Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit

We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter at the second derivative with respect to the"time"variable corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation. Moreover, in case where the limit attractor is regular, we give the detailed description of the structure of these uniform attractors when the perturbation parameter is small enough, and estimate the symmetric distance between the perturbed and non-perturbed attractors.


Introduction
We consider the following semi-linear elliptic boundary value problem in an infinite cylinder Ω := R × ω: , (t, x) ∈ Ω, u ∂ω = 0, where ω ⊂⊂ R n is a bounded domain of R n , u = (u 1 , · · · , u k ) is an unknown vector-valued function, a and γ are given constant matrices which satisfy a + a * > 0 and γ = γ * > 0, f and g are given nonlinear interaction function and the external forces respectively which satisfy some natural assumptions (formulated in Section 2) and ε > 0 is a small parameter.
Elliptic boundary problems of the form (1.1) appear, e.g. under studying the equilibria or the traveling waves for the corresponding evolution equations of mathematical physics.
For instance, let us consider the following reaction-diffusion system in the unbounded cylindrical domain Ω: where the variable t ∈ R remains to be spatial, the variable η plays the role of physical time and γ is a diagonal matrix. Thus, the external forces g(η, t, x) := g(t − ε −1 γη, x) in (1.2) have the form of a fast traveling (along the axis of the cylinder) wave (with the wave speed ε −1 γ ≫ 1). Then, the problem of finding the traveling wave solution v(η, t, x) := u(t − ε −1 γη, x) of equation (1.2) which is modulated by the traveling wave external forcing, obviously, reduces to the study of elliptic problem (1.1). Another natural example is the following reaction-diffusion system in the cylinder Ω: with the strong drift along the axis of the cylinder (which is described by the transport term ε −1 γ∂ t v). Then, (1.1) is the equation on equilibria for problem (1.3).
It is convenient to scale from the very beginning the variable t as follows: t ′ := ε −1 t. Then, problem (1.1) reads where we denote the new variable t ′ by t again for simplicity. We are interested in the global structure of the set of bounded (with respect to t → ±∞) solutions of problem (1.4). To this end, we use the so-called dynamical approach for the study of elliptic boundary value problems in cylindrical domains which has been initiated in [6] and [16], see also [2,3], [7], [10], [11], [19,20,21,22,23,24], [27,28,29] and the references therein for its further development. Following this approach, we introduce, for every τ ∈ R, the auxiliary elliptic boundary value problem: (1.5) a(ε 2 ∂ t u + ∆ x u) − γ∂ t u = g ε (t), (t, x) ∈ Ω τ + , u ∂ω = 0, u t=τ = u τ , in the half-cylinder Ω τ + := (τ, +∞) × ω equipped by the additional boundary condition u| t=τ = u τ at the origin of the half-cylinder Ω τ + and the function u τ is assumed to belong to the appropriate functional space V p ε (ω) which will be specified in Section 2. If problem (1.5) possesses a unique (bounded as t → +∞) solution (in certain functional class), for every u τ ∈ V p ε (ω), then (1.5) defines a dynamical process {U ε gε (t, τ ), t, τ ∈ R, t ≥ τ } via (1.6) U ε gε (t, τ )u τ := u(t), where u(t) solves (1.5), U ε gε (t, τ ) : V p ε (ω) → V p ε (ω). Moreover, if this dynamical process possesses a global (uniform) attractor A ε , then this attractor is generated by all bounded (with respect to t → ±∞) solutions of the initial problem (1.4) (and all its shifts along the t axis, together with their closure in the corresponding topology, see Section 4 for the details). Thus, studying of the bounded solutions of (1.4) is, in a sense, equivalent to the study of the attractor A ε of auxiliary dynamical process (1.6).
In the present paper, we give a detailed study of auxiliary problems (1.5) in case ε is small enough and investigate their behavior as ε → 0. The paper is organized as follows. The existence of a bounded solution u(t) of problem (1.5) and several important estimates are derived in Section 2. The uniqueness of this solution is verified in Section 3 under the assumption that ε is small enough. Moreover, we show there that the dynamical process (1.6) associated with problem (1.5) is uniformly (with respect to ε) Frechet differentiable with respect to the 'initial data' u τ ∈ V p ε (ω). The existence of the uniform attractor A ε for the process (1.6) is established in Section 4. Moreover, we prove there that, for rather wide class of the external forces g, the attractors A ε converge as ε → 0 (in the sense of upper semicontinuity) to the attractor A 0 of the limit parabolic problem (1.7) γ∂ t u − a∆ x u + f (u) = g 0 (t), u ∂ω = 0, u t=τ = u τ , where the limit external forces g 0 (t) average the external forces g ε (t) := g(ε −1 t) of problems (1.5). In particular, the class of admissible external forces g contains the autonomous external forces: g(t) ≡ g 0 , heteroclinic profiles: (1.8) g(t) → g ± as t → ±∞ and g ± are independent of t, solitary waves (g + = g − in (1.8)), periodic, quasiperiodic and almost-periodic with respect to t external forces g and even some classes of non almost-periodic oscillations, see Examples 4.9-4.12. Furthermore, in Section 5, we prove that dynamical processes (1.6) tend as ε → 0 to the process U 0 g 0 (t, τ ) associated with limit parabolic problem (1.7) and obtain the quantitative bounds for that convergence in terms of the parameter ε.
In Sections 6 and 7, we restrict ourselves to consider only the case of almost-periodic external forces g(t) in the right-hand side of equation (1.5). In this case, limit parabolic equation (1.7) is autonomous (1.9) g 0 (t) :=ḡ, whereḡ is the mean of almost-periodic functionḡ. We also assume that the global attractor A 0 of the limit parabolic equation is regular (it will be so if this equation possesses a global Lyapunov function and all of the equilibria are hyperbolic, see Section 7 for the details). Then, using the theory of non-autonomous perturbations of regular attractors developed in [12,13,30], we establish the existence of the non-autonomous regular attractor for problems (1.6) if ε is small enough. In this case, the attractors A ε are occurred not only upper semicontinuous, but also lower semicontinuous as ε → 0 and we give the quantitative bounds for the symmetric distance between them in terms of the perturbation parameter ε. In particular, we prove there that equation (1.4) possesses the finite number of different almost-periodic (with respect to t) solutions and that every other bounded solution of that equation is a heteroclinic orbit between two different almost-periodic solutions. We also recall that the regular attractor for system (1.5) with ε = 1, γ ≫ 1 and autonomous external forces g ε has been considered in our previous paper [28]. Moreover, the estimates for the nonsymmetric Hausdorff distance between the attractors A ε and A 0 in terms of the parameter ε have been obtained in [29] for the case where the attractor A 0 of the limit parabolic equation is regular and the external forces g ε (t) = g(ε −1 t) are almost-periodic with respect to t. Finally, several uniform (with respect to ε) estimates for the linear equation of the form (1.4) which are systematically used throughout of the paper are gathered in Appendix. Acknowledgements: This research was partially supported by the Russian Foundation of Basic Researches (projects 11-01-00339 and 10-01-00293).
2. Uniform (with respect to ε → 0) a priori estimates In this section, we consider the following nonlinear elliptic boundary value problem in a half cylinder Ω τ + := [τ, +∞) × ω, τ ∈ R: where ω ⊂⊂ R n is a bounded domain of R n with a sufficiently smooth boundary, u = (u 1 , · · · , u k ) is an unknown vector-valued function, ∆ x is the Laplacian with respect to x, a and γ are given constant k × k-matrices satisfying a + a * > 0 and γ = γ * > 0, f (u) is a given nonlinear function which satisfies the following assumptions: where v.w stands for the inner product of the vectors v ∈ R k and w ∈ R k and the critical growth exponent q max equals infinity for n = 1 or n = 2 and q max := n+2 n−2 for n ≥ 3. In order to formulate our assumptions on the solution u(t), the external forces g(t) and the initial data u 0 , we need to define the appropriate functional spaces.
Definition 2.1. For every l ∈ R + and s ∈ [2, ∞), we define the following spaces: where Ω T := (T, T + 1) × ω and W l,s denotes the ordinary Sobolev space of functions whose derivatives up to order l belong to L s , see [26]. In particular, we write in the sequel L s b (Ω τ + ) instead of W 0,s b (Ω τ + ). Moreover, we also introduce the following spaces associated with the linear part of equation (2.1): T +1],W 2,s (ω)) < ∞, u ∂ω = 0} and, analogously to (2.3) We also introduce the uniform with respect to ε trace space (at t = τ ) of functions belonging to the space (2.5): . We note that, for ε > 0, the space W (Ω τ + ) is equivalent to W 2,s b (Ω τ + ) and, for ε = 0 this space coincides with the anisotropic Sobolev-Slobodetskij space W (Ω τ + ) which corresponds to a second order parabolic operator. Analogously, for ε > 0, the space V s ε (ω) coincides with the trace space at t = τ of of the classical Sobolev space W 2,s (Ω τ ) and, for ε = 0, we have the trace space at t = τ , for the anisotropic space W (1,2),s (Ω τ ), see e.g. [5]. The dependence of the norms (2.4) and (2.6) on the parameter ε is chosen in such way that the constants in the proper maximal regularity estimates and the trace theorems will be independent of ε → 0, see Appendix below. Definition 2.2. We restrict ourselves to consider only such solutions u(t) of problem (2.1) which remain bounded as t → +∞. To be more precise, a function u will be a solution of problem (2.1) if u belongs to the space W (1,2),p ε,b (Ω τ + ) for some p > p min := max{2, (n + 2)/2}, and satisfies the equation and the boundary conditions in the sense of distributions. Then, we should require the initial data u 0 belongs to the trace space V p ε (ω) and the external forces g ∈ L p b (Ω τ + ). Note that the assumption p > p max guarantees that u ∈ C b (Ω τ + ) for all ε (including the limit case ε = 0) and, therefore, the non-linearity f (u) is well-defined and belongs to (Ω τ + ). The main result of this section is the following theorem. Theorem 2.3. Let the above assumptions hold. Then, for every ε ∈ [0, 1] and u τ ∈ V p ε (ω), problem (2.1) has at least one solution u ∈ W (1,2),p ε,b (Ω τ + ) and the following estimate hold, for every such solution: , where the constant α > 0 and the monotonic function Q : We first prove the analogue of estimate (2.7) for p = 2.
Lemma 2.4. Let u(t) be a solution of (2.1) (in the sense of Definition 2.2). Then, the following estimate holds: , where the constant α > 0 and the monotonic function Q are independent of τ ∈ R, T ≥ τ , ε ∈ [0, 1], u 0 and g.
Proof. We set φ T (t) := e −α|t−T | , where T ∈ R and α > 0 is a small parameter which will be specified below, multiply equation (2.1) by φ T (t)u(t) and integrate over Ω τ + . Then, after integrating by parts and using that γ = γ * , we have .v ≥ −C and the obvious inequality together with the Cauchy-Schwartz inequality, we have where the constant C is independent of ε, τ , T and α (we also note that all of the integrals in (2.11) have a sense since the solution u is assumed to belong to W (1,2),p ε,b (Ω τ + )). Estimate (2.11) implies that, for sufficiently small (but independent of ε), α > 0 where the constant C 1 is independent of ε, τ and T . We now set , multiply equation (2.1) byL ε u and integrate over Ω τ + . Then, integrating by parts, using that γ = γ * , we have and using that a + a * > 0, γ > 0 and f ′ (v) ≥ −K, together with estimate (2.10), after the straightforward estimates, we end up with where the constant C 2 is independent of ε, τ and T . Applying estimate (2.12) in order to estimate the first term in the right-hand side of (2.14), we obtain, after the obvious estimates that where the constant C 3 is independent of ε, τ and T . We now claim that where C 4 is independent of ε, τ and T ≥ τ . Indeed, let ϕ(t) ∈ C ∞ 0 (R) be a cut-off function such that ϕ(t) = 1, for t ∈ [0, 1], and ϕ(t) = 0, for t / ∈ [−1, 2]. For every T ≥ τ , we set ϕ T (t) := ϕ(t − T ) and u T (t) := ϕ T (t)u(t). Then, the last function satisfies the following equation: and estimate (2.16) is an immediate corollary of this estimate. Inserting now estimate (2.15) into the right-hand side of (2.16) and using the embedding W 2(1−1/p),p (ω) ⊂ C(ω) (due to our choice of the exponent p), we have , where the constant C 5 and the monotonic function Q are independent of ε, τ and T ≥ τ .
Thus, there only remains to estimate the L 2 -norm of ∂ t u. In order to do so, we rewrite elliptic system (2.1) in the following form: Equation (2.18) has the form of a nonlinear reaction-diffusion system in the bounded domain ω with the non-autonomous external forcesh u (t) belonging to L 2 b ((τ, ∞) × ω) (due to estimate (2.17)). Moreover, the nonlinearity f (u) satisfies the quasimonotonicity assumption f ′ (v) ≥ −K. Consequently, multiplying (2.18) by ∆ x u(t), integrating over x and applying the Gronwall's inequality, we derive (in a standard way, see e.g. [9]) that where the positive constants α and C are independent ofh u . Using estimate (2.17) for estimating the last term in the right-hand side of (2.19), we have where the constant C 6 and the monotonic function Q are independent of ε, τ and T .
We now recall that, according to the embedding theorem, see e.g. [17] and [26] (2.21) where the exponent q max is the same as in (2.2). Estimates (2.20) and (2.21), together with the growth restriction (2.2), imply that (Ω τ + ) ), where the constant α > 0 and the monotonic function Q 1 are independent of ε, τ and T ≥ τ . Expressing now ∂ t u from equation (2.1) and using estimates (2.20) and (2.22), we obtain the desired estimate for ∂ t u and finish the proof of Lemma 2.4.
We are now ready to prove estimate (2.7), for p > 2. We consider only the more complicated case n ≥ 3 and rest the simpler case n ≤ 2 to the reader.
Remind that the nonlinearity f (u) satisfies growth restriction given by the third formula of (2.2) where the exponent q is strictly less than q max and, due to the embedding theorem for the anisotropic spaces see [26], and, consequently, due to (2.8), estimate (2.22) can be improved as follows: q > 0 and the constant α > 0 and the monotone function Q are independent of ε, τ and T . We now rewrite equation (2.1) in the following way: , u ∂ω = 0, u t=τ = u τ and apply the maximal elliptic regularity estimate of Corollary 8.6 to this linear equation which reads see Corollary 8.6 of Appendix.
Then, using (2.23) in order to estimate the integral into the right-hand side of this formulae, we get (Ω τ + ) ), where the constant C and the function Q 1 are independent of ε, τ and T . We now recall that ) and, due to the embedding theorem for anisotropic Sobolev spaces see [26]. Consequently, analogously to (2.23), we have (Ω τ + ) ), where α > 0 and Q 2 are independent of ε, τ and T and Iterating the above procedure, we finally derive estimates (2.25) and (2.27) with the exponent 2 + δ l ≡ p. Indeed, formulae (2.26) and (2.28) guarantee that the number l of the iterations will be finite. Thus, estimate (2.7) is proved. In order to finish the proof of Theorem 2.3, there remains to note that the existence of a solution u ∈ W (1,2),p ε,b (Ω τ + ) of problem (2.1) can be proved in a standard way based on a priori estimate (2.7). Indeed, for instance, the existence of a solution of the analogue of (2.1) in a finite cylinder Ω τ,N := (τ, τ + N) × Ω can be proved using the Leray-Schauder fixed point principle and the existence of a solution u in the infinite cylinder Ω τ + can be obtained then by passing to the limit N → ∞, see e.g. [27,28] for the details. Theorem 2.3 is proved.
Remark 2.5. We note that estimate (2.7) is valid for every p ≥ 2 although we have formally proved it only for p > p min . Indeed, we have used the last assumption only in order to estimate the term ε 2 φ(τ ) f (u(τ ) 2 L 2 (ω) in (2.14) which appears after applying the Schwartz inequality to the term ε 2 φ(τ ) (∂ t u(τ ), f (u(τ ))) L 2 (ω) . But the growth restriction of (2.2), Lemma 8.2 and the appropriate interpolation inequality allow to estimate this term in more accurate way: , where the parameter µ can be arbitrarily small and a function Q µ depends on µ, but is independent of ε (see [24] for the details). Inserting this estimate to the right-hand side of (2.14), we can easily derive (2.7) with p = 2.
Indeed, (2.29) is an immediate corollary of (2.7) and the fact that V p ε (ω) is the uniform (with respect to ε) trace space of functions belonging to W (1,2),p ε,b (Ω τ + ), see Apendix. Corollary 2.8. Let the assumptions of Theorem 2.3 hold and let, in addition, the external forces g belong to L p 1 b (Ω τ + ), for some p 1 > p. Then, every solution u ∈ W (1,2),p ε,b (Ω τ + ) of problem (2.1) satisfies the following estimate: where the constant α > 0 and the function Q are independent of ε, τ , T and u.
Indeed, since W (1,2),p (Ω T ) ⊂ C(Ω T ) (due to our choice of the exponent p) then, estimate (2.7) implies that , where the constant α > 0 and the function q are independent of ε, τ , T and u. Rewriting now equation (2.1) in the form of (2.24) and applying the uniform (with respect to ε) interior L p 1 -regularity estimate to this equation (see Corollary 8.7 and estimate (8.28)), we derive estimate (2.30).

Uniqueness of the solutions
In this section, we prove that the solution u(t) of problem (2.1) which is constructed in Theorem 2.3 is unique if ε > 0 is small enough. Moreover, we also verify the differentiability of that solution with respect to the initial data u τ ∈ V p ε (ω) in the corresponding functional spaces. We start with the following theorem.
Theorem 3.1. Let the assumptions of Theorem 2.3 hold and let, in addition, ε ≤ ε 0 := ε 0 (a, f, γ) is small enough. Then, for every two solutions u 1 (t) and u 2 (t) of problem (2.1), the following estimate holds: , but is independent of ε, τ and T . In particular, the solution of (2.1) is unique if ε ≤ ε 0 .
Proof. We set v(t) := u 1 (t) − u 2 (t). Then, this function satisfies the following equation: Moreover, due to the second assumption of (2.2), estimate (2.7) and the embedding , but is independent of ε and τ . It is however convenient to consider more general (than (3.2)) problem where the given matrix-valued function l(t) satisfies (3.3) and h(t) = h(t, x) are given external forces.
Lemma 3.2. Let Λ 0 be a nonnegative number which satisfies the following condition: where a + := 1/2(a + a * ) and a − = 1/2(a − a * ). Then, for every w τ ⊂ V p ε (ω) and every external forces h satisfying (Ω τ + ) and the following estimate holds: where the positive constants α and C depend on M and Λ 0 , but are independent of ε, τ and T ≥ τ .
Proof. We first note that, due to the fact that V p ε (ω) is the uniform (with respect to ε) trace space for functions belonging to W (1,2),p ε (Ω τ ), it is sufficient to verify Lemma 3.2 for the case w τ = 0 only. In order to do so, we set θ(t) := e −Λ 0 t w(t). Then, this function belongs to W (Ω τ + ) (due to assumption (3.7)) and satisfies the following equation: , θ ∂ω = 0, θ t=τ = 0. Multiplying now this equation by φ T (t)θ(t) (where the weight function φ T (t) := e −α|T −t| ) and integrating over Ω τ + , we obtain after the standard transformations (integrating by parts and using that γ = γ * , l(t) ≥ −K and estimate (2.10)) that where the constant C depends only on a and γ and ·, · τ stands for the inner product in L 2 (Ω τ + ). Estimating the last term in the right-hand side of (3.10) as follows: θ, fixing the parameter α > 0 to be small enough and using the Friedrichs and Schwartz inequalities, we have Using assumption (3.5), positivity of a + and the obvious inequality φ T (t) ≥ e −α for t ∈ [T, T + 1], we have Estimate (3.8) (with w τ = 0) can be now derived from (3.11) iterating the maximal elliptic regularity estimate (8.27) exactly as in the end of the proof of Theorem 2.3. The existence of the solution can be then verified in a standard way based on a priori estimate (3.8), see e.g. [27,28]. Lemma 3.2 is proved.
We are now ready to finish the proof of Theorem 3.1. To this end, we note that the left-hand side of (3.5) tends to Λ 0 γ − K as ε → 0 and, consequently, for every sufficiently large Λ 0 > 0, we may fix (due to positivity of the matrix γ) ε 0 = ε 0 (Λ 0 , K, a, γ) such that (3.5) is satisfied, for every ε ≤ ε 0 . Applying then estimate (3.8) (with h ≡ 0) to equation (3.2), we finish the proof of Theorem 3.1.
Let us assume from now on that where Ω := R × ω. Then, under the assumptions of Theorem 3.1, problem (2.1) defines a two-parametrical family of solving operators {U ε Moreover, Theorem 3.1 shows that these operators are uniformly (with respect to ε) Lipschitz continuous in V p ε (ω). Our next task is to prove their Frechet differentiability with respect to the initial data u τ ∈ V p ε (ω). To this end, we consider the following formal equation of variations associated with a solution u(t) := U ε g (t, τ )u τ : , where the solution v(t) satisfies (3.7) and the constants α > 0 and C are independent of ε, τ and T . The following theorem shows that (3.15) defines indeed the Frechet derivative of the process U ε g (t, τ ) at u τ . Theorem 3.3. Let the assumptions of Theorem 3.1 hold. Let also u(t) and u 1 (t) be two solutions of (2.1) and v(t) be a solution of (3.15) with v τ := u(τ ) − u 1 (τ ) (associated with u(t)). Then, there exists ε ′ 0 = ε ′ 0 (f, a, γ) > 0 such that ε ′ 0 ≤ ε 0 and for every ε ≤ ε ′ 0 the following estimate is valid: where the constants C and α > 0 depend on u(τ ) V p ε (ω) and u 1 (τ ) V p ε (Ω) , but are independent of ε, τ and T .
Corollary 3.4. Let the assumptions of Theorem 3.3 hold and let u(t), u 1 (t) and v(t) be the same as in Theorem 3.3. Then, for every q ≥ p, q < ∞ and every T ≥ τ + 1, the following estimate holds: and q, but are independent of ε, τ and T .
Corollary 3.5. Let the assumptions of Theorem 3.3 hold. Then, the operators U ε g (t, τ ) are Frechet differentiable with respect to the initial data, their Frechet derivative is defined by where v(t) is the solution of (3.15) with v τ = ξ, and the following estimates hold: , but is independent of ε, τ and t.
Arguing analogously, but using estimate (3.20) instead of (3.17), we derive the following result.
Corollary 3.6. Under the assumptions of Corollary 3.5 the following estimates hold, for every q ≥ p and T ≥ τ + 1: for every u 1 τ , u 2 τ ∈ V p ε (ω) and, consequently where the constant C q depends on q, u 1 , but is independent of ε, τ and t.
We now recall that, for ε > 0, operators U ε g (t, τ ) are defined on the space V p ε (ω) ∼ W 2−1/p,p (ω) and, for ε = 0, the limit process U 0 which is not convenient for the study of the limit ε → 0. In order to overcome this difficulty, we consider the following discrete analogue of process (3.13): Moreover, we assume, in addition, that the exponent p satisfies p ≥ 2p min and use the following obvious embeddings: (Ω) ), for every u m ∈ V p 0 (ω), and the constant α > 0 and the monotonic function Q are independent of 0 ≤ ε ≤ ε ′ 0 , l, m ∈ Z and l ≥ m. Moreover, using Corollaries 3.5-3.6 and arguing analogously, we derive the following result.
Corollary 3.7. Let the assumptions of Theorem 3.3 hold and let, in addition, p > 2p min . Then the following estimates hold: , but is independent of ε, l and m.
To conclude, we formulate the result on injectivity of operators U ε g (t, τ ).
Theorem 3.8. Let the assumptions of Theorem 3.1 hold and let for some τ ∈ R, t ≥ τ and u 1 τ , u 2 τ ∈ V p ε (ω). Then, necessarily, u 1 τ = u 2 τ . The proof of this Theorem is based on the logarithmic convexity results (see [1]) for solutions of (2.1) and can be found, e.g. in [28]. 4. Uniform attractors and their convergence as ε → 0 In this section, we construct the global attractors A ε for problems (2.1) and investigate their behavior as ε → 0. Since the external forces g(t) in (2.1) (which are assumed from now on to be defined on the whole cylinder Ω and to belong to the space L p b (Ω)) depend explicitly on t, then we use below the skew-product technique in order to reduce the nonautonomous dynamical process (3.13) associated with problem (2.1) to the autonomous semigroup on the extended phase space. Following the general procedure (see [9] and [14]), we define a hull H(g) of the external forces g as follows: loc,w (Ω) stands for the closure in the space L p loc,w (Ω) which is the space L p loc (Ω) endowed by the weak topology. We recall that a sequence g k → g in L p loc,w (Ω) as k → ∞ if and only if g k Ω T → g Ω T weakly in L p (Ω T ), for every T ∈ R. It is also well-known, that every bounded subset of L p loc,w (Ω) is precompact and metrizable and, consequently (due to the assumption g ∈ L p b (Ω)), hull (4.1) is a compact metrizable subset of L p loc,w (Ω). Thus, a function ξ(t) belongs to H(g) if and only if there exists a sequence {h n } ∞ n=1 ∈ R such that (4.2) ξ = lim n→∞ T hn g in the space L p loc,w (Ω). Moreover, it also obvious that the group {T h , h ∈ R} of temporal translations acts on H(g), i.e.
In order to construct the attractor of (2.1), we consider the following family of equations of type (2.1) which correspond to all external forces ξ ∈ H(g): , u t=τ = u τ , u ∂ω = 0, ξ ∈ H(g) which generates the family {U ε ξ (t, τ ), ξ ∈ H(g)} of dynamical processes in V p ε (ω) (under the assumptions of Theorem 3.1). This family of processes generates a semigroup {S ε t , t ≥ 0} on the extended phase space Φ ε := V p ε,w (ω) × H(g) (as usual V p ε,w (ω) denotes the space V p ε (ω) endowed by the weak topology) by the following expression: [9] for the details). Thus, we describe the 'longtime' behavior of solutions of (4.4) in terms of the global attractor of semigroup (4.5) in the extended phase space Φ ε . For the convenience of the reader, we recall below the definition of the attractor adapted to our case, see e.g. [4], [9] and [25] for the detailed exposition.  The next theorem establishes the existence of the attractor described above.
Theorem 4.2. Let the assumptions of Theorem 3.1 hold and let g ∈ L p b (Ω). Then, semigroup (4.5) possesses a global attractor A ε in the phase space Φ ε and, consequently, family of problems (4.4) possesses a uniform attractor A ε which can be described as follows: ξ is a set of all solutions of problem (4.4) (with the right-hand side ξ ∈ H(g)) which are defined for all t ∈ R and belong to C b (R, V p ε (ω)) and Π 0 u := u(0). Proof. According to the abstract theorem on the global (and uniform) attractors existence (see [4], [9] and [25]), it is sufficient to verify the following conditions: 1. The semigroup S ε t possesses a compact absorbing set B in Φ ε . 2. The operators S ε t are continuous on B, for every fixed t ≥ 0. Let us verify these conditions. It follows from estimate (2.29) that the set (Ω) ), ξ ∈ H(g)} is an absorbing set for the semigroup S ε t (here we have implicitly used the obvious fact that (Ω) , for every ξ ∈ H(g)). Moreover, since the space V p ε (ω) is reflexive, then bounded subsets of it are precompact in a weak topology. Using the fact that H(g) is also compact, we derive that set (4.8) is compact in Φ ε . Thus, the first condition is verified.
In order to verify the second one, we first note that the set B is metrizable, consequently, it is sufficient to verify only the sequential continuity of S ε t on B. Indeed, let (u n 0 , ξ n ) ∈ B be an arbitrary (weakly) convergent sequence in B and let (u 0 , ξ 0 ) ∈ B be its (weak) limit. We set u n (t) := U ε ξn (t, 0)u n 0 . Then, by definition, these functions satisfy the equations: (4.9) a(ε 2 ∂ 2 t u n (t)+∆ x u n (t))−γ∂ t u n (t)−f (u n (t)) = ξ n (t), u n t=0 = u n 0 , u n ∂ω = 0. In order to verify the desired continuity, we need to prove that u n (t) → u 0 (t) weakly in V p ε (ω), for every t ≥ 0, where u 0 (t) := U ε ξ (t, 0)u 0 is a solution of the limit (as n → ∞) equation of (4.9). We note that the sequence u n 0 is uniformly bounded in V p ε (ω) (since it converges weakly to u 0 ), consequently, due to Theorem 2.3, we have (4.10) u n W (1,2),p ε (Ω T ) ≤ C, where C is independent of T ≥ 0 and n ∈ N. Therefore, the sequence of the solutions u n (t) is precompact in a weak topology of the space W (1,2),p ε,loc (Ω 0 + ) (since this space is reflexive). Letū :=ū(t) ∈ W (1,2),p ε,loc (Ω 0 + ) be an arbitrary limit point of this sequence. Then, due to estimate (4.10), the functionū(t) belongs to W (1,2),p ε,b (Ω 0 + ). Moreover, due to compactness of the embedding W for the appropriate subsequence {n k } ∞ k=1 ∈ N. Passing now to the limit k → ∞ in equations (4.9) and using (4.11) and that ξ n → ξ weakly in L p loc (Ω), we derive thatū is a bounded solution of the limit equation of (4.9). Since, due to Theorem 3.1, this solution is unique, then, necessarily,ū(t) ≡ u 0 (t) := U ε ξ (t, 0)u 0 . Moreover, since the limit pointū is arbitrary, then we have proved that u n → u 0 weakly in W (1,2),p ε (Ω T ), for every T ∈ R + and, consequently, u n (t) → u 0 (t) weakly in V p ε (ω), for every t ∈ R + . Thus, the second condition of the abstract theorem on the attractors existence is also verified and, therefore, according to this theorem, the semigroup S ε t possesses indeed the global attractor A ε in Φ ε and the family of problems (4.4) possesses the uniform attractor A ε := Π 1 A ε ∈ V p ε (ω).
Description (4.7) is also a standard corollary of that theorem, see [4] and [9]. Theorem 4.2 is proved.
Remark 4.3. There exists an alternative way to introduce the concept of the uniform attractor of equation (2.1) without using the skew-product flow on the extended phase space Φ ε . Namely, the set A ε is a uniform attractor for equation (2.1) if the following conditions are satisfied: , for every τ ∈ R and t ≥ T . 3. The set A ε is a minimal set which satisfies 1) and 2). The equivalence of this definition to Definition 4.1 is proved in [9].  H(g) is compact in a strong topology of L p loc (Ω), then, arguing in a standard way (see, e.g. [9] and [27]), we can prove that the attractor A ε attracts the bounded subsets of Φ ε not only in a weak topology, but also in more natural strong topology of Φ ε and A ε is compact in a strong topology of V p ε (ω). Nevertheless, we prefer to use the weak topology in Definition 4.1, since the choice of the weak topology is more convenient for what follows.
(ω), δ > 0 and V p ε (ω) ⊂ C(ω) are compact, then (4.12) implies the following convergence: for every bounded set B ⊂ V p ε (ω) and every δ > 0. Here an below dist V (X, Y ) denotes the nonsymmetric Hausdorff distance between sets X and Y in the space V : The rest of this section is devoted to study the behavior of the attractors A ε as ε → 0. To this end, keeping in mind equation (1.4), it is convenient to consider slightly more general family of equations of the form (2.1): where the external forces depend explicitly on ε. We assume (following [8]) that these external forces are uniformly bounded in L p b (Ω): (4.17) g ε L p b (Ω) ≤ C, where C is independent of ε, and converge to the limit external forces g 0 ∈ L p b (Ω) in the following weak sense: for every φ ∈ L q (Ω 0 ), 1 q + 1 p = 1, there exists a function α φ : R + → R + , such that for all h ∈ R (4.18) Ω 0 (g ε (t + h, x) − g 0 (t + h, x)).φ(t, x) dx dt ≤ α φ (ε) and lim ε→0 + α φ (ε) = 0.
The main result of this section is the following theorem.
Theorem 4.6. Let the assumptions of Theorem 4.2 hold and let, in addition, the external forces g ε (t) satisfy (4.17) and (4.18). Let also A ε , 0 ≤ ε ≤ ε 0 , be the uniform attractors of equations (4.16). Then, A ε tends to A 0 in the following sense: for every neighborhood Proof. The proof of this theorem is based on the following lemma which clarifies the nature of convergence (4.18).
Lemma 4.7. Let functions g ε , 0 ≤ ε ≤ 1, belong to L p b (Ω) and satisfy (4.17) and (4.18). Then, the following conditions hold: 1. The functions g ε (t) converges to g 0 (t) weakly in L p loc (Ω) and the set ∪ ε H(g ε ) is weakly precompact in L p loc (Ω). 2. For every sequences ε n → 0 and ξ n ∈ H(g εn ) such that ξ n → ξ weakly in L p loc (Ω), the function ξ necessarily belongs to H(g 0 ) and The convergence (4.18) is uniform with respect to φ belonging to compact sets in L p ′ (Ω 0 ), i.e. the function α φ in (4.18) can be chosen in such way that The assertion of the lemma can be proved in a standard way, using the representation (4.2) for functions belonging to the halls H(g ε ) and basic properties of the weak convergence in reflexive spaces, see [8] and [9].
We are now ready to prove Theorem 4.6. We first note that, due to Theorem 2.3, Corollary 2.3 and estimate (4.17), we have where the constant C is independent of ε. Thus, in order to prove the theorem, it is sufficient to verify that, if u 0 n ∈ A εn , ε n → 0 as n → ∞, be an arbitrary sequence which converges weakly in V p 0 (ω) to some u 0 ∈ V p 0 (ω), then u 0 ∈ A 0 (see [9]). Taking into account description (4.7), estimates (4.17) and (4.22) and the weak compactness of bounded sets in reflexive spaces, this assertion can be reformulated as follows: if ε n → 0, ξ n ∈ H(g εn ) and u n ∈ K εn ξn be arbitrary sequences such that u n → u weakly in W (1,2),p 0,loc (Ω) and ξ n → ξ weakly in L p loc (Ω), then ξ ∈ H(g 0 ) and u ∈ K 0 ξ . Let us verify this assertion. Indeed, the fact that ξ ∈ H(g 0 ) is an immediate corollary of Lemma 4.7. Thus, there only remains to pass to the weak limit in the following equations: a(ε 2 n u n (t) + ∆ x u n (t)) − γ∂ t u n (t) − f (u n (t)) = ξ n (t), t ∈ R, u n ∂ω = 0. We recall that the embedding W (Ω) implies the strong convergence u n → u in C loc (Ω). Passing now to the limit n → ∞ in (4.23), we derive that the function u ∈ W (1,2),p 0,b (Ω) and satisfies a∆ x u(t) − γ∂ t u(t) − f (u(t)) = ξ(t), t ∈ R and, consequently u ∈ K 0 ξ and Theorem 4.2 is proved. To conclude this section, we consider the applications of Theorem 4.6 to equation (1.4) and, consequently, we assume from now on that (4.25) g ε (t) := g(ε −1 t), for some g ∈ L p b (Ω). Example 4.9. Let the assumptions of Theorem 4.2 hold, (4.25) be satisfied and the function g ∈ L p b (Ω) have the following heteroclinic profile structure: there exist g ± := g ± (x) ∈ L p (ω) such that and, consequently, (4.17) and (4.18) are also satisfied. Thus, due to Theorem 4.6, the uniform attractors A ε of equations (4.16) (or, which is the same, the attractors of (1.4)) tend as ε → 0 (in the sense of (4.19)) to the uniform attractor of the limit parabolic equation with the external forces (4.27). In particular, if g + = g − then the limit parabolic problem is autonomous.
In order to consider the next examples, we need the following proposition which is adopted to the study of oscillating in time external forces g ε in (4.16).
Example 4.11. Let the assumptions of Theorem 4.2 hold, (4.25) be satisfied and the function g belong to C b (R, L p (ω)) and be almost-periodic with respect to t with values in L p (ω) (the latter means that the hull H(g) is compact in C b (R, L p (ω)), according to the Bochner-Amerio criterium, see [18]). Then, assumption (4.28) is satisfied, due to the Kronecker-Weyl theorem, see [18]. Thus, the uniform attractors A ε of elliptic problems (4.16) with the rapidly oscillating external forces g ε (t) := g(ε −1 t) (g is now almost-periodic) converge as ε → 0 to the global attractor A 0 of the limit autonomous parabolic equation with the averaged external forces g 0 ≡ḡ.
In conclusion, we give an example of oscillating external forces g ∈ L p b (Ω) which are not almost-periodic with respect to time, but satisfy the assumptions of Proposition 4.10 (see [9] for further examples).
Then, obviously, this function is not almost-periodic with respect to t (even in the case where g 1 and g 2 are smooth), but condition (4.28) is obviously satisfied withḡ = 0, since the periodic functions g 1 and g 2 have zero mean. Thus, in this case, the attractors A ε of equations (4.16) with non almost-periodic rapidly oscillating external forces g ε (t) := g(ε −1 t) converge as ε → to the attractor A 0 of the limit parabolic equation with zero external forces.
The following corollary reformulates estimate (5.2) in terms of discrete cascades (3.26) acting on the phase space V p 0 (ω). Corollary 5.2. Let the assumptions of Theorem 5.1 hold and let, in addition, the external forces g ε be uniformly bounded in L p+δ b (Ω), for some δ > 0, i.e. (5.9) g ε L p+δ b (Ω) ≤ C, where C is independent of ε. Then, the following estimate is valid: and l, m ∈ Z (with l ≥ m). Proof. Since the functions g ε are assumed to be uniformly bounded in L p+δ b (ω), then (replacing the exponent p by p+δ) we derive from Theorem 2.3 and Corollary 2.8 (analogously to (3.28)) that where the constant C R is independent of ε, h ε , l, m and u m . Estimate (5.10) is an immediate corollary of (5.2), (5.11) and the following interpolation inequality: (ω) , for the appropriate 0 < κ δ < 1 (see, e.g. [26]) and Corollary 5.2 is proved. Our next task is to obtain the analogue of Theorem 5.1 and Corollary 5.2 for the Frechet derivatives of the processes U ε gε (t, τ ). Theorem 5.3. Let the assumptions of Theorem 5.1 hold. Then, for every R ∈ R + and u τ , u τ V p 0 (ω) ≤ R, the following estimate is valid: is an arbitrary vector. Then, according to Theorem 3.3, these functions satisfy the equations where u 0 (t) and u ε (t) are the same as in the proof of Theorem 5.1. Then, according to Lemma 3.2,embeddings (3.27) and Corollary 8.7, analogously to (5.5), we have Then, this function satisfies , θ ε t=τ = 0. Multiplying this equation by θ ε (t), integrating over (T, τ )×ω and using that f ′ ≥ −K and the functions u ε (t) and u 0 (t) are uniformly bounded in the L ∞ -norm, we have (analogously to (5.4)) where the constants C, K 3 and C R are independent of ε and T . Inserting estimates (5.2) and (5.15) to the right-hand side of (5.16), we have . Applying the Gronwall inequality to this estimate, we derive estimate (5.13) (with the appropriate new function α R ) and finish the proof of Theorem 5.3.
The following corollary is the analogue of Corollary 5.2 for the Frechet derivatives.
Corollary 5.4. Let the assumptions of Theorem 5.1 hold and let, in addition, the external forces g ε be uniformly bounded in L p+δ b (Ω), for some δ > 0 (i.e., (5.9) be satisfied) Then, the following estimate is valid, for every R ∈ R + and u m ∈ V p 0 (ω) with u m V p 0 (ω) ≤ R: where the constant K ′′ and the functionᾱ R (ᾱ R (ε) → 0 as ε → 0 + ) are independent of ε ≤ ε 0 , h ε ∈ H(g ε ), u m ∈ V p 0 and l, m ∈ Z (with l ≥ m). Indeed, it follows from (5.15) and Corollary 8.7 (where the exponent p is replaced by p + δ) that where l ≥ m + 1 and the constant C ′ R is independent of ε, l and m. Estimate (5.18) is an immediate corollary of (5.19), (5.13) and (5.12).
To conclude this section, we investigate the dependence of functions α R (ε) andᾱ R (ε) which are introduced in Theorems 5.1-5.3 and Corollaries 5.2-5.4 respectively on ε → 0. For simplicity, we assume that the external forces g ε (t) satisfy (4.25) (i.e., g ε (t) := g(ε −1 t)) and and g ∈ L p+δ b (Ω). Theorem 5.5. Let the assumptions of Theorem 5.1 hold and (4.25) be satisfied for some g ∈ L p+δ b (Ω), δ > 0. Assume also that the function g(t) −ḡ has a bounded primitive, i.e. (5.20) for some G ∈ L 2 b (ω). Then, the functions α R (ε) in Theorems 5.1 and 5.3 and the functions α R (ε) in Corollaries 5.2 and 5.4 have the following structure: where κ δ is the same as in (5.12) and the constants C R andC R depend on R, but are independent of ε.
Proof. We first note that, due to representation (4.2) and the weak compactness of bounded subsets of L p loc (Ω), it follows from (5.20) that, for every h ∈ H(g), there exists H ∈ H(G) such that We also note that, according to (5.4)-(5.5) and (5.17), it is sufficient to verify that where h ∈ H(g), h ε (t) := h(ε −1 t) and the constant C R is independent of ε. Let us verify this inequality. Indeed, it follows from (5.22) that consequently, integrating by parts in (5.23), we have (Ω) ). Estimate (5.23) is an immediate corollary of (5.25) and (5.5). Theorem 5.5 is proved.

Nonautonomous unstable manifolds of the nonlinear elliptic equation
In this section, using the standard perturbation technique, we define, for a sufficiently small ε > 0, the nonautonomous unstable manifold of nonautonomous elliptic system (2.1) which corresponds to the hyperbolic equilibrium of the limit parabolic equation (5.1). This result will be used in the next section in order to construct the nonautonomous regular attractor for system (2.1). For simplicity, we restrict ourselves to consider only the case of rapidly oscillating external forces g ε (t) := g(ε −1 t) where g is an almost-periodic function with respect to t with values in L p+δ (ω): although all of the results formulated below remain true (after minor changing) under assumptions of previous section. Moreover, since the results of this section are more or less standard, we give below only schematic proofs resting the details to the reader. Our main assumption is that equation (5.1) possesses a hyperbolic equilibrium z 0 ∈ V p 0 (ω), i.e.
) and σ(L) denotes the spectrum of the operator L. It is well known that hyperbolicity assumption (6.2) implies existence of the spectral decomposition , where the linear subspaces V ± are invariant with respect to the operator L z 0 and satisfy the following properties: for a sufficiently small positive ν. Moreover, the dimension of the unstable subspace V + is finite and is called the instability index of the equilibrium z 0 : see e.g. [4] and [26]. It is also well known (see [4], [25]) that spectral decomposition (6.3) generates two invariant manifolds M + z 0 and M − z 0 for nonlinear problem (5.1) in a sufficiently small neighborhood of z 0 which correspond to linear subspaces V + and V − respectively. Since only unstable manifolds are important for the attractors theory, we formulate below the rigorous result on the unstable manifold M + z 0 only. Theorem 6.1. Let the assumptions of Theorem 5.1 hold and let, in addition, (6.2) be satisfied. Then, there exists a small neighborhood W z 0 of the equilibrium z 0 (in the V p 0metric) such that the set (6.5) M +,loc The main task of this Section is to construct the analogue of unstable manifold (6.5) for nonlinear elliptic equation (4.16) if ε > 0 is small enough. Since equation (4.16) is 'nonautonomous' then we first need to find the analogue of the hyperbolic equilibrium z 0 . Proposition 6.2. Let the assumptions of Theorem 5.1 hold and (6.1)-(6.2) be satisfied. Then there existε 0 > 0 and δ 0 > 0 such that, for every ε ≤ε 0 , h ∈ H(g) there exists a unique solution u = u ε hε,z 0 (t) of the problem Moreover, this solution is almostperiodic with respect to t: , where the constants R 0 and C are independent of ε and h and the functionᾱ R 0 (ε) is the same as in Corollaries 5.2 and 5.4. In particular, u ε hε,z 0 → z 0 as ε → 0. Sketch of proof. Instead of solving (6.6), we first solve the following discrete analogue of this equation: (6.8) u(n + 1) = U ε hε (n + 1, n)u(n) in the space of sequences u ∈ l ∞ (Z, V p 0 (ω)). We are going to solve (6.8) near the constant sequence u(n) ≡ z 0 using the implicit function theorem. Indeed, due Corollaries 3.7, 5.2 and 5.4, the operators U ε hε (n + 1, n) are close to S 1 together with their Frechet derivatives as ε → 0 (uniformly with respect to n and h). Moreover, the linearized problem (which corresponds to (6.8) at ε = 0 and u = z 0 ) is uniquely solvable in l ∞ (Z, V p 0 (ω)) for everyh ∈ l ∞ (Z, V p 0 (ω)) (due to hyperbolicity of the equilibrium z i . Thus, the implicit function theorem is indeed applicable to equation (6.8) and gives the existence and uniqueness of the solutionū ε hε,z 0 ∈ l ∞ (Z, V p 0 (ω)) of (6.8), for sufficiently small ε > 0, which belongs to a small neighborhood of the equilibrium z 0 . Moreover, it follows in a standard way from (3.29), (5.10) and (5.18) that this solution satisfies (6.10) ū ε hε,z 0 (n) − z 0 V p 0 (ω) ≤ Cᾱ R 0 (ε), ∀n ∈ Z, where C is independent of h, i and ε and R 0 is the radius of the uniform (with respect to ε and h) absorbing ball in V p 0 (ω) for the discrete processes U ε hε (l, m), l, m ∈ Z, l ≥ m, and h ∈ H(g) (which exists due to estimate (3.28)). The desired continuous function u ε hε,z 0 (t), t ∈ R, can be now defined as follows: (6.11) u ε hε,z 0 (t) := u ε Tthε,z 0 (0). Obviously, (6.11) is a solution of (6.6) which belongs to the space C b (R, V p+δ ε (ω)) (due to Corollary 2.8) and satisfies (6.12) u ε hε,z 0 C b (R,V p+δ ε (ω)) ≤ C, where the constant C is independent of ε, i and h. The uniqueness of this solutions (in a small neighborhood of z 0 ) is an immediate corollary of the uniqueness of the discrete solutionū ε hε,z 0 (m). The almost-periodicity of this function is a standard corollary of that uniqueness, see e.g. [18]). Proposition 6.2 is proved. Now we are ready to define the analogues of the unstable sets M + z 0 for problem (6.6). Since this problem is 'nonautonomous' then these manifolds also depend on t. Definition 6.3. Let the assumptions of Proposition 6.2 hold. For every ε ≤ ε 0 , h ∈ H(g) and t ∈ R, we define the set M +,loc ε,hε,z 0 (τ ) as follows: (6.13) M +,loc ε,hε,z 0 (τ ) := u τ ∈ V p ε (ω), ∃u ∈ C b (R, V p ε (ω)), which solves (6.6), is the solution of (6.6) constructed in Proposition 6.2 and W z 0 is the same as in Theorem 6.1. Thus, set (6.13) consists of the values u(τ ) at moment τ of all solutions u ∈ C b (R, V p ε (ω)) of (6.6) which tend to u ε hε,z 0 (t) as t → −∞ and belong to W z 0 for t ≤ τ . The following theorem is the 'nonautonomous' analogue of Theorem 6.1.
Remark 6.5. It is not difficult to verify that the manifolds M +,loc ε,hε,z 0 (τ ) are almost periodic with respect to τ and tend to the unstable manifold M +,loc z 0 of the limit parabolic problem (5.1) as ε → 0 uniformly with respect to h ∈ H(g) and τ ∈ R, [12,13] and [15] for the details.
Remark 6.6. Arguing in a standard way, it is not difficult to verify that expression (6.18) together with the backward uniqueness proved in Theorem 3.8 allows indeed to endow the sets M + ε,hε,z 0 (τ ) by the structure of a C 1 -manifold diffeomorphed to V + , see [4]. We however note that, in contrast to local unstable manifolds, the global ones are not, in general, submanifolds of V p ε (ω) (even in the limit autonomous case ε = 0). Indeed, if the limit equation (5.1) possesses a homoclinic orbit to the equilibrium z 0 , then the corresponding global unstable manifold M + z 0 cannot be a submanifold of V p 0 (ω). Nevertheless, in particular case where the limit parabolic equation possesses a global Lyapunov function, the sets M + ε,hε,z 0 (τ ) are occurred to be submanifolds of the phase space V p ε for a sufficiently small ε, see [4] for the details.

The nonautonomous regular attractor
In this section, using the theory of nonautonomous perturbations of regular attractors (see [12]), we obtain the detailed description of the structure of attractors A ε , ε ≪ 1, of equations (4.16) in case where the limit parabolic equation (5.1) is autonomous and possesses a global Lyapunov function. In order to have the explicit expression for the Lyapunov function, we assume that (7.1) a = a * and f (u) := ∇ u F (u), for some F ∈ C 1 (R k , R).
Then, system (5.1) possesses the following global Lyapunov function: Let R ⊂ V p 0 (ω) be the set of equilibria of problem (5.1), i.e. (7.3) R := {z ∈ V p 0 (ω), a∆ x z − f (z) +ḡ = 0}. Our main assumption is that all of the equilibria of R are hyperbolic, i.e. 2. For every fixed τ ∈ R, the sets M + ε,hε,z i (τ ) are C 1 -submanifolds of V p ε (ω) C 1diffeomorphic to the unstable manifolds M + z i of the limit autonomous parabolic problem (5.1) (which are independent of τ and h).
3. The sets A ε,hε (τ ), τ ∈ R, attract exponentially the images of all bounded subsets of V p 0 (ε), i.e, there exist a positive number α and a monotonic function Q (which are independent of ε, h, t and τ ) such that , for every bounded subset B ⊂ V p ε (ω), h ∈ H(g) and t, τ ∈ R, t ≥ τ . 4. The attractors A ε,hε (τ ) tend to A 0 as ε → 0 in the following sense: where the constantsC, R 0 and 0 < κ < 1 are independent of ε,ᾱ R 0 (ε) is the same as in Corollaries 5.2 and 5.4 and Sketch of proof. The result of Theorem 7.2 is a corollary of the nonautonomous perturbation theory of regular attractors developed in [12,30]. In order to apply this theory to equation (6.6), we consider the discrete processes (7.14) U ε hε (l, m) : V p 0 (ω) → V p 0 (ω) associated with this problem in phase space V p 0 (ω) which is independent of ε. Then, it follows Corollaries 3.7, 5.2 and 5.4 then these processes tend (together with their Frechet derivative) as ε → 0 to the semigroup S l−m associated with the limit parabolic equation (5.1) in the phase space V p 0 (ω) and this convergence is uniform with respect to h ∈ H(g). Moreover, estimate (3.28) guarantees the uniform (with respect to ε and h ∈ H(g)) dissipativity of these processes and Theorem 3.8 gives the injectivity of all the operators (7.14). We also recall that the limit discrete semigroup S n , n ∈ N, possesses the regular attractor A 0 (due to Theorem 7.1). Then, arguing in a standard way (see [12]), we derive that discrete processes (7.14) possess the nonautonomous regular attractors A ε,hε (l), l ∈ Z (if ε > 0 is small enough) in V p 0 (ω) which satisfy the discrete analogue of Theorem 7.2 (since all of the estimates formulated in Corollaries 3.7, 5.2 and 5.4 are uniform with respect to h ∈ H(g) then estimates (7.11) and (7.12) also hold for A ε,hε (l) uniformly with respect to h ∈ H(g)).
When the discrete nonautonomous attractors A ε,hε (l), l ∈ Z, for processes (7.14) (which satisfy all of the assertions of Theorem 7.2) are already constructed, we can extend in a standard way this result to the continuous case by the following expression: which is an immediate corollary of (7.20), see [12] for the details. Theorem 7.2 is proved.
Corollary 7.4. Let the assumptions of Theorem 7.2 hold. Then the nonautonomous regular attractor A ε,g (t), t ∈ R, of (6.6) and its uniform attractor A ε (constructed in Theorem 4.2) satisfy the following relation: and, consequently In particular, the uniform attractors A ε tend to A 0 (upper and lower semicontinuous) as ε → 0.
Indeed, the first equality in (7.18) is an immediate corollary the first assertion of Theorem 7.2 and description (4.7) of uniform attractor A ε . The second inequality in (7.18) can be easily verified using the exponential attraction property (7.11) and the alternative definition of the uniform attractor A ε which is formulated in Remark 4.3. Estimate (7.19) follows immediately from (7.17) and (7.18).
Remark 7.5. The first assertion of Theorem 7.2 can be reformulated as follows: problem (6.6) has exactly N almost-periodic solutions u ε hε,z i (t) which are localized near the equilibria z i ∈ R, i = 1, · · · , N, and every other bounded solution u ∈ C b (R, V p 0 (ω)) is a heteroclinic connection between two different almost periodic solutions of this problem.
Remark 7.6. We note that condition (7.16) is, obviously, always satisfied if the external force g(t) is periodic with respect to t. Thus, in case of periodic g, we have estimate (7.17) for the symmetric distance between the perturbed (A ε,h (t)) and nonperturbed (A 0 ) regular attractors without any additional assumptions and (as a corollary) the following estimate is satisfied for the uniform attractors: Unfortunately, in more general case of quasiperiodic or almost-periodic external forces, condition (7.16) is not satisfied automatically and should be verified, see e.g. [9], and [18] for various sufficient conditions.
The main result of this appendix is the following uniform (with respect to ε) maximal regularity estimate for the solutions of (8.1).
Proof. The proof of estimate (8.2) is based on the classical localization technique and on the multiplicators theorems in Fourier spaces and is more or less standard (see e.g. [17], [26]). That is the reason why, in order to show that constant C is indeed independent of ε, we discuss below only the principal points of this proof resting the details to the reader. We start with the most simple case γ = 0. Lemma 8.2. Let u be a solution of (8.1) with γ = 0. Then, the following estimate holds: , where the constants C and C ′ are independent of ε.
We are now ready to consider the general case p > 2. We first note that, due to the classical localization technique and estimate (8.2) for p = 2 (which is necessary in order to estimate the subordinated terms appearing under the localization technique), it is sufficient to verify estimate (8.2) only for equation (8.6) a(ε 2 ∂ 2 t u + ∆ x u − u) − γ∂ t u = h, u t=0 = u 0 , u 0 ∂ω = 0 and only for two choices of the domain Ω, namely, for 1) ω = R n and 2) ω + = R x 1 + ×R n−1 x 2 ,··· ,xn (see e.g. [17] and [26]). Moreover, we also note that the second case of semi-space ω + can be easily reduced to the first one of the the whole space ω = R n by considering the odd (with respect to x 1 ) solutions of (8.6) in ω = R n . Thus, there only remains to verify estimate (8.2) for solutions of (8.6) in ω = R n . In the next step, we reduce the problem of studying the elliptic system of equations (8.6) to the analogous problem for the scalar equation. In order to do so, it is convenient to extend the class of admissible solutions of (8.6) and consider also the complex-valued solutions u(t, x) = Re u(t, x) + i Im u(t, x) ∈ C k , for every (t, x) ∈ Ω + . Then, equation (8.6) is equivalent to the following one: ε 2 ∂ 2 t u + ∆ x u − u − γ ′ ∂ t u = h, u ∂ω = 0, u t=0 = u 0 , where γ ′ := a −1 γ. Moreover, without loss of generality we may assume that the matrix γ ′ is reduced to its Jordan normal form. Then, our conditions on matrices a and γ guaranties that the real parts of all eigenvalues of γ ′ are strictly positive: (8.8) σ(γ ′ ) ⊂ {λ ∈ C, Re λ > 0}.
Thus, (8.7) is a cascade system of scalar elliptic equations coupled by the terms γ ′ ∂ t u and γ ′ is in Jordan normal form. That is why, it is sufficient to verify estimate (8.2) only for scalar complex-valued elliptic equations of the form (8.9) ε 2 ∂ 2 t u + ∆ x u − u − 2(α + iβ)∂ t u = h, u ∂ω = 0, u t=0 = u 0 , where α, β ∈ R and α > 0. We start with the case h = 0.
Lemma 8.4. Let u 0 ∈ V p ε (R n ) and let u be a solution of (8.9) with h = 0. Then, it satisfies uniform estimate (8.2).
Proposition 8.5. The solution of (8.10) satisfies where C is independent of ε.
Let us now prove estimate (8.12). To this end, we fix an extension v(t) of the initial data u 0 inside of Ω + in such way that (8.21) ∂ t v L p (Ω + ) + v L p (R + ,W 2,p (ω)) ≤ C 1 u 0 W 2(1−1/p),p (R n ) , where C 1 is independent of u 0 (such an extension exists due to the classical trace theorems, see [26]) and introduce a function w(t) := u(t) − v(t) which, obviously, satisfies equation (8.13) with h(t) := ∂ t v(t) + A ε (1 − ∆ x )v(t). Thus, thanks to (8.14) and (8.21), it is sufficient to verify that where C 2 is independent of ε and t. But this estimate can be easily verified using the multiplicators theorem and estimates (8.18) and (8.20) (in the same way as it was done in the proof of estimate (8.14)). Proposition 8.5 is proved.
Indeed, the prove of (8.28) is based on multiplication of equation (8.1) by the special cutoff function ψ T (t) which vanishes for t / ∈ [T −1/2, T +3/2] and equals one for t ∈ [T, T +1] and on application of Theorem 8.1 to the function u T (t) := ψ T (t)u(t) and can be derived in a standard way.