Dynamics and stability of the 3D Brinkman–Forchheimer equation with variable delay (I)

Abstract

The tempered pullback dynamics of the 3D Brinkman–Forchheimer equation with variable delay has been studied in this paper. With the different universes which has some topology property, the existence of minimal and unique family of pullback attractors were obtained. Moreover, the convergence of pullback attractors for the 3D Brinkman–Forchheimer equation as delay term vanishes is also been proved, i.e., the upper semi-continuity of attractors.

Keywords

Brinkman–Forchheimer equation delay upper semi-continuity universe tempered pullback attractors

1. Introduction

The Darcy law is an equation that describes the flow of a fluid through a porous medium, which was determined experimentally by Darcy based on the results of experiments, and derived from the Navier–Stokes equations via homogenization, see [6,25,26,29]. The Brinkman equation is a generalization of Darcy’s law that facilitates the matching of boundary conditions at an interface between the larger pores and the permeable medium. For flows in porous media with Reynolds numbers greater than about 1 to 10, inertial effects can also become significant. Sometimes an inertial term is added to the Darcy law, known as Forchheimer term. The governing equation (the momentum equation in a porous medium) for the above porous medium fluid flow is known as the nonlinear Brinkman–Forchheimer (BF in short) equation, see Gilver and Altobelli [8], Nield [16], which dealt with an important and classical problem involving the fluid mechanics of the interface region between a porous medium and a fluid layer.

The delay influence on differential equations was investigated in past decades which is also used in control theory and engineer especially from the mathematical analysis in physics, non-instant transmission phenomena and specially biological motivations. In this paper, we consider the tempered pullback dynamics and convergence (upper semi-continuity) of pullback attractors as delay vanishes for the 3D delayed Brinkman–Forchheimer equation: $\begin{array}{rcl} (1.1) & \{\begin{matrix} u_{t} - ν Δ u + α u + β | u | u + γ | u |^{2} u + \nabla p = f (x, u_{t}) + g (x, t), (x, t) \in Ω \times R^{+}, \\ \nabla \cdot u = 0, (x, t) \in Ω \times R^{+}, \\ u (t, x) |_{\partial Ω} = 0, t \in R^{+} \\ u_{τ} (θ, x) = u (τ + θ, x) = ϕ (θ, x), θ \in [- h, 0], x \in Ω, \end{matrix} \end{array}$ here $Ω \subset R^{3}$ be a bounded domain with sufficiently smooth boundary $\partial Ω$ , $u = (u_{1}, u_{2}, u_{3})$ is the velocity vector field, p is the pressure, $ν > 0$ is the Brinkman kinematic viscosity coefficient, $α > 0$ is the Darcy coefficient, $β > 0$ and $γ > 0$ are the Forchheimer coefficients, $f (x, u_{t})$ is the delay term, the external force $g (x, t)$ is locally square integrable in time for $(x, t) \in Ω \times R$ , i.e., for any $t \in R$ , $g (x, t) \in L_{loc}^{2} (R; H)$ . The function $ϕ (θ, x)$ be the initial data, which contains $ϕ (0, x) = φ (x) = u (t = τ, x)$ , $u_{t}$ is defined as $u_{t} (θ) = u (t + θ)$ with $θ \in [- h, 0]$ .

Let us recall some known results for the Brinkman–Forchheimer equation.

From the global well-posedness of 3D BF model in mathematical analysis, Vafai and Kim [25,26] obtained an exact solution to this problem using a Brinkman–Forchheimer-extended Darcy equation. Whitaker [30] investigated the theoretical development of the Forchheimer equation.

From physical viewpoint, what about the continuous dependence on the Darcy and Forchhrimer coefficients α, β, γ, and the convergence as these quantities tends to 0, even though the variable viscosity. One can refer to Celebi, Kalantarov and Ugurlu [4,5] (continuous dependence on coefficients in $H^{1}$ -norm which reflects the effect of small changes in coefficients of equations on the solutions), Liu [14], Payne, Song and Straughan [18], Payne and Straughan [19], Vafai and Kim [26], Zhao and You [32].

For the stability and infinite dimensional dynamic systems that we concern, Qin, Kaloni [20] studied the spatial decay estimate for plane flow of BF model. Uğurlu [24], Yan and Ouyang [17] gave the existence of global attractor in $H_{0}^{1} (Ω)$ respectively. [24] and [28] obtained the existence of global attractor for BF flow model by different methods. Song [21] deduced the existence of $D$ -pullback attractors family for 3D non-autonomous BF equation. You, Zhao and Zhou [31], Song and Hou [22] presented the existence of uniform attractors for non-autonomous model. With slip boundary condition of friction type, Djoko and Mbehoua [7] also show the global attractor for BF equation. Kalantarov and Zelik [9] investigated smoothness and regularity of global attractor for BF model with fast growing nonlinearities.

Although there are fruitful results about the dynamic systems for (1.1) presented above, till we know now, the tempered dynamic systems and convergence of pullback attractors as delay vanishes is still an open problem. Our objective in this paper is to investigate these issues. The main features and crucial technique can be summarized as following:

Based on different universes which has some topology property such as inclusion closed, the minimal and unique families of tempered pullback attractors has been achieved by the existence of more regular absorbing set and its compact embedding. This is an improvement of compact and bounded $D$ pullback attractors in some references Kang and Park [10], Song [21], i.e., we are no need the absorbing set compact or bounded.

By defining a projector as skew product flow which mapping $M_{H} = C_{H} \times H$ to $C_{H}$ and H, using the theory of dynamic systems, we can obtained the existence of global attractors in $M_{H}$ , its projection to $C_{H}$ is the family of pullback attractors. This is different from the injection mapping in [15].

The literatures [1,2,15] and its expansions have studied the pullback attractors for 2D Navier–Stokes equation with delays, but there is no result on the upper semi-continuity as delay effect vanishes, i.e., asymptotic stability as delay disappears. Inspired by [3,11] and [13], we presented the convergence of pullback attractors as delay tends to 0. The most difficult here is to deal with uniformly bounded estimate of nonlinear terms which is independent on delay parameter h.

The plan of this paper is arranged as following. In Section 2, we shall give some preliminary which is used for showing the main results in sequel. Then the proof of main theorems are presented in Section 3. The conclusion and further research is stated in Section 4.

2. Main results

2.1. Some notations

Throughout this paper, C will stand for all positive constants, which depend on Ω and some constants, but independent of the choice of τ and t. The set $L^{p} (Ω)$ and $H^{m} (Ω)$ are the general Lebesgue and Sobolev spaces respectively. The Hausdorff semi-distance ${dist}_{X} (O_{1}, O_{2})$ in X between two sets $O_{1}$ and $O_{2}$ is defined as $\begin{matrix} {dist}_{X} (O_{1}, O_{2}) = sup_{x \in O_{1}} inf_{y \in O_{2}} d_{X} (x, y), for O_{1}, O_{2} \subset X . \end{matrix}$

We set $\begin{array}{l} E : = {u | u \in {(C_{0}^{\infty} (Ω))}^{3}, div u = 0}, \\ H = {cl}^{{(L^{2} (Ω))}^{3}} E, V = {cl}^{{(H_{0}^{1} (Ω))}^{3}} E, \end{array}$ where ${cl}^{X}$ denotes the closure in the space X. H and V endowed with the inner products $\begin{array}{rcl} (u, v) = \int_{Ω} u \cdot v d x, for all u, v \in H, \end{array}$ and $\begin{array}{rcl} ((u, v)) = \sum_{i = 1}^{3} \int_{Ω} \nabla u_{i} \cdot \nabla v_{i} d x, for all u, v \in V \end{array}$ respectively. Denoting $V^{*}$ as the dual space of V, then it yields $V ↪ ↪ H \equiv H^{*} ↪ V^{*}$ , where the embedding is dense and continuous.

In this paper, we denote $L^{p} (Ω) = {(L^{p} (Ω))}^{3}$ and use $‖ \cdot ‖_{p}$ to denote the norm in $L^{p} (Ω)$ , especially use $‖ \cdot ‖$ for $L^{2} (Ω)$ . The notation $‖ \cdot ‖_{*}$ and $⟨ \cdot, \cdot ⟩$ denote the norm in $V^{*}$ and inner product between dual spaces V and $V^{*}$ respectively.

For simplicity, let P be the Helmholtz-Leray orthogonal projection operator from $L^{2} (Ω)$ onto H. We define here the operator $A : V \to V^{*}$ by $A u = - P Δ u$ , and so $\begin{matrix} ⟨ A u_{1}, u_{2} ⟩ = \int_{Ω} \nabla u_{1} \cdot \nabla u_{2} d x, for all u_{1}, u_{2} \in V, \end{matrix}$ where $A = P (- Δ)$ is the Stokes operator with the domain $D (A) = {(H^{2} (Ω))}^{3} \cap V$ , then the operator A has the property $⟨ A u, v ⟩ = ((u, v))$ for all $u, v \in V$ . Observe that on $D (A)$ the norms $‖ \cdot ‖_{{(H^{2} (Ω))}^{3}}$ and $‖ \cdot ‖_{D (A)}$ are equivalent, and $D (A)$ is compactly and densely embedding in V, $λ_{1} > 0$ be the first eigenvalue of the Stokes operator A.

2.2. Some assumption

We impose the following hypotheses on system (1.1): $(H_{f})$

The function $f : Ω \times R \to R$ satisfies the following:

For any $y \in R$ , the function $f (\cdot, y)$ is measurable and $f (\cdot, 0) \equiv 0$ .

For any $x \in Ω$ , $f (x, \cdot)$ is Lipschitz continuous, that is, there is $C_{f} > 0$ such that $\begin{matrix} | f (x, y_{2}) - f (x, y_{1}) | ⩽ C_{f} | y_{2} - y_{1} |, for y_{2}, y_{1} \in R . \end{matrix}$

From

(b)

, it is easy to observe that, there exists

L_{f} > 0

, such that for all

x \in Ω

\begin{array}{rcl} (2.1) & {‖ f (x, φ_{2}) - f (x, φ_{1}) ‖}_{C_{H}} ⩽ L_{f} ‖ φ_{2} - φ_{1} ‖_{C_{H}}, for φ_{2}, φ_{1} \in C_{H} . \end{array}

(H_{g})

The function $g (\cdot, \cdot) \in L_{loc}^{2} (R, L^{2} (Ω)) \cap L_{loc}^{4} (R, L^{4} (Ω))$ , there exists $m > 0$ such that, for any $t \in R$ , $\begin{array}{rcl} (2.2) & \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}^{2} < \infty . \end{array}$

(H_{0})

The coefficients satisfy $\begin{array}{rcl} (2.3) & α + 3 β + 2 γ - m - \frac{2 L_{f}^{2}}{α} e^{m h} ⩾ 0 . \end{array}$ Moreover, if $α + 3 β + 2 γ = m + \frac{2 L_{f}^{2}}{α} e^{m h}$ , we denote it’s solution as index $m_{0}$ .

2.3. Well-posedness: Global existence, uniqueness and continuous dependence on initial data of weak solution

Defining the space $X_{H} = C ([τ - h, T]; H) \times C ([τ, T]; H)$ and $M_{H} = C_{H} \times H$ , where $C_{H} = C ([- h, 0]; H)$ with the norm $‖ φ ‖_{C_{H}} : = {sup}_{θ \in [- h, 0]} ‖ φ (θ) ‖_{H}$ , for any $φ \in C_{H}$ , then we can give the definition of weak solutions for (1.1).

Definition 2.1.
A global weak solution of equation (1.1) is a function $u \in C ([τ - h, T]; H) \cap L^{2} (τ, T; V) \cap L^{4} (τ, T; L^{4} (Ω))$ for some $T > τ$ satisfying that $u_{τ} (θ, x) = u (τ + θ, x) = ϕ (θ, x)$ with $θ \in [- h, 0]$ , and for any $w \in V$ , the formula $\begin{matrix} (\frac{d}{d t} u (t), w) + ν (A u, w) + α (u, w) + β (| u | u, w) + γ (| u |^{2} u, w) = (f (x, u_{t}), w) + (g, w), \end{matrix}$ holds a.e. in $[τ - h, T]$ , or equivalently as the following sense $\begin{matrix} \frac{d}{d t} u (t) + ν A u + α u + β | u | u + γ | u |^{2} u = f (x, u_{t}) + g (t, x) in D^{'} (τ, T; V^{}) . \end{matrix}$

The existence of global weak solution can be presented as the following theorem.
Theorem 2.2.
Assume that the external forces* $g (\cdot, \cdot)$ and $f (\cdot, \cdot)$ satisfy the hypotheses $(H_{f})$ and $(H_{g})$ , the initial data $(ϕ, φ) \in M_{H} = C_{H} \times H$ and $(H_{0})$ is also true. Then there exists a unique global weak solution $u = u (\cdot, τ, ϕ, φ)$ of equation ( 1.1 ) on $[τ - h, T]$ .
Proof.
See Section 3.2. □

The continuous dependence on initial data of solutions can be stated in the following theorem. Theorem 2.3.
Under the assumptions of Theorem 2.2 , then the global weak solution of equation ( 1.1 ) is continuous dependent on the initial data ϕ, i.e., if $u^{i}$ , $i = 1, 2$ , be two solutions corresponding to the initial data $ϕ^{i} \in C_{H}$ , $i = 1, 2$ , then the following estimate holds $\begin{array}{rcl} (2.4) & {‖ u_{t}^{1} - u_{t}^{2} ‖}_{C_{H}}^{2} ⩽ {‖ ϕ^{1} - ϕ^{2} ‖}_{C_{H}}^{2} e^{2 L_{f} (t - τ)} . \end{array}$
Proof.
See Section 3.3. □
Remark 2.4.

Based on the well-posedness, we can define the semi-flow in space $M_{H}$ as $S (t, τ) = (U^{h} (t, t - τ), U (t, t - τ)) : M_{H} \to M_{H}$ , i.e., $S (t, τ) (ϕ, φ) = (u_{t}, u)$ .

Defining the projectors $Π_{1}$ and $Π_{2}$ from $M_{H}$ to $C_{H}$ and H respectively, then we have $\begin{array}{rcl} (2.5) & Π_{1} M_{H} = C_{H}, Π_{2} M_{H} = H . \end{array}$

2.4. Pullback dynamics of 3D B-F equation with delay

Based on the continuity of semiflow, the existence of absorbing ball in $M_{H}$ and $M_{V}$ and the compact embedding $V ↪ ↪ H$ , we can obtain the existence of global pullback attractor for semiflow $S (t, t - τ)$ as following.

Theorem 2.5.
Assume that the external forces $g (\cdot, \cdot)$ and $f (\cdot, \cdot)$ satisfy the hypotheses $(H_{f})$ and $(H_{g})$ , the initial data $(ϕ, φ) \in M_{H} = C_{H} \times H$ . Then the semiflow generated by problem ( 1.1 ) possesses a compact family of global pullback attractors $A (t)$ in $M_{H}$ . Moreover, the family of pullback attractors for process $U^{h} (t, t - τ)$ is stated as $Π_{1} A = A^{h}$ for $h > 0$ , and $Π_{2} A = A^{0}$ for the process $U (t, t - τ) = U^{0} (t, t - τ)$ .

∙ The universes

Considering any $m \in (0, m_{0}]$ fixed. Then, the solution u to (1.1) satisfies that $\begin{array}{rcl} (2.6) & ‖ u_{t} ‖_{C_{H}}^{2} ⩽ e^{- m σ} \frac{2 L_{f}^{2} e^{m h}}{α} e^{m h} ‖ φ ‖_{C_{H}}^{2} + C_{0} e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} (1 + {‖ g (s, \cdot) ‖}^{2}) d s \end{array}$ for all $t ⩾ τ$ .
Remark 2.6.
Assume that $g \in L_{loc}^{2} (R, H)$ , if we fix $m \in (0, m_{0}]$ , the inequality (2.6) still holds from the Poincaré inequality and different estimates, here we omit the detail.

Based on the estimate (2.6), we can define the following universes for tempered dynamics.
Definition 2.7 (Universe).

We will denote by $D_{m}^{H}$ the class of all families of nonempty subsets $\hat{D} = {D (t) : t \in R} \subset P (H)$ such that $\begin{matrix} (2.7) & lim_{τ \to - \infty} (e^{m τ} sup_{v \in D (τ)} e^{- m t} (1 + \frac{2 L_{f}^{2} e^{m h}}{α}) e^{m h} ‖ v ‖_{C_{H}}^{2}) = 0 . \end{matrix}$

$D_{F}^{H}$ will denote the class of families $\hat{D} = {D (t) = D : t \in R}$ with D a fixed nonempty bounded subset in $C_{H}$ .

Remark 2.8.
We observe that $D_{F}^{H} \subset D_{m}^{H}$ and that both of them are inclusion-closed which means the universes has some topology properties.

∙ The $D_{m}^{H}$ and $D_{F}^{H}$ pullback absorbing families in $C_{H}$

For $m \in (0, m_{0}]$ , the family ${\hat{D}}_{0} = {D_{0} (t) : t \in R}$ defined by $D_{0} (t) = {\overline{B}}_{C_{H}} (0, ρ_{C_{H}} (t))$ , the closed ball in $C_{H}$ with center zero and radius $ρ_{C_{H}} (t)$ , where $\begin{matrix} ρ_{C_{H}}^{2} (t) = 1 + C_{0} e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}^{2} d s, \end{matrix}$ then ${\overline{B}}_{C_{H}} (0, ρ_{C_{H}} (t))$ is $D_{m}^{H}$ -pullback absorbing families in $C_{H}$ for the process $U^{h} (t, t - τ)$ , and ${\hat{D}}_{0} \in D_{m}^{H}$ . Theorem 2.9.
Assume that the external forces $g (\cdot, \cdot)$ and $f (\cdot, \cdot)$ satisfy the hypothesis $(H_{f})$ and $(H_{g})$ hold, the initial data $(ϕ, φ) \in M_{H} = C_{H} \times H$ and $(H_{0})$ is also true for some $m \in (0, m_{0}]$ . Then, the process $U^{h} (t, t - τ)$ for problem ( 1.1 ) possesses a minimal family of $D_{m}^{H}$ -pullback attractors $\begin{matrix} A_{D_{m}^{H}} = {A_{D_{m}^{H}} (t) : t \in R}, \end{matrix}$ and $D_{F}^{H}$ -pullback attractors $\begin{matrix} A_{D_{F}^{H}} = {A_{D_{F}^{H}} (t) : t \in R} \end{matrix}$ which has the relationship $\begin{array}{rcl} (2.8) & A_{D_{F}^{H}} \subset A_{D_{m}^{H}} . \end{array}$ Moreover, if $m \in (0, m_{0})$ holds, then the process $U^{h} (t, t - τ)$ possesses a unique family of pullback attractors $A_{D_{m}^{H}}^{u}$ .
Proof.
See Section 4.4. □
Remark 2.10.
For the more strict index $m \in (0, 2 ν λ_{1} + 2 α + 3 β + 2 γ]$ , we can also derive the same results in Theorem 2.9, but it is nor clear for the relation between the families of pullback attractors with different indexes.
Remark 2.11.
In Theorem 2.9, the absorbing set is no need to be bounded or compact, the pullback attractors we obtained have tempered property. Moreover, if the absorbing sets are bounded or compact with pullback attractors $A_{b}$ and $A_{c}$ which means $A_{c} \subset A_{b} \subset A_{D_{F}^{H}} \subseteq A_{D_{m}^{H}}$ .

2.5. The convergence of pullback attractors as delay effect vanishes

This section is devoted to the upper semi-continuity of pullback attractors with topology of H. Considering $h \in (0, 1]$ , then we can write the solutions and continuous process of problem (1.1) as $u^{h}$ and $U^{h} (\cdot, \cdot)$ , and for $h = 0$ as $u^{0}$ and $U^{0} (\cdot, \cdot)$ respectively.

∙ The BF model with parameter $h > 0$ : pullback attractors

For given $t \in R$ , let $\begin{array}{rcl} (2.9) & B_{h} (t) = {u \in C_{H} : ‖ u ‖_{C_{H}}^{2} ⩽ ρ_{C_{H}}^{2} (t)}, \end{array}$ where $ρ_{C_{H}}^{2} (t) = 1 + C_{0} e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} ‖ g (s, \cdot) ‖^{2} d s$ . Since the estimates in Section 4 are uniform with respect to h in $(0, 1]$ , by (2.9) and Lemma 4.14, we find that $B_{h} = {B_{h} (t) : t \in R}$ is a family of pullback absorbing sets for the process $U^{h}$ in $C_{H}$ for all $h \in (0, 1]$ . Let $A_{h} (t)$ be the pullback attractors of $U^{h} (\cdot, \cdot)$ . Then we have that for every $t \in R$ , $\begin{array}{rcl} (2.10) & A_{D_{m}^{H}} (t) = A_{h} (t) \subset B_{h} (t) . \end{array}$

∙ The BF equation with parameter $h = 0$ : pullback attractors with delay term vanishes

For $h = 0$ , the system (1.1) reduces to $\begin{array}{rcl} (2.11) & \frac{\partial u}{\partial t} - ν Δ u + α u + β | u | u + γ | u |^{2} u + \nabla p = f (x, u) + g (t, x), \end{array}$ with boundary condition $\begin{array}{rcl} (2.12) & u (t, x) = 0, x \in \partial Ω, t > τ, \end{array}$ and initial data $\begin{array}{rcl} (2.13) & u (τ, x) = φ (x), x \in Ω . \end{array}$ We use $U^{0}$ to denote the continuous process generated by problem (2.11)–(2.13).

Since all the estimates in Section 4 are still valid for $h = 0$ , we deduce that $U^{0} (\cdot, \cdot)$ possesses a pullback attractor $A_{0} = {A_{0} (t) : t \in R}$ in H and a compact absorbing set $B_{0} = {B_{0} (t) : t \in R}$ given by $\begin{array}{rcl} (2.14) & B_{0} (t) = {u \in H : ‖ u ‖_{H}^{2} ⩽ ρ_{0}^{2} (t)}, \end{array}$ where $ρ_{0}^{2} (t) = 1 + c_{1} e^{- m t} \int_{- \infty}^{t} e^{m s} ‖ g (s, \cdot) ‖^{2} d s$ . It is trivial from (2.9) and (2.14) that for all $t \in R$ , $\begin{array}{rcl} (2.15) & lim_{h \to 0} sup ρ_{C_{H}}^{2} (t) = ρ_{0}^{2} (t) . \end{array}$

∙ The convergence from pullback attractors $A_{h}$ to $A_{0}$ as h tends to 0

For some $m > 0$ , we can conclude the convergence (the upper semi-continuity) of pullback attractors as $h \to 0^{+}$ .

Theorem 2.12.
Assume that the external forces $g (\cdot, \cdot)$ and $f (\cdot, \cdot)$ satisfy the hypotheses $(H_{f})$ and $(H_{g})$ hold, the initial data $(ϕ, φ) \in M_{H} = C_{H} \times H$ and $(H_{0})$ is also true for some $m \in (0, m_{0}]$ . Then for every $t \in R$ , we have the upper semi-continuity of pullback attractors $A_{h}$ and $A_{0}$ in H-topology as following $\begin{matrix} lim_{h \to 0^{+}} {dist}_{H} (A_{h} (t), A_{0} (t)) = 0, \end{matrix}$ where ${dist}_{H}$ is the Hausdorff semi-distance defined in ( 5.1 ).
Proof.
See Section 5.2. □

3. Proof of Theorems 2.2 and 2.3: Well-posedness

3.1. Some lemmas

The following Aubin–Lions Compactness Lemma is needed in order to construct solutions for our problem.

Lemma 3.1 (See [12,23]).

Let X, Y, and Z be separable, reflexive Banach spaces, where X is compactly embedded in Y, and Y is continuously embedded in Z. Let $T > 0$ , $p \in (1, \infty)$ and let ${f_{n} (t, \cdot)}_{n = 1}^{\infty}$ be a bounded sequence in $L^{p} ([0, T]; X)$ such that ${\frac{\partial f_{n}}{\partial t}}_{n = 1}^{\infty}$ is bounded in $L^{p} ([0, T]; Z)$ . Then ${f_{n}}_{n = 1}^{\infty}$ has a strongly convergent subsequence in $C ([0, T]; Y)$ .

We will also use the following lemma, which will be useful in the proof of the existence and uniqueness of global solution.

Lemma 3.2 (See [12]).

Assume that $p ⩾ 2$ , then for any $a, b \in R^{n}$ , $\begin{matrix} (| a |^{p - 2} a - | b |^{p - 2} b) \cdot (a - b) ⩾ γ_{0} | a - b |^{p}, \end{matrix}$ where $γ_{0} > 0$ is a constant, and $min γ_{0} = \frac{1}{2} 6^{- \frac{p}{2}}$ .

3.2. Existence of global solution

Proof of Theorem 2.2.
We divide the proof into several steps.

Step 1. The Galerkin approximation.

Let ${e_{j} : j \in N} \subset V$ be an orthonormal basis of $L^{2} (Ω)$ such that span ${e_{j} : j \in N}$ is dense in V. Given $n \in N$ , denote by $X_{n} = span {e_{j} : j = 1, 2, \dots, n}$ and the projector $P_{n} u = \sum_{j = 1}^{n} (u, e_{j}) e_{j}$ , for $u \in L^{2} (Ω)$ .

Define also $\begin{matrix} u_{n} (t) = \sum_{j = 1}^{n} α_{n, j} (t) e_{j}, \end{matrix}$ where $α_{n, j} (t)$ are the coefficients so that $u_{n} (t)$ satisfies the following systems of functional differential equations, for each $1 ⩽ j ⩽ n$ , $\begin{array}{l} (\frac{d}{d t} u_{n} (t), e_{j}) + ν (A u_{n}, e_{j}) + α (u_{n}, e_{j}) + β (| u_{n} | u_{n}, e_{j}) + γ (| u_{n} |^{2} u_{n}, e_{j}) \\ (3.1) & = (f (x, u_{n, t}), e_{j}) + (g (t, x), e_{j}), \end{array}$ which holds a.e. $t > τ$ , with the initial conditions $u_{n, τ} (θ) = u_{n} (τ + θ) = P_{n} φ (θ)$ for $θ \in [- h, 0]$ , and the equations are understood in the sense of distribution. Note that the above systems of functional differential equations (3.1) possess a unique local solution defined on $[τ - h, t_{n})$ , with $τ < t_{n} < \infty$ .

Step 2. A priori estimates.

We will perform some estimates ensuring that the solutions do exist for the whole interval $[τ - h, \infty)$ . Let $τ < T < t_{n}$ be fixed. Multiplying (3.1) by $α_{n, j} (t)$ and summing in j, we obtain, for a.e. $t \in [τ, T]$ , that $\begin{matrix} \frac{d}{d t} ‖ u_{n} ‖^{2} + 2 ν ‖ \nabla u_{n} ‖^{2} + 2 α ‖ u_{n} ‖^{2} + 2 β ‖ u_{n} ‖_{3}^{3} + 2 γ ‖ u_{n} ‖_{4}^{4} = 2 (f (x, u_{n, t}), u_{n}) + 2 (g (t, x), u_{n}) . \end{matrix}$ Using Young’s inequality, (2.1), we obtain $\begin{array}{rcl} (3.2) & 2 (g (t, x), u_{n}) ⩽ \frac{2}{α} {‖ g (t, \cdot) ‖}^{2} + \frac{α}{2} ‖ u_{n} ‖^{2}, \end{array}$ and $\begin{array}{rcl} (3.3) & 2 (f (x, u_{n, t}), u_{n}) ⩽ \frac{2 L_{f}^{2}}{α} ‖ u_{n, t} ‖_{C_{H}}^{2} + \frac{α}{2} ‖ u_{n} ‖^{2} . \end{array}$ Hence, it follows from (3.2) and (3.3) that $\begin{matrix} \frac{d}{d t} ‖ u_{n} ‖^{2} + 2 ν ‖ \nabla u_{n} ‖^{2} + α ‖ u_{n} ‖^{2} + 2 β ‖ u_{n} ‖_{3}^{3} + 2 γ ‖ u_{n} ‖_{4}^{4} ⩽ \frac{2 L_{f}^{2}}{α} ‖ u_{n, t} ‖_{C_{H}}^{2} + \frac{2}{α} {‖ g (t, \cdot) ‖}^{2} . \end{matrix}$ Integrating the above inequality from τ to t, we get $\begin{array}{l} {‖ u_{n} (t) ‖}^{2} - {‖ u_{n} (τ) ‖}^{2} + 2 ν \int_{τ}^{t} {‖ \nabla u_{n} (s) ‖}^{2} d s + α \int_{τ}^{t} {‖ u_{n} (s) ‖}^{2} d s \\ + 2 β \int_{τ}^{t} {‖ u_{n} (s) ‖}_{3}^{3} d s + 2 γ \int_{τ}^{t} {‖ u_{n} (s) ‖}_{4}^{4} d s \\ (3.4) & ⩽ \frac{2 L_{f}^{2}}{α} \int_{τ}^{t} ‖ u_{n, s} ‖_{C_{H}}^{2} d s + \frac{2}{α} \int_{τ}^{t} {‖ g (s, \cdot) ‖}^{2} d s . \end{array}$ Replacing t by $t + θ$ in (3.4) with $θ \in [- h, 0]$ , we deduce that $\begin{array}{l} ‖ u_{n, t} ‖_{C_{H}}^{2} + 2 ν \int_{τ}^{t + θ} {‖ \nabla u_{n} (s) ‖}^{2} d s + α \int_{τ}^{t + θ} {‖ u_{n} (s) ‖}^{2} d s \\ + 2 β \int_{τ}^{t + θ} {‖ u_{n} (s) ‖}_{3}^{3} d s + 2 γ \int_{τ}^{t + θ} {‖ u_{n} (s) ‖}_{4}^{4} d s \\ ⩽ ‖ φ ‖_{C_{H}}^{2} + \frac{2 L_{f}^{2}}{α} \int_{τ}^{t} ‖ u_{n, s} ‖_{C_{H}}^{2} d s + \frac{2}{α} \int_{τ}^{t} {‖ g (s, \cdot) ‖}^{2} d s . \end{array}$ and by Gronwall’s inequality, it yields $\begin{array}{rcl} (3.5) & ‖ u_{n, t} ‖_{C_{H}}^{2} ⩽ R (t, φ) : = (‖ φ ‖_{C_{H}}^{2} + \frac{2}{α} \int_{τ}^{t} {‖ g (s, \cdot) ‖}^{2} d s) e^{\frac{2 L_{f}^{2}}{α} (t - τ)}, for τ ⩽ t ⩽ T . \end{array}$ from which and (3.4), we obtain that $\begin{array}{rcl} (3.6) & \int_{τ}^{t} {‖ u_{n} (s) ‖}^{2} d s ⩽ \frac{1}{α} ‖ φ ‖_{C_{H}}^{2} + \frac{2 L_{f}^{2}}{α^{2}} \int_{τ}^{t} {‖ R (s, φ) ‖}^{2} d s + \frac{2}{α^{2}} \int_{τ}^{t} {‖ g (s, \cdot) ‖}^{2} d s \end{array}$

Hence, we obtain that there exists a constant $C = C (T, φ) > 0$ , such that $\begin{matrix} ‖ u_{n, t} ‖_{C_{H}}^{2} ⩽ C (T, φ), for all t \in [τ, T], n ⩾ 1, \end{matrix}$ which implies that $t_{n} = \infty$ for all n. And since $u_{n, τ} (θ) = P_{n} φ (θ)$ for $θ \in [- h, 0]$ , we obtain that $\begin{matrix} the sequence {u_{n}} is bounded in L^{\infty} (τ - h, T; H) . \end{matrix}$ Moreover, it follows from (3.4) and (3.5) that $\begin{array}{rcl} (3.7) & the sequence {u_{n}} is bounded in L^{\infty} (τ - h, T; H) \cap L^{2} (τ, T; V) \cap L^{4} (τ, T; L^{4} (Ω)) . \end{array}$ On the other hand, by the definition of operator A and (3.7), we observe that $\begin{array}{rcl} (3.8) & the sequence {A u_{n}} is bounded in L^{2} (τ, T; V^{}) . \end{array}$ Thus, we infer from (3.8) and (3.7) that $\begin{matrix} the sequence {\frac{d u_{n}}{d t}} is bounded in L^{2} (τ, T; V^{}) . \end{matrix}$

Step 3. Compactness results.

There exist functions $u \in L^{\infty} (τ - h, T; H) \cap L^{2} (τ, T; V) \cap L^{4} (τ, T; L^{4} (Ω))$ , $χ \in L^{2} (τ, T; V^{})$ and $ζ \in L^{2} (τ, T; V^{})$ such that $\begin{array}{l} u_{n} \to u weakly star in L^{\infty} (τ - h, T; H), \\ (3.9) & u_{n} \to u weakly in L^{2} (τ, T; V), \\ u_{n} \to u weakly in L^{4} (τ, T; L^{4} (Ω)), \\ (3.10) & A u_{n} \to χ weakly in L^{2} (τ, T; V^{}), \\ (3.11) & \frac{d u_{n}}{d t} \to ζ weakly in L^{2} (τ, T; V^{}) . \end{array}$ Note that $u_{n}, u \in C (τ, T; H)$ . Furthermore, note that $V ↪ H ↪ V^{}$ and the embedding is compact, then by Aubin–Lions compactness result we observe that there is a subsequence of $u_{n}$ converging strongly to u in $L^{2} (τ, T; H)$ .

We next verify $ζ = \frac{d u}{d t}$ in (3.11). Since the approximate solutions $u_{n}$ satisfy $\begin{matrix} u_{n} (s) = P_{n} φ (0) + \int_{τ}^{s} \frac{d u_{n}}{d t} d t, for s \in [τ, T], n = 1, 2, \dots . \end{matrix}$ Evidently, $P_{n} φ (0) \to φ (0)$ as $n \to + \infty$ , then by taking the weak limits as $n \to + \infty$ in the above equality and using (3.11), we get readily that $\begin{matrix} u (s) = φ (0) + \int_{τ}^{s} ζ d t, for s \in [τ, T], n = 1, 2, \dots \end{matrix}$ which means $ζ = \frac{d u}{d t}$ . So we can write $\begin{array}{rcl} (3.12) & \frac{d u_{n}}{d t} \to \frac{d u}{d t} weakly in L^{2} (τ, T; V^{}) . \end{array}$

Step 4. Strong convergence.

We then prove $u_{n, t} \to u_{t}$ in $‖ \cdot ‖_{C_{H}}$ . For this purpose, it suffices to verify $\begin{array}{rcl} (3.13) & u_{n} \to u in C ([τ, T]; H), \end{array}$ since $P_{n} φ \to φ$ in $C_{H}$ . Due to the strong convergence of ${u_{n}}$ to u in $L^{2} (τ, T; H)$ , we infer that there is a subsequence (relabeled as $u_{n}$ ) converging to $u (t)$ in $L^{2} (Ω)$ a.e. $t \in [τ, T]$ .

Let $u_{n}$ and $u_{m}$ satisfy the approximated equation and set $w (t) = u_{n} (t) - u_{m} (t)$ , then we have $\begin{array}{l} \frac{\partial w}{\partial t} + A u_{n} - A u_{m} + α w + β (| u_{n} | u_{n} - | u_{m} | u_{m}) + γ (| u_{n} |^{2} u_{n} - | u_{m} |^{2} u_{m}) \\ = f (x, u_{n, t}) - f (x, u_{m, t}) . \end{array}$ Multiplying the above equality by $w (t)$ and using (2.1) and the property of operator A and Lemma 3.2, we derive that $\begin{matrix} \frac{d}{d t} {‖ u_{n} (t) - u_{m} (t) ‖}^{2} ⩽ 2 L_{f} ‖ u_{n, t} - u_{m, t} ‖_{C_{H}}^{2} . \end{matrix}$ Integrating this inequality over $[τ, t]$ , shows that $\begin{matrix} {‖ u_{n} (t) - u_{m} (t) ‖}^{2} ⩽ {‖ u_{n} (τ) - u_{m} (τ) ‖}^{2} + 2 L_{f} \int_{τ}^{t} ‖ u_{n, s} - u_{m, s} ‖_{C_{H}}^{2} d s, \end{matrix}$ for all $t ⩾ τ$ . Hence, $\begin{matrix} ‖ u_{n, t} - u_{m, t} ‖^{2} ⩽ ‖ P_{n} φ - P_{m} φ ‖^{2} + 2 L_{f} \int_{τ}^{t} ‖ u_{n, s} - u_{m, s} ‖_{C_{H}}^{2} d s . \end{matrix}$ By Gronwall’s inequality again, we obtain $\begin{matrix} ‖ u_{n, t} - u_{m, t} ‖^{2} ⩽ e^{2 L_{f} (t - τ)} ‖ P_{n} φ - P_{m} φ ‖_{C_{H}}^{2} . \end{matrix}$ Since $φ_{n} \to φ$ in $C_{H}$ , this inequality indicates that ${u_{n}}$ is a Cauchy sequence in $C ([τ - h, T]; H)$ . Then (3.13) is proved.

Now we can prove the existence of solutions of equation (1.1). From (3.13), it follows that $\begin{matrix} f (\cdot, u_{n, t}) \to f (\cdot, u_{t}) in L^{2} (τ, T; H), for all T > 0 . \end{matrix}$ Then in view of (3.10) and (3.12), we deduce by passing the limit in (3.1) that u is a solution of $\begin{matrix} (\frac{d u}{d t}, v) + ν (χ, v) + α (u, v) + β (| u | u, v) + γ (| u |^{2} u, v) = (f (x, u_{t}), v) + (g (t, x), v), \end{matrix}$ for all $v \in V$ and $t \in [τ, T]$ , which implies that $\begin{array}{rcl} (3.14) & \frac{d}{d t} u + ν χ + α u + β | u | u + γ | u |^{2} u = f (x, u_{t}) + g (t, x) . \end{array}$ By a similar technique as proving Lemma 3.2 in [27], we can find $\begin{array}{rcl} (3.15) & \underset{n \to \infty}{lim sup} \int_{τ}^{t} {(A u_{n}, u_{n})}_{V^{}, V} d s ⩽ \int_{τ}^{t} {(χ, u)}_{V^{}, V} d s . \end{array}$ Since $A : L^{2} (τ, T; V) \to L^{2} (τ, T; V^{*})$ is continuous and monotone, by (3.9), (3.10) and (3.15), we infer that $\begin{array}{rcl} (3.16) & χ = A u . \end{array}$ Therefore, it follows from (3.14) and (3.16) that u is a solution of (1.1).

Step 5. Uniqueness of solutions.

Let u, v be two weak solutions with the same initial condition and set $w = u - v$ , then $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ w ‖^{2} + ν (A u - A v, w) + α (w, w) + β (| u | u - | v | v, w) + γ (| u |^{2} u - | v |^{2} v, w) \\ = (f (x, u_{t}) - f (x, v_{t}), w) . \end{array}$ In view of the definition of A and Lemma 3.2, and $\begin{matrix} (f (x, u_{t}) - f (x, v_{t}), w) ⩽ L_{f} ‖ w_{s} ‖_{C_{H}}^{2} . \end{matrix}$ It follows that $\begin{matrix} \frac{d}{d t} ‖ w ‖^{2} ⩽ 2 L_{f} ‖ w_{t} ‖_{C_{H}}^{2}, for all t \in [τ, T] . \end{matrix}$ Observe that $w (s) = 0$ if $s ⩽ τ$ we have, for $t \in [τ, T]$ , $\begin{matrix} ‖ w_{t} ‖_{C_{H}}^{2} ⩽ 2 L_{f} \int_{τ}^{t} ‖ w_{s} ‖_{C_{H}}^{2} d s, \end{matrix}$ which shows that $w \equiv 0$ , i.e., the solution is unique. □
3.3. Proof of Theorem 2.2: Continuous dependence on initial data

We denote $w = u^{1} - u^{2}$ . It follows easily that $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ w ‖^{2} + ν (A u^{1} - A u^{2}, w) + α (w, w) + β (| u^{1} | u^{1} - | u^{2} | u^{2}, w) + γ ({| u^{1} |}^{2} u^{1} - {| u^{2} |}^{2} u^{2}, w) \\ = (f (x, u_{t}^{1}) - f (x, u_{t}^{2}), w) . \end{array}$ Then by (2.1), Lemma 3.2 and Young’s inequality, we find that, for all $t ⩾ τ$ , $\begin{matrix} {‖ w (t) ‖}^{2} ⩽ {‖ w (τ) ‖}^{2} + 2 L_{f} \int_{τ}^{t} ‖ w_{s} ‖_{C_{H}}^{2} d s . \end{matrix}$ Thus, $\begin{matrix} ‖ w_{(} t) ‖_{C_{H}}^{2} ⩽ {‖ φ^{1} - φ^{2} ‖}^{2} + 2 L_{f} \int_{τ}^{t} ‖ w_{s} ‖_{C_{H}}^{2} d s . \end{matrix}$ So, thanks to the Gronwall lemma, we obtain (2.4).

4. Proof of Theorem 2.9: The minimal and unique family of pullback attractors

4.1. Abstract theory on pullback attractors

Denoting $P (X)$ as the family of all nonempty subsets of X, and considering a family of nonempty sets ${\hat{D}}_{0} = D_{0} (t) : t \in R \subset P (X)$ , we have some preliminary definitions and theorem about tempered pullback dynamic theory in following, which can be founded in [3,15].

∙ Definition of pullback ${\hat{D}}_{0}$ -asymptotic compactness

Definition 4.1.
We call a process U on X is pullback ${\hat{D}}_{0}$ -asymptotically compact if for any $t \in R$ and any sequences $τ_{n} \subset (- \infty, t]$ and $x_{n} \subset X$ satisfying $τ_{n} \to - \infty$ and $x_{n} \in D_{0} (τ_{n})$ for all n, the sequence $U (t, τ_{n}) x_{n}$ is relatively compact in X.
Proposition 4.2.
If the process U on X is pullback ${\hat{D}}_{0}$ -asymptotically compact, then, for all $t \in R$ , the set $Λ ({\hat{D}}_{0}, t)$ given by $\begin{array}{rcl} (4.1) & Λ ({\hat{D}}_{0}, t) : = ⋂_{s ⩽ t} \overline{⋃_{τ ⩽ s} U (t, τ) D_{0} {(τ)}^{X}}, \forall t \in R \end{array}$ is a nonempty compact subset of X, and $\begin{array}{rcl} (4.2) & lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D_{0} (τ), Λ ({\hat{D}}_{0}, t)) = 0, \end{array}$ Moreover, it is the minimal family of closed sets satisfying ( 4.2 ).

∙ Dissipation defined via absorbing set and universe
Definition 4.3.
The class $D$ will be called a universe in $P (X)$ if $D$ is a nonempty class of families parameterized in time $\hat{D} = {D (t) : t \in R} \subset P (X)$ .
Definition 4.4.
It is said that ${\hat{D}}_{0} = {D_{0} (t) : t \in R} \subset P (X)$ is pullback $D$ -absorbing for the process U on X if for any $t \in R$ and any $\hat{D} \in D$ , there exists a $τ_{0} (t, \hat{D}) ⩽ t$ such that $\begin{matrix} U (t, τ) D (τ) \subset D_{0} (t), for all τ ⩽ τ_{0} (t, \hat{D}) . \end{matrix}$
Remark 4.5.

Observe that in Definition 4.4 above, the set ${\hat{D}}_{0}$ is no need to belong the class $D$ necessarily.

If the pullback $D$ -absorbing sets in some class of universes, then it is no need to be bounded in X.

If ${\hat{D}}_{0}$ is pullback $D$ -absorbing for a process U, then $\begin{matrix} Λ (\hat{D}, t) \subset Λ ({\hat{D}}_{0}, t), for all \hat{D} \in D, t \in R . \end{matrix}$ In addition, if ${\hat{D}}_{0} \in D$ , then $\begin{matrix} Λ ({\hat{D}}_{0}, t) \subset \overline{D_{0} (t)}, for all t \in R . \end{matrix}$

∙ Definition of pullback $D$ -asymptotic compactness
Definition 4.6.
A process U on X is said to be pullback $D$ -asymptotically compact if it is $\hat{D}$ -asymptotically compact for any $\hat{D} \in D$ , i.e. if for any $t \in R$ , any $\hat{D} \in D$ , and any sequences $τ_{n} \subset (- \infty, t]$ and $x_{n} \subset X$ satisfying $τ_{n} \to - \infty$ and $x_{n} \in D (τ_{n})$ for all n, the sequence ${U (t, τ_{n}) x_{n}}$ is relatively compact in X.
Proposition 4.7.
Assume that the process $U (\cdot, \cdot)$ is closed (strong or norm-to-weak continuous) and pullback $D$ -asymptotically compact. Then, for each $\hat{D} \in D$ and any $t \in R$ , the set $Λ (\hat{D}, t)$ is a nonempty compact subset of X, invariant for U, that attracts $\hat{D}$ in the pullback sense, i.e. $\begin{array}{rcl} (4.3) & lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), Λ (\hat{D}, t)) = 0, \end{array}$ Moreover, it is the minimal family of closed sets satisfying ( 4.3 ).

∙ Existence theorem of minimal and unique families of $D$ -pullback attractors
Theorem 4.8 (See Carvalho, Langa and Robinson [3], Marín-Rubio and Real [15]).

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

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

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

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

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

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

Definition 4.9.
The family $A_{D}$ is minimal in the sense that if $\hat{C} = {C (t) : t \in R} \subset P (X)$ is a family of closed sets such that for any $\hat{D} = {D (t) : t \in R} \in D$ , $\begin{matrix} lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), C (t)) = 0, \end{matrix}$ then $A_{D} (t) \subset C (t)$ .
Remark 4.10.

Under the assumptions of Theorem 4.8, the family $D$ is called the minimal pullback $D$ -attractor for the process U.

If $A_{D} \in D$ and the family $D$ is inclusion-closed, then the family $D$ -pullback attractors is unique.

∙ The connection among the families of pullback attractors based on different universes

Denoting $D_{F}^{X}$ as the universe of fixed nonempty bounded subsets of X, i.e. the class of all families $\hat{D}$ of the form $\hat{D} = {D (t) = D : t \in R}$ with D a fixed nonempty bounded subset of X. For the fixed universe $D_{F}^{X}$ , the corresponding minimal pullback $D_{F}^{X}$ -attractor for the process $U (\cdot, \cdot)$ is the pullback attractor defined by Crauel, Debussche, and Flandoli, and will be denoted by $A_{D_{F}^{X}}$ . Proposition 4.11.
Under the assumption of Theorem 4.8 , if the universe $D$ contains the universe $D_{F}^{X}$ , then both attractors, $A_{D_{F}^{X}}$ and $A_{D}$ , exist, and the following relation $A_{D_{F}^{X}} (t) \subset A_{D} (t)$ , $\forall t \in R$ holds.
Theorem 4.12 (See Marín-Rubio and Real [15]).

Let ${(X_{i}, d_{X_{i}})}_{i = 1, 2}$ be two metric spaces such that $X_{1} \subset X_{2}$ with continuous injection, and for $i = 1, 2$ , let $D_{i}$ be a universe in $P (X_{i})$ , with $D_{1} \subset D_{2}$ . Assume that we have a map U that acts as a process in both case, i.e. $U : R_{d}^{2} \times X_{i} \to X_{i}$ for $i = 1, 2$ is a process.

For each $t \in R$ , let us denote $\begin{matrix} A_{i} (t) = \overline{⋃_{{\hat{D}}_{i} \in D_{i}} Λ_{i} {({\hat{D}}_{i}, t)}^{X_{i}}}, i = 1, 2, \end{matrix}$ where the subscript i in the symbol of the omega-limit set $Λ_{i}$ is used to denote the dependence of the respective topology.

Then, $\begin{matrix} A_{1} (t) \subset A_{2} (t), \forall t \in R . \end{matrix}$

4.2. Continuity of processes

We have showed the existence and uniqueness of solution for (1.1), and continuity with respect to initial datum. Therefore, when $g \in L_{loc}^{2} (R; V^{'})$ , we can define the continuity of semiflow as following.

Proposition 4.13 (Continuity of the process).

If $g \in L_{loc}^{2} (R; V^{'})$ , for any pair $(t, τ) \in R_{d}^{2}$ , the mapping $S (\cdot, \cdot)$ is continuous from $M_{H}$ into $M_{H}$ . Moreover, if $g \in L_{loc}^{2} (R; H)$ , then $S (\cdot, \cdot)$ is also continuous from $M_{V}$ into $M_{V}$ .

4.3. Existence of absorbing family of sets in $C_{H}$ and $C_{V}$

In this subsection, we present the uniform estimates on the weak solutions of Eq. (1.1) which will indicate the existence of absorbing families of sets in $C_{H}$ for the associated process $U (\cdot, \cdot)$ obtained above.

Lemma 4.14.
Assume that $(H_{f})$ , $(H_{g})$ , and $(H_{0})$ are satisfied. Then the solution u of ( 1.1 ) satisfies that, for any $K > 0$ , there exists $t_{K} > 0$ such that, for $φ \in C_{H}$ , $‖ φ ‖_{C_{H}} ⩽ K$ implies, for all $σ ⩾ t_{K}$ , $\begin{array}{rcl} ‖ u_{t} ‖_{C_{H}}^{2} & = & sup_{- h ⩽ θ ⩽ 0} {‖ u (t + θ; t - τ, φ) ‖}^{2} \\ ⩽ & ρ_{C_{H}}^{2} (t) : = 1 + C_{0} e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}^{2} d s, \end{array}$ where $C_{0} = max {\frac{2}{α}, β}$ .
Proof.
Taking the inner product of (1.1) with u in $L^{2} (Ω)$ using Young’s inequality, we arrive at $\begin{array}{l} \frac{d}{d t} ‖ u ‖^{2} + 2 ν ‖ \nabla u ‖^{2} + (α + 3 β + 2 γ) ‖ u ‖^{2} \\ (4.4) & ⩽ \frac{2 L_{f}^{2}}{α} sup_{- h ⩽ θ ⩽ 0} {‖ u (t + θ) ‖}^{2} + C_{0} (1 + {‖ g (t, \cdot) ‖}^{2}), \end{array}$ where $C_{0} = max {\frac{α}{2}, β + γ}$

Multiplying the above inequality by $e^{m t}$ and integrating from τ to t, we obtain $\begin{array}{l} e^{m t} {‖ u (t) ‖}^{2} - e^{m τ} {‖ u (τ) ‖}^{2} - m \int_{τ}^{t} e^{m s} {‖ u (s) ‖}^{2} d s + 2 ν \int_{τ}^{t} e^{m s} {‖ \nabla u (s) ‖}^{2} d s \\ + (α + 3 β + 2 γ) \int_{τ}^{t} e^{m s} {‖ u (s) ‖}^{2} d s \\ (4.5) & ⩽ \frac{2 L_{f}^{2}}{α} \int_{τ}^{t} e^{m s} sup_{- h ⩽ θ ⩽ 0} {‖ u (s + θ) ‖}^{2} d s + C_{0} \int_{τ}^{t} e^{m s} (1 + {‖ g (s, \cdot) ‖}^{2}) d s . \end{array}$ Since $\begin{array}{l} \frac{2 L_{f}^{2}}{α} \int_{τ}^{t} e^{m s} sup_{- h ⩽ θ ⩽ 0} {‖ u (s + θ) ‖}^{2} d s \\ = \frac{2 L_{f}^{2}}{α} \int_{τ + θ}^{t + θ} sup_{- h ⩽ θ ⩽ 0} e^{m (s - θ)} {‖ u (s) ‖}^{2} d s \\ ⩽ \frac{2 L_{f}^{2}}{α} \int_{τ - h}^{τ} e^{m (s + h)} {‖ u (s) ‖}^{2} d s + \frac{2 L_{f}^{2}}{α} \int_{τ}^{t} e^{m (s + h)} {‖ u (s) ‖}^{2} d s \\ ⩽ \frac{2 L_{f}^{2}}{α} e^{m (τ + h)} ‖ φ ‖_{C_{H}}^{2} + \frac{2 L_{f}^{2}}{α} e^{m h} \int_{τ}^{t} e^{m s} {‖ u (s) ‖}^{2} d s . \end{array}$ and $(H_{0})$ , (4.5) becomes $\begin{array}{l} {‖ u (t) ‖}^{2} + 2 ν \int_{τ}^{t} e^{m (s - t)} {‖ \nabla u (s) ‖}^{2} d s + 2 γ \int_{τ}^{t} e^{m (s - t)} {‖ u (s) ‖}_{4}^{4} d s \\ ⩽ e^{m (τ - t)} {‖ φ (0) ‖}^{2} + \frac{2 L_{f}^{2}}{α} e^{m (τ + h - t)} ‖ φ ‖_{C_{H}}^{2} + C_{0} \int_{τ}^{t} e^{m (s - t)} (1 + {‖ g (s, \cdot) ‖}^{2}) d s, t ⩾ τ . \end{array}$ Let $t ⩾ τ + h$ , we have for $θ \in [- h, 0]$ , $\begin{array}{l} {‖ u (t + θ) ‖}^{2} + 2 ν \int_{τ}^{t + θ} e^{m (s - t - θ)} {‖ \nabla u (s) ‖}^{2} d s + 2 γ \int_{τ}^{t + θ} e^{m (s - t - θ)} {‖ u (s) ‖}_{4}^{4} d s \\ ⩽ e^{m (τ - t - θ)} {‖ φ (0) ‖}^{2} + \frac{2 L_{f}^{2}}{α} e^{m (τ + h - t - θ)} ‖ φ ‖_{C_{H}}^{2} + C_{0} \int_{τ}^{t + θ} e^{m (s - t - θ)} (1 + {‖ g (s, \cdot) ‖}^{2}) d s, \\ (4.6) & ⩽ e^{- m (t - τ)} (1 + \frac{2 L_{f}^{2} e^{m h}}{α}) e^{m h} ‖ φ ‖_{C_{H}}^{2} + C_{0} e^{- m (t - h)} \int_{τ}^{t} e^{m s} (1 + {‖ g (s, \cdot) ‖}^{2}) d s . \end{array}$ Now, we replace τ by $t - σ$ in the above inequality, that is, $u (\cdot)$ denotes now $u (\cdot; t - σ, φ)$ , so that we can use the definition of the absorbing set more conveniently. Then we get $\begin{matrix} {‖ u (t + θ) ‖}^{2} ⩽ e^{- m σ} \frac{2 L_{f}^{2} e^{m h}}{α} e^{m h} ‖ φ ‖_{C_{H}}^{2} + C_{0} e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} (1 + {‖ g (s, \cdot) ‖}^{2}) d s . \end{matrix}$ So, for any $φ \in C_{H}$ with $‖ φ ‖_{C_{H}} ⩽ K$ , there exists $t_{K} > 0$ such that for any $σ ⩾ t_{K}$ , $\begin{matrix} ‖ u_{t} ‖_{C_{H}}^{2} = sup_{- h ⩽ θ ⩽ 0} {‖ u (t + θ) ‖}^{2} ⩽ 1 + C_{0} e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}^{2} d s = ρ_{C_{H}}^{2} (t) . \end{matrix}$ Thus the Lemma is proved. □

We now prove the existence of a family of absorbing sets in $C_{V}$ . We proceed in a similar way as we have already done in previous Lemma 4.14.
Lemma 4.15.
Suppose that all the assumptions of Lemma 4.14 are fulfilled. Then for any $K > 0$ and $φ \in C_{H}$ with $‖ φ ‖_{C_{H}} ⩽ K$ , there is a $t_{k} > 0$ such that for all $σ ⩾ t_{k}$ , there holds the following inequality: $\begin{array}{rcl} ‖ u_{t} ‖_{C_{V}}^{2} & = & sup_{- h ⩽ θ ⩽ 0} {‖ \nabla u (t + θ; t - τ, φ) ‖}^{2} \\ ⩽ & ρ_{C_{V}}^{2} (t) : = C ρ_{C_{H}}^{2} (t) + C e^{- m (t - 1)} \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}^{2} d s, \end{array}$ where $ρ_{C_{H}}^{2}$ is given in Lemma 4.14 .
Proof.
Denote $u (\cdot) = u (\cdot; t_{0} - σ, φ)$ for $φ \in C_{H}$ with $‖ φ ‖_{C_{H}} ⩽ K$ , where $t_{0} \in R$ is a fixed, but arbitrary, number, and let us take $σ > t_{K}$ with $t_{K}$ from Lemma 4.14. We can then integrate (4.4) over $[t - 1, t]$ for $t ⩽ t_{0} + 1$ and $σ > t_{K}$ , $\begin{array}{l} {‖ u (t) ‖}^{2} - {‖ u (t - 1) ‖}^{2} + 2 ν \int_{t - 1}^{t} {‖ \nabla u (s) ‖}^{2} d s \\ ⩽ \frac{2 L_{f}^{2}}{α} \int_{t - 1}^{t} {‖ u (s) ‖}_{C_{H}}^{2} d s + C_{0} \int_{t - 1}^{t} (1 + {‖ g (s, \cdot) ‖}^{2}) d s, \end{array}$ from which and the fact $σ > t_{K}$ , we have $\begin{array}{rcl} \int_{t - 1}^{t} ‖ \nabla u_{s} ‖^{2} d s & ⩽ & C ρ_{C_{H}}^{2} (t) + C \int_{t - 1}^{t} (1 + {‖ g (s, \cdot) ‖}^{2}) d s + \frac{1}{2 ν} {‖ u (t - 1) ‖}^{2} \\ (4.7) & = & C ρ_{C_{H}}^{2} (t) + C \int_{t - 1}^{t} (1 + {‖ g (s, \cdot) ‖}^{2}) d s . \end{array}$ On the other hand, taking inner of (1.1) with $\frac{d u}{d t}$ in $L^{2} (Ω)$ , we obtain that $\begin{matrix} {‖ \frac{d u}{d t} ‖}^{2} + \frac{1}{2 ν} \frac{d}{d t} ‖ \nabla u ‖^{2} ⩽ (f (x, u_{t}), \frac{d u}{d t}) + (g (t, x), \frac{d u}{d t}) . \end{matrix}$ By (2.1) and Young’s inequality, we observe $\begin{array}{l} (4.8) & \int_{Ω} f (x, u_{t}) \frac{d u}{d t} d x ⩽ L_{f}^{2} ‖ u_{t} ‖_{C_{H}}^{2} + \frac{1}{4} {‖ \frac{d u}{d t} ‖}^{2}, \\ (4.9) & \int_{Ω} g (t, x) \frac{d u}{d t} d x ⩽ {‖ g (t, \cdot) ‖}^{2} + \frac{1}{4} {‖ \frac{d u}{d t} ‖}^{2} . \end{array}$ It then follows from (4.8) and (4.9) that $\begin{array}{rcl} (4.10) & ν {‖ \frac{d u}{d t} ‖}^{2} + \frac{d}{d t} ‖ \nabla u ‖^{2} ⩽ 2 ν L_{f}^{2} ‖ u_{t} ‖_{C_{H}}^{2} + 2 ν {‖ g (t, \cdot) ‖}^{2} . \end{array}$ Integrating it over $[s, t]$ with $s \in [t - 1, t]$ gives $\begin{array}{rcl} {‖ \nabla u (t) ‖}^{2} ⩽ {‖ \nabla u (s) ‖}^{2} + 2 ν L_{f}^{2} \int_{s}^{t} ‖ u_{r} ‖_{C_{H}}^{2} d r + 2 ν \int_{s}^{t} {‖ g (r, \cdot) ‖}^{2} d r . \end{array}$ Then integrating this inequality with respect to s on $[t - 1, t]$ and making use of Lemma 4.14 and (4.7), we obtain $\begin{array}{rcl} {‖ \nabla u (t) ‖}^{2} & ⩽ & \int_{t - 1}^{t} {‖ \nabla u (s) ‖}^{2} d s + 2 ν L_{f}^{2} \int_{t - 1}^{t} ‖ u_{s} ‖_{C_{H}}^{2} d s + 2 ν \int_{t - 1}^{t} {‖ g (s, \cdot) ‖}^{2} d s \\ ⩽ & c ρ_{C_{H}}^{2} (t) + c \int_{t - 1}^{t} (1 + {‖ g (s, \cdot) ‖}^{2}) d s \\ ⩽ & c ρ_{C_{H}}^{2} (t) + c \int_{t - 1}^{t} e^{m (s - t + 1)} (1 + {‖ g (s, \cdot) ‖}^{2}) d s . \end{array}$ Substituting t by $t + θ$ yields readily that $\begin{array}{rcl} ‖ u_{t} ‖_{C_{V}}^{2} & = & sup_{- h ⩽ θ ⩽ 0} {‖ \nabla u (t + θ; t - τ, φ) ‖}^{2} \\ ⩽ & ρ_{C_{V}}^{2} (t) : = c ρ_{C_{H}}^{2} (t) + c e^{- m (t - 1)} \int_{- \infty}^{t} e^{m s} {‖ g (s, \cdot) ‖}^{2} d s, \end{array}$ which is the desired inequality. The proof is completed. □

The following result will be applied to verify the compactness property of the (uniformly) absorbing sets for the process $U (\cdot, \cdot)$ in $C_{H}$ . Lemma 4.16.
Assume that $(H_{f})$ , $(H_{g})$ , and $(H_{0})$ hold. Then for any $K > 0$ and any $t \in R$ , there is $σ_{0} ⩾ t_{K} + h + 2$ , such that for all $σ ⩾ σ_{0}$ and $φ^{σ} \in C_{H}$ with $‖ φ^{σ} ‖_{C_{H}} ⩽ K$ , one has $\begin{array}{l} {‖ u (r; t - σ, φ^{σ}) ‖}^{2} ⩽ ρ_{1}^{2} (t), for r \in [t - h - 2, t], \\ (4.11) & \int_{r - 1}^{r} {‖ \nabla u (l; t - σ, φ^{σ}) ‖}^{2} d l ⩽ ρ_{2}^{2} (t), for r \in [t - h, t], \\ \int_{r - 1}^{r} {‖ u^{'} (r; t - σ, φ^{σ}) ‖}^{2} d l ⩽ ρ_{3}^{2} (t), for r \in [t - h, t], \end{array}$ where $\begin{array}{l} ρ_{1}^{2} (t) = C (1 + e^{- m (t - h)} \int_{- \infty}^{t} e^{m s} (1 + {‖ g (s, \cdot) ‖}^{2}) d s), \\ ρ_{2}^{2} (t) = C (1 + ρ_{1}^{2} (t) + \int_{r - 1}^{r} {‖ g (l, \cdot) ‖}^{2} d l), \\ ρ_{3}^{2} (t) = C (ρ_{C_{V}}^{2} (t) + ρ_{1}^{2} (t) + \int_{r - 1}^{r} {‖ g (l, \cdot) ‖}^{2} d l), \end{array}$ for some constant $C > 0$ . Here $t_{K} > 0$ and $ρ_{C_{V}}^{2}$ are from Lemma 4.15 .
Proof.
The first estimate follows directly from Lemma 4.14, using the definition of the norm $‖ \cdot ‖_{C_{H}}$ . Now we estimate the rest inequalities in (4.11). We denote, for short, $u (r) = u (r; t - σ, φ^{σ})$ .

Multiplying (1.1) by u and integrating over $[r - 1, r]$ , we find $\begin{array}{rcl} \int_{r - 1}^{r} {‖ \nabla u (t) ‖}^{2} d l ⩽ \frac{1}{2 ν} {‖ u (r - 1) ‖}^{2} + \frac{L_{f}^{2}}{ν α} \int_{r - 1}^{r} ‖ u_{l} ‖_{C_{H}}^{2} d l + \frac{1}{ν α} \int_{r - 1}^{r} {‖ g (l, \cdot) ‖}^{2} d l + \frac{β}{2 ν} | Ω | . \end{array}$ From this, together with the first estimate in (4.11), one deduces that $\begin{matrix} \int_{r - 1}^{r} {‖ \nabla u (l) ‖}^{2} d l ⩽ C (1 + ρ_{1}^{2} (t) + \int_{r - 1}^{r} {‖ g (l, \cdot) ‖}^{2} d l) = ρ_{2}^{2} (t) . \end{matrix}$

On the other hand, integrating (4.10) over $[r - 1, r]$ and using Lemma 4.14, we conclude that there exits $σ_{0} ⩾ t_{K} + h + 2$ such that, for all $σ ⩾ σ_{0}$ , there holds $\begin{array}{rcl} \int_{r - 1}^{r} {‖ u^{'} (l) ‖}^{2} d l & ⩽ & {‖ \nabla u (r - 1) ‖}^{2} + 2 L_{f}^{2} \int_{r - 1}^{r} ‖ u_{l} ‖_{C_{H}}^{2} d l + C \int_{r - 1}^{r} {‖ g (l, \cdot) ‖}^{2} d l \\ ⩽ & C ({‖ \nabla u (r - 1) ‖}^{2} + ρ_{1}^{2} (t) + \int_{r - 1}^{r} {‖ g (l, \cdot) ‖}^{2} d l) . \end{array}$ Hence, we obtain from Lemma 4.15 that $\begin{array}{rcl} (4.12) & \int_{r - 1}^{r} {‖ u^{'} (l) ‖}^{2} d l ⩽ ρ_{3}^{2} (t), for r \in [t - h, t] . \end{array}$ This proves the assertion. □

4.4. Proof of Theorems 2.5 and 2.9: Pullback attractors in $M_{H}$ and $C_{H}$

Proof of Theorem 2.5.
Combining the continuous of semiflow in Section 4.2, existence of absorbing set in $M_{H} = C_{H} \times H$ and $M_{V} = C_{V} \times V$ and asymptotic compactness via the compact embedding of V to H for the processes in Section 4.3, it is easy to achieve the global pullback attractor by the theory in Section 4.1. □
Proof of Theorem 2.9.
By the theory in Section 4.1 and the definition of projectors $Π_{1}$ and $Π_{2}$ , from the choice of universes, the minimal family of pullback attractors can be derived easily.

For the inclusion closed, since $m \in (0, m_{0})$ , and the closed property absorbing set, it is easy to see the closed property of universe. Choosing a new subset $C (t)$ satisfies (2.6), it is easy to verify that all elements in $C (t)$ has the form (2.7), i.e., $C (t)$ is inclusion closed. By the theory in Section 4.1, we conclude that the family of pullback attractors in unique. □
5. Proof of Theorem 2.12: Convergence of pullback attractors as delay vanishes

5.1. The abstract scheme of upper semi-continuity of pullback attractors

Definition 5.1.
Let I be a metric space. A family of sets ${A_{h} (t)}_{h \in I}$ in $C_{Y}$ is said to be upper semi-continuous if for every $t \in R$ , there holds $\begin{matrix} lim_{h \to 0} {dist}_{Y} (A_{h} (t), A_{0} (t)) = 0, \end{matrix}$ where the distance ${dist}_{Y}$ for $E \subset C ([- h, 0]; Y)$ and $S \subset Y$ is defined as $\begin{array}{rcl} (5.1) & {dist}_{Y} (E, S) = sup_{φ \in E} inf_{ϕ \in S} sup_{- h ⩽ θ ⩽ 0} {‖ φ (θ) - ϕ ‖}_{Y} . \end{array}$

Next, we present an abstract result for verifying the upper semi-continuity of a family of pullback attractors for delayed equations. For this, let $(Y, ‖ \cdot ‖_{Y})$ be a Banach space and suppose that ${U_{h} (t, τ)}$ and ${U_{0} (t, τ)}$ are continuous processes on $C ([- h, 0]; Y)$ and Y, respectively. We assume the following:

For every $t \in R^{+}$ , $τ \in R$ with $τ ⩽ t$ , $\begin{matrix} lim_{n \to \infty} sup_{- h_{n} ⩽ θ ⩽ 0} {‖ U_{h_{n}} (t, τ) φ_{n} (θ) - U_{0} (t, τ) y^{0} ‖}_{Y} = 0, \end{matrix}$ for any $φ_{n} (θ) \in C ([- h_{n}, 0]; Y)$ , $y^{0} \in Y$ and $h_{n} \to 0$ , with ${sup}_{- h_{n} ⩽ θ ⩽ 0} ‖ φ_{n} (θ) - y^{0} ‖_{Y} \to 0$ as $n \to \infty$ .

For each $h > 0$ , ${U_{h} (t, τ)}$ has a pullback attractor $A_{n}$ and a compact (pullback) absorbing set $B_{h}$ in $C ([- h, 0]; Y)$ , while ${U_{0} (t, τ)}$ has a pullback attractor $A_{0}$ and a compact (pullback) absorbing set $B_{0}$ in Y such that for all $t \in R$ , $\begin{matrix} lim sup_{h \to 0} sup_{φ \in B_{h}} {‖ φ (θ) ‖}_{C_{Y}} ⩽ ρ_{0} (t), \end{matrix}$ where $ρ_{0} : R \to R$ verifies $\begin{matrix} B_{0} (t) = {y \in Y; ‖ y ‖_{Y} ⩽ ρ_{0} (t)} is the absorbing set of U_{0} (t, τ) . \end{matrix}$

For every $t \in R$ , if $h_{n} \to 0$ and $φ_{n} \in A_{h_{n}} (t)$ , then there exists $y \in Y$ such that $\begin{matrix} lim_{n \to \infty} sup_{- h_{n} ⩽ θ ⩽ 0} {‖ φ_{n} (θ) - y ‖}_{Y} = 0 . \end{matrix}$

Then, under the above assumptions, we can establish the following result. Theorem 5.2.
Assume that $(a_{1})$ – $(a_{3})$ hold, then for every $t \in R$ , $\begin{matrix} {dist}_{Y} (A_{h} (t), A_{0} (t)) \to 0, as h \to 0 . \end{matrix}$

5.2. Proof of Theorem 2.12: The convergence of pullback attractors as delay effect vanishes

The following lemma is the truth for convergence of solution as $h \to 0$ .

Lemma 5.3.
Suppose $(H_{f})$ , $(H_{g})$ and $(H_{0})$ hold. Let $u^{h}$ and u be the solutions of Eqs ( 1.1 ) and ( 2.11 ) with initial data $φ^{h} \in C_{H}$ and $φ^{0} \in H$ , respectively. Then for $t, τ \in R$ ( $τ ⩽ t$ ) and for and $ϵ > 0$ , there exists $h_{0} \in (0, 1]$ depending on t and ϵ such that for all $h ⩽ h_{0}$ , it holds $\begin{matrix} sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ u^{h} (t + \tilde{θ}, τ, φ^{h}) - u (t, τ, φ^{0}) ‖}^{2} ⩽ sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} (\tilde{θ}) - φ^{0} ‖}^{2} + C ϵ, \end{matrix}$ where C is independent of h and ϵ.
Proof.
Fix $\tilde{θ} \in [- h, 0]$ and set $w (t) = u^{h} (t + \tilde{θ}, τ, φ^{h}) - u (t, τ, φ^{0})$ . Then for $t ⩾ τ - \tilde{θ}$ , w satisfies $\begin{array}{l} \frac{\partial w}{\partial t} + A u^{h} - A u + α w + β (| u^{h} | u^{h} - | u | u) + γ ({| u^{h} |}^{2} u^{h} - | u |^{2} u) \\ (5.2) & = f (x, u^{h} (t + θ + \tilde{θ}, x)) - f (x, u) + g (t + \tilde{θ}, x) - g (t, x) . \end{array}$ Multiplying (5.2) with w and integrating on Ω, we get for $\tilde{θ} \in [- h, 0]$ that $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ w ‖^{2} + (A u^{h} - A u, w) + α ‖ w ‖^{2} + β (| u^{h} | u^{h} - | u | u, w) + γ ({| u^{h} |}^{2} u^{h} - | u |^{2} u, w) \\ = (f (x, u^{h} (t + θ + \tilde{θ})) - f (x, u), w) + (g (t + \tilde{θ}, x) - g (t, x), w) . \end{array}$ Since $\begin{array}{rcl} (f (x, u^{h} (t + θ + \tilde{θ})) - f (x, u), w) & ⩽ & C_{f} \int_{Ω} | u^{h} (t + \tilde{θ} + θ) - u | | w | d x \\ ⩽ & \frac{1}{2} C_{f}^{2} {‖ u^{h} (t + \tilde{θ} + θ) - u ‖}^{2} + \frac{1}{2} ‖ w ‖^{2}, \end{array}$ and $\begin{array}{rcl} (g (t + \tilde{θ}, x) - g (t, x), w) ⩽ \frac{1}{2} {‖ g (t + \tilde{θ}, \cdot) - g (t, \cdot) ‖}^{2} + \frac{1}{2} ‖ w ‖^{2} . \end{array}$ and Lemma 3.1, we deduce that $\begin{array}{rcl} (5.3) & \frac{d}{d t} ‖ w ‖^{2} ⩽ 2 ‖ w ‖^{2} + {‖ g (t + \tilde{θ}, x) - g (t, x) ‖}^{2} + C_{f}^{2} {‖ u^{h} (t + \tilde{θ} + θ, x) - u ‖}^{2} . \end{array}$ Integrating (5.3) over $[τ - \tilde{θ}, t]$ with $t \in [τ, τ + T]$ , we get $\begin{array}{rcl} {‖ w (t) ‖}^{2} & ⩽ & {‖ w (τ - \tilde{θ}) ‖}^{2} + 2 \int_{τ - \tilde{θ}}^{t} {‖ w (r) ‖}^{2} d r + \int_{τ - \tilde{θ}}^{t} {‖ g (r + \tilde{θ}, x) - g (r, x) ‖}^{2} d r \\ (5.4) & + C_{f}^{2} \int_{τ - \tilde{θ}}^{t} {‖ u^{h} (r + \tilde{θ} + θ, x) - u (r) ‖}^{2} d r . \end{array}$ Noting that $\begin{array}{l} \int_{τ - \tilde{θ}}^{t} {‖ u^{h} (r + \tilde{θ} + θ, x) - u (r) ‖}^{2} d r \\ = \int_{τ - \tilde{θ}}^{τ - \tilde{θ} - θ} {‖ u^{h} (r + \tilde{θ} + θ, x) - u (r) ‖}^{2} d r + \int_{τ - \tilde{θ} - θ}^{t} {‖ u^{h} (r + \tilde{θ} + θ, x) - u (r) ‖}^{2} d r \\ ⩽ 2 \int_{τ - \tilde{θ}}^{τ - \tilde{θ} - θ} {‖ u^{h} (r + \tilde{θ} + θ, x) - φ^{0} ‖}^{2} d r + \int_{τ - \tilde{θ}}^{τ - \tilde{θ} - θ} {‖ u (r) - φ^{0} ‖}^{2} d r \\ + \int_{τ - \tilde{θ}}^{t} {‖ u^{h} (r + \tilde{θ}) - u (r - θ) ‖}^{2} d r \\ ⩽ 2 \int_{τ - h}^{τ} {‖ u^{h} (r) - φ^{0} ‖}^{2} d r + 2 \int_{τ - \tilde{θ}}^{τ - \tilde{θ} - θ} {‖ u (r) - φ^{0} ‖}^{2} d r \\ + \int_{τ - \tilde{θ}}^{t} {‖ u^{h} (r + \tilde{θ}) - u (r - θ) ‖}^{2} d r \\ ⩽ 2 h sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} - φ^{0} ‖}^{2} + 2 \int_{τ}^{τ + 2 h} {‖ u (r) - φ^{0} ‖}^{2} d r + 2 \int_{τ - \tilde{θ}}^{t} {‖ w (r) ‖}^{2} d r \\ (5.5) & + 2 \int_{τ - \tilde{θ}}^{t} {‖ u (r - θ) - u (r) ‖}^{2} d r, \end{array}$ from (5.4) and (5.5), we obtain that for $t ⩾ τ - \tilde{θ}$ and $t \in [τ, τ + T]$ , $\begin{array}{rcl} {‖ w (t) ‖}^{2} & ⩽ & {‖ w (τ - \tilde{θ}) ‖}^{2} + 2 (1 + C_{f}^{2}) \int_{τ - \tilde{θ}}^{t} {‖ w (r) ‖}^{2} d r + 2 h C_{f}^{2} sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} - φ^{0} ‖}^{2} \\ + 2 C_{f}^{2} \int_{τ}^{τ + 2 h} {‖ u (r) - φ^{0} ‖}^{2} d r + 2 C_{f}^{2} \int_{τ}^{τ + T} {‖ u (r - θ) - u (r) ‖}^{2} d r \\ (5.6) & + \int_{τ}^{τ + T} {‖ g (r + \tilde{θ}, x) - g (r, x) ‖}^{2} d r . \end{array}$ Clearly, there exists $h_{1} \in (0, 1]$ such that for all $h ⩽ h_{1}$ , $\begin{array}{rcl} (5.7) & 2 C_{f}^{2} \int_{τ}^{τ + 2 h} {‖ u (r) - φ^{0} ‖}^{2} d r ⩽ ϵ . \end{array}$ Since $u : [τ, τ + 1 + T] \to H$ is uniformly continuous, there exists $h_{2} ⩽ h_{1}$ such that for all $h ⩽ h_{2}$ and $θ \in [- h, 0]$ , $r \in [τ, τ + T]$ , $\begin{array}{rcl} (5.8) & 2 C_{f}^{2} \int_{τ}^{τ + T} {‖ u (r - θ) - u (r) ‖}^{2} d r ⩽ ϵ . \end{array}$ Due to the fact $g \in L_{loc}^{2} (R, H)$ , we know $\begin{array}{rcl} lim_{\tilde{θ} \to 0} \int_{τ}^{τ + T} {‖ g (r + \tilde{θ}, x) - g (r, x) ‖}^{2} d r = 0, \end{array}$ and thus there exists $h_{3} ⩽ h_{2}$ such that for all $h ⩽ h_{3}$ and $\tilde{θ} \in [- h, 0]$ , $\begin{array}{rcl} (5.9) & \int_{τ}^{τ + T} {‖ g (r + \tilde{θ}, x) - g (r, x) ‖}^{2} d r ⩽ ϵ . \end{array}$ It then follows from (5.6)–(5.9) that for all $h ⩽ h_{3}$ , $t \in [τ - \tilde{θ}, τ + T]$ , $\begin{array}{rcl} {‖ w (t) ‖}^{2} ⩽ {‖ w (τ - \tilde{θ}) ‖}^{2} + 2 (1 + C_{f}^{2}) \int_{τ}^{t} {‖ w (r) ‖}^{2} d r + 2 h C_{f}^{2} sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} - φ^{0} ‖}^{2} + C ϵ . \end{array}$ By the Gronwall lemma, we obtain, for all $h ⩽ h_{3}$ , $t \in [τ - \tilde{θ}, τ + T]$ with $\tilde{θ}, θ \in [- h, 0]$ , $\begin{array}{rcl} (5.10) & {‖ w (t) ‖}^{2} ⩽ C {‖ w (τ - \tilde{θ}) ‖}^{2} + C ϵ + C sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} - φ^{0} ‖}^{2} . \end{array}$ Furthermore, it yields $\begin{array}{rcl} {‖ w (τ - \tilde{θ}) ‖}^{2} = {‖ u^{h} (τ) - u (τ - \tilde{θ}) ‖}^{2} ⩽ 2 {‖ u^{h} (τ) - φ^{0} ‖}^{2} + 2 {‖ u (τ - \tilde{θ}) - φ^{0} ‖}^{2} . \end{array}$ From the continuous of u at τ, it yields that there exists $h_{4} ⩽ h_{3}$ such that for all $h ⩽ h_{4}$ , $\begin{array}{rcl} {‖ w (τ - \tilde{θ}) ‖}^{2} ⩽ ϵ + 2 sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} - φ^{0} ‖}^{2} . \end{array}$ Thus, from (5.10), we obtain that, for all $h ⩽ h_{4}$ , and $t \in [τ - \tilde{θ}, τ + T]$ , $\begin{array}{rcl} (5.11) & {‖ u^{h} (t + \tilde{θ}) - u (t) ‖}^{2} ⩽ C ϵ + C sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} - φ^{0} ‖}^{2} . \end{array}$ Next we consider the case $τ ⩽ τ - \tilde{θ}$ . Let $r = t - τ$ . Then $t = r + τ$ with $0 ⩽ r ⩽ h$ , and hence, $\begin{array}{rcl} {‖ u^{h} (t + \tilde{θ}) - u (t) ‖}^{2} & ⩽ & 2 {‖ u^{h} (t + \tilde{θ}) - φ^{0} ‖}^{2} + 2 {‖ φ^{0} - u (t) ‖}^{2} \\ ⩽ & 2 sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} (\tilde{θ}) - φ^{0} ‖}^{2} + 2 {‖ u (r + τ) - φ^{0} ‖}^{2} . \end{array}$ By the continuity of u at τ again, and the fact that $0 ⩽ r ⩽ h$ , there exists $h_{5} ⩽ h_{4}$ such that for all $h ⩽ h_{5}$ and $τ ⩽ t ⩽ τ - \tilde{θ}$ , $\begin{matrix} {‖ u^{h} (t + \tilde{θ}) - u (t) ‖}^{2} ⩽ C sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} (\tilde{θ}) - φ^{0} ‖}^{2} + C ϵ, \end{matrix}$ with together with (5.11) indicates that for all $h ⩽ h_{5}$ and $t ⩾ τ$ , $\begin{matrix} {‖ u^{h} (t + \tilde{θ}) - u (t) ‖}^{2} ⩽ C sup_{- h ⩽ \tilde{θ} ⩽ 0} {‖ φ^{h} (\tilde{θ}) - φ^{0} ‖}^{2} + C ϵ, \end{matrix}$ as desired. □

The next theorem will show the uniform compactness of attractors with respect to h. Lemma 5.4.
Suppose that $(H_{f})$ , $(H_{g})$ and $(H_{0})$ hold. If $h_{n} \to 0$ and $u_{n} \in A_{h_{n}} (t)$ , then there exist a subsequence ${u_{n_{m}}}$ of ${u_{n}}$ and $u \in H$ such that $\begin{matrix} lim_{m \to \infty} sup_{- h_{n_{m}} ⩽ θ ⩽ 0} {‖ u_{n_{m}} (θ) - u ‖}^{2} = 0 . \end{matrix}$
Proof.
Take a sequence $τ_{n} \to + \infty$ and $u_{n} \in A_{h_{n}} (t)$ , then by the invariance of $A_{h_{n}} (t)$ , there exists ${\tilde{u}}_{n} \in A_{h_{n}} (t - τ_{n})$ such that $\begin{matrix} u_{n} = U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} . \end{matrix}$ From (2.10), we know ${\tilde{u}}_{n} \in B_{h_{n}} (t - τ_{n})$ . Since all estimates in Section 4 are uniform with respect to bounded h, using the Sobolev compact embedding $V ↪ ↪ H$ , we can verify the following facts:
$U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (0)$ is precompact in H.

Given any $ϵ > 0$ , there exists $N_{1} ⩾ 1$ such that for all $n ⩽ N_{1}$ and $θ \in [- h_{n}, 0]$ , $\begin{matrix} {‖ U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (θ) - U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (0) ‖}^{2} ⩽ \frac{ϵ}{4} . \end{matrix}$
Assertion $(i)$ implies that there exists $u \in H$ such that, up to a subsequence, $\begin{matrix} U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (0) \to u in H . \end{matrix}$ Therefore, there exists $N_{2} ⩾ N_{1}$ , such that for all $n ⩾ N_{2}$ , $\begin{matrix} {‖ U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (0) - u ‖}^{2} ⩽ \frac{ϵ}{4}, \end{matrix}$ from which and $(ii)$ , it follows that for all $n ⩾ N_{2}$ and $θ \in [- h_{n}, 0]$ , $\begin{array}{l} {‖ U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (θ) - u ‖}^{2} \\ ⩽ 2 {‖ U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (θ) - U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (0) ‖}^{2} + 2 {‖ U_{h_{n}} (t, t - τ_{n}) {\tilde{u}}_{n} (0) - u ‖}^{2} \\ ⩽ ϵ . \end{array}$ So we get that $‖ u_{n} (θ) - u ‖^{2} ⩽ ϵ$ for all $n ⩾ N_{2}$ and $θ \in [- h_{n}, 0]$ . The proof is completed. □
Proof of Theorem 2.12: the convergence of pullback attractors as delay vanishes.
Let $h_{n} \to 0$ , $φ^{h_{n}} \in C_{H}$ , and $φ^{0} \in H$ with ${sup}_{- h_{n} ⩽ θ ⩽ 0} ‖ φ^{h_{n}} (θ) - φ^{0} ‖^{2} \to 0$ as $n \to \infty$ . It follows from Lemma 5.3 that, for $t, τ \in R$ ( $τ ⩽ t$ ), $\begin{matrix} sup_{- h_{n} ⩽ θ ⩽ 0} {‖ U_{h_{n}} (t, t - τ) φ^{h_{n}} (θ) - U_{0} (t, t - τ) φ^{0} ‖}^{2} \to 0, as n \to \infty . \end{matrix}$ In view of (2.15), Lemma 5.4 and Theorem 5.2, the result follows immediately. □

6. Conclusion and further research

The 3D Brinkman–Forchheimer equation is a homogenization of Navier–Stokes model, its dynamics is also important to understand turbulence in the domain through a porous medium. However, the structure of attractors for BF equation with delay is unknown, and the physical interpret of pullback dynamics also not clear. Moreover, the BF equation coupled with some other systems is also interesting in hydrodynamics, such as the wave equations, this issue is what we want to pay attention next.

In this paper, the delay in finite interval has been investigated, but what about the infinite delay, and its stability as finite delay tends to opposite case is open.

Footnotes

Acknowledgements

This work was partially supported by NSFC of China (Grants No. 11726626 & 11771017 & 11501560), the Key Project of Science and Technology of Henan Province (Grant No. 182102410069 & 17A120003) and the Project of Science and Technology in Henan Province (Grant No. 172102210342).

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

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Dynamics and stability of the 3D Brinkman–Forchheimer equation with variable delay (I)

Abstract

Keywords

1. Introduction

2. Main results

2.1. Some notations

2.2. Some assumption

2.3. Well-posedness: Global existence, uniqueness and continuous dependence on initial data of weak solution

3.1. Some lemmas

Lemma 3.1 (See [12,23]).

Lemma 3.2 (See [12]).

3.2. Existence of global solution

4. Proof of Theorem 2.9: The minimal and unique family of pullback attractors

4.1. Abstract theory on pullback attractors

4.2. Continuity of processes

Proposition 4.13 (Continuity of the process).

4.3. Existence of absorbing family of sets in C H and C V

5.1. The abstract scheme of upper semi-continuity of pullback attractors

Footnotes

Acknowledgements

References

4.3. Existence of absorbing family of sets in $C_{H}$ and $C_{V}$