Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains

Abstract

This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.

Keywords

Navier–Stokes equations multi-delays Lipschitz-like domain robustness

1. Introduction

The 2D incompressible Navier–Stokes equations govern the motion of an incompressible fluid. We refer the reader to, e.g., [14,19,22,35] for the well-posedness and [10,11,14,31,34] for the asymptotic behavior, including the existence of attractors with finite fractal/Hausdorff dimension. Delays on differential equations originate from the control on the boundary in engineering, which can be described by evolutionary partial differential equations with delay terms, and was first investigated for ordinary differential equations. The Navier–Stokes equations with delay have also become an interesting topic in the last two decades. These are important models in fluid mechanics; we can mention, e.g., the wind tunnel model. The research on the well-posedness and dynamics of the Navier–Stokes equations with delay can be seen in [2,7–9,15–17,23,25,26,30,33] and references therein. Also, robustness results, especially the upper semi-continuity of attractors of the system as the delays vanish, can be found in [21,24]. However, many known results concentrate on smooth domain with at least a $C^{2}$ boundary. Here, we are concerned with the pullback dynamics and its robustness for the 2D incompressible Navier–Stokes equations with delays defined in Lipschitz-like domains.

A bounded domain $Ω \subset R^{d}$ is said to be Lipschitz-like if $\partial Ω$ can be covered by a finite number of balls $B_{i} = B (Q_{i}, r_{0})$ with $Q_{i} \in \partial Ω$ such that for any ball $B_{i}$ , there are a rectangular coordinate system and a Lipschitz function $Ψ_{i} : R^{d - 1} \to R$ with $\begin{matrix} B (Q_{i}, 3 r_{0}) \cap Ω = {(x_{1}, x_{2}, \dots, x_{d}) | x_{d} > Ψ_{i} (x_{1}, x_{2}, \dots, x_{d - 1})} \cap Ω \end{matrix}$ (see [3,13,32]). Inspired from [3,15,16,20,27,28,32,38] and [39], the aim of this paper is to study the 2D incompressible Navier–Stokes system with multi-delays in a Lipschitz-like domain $Ω \subset R^{2}$ with inhomogeneous Dirichlet boundary conditions, $\begin{array}{rcl} (1.1) & \{\begin{matrix} \frac{\partial u}{\partial t} - ν Δ u + (u (t - ρ (t)) \cdot \nabla) u + \nabla p = f (t, u (t - ρ (t))) + g (t, x), & (t, x) \in Ω_{τ}, \\ div u = 0, & (t, x) \in Ω_{τ}, \\ u (t, x) = φ, φ \cdot n = 0, & (t, x) \in \partial Ω_{τ}, \\ u (τ, x) = u (τ), & x \in Ω, \\ u (τ + θ, x) = ϕ (θ, x), & (θ, x) \in Ω_{h}, \end{matrix} \end{array}$ where $Ω_{τ} = (τ, + \infty) \times Ω$ , $\partial Ω_{τ} = (τ, + \infty) \times \partial Ω$ , $Ω_{h} = [- h, 0] \times Ω$ and $τ \in R$ is the initial time. Here, ν is the kinematic viscosity of the fluid, $u = (u_{1} (t, x), u_{2} (t, x))$ is the unknown velocity field and p is the unknown pressure. The non-autonomous external forces contain $g (x, t)$ and the time-varying delay $f (t, u (t - ρ (t)))$ , where $h > 0$ is a constant and $ρ : [0, + \infty) \to [0, h]$ is the delay function. In addition, $ϕ (θ, x)$ denotes the initial state in $[- h, 0]$ , $u (τ) = ϕ (0, x)$ , $φ \in L^{\infty} (\partial Ω)$ , and n is the outward unit normal to the boundary $\partial Ω$ .

If the delays disappear, then problem (1.1) reduces to the case studied in Brown, Perry and Shen [3] and Miranville and Wang [27,28] if the domain is smooth. The idea to deal with inhomogeneous boundary conditions consists in defining a stream function (also called backward flow), introduced in [27,28] for smooth domains, and in [3] for non smooth ones. Inspired by [21], we aim to study the tempered pullback dynamics and its robustness for (1.1). The main results and features can be summarized as follows.

Problem (1.1) contains inhomogeneous boundary conditions in a Lipschitz-like domain. Using the stream function ψ for the corresponding Stokes equations subject to the same boundary conditions, the inhomogeneous problem (1.1) can be transformed into an equivalent homogeneous system (3.4). Based on the existence of and estimates on the stream function, systems (1.1) and (3.4) are equivalent. The global weak solution for (3.4) is obtained by using a Galerkin scheme as in [22], while the uniqueness and regularity of solutions are based on some delicate estimates for Lipschitz-like domains and a retarded Gronwall inequality from [21], which is different from [3] since the delays exist in the convective term and external forces, see Theorems 3.3, 3.4 and 3.5 in Section 3.

Based on the global well-posedness of weak solutions, using the concept of universe and the theory of pullback attractors in [10,16,17], we obtain the minimal families of tempered and $D$ -pullback attractors for the process ${S (t, τ)}$ in $H \times C_{H}$ and $M_{H}$ respectively, see Theorems 3.6 and 3.7. The main difficult here is to derive an estimate on the convective term with delay on a Lipschitz-like domain, while deriving the $D$ -pullback asymptotic compactness is based on an energy equation approach as in [1,29]. Furthermore, when the delay, considered as a small perturbation, disappears, the robustness of the pullback attractors for (1.1) and the classical Navier–Stokes equations, in particular the upper semi-continuity property, was studied in [40] (see Theorem 3.9).

Since the Navier–Stokes equations contain a nonlinear convective term and a singularity on the boundary of the domain, the regularity of the global solution is difficult to obtain. Furthermore, as they also contain a delay defined in a Lipschitz-like domain, deriving uniform estimates is more delicate. Four bilinear operators in the equivalent system (3.4) need to be estimated, which is obtained by using Hardy’s inequality and choosing proper parameters. Also, the assumption on $f (t, u (t - ρ (t)))$ makes a pullback absorbing set difficult to be easily proved by simple hypotheses on $α (t)$ and $β (t)$ ; we introduce a variable index $κ_{δ} (t, τ)$ to overcome this difficulty, yielding the minimal family of tempered pullback attractors in Theorem 3.6.

The study of the continuity of attractors has attracted the attention of many mathematicians. We can find many interesting results for evolutionary equations with perturbations, as the perturbed parameter vanishes, in [4,5,10,36]. The regularity of the solutions for (1.1) and (3.4) can not go beyond $D (A^{1 / 4})$ , which is different from parabolic equations with delay in [24]. This implies that the techniques to achieving the upper semi-continuity of pullback attractors in [24] can no longer be used. Inspired from [40] for the 3D Navier–Stokes-Voigt equations, we use the weak topology of $L^{2}$ to study the convergence of the $D$ -pullback attractors of systems (1.1) and (3.4) as the delay vanishes, see the details in Section 6.2.

The results in this paper are different from those in [3,16] and [26] which contain the uniform boundedness of the stream function ψ via estimates on φ on the boundary and the influence of the delay in the topology of $C_{H}$ and $L_{V}^{2}$ . The regularity and pullback dynamics in Lipschitz-like domains are more complicated to achieve than the smooth case; see in particular Dahlberg and Kenig [12] for techniques to deal with bad boundary conditions for elliptic partial differential equations.

This paper is organized as follows. Some preliminaries are given in Section 2. Then, the main results are stated in Section 3, which contains the global well-posedness and regularity of solutions, the pullback attractor and its robustness. Finally, the results are proved in the last three sections.

2. Preliminaries

∙ The generic functional spaces

Denote ${(C_{0}^{\infty} (Ω))}^{2}$ as $D (Ω)$ , set $E : = {u | u \in D (Ω), div u = 0}$ and let H be the closure of $E$ in the ${(L^{2} (Ω))}^{2}$ topology; $‖ \cdot ‖_{H}$ and $(\cdot, \cdot)$ denote the norm and inner product in H respectively, $\begin{matrix} | u | = ‖ u ‖_{H} = {(\int_{Ω} | u |^{2} d x)}^{1 / 2}, (u, v) = \sum_{j = 1}^{2} \int_{Ω} u_{j} (x) v_{j} (x) d x, \forall u, v \in H . \end{matrix}$ The space V is the closure of $E$ in ${(H^{1} (Ω))}^{2}$ topology; $‖ \cdot ‖_{V}$ and $((\cdot, \cdot))$ denote the norm and inner product in V respectively, $\begin{matrix} ‖ u ‖ = ‖ u ‖_{V} = {((u, u))}^{1 / 2}, ((u, v)) = \sum_{i, j = 1}^{2} \int_{Ω} \frac{\partial u_{j}}{\partial x_{i}} \frac{\partial v_{j}}{\partial x_{i}} d x, \forall u, v \in V . \end{matrix}$ It is easy to check that H and V are Hilbert spaces with duals H and $V^{'}$ respectively, $V ↪ H \equiv H^{'} ↪ V^{'}$ , the injections being dense and continuous; $‖ \cdot ‖_{*}$ and $⟨ \cdot, \cdot ⟩$ denote the norm in $V^{'}$ and the dual product between V and $V^{'}$ respectively, also between H and itself.

∙ The fractional power functional spaces

Let $P_{L}$ be the Helmholz-Leray orthogonal projection in ${(L^{2} (Ω))}^{2}$ onto H (see [14,34,35]), and $\begin{matrix} P_{L} : H \oplus G (Ω) \to H, \end{matrix}$ where $\begin{matrix} G (Ω) = H^{⊥} : = {u \in {(L^{2} (Ω))}^{2}; \exists ς \in {(L_{loc}^{2} (Ω))}^{2} : u = \nabla ς} . \end{matrix}$ Let $A : = - P_{L} Δ$ be the Stokes operator, ${λ_{j}}_{j = 1}^{\infty}$ ( $0 < λ_{1} ⩽ λ_{2} ⩽ \dots$ , $λ_{j} \to + \infty$ as $j \to + \infty$ ) being the eigenvalues of A corresponding to the orthonormal eigenfunctions ${ω_{j}}_{j = 1}^{\infty}$ .

Define the fractional operator $A^{s}$ by $\begin{matrix} A^{s} u = \sum_{j} λ_{j}^{s} (u, ω_{j}) ω_{j}, s \in R, j \in Z^{+} \end{matrix}$ for $u = \sum_{j} (u, ω_{j}) ω_{j}$ with the domain $D (A^{s}) = {u | A^{s} u \in H}$ ; we define the norm of $D (A^{s})$ as $\begin{matrix} ‖ u ‖_{2 s}^{2} = {| A^{s} u |}^{2} = \sum_{j} λ_{j}^{2 s} {| (u, ω_{j}) |}^{2} . \end{matrix}$ Especially, $V = D (A^{1 / 2})$ with norm $‖ u ‖_{1} = | A^{1 / 2} u | = ‖ u ‖$ for any $u \in V$ , see [14,34].

∙ The delayed functional spaces

For any $t \in (τ, T)$ , $u_{t}$ is defined by $u_{t} (s) = u (t + s)$ , and hence the delayed functional spaces are defined as $\begin{matrix} C_{X} = C ([- h, 0]; X), ‖ ϕ ‖_{C_{X}} = sup_{θ \in [- h, 0]} {‖ ϕ (θ) ‖}_{X}, X = H, V, D (A^{1 / 4}) . \end{matrix}$ Moreover, the pth-power delayed Lebesgue spaces can be defined as $\begin{matrix} L_{X}^{p} = L^{p} (- h, 0; X) (1 < p ⩽ + \infty), \end{matrix}$ and the norm is similar to the one for general Lebesgue spaces in the delayed interval $[- h, 0]$ . Also, the space $M_{H} = H \times (C_{H} \cap L_{V}^{2})$ is equipped with the topology $\begin{matrix} {‖ (u (t), u_{t}) ‖}_{M_{H}}^{2} = {| u (t) |}^{2} + ‖ u_{t} ‖_{C_{H}}^{2} + ‖ u_{t} ‖_{L_{V}^{2}}^{2} . \end{matrix}$ Let finally $P_{1}$ and $P_{2}$ be the projections from $M_{H}$ to H and $C_{H} \cap L_{V}^{2}$ respectively, that is $\begin{matrix} P_{1} M_{H} = H, P_{2} M_{H} = C_{H} \cap L_{V}^{2} . \end{matrix}$

∙ The bilinear and trilinear operators

The bilinear and trilinear operators are defined as follows (see [14,19,35]): $\begin{array}{l} (2.1) & B (u, v) : = P_{L} ((u \cdot \nabla) v), \forall u, v \in V, \\ (2.2) & b (u, v, w) = ⟨ B (u, v), w ⟩ = \sum_{i, j = 1}^{2} \int_{Ω} u_{i} \frac{\partial v_{j}}{\partial x_{i}} w_{j} d x \end{array}$ with $B (u) = B (u, u)$ for short, where $B (u, v)$ is a continuous operator from $V \times V$ to $V^{'}$ , and $b (u, v, w)$ satisfies $\begin{matrix} (2.3) & \{\begin{matrix} b (u, v, v) = 0, & \forall u, v \in V, \\ b (u, v, w) = - b (u, w, v), & \forall u, v, w \in V, \\ | b (u, v, w) | ⩽ C | u |^{\frac{1}{2}} ‖ u ‖^{\frac{1}{2}} ‖ v ‖ | w |^{\frac{1}{2}} ‖ w ‖^{\frac{1}{2}}, & \forall u, v, w \in V . \end{matrix} \end{matrix}$

∙ Some inequalities

Lemma 2.1 (See [3]).

The Gagliardo–Nirenberg interpolation inequality: $\begin{array}{rcl} (2.4) & {| A^{1 / 2} u |}^{2} ⩽ C | A^{1 / 4} u | | A^{3 / 4} u |, \forall u \in D (A^{3 / 4}) . \end{array}$

Hardy’s inequality: $\begin{array}{rcl} (2.5) & \int_{Ω} \frac{| u (x) |^{2}}{{[dist (x, \partial Ω)]}^{2}} d x ⩽ C \int_{Ω} {| \nabla u (x) |}^{2} d x, \forall u \in V . \end{array}$

For $u \in D (A^{1 / 4})$ , there exists a constant C independent of the domain Ω such that $\begin{matrix} (2.6) & ‖ u ‖_{L^{4} (Ω)} ⩽ C | A^{1 / 4} u | and \int_{Ω} \frac{| u (x) |^{2}}{dist (x, \partial Ω)} d x ⩽ C {| A^{1 / 4} u |}^{2} . \end{matrix}$

Moreover, there holds for any $u \in D (A^{α + \frac{1}{4}})$ that $\begin{matrix} (2.7) & \int_{Ω} \frac{| A^{α} u (x) |^{2}}{dist (x, \partial Ω)} d x ⩽ C {| A^{α + 1 / 4} u |}^{2} . \end{matrix}$

Lemma 2.2 (See [20]).

Let X be a Banach space with spatial variable, based on the retarded Banach space above, and let $‖ \cdot ‖_{X}$ denote the norm of the space $C ([- h, 0]; X)$ for some $h ⩾ 0$ ; $y (t) ⩾ 0$ is a continuous function defined on $C ([- h, T]; X)$ , $y_{t} (s) = y (t + s)$ for $s \in [- h, 0]$ .

Consider the following retarded integral inequality: $\begin{array}{rcl} (2.8) & y (t) ⩽ E (t, τ) ‖ y_{τ} ‖_{X} + \int_{τ}^{t} K_{1} (t, s) ‖ y_{s} ‖_{X} d s + \int_{t}^{\infty} K_{2} (t, s) ‖ y_{s} ‖_{X} d s + \bar{ρ}, \forall t ⩾ τ ⩾ 0, \end{array}$ where E, $K_{1}$ and $K_{2}$ are non-negative measurable functions on $R^{2}$ ; here $\bar{ρ} ⩾ 0$ denotes a constant.

Let $L (E, K_{1}, K_{2}, \bar{ρ}) = {y \in C ([- h, T]; X) | y ⩾ 0 and satisfies the inequality (2.8)}$ , and $\begin{matrix} \bar{κ} (K_{1}, K_{2}) = sup_{t ⩾ τ} (\int_{τ}^{t} K_{1} (t, s) d s + \int_{t}^{\infty} K_{2} (t, s) d s) \end{matrix}$ with $\bar{κ} (K_{1}, K_{2}) < + \infty$ . Assume that $\begin{array}{rcl} (2.9) & lim_{t \to + \infty} E (t + s, s) = 0 \end{array}$ uniformly with respect to $s \in R^{+}$ , and set $ϑ = {sup}_{t ⩾ s ⩾ τ} E (t, s)$ and $\bar{κ} = \bar{κ} (K_{1}, K_{2})$ . Then we have the following estimate:

If $\bar{κ} < 1$ , then for any $R, ε > 0$ , there exists $\tilde{T} > 0$ such that $\begin{array}{rcl} (2.10) & ‖ y_{t} ‖_{X} < μ \bar{ρ} + ε \end{array}$ for $t > \tilde{T}$ and all bounded functions $y \in L (E, K_{1}, K_{2}, \bar{ρ})$ with $‖ y_{0} ‖ ⩽ R$ , where $μ = \frac{1}{1 - \bar{κ}}$ .

If $\bar{κ} < \frac{1}{1 + ϑ}$ , then there exist $M, λ > 0$ which are independent on $\bar{ρ}$ such that $\begin{array}{rcl} (2.11) & ‖ y_{t} ‖_{X} ⩽ M ‖ y_{0} ‖_{X} e^{- λ t} + γ \bar{ρ}, t ⩾ τ \end{array}$ for all bounded functions $y \in L (E, K_{1}, K_{2}, \bar{ρ})$ , where $γ = \frac{μ + 1}{1 - \bar{κ} c}$ and $c = max {\frac{ϑ}{1 - \bar{κ}}, 1}$ .

If $\bar{κ} < \frac{1}{1 + ϑ}$ , then the solution reduces to the trivial one when $\bar{κ} c < 1$ .

∙ A useful lemma

Lemma 2.3.
Let X and $X_{0}$ be two Banach spaces, $X_{0} = X^{'}$ , and $1 < p < \infty$ . If $\begin{matrix} v_{n} ⇀ v, weakly in L^{p} (0, T; X) \end{matrix}$ and $\begin{matrix} \frac{\partial}{\partial t} v_{n} ⇀ \frac{\partial}{\partial t} v, weakly in L^{p} (0, T; X_{0}) \end{matrix}$ hold, then $v_{n} (0) ⇀ v (0)$ weakly in X.

3. Main results

3.1. The Stokes problem in a Lipschitz-like domain

From [3,12,13] and [32], we know that the stream function ψ solves the following Stokes system in a Lipschitz-like domain: $\begin{matrix} (3.1) & \{\begin{matrix} - Δ u + \nabla q = 0, & x \in Ω, \\ div u = 0, & x \in Ω, \\ u = φ a.e. & x \in \partial Ω in the sense of nontangential convergence . \end{matrix} \end{matrix}$

Assume that $u = (u_{1}, u_{2})$ is the solution to (3.1) with $φ \in L^{\infty} (\partial Ω)$ and $φ \cdot n = 0$ . Therefore, we define the stream function ψ satisfying $\begin{matrix} \{\begin{matrix} div ψ = 0, x \in Ω, \\ ψ = u in {x \in Ω; dist (x, \partial Ω) ⩽ C_{1}^{'} ε}, \\ supp ψ \subset {x \in \bar{Ω}; dist (x, \partial Ω) ⩽ C_{2}^{'} ε}, \\ ψ = φ a.e. x \in \partial Ω in the sense of nontangential convergence \end{matrix} \end{matrix}$ and $\begin{matrix} \{\begin{matrix} ‖ ψ ‖_{L^{\infty} (Ω)} ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)}, \\ {sup}_{x \in Ω} | ψ (x) | + {sup}_{x \in Ω} | \nabla ψ (x) | dist (x, \partial Ω) ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)}, \\ ‖ | \nabla ψ | dist {(\cdot, \partial Ω)}^{1 - \frac{1}{p}} ‖_{L^{p} (Ω)} ⩽ C ‖ φ ‖_{L^{p} (\partial Ω)}, 2 ⩽ p ⩽ \infty . \end{matrix} \end{matrix}$ In addition, the stream function ψ can be written as follows: $\begin{matrix} Δ ψ = \nabla (q η_{ε}) + F, \end{matrix}$ where $supp F \subset {x \in Ω; C_{1}^{'} ε ⩽ dist (x, \partial Ω) ⩽ C_{2}^{'} ε}$ , $| F | ⩽ \frac{C}{ε^{3 / 2}} ‖ φ ‖_{L^{2} (\partial Ω)}$ and $η_{ε} \in C_{0}^{\infty} (R^{2})$ satisfies $0 ⩽ η_{ε} ⩽ 1$ and $\begin{matrix} \{\begin{matrix} η_{ε} = 1 & in {x \in R^{2}; dist (x, \partial Ω) ⩽ C_{1} ε}, \\ η_{ε} = 0 & in {x \in R^{2}; dist (x, \partial Ω) ⩾ C_{2} ε} . \end{matrix} \end{matrix}$

3.2. Assumptions

For the global well-posedness and pullback dynamics of problem (1.1), we assume that $g (\cdot, \cdot)$ and $f (\cdot, \cdot)$ satisfy the following hypotheses.

There exists $m > 0$ such that the external force $g (\cdot, \cdot) \in L_{loc}^{2} (R, H^{- 1} (Ω))$ satisfies $\begin{array}{rcl} (3.2) & \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}_{H^{- 1}}^{2} d s < \infty, \forall t \in R . \end{array}$

The function $f (\cdot, u) : [τ, + \infty) \to H$ is measurable for all $u \in H$ , and $f (t, \cdot) : C_{H} \to H$ is continuous for all $t ⩾ τ$ . Let $ρ \in C^{1} ([0, + \infty); [0, h])$ , and there exists a positive constant $ρ^{*} < 1$ such that $\begin{matrix} | \frac{d ρ}{d t} | ⩽ ρ^{*}, \forall t ⩾ 0 . \end{matrix}$

Assume that $\begin{array}{rcl} (3.3) & {| f (t, u) |}^{2} ⩽ α (t) | u |^{2} + β (t), \forall t ⩾ τ, \end{array}$ where $α (t)$ and $β (t)$ are defined as $α (\cdot), β (\cdot) : R \to [0, + \infty)$ for $α (\cdot) \in L^{\infty} (- \infty, T)$ and $β (\cdot) \in L^{1} (- \infty, T)$ with ${lim sup}_{τ \to - \infty} \frac{1}{t - τ} \int_{τ}^{t} β (s) d s = {\tilde{β}}_{0} \in (0, + \infty)$ .

$ν λ_{1} - \frac{1}{ν (1 - ρ^{*})} ‖ α (t) ‖_{L^{\infty}} > 0$ .

3.3. Main results for the equivalent system

Let $v = u - ψ$ , $v (τ) \in H$ , $η \in C_{H} \cap L_{V}^{2}$ . Then (1.1) can be transformed into the following equivalent homogeneous problem in abstract form: $\begin{matrix} (3.4) & \{\begin{matrix} \frac{\partial v}{\partial t} + ν A v + B (v (t - ρ (t)), v) + B (v (t - ρ (t)), ψ) + B (ψ, v) \\ = P_{L} (g (t, x) + f (t, u (t - ρ (t))) + ν F) - B (ψ), & (t, x) \in Ω_{τ}, \\ div v = 0, & (t, x) \in Ω_{τ}, \\ v = 0, & (t, x) \in \partial Ω_{τ}, \\ v (τ, x) = v (τ), & x \in Ω, \\ v (θ) = η (θ, x) = η (θ), & (θ, x) \in Ω_{h} . \end{matrix} \end{matrix}$

∙ Existence and uniqueness of the global weak solution

Definition 3.1.
A function $v : [τ, \infty) \to H$ is called the weak solution to (3.4) if for any $T > 0$ there hold
$v \in C ([τ - h, T]; H) \cap L^{2} (τ, T; V)$ , $\frac{\partial v}{\partial t} \in L^{2} (τ, T; V^{'})$ ,

$v (τ + θ, x) = η (θ)$ , $θ \in [- h, 0]$ ,

for any $w \in V$ and $s, t \in [τ, T)$ with $s ⩽ t$ , there holds $\begin{array}{l} (v (t), w) + ν \int_{s}^{t} ⟨ A v (l), w ⟩ d l + \int_{s}^{t} ⟨ B (v (l - ρ (l)), v (l)), w ⟩ d l \\ + \int_{s}^{t} ⟨ B (v (l - ρ (l)), ψ), w ⟩ d l + \int_{s}^{t} ⟨ B (ψ, v), w ⟩ d l \\ = (v (s), w) + \int_{s}^{t} ⟨ P_{L} g (t, x), w ⟩ d l + \int_{s}^{t} (P_{L} (f (t, u (t - ρ (t))) + ν F), w) d l \\ (3.5) & - \int_{s}^{t} ⟨ B (ψ), w ⟩ d l, \end{array}$

for any $T ⩾ τ$ , the energy inequality holds in the distribution sense, $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {| v (t) |}^{2} + ν {‖ v (t) ‖}^{2} + b (v (t - ρ (t)), ψ, v) + b (ψ, ψ, v) \\ (3.6) & ⩽ ⟨ g (t, x), v (t) ⟩ + (f (t, u (t - ρ (t))) + ν F, v (t)) . \end{array}$

Remark 3.2.
The inequality (3.6) can be understood in the sense that there holds for any test function $ζ \in C_{0}^{\infty} [τ, T]$ that $\begin{array}{l} - \frac{1}{2} \int_{τ}^{T} {| v (t) |}^{2} ζ^{'} (t) d t + ν \int_{τ}^{T} {‖ v (t) ‖}^{2} ζ (t) d t + \int_{τ}^{T} b (v (t - ρ (t)), ψ, v) ζ (t) d t \\ ⩽ \int_{τ}^{T} ⟨ g (t, x), v (t) ⟩ ζ (t) d t + \int_{τ}^{T} (f (t, u (t - ρ (t))) + ν F, v (t)) ζ (t) d t \\ - \int_{τ}^{T} b (ψ, ψ, v) ζ (t) d t . \end{array}$
Theorem 3.3 (Global weak solution).

Let $(v (τ), η) \in M_{H}$ , and the hypotheses (H-a)–(H-d) hold. Then there exists at least one global weak solution $v (t, x)$ to system ( 3.4 ) on $[τ - h, \infty)$ in the sense of Definition 3.1 .

Proof.
See Section 4.1. □
Theorem 3.4 (Uniqueness).

Assume that the hypotheses in Theorem 3.3 hold. Moreover, we assume that for any $r > 0$ , there exists a constant $L (r) > 0$ such that $\begin{array}{rcl} (3.7) & | f (t, w_{1}) - f (t, w_{2}) | ⩽ L (r) γ^{1 / 2} (t) | w_{1} - w_{2} |, \forall t ⩾ τ, | w_{1} | ⩽ r, | w_{2} | ⩽ r, \end{array}$ where $γ \in L^{\infty} (τ, T) : [τ, T) \to [0, + \infty)$ . Then the global weak solution in Theorem 3.3 is unique, which generates a continuous process ${S (t, τ)}$ in $M_{H}$ .

Proof.
See Section 4.2. □

∙ Regularity

In this part, we shall use the technique in [3] to study the regularity of the solution v to system (3.4) obtained in Theorems 3.3 and 3.4.
Theorem 3.5 (Regularity of the global solution).

Let $(v (τ), η) \in M_{H}$ , $g (t, x) \in L^{2} (τ, T; H)$ and $v (t)$ be the solution to system ( 3.4 ) in Theorem 3.3 . Then $v (t) \in L^{\infty} (τ, T; D (A^{1 / 4}))$ .

Proof.
See Section 4.3. □

∙ Existence of tempered pullback attractors

Denoting $\begin{array}{rcl} (3.8) & \underset{τ \to - \infty}{lim sup} \frac{1}{t - τ} \int_{τ}^{t} α (r) d r = α_{0} \in [0, + \infty), \end{array}$ for arbitrary $t \in R$ , then there exists some $δ > 0$ such that $\begin{array}{rcl} (3.9) & \frac{e^{ν λ_{1} h} α_{0}}{ν λ_{1} (1 - ρ^{})} + \frac{C e^{\frac{ν λ_{1} h}{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} + δ < ν λ_{1} . \end{array}$

Assume that $\begin{array}{rcl} (3.10) & κ_{δ} (t, s) = (ν λ_{1} - \frac{C e^{\frac{ν λ_{1} h}{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} - δ) (t - s) - \frac{e^{ν λ_{1} h}}{ν λ_{1} (1 - ρ^{})} \int_{s}^{t} α (r) d r, \end{array}$ where $\begin{array}{rcl} (3.11) & κ_{δ} (0, t) - κ_{δ} (0, s) = - κ_{δ} (t, s) \end{array}$ and $\begin{array}{rcl} (3.12) & \begin{matrix} κ_{δ} (0, r) ⩽ κ_{δ} (0, t) + (ν λ_{1} - \frac{C e^{\frac{ν λ_{1} h}{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} - δ) h, \\ if ν λ_{1} - \frac{C e^{\frac{ν λ_{1} h}{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} - δ > 0 \end{matrix} \end{array}$ for $r \in [t - h, t]$ .

The function $β (\cdot)$ satisfies the pullback tempered condition $\begin{array}{rcl} (3.13) & \int_{- \infty}^{t} e^{- κ_{δ} (t, s)} β (s) d s < + \infty . \end{array}$

The existence of a minimal family of pullback attractors for problem (3.4) can be stated as follows.
Theorem 3.6 (Tempered pullback dynamics).

Suppose that $f : R \times C_{H} \to H$ satisfies the hypotheses (H-a)–(H-d), and let the functions $α (\cdot)$ and $β (\cdot)$ satisfy (H-e)–(H-f). Then for any $(v (τ), η) \in M_{H}$ , the process $(S (t, τ); M_{H})$ generated by the global weak solutions of problem ( 3.4 ) possesses a minimal family of tempered pullback attractors $A_{κ_{δ}}$ in $H \times C_{H}$ , for all $κ_{δ} \in (0, κ_{δ} (t, τ)]$ . Moreover, if we choose fixed $κ_{δ}^{F}$ for a fixed universe to obtain the pullback attractors $A_{κ_{δ}^{F}}^{F}$ , then we have the relation $A_{κ_{δ}^{F}}^{F} \subset A_{κ_{δ}} \subset A_{κ_{δ} (t, τ)} \subset D_{κ_{δ} (t, τ)}^{H \times C_{H}}$ .

Proof.
Combining the existence of tempered pullback absorbing sets with the asymptotic compactness of the process in Theorem 5.10 and using the abstract theory of pullback attractors in [10], since the universe $D_{κ_{δ}}^{M_{H}}$ is inclusion closed, we can conclude on the existence of a minimal family of pullback attractors in $H \times C_{H}$ . The detail can be seen in Section 5.2. □

∙ Existence of $D$ -pullback attractors
Theorem 3.7.
Assume that $(v (τ), η) \in M_{H}$ and $\frac{ν}{4} > \frac{C e^{\frac{ν λ_{1}}{2} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} + \frac{4 e^{\frac{ν λ_{1}}{2} h} ‖ α (t) ‖_{L^{\infty} (τ, T)}}{ν (1 - ρ^{})}$ . Then the process $S (t, τ) : M_{H} \to M_{H}$ generated by the system ( 3.4 ) possesses a minimal family of $D$ -pullback attractors $A = {A (t)}_{t \in R}$ in $M_{H}$ .
Proof.
See Section 5.2. □
Remark 3.8.
The process $S^{h} (t, τ)$ generated by the system (3.4) possesses the $D$ -pullback attractor $A^{h} = P_{2} A = {A^{h} (t)}_{t \in R}$ in $C_{H} \cap L_{V}^{2}$ for (3.4) with $h > 0$ . Recalling the results for 2D classical incompressible Navier–Stokes equations in [38], i.e., $h = 0$ in (3.4), the existence of a minimal family of pullback attractors $A^{0} = {A^{0} (t)}_{t \in R}$ in H for the process $S^{0} (t, τ)$ follows.

∙ Robustness: continuity of $D$ -pullback attractors

Let Λ be a metric space, and ${A^{λ}}_{λ \in Λ}$ be a family of subsets in X. Then it is said that ${A^{λ}}$ has the property of upper semi-continuity as $λ \to λ_{0}$ in X if $\begin{matrix} lim_{λ \to λ_{0}} {dist}_{X} (A^{λ}, A^{λ_{0}}) = 0 . \end{matrix}$ The upper semi-continuity of pullback attractors and related conclusions can be seen in [10] for more details. Inspired from [4,5,10,24,36], we state the robustness of pullback attractors for problem (3.4) as follows.

We intend to establish some results on the convergence between the $D$ -pullback attractors $A^{h}$ to system (3.4) and $A^{0}$ to system (3.14) as $h \to 0$ .

When $h = 0$ , the system (3.4) can be reduced to the following system: $\begin{matrix} (3.14) & \{\begin{matrix} \frac{\partial v}{\partial t} + ν A v + B (v) + B (v, ψ) + B (ψ, v) \\ = P_{L} (g (t, x) + f (t, u) + ν F) - B (ψ), & (t, x) \in Ω_{τ}, \\ div v = 0, & (t, x) \in Ω_{τ}, \\ v = 0, & (t, x) \in \partial Ω_{τ}, \\ v (τ, x) = v (τ), & x \in Ω . \end{matrix} \end{matrix}$ Theorem 3.9 (Robustness of pullback dynamics).

Let $(v (τ), η) \in M_{H}$ , $A^{h}$ be the family of $D$ -pullback attractors of $S^{h} (t, τ)$ in $C_{H} \cap L_{V}^{2}$ to system ( 3.4 ), and $A^{0}$ be the $D$ -pullback attractor of $S^{0} (t, τ)$ in H to system ( 3.14 ). Then we have the upper semi-continuity of the pullback attractors as $\begin{matrix} lim_{h \to 0} {dist}_{H} (A^{h}, A^{0}) = 0 . \end{matrix}$

Proof.
See Section 6. □

3.4. Back to the original problem (1.1)

Considering (1.1), by using the estimates on the stream function for the Stokes problem in a Lipschitz-like domain, we can give the corresponding well-posedness and pullback dynamics results.

Definition 3.10.
Let $(u (τ), ϕ) \in M_{H}$ , $φ \in L^{\infty} (\partial Ω)$ and $φ \cdot n = 0$ on $\partial Ω$ . Then u is called a global weak solution of (1.1) if
$\forall T > 0$ , $u \in C ([τ - h, T]; H)$ , $u (τ, \cdot) = u (τ)$ , and $d u / d t \in L^{2} (τ, T; V^{'})$ .

For all $χ \in C_{0}^{\infty} (Ω)$ with $div χ = 0$ , one has $\begin{matrix} \frac{d}{d t} (u, χ) + ν ⟨ A u, χ ⟩ + b (u, u, v) = ⟨ g (t), χ ⟩ + (f (u (t - ρ (t))), χ) for a.e. [τ, + \infty) . \end{matrix}$

There exist $ψ \in C^{2} (Ω) \cap L^{\infty} (Ω)$ and $q \in C^{1} (Ω)$ such that $\begin{matrix} \{\begin{matrix} - △ ψ + \nabla q = 0, & in Ω, \\ div ψ = 0, & in Ω, \\ ψ = φ & on \partial Ω, \end{matrix} \end{matrix}$ where we assume that ψ can reach its boundary values in the sense of non-tangential convergence and $u - ψ \in L^{2} (τ, T; V)$ .

Theorem 3.11 (Well-posedness).

Let $(u (τ), ϕ) \in M_{H}$ , $ϕ \in L_{V}^{2}$ , $φ \in L^{\infty} (\partial Ω)$ , $φ \cdot n = 0$ on $\partial Ω$ and the assumptions of Theorem 3.3 hold. Then ( 1.1 ) has a unique global weak solution which generates a continuous process ${U (t, τ)}$ in $M_{H}$ .

Proof.
By using the same procedure as in [37,38] and [39], we can easily derive the conclusion. □
Theorem 3.12.
Suppose that $f : R \times C_{H} \to H$ satisfies the hypotheses (H-a)–(H-d). Moreover, let the functions $α (\cdot)$ and $β (\cdot)$ satisfy (H-e)–(H-f). Then for any $(u (τ), ϕ) \in M_{H}$ , the family of process ${U (t, τ)}$ generated by the global weak solutions of problem ( 1.1 ) possesses minimal families of $D$ -pullback attractors $\tilde{A}$ in $M_{H}$ , which consist of all bounded complete trajectories and especially the equilibria of the Stokes problem in a Lipschitz-like domain.
Proof.
Since we use a linear transformation to deal with non-homogeneous boundary conditions in Lipschitz-like domains and the background function $ψ (x)$ is a solution of the Stokes problem which is bounded in H, then we easily verify the existence of pullback attractor for problem (1.1). □
Remark 3.13.
By using the theory of upper semi-continuity for pullback attractors, from Theorems 3.12 and 3.9, the robustness of the pullback attractors $\tilde{A}$ still holds, i.e., the two systems have corresponding well-posedness and robustness properties.

4. Global well-posedness and regularity

4.1. Existence of a global weak solution: Proof of Theorem 3.3

Step 1: Existence of a local approximated solution

Let ${e_{1}, e_{2}, \dots, e_{j}, \dots}$ be the basis of H, which are also the eigenfunctions of the Stokes operator A corresponding to the eigenvalues ${λ_{j}}_{j = 1}^{\infty}$ , $H_{m} = span {e_{1}, e_{2}, \dots, e_{m}}$ , and $P_{m} : H \to H_{m}$ be the Helmhotz–Galerkin projection. We construct the approximated solution to (3.4) in H as $\begin{matrix} v_{m} (t) = \sum_{j = 1}^{m} γ_{m j} (t) e_{j}, \end{matrix}$ where $γ_{m j} (t) = (v_{m} (t), e_{j})$ is to be determined. Then we see that $v_{m} (t)$ satisfies the following Cauchy problem for a retarded differential equation: $\begin{array}{rcl} (4.1) & \{\begin{matrix} \frac{d}{d t} (v_{m} (t), e_{j}) + ν ⟨ A v_{m}, e_{j} ⟩ + b (v_{m} (s - ρ (s)), v_{m}, e_{j}) \\ + b (v_{m} (s - ρ (s)), ψ, e_{j}) + b (ψ, v_{m}, e_{j}) \\ = ⟨ P_{m} P_{L} g (t, x), e_{j} ⟩ + (P_{m} P_{L} (f (t, u_{m} (t - ρ (t))) + ν F), e_{j}) - b (ψ, ψ, e_{j}), \\ v_{m} (τ + θ, x) = η_{m} (θ), (θ, x) \in Ω_{h}, \end{matrix} \end{array}$ where $\begin{matrix} η_{m} (θ) = \sum_{j = 1}^{m} γ_{m j} e_{j} \to η (θ) \end{matrix}$ and $\begin{matrix} P_{m} P_{L} (g (t, x) + f (t, u (t - ρ (t))) + ν F) \to P_{L} (g (t, x) + f (t, u (t - ρ (t))) + ν F) (m \to + \infty) . \end{matrix}$ Since $f (\cdot, \cdot)$ is Lipschitz continuous, hence from the local existence theory of functional differential equations in [18], we know that the system (4.1) has a local solution on the interval $[τ - h, t_{m})$ for $t_{m} \in (τ, + \infty)$ .

Step 2: A priori estimates

Multiplying (4.1) by $γ_{m j} (t)$ and then summing from $j = 1$ to m, we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} | v_{m} |^{2} + ν ‖ v_{m} ‖^{2} \\ ⩽ | b (v_{m} (t - ρ (t)), ψ, v_{m}) | + | b (ψ, ψ, v_{m}) | + | ⟨ g, v_{m} ⟩ | \\ (4.2) & + | (f (t, u_{m} (t - ρ (t))) + ν F, v_{m}) | . \end{array}$ Using the Hardy and Hölder inequalities, we can derive that $\begin{array}{l} (4.3) & \begin{matrix} | b (v_{m} (t - ρ (t)), ψ, v_{m}) | & ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)} \int_{dist (x, \partial Ω) ⩽ C_{2}^{'} ε} \frac{| v_{m} (t) | | v_{m} (t - ρ (t)) |}{dist (x, \partial Ω)} d x \\ ⩽ \frac{ν}{10} ‖ v_{m} ‖^{2} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν} {| v_{m} (t - ρ (t)) |}^{2}, \end{matrix} \\ (4.4) & \begin{matrix} | b (ψ, ψ, v_{m}) | & ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)} \int_{Ω} \frac{| v_{m} |}{dist (x, \partial Ω)} | ψ | d x \\ ⩽ C ε^{1 / 2} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} | \partial Ω |^{1 / 2} ‖ v_{m} ‖ \\ ⩽ \frac{ν}{10} ‖ v_{m} ‖^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν} \end{matrix} \end{array}$ and $\begin{array}{rcl} ν | ⟨ F, v_{m} ⟩ | & ⩽ & ν | F | \cdot {(\int_{Ω} \frac{| v_{m} |^{2}}{{[dist (x, \partial Ω)]}^{2}} d x)}^{1 / 2} \cdot C ε \\ ⩽ & \frac{C ν}{\sqrt{ε}} ‖ φ ‖_{L^{2} (\partial Ω)} ‖ v_{m} ‖ \\ (4.5) & ⩽ & \frac{ν}{10} ‖ v_{m} ‖^{2} + \frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} \end{array}$ as $‖ φ ‖_{L^{2} (\partial Ω)} ⩽ C | \partial Ω |^{1 / 2} ‖ φ ‖_{L^{\infty} (\partial Ω)}$ from [3]. Moreover, $\begin{matrix} (4.6) & | ⟨ g (t), v_{m} ⟩ | ⩽ \frac{ν}{10} ‖ v_{m} ‖^{2} + \frac{C}{ν} {‖ g (t) ‖}_{H^{- 1}}^{2} . \end{matrix}$ By hypotheses (H-a)–(H-d) and the Minkowski inequality, we obtain that $\begin{array}{l} (f (t, u_{m} (t - ρ (t))), v_{m} (t)) \\ ⩽ | f (t, v_{m} (t - ρ (t)) + ψ) | | v_{m} (t) | \\ ⩽ α^{\frac{1}{2}} (t) | v_{m} (t - ρ (t)) | | v_{m} (t) | + α^{\frac{1}{2}} (t) | ψ | | v_{m} (t) | + β^{\frac{1}{2}} (t) | v_{m} (t) | \\ ⩽ \frac{ν (1 - ρ^{*})}{2} {| v_{m} (t - ρ (t)) |}^{2} + \frac{α (t)}{2 ν (1 - ρ^{*})} {| v_{m} (t) |}^{2} + \frac{ν}{10} {‖ v_{m} (t) ‖}^{2} \\ (4.7) & + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} α (t) + \frac{10 | Ω |}{ν λ_{1}} β (t) \end{array}$ and $\begin{array}{rcl} \int_{τ}^{t} {| v_{m} (s - ρ (s)) |}^{2} d s & ⩽ & \frac{1}{(1 - ρ^{*})} \int_{- ρ (0)}^{t - ρ (t)} {| v_{m} (s) |}^{2} d s \\ (4.8) & ⩽ & \frac{1}{λ_{1} (1 - ρ^{*})} {‖ η_{m} (t) ‖}_{L_{V}^{2}}^{2} + \frac{1}{(1 - ρ^{*})} \int_{τ}^{t} {| v_{m} (s) |}^{2} d s . \end{array}$ Combining (4.2)–(4.7), we get $\begin{array}{l} \frac{1}{2} \frac{d}{d t} | v_{m} |^{2} + ν ‖ v_{m} ‖^{2} \\ ⩽ [\frac{ν (1 - ρ^{*})}{2} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν}] {| v_{m} (t - ρ (t)) |}^{2} + \frac{α (t)}{2 ν (1 - ρ^{*})} {| v_{m} (t) |}^{2} + \frac{C}{ν} {‖ g (t) ‖}_{H^{- 1}}^{2} \\ (4.9) & + \frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν} + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} α (t) + \frac{10 | Ω |}{ν λ_{1}} β (t) . \end{array}$ Since $ν λ_{1} - \frac{1}{ν (1 - ρ^{*})} ‖ α (t) ‖_{L^{\infty}} > 0$ , integrating (4.9) over $[τ, t]$ and using the Poincaré and Gronwall inequalities, we have $\begin{array}{rcl} (4.10) & {| v_{m} (t) |}^{2} ⩽ C ({| v_{m} (τ) |}^{2} + ‖ η_{m} ‖_{L_{V}^{2}}^{2} + {\tilde{β}}_{0}) + C_{1} N_{2} \end{array}$ and $\begin{array}{rcl} (4.11) & N_{1} \int_{τ}^{t} {‖ v_{m} (s) ‖}^{2} d s ⩽ C ({| v_{m} (τ) |}^{2} + ‖ η_{m} ‖_{L_{V}^{2}}^{2} + {\tilde{β}}_{0}) + N_{2} (T - τ + C_{1}), \end{array}$ where $N_{1} = ν - \frac{‖ α (t) ‖_{L^{\infty}}}{ν λ_{1} (1 - ρ^{*})}$ and $N_{2} = \frac{C ν | \partial Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ε} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν} + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} ‖ α (t) ‖_{L^{\infty}}}{ν λ_{1}}$ . Hence, we can conclude that $\begin{matrix} (4.12) & {v_{m}} is uniformly bounded in L^{\infty} (τ - h, T; H) \cap L^{2} (τ - h, T; V) \end{matrix}$ for any $T ⩾ τ$ , i.e., $\begin{matrix} (4.13) & v_{m} \in L^{\infty} (τ - h, T; H) \cap L^{2} (τ - h, T; V), \forall T > τ . \end{matrix}$

Step 3: Compactness argument.

From (3.4), we know that the approximated solution satisfies $\begin{array}{rcl} \frac{\partial v_{m}}{\partial t} & = & - ν A v_{m} - B (v_{m} (t - ρ (t)), v_{m}) - B (v_{m} (t - ρ (t)), ψ) - B (ψ, v_{m}) \\ (4.14) & + P_{m} P_{L} (g (t, x) + f (t, u_{m} (t - ρ (t))) + ν F) - B (ψ) . \end{array}$ Since ${v_{m}} \in L^{\infty} (τ - h, T; H) \cap L^{2} (τ - h, T; V)$ , we deduce that $- ν A v_{m} \in L^{2} (τ, T; V^{'})$ , $\begin{array}{rcl} {‖ B (v_{m} (t - ρ (t)), v_{m}) ‖}_{L^{2} (τ, T; V^{'})}^{2} & = & \int_{τ}^{T} sup_{‖ w ‖ = 1} {| b (v_{m} (t - ρ (t)), v_{m}, w) |}^{2} d s \\ ⩽ & C {‖ v_{m} (t - ρ (t)) ‖}_{L^{\infty} (τ, T; H)}^{2} ‖ v_{m} ‖_{L^{2} (τ, T; V)}^{2} ‖ w ‖_{L^{2} (τ, T; H)}^{2} \\ (4.15) & ⩽ & C {‖ v_{m} (t - ρ (t)) ‖}_{L^{\infty} (τ, T; H)}^{2} ‖ v_{m} ‖_{L^{2} (τ, T; V)}^{2} \end{array}$ and $\begin{array}{rcl} {‖ B (v_{m} (t - ρ (t)), ψ) ‖}_{L^{2} (τ, T; V^{'})}^{2} & = & \int_{τ}^{T} sup_{‖ w ‖ = 1} {| b (v_{m} (t - ρ (t)), ψ, w) |}^{2} d s \\ ⩽ & C ε^{2} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} {‖ v_{m} (t - ρ (t)) ‖}_{L^{2} (τ, T; V)}^{2} ‖ w ‖_{L^{2} (τ, T; V)}^{2} \\ (4.16) & ⩽ & C ε^{2} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} {‖ v_{m} (t - ρ (t)) ‖}_{L^{2} (τ, T; V)}^{2}, \end{array}$ which implies that $B (v_{m} (t - ρ (t)), v_{m}), B (v_{m} (t - ρ (t)), ψ) \in L^{2} (τ, T; V^{'})$ . Similarly we have $\begin{array}{rcl} (4.17) & B (ψ, v_{m}), B (ψ) \in L^{2} (τ, T; V^{'}) . \end{array}$ Concerning the term $f (t, u_{m} (t - ρ (t)))$ , we have $\begin{array}{l} \int_{τ}^{T} {‖ f (t, u_{m} (t - ρ (t))) ‖}_{V^{'}}^{2} d t \\ = \int_{τ}^{T} {(sup_{‖ w ‖ = 1} | ⟨ f (t, u_{m} (t - ρ (t))), w ⟩ |)}^{2} d t \\ ⩽ C [\int_{τ}^{T} α (t) {| u_{m} (t - ρ (t)) |}^{2} d t + ‖ β ‖_{L^{1} (τ, T)}] \\ (4.18) & ⩽ C [‖ α ‖_{L^{\infty} (τ, T)} ‖ v_{m} ‖_{L^{2} (τ - h, T; H)}^{2} + ‖ α ‖_{L^{\infty} (τ, T)} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + ‖ β ‖_{L^{1} (τ, T)}], \end{array}$ which means that $f (t, u_{m} (t - ρ (t))) \in L^{2} (τ, T; V^{'})$ . Combining (4.15)–(4.18) with (4.14), it follows that $\frac{\partial v_{m}}{\partial t} \in L^{2} (τ, T; V^{'})$ . By the compact embedding $V ↪ ↪ H ↪ V^{'}$ , using the Aubin–Lions Lemma, we obtain that ${v_{m}}$ has a strongly convergent subsequence (also called ${v_{m}}$ without confusion) in $L^{2} (τ, T; H)$ and $\begin{matrix} (4.19) & v_{m} (t) \to v (t), in H a.e. t ⩾ τ . \end{matrix}$

Step 4: Passage to the limit

The uniform priori estimates together with the Aubin–Lions lemma lead to the existence of functions $v \in L^{\infty} (τ - h, T; H) \cap L^{2} (τ - h, T; V)$ , $v_{1} \in L^{2} (τ, T; V^{'})$ , and $v_{2} \in L^{2} (τ, T; V^{'})$ such that $\begin{array}{rcl} (4.20) & \{\begin{matrix} v_{m} ⇀ v, & weakly star in L^{\infty} (τ - h, T; H), \\ v_{m} ⇀ v, & weakly in L^{2} (τ - h, T; V), \\ A v_{m} ⇀ v_{1}, & weakly in L^{2} (τ - h, T; V^{'}), \\ \frac{\partial}{\partial t} v_{m} ⇀ v_{2}, & weakly in L^{2} (τ, T; V^{'}), \\ v_{m} \to v, & strongly in L^{2} (τ, T; H), \\ v_{m} (s) \to v (s) & a.e. in Ω_{T} . \end{matrix} \end{array}$ From the compactness argument, we can show that $v_{1} = A v$ , $v_{2} = \frac{\partial}{\partial t} v$ . Moreover, the Lions–Magenes theorem together with the above convergences implies that $v_{m} \in C ([τ, T]; H)$ .

Then, from (4.20) and the Sobolev interpolation inequality we can obtain that $\begin{matrix} {‖ v_{m} (t) ‖}_{L^{4} (Ω)} ⩽ C {‖ v_{m} (t) ‖}^{1 / 2} \cdot {| v_{m} (t) |}^{1 / 2} ⩽ C {‖ v_{m} (t) ‖}^{1 / 2}, \end{matrix}$ and $v_{m} (t) \in L^{4} (τ - h, T; L^{4} (Ω))$ , i.e., $v_{m} (t - ρ (t)) v_{m} (t)$ is bounded in $L^{2} (τ, T; H)$ , which means that there exists a subsequence (also relabelled as ${v_{m}}$ ) such that $\begin{matrix} v_{m} (t - ρ (t)) v_{m} (t) ⇀ v (t - ρ (t)) v (t), weakly in L^{2} (τ, T; H) . \end{matrix}$ In addition, $v_{m} (t - ρ (t)) v_{m} (t) ⇀ v (t - ρ (t)) v (t)$ weakly in $L^{2} (Ω_{T})$ , and for any $w \in V$ there holds $\begin{array}{rcl} \int_{τ}^{T} b (v_{m} (t - ρ (t)), v_{m}, w) d t & \to & - \sum_{i, j = 1}^{2} \int_{τ}^{T} \int_{Ω} v_{i} (t - ρ (t)) D_{i} w_{j} v_{j} d x d t \\ (4.21) & = & \int_{τ}^{T} b (v (t - ρ (t)), v, w) d t . \end{array}$ Since $\begin{array}{rcl} {‖ v_{m} (t - ρ (t)) ψ ‖}_{L^{2} (τ, T; H)}^{2} & ⩽ & C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} \int_{τ}^{T} {| v_{m} (t - ρ (t)) |}^{2} d t \\ (4.22) & ⩽ & C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} {‖ v_{m} (t - ρ (t)) ‖}_{L^{2} (τ, T; H)}^{2}, \end{array}$ we have $v_{m} (t - ρ (t)) ψ \in L^{2} (τ, T; H)$ , and hence $ψ v_{m} (t), ψ ψ \in L^{2} (τ, T; H)$ . Using the same technique, we can also obtain that for any $w \in V$ $\begin{matrix} \{\begin{matrix} \int_{τ}^{T} b (v_{m} (t - ρ (t)), ψ, w) d t \to \int_{τ}^{T} b (v (t - ρ (t)), ψ, w) d t, \\ \int_{τ}^{T} b (ψ, v_{m}, w) d t \to \int_{τ}^{T} b (ψ, v, w) d t, \\ \int_{τ}^{T} ⟨ f (t, u_{m} (t - ρ (t))), w ⟩ d t \to \int_{τ}^{T} ⟨ f (t, u (t - ρ (t))), w ⟩ d t . \end{matrix} \end{matrix}$

From (4.20), we can also show that for any $w \in V$ $\begin{array}{rcl} (4.23) & (v_{m} (t), w) + ν \int_{τ}^{T} ⟨ A v_{m} (t), w ⟩ d t \to (v_{m} (t), w) + ν \int_{τ}^{T} ⟨ A v_{m} (t), w ⟩ d t, \end{array}$ and passing to the limit in (4.1), we can conclude that v satisfies (3.5). Moreover, from Lemma 2.3 we can also derive that $\begin{matrix} v_{m} (τ) = P_{m} v (τ) ⇀ v (τ), weakly in V . \end{matrix}$

4.2. Uniqueness: Proof of Theorem 3.4

Denote by $v_{1} (t; τ, η)$ , $v_{2} (t; τ, η)$ two global weak solutions to system (3.4) corresponding to the initial data $(v_{1} (τ), η_{1})$ and $(v_{2} (τ), η_{2})$ respectively. Let $w (t) = v_{1} (t) - v_{2} (t)$ . Then $w (t)$ satisfies the system $\begin{array}{l} \frac{\partial w}{\partial t} + ν A w + B (w (t - ρ (t)), v_{1}) + B (v_{2} (t - ρ (t)), w) + B (w (t - ρ (t)), ψ) + B (ψ, w) \\ (4.24) & = P_{L} (f (t, v_{1} (t - ρ (t)) + ψ) - f (t, v_{2} (t - ρ (t)) + ψ)), \end{array}$ with the initial data $w (τ) = v_{1} (τ) - v_{2} (τ)$ and $\tilde{η} (θ) = η_{1} (θ) - η_{2} (θ)$ .

Multiplying (4.24) by w, we derive $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} | w |^{2} + ν ‖ w ‖^{2} & ⩽ & | b (w (t - ρ (t)), v_{1}, w) | + | b (w (t - ρ (t)), ψ, w) | \\ (4.25) & + | (f (t, v_{1} (t - ρ (t)) + ψ) - f (t, v_{2} (t - ρ (t)) + ψ), w) | . \end{array}$

Case 1: For the interval $[t - h, t]$ .

By a similar technique as in the proof of Theorem 3.3, we have $\begin{array}{l} | b (w (t - ρ (t)), v_{1}, w) | \\ ⩽ C {| w (t - ρ (t)) |}^{1 / 2} {‖ w (t - ρ (t)) ‖}^{1 / 2} ‖ v_{1} ‖ ‖ w ‖^{1 / 2} | w |^{1 / 2} \\ ⩽ C \underset{r \in [t - h, t]}{esssup} | w (r) | ‖ v_{1} ‖ {‖ w (t - ρ (t)) ‖}^{1 / 2} {‖ w (t) ‖}^{1 / 2} \\ ⩽ \frac{C}{ε} \underset{r \in [t - h, t]}{esssup} {| w (r) |}^{2} ‖ v_{1} ‖^{2} + ε ‖ w (t - ρ (t)) ‖ ‖ w (t) ‖ \\ (4.26) & ⩽ \frac{C}{ε} \underset{r \in [t - h, t]}{esssup} {| w (r) |}^{2} ‖ v_{1} ‖^{2} + \frac{5 ε^{2}}{2 ν} {‖ w (t - ρ (t)) ‖}^{2} + \frac{ν}{10} {‖ w (t) ‖}^{2} \end{array}$ and $\begin{array}{rcl} | b (w (t - ρ (t)), ψ, w) | & ⩽ & C ε ‖ φ ‖_{L^{\infty} (\partial Ω)} \int_{dist (x, \partial Ω) ⩽ C_{2}^{'} ε} \frac{| w (t) | | w (t - ρ (t)) |}{{[dist (x, \partial Ω)]}^{2}} d x \\ ⩽ & C ε ‖ φ ‖_{L^{\infty} (\partial Ω)} (\int_{Ω} \frac{| w |^{2}}{{[dist (x, \partial Ω)]}^{2}} d x + \int_{Ω} \frac{| w (t - ρ (t)) |^{2}}{{[dist (x, \partial Ω)]}^{2}} d x) \\ (4.27) & ⩽ & C ε ‖ φ ‖_{L^{\infty} (\partial Ω)} (‖ w ‖^{2} + {‖ w (t - ρ (t)) ‖}^{2}) . \end{array}$ Hence, it follows that $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {| w (t) |}^{2} + ν {‖ w (t) ‖}^{2} \\ ⩽ \frac{C}{ε} \underset{r \in [t - h, t]}{esssup} {| w (r) |}^{2} ‖ v_{1} ‖^{2} + \frac{5 ε^{2}}{2 ν} {‖ w (t - ρ (t)) ‖}^{2} + \frac{ν}{10} {‖ w (t) ‖}^{2} \\ + C ε ‖ φ ‖_{L^{\infty} (\partial Ω)} (‖ w ‖^{2} + {‖ w (t - ρ (t)) ‖}^{2}) \\ (4.28) & + \frac{ν}{10} {‖ w (t) ‖}^{2} + \frac{10}{ν λ_{1}} L^{2} (r) γ (t) {| w (t - ρ (t)) |}^{2} . \end{array}$ Integrating (4.28) from τ to t, this yields $\begin{array}{l} \frac{1}{2} ({| w (t) |}^{2} - {| w (τ) |}^{2}) + ν \int_{τ}^{t} {‖ w (s) ‖}^{2} d s \\ ⩽ \frac{C}{ε} \int_{τ}^{t} \underset{r \in [s - h, s]}{esssup} {| w (r) |}^{2} {‖ v_{1} (s) ‖}^{2} d s + \frac{5 ε^{2}}{2 ν} \int_{τ}^{t} {‖ w (s - ρ (s)) ‖}^{2} d s + \frac{ν}{10} \int_{τ}^{t} {‖ w (s) ‖}^{2} d s \\ + C ε ‖ φ ‖_{L^{\infty} (\partial Ω)} \int_{τ}^{t} {‖ w (s) ‖}^{2} d s + C ε ‖ φ ‖_{L^{\infty} (\partial Ω)} \int_{τ}^{t} {‖ w (s - ρ (s)) ‖}^{2} d s \\ (4.29) & + \frac{ν}{10} \int_{τ}^{t} {‖ w (s) ‖}^{2} d s + \frac{10}{ν λ_{1}} L^{2} (r) \int_{τ}^{t} γ (s) {| w (s - ρ (s)) |}^{2} d s . \end{array}$ Choosing ε satisfying $max {C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}, \frac{5 ε^{2}}{2 ν (1 - ρ^{*})}, \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}}{1 - ρ^{*}}} ⩽ \frac{ν}{10}$ , we get $\begin{array}{l} {| w (t) |}^{2} + ν \int_{τ}^{t} {‖ w (s) ‖}^{2} d s \\ (4.30) & ⩽ {| w (τ) |}^{2} + C ‖ η_{1} - η_{2} ‖_{L_{V}^{2}}^{2} + \int_{τ}^{t} (C L^{2} (r) γ (s) + C {‖ v_{1} (s) ‖}^{2}) \underset{σ \in [s - h, s]}{esssup} {| w (σ) |}^{2} d s \end{array}$ and $\begin{array}{l} \underset{σ \in [t - h, t]}{esssup} {| w (σ) |}^{2} \\ (4.31) & ⩽ | w_{τ} |_{C_{H}}^{2} + C ‖ η_{1} - η_{2} ‖_{L_{V}^{2}}^{2} + \int_{τ}^{t} (C L^{2} (r) γ (s) + C {‖ v_{1} (s) ‖}^{2}) \underset{σ \in [s - h, s]}{esssup} {| w (σ) |}^{2} d s . \end{array}$ The Gronwall inequality shows that $\begin{array}{rcl} (4.32) & \underset{r \in [t - h, t]}{esssup} {| w (r) |}^{2} ⩽ (| w_{τ} |_{C_{H}}^{2} + C ‖ η_{1} - η_{2} ‖_{L_{V}^{2}}^{2}) e^{\int_{τ}^{t} (C L^{2} (r) γ (s) + C ‖ v_{1} (s) ‖^{2}) d s}, \end{array}$ which implies the uniqueness and the continuous dependence with respect to the initial data.

Case 2: When t is large.

By using the estimate of the trilinear operators, we have $\begin{array}{l} (4.33) & \begin{matrix} | b (w (t - ρ (t)), v_{1}, w) | & ⩽ C | w (t - ρ (t)) | ‖ w ‖ ‖ v_{1} ‖_{L^{\infty}} \\ ⩽ \frac{C}{ν} ‖ v_{1} ‖^{2} {| w (t - ρ (t)) |}^{2} + \frac{ν}{6} {‖ w (t) ‖}^{2}, \end{matrix} \\ (4.34) & | b (w (t - ρ (t)), ψ, w) | ⩽ \frac{ν}{6} ‖ w ‖^{2} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν} {| w (t - ρ (t)) |}^{2} \end{array}$ and $\begin{array}{l} | (f (t, v_{1} (t - ρ (t)) + ψ) - f (t, v_{2} (t - ρ (t)) + ψ), w) | \\ (4.35) & ⩽ \frac{ν}{6} {‖ w (t) ‖}^{2} + \frac{3}{2 ν λ_{1}} L^{2} (r) γ (t) {| w (t - ρ (t)) |}^{2} . \end{array}$ Hence, it follows that $\begin{array}{rcl} (4.36) & \frac{d}{d t} {| w (t) |}^{2} + ν {‖ w (t) ‖}^{2} ⩽ [\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{3}{ν λ_{1}} L^{2} (r) γ (t) + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν}] {| w (t - ρ (t)) |}^{2} . \end{array}$ By using the Poincaré inequality, we have $\begin{array}{rcl} {| w (t) |}^{2} & ⩽ & {| w (τ) |}^{2} e^{- ν λ_{1} (t - τ)} \\ (4.37) & + \int_{τ}^{t} e^{- ν λ_{1} (t - s)} [\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{3}{ν λ_{1}} L^{2} (r) γ (t) + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν}] {| v_{m} (s - ρ (s)) |}^{2} d s . \end{array}$ Considering the retarded integral-type Gronwall inequality in Lemma 2.2, setting $\begin{array}{l} (4.38) & E (t, τ) = e^{- ν λ_{1} (t - τ)}, \\ (4.39) & K_{1} (t, s) = e^{- ν λ_{1} (t - s)} [\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{3}{ν λ_{1}} L^{2} (r) γ (t) + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν}], \\ (4.40) & Θ = sup_{t ⩾ s ⩾ τ} E (t, s), \bar{κ} (K_{1}, 0) = sup_{t ⩾ τ} \int_{τ}^{t} K_{1} (t, s) d s, \bar{ρ} = 0, \end{array}$ and choosing a suitable large time leads to $\int_{τ}^{t} e^{- ν λ_{1} (t - s)} [\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{3}{ν λ_{1}} L^{2} (r) γ (t) + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν}] d s < \frac{1}{2}$ such that $\bar{κ} (K_{1}, 0) < \frac{1}{1 + Θ}$ . Then by virtue of Lemma 2.2, we derive that there exist $M, λ > 0$ such that $\begin{array}{rcl} (4.41) & {| w (t - ρ (t)) |}^{2} & ⩽ & M {| \tilde{η} (θ) |}^{2} e^{- λ t} . \end{array}$ Substituting (4.41) into (4.37), we can obtain that $\begin{array}{rcl} {| w (t) |}^{2} & ⩽ & {| v_{1} (τ) - v_{2} (τ) |}^{2} e^{- ν λ_{1} (t - τ)} \\ + M ‖ η_{1} - η_{2} ‖_{C_{H} \cap L_{V}^{2}}^{2} [\frac{C}{ν} ‖ v_{1} ‖_{L^{2} (τ, T; V)}^{2} + \frac{3 (T - τ)}{ν λ_{1}} L^{2} (r) {‖ γ (t) ‖}_{L^{\infty}} \\ (4.42) & + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} (T - τ)}{ν}], \end{array}$ which implies the uniqueness and continuous dependence on the initial data for large times.

In conclusion, combining the uniqueness of local solutions and the above continuous dependences on the initial data, the proof is completed.

4.3. Regularity of global solutions: Proof of Theorem 3.5

Step 1: A priori estimate.

From (4.1), we get $\begin{array}{l} \frac{1}{2} \frac{\partial}{\partial t} ⟨ v_{m}, A^{\frac{1}{2}} v_{m} ⟩ + ν ⟨ A v_{m}, A^{\frac{1}{2}} v_{m} ⟩ d s + b (v_{m} (t - ρ (t)), v_{m}, A^{\frac{1}{2}} v_{m}) \\ + b (v_{m} (t - ρ (t)), ψ, A^{\frac{1}{2}} v_{m}) + b (ψ, v_{m}, A^{\frac{1}{2}} v_{m}) + b (ψ, ψ, A^{\frac{1}{2}} v_{m}) \\ (4.43) & = ⟨ P_{m} P_{L} (f (t, u_{m} (t - ρ (t))) + ν F + g), A^{\frac{1}{2}} v_{m} ⟩ . \end{array}$ From the Sobolev inequality, Hölder and Young inequalities, we derive $\begin{array}{l} \frac{d}{d t} {| A^{\frac{1}{4}} v_{m} (t) |}^{2} + 2 ν {| A^{\frac{3}{4}} v_{m} |}^{2} \\ ⩽ 2 \int_{Ω} | v_{m} (t - ρ (t)) | | \nabla v_{m} | | A^{\frac{1}{2}} v_{m} | d x + 2 \int_{Ω} | v_{m} (t - ρ (t)) | | \nabla ψ | | A^{\frac{1}{2}} v_{m} | d x \\ + 2 \int_{Ω} | ψ | | \nabla v_{m} | | A^{\frac{1}{2}} v_{m} | d x + 2 \int_{Ω} | ψ | | \nabla ψ | | A^{\frac{1}{2}} v_{m} | d x \\ (4.44) & + 2 | f | | A^{\frac{1}{2}} v_{m} | + 2 ν | F | | A^{\frac{1}{2}} v_{m} | + 2 | g | | A^{\frac{1}{2}} v_{m} |, \end{array}$ where $\begin{array}{l} (4.45) & \begin{matrix} 2 \int_{Ω} | v_{m} (t - ρ (t)) | | \nabla v_{m} | | A^{\frac{1}{2}} v_{m} | d x \\ ⩽ C {‖ v_{m} (t - ρ (t)) ‖}_{L^{4}} ‖ v_{m} ‖ {‖ A^{\frac{1}{2}} v_{m} ‖}_{L^{4}} \\ ⩽ C | A^{\frac{1}{4}} v_{m} (t - ρ (t)) | ‖ v_{m} ‖ | A^{\frac{3}{4}} v_{m} | \\ ⩽ \frac{C}{ν} {| A^{\frac{1}{4}} v_{m} (t - ρ (t)) |}^{2} ‖ v_{m} ‖^{2} + \frac{ν}{7} {| A^{\frac{3}{4}} v_{m} |}^{2}, \end{matrix} \\ (4.46) & \begin{matrix} 2 \int_{Ω} | v_{m} (t - ρ (t)) | | \nabla ψ | | A^{\frac{1}{2}} v_{m} | d x \\ ⩽ C \sqrt{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)} \int_{dist (x, \partial Ω) ⩽ C_{2}^{'} ε} \frac{| A^{\frac{1}{2}} v_{m} (t) | | v_{m} (t - ρ (t)) |}{{[dist (x, \partial Ω)]}^{3 / 2}} d x \\ ⩽ C \sqrt{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)} (\int_{Ω} \frac{| A^{\frac{1}{2}} v_{m} |^{2}}{[dist (x, \partial Ω)]} d x + \int_{Ω} \frac{| v_{m} (t - ρ (t)) |^{2}}{{[dist (x, \partial Ω)]}^{2}} d x) \\ ⩽ C \sqrt{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)} ({‖ A^{\frac{3}{4}} v_{m} ‖}^{2} + {‖ v_{m} (t - ρ (t)) ‖}^{2}), \end{matrix} \\ (4.47) & \begin{matrix} 2 \int_{Ω} | ψ | | \nabla v_{m} | | A^{\frac{1}{2}} v_{m} | d x & ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)} {| A^{\frac{1}{2}} v_{m} |}^{2} \\ ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)} | A^{\frac{1}{4}} v_{m} | | A^{\frac{3}{4}} v_{m} | \\ ⩽ \frac{ν}{7} {| A^{\frac{3}{4}} v_{m} |}^{2} + \frac{C}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} {| A^{\frac{1}{4}} v_{m} |}^{2}, \end{matrix} \\ (4.48) & \begin{matrix} 2 \int_{Ω} | ψ | | \nabla ψ | | A^{\frac{1}{2}} v_{m} | d x & ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)} ‖ φ ‖_{L^{2} (\partial Ω)} | A^{\frac{3}{4}} v_{m} | \\ ⩽ \frac{ν}{7} {| A^{\frac{3}{4}} v_{m} |}^{2} + \frac{C | \partial Ω |}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4}, \end{matrix} \\ (4.49) & \begin{matrix} 2 ν | ⟨ F, A^{\frac{1}{2}} v_{m} ⟩ | & ⩽ 2 ν \cdot \int_{Ω} | F | | A^{\frac{1}{2}} v_{m} | d x \\ ⩽ ν | F | \cdot {(\int_{Ω} \frac{| A^{\frac{1}{2}} v_{m} |^{2}}{dist (x, \partial Ω)} d x)}^{1 / 2} \cdot C \sqrt{ε} \\ ⩽ \frac{C ν}{ε} ‖ φ ‖_{L^{2} (\partial Ω)} | A^{\frac{3}{4}} v_{m} | \\ ⩽ \frac{ν}{7} {| A^{\frac{3}{4}} v_{m} |}^{2} + \frac{C ν | \partial Ω |}{ε^{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}, \end{matrix} \\ (4.50) & 2 | g | | A^{\frac{1}{2}} v_{m} | ⩽ \frac{ν}{7} {| A^{\frac{3}{4}} v_{m} |}^{2} + \frac{C}{ν λ_{1}} | g |^{2}, \end{array}$ and $\begin{array}{l} 2 | (f (t, u_{m} (t - ρ (t))), A^{\frac{1}{2}} v_{m} (t)) | \\ ⩽ α^{\frac{1}{2}} (t) | v_{m} (t - ρ (t)) | | A^{\frac{1}{2}} v_{m} (t) | + α^{\frac{1}{2}} (t) | ψ | | A^{\frac{1}{2}} v_{m} (t) | + β^{\frac{1}{2}} (t) | A^{\frac{1}{2}} v_{m} (t) | \\ (4.51) & ⩽ \frac{ν}{7} {| A^{\frac{3}{4}} v_{m} |}^{2} + \frac{C α (t)}{ν λ_{1}} {| v_{m} (t - ρ (t)) |}^{2} + \frac{C α (t) | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} + \frac{C}{ν λ_{1}} β (t) . \end{array}$ Choosing suitably ε such that $C \sqrt{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)} ⩽ \frac{ν}{7}$ , and combining (4.45)–(4.51) with (4.44), we have $\begin{array}{l} \frac{d}{d t} {| A^{\frac{1}{4}} v_{m} |}^{2} + ν {| A^{\frac{3}{4}} v_{m} |}^{2} \\ ⩽ \frac{C}{ν} ‖ v_{m} ‖^{2} {| A^{\frac{1}{4}} v_{m} (t - ρ (t)) |}^{2} + \frac{ν}{7} {‖ v_{m} (t - ρ (t)) ‖}^{2} + \frac{C}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} {| A^{\frac{1}{4}} v_{m} |}^{2} \\ + \frac{C | \partial Ω |}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} + \frac{C ν | \partial Ω |}{ε^{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} \\ (4.52) & + \frac{C α (t)}{ν λ_{1}} {| v_{m} (t - ρ (t)) |}^{2} + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} α (t) + \frac{C}{ν λ_{1}} β (t) + \frac{C}{ν λ_{1}} | g |^{2} . \end{array}$ Integrating (4.52) over $[s, t]$ , we obtain that $\begin{array}{l} {| A^{\frac{1}{4}} v_{m} (t) |}^{2} + ν \int_{s}^{t} {| A^{\frac{3}{4}} v_{m} |}^{2} d l - {| A^{\frac{1}{4}} v_{m} (s) |}^{2} \\ ⩽ \frac{ν}{7} \int_{s}^{t} {‖ v_{m} (l - ρ (l)) ‖}^{2} d l + \int_{s}^{t} (\frac{C}{ν} ‖ v_{m} ‖^{2} + \frac{C}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) \underset{l_{1} \in [l - h, l]}{esssup} {| A^{\frac{1}{4}} v_{m} (l_{1}) |}^{2} d l \\ + (\frac{C | \partial Ω |}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} + \frac{C ν | \partial Ω |}{ε^{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) (t - s) \\ (4.53) & + \frac{C}{ν λ_{1}} \int_{s}^{t} (α (l) {| v_{m} (l - ρ (l)) |}^{2} + | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} α (l) + β (l)) d l + \frac{C}{ν λ_{1}} ‖ g ‖_{L^{2} (τ, T; H)}^{2}, \end{array}$ which means that $\begin{array}{l} \underset{l \in [t - h, t]}{esssup} {| A^{\frac{1}{4}} v_{m} (l) |}^{2} - {| A^{\frac{1}{4}} v_{m} (s) |}^{2} \\ ⩽ \frac{ν}{7} \int_{s}^{t} {‖ v_{m} (l - ρ (l)) ‖}^{2} d l + \int_{s}^{t} (\frac{C}{ν} ‖ v_{m} ‖^{2} + \frac{C}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) \underset{l_{1} \in [l - h, l]}{esssup} {| A^{\frac{1}{4}} v_{m} (l_{1}) |}^{2} d l \\ + (\frac{C | \partial Ω |}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} + \frac{C ν | \partial Ω |}{ε^{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) (t - s) \\ (4.54) & + \frac{C}{ν λ_{1}} \int_{s}^{t} (α (l) {| v_{m} (l - ρ (l)) |}^{2} + | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} α (l) + β (l)) d l + \frac{C}{ν λ_{1}} ‖ g ‖_{L^{2} (τ, T; H)}^{2} . \end{array}$ The Gronwall lemma leads to $\begin{array}{l} \underset{l \in [t - h, t]}{esssup} {| A^{\frac{1}{4}} v_{m} (l) |}^{2} \\ ⩽ [{| A^{\frac{1}{4}} v_{m} (s) |}^{2} + (\frac{C | \partial Ω |}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} + \frac{C ν | \partial Ω |}{ε^{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) (t - s) \\ + \frac{ν}{7} \int_{s}^{t} {‖ v_{m} (l - ρ (l)) ‖}^{2} d l \\ + \frac{C}{ν λ_{1}} \int_{s}^{t} (α (l) {| v_{m} (l - ρ (l)) |}^{2} + | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} α (l) + β (l)) d l] \\ (4.55) & \times e^{[\frac{C}{ν} \int_{s}^{t} (‖ v_{m} ‖^{2} + ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) d l + \frac{C}{ν λ_{1}} ‖ g ‖_{L^{2} (τ, T; H)}^{2}]} . \end{array}$ Moreover, proceeding as in the proof of the existence of a solution, we have $\begin{array}{rcl} (4.56) & N_{1} \int_{s}^{t} ‖ v_{m} (l) ‖ d l ⩽ C ({| v (τ) |}^{2} + ‖ η ‖_{L_{V}^{2}}^{2} + {\tilde{β}}_{0}) + N_{2} (t - s + C_{1}) . \end{array}$ Let $t - s = C_{1}$ . Using the same technique as in Theorem 4.11 in [3], we can show that, in any interval with length $C_{1}$ , there exists l such that $\begin{array}{rcl} (4.57) & {‖ v_{m} (l) ‖}^{2} ⩽ \frac{C}{N_{1} C_{1}} ({| v (τ) |}^{2} + ‖ η ‖_{L_{V}^{2}}^{2} + {\tilde{β}}_{0}) + \frac{2 N_{2}}{N_{1}} . \end{array}$ Combining (4.57) with (4.55)–(4.56), we conclude that $\begin{array}{l} sup_{t > C_{1}} {| A^{\frac{1}{4}} v_{m} (t) |}^{2} \\ ⩽ [\frac{C}{\sqrt{λ_{1}}} ({| v (τ) |}^{2} + ‖ η ‖_{L_{V}^{2}}^{2} + {\tilde{β}}_{0}) + \frac{2 N_{2}}{N_{1} \sqrt{λ_{1}}} \\ + C_{1} (\frac{C | \partial Ω |}{ν} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} + \frac{C ν | \partial Ω |}{ε^{2}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) + \frac{ν}{7} \int_{s}^{t} {‖ v_{m} (l - ρ (l)) ‖}^{2} d l \\ + \frac{C}{ν \sqrt{λ_{1}}} \int_{s}^{t} (α (l) {| v_{m} (l - ρ (l)) |}^{2} + | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} α (l) + β (l)) d l] \\ (4.58) & \times e^{[\frac{C}{ν} \int_{s}^{t} (‖ v_{m} ‖^{2} + ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) d l + \frac{C}{ν λ_{1}} ‖ g ‖_{L^{2} (τ, T; H)}^{2}]} . \end{array}$

Step 2: Passage to the limit.

Since $\begin{matrix} v_{m} ⇀ v weakly in L^{2} (τ - h, T; V), \end{matrix}$ the compact embedding $V ↪ D (A^{\frac{1}{4}})$ leads to the existence of a subsequence (still rewritten as ${v_{m}}$ ) satisfying $\begin{matrix} v_{m} \to v strongly in L^{2} (τ, T; D (A^{\frac{1}{4}})) . \end{matrix}$ Thus, there exists a subsequence ${v_{m_{j}}}$ such that $\begin{matrix} v_{m_{j}} \to v in D (A^{\frac{1}{4}}) for a.e. t, \end{matrix}$ which implies $\begin{matrix} sup_{t > C_{1}} | A^{\frac{1}{4}} v (t) | ⩽ C . \end{matrix}$ Therefore, the conclusion holds.

5. Pullback dynamics

5.1. Theory of tempered pullback dynamics

In this section, the fundamental theory for existence of attractors for non-autonomous systems will be presented; details can be founded in [6,10,11,14,31,34] and references therein.

∙ Dissipation defined via absorbing sets and universes

Definition 5.1.
The family ${\hat{D}}_{0} = {D_{0} (t) : t \in R} \subset P (X)$ is $D$ -pullback absorbing for the process U on X if for any $t \in R$ and any $\hat{D} \in D$ , there exists $τ_{0} (t, \hat{D}) ⩽ t$ 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 ${\hat{D}}_{0}$ -pullback asymptotic compactness
Definition 5.2.
A process U on X is ${\hat{D}}_{0}$ -pullback 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 $D$ -pullback asymptotic compactness
Definition 5.3.
A process U on X is said to be $D$ -pullback 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.

∙ Existence of minimal and unique families of $D$ -pullback attractors Definition 5.4.
A universe $D$ defined in a metric space X is a class of families $\hat{D} \subset 2^{X}$ of the form $\hat{D} = {D (t) | t \in R}$ , where each $D (t)$ is a nonempty bounded subset of X. Given a universe $D$ defined on X, a family $A = {A (t)}_{t \in R}$ is called a $D$ -pullback attractor of a process $U (t, τ) : X \to X$ if the following properties hold.
Compactness: $A (t)$ is a nonempty compact set of X, $t \in R$ .

Invariance: $U (t, τ) A (τ) = A (t)$ , $t ⩾ τ$ .

$D$ -pullback attraction: ${lim}_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), A (t)) = 0$ , $t \in R$ , ${D (t)}_{t \in R} \in D$ .
In addition, a $D$ -pullback attractor $A$ is said to be minimal if whenever $\hat{C}$ is another $D$ -attracting family of closed sets, then $A (t) \subset C (t)$ for all $t \in R$ .
Theorem 5.5 (See [10,26]).

Consider a 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 $D$ -pullback absorbing for $U (\cdot, \cdot)$ , and assume also that $U (\cdot, \cdot)$ is ${\hat{D}}_{0}$ -pullback asymptotically compact.

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

Remark 5.6.

For any family of closed sets $C = {C (t)}_{t \in R}$ satisfying $\begin{matrix} lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), C (t)) = 0, \end{matrix}$ if there always holds that $A \subset C (t)$ , then it is said that the $D$ -pullback attractor $A$ is minimal.

If $A \subset D$ , then the family of $D$ -pullback attractors $A = {A (t)}_{t \in R}$ is unique provided that ${\hat{D}}_{0} \in D$ , $D_{0} (t)$ is closed for any $t \in R$ , and $D$ is inclusion closed.

5.2. Pullback dynamics: Proof of Theorem 3.7

From Theorems 3.3 and 3.4, we know that the system (3.4) generates a continuous process $S (t, τ)$ in $M_{H}$ by $\begin{matrix} S (t, τ) (v (τ), η) = (v (t), v_{t}) . \end{matrix}$ To obtain the $D$ -pullback attractor, we need to establish the existence of a $D$ -pullback absorbing set and the $D$ -pullback asymptotic compactness of ${S (t, τ)}$ .

∙ Tempered pullback absorbing set in $H \times C_{H}$

Let $D (τ, v (τ), η) = {D (t)}_{t \in R}$ denote the family consisting of all global weak solutions for $(v (τ), η) \in M_{H}$ .

Lemma 5.7.
Assume that $(v (τ), η) \in M_{H}$ and that $f : R \times C_{H} \to H$ satisfies the hypotheses (H-a)–(H-d). Let the functions $α (\cdot)$ and $β (\cdot)$ satisfy (H-e)–(H-f). Then it holds that $\begin{array}{rcl} {| v (t) |}^{2} & ⩽ & (e^{ν λ_{1} τ} {| v (τ) |}^{2} + \frac{e^{ν λ_{1} h}}{λ_{1} (1 - ρ^{})} \int_{- h}^{0} e^{ν λ_{1} θ} {‖ η (θ) ‖}^{2} d θ) e^{- κ_{δ} (t, τ)} \\ + [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \\ + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} \int_{τ}^{t} e^{- κ_{δ} (t, s)} α (s) d s + \frac{10}{ν λ_{1}} δ^{- 1} \int_{τ}^{t} e^{- κ_{δ} (t, s)} β (s) d s \\ (5.1) & + \frac{C}{ν} \int_{τ}^{t} e^{- κ_{δ} (t, s)} {‖ g (s) ‖}_{H^{- 1}}^{2} d s . \end{array}$
Proof.
Multiplying (3.4) by $e^{ν λ_{1} t} v$ , by using similar techniques as in Section 4.1, we have $\begin{array}{l} \frac{d}{d t} (e^{ν λ_{1} t} | v |^{2}) + ν e^{ν λ_{1} t} ‖ v ‖^{2} \\ ⩽ [\frac{ν λ_{1} (1 - ρ^{})}{e^{ν λ_{1} h}} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν}] e^{ν λ_{1} t} {| v (t - ρ (t)) |}^{2} + \frac{e^{ν λ_{1} h} α (t)}{ν λ_{1} (1 - ρ^{})} e^{ν λ_{1} t} {| v (t) |}^{2} \\ + \frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} e^{ν λ_{1} t} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν} e^{ν λ_{1} t} \\ (5.2) & + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} α (t) e^{ν λ_{1} t} + \frac{10 | Ω |}{ν λ_{1}} β (t) e^{ν λ_{1} t} + \frac{C}{ν} e^{ν λ_{1} t} ‖ g ‖_{H^{- 1}}^{2}, \end{array}$ where $\begin{array}{rcl} (5.3) & \begin{matrix} \int_{τ}^{t} e^{ν λ_{1} s} {| v (s - ρ (s)) |}^{2} d s \\ ⩽ \frac{e^{ν λ_{1} h}}{(1 - ρ^{})} \int_{τ}^{t} e^{ν λ_{1} s} {| v (s) |}^{2} d s + \frac{e^{ν λ_{1} h}}{λ_{1} (1 - ρ^{})} \int_{- h}^{0} e^{ν λ_{1} θ} {‖ η (θ) ‖}^{2} d θ . \end{matrix} \end{array}$ Integrating (5.2) on $[τ, t]$ and using the notation in (3.10)–(3.12), we have $\begin{array}{l} e^{ν λ_{1} t} {| v (t) |}^{2} + \frac{ν}{2} \int_{τ}^{t} e^{ν λ_{1} s} {‖ v (s) ‖}^{2} d s \\ ⩽ e^{ν λ_{1} τ} {| v (τ) |}^{2} + \frac{e^{ν λ_{1} h}}{λ_{1} (1 - ρ^{})} \int_{- h}^{0} e^{ν λ_{1} θ} {‖ η (θ) ‖}^{2} d θ \\ + \int_{τ}^{t} [(ν λ_{1} + \frac{C e^{ν λ_{1} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})}) + (\frac{α (s) e^{ν λ_{1} h}}{ν λ_{1} (1 - ρ^{})} + δ)] e^{ν λ_{1} s} {| v (s) |}^{2} d s \\ + [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \int_{τ}^{t} e^{ν λ_{1} s} d s \\ + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} \int_{τ}^{t} e^{ν λ_{1} s} α (s) d s + \frac{10}{ν λ_{1}} δ^{- 1} \int_{τ}^{t} e^{ν λ_{1} s} β (s) d s \\ (5.4) & + \frac{C}{ν} \int_{τ}^{t} e^{ν λ_{1} s} {‖ g (s) ‖}_{H^{- 1}}^{2} d s . \end{array}$ Applying the Poincaré and Gronwall inequalities to (5.4), we find (5.1). □

Our results are concerned with families of universes determined by $β (\cdot)$ and some other terms.
Definition 5.8 (The tempered universe in $H \times C_{H}$ ).

Denote by $P (H \times C_{H})$ the set consisting of all nonempty subsets in $H \times C_{H}$ , $\hat{D} = {D (t)} \subset P (H \times C_{H})$ , and let $B (0, {\tilde{ρ}}_{\hat{D}} (t))$ be a family of balls of center 0 with radius ${\tilde{ρ}}_{\hat{D}} (t)$ satisfying $| {\tilde{ρ}}_{\hat{D}} (t) |^{2} e^{- κ_{δ} (t, τ)} \to 0$ as $τ \to - \infty$ .

The $κ_{δ} (t, τ)$ -indexed universes are defined by $\begin{array}{l} (5.5) & D_{κ_{δ} (t, τ)}^{H \times C_{H}} = {{\hat{D}}_{κ_{δ} (t, τ)} | D_{κ_{δ} (t, τ)} (t) \subset B (0, {\tilde{ρ}}_{{\hat{D}}_{κ_{δ} (t, τ)}} (t)) with lim_{τ \to - \infty} {| {\tilde{ρ}}_{{\hat{D}}_{κ_{δ} (t, τ)}} (t) |}^{2} e^{- κ_{δ} (t, τ)} = 0}, \end{array}$ where ${\tilde{ρ}}_{{\hat{D}}_{κ_{δ} (t, τ)}} : R \to R^{+}$ is a continuous function and $B (0, {\tilde{ρ}}_{{\hat{D}}_{κ_{δ} (t, τ)}} (t))$ denotes a family of closed ball in $H \times C_{H}$ with radius ${\tilde{ρ}}_{{\hat{D}}_{κ_{δ} (t, τ)}} (t)$ .

Remark 5.9.
By the properties of universe $D_{κ_{δ} (t, τ)}^{H \times C_{H}}$ in (5.5), we can easily verify that the above universes are inclusion closed and contain any fixed universe and uniformly bounded absorbing set.
Lemma 5.10 (The tempered pullback absorbing set in $H \times C_{H}$ ).

Suppose that $g (t, x)$ and $f : R \times C_{H} \to H$ satisfy the hypotheses (H-a)–(H-d) and let the functions $α (\cdot)$ and $β (\cdot)$ satisfy (H-e)–(H-f). Then for any $v \in D (τ, v (τ), η)$ and $(v (τ), η) \in M_{H}$ , there exits a time $τ ⩽ τ_{0} (B)$ for any family $B = {B (t)} \subset D_{κ_{δ} (t, τ)}^{H \times C_{H}}$ such that we have the tempered pullback absorbing set $B_{H \times C_{H}} (0, R_{H \times C_{H}} (t))$ with radius $R_{H \times C_{H}} (t)$ defined by $\begin{array}{rcl} R_{H \times C_{H}}^{2} (t) & = & 1 + [(C ν λ_{1} | Ω | (1 - ρ^{*}) + \frac{C ν | \partial Ω |}{ε}) ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \\ (5.6) & + \frac{10}{ν λ_{1}} δ^{- 1} \int_{\infty}^{t} e^{- κ_{δ} (t, s)} β (s) d s + \frac{C}{ν} \int_{\infty}^{t} e^{- κ_{δ} (t, s)} {‖ g (s) ‖}_{H^{- 1}}^{2} d s \end{array}$ for the process $S (t, τ)$ , which implies that $| v (t) |^{2} ⩽ R_{H \times C_{H}}^{2} (t)$ .

Proof.
Since $κ_{δ} (t, τ)$ and $κ_{δ} (t, s)$ are positive, then from the definition of the universe $D_{κ_{δ} (t, τ)}^{H \times C_{H}}$ , we can easily derive the tempered pullback absorbing ball $B_{H \times C_{H}} (0, R_{H \times C_{H}} (t))$ from Lemma 5.7. □

∙ $D$ -pullback absorbing set in $M_{H}$

Let $D$ denote the family of all ${D (t)}_{t \in R} \subset P (M_{H})$ , where $\begin{array}{l} lim_{τ \to - \infty} e^{\tilde{r} τ} sup_{(v (τ), η) \in D (τ)} {| v (τ) |}^{2} = 0, \\ \frac{ν}{4} - \frac{C e^{\frac{ν λ_{1}}{2} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} - \frac{4 e^{\frac{ν λ_{1}}{2} h} ‖ α (t) ‖_{L^{\infty} (- \infty, T)}}{ν (1 - ρ^{})} - \tilde{r} ⩾ 0, \tilde{r} > 0 . \end{array}$
Lemma 5.11.
Assume that $(v (τ), η) \in M_{H}$ and $\frac{ν}{4} > \frac{C e^{\frac{ν λ_{1}}{2} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} + \frac{4 e^{\frac{ν λ_{1}}{2} h} ‖ α (t) ‖_{L^{\infty} (- \infty, T)}}{ν (1 - ρ^{})}$ . Then the process ${S (t, τ)}$ to system ( 3.4 ) possesses a $D$ -pullback absorbing set $B = {B (t)}_{t \in R}$ in $M_{H}$ , where $\begin{matrix} B (t) = {\bar{B}}_{H} (0, {\tilde{ρ}}_{H} (t)) \times ({\bar{B}}_{C_{H}} (0, {\tilde{ρ}}_{H} (t)) \cap {\bar{B}}_{L_{V}^{2}} (0, {\tilde{ρ}}_{L_{V}^{2}} (t))) \end{matrix}$ with radius $\begin{array}{rcl} {\tilde{ρ}}_{H}^{2} (t) & = & 1 + \frac{2}{ν λ_{1}} [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \\ + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν^{2} λ_{1}^{2}} {‖ α (s) ‖}_{L^{\infty} (- \infty, T)} + \frac{10 | Ω |}{ν λ_{1}} {‖ β (s) ‖}_{L^{1} (- \infty, T)} \\ (5.7) & + \frac{C}{ν} \int_{- \infty}^{t} e^{\frac{ν λ_{1}}{2} s} ‖ g ‖_{H^{- 1}}^{2} d s \end{array}$ and $\begin{array}{rcl} {\tilde{ρ}}_{L_{V}^{2}}^{2} (t) & = & [\frac{ν}{λ_{1}} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν λ_{1} (1 - ρ^{})}] {‖ η (t) ‖}_{L_{V}^{2}}^{2} \\ + \frac{‖ α (t) ‖_{L^{\infty}}}{2 ν (1 - ρ^{})} {\tilde{ρ}}_{H}^{2} + [\frac{ν (1 - ρ^{})}{2} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν}] \frac{h}{(1 - ρ^{})} {\tilde{ρ}}_{H}^{2} \\ + \frac{C ν | \partial Ω | h}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω | h}{ν} \\ (5.8) & + \frac{C | Ω | h ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} {‖ α (t) ‖}_{L^{\infty} (- \infty, T)} + \frac{10 | Ω | h}{ν λ_{1}} {‖ β (t) ‖}_{L^{1} (- \infty, T)} . \end{array}$
Proof.
Multiplying (3.4) by $e^{\frac{ν λ_{1}}{2} t} v$ and proceeding as in Theorem 3.4, we have $\begin{array}{l} \frac{d}{d t} (e^{\frac{ν λ_{1}}{2} t} | v |^{2}) + \frac{ν}{2} e^{\frac{ν λ_{1}}{2} t} ‖ v ‖^{2} \\ ⩽ [\frac{ν (1 - ρ^{})}{4 e^{\frac{ν λ_{1}}{2} h}} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν}] e^{\frac{ν λ_{1}}{2} t} {| v (t - ρ (t)) |}^{2} + \frac{4 e^{\frac{ν λ_{1}}{2} h} α (t)}{ν (1 - ρ^{})} e^{\frac{ν λ_{1}}{2} t} {| v (t) |}^{2} \\ + [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] e^{\frac{ν λ_{1}}{2} t} \\ (5.9) & + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} e^{\frac{ν λ_{1}}{2} t} α (t) + \frac{10 | Ω |}{ν λ_{1}} e^{\frac{ν λ_{1}}{2} t} β (t) + \frac{C}{ν} e^{\frac{ν λ_{1}}{2} t} ‖ g ‖_{H^{- 1}}^{2} . \end{array}$ Since $\begin{array}{rcl} (5.10) & \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} {| v (s - ρ (s)) |}^{2} d s & ⩽ & \frac{e^{\frac{ν λ_{1}}{2} h}}{(1 - ρ^{})} \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} {| v (s) |}^{2} d s + \frac{e^{\frac{ν λ_{1}}{2} h} e^{\frac{ν λ_{1}}{2} τ}}{λ_{1} (1 - ρ^{})} ‖ η ‖_{L_{V}^{2}}^{2}, \end{array}$ integrating (5.9) on $[τ, t]$ , we have $\begin{array}{l} e^{\frac{ν λ_{1}}{2} t} {| v (t) |}^{2} + \frac{ν}{4} \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} {‖ v (s) ‖}^{2} d s \\ ⩽ e^{\frac{ν λ_{1}}{2} τ} {| v (τ) |}^{2} + e^{\frac{ν λ_{1}}{2} τ} [\frac{ν}{λ_{1}} + \frac{C e^{\frac{ν λ_{1}}{2} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν λ_{1} (1 - ρ^{})}] ‖ η ‖_{L_{V}^{2}}^{2} \\ + [\frac{C e^{\frac{ν λ_{1}}{2} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν (1 - ρ^{})} + \frac{4 e^{\frac{ν λ_{1}}{2} h} α (t)}{ν (1 - ρ^{})}] \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} {| v (s) |}^{2} d s \\ + [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} d s \\ + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} α (s) d s + \frac{10 | Ω |}{ν λ_{1}} \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} β (s) d s \\ (5.11) & + \frac{C}{ν} \int_{τ}^{t} e^{\frac{ν λ_{1}}{2} s} ‖ g ‖_{H^{- 1}}^{2} d s . \end{array}$ Hence, This yields $\begin{array}{rcl} {| v (t) |}^{2} & ⩽ & e^{- \frac{ν λ_{1}}{2} (t - τ)} [{| v (τ) |}^{2} + (\frac{ν}{λ_{1}} + \frac{C e^{\frac{ν λ_{1}}{2} h} ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν λ_{1} (1 - ρ^{})}) ‖ η ‖_{L_{V}^{2}}^{2}] \\ + \frac{2}{ν λ_{1}} [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \\ + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν^{2} λ_{1}^{2}} {‖ α (s) ‖}_{L^{\infty} (τ, T)} + \frac{10 | Ω |}{ν λ_{1}} {‖ β (s) ‖}_{L^{1} (τ, T)} \\ (5.12) & + \frac{C}{ν} \int_{- \infty}^{t} e^{\frac{ν λ_{1}}{2} s} ‖ g ‖_{H^{- 1}}^{2} d s, \end{array}$ which means that there exists $τ (D, t) < t - h$ such that, for $τ < τ (D, t)$ , $\begin{array}{rcl} {| v (t) |}^{2} & ⩽ & 1 + \frac{2}{ν λ_{1}} [\frac{C ν | \partial Ω |}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω |}{ν}] \\ + \frac{C | Ω | ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν^{2} λ_{1}^{2}} {‖ α (s) ‖}_{L^{\infty} (τ, T)} + \frac{10 | Ω |}{ν λ_{1}} {‖ β (s) ‖}_{L^{1} (τ, T)} + \frac{C}{ν} \int_{- \infty}^{t} e^{\frac{ν λ_{1}}{2} s} ‖ g ‖_{H^{- 1}}^{2} d s \\ (5.13) & = & {\tilde{ρ}}_{H}^{2} (t) . \end{array}$ Moreover, repeating the above integration over $[s, t]$ and putting $s = t - h$ , using the above estimate, we can derive $\begin{array}{rcl} (5.14) & ‖ v ‖_{C_{H}}^{2} ⩽ {\tilde{ρ}}_{H}^{2} (t), τ < τ (D, t) . \end{array}$ By virtue of a similar procedure as in (4.7)–(4.9), integrating over $[t - h, t]$ and from (5.13)–(5.14), we also obtain $\begin{array}{rcl} ν \int_{t - h}^{t} {‖ v (s) ‖}^{2} d s & ⩽ & [\frac{ν}{λ_{1}} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν λ_{1} (1 - ρ^{})}] {‖ η (t) ‖}_{L_{V}^{2}}^{2} \\ + \frac{‖ α (t) ‖_{L^{\infty}}}{2 ν (1 - ρ^{})} {\tilde{ρ}}_{H}^{2} + [\frac{ν (1 - ρ^{})}{2} + \frac{C ‖ φ ‖_{L^{\infty} (\partial Ω)}}{ν}] \frac{h}{(1 - ρ^{})} {\tilde{ρ}}_{H}^{2} \\ + \frac{C ν | \partial Ω | h}{ε} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + \frac{C ε ‖ φ ‖_{L^{\infty} (\partial Ω)}^{4} | \partial Ω | h}{ν} \\ (5.15) & + \frac{C | Ω | h ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν λ_{1}} {‖ α (t) ‖}_{L^{\infty} (- \infty, T)} + \frac{10 | Ω | h}{ν λ_{1}} {‖ β (t) ‖}_{L^{1} (- \infty, T)}, \end{array}$ and there holds $\begin{matrix} ‖ v ‖_{L_{V}^{2}}^{2} ⩽ {\tilde{ρ}}_{L_{V}^{2}}^{2} (t), τ < τ (D, t) - h, \end{matrix}$ so that the conclusion holds. □
Remark 5.12.
Concerning the process $S^{0} (t, τ)$ , one has the $D$ -pullback absorbing set $B^{0} = {B^{0} (t)}_{t \in R}$ in H. Also, concerning the process $S^{h} (t, τ)$ , one has the $D$ -pullback absorbing set $B^{h} = {B^{h} (t)}_{t \in R}$ in $C_{H}$ , where $\begin{array}{rcl} (5.16) & B^{h} (t) = {v_{t} \in C_{H}; ‖ v_{t} ‖^{2} ⩽ {\tilde{ρ}}_{H}^{2} (t)} \end{array}$ and $\begin{array}{rcl} (5.17) & \underset{h \to 0}{lim sup} {\tilde{ρ}}_{H}^{2} (t) = {\tilde{ρ}}_{0}^{2} (t) . \end{array}$

∙ $D$ -pullback asymptotic compactness of $S (t, τ)$ in $M_{H}$

Since the embedding $C_{V}$ into $C_{H}$ is not compact, we will use the energy equation approach to achieve the compactness of the process. This approach originates from [1,29] and references therein. Lemma 5.13.
Let $(v (τ), η) \in M_{H}$ . Then the process $S (t, τ) : M_{H} \to M_{H}$ generated by the system ( 3.4 ) is $D$ -pullback asymptotically compact.
Proof.
For any family $\hat{D} = {D (t)}_{t \in R}$ in $D$ , the sequence ${τ_{n}} \subset (- \infty, t]$ satisfies $τ_{n} \to - \infty$ as $n \to \infty$ , ${(v (τ_{n}), η^{n})} \subset D (τ_{n})$ , and we write $v^{n} (\cdot) = v (\cdot; τ_{n}, v (τ_{n}), η^{n})$ .

Step 1: The weak convergence of ${v^{n} (t, x)}$ in $[t - h, t]$ and in H for any arbitrary fixed $t ⩾ τ$ .

From the conclusions and techniques in Theorems 3.3 and 5.7, we know that there exists $τ (\hat{D}, t) ⩽ t - 3 h - 1$ such that, for $τ ⩽ τ (\hat{D}, t)$ , $\begin{matrix} {v^{n}} \subset L^{\infty} (t - 3 h - 1, t; H) \cap L^{2} (t - 2 h - 1, t; V), {\frac{\partial}{\partial t} v^{n}} \subset L^{2} (t - h - 1, t; V^{'}) . \end{matrix}$ This together with the compactness Lemma and the diagonal procedure can lead to the existence of a subsequence (written also as ${v^{n}}$ ) satisfying $\begin{array}{rcl} (5.18) & \{\begin{matrix} v^{n} ⇀ v & weakly star in L^{\infty} (t - 3 h - 1, t; H), \\ v^{n} ⇀ v & weakly in L^{2} (t - 2 h - 1, t; V), \\ \frac{\partial}{\partial t} v^{n} ⇀ \frac{\partial}{\partial t} v & weakly in L^{2} (t - h - 1, t; V^{'}), \\ v^{n} \to v & strongly in L^{2} (t - h - 1, t; H), \\ v^{n} \to v & in H, a.e. s \in (t - h - 1, t) . \end{matrix} \end{array}$ Thus, we can conclude that $v \in C ([t - h - 1, t]; H)$ is a weak solution of system (3.4) with the initial datum $v (t - h - 1)$ .

Since ${v^{n}} \subset L^{\infty} (t - h - 1, t; D (A^{1 / 4}))$ and ${\frac{\partial}{\partial t} v^{n}} \subset L^{2} (t - h - 1, t; V^{'})$ , the Aubin–Lions–Simon lemma leads to $\begin{array}{rcl} (5.19) & v^{n} \to v strongly in C ([t - h - 1, t]; H) . \end{array}$ It thus follows that for any ${s_{n}} \subset [t - h - 1, t]$ with $s_{n} \to s \in [t - h - 1, t]$ as $n \to \infty$ $\begin{array}{rcl} (5.20) & v^{n} (s_{n}) ⇀ v (s) weakly in H, \end{array}$ and there holds that $\begin{array}{rcl} (5.21) & \underset{n \to \infty}{lim inf} | v^{n} (s_{n}) | ⩾ | v (s) | . \end{array}$

Step 2: The strong convergence of the sequence ${v^{n} (s_{n})}$ in $C ([t - h, t]; H)$ for any sequence ${s_{n}} \subset [t - h, t]$ with $s_{n} \to s$ as $n \to + \infty$ .

According to the energy equality for $v^{n}$ and v, we define the following two functionals in $[t - h - 1, t]$ : $\begin{array}{rcl} J_{n} (s) & = & \frac{1}{2} {| v^{n} (s) |}^{2} + \int_{t - h - 1}^{s} b (v^{n} (σ - ρ (σ)), ψ, v^{n} (σ)) d σ \\ + \int_{t - h - 1}^{s} b (ψ, ψ, v^{n} (σ)) d σ - \int_{t - h - 1}^{s} ⟨ P_{L} g (σ, x), v^{n} (σ) ⟩ d σ \\ (5.22) & - \int_{t - h - 1}^{s} (P_{L} (f (σ, u^{n} (σ - ρ (σ))) + ν F), v^{n} (σ)) d σ \end{array}$ and $\begin{array}{rcl} J (s) & = & \frac{1}{2} {| v (s) |}^{2} + \int_{t - h - 1}^{s} b (v (σ - ρ (σ)), ψ, v (σ)) d σ + \int_{t - h - 1}^{s} b (ψ, ψ, v (σ)) d σ \\ (5.23) & - \int_{t - h - 1}^{s} ⟨ P_{L} g (σ, x), v (σ) ⟩ d σ - \int_{t - h - 1}^{s} (P_{L} (f (σ, u (σ - ρ (σ))) + ν F), v (σ)) d σ, \end{array}$ which are continuous and non-increasing in $[t - h - 1, t]$ . Proceeding as in Theorem 3.3, from (5.18), when $n \to + \infty$ , we have $\begin{array}{l} (5.24) & \begin{matrix} \int_{t - h - 1}^{s} ⟨ P_{L} g (σ, x), v^{n} (σ) ⟩ d σ + \int_{t - h - 1}^{s} (P_{L} (f (σ, u^{n} (σ - ρ (σ))) + ν F), v^{n} (σ)) d σ \\ \to \int_{t - h - 1}^{s} ⟨ P_{L} g (σ, x), v (σ) ⟩ d σ + \int_{t - h - 1}^{s} (P_{L} (f (σ, u (σ - ρ (σ))) + ν F), v (σ)) d σ \end{matrix} \\ (5.25) & \int_{t - h - 1}^{s} b (v^{n} (σ - ρ (σ)), ψ, v^{n} (σ)) d σ \to \int_{t - h - 1}^{s} b (v (σ - ρ (σ)), ψ, v (σ)) d σ \end{array}$ and $\begin{array}{rcl} (5.26) & \int_{t - h - 1}^{s} b (ψ, ψ, v^{n} (σ)) d σ \to \int_{t - h - 1}^{s} b (ψ, ψ, v (σ)) d σ . \end{array}$ Hence, we conclude that $\begin{array}{rcl} (5.27) & J_{n} (s) \to J (s) a.e. s \in (t - h - 1, t), \end{array}$ which means that, for any $ε > 0$ , there exists a constant $\hat{N}$ such that, for any $n ⩾ \hat{N}$ and any sequence ${s_{n}} \subset [t - h - 1, t]$ , $\begin{array}{rcl} (5.28) & | J_{n} (s_{n}) - J (s_{n}) | ⩽ \frac{ε}{2} . \end{array}$ Furthermore, the uniform continuity of $J (s)$ with respect to s can lead to the fact that, for $ε > 0$ in (5.28), there exists a constant $\tilde{N}$ such that, for any $n ⩾ \tilde{N}$ and any sequence ${s_{n}} \subset [t - h - 1, t]$ satisfying $s_{n} \to s$ , $\begin{array}{rcl} (5.29) & | J (s_{n}) - J (s) | ⩽ \frac{ε}{2} . \end{array}$ Choosing $N = max {\hat{N}, \tilde{N}}$ and using (5.28) and (5.29), we can obtain that for $n ⩾ N$ $\begin{array}{rcl} (5.30) & | J_{n} (s_{n}) - J (s) | ⩽ | J_{n} (s_{n}) - J (s_{n}) | + | J (s_{n}) - J (s) | ⩽ ε . \end{array}$ It thus follows that for any ${s_{n}} \subset [t - h - 1, t]$ $\begin{array}{rcl} (5.31) & \underset{n \to \infty}{lim sup} J_{n} (s_{n}) ⩽ J (s), \end{array}$ and the strong convergence of ${v^{n} (s_{n})}$ in $C ([t - h, t]; H)$ follows by using (5.21).

Step 3: The strong convergence of ${v^{n} (s_{n})}$ in $L^{2} (t - h, t; V)$ for any sequence ${s_{n}} \subset [t - h, t]$ with $s_{n} \to s$ as $n \to + \infty$ .

For $v^{n}$ and v, there hold $\begin{array}{l} {| v^{n} (t) |}^{2} + 2 ν \int_{t - h}^{t} {‖ v^{n} (σ) ‖}^{2} d σ \\ + 2 \int_{t - h}^{t} b (v^{n} (σ - ρ (σ)), ψ, v^{n} (σ)) d σ + 2 \int_{t - h}^{t} b (ψ, ψ, v^{n} (σ)) d σ \\ = {| v^{n} (t - h) |}^{2} + 2 \int_{t - h}^{t} ⟨ P_{L} g (σ, x), v^{n} (σ) ⟩ d σ \\ (5.32) & + 2 \int_{t - h}^{t} (P_{L} (f (σ, u^{n} (σ - ρ (σ))) + ν F), v^{n} (σ)) d σ \end{array}$ and $\begin{array}{l} {| v (t) |}^{2} + 2 ν \int_{t - h}^{t} {‖ v (σ) ‖}^{2} d r + 2 \int_{t - h}^{t} b (v (σ - ρ (σ)), ψ, v (σ)) d σ + 2 \int_{t - h}^{t} b (ψ, ψ, v (σ)) d σ \\ = {| v (t - h) |}^{2} + 2 \int_{t - h}^{t} ⟨ P_{L} g (σ, x), v (σ) ⟩ d σ \\ (5.33) & + 2 \int_{t - h}^{t} (P_{L} (f (σ, u (σ - ρ (σ))) + ν F), v (σ)) d σ, \end{array}$ Combining (5.18), (5.32) and (5.33), we can prove that as $n \to + \infty$ $\begin{array}{l} (5.34) & 2 ν \int_{t - h}^{t} {‖ v^{n} (σ) ‖}^{2} d σ \to 2 ν \int_{t - h}^{t} {‖ v (σ) ‖}^{2} d σ, \end{array}$ and the strong convergence of ${v^{n} (s_{n})}$ in $L^{2} (t - h, t; V)$ can be derived by using (5.18) and (5.34).

In conclusion, from the above three steps we can obtain that the process ${S (t, τ)} : M_{H} \to M_{H}$ is $D$ -pullback asymptotically compact. □
Proof of Theorems 3.6 and 3.7.
Combining Lemmas 5.10-5.13, by using the theory of tempered and $D$ -pullback attractors, we can prove Theorems 3.6 and 3.7. □

6. Robustness: Convergence between $A^{0}$ and $A^{h}$

6.1. The theory of robustness

By definition of the upper semi-continuity, the following lemma can be used to prove the robustness of pullback attractors for evolutionary equations.

Lemma 6.1 (See [10]).

Let $λ \in [λ_{0}, λ_{0} + 1)$ , and ${U_{λ} (\cdot, \cdot)}$ be families of processes such that

$U_{λ} (\cdot, \cdot)$ has the pullback attractor $A_{λ} (\cdot)$ for any $λ \in [λ_{0}, λ_{0} + 1)$ .

For any $t \in R$ , $T ⩾ 0$ , and any bounded set $D \subset X$ , there holds that $\begin{matrix} sup_{s \in [0, T], u_{0} \in D} {dist}_{X} (U_{λ} (t + s, t) u_{0}, U_{λ_{0}} (t + s, t) u_{0}) \to 0 as λ \to λ_{0} . \end{matrix}$

There are $δ > 0$ and $t_{0} \in R$ such that $\begin{matrix} ⋃_{λ \in (λ_{0}, λ_{0} + δ)} ⋃_{s ⩽ t_{0}} A_{λ} (s) \end{matrix}$ is bounded.

Then there holds for each $t \in R$ that $\begin{matrix} lim_{λ \to λ_{0}} {dist}_{X} (A_{λ}, A_{λ_{0}}) = 0 . \end{matrix}$

Since our problem contains time varying delays in the convective term and external force, which yields additional difficulties when considering the strong topology of H, we use the weak topology of $L^{2}$ for the convergence of pullback attractors. We denote the metric in H by $d_{H}^{\circ}$ in which the weak topology on bounded sets is metrizable, then give the corresponding definition and theory of upper semi-continuity as follows.

Definition 6.2.
It is said that a family of subsets ${A^{h} (t)}_{t \in R}$ in H has the property of upper semi-continuity if $\begin{matrix} lim_{h \to 0} sup_{- h ⩽ θ ⩽ 0} {dist}_{H} (A^{h} (t), A^{0} (t)) = 0, \forall t \in R, \end{matrix}$ where $\begin{matrix} {dist}_{H} (E, S) = sup_{ϕ \in E} inf_{φ \in S} {‖ ϕ (θ) - φ ‖}_{H}, \forall E \subset H, S \subset H . \end{matrix}$

Moreover, we will give the upper semi-continuity of pullback attractor $A^{h} (t)$ for our model with time varying delays for any $t \in R$ provided that:
For any $τ ⩽ t \in R^{+}$ , $\begin{matrix} lim_{n \to \infty} sup_{- h_{n} ⩽ θ ⩽ 0} {‖ S^{h_{n}} (t, τ) ϕ_{n} (θ) - S^{0} (t, τ) φ ‖}_{H} = 0, \end{matrix}$ for any $ϕ_{n} (θ) \in C ([- h_{n}, 0]; H) \cap L^{2} (- h_{n}, 0; V)$ , $φ \in H$ , $h_{n} \to 0$ and ${sup}_{- h_{n} ⩽ θ ⩽ 0} ‖ ϕ_{n} (θ) - φ ‖_{H} \to 0$ as $n \to \infty$ .

For any $t \in R$ , $\begin{matrix} \underset{h \to 0}{lim sup} sup_{ϕ \in B^{h}} {‖ ϕ (θ) ‖}_{C_{H}} ⩽ ρ_{0} (t), \end{matrix}$ where $ρ_{0} (t)$ is the radius of the pullback absorbing ball of ${S^{0} (t, τ)$ in H.

For any $t \in R$ , $ϕ_{n} (θ) \in A^{h_{n}}$ , and $h_{n} \to 0$ as $n \to \infty$ , there is $φ \in H$ such that $\begin{matrix} lim_{n \to \infty} sup_{- h_{n} ⩽ θ ⩽ 0} {‖ ϕ_{n} (θ) - φ ‖}_{H} = 0 . \end{matrix}$
Theorem 6.3.
For any $t \in R$ , if the assumptions (I)∽(III) hold, then we have the robustness of pullback attractors in terms of the upper semi-continuity, $\begin{matrix} lim_{h \to 0} {dist}_{H} (A^{h}, A^{0}) = 0 . \end{matrix}$
Proof.
Using a similar technique as in Theorem 2 in [5], except for revising the general functional space by including delays, we can derive the conclusion; we refer the reader to [4,5] and [36] for more details. □

6.2. Proof of Theorem 3.9

Since the model that we study in this paper needs different techniques from reaction-diffusion equations, inspired by Zelati and Gal [40], we give the following procedure to verify Theorem 6.3.

Lemma 6.4.
Let ${h_{n}} \subset (0, h]$ be any sequence satisfying $h_{n} \to 0$ as $n \to + \infty$ , $(v_{n} (τ), η_{n}) \in M_{H}$ with $v_{n} (τ) = v (τ)$ , and $S^{h_{n}} v_{n} (τ)$ be the solution to the system ( 3.4 ) with $h = h_{n}$ . Assume that $v : [τ, + \infty) \to H$ is the solution to system ( 3.14 ). Then there exists a subsequence ${n_{k}}_{k \in N}$ such that $\begin{matrix} lim_{k \to + \infty} d_{H} (S^{h_{n_{k}}} v_{n} (τ), v (t)) = 0, \forall t ⩾ τ . \end{matrix}$
Proof.
Similar to the proof of Theorem 5.13, we can show that, for every $T > τ$ , the sequence of solutions ${v_{n} (t); v_{n} (t) = S^{h_{n}} v_{τ, n}}$ satisfies $\begin{array}{rcl} \{\begin{matrix} {v_{n}} \subset L^{\infty} (τ - h, T; H) \cap L^{2} (τ - h, T; V), \\ {\frac{\partial}{\partial t} v_{n}} \subset L^{2} (τ, T; V^{'}) \end{matrix} \end{array}$ and there exists a function $v : [τ, + \infty) \to H$ such that $\begin{array}{rcl} \{\begin{matrix} {v (t)} \subset L^{\infty} (τ - h, T; H) \cap L^{2} (τ - h, T; V), \\ {\frac{\partial}{\partial t} v (t)} \subset L^{2} (τ, T; V^{'}) \end{matrix} \end{array}$ with $\begin{array}{rcl} (6.1) & \{\begin{matrix} v_{n} \to v, & strongly in L^{2} (τ, T; H), \\ v_{n} ⇀ v, & weakly in L^{2} (τ - h, T; V), \\ v_{n} (t) \to v (t) & a.e. in [τ, T] . \end{matrix} \end{array}$ For each $n \in N$ and any $s, t \in [τ, T]$ , we have $\begin{array}{l} (v_{n} (t), w) + ν \int_{s}^{t} (A^{1 / 2} v_{n} (l), A^{1 / 2} w) d l + \int_{s}^{t} ⟨ B (v_{n} (l - ρ (l)), v_{n} (l)), w ⟩ d l \\ + \int_{s}^{t} ⟨ B (v_{n} (l - ρ (l)), ψ), w ⟩ d l + \int_{s}^{t} ⟨ B (ψ, v_{n} (l)), w ⟩ d l + \int_{s}^{t} ⟨ B (ψ, ψ), w ⟩ d l \\ = (v_{n} (s), w) + \int_{s}^{t} ⟨ P_{L} g (l, x), w ⟩ d l \\ (6.2) & + \int_{s}^{t} (P_{L} (f (l, u_{n} (l - ρ (l))) + μ F), w) d l, \forall w \in V . \end{array}$ From (6.1), we know that $\begin{matrix} (v_{n} (t), w) \to (v (t), w) for any w \in H as n \to + \infty \end{matrix}$ and $\begin{matrix} ν \int_{s}^{t} (A^{1 / 2} v_{n} (l), A^{1 / 2} w) d l \to ν \int_{s}^{t} (A^{1 / 2} v (l), A^{1 / 2} w) d l as n \to + \infty . \end{matrix}$ Then the hypotheses (H-b) and (H-c) lead to $\begin{matrix} \int_{s}^{t} (P_{L} (g + f (l, u_{n} (l - ρ (l))) + μ F), w) d l \to \int_{s}^{t} (P_{L} (g + f (l, u (l)) + μ F), w) d l, as n \to + \infty . \end{matrix}$ Also, using a similar technique as in Theorem 3.3, we have $\begin{matrix} \int_{s}^{t} ⟨ B (ψ, v_{n} (l)), w ⟩ d l \to \int_{s}^{t} ⟨ B (ψ, v (l)), w ⟩ d l, as n \to + \infty . \end{matrix}$ Thus, it follows that $\begin{array}{l} (v (t), w) + ν \int_{s}^{t} (A^{1 / 2} v (l), A^{1 / 2} w) d l + \int_{s}^{t} ⟨ B (v (l), v (l)), w ⟩ d l \\ + \int_{s}^{t} ⟨ B (v (l), ψ), w ⟩ d l + \int_{s}^{t} ⟨ B (ψ, v (l)), w ⟩ d l + \int_{s}^{t} ⟨ B (ψ, ψ), w ⟩ d l \\ (6.3) & = (v (s), w) + \int_{s}^{t} ⟨ P_{L} g, w ⟩ d l + \int_{s}^{t} (P_{L} (f (t, u (l)) + μ F), w) d l, \forall w \in V, \end{array}$ which means that (3.5) holds, and v is the weak solution to system (3.14).

For any $n \in N$ , since $v_{n}$ is the weak solution to system (3.4) with $h = h_{n}$ , there holds for any $T > τ$ and any positive function $ζ (t) \in C_{0}^{\infty} (τ, T)$ that $\begin{array}{l} - \frac{1}{2} \int_{τ}^{T} {| v_{n} (t) |}^{2} ζ^{'} (t) d t + ν \int_{τ}^{T} {‖ v_{n} (t) ‖}^{2} ζ (t) d t + \int_{τ}^{T} b (v_{n} (t - ρ (t)), ψ, v_{n}) ζ (t) d t \\ ⩽ \int_{τ}^{T} ⟨ g (t, x), v_{n} (t) ⟩ ζ (t) d t + \int_{τ}^{T} (f (t, u_{n} (t - ρ (t))) + ν F, v_{n} (t)) ζ (t) d t \\ (6.4) & - \int_{τ}^{T} b (ψ, ψ, v_{n}) ζ (t) d t . \end{array}$ To reach our goals, we need to get the convergence of each term in (6.4). From (6.1) we know that $| v_{n} (t) |^{2} \to | v (t) |^{2}$ as $n \to + \infty$ for almost all $t \in [τ, T]$ and the sequence ${| v_{n} (t) |^{2} ζ^{'} (t)} \subset L^{1} (τ, T)$ . The Lebesgue dominated convergence theorem then leads to $\begin{matrix} \int_{τ}^{T} {| v_{n} (t) |}^{2} ζ^{'} (t) d t \to \int_{τ}^{T} {| v (t) |}^{2} ζ^{'} (t) d t, as n \to + \infty . \end{matrix}$ From (6.1) together with the property of weak convergence, we get $\begin{array}{l} ν \int_{τ}^{T} {‖ v (t) ‖}^{2} ζ (t) d t ⩽ \underset{n \to + \infty}{lim inf} ν \int_{τ}^{T} {‖ v_{n} (t) ‖}^{2} ζ (t) d t, \\ \int_{τ}^{T} ⟨ g, v_{n} (t) ⟩ ζ (t) d t + \int_{τ}^{T} (f (t, u_{n} (t - ρ (t))) + ν F, v_{n} (t)) ζ (t) d t \\ \to \int_{τ}^{T} ⟨ g, v (t) ⟩ ζ (t) d t + \int_{τ}^{T} (f (t, u (t)) + ν F, v (t)) ζ (t) d t, as n \to + \infty \end{array}$ and $\begin{matrix} lim_{n \to + \infty} \int_{τ}^{T} (B (ψ), v_{n}) ζ (t) d t = \int_{τ}^{T} (B (ψ), v) ζ (t) d t . \end{matrix}$ It thus follows that $\begin{array}{l} - \frac{1}{2} \int_{τ}^{T} {| v (t) |}^{2} ζ^{'} (t) d t + ν \int_{τ}^{T} {‖ v (t) ‖}^{2} ζ (t) d t + \int_{τ}^{T} b (v, ψ, v) ζ (t) d t \\ (6.5) & ⩽ \int_{τ}^{T} ⟨ g, v (t) ⟩ ζ (t) d t + \int_{τ}^{T} (f (t, u (t)) + ν F, v (t)) ζ (t) d t - \int_{τ}^{T} b (ψ, ψ, v) ζ (t) d t, \end{array}$ which is the energy inequality for v to system (3.14), and v is the weak solution. □
Lemma 6.5.
Let $(v (τ), η) \in M_{H}$ , $B^{h} = {B^{h} (t)}_{t \in R}$ be the $D$ -pullback absorbing set of $S^{h} (t, τ)$ in $C_{H} \times L_{V}^{2}$ for system ( 3.4 ), and $B^{0} = {B^{0} (t)}_{t \in R}$ be the $D$ -pullback absorbing set of $S^{0} (t, τ)$ in H for system ( 3.14 ). Then $\begin{matrix} lim_{h \to 0} {dist}_{H} (S^{h} (t, τ) B^{h}, S^{0} (t, τ) B^{0}) = 0 . \end{matrix}$
Proof.
We proceed by contradiction. Assume that there are $t > τ$ , a sequence ${h_{n}} \subset (0, 1)$ , where $h_{n} \to 0$ as $n \to + \infty$ , and $v_{n} (τ) \in B^{h_{n}} (τ) \in B^{h_{n}}$ such that $\begin{matrix} inf_{v \in S^{0} (t, τ) B^{0}} {dist}_{H} (S^{h_{n}} (t, τ) v_{τ, n}, v) ⩾ ε, \forall n \in N . \end{matrix}$ Then, for any $v \in S^{0} (t, τ) B^{0}$ , we obtain $\begin{matrix} {dist}_{H} (S^{h_{n}} (t, τ) v_{n} (τ), v) ⩾ ε, \forall n \in N, \end{matrix}$ which is a contradiction to Lemma 6.4. □
Proof of Theorem 3.9.
It is known that $A^{0} = {A^{0} (t)}_{t \in R}$ is the $D$ -pullback attractor of $S^{0} (t, τ)$ in H for system (3.14). From the definition of $D$ -pullback attractors we know that, for any arbitrarily fixed constant $ε > 0$ , there exists $τ_{ε} < 0$ such that $\begin{matrix} {dist}_{H} (S^{0} (t, τ_{ε}) B^{0}, A^{0}) ⩽ \frac{ε}{2} . \end{matrix}$ From Lemma 6.5 we know that there exists $0 < h_{ε} < 1$ such that $\begin{matrix} {dist}_{H} (S^{h} (t, τ_{ε}) B^{h}, S^{0} (t, τ_{ε}) B^{0}) ⩽ \frac{ε}{2}, \forall h < h_{ε} . \end{matrix}$ And the conclusion that $\begin{matrix} S^{h} (t, τ) A (τ) = A (t), A^{h} \subset B^{h}, \end{matrix}$ leads to $\begin{array}{rcl} {dist}_{H} (A^{h}, S^{0} (t, τ_{ε}) B^{0}) & = & {dist}_{H} (S^{h} (t, τ_{ε}) A^{h}, S^{0} (t, τ_{ε}) B^{0}) \\ (6.6) & ⩽ & {dist}_{H} (S^{h} (t, τ_{ε}) B^{h}, S^{0} (t, τ_{ε}) B^{0}), \end{array}$ i.e., $\begin{matrix} {dist}_{H} (A^{h}, S^{0} (t, τ_{ε}) B^{0}) ⩽ \frac{ε}{2} . \end{matrix}$ In addition, the triangle inequality $\begin{matrix} {dist}_{H} (A^{h}, A^{0}) ⩽ {dist}_{H} (A^{h}, S^{0} (t, τ_{ε}) B^{0}) + {dist}_{H} (S^{0} (t, τ_{ε}) B^{0}, A^{0}) \end{matrix}$ ensures that the conclusion holds. □

7. Conclusion and further research

This work has investigated the well-posedness and pullback dynamics for the 2D Navier–Stokes equations with double time varying delays in a Lipschitz-like domain, which is an extension of Yang, Wang, Yan and Miranville in DCDS 41(7) (2021), 3343–3366. However, when the delay is infinite, the dynamics and asymptotic stability is still open. This will be addressed in future works.

Footnotes

Acknowledgements

This work is inspired by discussions at the Université de Poitiers during a visit of Xin-Guang Yang, which was partially supported by the Key project of Henan Education Department (No. 22A110011), Incubation Fund Project of Henan Normal University (No. 2020PL17) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003).

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Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains

Abstract

Keywords

1. Introduction

2. Preliminaries

Lemma 2.1 (See [3]).

Lemma 2.2 (See [20]).

Lemma 2.3. Let X and X 0 be two Banach spaces, X 0 = X ′ , and 1 < p < ∞ . If v n ⇀ v , weakly in L p ( 0 , T ; X ) and ∂ ∂ t v n ⇀ ∂ ∂ t v , weakly in L p ( 0 , T ; X 0 ) hold, then v n ( 0 ) ⇀ v ( 0 ) weakly in X. 3. Main results

3.1. The Stokes problem in a Lipschitz-like domain

3.2. Assumptions

3.3. Main results for the equivalent system

Proof. See Section 4.1. □ Theorem 3.4 (Uniqueness).

Proof. See Section 4.2. □ ∙ Regularity In this part, we shall use the technique in [3] to study the regularity of the solution v to system (3.4) obtained in Theorems 3.3 and 3.4. Theorem 3.5 (Regularity of the global solution).

Proof. See Section 6. □ 3.4. Back to the original problem (1.1)

4.1. Existence of a global weak solution: Proof of Theorem 3.3

4.2. Uniqueness: Proof of Theorem 3.4

4.3. Regularity of global solutions: Proof of Theorem 3.5

5. Pullback dynamics

5.1. Theory of tempered pullback dynamics

Remark 5.9. By the properties of universe D κ δ ( t , τ ) H × C H in (5.5), we can easily verify that the above universes are inclusion closed and contain any fixed universe and uniformly bounded absorbing set. Lemma 5.10 (The tempered pullback absorbing set in H × C H ).

6.1. The theory of robustness

Lemma 6.1 (See [10]).

Footnotes

Acknowledgements

References

Proof.
See Section 4.1. □
Theorem 3.4 (Uniqueness).

Proof.
See Section 4.2. □

∙ Regularity

In this part, we shall use the technique in [3] to study the regularity of the solution v to system (3.4) obtained in Theorems 3.3 and 3.4.
Theorem 3.5 (Regularity of the global solution).

Proof.
See Section 6. □

3.4. Back to the original problem (1.1)

Remark 5.9.
By the properties of universe $D_{κ_{δ} (t, τ)}^{H \times C_{H}}$ in (5.5), we can easily verify that the above universes are inclusion closed and contain any fixed universe and uniformly bounded absorbing set.
Lemma 5.10 (The tempered pullback absorbing set in $H \times C_{H}$ ).