Abstract
The aim of this paper is to study the regular dynamics for the 3D delay Brinkman–Forchheimer (BF) equations. We first prove the existence, uniqueness and time-dependent property of regular tempered pullback attractors as well as the existence of invariant measures for the 3D BF equations with non-autonomous abstract delay. We then study the asymptotic autonomy of regular pullback attractors for the 3D BF equations with autonomous abstract delay. Finally, we discuss the upper semicontinuity of regular pullback attractors as the delay time approaches to zero for the 3D BF equations with variable delay and distributed delay.
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1 Introduction
The attractors play an important role in the study of long-term behaviour for evolution equations, see [4, 6, 17, 30, 31] and the references therein. As we know, the pullback attractor is a collection of a family of time-dependent compact sets. Hence, a natural problem is to consider the time-dependent property of pullback attractors. Although this topic has been studied by several authors in [7,8,9, 18, 19, 21,22,23, 38], there is no paper on this subject in the regular space. On the other hand, the long-term behaviour of delay PDEs has wide attention [1,2,3, 11, 16, 24, 34, 40] due to it being able to control the system by the history of solutions. In [24, 40], the authors investigated the time-dependent property of pullback attractors in the initial space for delay PDEs. As far as we know, this topic in the regular space for the delay BF equation has not been studied.
The BF equation depicts the fluid flow in a saturated porous medium, see [12, 29]. Recently, the long-term behaviour of BF equations has been studied by several authors in [13, 14, 32, 33, 36, 39, 42, 43] for the non-delay case and in [15, 20, 25, 37] for the delay case. In [15], the authors proved the existence of uniform attractors for the BF equation with autonomous delay. Li et al. [20] studied the existence and non-delay stability of pullback attractors for the BF equation with autonomous delay. In [37], the authors investigated the structure and asymptotic stability of pullback attractors for the BF equation with non-autonomous delay. As far as we know, the existence, time-dependent property, asymptotic autonomy and non-delay stability of regular pullback attractors for the BF equation with delays have not been studied.
We first consider the following BF equation with non-autonomous delay:
where \(t_{0}\in \mathbb {R}\), \(\varOmega \subset \mathbb {R}^{3}\) is a bounded domain with smooth boundary \(\partial \varOmega \), \(u=(u_{1}, u_{2}, u_{3})\) is the velocity of the fluid, and p is the pressure of the fluid. \(\nu >0\) denotes the Brinkman effective viscosity, \(\alpha >0\) denotes the Darcy coefficient, and \(\beta >0\) and \(\gamma >0\) denote the Forchheimer coefficients. \(\varrho >0\) is the delay time of Eq. (1), g is the given non-autonomous forcing term, and the delay term f will be specified later. For each \(t\ge t_{0}\), \(u_{t}(\cdot )\) is a delay-shift function defined on \([-\varrho , 0]\) given by \(u_{t}(\theta )=u(t+\theta )\) with \(\theta \in [-\varrho , 0]\).
Recently, the authors in [7, 21, 23, 38] studied that the backward compactness of pullback attractors \(\mathcal {A}=\{\mathcal {A}(t): t\in \mathbb {R}\}\) (\(\bigcup \nolimits _{s\le t}\mathcal {A}(s)\) is pre-compact with \(t\in \mathbb {R}\)) when the basin of attraction of pullback attractors be composed of all fixed bounded sets in the initial space. However, we consider two different universes to show that the backward compactness of pullback attractors in this paper. One universe is made up of all tempered sets, and the other is composed of all backward tempered sets. It is worth pointing out that the backward compactness of pullback attractors on tempered universe is not easily proved. Fortunately, we can prove the attractors on two different universes are identical. On the other hand, we apply the spectrum decomposition technique to prove the backward asymptotic compactness of solutions due to the solution of Eq. (1) has no higher regularity, and then obtain a unique backward compact regular tempered pullback attractor for Eq. (1).
In [10], the authors show that the invariant measure plays an important role in understanding turbulence. Hence, the invariant measure of evolution equations has been studied by several authors in [5, 26,27,28, 35]. Inspire by [28], we establish the existence of a unique family of invariant Borel probability measures for Eq. (1), which are supported by the tempered pullback attractor.
We then discuss the following BF equation with autonomous delay:
In [8, 9, 18, 19, 22], the authors studied the forward asymptotic autonomy of pullback attractors in the initial space for PDEs without delay, that is,
where X is a Banach space with norm \(\Vert \cdot \Vert _{X}\), \(dist_{X}(\cdot , \cdot )\) denotes the Hausdorff semi-distance and \(\mathcal {A}_{\infty }\) is a global attractor of the autonomous equation corresponding to (2). As far as we know, the asymptotic autonomy of pullback attractors in the regular space (the regularity of the initial space is low) has not been considered. In this paper, we study the backward asymptotically autonomous dynamics for Eq. (2) in the regular space:
where Y is a Banach space with norm \(\Vert \cdot \Vert _{Y}\), \(X\hookleftarrow Y\) and \(Y_{\varrho }=C([-\varrho , 0]; Y)\).
Finally, we study the following BF equation with variable delay and distributed delay:
Zhao et al. [41] studied the upper semicontinuity of the global attractor in the initial space as the delay time tends to zero for retarded lattice systems. Wang et al. [34] proved the upper semicontinuity of pullback random attractors as the memory time approaches zero in the initial space for stochastic reaction–diffusion equation. In this paper, we study the upper semicontinuity of tempered pullback attractors as the delay time tends to zero in the regular space for Eq. (3):
where
In the next section, we establish the existence of an evolution process for the retarded 3D BF equation. Section 3 is devoted to the backward uniform estimates, continuity with respect to initial time and backward flattening of solutions, and then obtain the existence, regularity and backward compactness of pullback attractors as well as the existence of invariant measures. In Sect. 4, we prove the backward regularly asymptotic autonomy of pullback attractors by the convergence of systems from non-autonomous to autonomous and the backward compactness of pullback attractors. The delay-free stability of regular pullback attractors is established by the convergence of solutions from delay to non-delay, the eventually compactness of pullback attractors and the recurrence of absorbing sets in the last section.
2 Existence of an Evolution Process for the 3D BF Equation with Delays
Suppose X is a Banach space with norm \(\Vert \cdot \Vert _{X}\). A family of maps \(\{S(t, r): t\ge r\}\) is called an evolution process if
for all \(t\ge r\ge s\). In this section, we mainly establish the existence of an evolution process for Eq. (1) and make some assumptions. Denote by
and
where \(\mathbf {n}\) is the unit outward normal vector at \(\partial \varOmega \). It is easy to see that H and V are separable Hilbert spaces with the inner products and norms given by
and
Notice that \(V\hookrightarrow H\equiv H'\hookrightarrow V'\) is dense and continuous. Moreover, we use \(\Vert \cdot \Vert _{p}\) to denote the norm in \(\mathbb {L}^{p}(\varOmega )\) and \(\langle \cdot , \cdot \rangle \) represents the duality pairing between V and \(V'\). Let \(\tilde{P}\) be the Leray orthogonal projection from \(\mathbb {L}^{2}(\varOmega )\) onto H. Applying \(\tilde{P}\) to Eq. (1), we obtain
where \(A=-\tilde{P}\varDelta \) is defined by \(\langle Au, v\rangle =((u, v))\). Let \(H_{\varrho }:=C([-\varrho , 0], H)\) with \(\Vert \vartheta \Vert _{H_{\varrho }}=\sup \nolimits _{\theta \in [-\varrho , 0]}\Vert \vartheta (\theta )\Vert \) and \(V_{\varrho }:=C([-\varrho , 0], V)\) with \(\Vert \vartheta \Vert _{V_{\varrho }}=\sup \nolimits _{\theta \in [-\varrho , 0]}\Vert \vartheta (\theta )\Vert _{V}\). The delay forcing \(f:\mathbb {R}\times H_{\varrho }\rightarrow H\) satisfies
-
(F1) For each \(\varphi \in H_{\varrho }\), \(r\rightarrow f(r, \varphi )\) is measurable from \(\mathbb {R}\) into H;
-
(F2) \(f(r, 0)=0\) for all \(r\in \mathbb {R}\);
-
(F3) There is a positive function \(L_{f}(\cdot )\) such that for all \(\varphi , \psi \in H_{\varrho }\)
$$\begin{aligned} \Vert f(r, \varphi )-f(r, \psi )\Vert \le L_{f}(r)\Vert \varphi -\psi \Vert _{H_{\varrho }}, \end{aligned}$$(5)where \(L_{f}(\cdot )\) satisfies, for all \(t\in \mathbb {R}\),
$$\begin{aligned} \int \limits _{-\infty }^{t}L_{f}^{2}(r)dr<+\infty , \ \ \ \lim _{\beta \rightarrow +\infty }\sup _{s\le t}\int \limits _{-\infty }^{s}e^{\beta (r-s)}L_{f}^{2}(r)dr=0. \end{aligned}$$(6)By the standard Galerkin method as in [11, 33], one can see that the well-posedness of (4).
Lemma 1
Suppose that (F1)–(F3) hold and \(g\in L^{2}_{loc}(\mathbb {R}, H)\). Then for all \(t_{0}\in \mathbb {R}\) and \(\phi \in H_{\varrho }\), Eq. (4) has a unique weak solution
such that \(u(t_{0}+\theta ; t_{0}, \phi )=\phi (\theta )\) for all \(\theta \in [-\varrho , 0]\). Furthermore, for any \(\varepsilon >0\) and \(T>t_{0}+\varepsilon \)
where \(D(A)=H^{1}_{0}(\varOmega )\cap H^{2}(\varOmega )\).
We define a mapping \(S(t, t_{0}): H_{\varrho }\rightarrow H_{\varrho }\) by
where u is the solution of (4). Based on Lemma 1, we obtain \(S(\cdot , \cdot )\) is an evolution process. In addition, we also need the following assumptions:
(G1) \(g\in L^{2}_{loc}(\mathbb {R}, H)\) is backward limitable:
which implies \(g\in L^{2}_{loc}(\mathbb {R}, H)\) is backward tempered:
On the other hand, we will frequently use the following inequalities:
where \(x, y\in \mathbb {R}^{n}\) and \(p\ge 2\).
Gagliardo–Nirenberg inequality: assume \(\varOmega \subset \mathbb {R}^{n}\), if \(0\le j<l, 1\le q, r\le +\infty , p\in \mathbb {R}, \frac{j}{l}<\eta \le 1\) and
Then,
where j, l be any integers and c depending only on \(n, l, j, q, r, \eta \).
Let \(\mathcal {P}(H_{\varrho })\) denotes the collection of all nonempty subsets in \(H_{\varrho }\). We provide two different universes of attraction. One is \(\mathfrak {D}\) given by, for all \(\beta >0\)
Observe that \(\mathfrak {D}\) is inclusion-closed. Another is \(\mathfrak {B}\) defined by, for all \(\beta >0\)
Then, \(\mathfrak {B}\) is backward-closed: \(\tilde{\mathcal {B}}\in \mathfrak {B}\) whenever \(\mathcal {B}\in \mathfrak {B}\) and \(\tilde{\mathcal {B}}(\tau , \omega )=\cup _{s\le t}\mathcal {B}(s, \omega )\). Notice that \(\mathfrak {B}\subset \mathfrak {D}\) and \(\mathfrak {B}\) is also inclusion-closed.
3 Dynamics of the 3D Non-Autonomous BF Equations with Delays
Throughout this paper, we denote by c a positive constant, which may change from line to line or even in the same line.
3.1 Backward Uniform Estimates of Solutions
Lemma 2
Suppose that (F1)–(F3) and (G1) hold. Let \(t\in \mathbb {R}\), then we have the following conclusions:
-
(i)
For each \(\mathcal {D}\in \mathfrak {D}\), there exists a \(\tau _{d}:=\tau _{d}(t, \mathcal {D})>0\) such that
$$\begin{aligned} \sup _{\sigma \in [t-\varrho -4, t]}\Vert u_{\sigma }(\cdot ; t-\tau , \phi )\Vert _{H_{\varrho }}^{2} \le e^{2\alpha (\varrho +2)}\left( 1+\frac{2}{\alpha }G_{d}(t)\right) e^{\frac{2}{\alpha }e^{\alpha \varrho }L(t)}, \end{aligned}$$(14)for all \(\tau \ge \tau _{d}\) and \(\phi \in \mathcal {D}(t-\tau )\), where
$$\begin{aligned} G_{d}(t)=\int \limits _{-\infty }^{t}e^{\alpha (r-t)}\Vert g(r)\Vert ^{2}dr, \quad L(t):=\int \limits _{-\infty }^{t}L^{2}_{f}(r)dr. \end{aligned}$$(15) -
(ii)
For each \(\mathcal {B}\in \mathfrak {B}\), there exists a \(\tau _{b}:=\tau _{b}(t, \mathcal {B})>0\) such that
$$\begin{aligned} \sup _{s\le t}\sup _{\sigma \in [s-\varrho -4, s]}\Vert u_{\sigma }(\cdot ; s-\tau , \phi )\Vert _{H_{\varrho }}^{2} \le e^{2\alpha (\varrho +2)}\left( 1+\frac{2}{\alpha }G_{b}(t)\right) e^{\frac{2}{\alpha }e^{\alpha \varrho }L(t)}, \end{aligned}$$(16)for all \(\tau \ge \tau _{d}\) and \(\phi \in \mathcal {B}(s-\tau )\) with \(s\le t\), where
$$\begin{aligned} G_{b}(t)=\sup _{s\le t}G_{d}(s). \end{aligned}$$(17)
Proof
Taking the inner product of (4) with \(u(r; s-\tau , \phi )\) in H, by the Young inequality, we have
which implies
Integrating (19) on \([s-\tau , \sigma +\theta ]\) with \(\theta \in [-\varrho , 0]\) and \(\tau >s-\sigma +\varrho \), by (5) we obtain
Using the Gronwall inequality (see [4, p. 167]) to (20), we obtain
Hence we obtain, for all \(\tau \ge 2\varrho +4\),
-
(i)
Let \(s=t\) in (21). If \(\phi \in \mathcal {D}(t-\tau )\), by (12), there exists a \(\tau _{d}:=\tau _{d}(t, \mathcal {D})\ge 2\varrho +4\) such that for all \(\tau \ge \tau _{d}\),
$$\begin{aligned} e^{-\alpha \tau }\Vert \phi \Vert _{H_{\varrho }}^{2} \le e^{-\alpha \tau }\Vert \mathcal {D}(t-\tau )\Vert _{H_{\varrho }}^{2} \le 1. \end{aligned}$$Hence, we obtain (14) holds.
-
(ii)
If \(\phi \in \mathcal {B}(s-\tau )\) with \(s\le t\), by (13), there is a \(\tau _{b}:=\tau _{b}(t, \mathcal {B})\ge 2\varrho +4\) such that for all \(\tau \ge \tau _{b}\),
$$\begin{aligned} e^{-\alpha \tau }\sup _{s\le t}\Vert \phi \Vert _{H_{\varrho }}^{2} \le e^{-\alpha \tau }\sup _{s\le t}\Vert \mathcal {B}(s-\tau )\Vert _{H_{\varrho }}^{2} \le 1. \end{aligned}$$Taking the supremum of (21) over the past time \(s\le t\) yields (16) holds.
\(\square \)
Corollary 1
Under the assumptions in Lemma 2, we have the following auxiliary estimate:
Proof
Integrating (18) on \([s-\varrho -3, s]\), by (5), we obtain
By (16), we have
Then, we obtain
The proof is complete. \(\square \)
Lemma 3
Suppose that (F1)–(F3) and (G1) hold. For each \(t\in \mathbb {R}\) and \(\mathcal {B}\in \mathfrak {B}\), we have, for all \(\tau \ge \tau _{b}\) (\(\tau _{b}\) is given in Lemma 2) and \(\phi \in \mathcal {B}(s-\tau )\) with \(s\le t\),
Proof
Multiplying (4) by \(\frac{\partial }{\partial r}u(r; s-\tau , \phi )\), by (9) and the Young inequality, we have
Integrating (25) on \([\zeta , s+\theta ]\) with \(\zeta \in [s+\theta -1, s+\theta ]\) and \(\theta \in [-\varrho -2, 0]\), and then integrating the result on \([s+\theta -1, s+\theta ]\) w.r.t. \(\zeta \) we obtain, for all \(\theta \in [-\varrho -2, 0]\),
By (16) and (22), we have, for all \(\tau \ge \tau _{b}\),
Substituting (23), (27) and (28) into (26) yields
On the other hand, integrating (25) on \([s-\varrho , s]\) yields
It follows from (16), (23) and (29) that
We infer from (29) and (31) that (24) as desired. \(\square \)
3.2 Existence and Backward Compactness of Pullback Attractors in the Initial Space
We first establish the existence of pullback absorbing sets for the evolution process \(S(\cdot , \cdot )\) in (7).
Lemma 4
Suppose that (F1)–(F3) and (G1) hold. Then, we have
-
(i)
\(S(\cdot , \cdot )\) has a pullback \(\mathfrak {D}\)-absorbing set \(\mathcal {K}_{d}=\{\mathcal {K}_{d}(t): t\in \mathbb {R}\}\in \mathfrak {D}\), defined by
$$\begin{aligned} \mathcal {K}_{d}(t)=\{\vartheta \in H_{\varrho }: \Vert \vartheta \Vert _{H_{\varrho }}^{2}\le e^{2\alpha (\varrho +2)}\left( 1+\frac{2}{\alpha }G_{d}(t)\right) e^{\frac{2}{\alpha }e^{\alpha \varrho }L(t)}\}. \end{aligned}$$(32) -
(ii)
\(S(\cdot , \cdot )\) has a pullback \(\mathfrak {B}\)-absorbing set \(\mathcal {K}_{b}=\{\mathcal {K}_{b}(t): t\in \mathbb {R}\}\in \mathfrak {B}\), defined by
$$\begin{aligned} \mathcal {K}_{b}(t)=\{\vartheta \in H_{\varrho }: \Vert \vartheta \Vert _{H_{\varrho }}^{2}\le e^{2\alpha (\varrho +2)}\left( 1+\frac{2}{\alpha }G_{b}(t)\right) e^{\frac{2}{\alpha }e^{\alpha \varrho }L(t)}\}=\overline{\cup _{s\le t}\mathcal {K}_{d}(s)}, \end{aligned}$$(33)where \(G_{d}(t)\), L(t) and \(G_{b}(t)\) are given in (15) and (17).
Proof
By (14) and (16), it is easy to know that \(\mathcal {K}_{d}\) and \(\mathcal {K}_{b}\) are pullback \(\mathfrak {D}\)-absorbing set and pullback \(\mathfrak {B}\)-absorbing set for \(S(\cdot , \cdot )\), respectively. Moreover, by \(G_{b}(t)=\sup \nolimits _{s\le t}G_{d}(t)\), we obtain \(\mathcal {K}_{b}(t)=\overline{\bigcup \nolimits _{s\le t}\mathcal {K}_{d}(s)}\).
We now prove that \(\mathcal {K}_{d}\in \mathfrak {D}\) and \(\mathcal {K}_{b}\in \mathfrak {B}\). Notice that \(\mathcal {K}_{b}(t_{1})\subseteq \mathcal {K}_{b}(t_{2})\) if \(t_{1}\le t_{2}\). Then, by \(t\rightarrow , G_{b}(t)\) and \(t\rightarrow L(t)\) are increasing, we obtain
Hence, we have \(\mathcal {K}_{b}\in \mathfrak {B}\). Since \(G_{d}(t)\le G_{b}(t)\) for all \(t\in \mathbb {R}\), we have \(\mathcal {K}_{d}\in \mathfrak {D}\). The proof is complete. \(\square \)
Next, we prove the pullback asymptotically compactness of the evolution process \(S(\cdot , \cdot )\) in (7).
Lemma 5
Suppose that (F1)–(F3) and (G1) hold. Then, we have \(S(\cdot , \cdot )\) is backward pullback \(\mathfrak {B}\)-asymptotically compact in \(H_{\varrho }\), more precisely, for each \(t\in \mathbb {R}\) and \(\mathcal {B}\in \mathfrak {B}\), the sequence \(\{S(s_{n}, s_{n}-\tau _{n})\phi _{n}\}\) is relatively compact in \(H_{\varrho }\) whenever \(s_{n}\le t\), \(\tau _{n}\rightarrow +\infty \) and \(\phi _{n}\in \mathcal {B}(s_{n}-\tau _{n})\).
Proof
Based on the Ascoli–Arzelà theorem, we divide the proof into two steps.
Step 1. \(\{S(s_{n}, s_{n}-\tau _{n})\phi _{n}\}_{n\in \mathbb {N}}\) in \(H_{\varrho }\) is equi-continuous from \([-\varrho , 0]\) to H.
Since \(\tau _{n}\rightarrow +\infty \), we assume \(\tau _{n}\ge \tau _{b}\) (\(\tau _{b}\) is given in Lemma 2) for all \(n\in \mathbb {N}\). Let \(\theta _{1}, \theta _{2}\in [-\varrho , 0]\) with \(\theta _{1}<\theta _{2}\). Due to \(s_{n}\le t\) for all \(n\in \mathbb {N}\), by (24), we obtain
Then for any \(\varepsilon >0\), there exists a \(\delta :=\delta (t, \varepsilon )\) such that \(\Vert S(s_{n}, s_{n}-\tau _{n})\phi _{n}(\theta _{1})-S(s_{n}, s_{n}-\tau _{n})\phi _{n}(\theta _{2})\Vert <\varepsilon \) when \(|\theta _{1}-\theta _{2}|<\delta \). The proof of the Step 1 is complete.
Step 2. For each \(\theta \in [-\varrho , 0]\), \(\{(S(s_{n}, s_{n}-\tau _{n})\phi _{n})(\theta )\}_{n\in \mathbb {N}}\) has a convergent subsequence in H.
By (24), we have \(\{(S(s_{n}, s_{n}-\tau _{n})\phi _{n})(\theta )\}_{n\in \mathbb {N}}\) is bounded in V. Then by the embedding \(V\hookrightarrow H\) is compact, we complete the proof of the Step 2.
Now, all conditions of Ascoli–Arzelà theorem are satisfied. Hence, we obtain the sequence \(\{S(s_{n}, s_{n}-\tau _{n})\phi _{n}\}\) is relatively compact in \(H_{\varrho }\). The proof is complete. \(\square \)
Finally, we state the main result of this subsection:
Theorem 1
Suppose that (F1)–(F3) and (G1) hold. Then, we obtain the following conclusions:
-
(i)
\(S(\cdot , \cdot )\) in (7) has a unique pullback \(\mathfrak {D}\)-attractor \(\mathcal {A}_{d}=\{\mathcal {A}_{d}(t): t\in \mathbb {R}\}\in \mathfrak {D}\), defined by
$$\begin{aligned} \mathcal {A}_{d}(t)=\bigcap _{T\ge 0}\overline{\bigcup _{\tau \ge T}S(t, t-\tau )\mathcal {K}_{d}(t-\tau )}^{H_{\varrho }}. \end{aligned}$$(35) -
(ii)
\(S(\cdot , \cdot )\) in (7) has a unique pullback \(\mathfrak {B}\)-attractor \(\mathcal {A}_{b}=\{\mathcal {A}_{b}(t): t\in \mathbb {R}\}\in \mathfrak {B}\), defined by
$$\begin{aligned} \mathcal {A}_{b}(t) =\bigcap _{T\ge 0}\overline{\bigcup _{\tau \ge T}S(t, t-\tau )\mathcal {K}_{b}(t-\tau )}^{H_{\varrho }}. \end{aligned}$$(36)Moreover, \(\mathcal {A}_{b}\) is backward compact in \(H_{\varrho }\), that is, \(\bigcup \nolimits _{s\le t}\mathcal {A}_{b}(s)\) is pre-compact.
-
(iii)
\(\mathcal {A}_{d}=\mathcal {A}_{b}\) and so \(\mathcal {A}_{d}\) is backward compact in \(H_{\varrho }\).
Proof
-
(i)
By the same method as in Lemma 5, we obtain \(S(\cdot , \cdot )\) is pullback \(\mathfrak {D}\)-asymptotically compact, which along with (i) of Lemma 4 implies all conditions of Carvalho et al. [4, Theorem 2.50] are fulfilled. Hence, we have \(S(\cdot , \cdot )\) has unique pullback \(\mathfrak {D}\)-attractor \(\mathcal {A}_{d}\), defined by (35). Since \(\mathfrak {D}\) is inclusion-closed and the pullback \(\mathfrak {D}\)-absorbing set \(\mathcal {K}_{d}\) is closed, we get \(\mathcal {A}_{d}\) is unique and \(\mathcal {A}_{d}\in \mathfrak {D}\).
-
(ii)
Similarly to (i), we only need to prove \(\mathcal {A}_{b}\) is backward compact in \(H_{\varrho }\). Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \bigcup \nolimits _{s\le t}\mathcal {A}_{b}(s)\). Then, for each \(u_{n}\), there is a \(s_{n}\le t\) such that \(u_{n}\in \mathcal {A}_{b}(s_{n})\). By the invariance of \(\mathcal {A}_{b}\), there exists a \(v_{n}\in \mathcal {A}_{b}(s_{n}-\tau _{n})\) with \(\tau _{n}\rightarrow +\infty \) such that \(S(s_{n}, s_{n}-\tau _{n})v_{n}=u_{n}\). Since \(\mathcal {A}_{b}\in \mathfrak {B}\), by Lemma 5, we obtain \(\{u_{n}\}_{n\in \mathbb {N}}\) has a convergent subsequence and so \(\cup _{s\le t}\mathcal {A}_{b}(s)\) is pre-compact.
-
(iii)
Since \(\mathcal {K}_{b}(t)=\overline{\bigcup \nolimits _{s\le t}\mathcal {K}_{d}(s)}\), we have \(\mathcal {K}_{b}(t)\supset \mathcal {K}_{d}(t)\) for all \(t\in \mathbb {R}\). Then by (35) and (36), we have \(\mathcal {A}_{b}(t)\supset \mathcal {A}_{d}(t)\) for all \(t\in \mathbb {R}\) and thus \(\mathcal {A}_{b}\supset \mathcal {A}_{d}\). On the other hand, notice that \(\mathcal {A}_{b}\in \mathfrak {D}\) because of \(\mathfrak {B}\subset \mathfrak {D}\). Since \(\mathcal {A}_{b}\) is a \(\mathfrak {D}\)-pullback attracting set, by the invariance of \(\mathcal {A}_{b}\) we find, for all \(t\in \mathbb {R}\),
$$\begin{aligned} dist_{H_{\varrho }}(\mathcal {A}_{b}(t), \mathcal {A}_{d}(t)) =dist_{H_{\varrho }}(S(t, t-\tau )\mathcal {A}_{b}(t-\tau ), \mathcal {A}_{d}(t))\rightarrow 0, \ \ \text {as}\ \tau \rightarrow +\infty , \end{aligned}$$which implies \(\mathcal {A}_{b}(t)\subset \mathcal {A}_{d}(t)\) for all \(t\in \mathbb {R}\) and so \(\mathcal {A}_{b}\subset \mathcal {A}_{d}\). Then, we obtain \(\mathcal {A}_{b}=\mathcal {A}_{d}\) and \(\mathcal {A}_{d}\) is backward compact in \(H_{\varrho }\).
\(\square \)
3.3 Existence of Invariant Measures
In the subsection, we consider the existence of invariant measures \(\{\mu _{t}: t\in \mathbb {R}\}\) on the pullback \(\mathfrak {D}\)-attractor \(\mathcal {A}_{d}\). To this end, we also need prove the evolution process \(S(\cdot , \cdot )\) in (7) is \(t_{0}\)-continuous in \(H_{\varrho }\), where \(S(\cdot , \cdot )\) is known as \(t_{0}\)-continuity if for every \(\phi \in H_{\varrho }\) and \(t\in \mathbb {R}\), the function \(t_{0}\rightarrow S(t, t_{0})\phi \) with values in \(H_{\varrho }\) is continuous and bounded on \((-\infty , t]\). We first prove the following auxiliary lemma.
Lemma 6
Suppose that (F1)–(F3) and (G1) hold. For any initial data \(\hat{\phi }\) and \(\tilde{\phi }\) in \(H_{\varrho }\), we have
Proof
Let \(\bar{u}=\hat{u}-\tilde{u}\). Then, \(\bar{u}\) satisfies
Taking the inner product of (38) with \(\bar{u}\) in H yields
and
Hence, we have
Integrating the above inequality on \([t_{0}, t]\) with \(t>t_{0}\) yields
Notice that \(\Vert \bar{u}(t)\Vert ^{2} \le \Vert \hat{\phi }-\tilde{\phi }\Vert ^{2}_{H_{\varrho }}\) for \(t\in [t_{0}-h, t_{0}]\). Then, we have
Using the Gronwall inequality, we obtain
The proof is complete. \(\square \)
Next, we prove the \(t_{0}\)-continuity of \(S(\cdot , \cdot )\) in (7).
Lemma 7
Suppose that (F1)–(F3) and (G1) hold. Then, the evolution process \(S(\cdot , \cdot )\) in (7) is \(t_{0}\)-continuous in \(H_{\varrho }\).
Proof
Based on the definition of \(t_{0}\)-continuous, we divide the proof into two steps.
Step 1. For all \(\phi \in H_{\varrho }\) and \(t\in \mathbb {R}\), \(t_{0}\rightarrow , S(t, t_{0})\phi \) is continuous on \((-\infty , t]\).
Let \(\tilde{t}_{0}\in (-\infty , t]\) and \(\tilde{t}_{0}\ge t_{0}\). By (37), we obtain, for all \(\phi \in H_{\varrho }\),
Notice that \(u: [t_{0}-\varrho , t_{0}+1]\rightarrow H\) is uniform continuous. Then for any \(\varepsilon >0\), there exists a positive constant \(\delta :=\delta (\varepsilon )<1\) with \(|t_{0}-\tilde{t}_{0}|<\delta \) such that for all \(\theta \in [-\varrho , 0]\),
which implies
Since \(e^{c\int \limits _{-\infty }^{t}L^{2}_{f}(r)dr}\) is finite, we obtain \(t_{0}\rightarrow S(t, t_{0})\phi \) is right continuous on \((-\infty , t]\). Similarly, we have \(t_{0}\rightarrow S(t, t_{0})\phi \) is left continuous on \((-\infty , t]\). Then, the proof of Step 1 is complete.
Step 2 For all \(\phi \in H_{\varrho }\) and \(t\in \mathbb {R}\), \(t_{0}\rightarrow S(t, t_{0})\phi \) is bounded on \((-\infty , t]\).
By the same method as Lemma 2, we have
Observe that the last line of the above inequality is a constant independent of \(t_{0}\). Then, we obtain
By Step 1, we know that \(t\in \mathbb {R}\), \(t_{0}\rightarrow S(t, t_{0})\phi \) is continuous on \((-\infty , t]\), and hence the proof of Step 2 is complete. \(\square \)
Now, we obtain all conditions of Łukaszewicz and Robinson [28, Theorem 3.1] are fulfilled by (i) of Theorem 1 and Lemma 7. Hence, we have the following result:
Theorem 2
Suppose that (F1)–(F3) and (G1) hold. Fix a generalized Banach limit \(LIM_{T\rightarrow +\infty }\) and let \(v: \mathbb {R}\rightarrow H_{\varrho }\) be a continuous map such that \(v(\cdot )\in \mathfrak {D}\). Then, there exists a unique family of Borel probability measures \(\{\mu _{t}: t\in \mathbb {R}\}\) in \(H_{\varrho }\) such that the support of the measure \(\mu _{t}\subset \mathcal {A}_{d}(t)\) and
for any real-valued continuous functional \(\varphi \) on \(H_{\varrho }\). In addition, \(\mu _{t}\) is invariant in the following sense:
where a generalized Banach limit \(LIM_{T\rightarrow +\infty }\) is any linear functional, which defined on the space of all bounded real-valued functions on \([0, +\infty )\) that satisfies
-
(1)
\(LIM_{T\rightarrow +\infty }h(T)\ge 0\) for nonnegative functions h;
-
(2)
\(LIM_{T\rightarrow +\infty }h(T)=\lim _{T\rightarrow +\infty }h(T)\) if the limit \(\lim _{T\rightarrow +\infty }h(T)\) exists.
3.4 Backward Flattening of Solutions
Lemma 8
Suppose that (F1)–(F3) and (G1) hold. For each \(t\in \mathbb {R}\) and \(\mathcal {B}\in \mathfrak {B}\), we have, for all \(\tau \ge \tau _{b}\) and \(\phi \in \mathcal {B}\),
Proof
Multiplying (4) by \(Au(r; s-\tau , \phi )\), by (5) and the Young inequality, we have
By the Young inequality, we obtain
Hence, by \(V\hookrightarrow L^{p}(\varOmega ) (2\le p\le 6)\), we obtain
Integrating the above inequality on \([s-\varrho -1, s]\) yields
By (16), we have
Inserting (23), (24) and (42) into (41) yields
The proof is complete. \(\square \)
Note that the Stokes operator A has a family of eigenfunctions \(\{e_{k}\}_{k=1}^{\infty }\subset V\) with the corresponding eigenvalues: \(\lambda _{1}\le \lambda _{2}\le \cdots \le \lambda _{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \). In addition, \(\{e_{k}\}_{k=1}^{\infty }\) be the orthonormal basis of H. Let \(P_{k}: H\mapsto H_{k}=span\{e_{1}, e_{2}, \cdots , e_{k}\}\), then \(P_{k}\) is an orthogonal projection. For each \(u\in V\), which has the following orthogonal decomposition:
Lemma 9
Suppose that (F1)–(F3) and (G1) hold. For each \(t\in \mathbb {R}\), \(\mathcal {B}\in \mathfrak {B}\) and any \(\varepsilon >0\), there exists a \(\delta :=\delta (t, k, \varepsilon )>0\) with \(|\theta _{1}-\theta _{2}|<\delta \) and \(\theta _{1}, \theta _{2}\in [-\varrho , 0]\) such that
for all \(\tau \ge \tau _{b}\) and \(\phi \in \mathcal {B}\).
Proof
Notice that \(\Vert A^{\frac{1}{2}}u_{k, 1}\Vert ^{2}\le \lambda _{k}\Vert u_{k, 1}\Vert ^{2}\) and suppose that \(\theta _{1}\le \theta _{2}\). Hence, by (24), we have
Then, for any \(\varepsilon >0\), there exists a \(\delta :=\delta (t, k, \varepsilon )>0\) with \(|\theta _{1}-\theta _{2}|<\delta \) such that
which implies (43) holds. \(\square \)
Lemma 10
Suppose that (F1)–(F3) and (G1) hold. For each \(t\in \mathbb {R}\), \(\mathcal {B}\in \mathfrak {B}\) and any \(\varepsilon >0\), there exists a \(K=K(\varepsilon , t)\in \mathbb {N}\) such that
for all \(k\ge K\), \(\tau \ge \tau _{b}\) and \(\phi \in \mathcal {B}\).
Proof
Taking the inner product of (4) with \(Au_{k, 2}(r; s-\tau , \phi )\) in H, we have
Therefore, by \(\Vert Au_{k, 2}\Vert ^{2}\ge \lambda _{k}\Vert A^{\frac{1}{2}}u_{k, 2}\Vert ^{2}\) we get
Let \(p=4, j=0, q=3, r=2, l=2\) in (11), we have
Again, let \(p=6, j=0, q=4, r=2, l=2\),
Hence, we have
Integrating (45) on \([\zeta , s+\theta ]\) with \(\zeta \in [s+\theta -1, s+\theta ]\) and \(\theta \in [-\varrho , 0]\), and then integrating this resulting on \([s+\theta -1, s+\theta ]\) w.r.t. \(\zeta \) yields
which implies
We now treat each term on the right-hand term of (46). For the first term, by (24), we have
For the second and third term, by (29) and (40), we get
For the delay term, by (6) and (16), we find
For the forcing term, by (8), we have
It follows from (46) to (50) that we obtain (44) as desired. \(\square \)
3.5 Existence and Backward Compactness of Pullback Attractors in the Regular Space
We review some basic concepts and theorem related to bi-spatial pullback attractors. Suppose that \((X, \Vert \cdot \Vert _{X})\) and \((Y, \Vert \cdot \Vert _{Y})\) be Banach spaces. An evolution process \(S(\cdot , \cdot )\) on X is said to take its values into Y if
We claim that the combination (X, Y) is limit-identical if \(X\cap Y\ne \emptyset \) and
Let \(\tilde{\mathfrak {D}}\) be a inclusion-closed universe of some sets \(\tilde{\mathcal {D}}=\{\tilde{\mathcal {D}}(t)\ne \emptyset : t\in \mathbb {R}\}\subset X\).
Definition 1
A family of sets \(\mathcal {A}=\{\mathcal {A}(t): t\in \mathbb {R}\}\) is called a pullback \(\tilde{\mathfrak {D}}\)-(X, Y)-attractor for \(S(\cdot , \cdot )\) if
-
(i)
\(\mathcal {A}(\cdot )\) is compact in \(X\cap Y\);
-
(ii)
\(\mathcal {A}(\cdot )\) is invariant, that is, \(S(t, s)\mathcal {A}(s)=\mathcal {A}(t)\) for all \(t\ge s\);
-
(iii)
\(\mathcal {A}(\cdot )\) attracts every \(\tilde{\mathcal {D}}\in \tilde{\mathfrak {D}}\) under the topology of Y, more precisely, for each \(\tilde{\mathcal {D}}\in \tilde{\mathfrak {D}}\), we have
$$\begin{aligned} \lim _{\tau \rightarrow +\infty }dist_{X\cap Y}(S(t, t-\tau )\tilde{\mathcal {D}}(t-\tau ), \mathcal {A}(t))=0, \end{aligned}$$where \(dist_{X\cap Y}(\cdot , \cdot )\) denotes the Hausdorff semi-distance.
Theorem 3
Assume that \(S(\cdot , \cdot )\) be an evolution process on X taking its values in Y and (X, Y) is limit-identical. Then, \(S(\cdot , \cdot )\) has a pullback \(\tilde{\mathfrak {D}}\)-(X, Y)-attractor \(\mathcal {A}(\cdot )\) if
-
(i)
\(S(\cdot , \cdot )\) has a closed pullback \(\tilde{\mathfrak {D}}\)-absorbing set \(\mathcal {K}:=\{\mathcal {K}(t): t\in \mathbb {R}\}\in \tilde{\mathfrak {D}}\);
-
(ii)
\(S(\cdot , \cdot )\) is pullback \(\tilde{\mathfrak {D}}\)-asymptotically compact in X;
-
(iii)
\(S(\cdot , \cdot )\) is pullback \(\tilde{\mathfrak {D}}\)-(X, Y)-asymptotically compact.
Theorem 4
Suppose that (F1)–(F3) and (G1) hold. Then, we have the following conclusions:
-
(i)
The backward compact pullback \(\mathfrak {B}\)-attractor \(\mathcal {A}_{b}\) as given in (36) is also a pullback \(\mathfrak {B}\)-(\(H_{\varrho }, V_{\varrho }\)) attractor, which is backward compact in \(V_{\varrho }\).
-
(ii)
The pullback \(\mathfrak {D}\)-attractor \(\mathcal {A}_{d}\) as given in (35) is also a pullback \(\mathfrak {D}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor, which is backward compact in \(V_{\varrho }\).
Proof
(i) We first show that \(\mathcal {A}_{b}\) is a pullback \(\mathfrak {B}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor. Since \(V_{\varrho }\hookrightarrow H_{\varrho }\), (\(H_{\varrho }, V_{\varrho }\)) is limit-identical. By Theorem 3 we also need to prove \(S(\cdot , \cdot )\) is backward pullback \(\mathfrak {B}\)-(\(H_{\varrho }, V_{\varrho }\))-asymptotically compact, that is, for each \(t\in \mathbb {R}\), \(\{S(s_{n}, s_{n}-\tau _{n})\phi _{n}\}_{n\in \mathbb {N}}\) has a convergent subsequence in \(V_{\varrho }\) whenever \(s_{n}\le t, \tau _{n}\rightarrow +\infty \) and \(\phi _{n}\in \mathcal {B}(s_{n}-\tau _{n})\). To this end, we split the proof into two steps based on Ascoli–Arzelà theorem.
Step 1. \(\{S(s_{n}, s_{n}-\tau _{n})\phi _{n}\}_{n\in \mathbb {N}}\) in \(V_{\varrho }\) is equi-continuous from \([-\varrho , 0]\) to V.
Notice that there exists a \(N\in \mathbb {N}\) such that \(\tau _{n}\ge \tau _{b}\) (\(\tau _{b}\) is given in Lemma 2) for all \(n\in \mathbb {N}\) due to \(\tau _{n}\rightarrow +\infty \). By (43) and (44), we have for all \(k_{0}\ge K\)
Hence, we have \(\{S_{\varrho }(s_{n}, s_{n}-\tau _{n})\phi _{n}\}_{n\in \mathbb {N}}\) is equi-continuous.
Step 2. For each fixed \(\theta \in [-\varrho , 0]\), the sequence \((S(s_{n}, s_{n}-\tau _{n})\phi _{n})(\theta )=u(s_{n}+\theta ; s_{n}-\tau _{n}, \phi _{n})\) is pre-compact in V.
By (24), we obtain that \(\{P_{k_{0}}u(s_{n}+\theta ; s_{n}-\tau _{n}, \phi _{n})\}_{n\in \mathbb {N}}\) is bounded in V and thus pre-compact in the \(k_{0}\)-dimensional subspace \(V_{k_{0}}\). Then, there is an index subsequence \(n^{*}\) of n such that \(\{P_{k_{0}}u(s_{n^{*}}+\theta ; s_{n^{*}}-\tau _{n^{*}}, \phi _{n^{*}})\}_{n^{*}\in \mathbb {N}}\) is a Cauchy sequence in \(V_{k_{0}}\). On the other hand, let \(n^{*}, m^{*}\) large enough, we have
Hence, \(\{u(s_{n^{*}}+\theta ; s_{n^{*}}-\tau _{n^{*}}, \phi _{n^{*}})\}_{n^{*}\in \mathbb {N}}\) is a Cauchy sequence in V and then the proof of Step 2 is complete.
Then, we obtain that \(S(\cdot , \cdot )\) is backward pullback \(\mathfrak {B}\)-(\(H_{\varrho }, V_{\varrho }\))-asymptotically compact, which implies \(S(\cdot , \cdot )\) is pullback \(\mathfrak {B}\)-(\(H_{\varrho }, V_{\varrho }\))-asymptotically compact. Hence, we have \(\mathcal {A}_{b}\) is a pullback \(\mathfrak {B}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor.
Next, we show that \(\mathcal {A}_{b}\) is backward compact in \(V_{\varrho }\). Indeed, it is easy to verify that the result by the same method as in (ii) of Theorem 1.
(ii) Similarly to (i), we have \(\mathcal {A}_{d}\) as given in (35) is also a pullback \(\mathfrak {D}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor. Since \(\mathcal {A}_{d}=\mathcal {A}_{b}\), we get \(\mathcal {A}_{d}\) is backward compact in \(V_{\varrho }\). \(\square \)
4 Asymptotically Autonomous Dynamics for the 3D BF Equation with Autonomous Delays
In this section, we consider the following non-autonomous equation:
Similarly to (4), Eq. (51) can be rewritten as
Since we consider the delay term is autonomous, the assumption conditions (F1)–(F3) will be changed, more precisely,
- (F1):
-
For each \(\varphi \in C_{H}\), \(x\rightarrow f(\varphi , x)\) is measurable from \(\varOmega \) into H;
- (F2):
-
\(f(0, x)=0\) for all \(x\in \varOmega \);
- (F3):
-
There is a positive constant \(L_{f}\) such that for all \(\varphi , \psi \in H_{\varrho }\),
$$\begin{aligned} \Vert f(\varphi , \cdot )-f(\psi , \cdot )\Vert \le L_{f}\Vert \varphi -\psi \Vert _{H_{\varrho }}. \end{aligned}$$(53) - (F4):
-
There exist two positive constants \(c_{f}\) and \(m_{f}\) such that for all \(u, v\in C([t_{0}-\varrho , t], H)\) with \(t_{0}\le t\),
$$\begin{aligned} \int \limits _{t_{0}}^{t}e^{m_{f}r}\Vert f(\varphi , \cdot )-f(\psi , \cdot )\Vert ^{2}dr \le c_{f}^{2}\int \limits _{t_{0}-\varrho }^{t}e^{m_{f}r}\Vert \varphi -\psi \Vert ^{2}dr. \end{aligned}$$
Hence, we can define an evolution process \(\tilde{S}(t, t_{0}): H_{\varrho }\rightarrow H_{\varrho }\) associated with (52), by
Notice that we add a condition (F4), which will be used to establish the existence of pullback absorbing sets. Similarly to Lemma 2, we can obtain the evolution process \(\tilde{S}\) has a pullback \(\mathfrak {D}\)-absorbing set \(\tilde{\mathcal {K}}_{d}\) and a pullback \(\mathfrak {B}\)-absorbing set \(\tilde{\mathcal {K}}_{b}\). Moreover, other results of Sect. 3 are hold for Eq. (52) due to \(L_{f}\) is a positive constant. Then, \(\tilde{S}(\cdot , \cdot )\) has a pullback \(\mathfrak {D}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor \(\tilde{\mathcal {A}}_{d}=\{\tilde{\mathcal {A}}_{d}(t): t\in \mathbb {R}\}\in \mathfrak {D}\), which is backward compact in \(V_{\varrho }\).
In order to study the upper semi-convergence of the pullback attractor \(\tilde{\mathcal {A}}_{d}\) from non-autonomous to autonomous, we give a further assumption for g:
(G2) There exists a function \(g_{\infty }\in H\) such that
In addition, we need to introduce the following autonomous equation:
Meanwhile, using the Leray orthogonal projection \(\tilde{P}\) to (55), we obtain
By the standard Galerkin method, we obtain the well-posedness of (56). Then, we obtain a semigroup \(T(t): H_{\varrho }\rightarrow H_{\varrho }\) generated by (56), given by
It follows from the same method as in Sect. 3 that T(t) has a (\(H_{\varrho }, V_{\varrho }\))-global attractor \(\mathcal {A}_{\infty }\).
4.1 Convergence of Solutions from Non-Autonomous to Autonomous
Lemma 11
Suppose (F1)–(F3), (G1)–(G2) hold. If \(u_{t_{0}}, v_{0}\in H_{\varrho }\) satisfy
then we have, for all \(\tilde{r}\ge 0\),
Proof
For each \(t_{0}\in \mathbb {R}\), we define a function \(w^{t_{0}}\) by
If follows from (52) and (56) that
Multiplying the above equality by \(w^{t_{0}}(r)\) yields
By (10), we have
By (53) and the Young inequality, we have
Inserting (60)–(62) into (59) yields
Integrating the above inequality on \([0, \tilde{r}]\) with \(\tilde{r}\in [0, R]\) and \(R>0\) yields
It is easy to see that \(\Vert w^{t_{0}}(\tilde{r})\Vert ^{2}\le \Vert u_{t_{0}}-v_{0}\Vert _{H_{\varrho }}^{2}\) when \(\tilde{r}\in [-\varrho , 0]\). Then, we have
Applying the Gronwall inequality to (63), by (54) and (57), we get
as \(t_{0}\rightarrow -\infty \). The proof is complete. \(\square \)
4.2 Upper Semi-Convergence of Regular Attractors from Non-Autonomous to Autonomous
Theorem 5
Suppose (F1)–(F4) and (G1)–(G3) hold. Then, we have
Proof
If (64) is false, then there are \(\delta >0\) and \(t_{n}\rightarrow -\infty \) such that
which implies for each \(n\in \mathbb {N}\), there exists a \(a_{n}\in \tilde{\mathcal {A}}_{d}(t_{n})\) such that
We assume that \(t_{n}\le 0\) for all \(n\in \mathbb {N}\) due to \(t_{n}\rightarrow -\infty \). Then, we have \(\{a_{n}\}_{n\in \mathbb {N}}\subset \overline{\bigcup \nolimits _{s\le 0}\tilde{\mathcal {A}}_{d}(s)}:=B\). Since \(\tilde{\mathcal {A}}_{d}\) is backward compact in \(V_{\varrho }\), \(\overline{\bigcup \nolimits _{s\le 0}\tilde{\mathcal {A}}_{d}(s)}\) is compact and so B is bounded in \(V_{\varrho }\). It follows from \(\mathcal {A}_{\infty }\) is a (\(H_{\varrho }, V_{\varrho }\))-global attractor and \(V_{\varrho }\hookrightarrow H_{\varrho }\) that there is a \(r:=r(B)>0\) such that
Moreover, there is a \(a_{\infty }\in B\) such that
By the invariance of \(\tilde{\mathcal {A}}_{d}\), we find a \(b_{n}\in \tilde{\mathcal {A}}_{d}(t_{n}-r)\subset B\) such that
By the backward compactness of \(\tilde{\mathcal {A}}_{d}\) in \(H_{\varrho }\), there exists a \(b_{\infty }\in B\) such that
which together with (58) implies
which along with (67) and (68) implies \(a_{\infty }=T(r)b_{\infty }\). Then by (66) and (67), there exists \(N:=N(\delta )\in \mathbb {N}\) such that for all \(n\ge N\),
which contradicts (65). The proof is complete. \(\square \)
5 Upper Semicontinuity of Regular Pullback Attractors as the Delay time Tends to Zero
In this section, we assume that \(\varrho \in (0, \varrho _{0}]\) for some \(\varrho _{0}>0\). Let \(f(t, u_{t})=f_{1}(u(t-\rho ^{\varrho }(t)), x)+\int \limits _{-\varrho }^{0}f_{2}(u(t+\theta ), \theta )d\theta \) in (1), and then applying the Leray orthogonal projection \(\tilde{P}\) to this result, we obtain
where \(\rho ^{\varrho }(\cdot )\) is a positive function such that \(\rho ^{\varrho }(\cdot )\in C^{1}(\mathbb {R})\) and
The
t variable delay \(f_{1}:\mathbb {R}\times \varOmega \rightarrow \mathbb {R}\) satisfies
where \(L_{f_{1}}\) is a positive constant satisfies \(L_{f_{1}}^{2}<\frac{\alpha ^{2}e^{-\alpha \varrho }}{24}(1-\rho _{*})\).
The distributed delay \(f_{2}:[-\varrho , 0]\times \mathbb {R}\rightarrow \mathbb {R}\) satisfies
where \(L_{f_{2}}(\cdot )\in L^{2}(-\varrho , 0)\) is a positive function satisfies \(\Vert L_{f_{2}}(\cdot )\Vert _{L^{2}(-\varrho , 0)}^{2}<\frac{\alpha ^{2}e^{-\alpha \varrho }}{24\varrho }\).
By the same method as in Sect. 3, we can obtain an evolution process \(S_{\varrho }(\cdot , \cdot )\) induced by (69) for each \(\varrho \in (0, \varrho _{0}]\), which has a pullback \(\mathfrak {D}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor \(\mathcal {A}_{d}^{\varrho }=\{\mathcal {A}^{\varrho }_{d}(t): t\in \mathbb {R}\}\in \mathfrak {D}\).
In this section, we consider the robustness of \(\mathcal {A}_{d}^{\varrho }\) as \(\varrho \rightarrow 0\). For this purpose, let \(\varrho =0\) in (69), we have
Similarly, we get an evolution process \(S_{0}(\cdot , \cdot )\) corresponding to (73), which has a pullback \(\mathfrak {D}_{0}\)-(\(H_{\varrho }, V_{\varrho }\))-attractor \(\mathcal {A}_{d}^{0}=\{\mathcal {A}^{0}_{d}(t): t\in \mathbb {R}\}\in \mathfrak {D}_{0}\), where
5.1 Convergence of Evolution Processes from Delay to Non-Delay
Lemma 12
Suppose \(\phi ^{\varrho }\in H_{\varrho }\) and \(\phi ^{0}\in H\) such that
Then, the solution \(u^{\varrho }\) of Eq. (69) converges to the solution \(u^{0}\) of Eq. (73) in the following sense:
for all \(t\ge t_{0}\) and \(t_{0}\in \mathbb {R}\).
Proof
Let
Subtracting (73) from (69), and then multiplying this result by \(\tilde{u}^{\varrho }_{\theta }(r+\theta )\) in H, we have
By (10), we have
Hence, by (71), (72) and the Young inequality, we have
For each \(T>\varrho \), integrating (76) on \([t_{0}-\theta , t]\) with \(t\in [t_{0}-\theta , t_{0}+T]\) yields
We now estimate each term on the right-hand side of (77). For the first term, we have
For the second term, let \(s=h(r)=r+\theta -\rho ^{\varrho }(r+\theta )\) for any \(r\in \mathbb {R}\) and fixed \(\theta \in [-\varrho , 0]\). Since \(h'(r)\ge 1-\rho _{*}>0\), it has an inverse function such that \(r=h^{-1}(s)\) for any \(s\in \mathbb {R}\). Then, we obtain
Note that
and
Hence, we obtain
For the third term, we obtain
Hence, we obtain
where we use \(h^{-1}(t_{0})\le t_{0}+2\varrho \). Applying the Gronwall inequality to (78) yields
where \(I_{3}(\varrho )=ce^{c(1+\varrho \Vert L_{f_{2}}(\cdot )\Vert ^{2}_{L^{2}(-\varrho , 0)})(T+\varrho )}\). We now treat the limit of each term on the right-hand side of (79) as \(\varrho \rightarrow 0\). For the first term and third term, by (74), we have
For the second term and forth term, it follows from the continuity of \(u^{0}(\cdot )\) at \(t_{0}\) that
For the fifth term, by \(u^{0}:[t_{0}, t_{0}+T+\varrho ]\) is uniform continuity, we obtain
where we use \(h^{-1}(r+\theta )=r+\rho ^{\varrho }(h^{-1}(r+\theta )+\theta )\). For the sixth term, by the same method as in Lemma 2, we obtain \(\int \limits _{t_{0}}^{t_{0}+T}\Vert u^{0}(r)\Vert ^{2}dr\) is finite and so
For the last term, we infer from \(g\in L^{2}_{loc}(\mathbb {R}, H)\) and \(\theta \in [-\varrho , 0]\) that
Substituting (80)–(84) into (79), we find
We now consider the case of \(t\in [t_{0}, t_{0}-\theta ]\).
By the continuity of \(u^{0}(\cdot )\) at \(t_{0}\) again, we have
It follows from (85) and (86) that
The proof is complete. \(\square \)
5.2 Upper Semi-Convergence of Regular Pullback Attractors from Delay to Non-Delay
We first prove the eventually compactness of pullback attractors.
Lemma 13
Let \(\vartheta _{n}\in \mathcal {A}_{d}^{\varrho _{n}}(t)\) with \(\varrho _{n}\rightarrow 0\) and \(t\in \mathbb {R}\), then there are \(\vartheta \in V\) and an index subsequence of \(\{n^{*}\}\) of \(\{n\}\) such that
Proof
By the invariance of \(\mathcal {A}_{d}^{\varrho _{n}}(\cdot )\), there exists a \(\tilde{\vartheta }_{n}\in \mathcal {A}_{d}^{\varrho _{n}}(t-\tau _{n})\) with \(\tau _{n}\rightarrow +\infty \) such that
Since \(\mathcal {A}_{d}^{\varrho _{n}}\in \mathfrak {D}\), by the same method as in Step 1 of Theorem 4, there exists \(\delta >0\) with \(|\theta _{1}-\theta _{2}|\le \delta \) such that for any \(\varepsilon >0\),
Since \(\varrho _{n^{*}}\rightarrow 0\) as \(n^{*}\rightarrow +\infty \), there exists a \(N_{1}\in \mathbb {N}\) such that \(\varrho _{n^{*}}<\delta \) for all \(n^{*}\ge N_{1}\). Then, we have
for all \(n^{*}\ge N_{1}\) and \(\theta \in [-\varrho _{n^{*}}, 0]\). On the other hand, by the same method as in Step 2 of Theorem 4, we obtain that \(\{(S_{\varrho _{n}}(t, t-\tau _{n})\tilde{\phi }_{n})(0)\}_{n\in \mathbb {N}}\) is pre-compact in V. Then, there exist a \(\vartheta \in V\) and an index subsequence of \(\{n^{*}\}\) of \(\{n\}\) such that
It follows from (88) to (90) that there exists a \(N_{2}\ge N_{1}\) such that
for all \(n^{*}\ge N_{2}\) and \(\theta \in [-\varrho _{n^{*}}, 0]\), which proves (87) as desired. \(\square \)
Next, we show the recurrence of absorbing sets. Similar to Lemma 4, for each \(\varrho \in (0, \varrho _{0}]\), we obtain that the evolution process \(S_{\varrho }(\cdot , \cdot )\) associated with (69) has a pullback \(\mathfrak {D}\)-absorbing set \(\mathcal {K}_{d}=\{\mathcal {K}_{d}(t): t\in \mathbb {R}\}\in \mathfrak {D}\), defined by
Define a new non-autonomous set \(\mathcal {K}^{0}=\{\mathcal {K}^{0}(t): t\in \mathbb {R}\}\) by
Similar to (34), we get \(\mathcal {K}^{0}\in \mathfrak {D}_{0}\). Observe that
Now, we state the main result of this section.
Theorem 6
Suppose that (71), (72) and (G1) hold. Then, we have, for each \(t\in \mathbb {R}\),
Proof
Suppose that (92) is not true, then there exist \(\delta >0\) and \(\varrho _{n}\rightarrow 0\) such that
which implies for each \(n\in \mathbb {N}\), there is a \(a_{n}\in \mathcal {A}_{d}^{\varrho _{n}}(t)\) such that
By (87) in Lemma 13, there exist a subsequence (still denoted by \(a_{n}\)) and an element \(a_{0}\in V\) such that
We claim \(a_{0}\in \mathcal {A}_{d}^{0}(t)\). Indeed, by the invariance of \(\mathcal {A}_{d}^{\varrho _{n}}\), there exists a \(a_{n}^{k}\in \mathcal {A}_{d}^{\varrho _{n}}(t-\tau _{k})\) such that
where \(\tau _{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \). Similar to (94), there exist a subsequence (denote by itself) of \(a_{n}^{k}\) and an element \(a_{0}^{k}\in V\) such that \(d_{\varrho _{n}}(a_{n}^{k}, a_{0}^{k})\rightarrow 0\) as \(n\rightarrow \infty \), that is,
which combine with (75) implies
Hence, by (94) and (95), we obtain
By the invariance of \(\mathcal {A}_{d}^{\varrho _{n}}\) again, when \(t_{0}\) small enough, we obtain for all \(t\in \mathbb {R}\)
Then, we have \(a_{n}^{k}\in \mathcal {A}_{d}^{\varrho _{n}}(t-\tau _{k})\subset \mathcal {K}_{d}^{\varrho _{n}}(t-\tau _{k})\). It follows from (91) and (96) that \(a_{0}^{k}\in \mathcal {K}^{0}(t-\tau _{k})\in \mathfrak {D}_{0}\). Since \(\mathcal {A}_{d}^{0}\) is a pullback \(\mathfrak {D}_{0}\)-attracting set, we obtain
which implies \(a_{0}\in \mathcal {A}_{d}^{0}(t)\). By (94), we have
as \(n\rightarrow +\infty \), which contradicts with (93). The proof is complete. \(\square \)
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Zhang, Q. Regular Dynamics for 3D Brinkman–Forchheimer Equations with Delays. Bull. Malays. Math. Sci. Soc. 45, 2959–2995 (2022). https://doi.org/10.1007/s40840-022-01315-y
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DOI: https://doi.org/10.1007/s40840-022-01315-y