ATTRACTORS POLYMER SOLUTIONS MOTION

. Existence of trajectory, global and pullback attractors for an incompressible non-Newtonian ﬂuid (namely, for the mathematical model which describes a weak aqueous polymer solutions motion) in 2D and 3D bounded domains is studied in this paper. For this aim the approximating topologi- cal method is eﬀectively combined with the theory of attractors of trajectory spaces.

1. Introduction. The motion of an incompressible fluid of constant density filling a bounded domain Ω ⊂ R n , n = 2, 3, on a time interval [0, T ], T > 0, is described by the Cauchy momentum equation [13]: div v = 0, (x, t) ∈ Ω × (0, T ), where v(x, t) = (v 1 , ..., v n ) is the velocity vector of the particle at the point x at the time t, p = p(x, t) is the fluid pressure at the point x at the time t and f = f (x, t) is the body force vector acting on the fluid. The symbol Div σ stands for the vector n j=1 ∂σ1j ∂xj , n j=1 ∂σ2j ∂xj , ..., n j=1 ∂σnj ∂xj , whose coordinates are divergences of rows of the matrix σ = (σ ij (v)) i=1,...,n j=1,...,n , where σ(v) is the deviator of the stress tensor. System (1)-(2) describes the motion of all kinds of fluids. However, it is incomplete. As a rule, one uses additional relations between the deviator of the stress tensor σ(v) and the strain velocity tensor E = (E ij (v)) i=1,...,n j=1,...,n , E ij (v) = 1 2 ∂vi ∂xj + ∂vj ∂xi . Such relations are known as constitutive or rheological laws. Choosing a constitutive law we specify a type of fluid. Note that these laws are hypotheses and have to be verified for specific fluids by experimental data.
The rheological relation that governs the motion of viscoelastic medium is the following σ = 2νE + 2κĖ,

ANDREY ZVYAGIN
where ν > 0 is the viscosity of fluid and κ > 0 is the retardation (delay) time. This model of fluid motion describes the motion of a viscous non-Newtonian fluid that needs time to start moving under the action of force instantly applied. As far as we know, this model was first practically considered by [21]. He named it the model of motion of weak aqueous polymer solutions. According to Pavlovsky, it is necessary to consider elastic properties as well as viscous ones in the case of such solutions. This is due to the fact that the stress depends both on the history of deformation and on the instantaneous value of the strain velocity tensor. The viscous properties of such a material are associated with the influence of the solvent. If the concentration of a polymer is low, this contribution is not neglible. This is confirmed by experimental research of solutions of polyethyleneoxide and polyacrylamide [1] and solutions of polyacrylamide and guar gum [2].
The constitutive law (3) involves a time derivativeĖ. The first mathematical treatment of (3) involved the partial derivative, in which case the law yields the Voigt model (a model of motion of linear viscoelastic non-Newtonian fluids [19]). This system of equations were also proposed in [4] as a regularization, for small values of the parameter α, of the 3D Navier-Stokes equations. Then the case of the total derivative was studied by [20]; however, the proofs of his results were incorrect [15]. A complete proof of weak solvability of (1)-(3) with the total derivative was first given in [45].
In the recent years rational mechanics [29] has influenced scientists in the way that they have started to investigate constitutive laws independent of the observer, i.e. that do not change under the Galilean transformation: where a is a time value, x 0 is a point in a space, x * 0 is a time function with values in space, Q is a time function with values in the set of orthogonal tensors. In other words, if the original tensor function changes according to the law (4), will the constitutive law be the same in different reference frames? In the case of partial and total derivatives the answer is negative. In order to answer this question positively one introduces objective derivatives [29].
Example of an objective derivative of a tensor is the regularized Jaumann's derivative [46]: where ρ : R n → R is a smooth function with compact support such that R n ρ(y)dy = 1 and ρ(x) = ρ(y) for x and y with the same Euclidean norm; W = (W ij (v)) i=1,...,n j=1,...,n , W ij (v) = 1 2 ( ∂vi ∂xj − ∂vj ∂xi ) is the vorticity tensor. Note that the constitutive law (3) with the regularized Jaumann's derivative is similar to a particular case of second grade fluids [22], [23]. Existence and uniqueness results for weak and classical solutions of both the stationary and time-dependent problems have been established under various restrictions on the normal stress modulus and the data in [9,12,26].
As we investigate the existence of attractors we consider the mathematical model (5)-(6) in a bounded domain Ω ⊂ R n , n = 2, 3, with the boundary ∂Ω of class C 2 on a time interval [0, +∞).
For the system (5)-(6) we consider the initial-boundary value problem with the initial condition and the boundary nonslip condition The problem (5)-(6) is investigated in [39]- [42]. In the case of this problem neither the global solvability in the strong sense nor the uniqueness of the weak solution have been proved. Consequently, it is impossible to use classical approach to attractors.
It is well known that limiting regimes for problems which have no uniqueness theorems can be investigated on the basis of the theory of trajectory attractors. This theory was constructed by M.I. Vishik and V.V. Chepyzhov [6,7,8,30]. A similar method was put forward independently by G.R. Sell [24,25]. In particular, it turns out that an attractor can be constructed for the Navier-Stokes system in the 3D case [8] or for similar our considered problem 2D non-Newtonian fluid [34].
However, for the model of polymer solutions motion (and others viscoelastic models) we cannot always find shift-invariant trajectory spaces, so a theory attractors in which trajectory spaces were not necessarily shift-invariant was developed in [31,47], whereas the construction used in [8] and [25] required this invariance. So why for prove of existence of attractors for this model in this paper the method which developed in [47] and which is based on approximating topological approach and on theory of attractors of trajectory spaces will be used. Given an autonomous system with an attractor, it is only a matter of elapsed time, when the initial data gets forgotten? In nonautonomous systems the absolute times of both start and check are to be taken into account. As a consequence, there is more than one way to generalise the notion of attractor to nonautonomous systems.
One approach is to consider pullback attractors. They were first considered in [10,14]. Initially, the theory of pullback attractors was naturally developed in the framework of processes (biparametric families of operators describing the evolution of nonautonomous systems). The infinite-dimensional setting of this theory has become quite rich both in abstract results and in applications. In particular, there are a number of results concerning pullback attractors of Newtonian fluids as well as certain non-Newtonian ones [3,5,33,35,36]. For these pullback attractors the invariant measures can be constructed [18,37,38]. However, typical lack of uniqueness impedes the use of processes in fluid mechanics. The notion of pullback attractor has recently been ported to trajectory spaces and constructed for the Navier-Stokes system by D.A. Vorotnikov [32].
The paper is organized as follows. In Section 2 we recall some auxiliary definitions and results and formulate main results. Section 3 is devoted to the proof of the main result on existence of weak solution. Section 4 is devoted to the proof of the main result on existence of trajectory and global attractors and in Section 5 the proof of the main result on existence of pullback attractor is described. 2. Preliminaries, notation and the main result.
2.1. Functional spaces. We use standard notation for the Lebesque and Sobolev spaces. Let V be the set of smooth divergence free functions Ω → R n , n = 2, 3, with compact support contained in Ω. Also let V 0 be the closure of V with respect to the norm of space L 2 (Ω), V 1 be the closure of V with respect to the norm of space W 1 2 (Ω) and V 2 = W 2 2 (Ω) ∩ V 1 . We will also use the well-known decomposition of L 2 (Ω) (see [28]): Let π : L 2 (Ω) → V 0 be the Leray projector. Consider the operator A = −π∆ defined on D(A). It can be extended to a closed self-adjoint operator in V 0 . We denote the extension by the same letter. The extended operator A is positive and has a compact inverse. Hence A has countably many eigenvalues 0 < λ 1 λ 2 . . .. Let e k denote associated eigenfunctions. Vector functions e k (k = 1, 2, . . .) are smooth (depends on boundary smoothness). Consider the set (here m depends on v) and for any α ∈ R define space V α as the closure of E ∞ with respect to the norm It can be shown that for α = 0, 1, 2 the construction described above leads to the same V 0 , V 1 and V 2 as introduced at the beginning.
In case α 0 we have continuous embedding V α ⊂ W α 2 (Ω) and the norm · V α is equivalent to the norm of W α 2 (Ω) (see [11,45]). Note that for α = 0, 1, 3 we have (for matrices A = (a ij ) and B = (b ij ) of order n we put A : B = a ij b ij ). For α > β 0 the embedding V α ⊂ V β is compact. Let α 0 and (V α ) * be the conjugate space of V α . Then the space (V α ) * is isometric to V −α . We identify these spaces.
We need the following Banach spaces in order to define weak solutions and Also let W loc 1 (R + ) be the class of function v : R + → V 1 (R + denote the nonnegative half-axis of the real axis R + ) such that the restriction of v to any segment [0, T ] belongs to W 1 [0, T ]; likewise, let W loc 2 (R + ) denote the class of functions v ∈ C(R + , V 3 ) such that the restriction of v to any segment [0, T ] belongs to W 2 [0, T ]. These classes are needed to defining solutions on the nonnegative semiaxis.
The following compactness theorem is very important. Let X 0 ⊂ F ⊂ X 1 be Banach spaces, where the first embedding is compact and X 0 is reflexive; further, let 0 < T < ∞ and 1 p i ∞ (i = 0, 1). Consider the space Theorem 2.1. If p 0 < ∞, the following embedding is compact: If p 0 = ∞ and p 1 > 1, the following embedding is compact: The proof can be found, e.g., in [27].

Statement of weak solution problem.
Let the body force f ∈ L 2 (Ω) be fixed.

Definition 2.2.
A weak solution of the initial-boundary value problem (5)-(8) on the interval [0, T ] is a function v ∈ W 1 [0, T ] such that for any ϕ ∈ V 3 and almost all t ∈ (0, T ) it satisfies the equality (9) and the initial condition . Consequently, all the integrals on the left-hand side of (9) exist. The identity (9) is derived from equations (5)-(8) in a standard way: under the assumption that a classical solution exists, multiply equation (5) by an arbitrary function ϕ ∈ V 3 and integrate by parts certain terms; since ϕ is solenoidal, the term grad p is eliminated.
The following existence theorem holds.
For any a ∈ V 1 , problem (5)-(8) has a solution on the semiaxis R + that satisfies the inequality where the constants R 0 > 0 and α > 0 are independent of v.
This theorem is proved in Section 3.

Statement of trajectory and global attractors problem.
We first introduce some definitions and theorems concerning the trajectory and global attractors.
The existence result of a trajectory and global attractors and its property for a non-Newtonian fluid process one can find at the book [47] and the review article [43]. Let E and E 0 be Banach spaces, E ⊂ E 0 (the embedding is assumed to be continuous); we also assume that E is reflexive. Let L ∞ (R + ; E) be the Banach space of essentially bounded functions on R + taking values in E. The linear space Now we look at the shift operators T (h): to each function g this operator assigns the function is obvious, as is the fact that T(0) is the identity operator; this allows us to say that the family {T (h) : h 0} is a semigroup, the so-called translation semigroup.

Definition 2.4. A nonempty set
is called a trajectory space for the problem; its elements are called trajectories for the problem.
The only requirement is that the trajectory space must be nonempty (and consist of functions in C(R + ; E 0 ) ∩ L ∞ (R + ; E)).
Of course, it follows from these definitions that each absorbing set is attracting.
is called a trajectory semi-attractor (of the trajectory space H + ), if the following conditions hold: (i) P is compact in C(R + ; E 0 ) and bounded in L ∞ (R + ; E); (ii) the inclusion T (t)P ⊂ P holds for all t 0; (iii) P is an attracting set in the sense of Definition 2.5.
is called a trajectory attractor (of a trajectory space H + ) if it satisfies the following conditions: (iii) the set U is attracting in the sense of Definition 2.5.
Definition 2.9. A minimal trajectory attractor (of a trajectory space H + ) is a trajectory attractor which is minimal with respect to inclusion (it lies in any other trajectory attractor of the space H + ).
(iii) A is the minimal set satisfying conditions (i) and (ii) in this definition (A lies in any set satisfying these two conditions).
Remark 3. It follows directly from definitions that if a trajectory space has a minimal trajectory attractor or a global attractor, then this attractor is unique.
Theorem 2.11. Suppose the trajectory space H + has a trajectory semi-attractor P .
Then H + has the minimal trajectory attractor U and the kernel of H + is compact in C(R; E 0 ) and bounded with respect to the norm of L ∞ (R; E).
Theorem 2.12. Suppose the trajectory space H + has the minimal trajectory attractor. Then the global attractor A of H + exists.
it is a solution of this problem with some a ∈ V 1 and the following inequality holds: The set of trajectories is called its trajectory space of the problem and is denoted by H + .
. Thus inequality (12) follows from inequality (11), and by Theorem 2.3 we see that any point a ∈ V 1 is the beginning of a trajectory.
Consider a number δ ∈ (0, 1) and suppose that f ∈ L 2 (Ω). These are main results concerning the existence of attractors.
Theorem 2.14. The trajectory space H + of problem (5)-(8) has the minimal trajectory attractor U. The attractor is bounded in L ∞ (R + ; V 1 ) and compact in C(R + ; V 1−δ ); it attracts sets of trajectories bounded in L ∞ (R + ; V 1 ) with respect to the topology of C(R + ; V 1−δ ). These theorems are proved in Section 4.
2.4. Statement of pullback attractors problem. We first introduce some definitions and theorems concerning the pullback attractors.
Let E and E 0 be Banach spaces, E ⊂ E 0 (the embedding is assumed to be continuous); we also assume that E is reflexive. For every τ ∈ R we assign the non-empty set . We start with basic definitions and results of the abstract theory of trajectory pullback attractors [32,44].
The sets H + τ are called the trajectory spaces and elements thereof are called trajectories. The family H + = {H + τ } τ ∈R is called the family of trajectory spaces. Fix a class of families of sets D over E assuming that for any family D = {D t } ∈ D we have D t = ∅ for any t ∈ R. For any D = {D τ } ∈ D we consider the family Definition 2.17. A family P = {P θ } (P θ ⊂ T ) is called pullback D-absorbing for H + , if for any family D ∈ D and for any θ ∈ R it exists the number τ D (θ) θ, such that for all τ τ D (θ) the inclusion T(θ − τ )H + τ (D) ⊂ P θ , holds, and the function τ D : R → R is nondecreasing.
(ii) for any θ ∈ R there exists a continuous function ϕ θ : R + → R such that for any trajectory v ∈ P θ the inequality v(t) E ϕ θ (t) holds for all t ∈ R + . This family is called T -compact, if in addition P θ is closed (and thus, compact) in C(R + ; E 0 ) for any θ ∈ R. (ii) for any D ∈ D and θ ∈ R the pullback attraction Remark 5. The minimal trajectory pullback attractor is unique, and so is the global pullback attractor [32].
Theorem 2.23. Suppose that H + admits a T -precompact pullback D-absorbing family P, and let P denote the closure of P with respect to the topology of C(R + ; E 0 ). Then there exists a minimal trajectory pullback D-attractor U ⊂ P.
Theorem 2.24. Suppose that H + has a trajectory pullback D-semiattractor P.
Then it also has the minimal trajectory pullback D-attractor U ⊂ P .
We use this approach to consider pullback attractors of the problem (5)- (8).
In this section we consider equalities (5) and (6) on Ω × (τ, +∞) with the initial condition v(x, τ ) = a(x), x ∈ Ω, (13) and the boundary condition We assume that the body force f in the equation (5) belongs to L loc 2 (R; V 0 ) and verifies for all t ∈ R (where α > 0). Fix δ ∈ (0, 1]. For the introduction of the class T we use E = V 1 and E 0 = V 1−δ . Theorem 2. 26. Suppose that f ∈ L loc 2 (R, V 0 ) satisfies (15). Then the family of trajectory spaces H + has a minimal trajectory pullback D-attractor U and a global pullback D-attractor A = U(0).
The proof of this theorem will be given below in the section 5.
and the initial condition where ε > 0, α = ν/(k 2 0 + κ), k 0 and b are constants. Note that (16) differs from (9)  In what follows we consider operator equations. Consider the following operators: It will be convenient to have a notation for the exponential function. By definition, for any β ∈ R put e β (t) = e βt .
Since ϕ ∈ V 3 is arbitrary in (16) this identity is equivalent to the following operator equation We also define the following operators: The problem of finding a solution of equation (18) satisfying the initial condition (17) is equivalent to the problem of finding a solution for the following operator equation:

Properties of operators.
Lemma 3.2. The following properties hold.

3.3.
A priori estimate. Consider the family of auxiliary problems depending on the parameter λ ∈ [0, 1]: The definition of a solution has the same sense for (19) as for (18).
We will use necessary estimates from [17]. Let v ∈ W 2 [0, T ] be a solution of (19) on [0, T ] for certain λ ∈ [0, 1]. Apply both sides (16) with some λ to v(t) and observe that Moreover, it is known [39] that B i (v(t)), v(t) = 0 (i = 1, 2, 3) and D(v(t)), v(t) = 0. Thus we obtain: Now we get from (20) a dissipative estimate with a decaying exponential. We estimate the right-hand side of the latter equation using the Cauchy inequality: Combining this with (20), we get Consider an auxiliary norm on V 1 defined by the formula u 2 = u 2 V 0 +κ u 2 V 1 . This norm is equivalent to · V 1 . We have: where α = ν k0+ε and k 0 is a constant which does not depend on v. Thus it follows from (21) that Substitute v(t) =v(t) exp(−λαt/2) in the first and third terms in the left-hand side of the last inequality. We get Multiplying both sides by exp(λαt), we obtain Integrating the last inequality, we have for all t (this is true both for λ > 0 and for λ = 0). Now multiply both parts of the last inequality by exp(−λαt), whence we obtain Since norms · and · V 1 are equivalent, it follows from the last equality that with a constant C independent of λ, ε, and v. Using (19) it is possible to estimate the derivative v in terms of v. Combining the estimate obtained in this way with (23), we obtain with a constant R 1 that does not depend on ε, λ, and v.

3.4.
Existence of solutions. Now we state the main existence theorem for the auxiliary problem.
a. e. on R + with a constant R 1 independent of ε, λ, and v.
The proof of Theorem 3.3. involves two steps. First we prove the solvability on a finite segment [0, T ] with an arbitrary T > 0 and then we prove that there exists a solution on R + .
Step I. Let T > 0. Let us prove that problem (18), (17)  Lv Equation (26) with λ = 1 corresponds to (18), (17). Note that it follows from (24) that solutions of (26) (if they exist) satisfy the following a priori estimate: where C does not depend on λ (but generally speaking, it can depend on other parameters of the equation). Indeed, it follows from (24) that for a. a. t ∈ [0, T ] the norms v(t) V 3 and e −αT v (t) V 3 do not exceed , and the right-hand part of the last inequality does not depend on t and λ. Also it follows from (27) that solutions of (26) satisfy where R does not depend on λ.
Apply L −1 to both sides of (26)  This equation is equivalent to equation (26) with λ = 1, and the latter equation is in turn equivalent to problem (18), (17). We have thus proved that the auxiliary problem (18), (17) has a solution on [0, T ].
Step II. Let v m be a solution of problem (18), (17)  It is obvious that the functionsṽ m belong to W loc 2 (R + ).
Suppose that 0 < δ < 1. Take an arbitrary T > 0. All but finitely many terms of the sequence {v m } are solutions of (18), (17) on [0, T ]. Since functionsv m take the same value b at 0, by Step I of Theorem 3.3 it follows that they satisfy the estimate v m L∞(0,T ; where C(ε, T ) does not depend on m.
Thus the sequence {v m } is bounded in L ∞ (0, T ; V 1 ) and the sequence of derivatives {v m } is bounded with respect to the norm of L ∞ (0, T ; V −1 ). By Theorem 2.1 we have that the sequence {v m } is precompact in C([0, T ]; V 1−δ ). Since this is true for arbitrary T , the sequence is precompact in C(R + ; V 1−δ ).
Thus the sequence {v m } has a subsequence {v m k } that converges to some function v * in the space C(R + , V 1−δ ). It can be proved [39,43] that this limit function is the the sought for solution of problem (18), (17) One can put Substitute ε m for ε in (18) and consider the initial condition v m (0) = b m for this equation. By Theorem 3.3 this initial-boundary value problem has a solution v m on R + . Inequalities (25) and (29) yield the following estimate: a.e. on R + . More precisely, for each m the last inequality holds for all t ∈ R + \ Q m , where Q m is a set of zero measure. Hence for any t ∈ R + \ Q, where Q = ∪ m Q m is a set of zero measure, inequality (30) holds for all m.
Suppose that 0 < δ 1. According to (30) we have that for any T > 0 the sequence {v m } is bounded in L ∞ (0, T ; V 1 ) and the sequence {v m } is bounded in L ∞ (0, T ; V −1 ). By Theorem 2.1 it follows that the sequence {v m } is compact in C([0, T ]; V 1−δ ). Since T is arbitrary, it follows that the latter sequence is precompact in C(R + , V 1−δ ) and thus has a subsequence {v m k } converging in C(R + , V 1−δ ) to a function v * . It can be proved in [39] and [43] that v * is a solution of problem (18), (17). Now we get (11). Discarding certain nonnegative terms in the left-hand side of (30) we obtain Given k, this inequality holds for any t belonging to a subset of R + of full measure that does not depend on k. Take such a t. First observe that v m k (t) → v * (t) in V 1−δ , since the convergence in C(R + , V 1−δ ) implies pointwise convergence. Further, it follows from (31) that the sequence {v m k (t)} is bounded in V 1 . Consequently, it has a subsequenceṽ µ (t) that converges to v * (t) weakly in V 1 . Therefore, Thus for a. a.
Moreover, one can use (18) and estimate v * in terms of v. Combining such an estimate with (32), we get (11).
4. Trajectory and global attractors for the initial-boundary value problem (5)- (8). In this subsection we fix a number δ ∈ (0, 1). Consider E = V 1 and E 0 = V 1−δ as the pair of Banach spaces needed to introduce a trajectory space. This choice is justified by the fact that V 1 is reflexive and is continuously embedded in V 1−δ .
By Remark 4 the trajectory space introduced by Definition 2.13 is nonempty. Thus it suffices to check the inclusion so as to make sure that the trajectory space is well defined.
The inclusion H + ⊂ L ∞ (R + ; E) directly follows from the definition of the trajectory space. We use Theorem 2.1 in order to prove that the trajectories are continuous. Consider three spaces Let v be an arbitrary trajectory. It follows from (12) that for any segment [0, T ] we have v ∈ L ∞ (0, T ; V 1 ) and v ∈ L ∞ (0, T ; V −1 ). Hence by Theorem 2.1 we obtain that v belongs to C([0, T ]; V 1−δ ). This is true for any T , so v ∈ C(R + ; V 1−δ ).
LetR 4R 0 . Consider the set Let us establish several properties of this set.
Lemma 4.1. The setP is bounded in L ∞ (R + ; V 1 ), compact in C(R + ; V 1−δ ), and the following inclusion holds: Proof. It follows from the definition ofP that it is bounded in L ∞ (R + ; V 1 ). It is not hard to prove that the setP is precompact in C(R + ; V 1−δ ). Indeed, take T > 0. It follows easily from the construction thatP is bounded in L ∞ (0, T ; V 1 ) and the set {v : v ∈P } is bounded in L ∞ (0, T ; V −1 ). By Theorem 2.1 the setP is precompact in C([0, T ]; V 1−δ ). Since T is arbitrary,P is precompact in C(R + ; V 1−δ ). Now let us show thatP is closed and therefore compact in C(R + ; V 1−δ ). Suppose that the sequence {v m } ⊂P converges to v 0 in C(R + ; V 1−δ ). This sequence is bounded in L ∞ (R + ; V 1 ), so it converges to its limit function * -weakly in L ∞ (R + ; V 1 ). Moreover, the sequence of derivatives {v m } converges to v 0 in the sense of distributions and also * -weakly in L ∞ (R + ; V −1 ), since it is bounded in This proves thatP contains the limit function v 0 . SoP is closed. Finally, let us prove the inclusion (33). Take h 0. For any v ∈P we have Proof of Theorem 2.14. Let us prove thatP is a semi-attractor of H + . By Lemma 4.1 we have thatP satisfies conditions (i) and (ii) of Definition 2.7. Let us prove that P is absorbing. Consider an arbitrary set B ⊂ H + bounded in L ∞ (R + ; V 1 ); to be definite, assume that v L∞(R+;V 1 ) R for any v ∈ B. Take h 0 0 such that R 2 e −αh0 1. Let v be an arbitrary function belonging to B. Since v satisfies inequality (12), for all h h 0 we have We have proved thatP is a semi-attractor of H + . By Theorem 2.11 the trajectory space H + has the minimal trajectory attractor.
Proof of Theorem 2.15. According to Theorem 2.12, the global attractor of a trajectory space exists if the trajectory space has the minimal trajectory attractor. Theorem 2.14 implies that the trajectory space H + satisfies this requirement.
Theorem 5.1. For any F ∈ L loc 2 (R + ; V 0 ) and a ∈ V 1 the problem (34)-(37) has a weak solution on the half-axis R + , which satisfies an inequality for almost all t > 0. Moreover, for any weak solution v on R + of the problem (34)- (37) we have that the derivative v ∈ L loc 2 (R + ; V −1 ) and the inequality v (t) holds for almost all t > 0 with the constance C which does not depend on t, v and f .
Remark 6. Estimates (38) and (39) are obtained similarly to Section 3.3 with addition that F depends on t. It means, when we integrate the inequality (22), we get Rest arguments remain to be the same.
We assume that the density of the external force f in the equation (5) belongs to the space L loc 2 (R; V 0 ) and satisfies the condition (15) for all t ∈ R. In this subsection we fix a number δ ∈ (0, 1). Consider E = V 1 and E 0 = V 1−δ as the pair of Banach spaces needed to introduce the class T .
Let τ ∈ R. As the trajectory space H + τ of the problem (5)- (6), (13)-(14) we consider the set of a weak solution v of the problem (34)-(37) with the right part F = T(τ )f and some initial condition a ∈ V 1 (each for every v), which is satisfying the estimate These trajectory spaces form a family of trajectory spaces H + = {H + τ }.
Remark 7. Note that for spaces H + τ the embedding H + τ ⊂ T holds. In fact, from the inequality (40) and condition (15) the uniform with respect to t estimate of the trajectory v ∈ H + τ on an arbitrary interval [0, T ] follows v(t) 2 where v ∈ L ∞ (0, T ; V 1 ). By virtue of the arbitrary of T we get v ∈ L loc ∞ (R + ; V 1 ). In addition, in view of Theorem 5.1 we have v ∈ L loc 2 (R + ; V −1 ). So by Theorem 2.1 applied in the case of triple spaces V 1 ⊂ V 1−δ ⊂ V −1 , we get v ∈ C(R + ; V 1−δ ). The inclusion H + τ ⊂ T is proved.
Proof. Theorem is a direct consequence of existence Theorem 5.1.
We describe the class D of attracting sets families. Let R denotes the set of functions r : R → R + , such that the function τ → e ατ (r(τ )) 2 increases and lim τ →−∞ e ατ (r(τ )) 2 = 0.
The class D consists of families of D = {D τ } (D τ ⊂ V 1 ) for which there exist functions r D ∈ R, such that w ∈ D τ and w 1 r D (τ ) for all τ ∈ R.
Proof of Theorem 2.26. We construct a family of sets P = {P θ } (P θ ⊂ T ) which is a T -precompact and pullback-absorbing. Then the theorem follows from Theorems 2.23 and 2.25.
Let a set P θ consists of functions v ∈ T which satisfy inequalities v(t) 2 (here C is a constant from the inequality (39)). Now we show that the family P = {P θ } is T -precompact. Condition (ii) of the definition 2.18 is performed with the function Thus, the set P θ is bounded in the norm of space L ∞ (0, T ; V 1 ) and the set {v |v ∈ P θ } is bounded in the norm L 2 (0, T ; V −1 ). By Theorem 2.1 applied for the case of spaces V 1 ⊂ V 1−δ ⊂ V −1 we get that the set P θ is precompact in C([0, T ]; V 1−δ ). By virtue of arbitrary of T we have that P θ is precompact in C(R + ; V 1−δ ), as it is required.
We have shown that the family P is T -precompact. Now we show that the family P satisfies conditions of Definition 2.17. Let D = {D τ } ∈ D. Take a number θ and show that there is τ D (θ) θ that for τ τ D (θ) the inclusion T(θ − τ )H + τ (D) ⊂ P θ (43) holds and the function τ D increases. By the definition of the class D for the family D there is a function r D : R → R + , such that for w ∈ D τ we have the estimate w 1 r D (τ ), and that the function χ D (τ ) = e ατ (r D (τ )) 2 increases and tends to 0 when τ → −∞. By the monotony of the function χ D the inverse function χ −1 D is increasing. Consider the inequality Due to the properties of χ D it holds either on the whole axis or on the line (−∞, χ −1 (e αθ )]. In the first case put τ D (θ) = θ and in the second case let τ D (θ) = min{χ −1 (e αθ ), θ}. Clearly, in any case the function τ D increases, satisfies τ D (θ) θ and for τ τ D (θ) the inequality (44) holds or, equivalently, e −α(θ−τ ) (r D (τ )) 2 1 (τ τ D (θ)).
Show that for the function T(θ − τ )v the estimate (41) holds. With the estimate (40) and inequality (45) we obtain: The estimate (41) for the function T(θ − τ )v is proved. Show that for the function T(θ − τ )v the estimate (42) holds. Since v is a weak solution of the problem (34)- (37) with the function F = T(τ )f , by Theorem 5.1 for almost all t > 0 the inequality . Thus, we have T(θ − τ )v ∈ P θ and the embedding (43) is proved. We have proved that a family of P is T -precompact and pullback D-absorbing, as it is required. According to Theorems 2.23 and 2.25 we finish our proof of Theorem 2.26.