THE STRUCTURE AND STABILITY OF PULLBACK ATTRACTORS FOR 3D BRINKMAN-FORCHHEIMER EQUATION WITH DELAY

. This paper concerns the stability of pullback attractors for 3D Brinkman-Forchheimer equation with delays. By some regular estimates and the variable index to deal with the delay term, we get the suﬃcient conditions for asymptotic stability of trajectories inside the pullback attractors for a ﬂuid ﬂow model in porous medium by generalized Grashof numbers.

1. Introduction. The delay effect originates from the boundary controllers in engineering. The dynamics of a system with boundary delay could be described mathematically by a differential equation with delay term subject to boundary value condition such as [20]. There are many results available in literatures on the wellposedness and pullback dynamics of fluid flow models with delays especially the 2D Navier-Stokes equations, which can be seen in [1], [2], [8] and references therein. Inspired by these works, in this paper, we study the stability of pullback attractors for 3D Brinkman-Forchheimer (BF) equation with delay, which is also a continuation of our previous work in [6]. The existence and structure of attractors are significant to understand the large time behavior of solutions for non-autonomous evolutionary equations. Furthermore, the asymptotic stability of trajectories inside invariant sets determines many important properties of trajectories. The 3D Brinkman-Forchheimer equation with delay is given below: ∂u ∂t − ν∆u + αu + β|u|u + γ|u| 2 u + ∇p = f (t, u t ) + g(x, t), ∇ · u = 0, u(t, x)| ∂Ω = 0, u| t=τ = u τ (x), x ∈ Ω, u τ (θ, x) = u(τ + θ, x) = φ(θ), θ ∈ (−h, 0), h > 0. (1) Here, (x, t) ∈ Ω × R + with Ω ⊂ R 3 be a bounded domain with sufficiently smooth boundary ∂Ω. u = (u 1 , u 2 , u 3 ) is the velocity vector field, p is the pressure, ν > 0 and α > 0 denotes the Brinkman kinematic viscosity and the Darcy coefficients respectively, β > 0 and γ > 0 are the Forchheimer coefficients, the external force g(x, t) ∈ L 2 loc (R; H) is a locally square integrable function and the delay term is considered either (1). a general delay f (t, u t ) with u t : [−h, 0] → H defined as u t = u(t + s) which denotes constant, variable and distributed delays, see Caraballo and Real [1], [2]. or (2). the special application of f (t, u t ) as a sub-linear operator for a smooth function ρ(·) defined in Section 4, which satisfies subadditive and positive homogeneous property with second variable component, see Marín-Rubio and Real [8].
The BF equation describes the conservation law of fluid flow in a porous medium that obeys the Darcy's law. The physical background of 3D BF model can be seen in [14], [9], [18], [19]. For the dynamic systems of problem (1) without delay, i.e., f (t, u t ) = 0, we can refer to [6], [10], [15], [16], [21] and literature therein for the existence of global weak solution and attractors. For the problem (1) with delay f (t, u t ), the global existence of mild solution, continuous dependence on initial data and the minimal family of pullback attractors have been obtained in [6]. In addition, the upper semi-continuous property of pullback attractors as delay vanishes has been proved by virtue of some regular estimates. Furthermore, as a special application of the delay f (t, u t ), the pullback dynamics of problem (1) with sub-linear operator (2) has also been shown. However, the asymptotic stability of trajectories inside pullback attractors is still open. Motivated by [3] and [17], applying regularity for weak solution and iteration technique with variable indices, we present some sufficient conditions with the generalized Grashof number to achieve the stability of pullback attractors in this paper. The main features and results can be summarized as follows.
(a) For problem (1) with delay f (t, u t ), we use the regular estimate to achieve an upper bound of Grashof number, which implies the exponential stability of trajectories inside pullback attractors. The proof does not depend on initial data with more regularity, see Section 3. Here we use, the delay f (t, u t ) in problem (1) can be the constant, variable and distributed delays F (u(t − h)), F (u(t−ρ(t))) and 0 −h k(t, s)u(t+s)ds respectively, here F (·) is an appropriate function, see [1], [2].
(b) For problem (1) with special application of f (t, u t ) as the sub-linear operator, the variable indices have been introduced to deal with nonlinear term β|u|u + γ|u| 2 u and sub-linear operator by iterative argument, see Section 4. (c) The asymptotic stability of trajectories inside pullback attractors is further research of the results established in [6]. However, the stability of pullback attractors for (1) with infinite delay is still unknown.
2. Preliminary. In this section, we give some notations and the equivalent abstract form of (1) in this section. Denoting E := {u|u ∈ (C ∞ 0 (Ω)) 3 , divu = 0}, H is the closure of E in (L 2 (Ω)) 3 topology, · H and (·, ·) denote the norm and inner product in H respectively. V is the closure of set E in (H 1 (Ω)) 3 topology, · V and ((·, ·)) denote the norm and inner product in V respectively. H and V are dual spaces of H and V respectively. Clearly, V → H ≡ H → V , H and V are dual spaces of H and V respectively, where the injection is dense, continuous. The norm · * denotes the norm in V , ·, · be the dual product in V and V . Let P be the Helmholz-Leray orthogonal projection from (L 2 (Ω)) 3 onto the space H, we define A := −P ∆ as the Stokes operator with domain D(A) = (H 2 (Ω)) 3 V and λ is the first eigenvalue of A, the sequence {ω j } ∞ j=1 is an orthonormal system of eigenfunctions of A, and {λ j } ∞ j=1 (0 < λ 1 ≤ λ 2 ≤ · · · ) are eigenvalues of A corresponding to the eigenfunctions {ω j } ∞ j=1 , see more details in [13].
By the Helmholz-Leray projection defined above, (1) can be transformed to the abstract equivalent form then we show our results for (3) with f (t, u t ) as either general case or its special case F (u(t − ρ(t))) in Sections 3 and 4, respectively. We also define some Banach spaces on delayed interval as respectively. The Lebesgue integrable spaces on delayed interval can be denoted as for purpose of phase space in next sections.
3. The asymptotic stability of trajectories inside pullback attractors for (1) with delay f (t, u t ).
3.1. Assumptions. Some assumptions on the external forces and parameters which will be imposed in our main results are the following: (H f ) The function f : R × C H → H satisfies: (a) For any ξ ∈ C H , the function f (·, ξ) is measurable and f (·, 0) ≡ 0. (b) There exists a L f > 0 such that (H g ) The function g(·, ·) ∈ L 2 loc (R, V ) satisfies that there exists η > 0 such that holds for any t ≥ τ . Then In this part, we shall present some retarded integral inequalities from Li, Liu and Ju [5]. Consider the following retarded integral inequalities: (8) where E, K 1 and K 2 are non-negative measurable functions on R 2 , ρ ≥ 0 denotes a constant. Let X be a Banach space with spatial variable, based on the retarded Banach space above, then we use · C X denotes the norm of space C([−h, 0]; X) for some h ≥ 0, y(t) ≥ 0 is a continuous function defined on C([−h, T ]; X), y t (s) = y(t + s) for s ∈ [−h, 0].
1+ϑ , then the solution reduces to trivial for the occasion κc < 1.
3.3. Well-posedness. The minimal family of pullback attractors will be stated here in preparation for our main result.
• Well-posedness Theorem 3.4. Assume that the external forces g(t, x) and f (t, u t ) satisfy the hypothesis (H g ) and (H f ), the initial data (u τ , φ) ∈ M H = H × (C H ∩ L 2 V ) and (H 0 ) are also true. Then there exists a unique global weak solution u = u(·, τ, u τ , φ) Proof.
Step 1. Existence of local approximate solution.
By the property of the Stokes operator A, the sequence of eigenfunctions {w i , i = 1, 2, · · · } of the Stokes operator is an orthonormal complete basis of H formed by elements of V ∩ (H 2 (Ω)) 3 such that Let H m = span{w 1 , w 2 , · · · , w m }, P m : H → H m be a projection, then the approximate solutions can be written as u Then it is easy to check that (13) is equivalent to an ordinary differential equations with unknown variable function h jm (t). By the Cauchy-Peano Theorem of ordinary differential equation, the problem (13) possesses a local solution over the time interval [0, t m ].
Step 2. Uniform estimates of approximate solutions.
Multiplying (13) by h jm (t), and then summing from j = 1 to m, it yields Integrating in time, using the hypotheses on f (·, ·) and g(t, x), by the Young inequality, we get Using the Gronwall Lemma of integrable form, we conclude that Step 3. Compact argument and passing to limit for deriving the global weak solutions.
In this step, we shall prove {u m } has a strong convergence subsequence by the Aubin-Lions Lemma along with the uniformly bounded estimate of dum dt in L 2 (0, T ; V ). By the estimates of u m above step and continuous embedding V → L p (Ω) with p ∈ [1, 6] for three dimension, we can obtained that |u m |u m ∈ L 2 (τ, T ; H) and |u m | 2 u m ∈ L 2 (τ, T ; H). From the equation and assumptions (H f ) and (H g ), we can see that {du m /dt} is bounded in L 2 (τ, T ; V ). By virtue of the Aubin-Lions Lemma, we obtain that {u m } has a strong convergent subsequence (also denoted as which coincides with the initial data u m (τ ) = P m u τ → u(τ ) = u τ and φ m (s) → φ(s).
For the purpose of passing to limit in (13), denoting v = u m − u, we point out that we can deal with the nonlinear terms as the following novelty. Since w j is an eigenfunction of Stokes operator, we claim that L 4 (τ,T ;L 4 (Ω)) ) (18) and the convergence of delayed external force f (t, u mt ) can be verified by the hypotheses.
Thus, passing to the limit of (13), we conclude that u is at least one of global weak solutions for problem (1).
• The regularity Proposition 1. Assume that the external forces g(t) and f (t, u t ) satisfy the hy- and (H 0 ) are also true. Then the global weak solution u in Theorem 3.4 has the regular boundedness in L ∞ (τ, T ; V ).
Proof. Taking inner product of (3) with Au, it yields 1 2 According to Lemma 3.3, the nonlinear terms have the following estimates and and hence, we conclude that d dt Letting t − 1 ≤ s ≤ t, neglecting the third and fourth terms on the left hand side of (23), integrating (23) with time variable from s to t, it yields and Then integrating with s from t − 1 to t, using the uniform boundedness of u in Theorem 3.4, we deduce that which means the uniform boundedness of the global weak solution u in L ∞ (τ, T ; V ). The proof has been finished.
• Uniqueness Proposition 2. Assume the hypotheses in Theorem 3.4 hold. Then the global weak solution u is unique.
Proof. Using the same energy estimates as above, we can deduce the uniqueness easily, here we skip the details.

Pullback attractors.
To description of pullback attractors, the functional Based on the well-posedness, we shall verify the pullback dissipation and asymptotic compactness for the process to achieve the existence of pullback attractors, which also needs the following assumption: for any t ≤ T .
• The continuous process which is a continuous process.
Proof. By the energy estimate of (1) and using Young's inequality, we arrive at where η ∈ (0, νλ 1 ). Multiplying the above inequality by e ηt , we obtain d dt (e ηt u 2 H ) + e ηt νλ 1 u 2 H + 2βe ηt u 3 L 3 (Ω) + 2γe ηt u 2 Thus integrating with respect to time variable, it yields and by the Gronwall Lemma, we can derive the estimate in our theorem. Using the energy estimate of (1) again, we can check that Integrating from s to t, using the estimate of u in H, we can derive the desired result. Based on Lemma 3.5, we can present the pullback dissipation based on the following universes for the tempered dynamics.
where the balls is defined as centered in the point zero and measured by the radius Moreover, the pullback D M H F -absorbing set can be defined as the same technique. Proof. Using the estimates in Lemma 3.5, choosing anyD ∈ D M H η (t), there exists a pullback time τ (D, t) ≤ t − h such that holds for any τ ≤ τ (D, t) and (u τ , φ) ∈ D(τ ). Moreover, in particular, it yields that u t 2 C H ≤ ρ 2 H (t). By the similar technique and estimate in Lemma 3.5, we derive that u t Combining the above estimate and the definition of universe, we conclude that D 0 ∈ D M H η . The proof has been finished.
• Pullback asymptotic compactness For arbitrary fixed t ≥ τ , consider a familyD ∈ D M H η , let {τ n } ⊂ (−∞, t] with τ n → −∞ and {(u τn , φ n )} with (u τn , φ n ) ∈ D(τ n ) be two sequences for all n, then we denote {(u n , u n t )} ∈D as a sequence with u n (·) = u(·; τ n , u τn , φ n ). By using the similar energy estimate in Theorem 3.4 and technique in Proposition 4, there exists a pullback time τ From the hypotheses (H f ), f (t, u n t ) is bounded in L 2 (t − h − 1, t; H), which implies {(u n ) } is bounded in L 2 (t−h−1, t; V ). Hence, by the Aubin-Lions Lemma and the diagonal procedure, there exists a subsequence (relabeled also as {u n }) such that u n (t) → u(t) strongly in L 2 (t − h − 1, t; H). Combining the uniform boundedness of sequence above, it yields that By Theorem 3.4, from the hypothesis on f , it follows that Thus, from (38) and (39), we can conclude that u ∈ C([t − h − 1, t]; H) is a weak solution for problem (1) with the initial data of u(·, x) at the initial time t − h − 1 denoted as u t−h−1 .
From the uniform bounded estimate of u n by Proposition 1 in L ∞ (t − h − 1, t; V ) and (u n ) is uniform bounded in L 2 (t − h − 1, t; V ), using the Aubin-Lions-Simon Lemma (see [12] ), we can derive that Therefore, we can conclude that Step 2. The strong convergence of corresponding sequences via energy equation method: u n (s n ) → u(s) strongly in C([t − h, t]; H).
The asymptotic compactness of sequence u n in H will be presented in sequel, i.e., which is equivalent to prove (42) combining with for a sequence {s n } ⊂ [t − h, t] and s n → s as n → +∞, which will be proved next. Using the energy estimate to all u n and u, we obtain that for all and Then, we define the functionals J n (s) and J(s) defined for s ∈ [t − h − 1, t] as following (f (r, u n r ), u n (r))dr (47) and Combining the convergence in (38), observing that J n (s) and J(s) are continuous and non-increasing in [t − h − 1, t], we derive that i.e., for ∀ ε > 0, there exists a n k ∈ N, for all n ≥ n k and s k ∈ [t − h − 1, t], such that Since J(s) is continuous and J n (s) is uniformly continuous with respect to time s, then for any ε > 0, there existsñ k ∈ N such that for the sequence {s k } ⊂ [t−h−1, t] with s k → s for all n ≥ñ k , Choosingn k = max{n k ,ñ k }, then for all n >n k , it yields that Therefore, for any which implies we conclude the strong convergence u n (s n ) → u(s) in C([t − h, t]; H).
Step 3. The strong convergence: u n (s n ) → u(s) strongly in L 2 (t − h, t; V ).
Combining the energy estimates in (45) and (46), noting the energy functionals J n (·) and J(·), using the convergence in (38), we can deduce the norm convergence Hence jointing with the weak convergence in (38), we can derive that u n (s n ) → u(s) strongly in L 2 (t − h, t; V ).
By using the results from Steps 2 to 4 and noting the definition of universe, we can conclude that the processes is D M H η -pullback asymptotic compact in M H , which means the proof has been finished.   pullback asymptotic compactness of the processes. Using the existence theory of pullback attractors in [3] or [4], we can conclude our desired results.
Based on the universes defined in Definition 3.6, the relation between

Main result:
The sufficient condition of asymptotic stability of trajectories inside pullback attractors of (1) with f (t, u t ).
Definition 3.9. The pullback attractors is asymptotically stable if the trajectories inside attractor reduces to a single orbit as τ → −∞.
is a generalized Grashof number for the fluid flow, and here C |Ω| > 0 is a constant which depends on the volume of Ω.
Proof. Let u(t) and v(t) be two weak solutions of problem (3) with delay f (t, u t ) which subject to initial data and respectively. Denoting as two trajectories inside the pullback attractors, letting w = u(t) − v(t) and w t = u t − v t , then it is easy to check that w satisfies the following problem Taking inner product of (62) with w in H, using Poincaré's inequality and Lemma 3.3, it follows Using the Poincaré inequality and Lemma 3.1, noting that if then we can obtain Denoting and and by virtue of Lemma 3.2, choosing κ(K 1 , 0) < 1 1+Θ , then there exists M > 0 and λ > 0, such that we can obtain the estimate Substituting (70) into (64), using Lemma 3.1 again, we can conclude the following estimate From (70) and (71), if we fixed u τ andũ τ and let τ → −∞, then we can conclude that the trajectories inside pullback attractors reduce to a point, which implies the pullback attractors is asymptotically stable provided that where Since u and v are the global weak solutions for problem (3), we will use some delicate estimates to make (72) more explicit next. Multiplying (3) with u and integrating by parts over Ω, we have Integrating (74) from τ to t, it follows By the estimate of (80) and (81), we derive Combining (72), (73) with (84), we conclude that and hence the asymptotic stability holds provided that If we define the generalized Grashof number as G(t) = , then a sufficient condition for the asymptotic stability of trajectories inside pullback attractors can be conclude as which completes the proof for our first result.
Remark 4. Theorem 3.10 is a further research for the existence of pullback attractor in [6].

4.1.
Assumptions. We first state some hypothesis on the external forces and sublinear operator.
4.2. Well-posedness and pullback attractors of (1) with f (t, u t ) = F (u(t − ρ(t))). In this part, the well-posedness and pullback attractors for problem (1) with sub-linear operator will be stated for our discussion in sequel.
• Well-posedness Assume that the initial date u τ ∈ H and φ ∈ C H ∩L 2q (−h, 0; H) with 1 q + 1 q = 1 and recall that (1) with sub-linear operator has the following abstract form: which possesses a global mild solution as the following theorem.
Theorem 4.1. Assume that the external forces g(t) and f (t, u t ) = F (u(t − ρ(t))) satisfy the hypothesis (H f ) and (H g ), the initial data u τ ∈ H and φ ∈ C H ∩ L 2q H and ( H 0 ) are also true. Then there exists a unique global mild solution u = u(·, τ, u τ , φ) ∈ L ∞ (τ − h, T ; H) ∩ L 2 (τ, T ; V ) ∩ L 4 (τ, T ; L 4 (Ω)) of problem (1) for special case of f (t, u t ) = F (t − ρ(t)), such that it satisfies (92) in distributed sense and the following energy equality Moreover, we can define a continuous process {U (t, τ )|t ≥ τ, τ ∈ R} as U (t, τ ) : Proof. Using the Galerkin method and compact argument as in Section 3.3, we can easily derive the result.

• The pullback dynamics
After obtaining the existence of the global well-posedness, we establish the existence of the pullback attractors to (1) with sub-linear operator.

Theorem 4.2. (The pullback attractors in H)
Assume that (H f ) and (H g ) hold, the initial data u τ ∈ H and φ ∈ C H ∩ L 2q H and ( H 0 ) are also true. then the process U (t, τ ) associated to problem (1) with f (t, u t ) = F (u(t − ρ(t))) has two families of pullback attractors A C H ×H (t) similar as in Theorem 3.7.
Proof. Using the similar technique as in Section3.3, we can obtain the existence of pullback attractors, here we skip the details.
Theorem 4.3. We assume that the external forces g(t) and f (t, u t ) = F (u(t−ρ(t))) satisfy the hypothesis (H f ) and (H g ), the initial data (φ, u τ ) ∈ (C H ∩ L 2q H ) × H and ( H 0 ) holds true.
Then the trajectories inside pullback attractors A C H is asymptotically stable if where G 2 (t) = is defined as the generalized Grashof number for the fluid here C |Ω| > 0 is a constant dependent on the volume of Ω. Proof.
Step 1. The inequality for asymptotic stability of trajectories.
Moreover, using the variable index introduced above, we can conclude that Step 3. The sufficient condition for asymptotic stability of trajectories inside pullback attractors.
Combining (107) with (116), we conclude that and hence the asymptotic stability holds provided that If we define the generalized Grashof number as G(t) = g 2 H | ≤t ν 2 λ1 1/2 , neglecting the positive terms on the left-hand side of (118), then a sufficient condition for the asymptotic stability of trajectories inside pullback attractors can be conclude as which completes the proof for our first result.
such that there exists some σ > 0, the following assumption holds. Then more precise sufficient condition for the asymptotic stability of pullback attractors is which has smaller upper boundedness than (119).
5. Further research. The structure and stability of 3D BF equations with delay are investigated in this paper. A future research in the pullback dynamics of (1) is to study the geometric property of pullback attractors, such as the fractal dimension.