on initial data, i.e., the proof has been achieved. □
Lemma 4.1.
The following estimate holds* $\begin{array}{l} \int_{τ}^{T} {‖ g (t, u (t - ρ (t))) ‖}_{H}^{2} d t \\ (4.28) & ⩽ \frac{1}{λ_{1} (1 - ρ^{})} [\int_{- ρ (τ)}^{0} \tilde{α} (t + τ) {‖ ϕ (t) ‖}_{V}^{2} d t + \int_{τ}^{T} \tilde{α} (t) {‖ u (t) ‖}_{V}^{2} d t] + \int_{τ}^{T} β (t) d t . \end{array}$
Proof.
Denoting $θ = s - ρ (s)$ , noting $\begin{matrix} (4.29) & d θ = (1 - ρ^{'} (s)) d s, α (t) \to \tilde{α} (\bar{t}) \in L^{p} (τ, T), \end{matrix}$ which means $\tilde{α} (t) \in L^{q} (τ - h, T)$ . By the hypotheses (a)–(c), we obtain that $\begin{array}{l} (g (t, u_{m} (t - ρ (t))), u_{m} (t)) \\ ⩽ | g (t, u_{m} (t - ρ (t))) | | u_{m} (t) | \\ ⩽ α^{\frac{1}{2}} (t) | u_{m} (t - ρ (t)) | | u_{m} (t) | + β^{\frac{1}{2}} (t) | u_{m} (t) | \\ (4.30) & ⩽ ν (1 - ρ^{}) {‖ u_{m} (t - ρ (t)) ‖}_{V}^{2} + (\frac{α (t)}{ν λ_{1}^{2} (1 - ρ^{})} + \frac{1}{λ_{1}}) {‖ u_{m} (t) ‖}_{V}^{2} + β (t) . \end{array}$ and $\begin{array}{rcl} \int_{τ}^{t} {‖ u_{m} (s - ρ (s)) ‖}_{V}^{2} d s & ⩽ & \frac{1}{1 - ρ^{}} \int_{- ρ (0)}^{t - ρ (t)} {‖ u_{m} (s) ‖}_{V}^{2} d s \\ (4.31) & ⩽ & \frac{1}{1 - ρ^{}} [\int_{τ - h}^{τ} {‖ ϕ (s) ‖}_{V}^{2} d s + \int_{τ}^{t} {‖ u_{m} (s) ‖}_{V}^{2} d s] . \end{array}$

Similarly, we have $\begin{array}{rcl} (4.32) & \int_{τ}^{t} {‖ u_{m} (s - ρ (s)) ‖}_{H}^{2} d s & ⩽ & \frac{1}{1 - ρ^{}} [\frac{1}{λ_{1}} \int_{τ - h}^{τ} {‖ ϕ (s) ‖}_{V}^{2} d s + \int_{τ}^{t} {‖ u_{m} (s) ‖}_{H}^{2} d s], \end{array}$ hence, $\begin{array}{l} \int_{τ}^{T} {‖ g (t, u (t - ρ (t))) ‖}_{H}^{2} d t \\ ⩽ \int_{τ}^{T} α (t) {‖ u (t - ρ (t)) ‖}_{H}^{2} d t + \int_{τ}^{T} β (t) d t \\ (4.33) & ⩽ \frac{1}{λ_{1} (1 - ρ^{*})} [\int_{- ρ (τ)}^{τ} \tilde{α} (t + τ) {‖ ϕ (t) ‖}_{V}^{2} d t + \int_{τ}^{T} \tilde{α} (t) {‖ u (t) ‖}_{V}^{2} d t] + \int_{τ}^{T} β (t) d t, \end{array}$ here $\tilde{α} (\cdot)$ is defined by $\tilde{α} (t) = h \circ θ^{- 1} (t)$ , $θ : [τ, + \infty) \to [- ρ (τ), + \infty)$ is the differentiable and nonnegative strictly increasing function given by $θ (s) = s - ρ (s)$ , $\tilde{α} \in L^{p} (- ρ (τ), T)$ for all $τ < T$ since $α \in L^{p} (τ, T)$ . Therefore, (4.33) implies that $g (t, u (t - ρ (t))) \in L^{2} (τ, T; H)$ . □
Remark 4.2.
The estimates in Lemma 4.1 are also true under hypotheses (a1), (b), (c1), (d) and (e).

5. Proof of Theorems 3.1 and 3.2: Pullback dynamics

Since the continuity of process, dissipation by absorbing set and asymptotic compactness are sufficient to existence of pullback attractor, The continuous dependence on initial data of global weak solution in Theorem 2.1 implies the process $U (t, τ)$ is continuous. Next, we only need to obtain the absorbing set based on tempered universes and verify asymptotic compactness.

5.1. Abstract theory on pullback attractors

Denoting $P (X)$ as the family of all nonempty subsets of X, and considering a family of nonempty sets ${\hat{D}}_{0} = D_{0} (t) : t \in R \subset P (X)$ , we have some preliminary definitions and theorem about tempered pullback dynamic theory in following, which can be founded in [15,23].

∙ Dissipation defined via absorbing set and universe

Definition 5.1.
The class $D$ will be called a universe in $P (X)$ if $D$ is a nonempty class of families parameterized in time $\hat{D} = {D (t) : t \in R} \subset P (X)$ .
Definition 5.2.
It is said that ${\hat{D}}_{0} = {D_{0} (t) : t \in R} \subset P (X)$ is pullback $D$ -absorbing for the process U on X if for any $t \in R$ and any $\hat{D} \in D$ , there exists a $τ_{0} (t, \hat{D}) ⩽ t$ such that $\begin{matrix} U (t, τ) D (τ) \subset D_{0} (t), for all τ ⩽ τ_{0} (t, \hat{D}) . \end{matrix}$

∙ Definition of pullback ${\hat{D}}_{0}$ -asymptotic compactness
Definition 5.3.
We call a process U on X is pullback ${\hat{D}}_{0}$ -asymptotically compact if for any $t \in R$ and any sequences $τ_{n} \subset (- \infty, t]$ and $x_{n} \subset X$ satisfying $τ_{n} \to - \infty$ and $x_{n} \in D_{0} (τ_{n})$ for all n, the sequence $U (t, τ_{n}) x_{n}$ is relatively compact in X.

∙ Definition of pullback $D$ -asymptotic compactness:
Definition 5.4.
A process U on X is said to be pullback $D$ -asymptotically compact if it is $\hat{D}$ -asymptotically compact for any $\hat{D} \in D$ , i.e. if for any $t \in R$ , any $\hat{D} \in D$ , and any sequences $τ_{n} \subset (- \infty, t]$ and $x_{n} \subset X$ satisfying $τ_{n} \to - \infty$ and $x_{n} \in D (τ_{n})$ for all n, the sequence ${U (t, τ_{n}) x_{n}}$ is relatively compact in X.
Remark 5.5.
Assume that the process $U (\cdot, \cdot)$ is closed (strong or norm-to-weak continuous) and pullback $D$ -asymptotically compact. Then, for each $\hat{D} \in D$ and any $t \in R$ , the set $Λ (\hat{D}, t)$ is a nonempty compact subset of X, invariant for U, that attracts $\hat{D}$ in the pullback sense, i.e. $\begin{array}{rcl} (5.1) & lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), Λ (\hat{D}, t)) = 0, \end{array}$ Moreover, it is the minimal family of closed sets satisfying (5.1).

∙ The definition of $D$ -pullback attractors:
Definition 5.6.
A compact subset $A = {A (t)}_{t \in R}$ in X is said to be $D$ -pullback attractor for the continuous process ${U (t, τ)}$ if
The set $A$ is pullback invariant;

The set $A$ is $D$ -pullback attracting in X, i.e., for any subset B belongs to the universe $D$ , we have $\begin{matrix} lim_{τ \to - \infty} {dist}_{X} (U (t, τ) B, A) = 0 . \end{matrix}$

∙ Existence theorem of minimal and unique families of $D$ -pullback attractors: Theorem 5.7 (See Carvalho, Langa and Robinson [15], Marín-Rubio and Real [23]).

Consider a closed (strong or norm-to-weak continuous) process $U (\cdot, \cdot) : R_{d}^{2} \times X \to X$ , a universe $D$ in $P (X)$ , and a family ${\hat{D}}_{0} = {D_{0} (t) : t \in R} \subset P (X)$ which is pullback $D$ -absorbing for $U (\cdot, \cdot)$ , and assume also that $U (\cdot, \cdot)$ is pullback ${\hat{D}}_{0}$ -asymptotically compact.

Then, the family $A_{D} = {A_{D} (t) : t \in R}$ is the family of $D$ -pullback attractors which is defined by $\begin{matrix} A_{D} (t) = \overline{⋃_{\hat{D} \in D} Λ {(\hat{D}, t)}^{X}}, t \in R, \end{matrix}$ and has the following properties:

for any $t \in R$ , the set $A_{D} (t)$ is a nonempty compact subset of X, and $A_{D} (t) \subset Λ ({\hat{D}}_{0}, t)$ ,

$A_{D}$ is pullback $D$ -attracting, i.e. $\begin{matrix} lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), A_{D} (t)) = 0, for all \hat{D} \in D, t \in R, \end{matrix}$

$A_{D}$ is invariant, i.e. $U (t, τ) A_{D} (τ) = A_{D} (t)$ for all $τ ⩽ t$ ,

if ${\hat{D}}_{0} \in D$ , then $A_{D} (t) = Λ ({\hat{D}}_{0}, t) \subset {\overline{D_{0} (t)}}^{X}$ , for all $t \in R$ .

Definition 5.8.
The family $A_{D}$ is minimal in the sense that if $\hat{C} = {C (t) : t \in R} \subset P (X)$ is a family of closed sets such that for any $\hat{D} = {D (t) : t \in R} \in D$ , $\begin{matrix} lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), C (t)) = 0, \end{matrix}$ then $A_{D} (t) \subset C (t)$ .
Remark 5.9.
Under the assumptions of Theorem 5.7, the family $D$ is called the minimal pullback $D$ -attractor for the process U.

5.2. Existence of pullback absorbing sets in $M_{V}^{2 q}$ and $C_{V}$

In the space $M_{V}^{2 q}$ , the bi-parameters operator as process is defined by $(U (t, τ), M_{V}^{2 q})$ in (3.5), and the process $(U (t, τ), C_{V})$ in (3.6) as following $\begin{array}{l} (5.2) & U (t, τ) (u_{τ}, ϕ) = u_{t} (\cdot; τ, (u_{τ}, ϕ)), \forall (u_{τ}, ϕ) \in M_{V}^{2 q}, t ⩾ τ, \\ (5.3) & U (t, τ) ϕ = U (t, τ) (ϕ (0), ϕ), t ⩾ τ, ϕ \in C_{V}, \end{array}$ then we will present the pullback absorbing set based on tempered universe.

Lemma 5.10.
Assume that the hypotheses (a1), (b), (c1), (d) and (e) hold, then for any $(u_{τ}, ϕ) \in M_{V}^{2 q}$ and any $u \in D (t, τ, u_{τ}, ϕ)$ , the global solution for problem ( 1.3 ) has the following estimate $\begin{array}{l} {‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2} \\ ⩽ [(‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} ‖ α ‖_{L^{q} (τ - h, τ)} ‖ ϕ ‖_{L^{2 q} (- h, 0; V)}] e^{- \int_{τ}^{t} μ_{0} (t) d t} \\ (5.4) & + \frac{1}{ν λ_{1}} \int_{τ}^{t} e^{- \int_{s}^{t} μ_{0} (r) d r} β (s) d s . \end{array}$
Proof.
Considering $u \in D (t, τ; u_{τ}, ϕ)$ , then multiplying (1.3) by u, from the energy equality (2.11), we obtain $\begin{array}{l} \frac{d}{d t} ({‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2}) + 2 ν ‖ u ‖_{V}^{2} \\ ⩽ 2 (g (t, u (t - ρ (t))), u (t)) \\ ⩽ 2 {‖ g (t, u (t - ρ (t))) ‖}_{H}^{2} ‖ u ‖_{H}^{2} \\ (5.5) & ⩽ ν ‖ u ‖_{V}^{2} + \frac{1}{ν λ_{1}} {‖ g (t, u (t - ρ (t))) ‖}_{H}^{2} . \end{array}$

Denoting $μ = min {ν / 2, ν λ_{1} / 2}$ , we have $\begin{array}{rcl} (5.6) & \frac{d}{d t} ({‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2}) + μ (‖ u ‖_{H}^{2} + ‖ u ‖_{V}^{2}) ⩽ \frac{1}{ν λ_{1}} {‖ g (t, u (t - ρ (t))) ‖}_{H}^{2}, \end{array}$ integrating (5.6) from τ to t, it follows $\begin{array}{l} ({‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2}) + μ \int_{τ}^{t} ({‖ u (s) ‖}_{H}^{2} + {‖ u (s) ‖}_{V}^{2}) d s \\ ⩽ (‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) \\ + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} [\int_{- ρ (τ)}^{τ} \tilde{α} (t + τ) {‖ ϕ (t) ‖}_{V}^{2} d t + \int_{τ}^{t} \tilde{α} (s) {‖ u (s) ‖}_{V}^{2} d s] + \frac{1}{ν λ_{1}} \int_{τ}^{t} β (s) d s \\ ⩽ (‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} ‖ α ‖_{L^{q} (τ - h, τ)} ‖ ϕ ‖_{L^{2 q} (- h, 0; V)} \\ (5.7) & + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} \int_{τ}^{t} \tilde{α} (s) {‖ u (s) ‖}_{V}^{2} d s + \frac{1}{ν λ_{1}} \int_{τ}^{t} β (s) d s . \end{array}$

From hypotheses (e), we deduce that $\begin{array}{l} ({‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2}) + μ_{0} \int_{τ}^{t} ({‖ u (s) ‖}_{H}^{2} + {‖ u (s) ‖}_{V}^{2}) d s \\ (5.8) & ⩽ (‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} ‖ \tilde{α} ‖_{L^{q} (τ - h, τ)} ‖ ϕ ‖_{L^{2 q} (- h, 0; V)} + \frac{1}{ν λ_{1}} \int_{τ}^{t} β (s) d s . \end{array}$

Using the Gronwall lemma, we conclude that $\begin{array}{l} {‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2} \\ ⩽ [(‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} ‖ \tilde{α} ‖_{L^{q} (τ - h, τ)} ‖ ϕ ‖_{L^{2 q} (- h, 0; V)}] e^{- \int_{τ}^{t} μ_{0} (t) d t} \\ (5.9) & + \frac{1}{ν λ_{1}} \int_{τ}^{t} e^{- \int_{s}^{t} μ_{0} (r) d r} β (s) d s \end{array}$ for all $τ ⩽ t$ . □
Remark 5.11.
In Lemma 5.10, the estimate not means the dissipation, we also need to assume that $\begin{array}{rcl} (5.10) & \int_{- \infty}^{t} e^{- \int_{s}^{t} μ_{0} (r) d r} β (s) d s < + \infty \end{array}$ or $\begin{array}{rcl} (5.11) & \int_{- \infty}^{τ} e^{- \int_{s}^{t} μ_{0} (r) d r} β (s) d s < + \infty, \end{array}$ which is preparation for dissipation based on tempered universe, $μ_{0} (t)$ is defined by hypothesis (e).
Remark 5.12.
Using the technique of Lemma 1 in [23], we can obtain a different parameter with $μ_{0} (t)$ which contains time variable, however, the restriction on this parameter such as hypothesis (e) is also necessary.

In addition, if the function $α (t)$ belongs to different space, then $μ_{0} (t)$ will be changed.

∙ Pullback absorbing set in $M_{V}^{2 q}$ :
Definition 5.13.
Let $P (M_{V}^{2 q})$ denotes all subsets in $M_{V}^{2 q}$ , which not need to be bounded. Let $D_{M_{V}^{2 q}} = {D}$ be the class of families $D = {D (t) | t \in R} \subset P (M_{V}^{2 q})$ , where $D (t) \subset {\bar{B}}_{M_{V}^{2 q}} (0, r_{D} (t))$ for radius $r_{D} (t)$ belongs to sets $R_{r_{D} (t)}$ .

The radius sets $R_{r_{D} (t)} = {r (t)}$ is defined as all functions $r (t) : R \to [0, + \infty)$ satisfying $\begin{array}{rcl} (5.12) & lim_{τ \to - \infty} e^{- \int_{τ}^{t} μ_{0} (s) d s} r^{2} (t) = 0 . \end{array}$

We call all families $D_{M_{V}^{2 q}}$ as tempered universe in $M_{V}^{2 q}$ .
Lemma 5.14.
Assume that the hypotheses (a1), (b), (c1), (d) and (e) hold, and ( 5.10 ) (or ( 5.11 )) also true, then the process $(U (t, τ); M_{V}^{2 q})$ generated by global weak solution possesses a family of sets $\bar{B} = {B (t) | B (t) = B (0, r_{V} (t)), t \in R}$ in $M_{V}^{2 q}$ for some $τ ⩽ τ (\bar{B}, t)$ , where $τ (\bar{B}, t) ⩽ t$ , i.e., for any $u \in D (t, τ; u_{τ}, ϕ)$ , we deduce that every orbit belongs to the ball $B (t)$ : $‖ u ‖_{V}^{2} ⩽ r_{V}^{2} (t)$ .
Proof.
By (5.4) in Lemma 5.10, we see that $\begin{array}{rcl} (5.13) & [(‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} ‖ \tilde{α} ‖_{L^{q} (τ - h, τ)} ‖ ϕ ‖_{L^{2 q} (- h, 0; V)}] e^{- \int_{τ}^{t} μ_{0} (t) d t} \to 0 \end{array}$ as $τ \to - \infty$ , then there exists a time $τ (B (t), t)$ , such that for $τ ⩽ τ (B (t), t)$ , $\begin{array}{rcl} (5.14) & [(‖ u_{τ} ‖_{H}^{2} + ‖ u_{τ} ‖_{V}^{2}) + \frac{1}{ν λ_{1}^{2} (1 - ρ^{})} ‖ \tilde{α} ‖_{L^{q} (τ - h, τ)} ‖ ϕ ‖_{L^{2 q} (- h, 0; V)}] e^{- \int_{τ}^{t} μ_{0} (t) d t} < 1 . \end{array}$

Using (5.4) and (5.10), we derive that $\begin{array}{rcl} (5.15) & {‖ u (t) ‖}_{H}^{2} + {‖ u (t) ‖}_{V}^{2} ⩽ 1 + \frac{1}{ν λ_{1}} \int_{- \infty}^{t} e^{- \int_{s}^{t} μ_{0} (r) d r} β (s) d s = r_{V}^{2} (t) . \end{array}$ Setting $B (t)$ as the ball $B (0, r_{V} (t))$ , by Lemma 5.10, it is easy to verify that for $(u_{τ}, ϕ) \in M_{V}^{2 q}$ , $‖ u (t) ‖_{V}^{2} ⩽ r_{V}^{2} (t)$ , i.e., $B = {B (t)}$ is the family of absorbing sets. □

∙ Pullback absorbing set in $C_{V}$ : Definition 5.15.
Let $P (C_{V})$ denotes all subsets in $C_{V}$ , which not need to be bounded. Let $D_{C_{V}} = {\tilde{D}}$ be the class of families $\tilde{D} = {\tilde{D} (t) | t \in R} \subset P (C_{V})$ , where $\tilde{D} (t) \subset {\bar{B}}_{C_{V}} (0, {\tilde{r}}_{\tilde{D}} (t))$ for radius ${\tilde{r}}_{\tilde{D}} (t)$ belongs to sets $R_{{\tilde{r}}_{\tilde{D}} (t)}$ . The radius sets $R_{{\tilde{r}}_{\tilde{D}} (t)} = {\tilde{r} (t)}$ is defined as all functions $\tilde{r} (t) : R \to [0, + \infty)$ satisfying $\begin{array}{rcl} (5.16) & lim_{τ \to - \infty} e^{- \int_{τ}^{t} μ_{0} (t) d t} {\tilde{r}}^{2} (t) = 0 . \end{array}$

We call all families $D_{C_{V}}$ as tempered universe in $C_{V}$ .
Lemma 5.16.
Assume that the hypotheses (a1), (b), (c1), (d) and (e) hold, and ( 5.10 ) (or ( 5.11 )) also true, then the process $(U (t, τ); C_{V})$ generated by global weak solution possesses a family of sets $\tilde{B} = {\tilde{B} (t) | \tilde{B} (t) = B (0, {\tilde{r}}_{V} (t)), t \in R}$ in $C_{V}$ for some $τ ⩽ τ (\tilde{B}, t)$ , where $τ (\tilde{B}, t) ⩽ t$ , i.e., for any $u \in D (t, τ; ϕ (0), ϕ)$ , we deduce that every orbit belongs to the ball $B (t)$ : $‖ u ‖_{V}^{2} ⩽ {\tilde{r}}_{V}^{2} (t)$ .
Proof.
Using the definition of $M_{V}^{2 q}$ and $C_{V}$ , noting that ${\tilde{r}}_{V} (t) = {sup}_{r \in [t - h, t]} r_{V} (\tilde{t})$ for arbitrary $t \in R$ , it is easily to conclude the result. □
Remark 5.17.
In Lemmas 5.14 and 5.16, if the universes we chosen is fixed bounded, then the $D$ -pullback absorbing sets family is also bounded, which leads the bounded pullback attractor belong to general $D$ -pullback case.

5.3. Pullback asymptotic compactness in $M_{V}^{2 q}$ and $C_{V}$

For overcoming the asymptotic compactness for some dissipative systems, Moise, Rosa and Wang [25] introduced the energy equation method, also see Ball [3]3], Yang, Li and Lu [40]. In this paper, we shall use this idea to deal with our problem.

Lemma 5.18.
Assume that the hypotheses (a1), (b), (c1), (d) and (e) hold, and ( 5.10 ) (or ( 5.11 )) also true, then the processes families $(U (t, τ); M_{V}^{2 q})$ and $(U (t, τ); C_{V})$ generated by global weak solution are pullback $D = {D_{0}}$ -asymptotically compact.
Proof.
Step 1: The convergence of sequence $u^{n}$ in $[t_{0} - h, t_{0}]$ which is independent on n.

For $t_{0} \in R$ , we assume that ${u^{n}} \in D (t, τ_{n}; ϕ_{n} (0), ϕ_{n})$ and ${ϕ_{n}} \in D_{0} (τ_{n})$ is a bounded sequence in $M_{V}^{2 q}$ with $τ_{n} \to - \infty$ as $n \to + \infty$ , here ${τ_{n}} \subset (- \infty, t_{0} - 2 h]$ . Using the Theorem 2.1 and using (2.9), one derive that $\begin{array}{rcl} (5.17) & {‖ {(u^{n} + A u^{n})}^{'} ‖}_{} ⩽ ν {‖ u^{n} ‖}_{V} + {‖ G (u^{n}) ‖}_{} + {‖ g (t - ρ (t), u_{m} (t - ρ (t))) ‖}_{H}, \end{array}$ by the similar technique in Section 5 and a diagonal procedure, there exists a subsequence (relabeled also as ${u^{n}}$ ) which converges to the function $u \in L^{\infty} (t_{0} - 2 h, t_{0}; V) \cap L^{2} (t_{0} - 2 h, t_{0}; V)$ and $\frac{d}{d t} u \in L^{2} (t_{0} - h, t_{0}; V^{'})$ such that $\begin{array}{l} (5.18) & u^{n} ⇀ u weakly star in L^{\infty} (t_{0} - 2 h; V), \\ (5.19) & u^{n} ⇀ u weakly in L^{2} (t_{0} - 2 h, t_{0}; V), \\ (5.20) & {(u^{n})}^{'} ⇀ {(u)}^{'} weakly in L^{2} (t_{0} - h, t_{0}; V^{'}), \end{array}$ by the Aubin–Lions Lemma, we see that $u^{n} (s) \to u (s)$ strongly in $V$ for almost all $s \in (t_{0} - 2 h, t_{0})$ . From the hypothesis on $G (\cdot)$ , it follows that $G (u^{n}) \to G (u)$ weakly in $L^{2} (t_{0} - 2 h, t_{0}; H)$ . By Lemma 4.1, we deduce that $\begin{array}{rcl} (5.21) & g (s, u^{n} (s - ρ (s))) \to g (s, u (s - ρ (s))) strongly in L^{2} (t_{0} - t, t_{0}; V) \end{array}$ and also in $L^{2} (t_{0} - t, t_{0}; H)$ . Thus, we conclude that $u \in C ([t_{0} - h, t_{0}]; V)$ and satisfies the equation in interval $[t_{0} - h, t_{0}]$ .

Step 2: Weak convergence of ${u^{n}}$ in $V$ .

Using the uniform bounded estimates of ${(u^{n})}^{'}$ in $L^{2} (t_{0} - h, t_{0}; V^{'})$ and $L^{\infty} (t_{0} - h, t_{0}; V)$ , we can see that ${u^{n}}$ is equi-continuous in $V^{'}$ in $[t_{0} - h, t_{0}]$ . Using the Ascoli–Arzelà Theorem, it is easily to derive $u^{n} \to u$ strongly in $C ([t_{0} - h, t_{0}]; V^{'})$ . Similarly to the uniform bounded estimate of $u^{n}$ in $C ([t_{0} - h, t_{0}]; V)$ , we have $\begin{array}{rcl} (5.22) & u^{n} (s) ⇀ u (s) weakly in V \end{array}$ for arbitrary $s \in [t_{0} - h, t_{0}]$ . Hence for any ${s_{n}} \subset [t_{0} - h, t_{0}]$ , $s_{n} \to s \in [t_{0} - h, t_{0}]$ , we conclude that $\begin{array}{rcl} (5.23) & u^{n} (s_{n}) ⇀ u (s) weakly in V . \end{array}$

Step 3: The strong convergence of corresponding sequences via energy equation method.

Next, we shall prove the asymptotic compactness in $V$ via energy equation method, i.e., the strong convergence of $u^{n}$ : $\begin{array}{rcl} (5.24) & {‖ u^{n} (t_{n}) - u (t^{}) ‖}_{V} \to 0 as n \to + \infty . \end{array}$ To this end, we shall prove $\begin{array}{l} (5.25) & \underset{n^{'} \to \infty}{lim inf} {‖ u^{n} (t_{n}) ‖}_{V} ⩾ {‖ u (t^{}) ‖}_{V}, \\ (5.26) & \underset{n^{'} \to \infty}{lim sup} {‖ u^{n} (t_{n}) ‖}_{V} ⩽ {‖ u (t^{}) ‖}_{V} \end{array}$ for a sequence ${t_{n}} \subset [t_{0} - h, t_{0}]$ and $t_{n} \to t^{}$ as $n \to + \infty$ .

The weak convergence (5.23) ensures that (5.25) is true, so we only need to prove (5.26) in following.

Using z replacing the unknown function and then multiplying equation (1.3) by z, we obtain $\begin{array}{rcl} (5.27) & \frac{d}{d t} ({‖ z (t) ‖}_{H}^{2} + {‖ z (t) ‖}_{V}^{2}) = - 2 ν {‖ z (t) ‖}_{V}^{2} + 2 (g (t, z (t - ρ (t))), z (t)) . \end{array}$

Integrating from s to t, it yields $\begin{array}{l} {‖ z (t) ‖}_{H}^{2} + {‖ z (t) ‖}_{V}^{2} \\ (5.28) & = ({‖ z (s) ‖}_{H}^{2} + {‖ z (s) ‖}_{V}^{2}) + \int_{s}^{t} (g (r, z (r - ρ (r))), z (r)) - 2 ν {‖ z (t) ‖}_{V}^{2} d r, \end{array}$ here z can be $u^{n} (t_{n})$ and $u (t)$ .

Considering the functionals defined for $t \in [t_{0} - h, t_{0}]$ , $\begin{array}{rcl} (5.29) & J (t) = ({‖ u (t_{0} - h) ‖}_{H}^{2} + {‖ u (t_{0} - h) ‖}_{V}^{2}) + \int_{t_{0} - h}^{t} (g (r, u (r - ρ (r))), u (r)) - 2 ν {‖ u (r) ‖}_{V}^{2} d r \end{array}$ and $\begin{array}{rcl} J_{n} (t) & = & ({‖ u^{n} (t_{0} - h) ‖}_{H}^{2} + {‖ u^{n} (t_{0} - h) ‖}_{V}^{2}) \\ (5.30) & + \int_{t_{0} - h}^{t} (g (r, u^{n} (r - ρ (r))), u^{n} (r)) - 2 ν {‖ u^{n} (t) ‖}_{V}^{2} d r, \end{array}$ since $G (\cdot, z)$ is Lipschitz continuous with respect to z, the other terms in (5.29) and (5.30) are linear, using the convergence of sequence ${u^{n}}$ in Step 1, we can obtain $\begin{matrix} (5.31) & J_{n} (t) \to J (t) \end{matrix}$ for almost every $t \in [t_{0} - h, t_{0}]$ , i.e., for $\forall ε > 0$ , there exists a $n_{k} \in N$ , for all $n ⩾ n_{k}$ and $t_{k} \in [t_{0} - h, t_{0}]$ , we have $\begin{array}{rcl} (5.32) & | J_{n} (t_{k}) - J (t_{k}) | ⩽ \frac{ε}{2} . \end{array}$

By the convergence if Steps 1 and 2, we know that $J (t)$ is continuous with respect to time t, $J_{n} (t)$ is uniformly continuous, this means for $\forall ε > 0$ , there exists ${\tilde{n}}_{k} \in N$ such that for a sequence ${t_{k}} \subset [t_{0} - h, t_{0}]$ with $t_{k} \to t^{}$ for all $n ⩾ {\tilde{n}}_{k}$ , then it follows $\begin{array}{rcl} (5.33) & | J (t_{k}) - J (t^{}) | ⩽ \frac{ε}{2}, \end{array}$

Choosing ${\bar{n}}_{k} = max {n_{k}, {\tilde{n}}_{k}}$ , then for all $n > {\bar{n}}_{k}$ , we have $\begin{array}{rcl} (5.34) & | J_{n} (t_{n}) - J (t^{}) | ⩽ | J_{n} (t_{n}) - J (t_{n}) | + | J (t_{n}) - J (t^{}) | < ε . \end{array}$

Hence, for any ${t_{n}} \subset [t_{0} - h, t_{0}]$ , we have Combining (4.21), (4.23) and (4.24), we obtain $\begin{array}{rcl} (5.35) & \underset{n \to \infty}{lim sup} ({‖ u^{n} (t_{n}) ‖}_{H}^{2} + {‖ u^{n} (t_{n}) ‖}_{V}^{2}) ⩽ {‖ u (t^{}) ‖}_{H}^{2} + {‖ u (t^{}) ‖}_{V}^{2}, \end{array}$ which means $\begin{array}{rcl} (5.36) & {‖ u^{n} (t_{n}) ‖}_{V} \to {‖ u (t^{*}) ‖}_{V} . \end{array}$ Combining the norm convergence (5.36), weak convergence (5.23), we can achieve (5.24), which gives the strong convergence of sequence $u^{n} (t_{n})$ in $V$ in interval $[t_{0} - h, t_{0}]$ for all $t_{0} \in R$ , i.e., the sequence $u^{n} (t_{n})$ is relative compact. We have completed the proof of our Lemma. □

5.4. Proof of Theorems 3.1 and 3.2: Existence of pullback attractors

Proof of Theorem 3.1.
The Lemmas 5.10 and 5.14 give the existence of pullback absorbing set in $M_{V}^{2 q}$ , Lemma 5.18 implies the $D_{0}$ -pullback asymptotic compactness for the process $(U (t, τ); M_{V}^{2 q})$ , it is easy to check that the universe $D_{M_{V}^{2 q}}$ is inclusion closed, using the theory in Theorem 5.5, we can obtain the result in Theorem 3.1. □
Proof of Theorem 3.2.
The Lemmas 5.10 and 5.16 give the existence of pullback absorbing set in $C_{V}$ , Lemma 5.18 implies the $D_{0}$ -pullback asymptotic compactness for the continuous process $(U (t, τ); C_{V})$ , it is easy to check that the universe $D_{C_{V}}$ is inclusion closed, using the theory in Theorem 5.5, we have completed the proof. □
5.5. Proof of Theorem 3.3

Proof.
Assume that $u (t)$ , $v (t)$ be two solutions of problem (1.3) with initial data $\begin{matrix} (5.37) & u (t) |_{t \in [τ - h, τ]} = ϕ_{1} (θ) |_{θ \in [- h, 0]} \end{matrix}$ and $\begin{matrix} (5.38) & v (t) |_{t \in [τ - h, τ]} = ϕ_{2} (θ) |_{θ \in [- h, 0]} \end{matrix}$ respectively, let $w = u (t) - v (t)$ , then the resulting difference function w satisfies the following equation $\begin{matrix} (5.39) & w_{t} - Δ w_{t} - ν Δ w + \nabla \cdot (\vec{F} (u) - \vec{F} (v)) = g (t, u (t - ρ (t))) - g (t, v (t - ρ (t))), \end{matrix}$ subject to the periodic boundary and initial data $\begin{matrix} (5.40) & w (x, t) |_{t \in [τ - h, τ]} = ϕ_{1} (θ) - ϕ_{2} (θ) = \tilde{ϕ} (θ), θ \in [- h, 0] . \end{matrix}$

Multiplying (5.39) with w, integrating by part with respect to x, it yields $\begin{array}{rcl} \frac{d}{d t} (| w |_{2}^{2} + ‖ w ‖^{2}) + 2 ν ‖ w ‖^{2} & ⩽ & 2 | \int_{Ω} \nabla \cdot (\vec{F} (u) - \vec{F} (v)) w d x | \\ (5.41) & + 2 | (g (t, u (t - ρ (t)) - g (t, v (t - ρ (t)), w) |, \end{array}$ hence, using the hypothesis (H.6), we derive that $\begin{array}{rcl} | \int_{Ω} \nabla \cdot (\vec{F} (u) - \vec{F} (v)) w d x | & = & | - \int_{Ω} \vec{F} (u) - \vec{F} (v) \cdot \nabla w d x | \\ ⩽ & \int_{Ω} | \vec{F} (u) - \vec{F} (v) | | \nabla w | d x \\ (5.42) & ⩽ & \tilde{L} \int_{Ω} | w | | \nabla w | d x ⩽ \frac{ν}{4} ‖ w ‖^{2} + \frac{\tilde{L}}{ν} | w |_{2}^{2} \end{array}$ and $\begin{array}{l} | (g (t, u (t - ρ (t))) - g (t, v (t - ρ (t))), w) | \\ (5.43) & ⩽ \frac{ν}{4} ‖ w ‖^{2} + \frac{L^{2} (R)}{ν λ_{1}} γ (t) {| w (t - ρ (t)) |}_{2}^{2} \end{array}$ and $\begin{array}{rcl} (5.44) & \int_{τ}^{t} γ (t) {| w (s - ρ (s)) |}_{2}^{2} d s & ⩽ & \frac{‖ γ ‖_{L^{\infty} (τ - h, T)}}{(1 - ρ^{})} [\int_{τ - h}^{τ} | \tilde{ϕ} |_{2}^{2} d s + \int_{τ}^{t} {| w (s) |}_{2}^{2} d s] . \end{array}$

Integrating (5.41) from τ to t, we derive $\begin{array}{l} (| w |_{2}^{2} + ‖ w ‖^{2}) + \frac{ν λ_{1}}{2} \int_{τ}^{t} {| w (s) |}_{2}^{2} d s + \frac{ν}{2} \int_{τ}^{t} {‖ w (s) ‖}^{2} d s \\ ⩽ {| ϕ_{1} (0) - ϕ_{2} (0) |}_{2}^{2} + \frac{2 \tilde{L}}{ν} \int_{τ}^{t} {| w (s) |}_{2}^{2} d s \\ + \frac{2 L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})} \int_{τ - h}^{τ} {| \tilde{ϕ} (θ) |}_{2}^{2} d s + \frac{2 L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})} \int_{τ}^{t} {| w (s) |}_{2}^{2} d s \\ ⩽ {| ϕ_{1} (0) - ϕ_{2} (0) |}_{2}^{2} + \frac{2 L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})} \int_{τ - h}^{τ} {| \tilde{ϕ} (θ) |}_{2}^{2} d s \\ + [\frac{\tilde{L}}{ν} + \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})}] \int_{τ}^{t} {| w (s) |}_{2}^{2} d s \\ (5.45) & + [\frac{\tilde{L}}{ν λ_{1}} + \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1}^{2} (1 - ρ^{})}] \int_{τ}^{t} {‖ w (s) ‖}^{2} d s \end{array}$ i.e., $\begin{array}{rcl} | w |_{2}^{2} + ‖ w ‖^{2} & ⩽ & [{| \tilde{ϕ} (0) |}_{2}^{2} + \frac{2 L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})} \int_{τ - h}^{τ} {| \tilde{ϕ} (θ) |}_{2}^{2} d s] \\ + \int_{τ}^{t} [\frac{\tilde{L}}{ν} + \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})} - \frac{ν λ_{1}}{2}] {| w (s) |}_{2}^{2} d s \\ (5.46) & + [\frac{\tilde{L}}{ν λ_{1}} + \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1}^{2} (1 - ρ^{})} - \frac{ν}{2}] \int_{τ}^{t} {‖ w (s) ‖}^{2} d s \end{array}$

Let the initial data $\tilde{ϕ} (0)$ and $\tilde{ϕ} (θ)$ are arbitrary and fixed, then by the Gronwall inequality, let τ tends to $- \infty$ , it follows that $w (\cdot) = 0$ , i.e., if $U (t, τ) (ϕ_{1} (0), ϕ_{1} (θ)) = U (t, τ) (ϕ_{2} (0), ϕ_{2} (θ))$ , then $u (t) = v (t)$ , which means the global solution has backward unique property and the pullback attractors reduces a single trajectory provided that $\begin{array}{rcl} max {\frac{\tilde{L}}{ν} + \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1} (1 - ρ^{})} - \frac{ν λ_{1}}{2}, \frac{\tilde{L}}{ν λ_{1}} + \frac{L^{2} (R) ‖ γ ‖_{L^{\infty} (τ - h, T)}}{ν λ_{1}^{2} (1 - ρ^{*})} - \frac{ν}{2}} < 0, \end{array}$ which is the sufficient condition that the pullback attractors reduces to a single trajectory but may be not optimal, which completes the proof. □
6. Further research and outlook

In this paper, under the global well-posedness especially uniqueness, we conclude the existence of minimal family of pullback attractors $A_{M_{V}^{2 q}} (t)$ and $A_{C_{V}} (t)$ in $M_{V}^{2 q}$ and $C_{V}$ respectively. However, since the regularity of global weak and strong solution has not been studied, the relation between families $A_{M_{V}^{2 q}} (t)$ and $A_{C_{V}} (t)$ is unknown. In addition, if the delay is infinite, the dynamic systems is also open. Moreover, if the boundedness of delay tends to infinity, can we derive the upper semi-continuity by the similar technique as in [21], this is our next objective.

Footnotes

Acknowledgements

This work was partly supported by the Key Project of Science and Technology of Henan Province (Grant No. 182102410069) and the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039).

The authors thank referees by his/her comments, which led to improvements in the presentation of this paper.

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