Global attractors for $p$-Laplacian differential inclusions in unbounded domains

In this work we consider a differential inclusion governed by a p-Laplacian 
operator with a diffusion coefficient depending on a parameter in which the 
space variable belongs to an unbounded domain. We prove the existence of a 
global attractor and show that the family of attractors behaves upper 
semicontinuously with respect to the diffusion parameter. Both autonomous and 
nonautonomous cases are studied.


1.
Introduction. During the last ten years, many researchers have spent much effort in obtaining results on global attractors for p-Laplacian problems (see for example [1,7,12,15,16,20,24,25,37,41,42,46,48,49,50,56,57,58,59]). It is worth noting that p-laplacian equations have applications in a variety of phenomena, such as nonlinear elasticity, flows in porous media, non-Newtonian fluids and many others (see [35], [36], [38] and the references therein). We also observe that such equations are often perturbed by a discontinuous nonlinear term, which leads to study a differential inclusion rather than a differential equation. Such models appear for example when studying processes of combustion in porous media [19] or conduction of electrical impulses in nerve axons [53], [54]. On the other hand, many parabolic problems in unbounded domains have been studied over the last years [2,5,13,17,18,24,25,34,45].
In this paper we consider a differential inclusion governed by a p-laplacian operator in an unbounded domain. We observe that, due to the absense of uniqueness of the Cauchy problem, to prove the existence of a global attractor for such partial differential inclusions it is necessary to use the theory of multivalued semiflows (or 3240 JACSON SIMSEN AND JOSÉ VALERO generalized semiflows as well) [6,11,29,39,47]. In this work, we use the theory developed in [39].
It is important to point out that one of the main difficulties when working with unbounded domains is the fact that the usual compact embeddings for Sobolev spaces fail. Another one is that p > q does not imply L p (R n ) ⊂ L q (R n ). The first problem is solved in many papers by obtaining suitable estimates of the tails of solutions (see e.g. [40], [45]), and the second one by adding a suitable linear dissipative term. In this paper we consider another aproach. Namely, under some assumptions on the p-laplacian operator we can define a related space in which a suitable embedding is true for the unbounded domain. Moreover, this condition allows us to prove also the existence of a bounded absorbing set, a crucial step in obtaining a global attractor.
Let us consider the problem where p > 2, u λ (0) = u 0,λ ∈ H := L 2 (R n ), n ≥ 1, D λ ∈ L ∞ (R n ), ∞ > M ≥ D λ (x) ≥ σ > 0 a.e. in R n , λ ∈ [0, ∞) and D λ → D λ1 in L ∞ (R n ) as λ → λ 1 , f : R n ×R → 2 R is a multivalued function with compact convex values and a (x) ≥ 1 in R n is a continuous function satisfying the following condition: The authors in [51] considered problem (1) on bounded domains when f is Lipschitz in the multivalued sense and a(x) ≡ 1 (see [31,32] for differential equations and inclusions generated by pseudo-monotone operators in the case where f ≡ 0).
In this paper we will extend the results in [51] by considering unbounded domains first in the case where f is Lipschitz in the multivalued sense in both autonomous and nonautonomous cases, and after that a more general situation where f just satisfies a suitable growth condition.
It is also worth mentioning that for differential inclusions and reaction-diffusion equations of a similar type the regularity of all weak solutions and the global attractors have been studied in [23,26,27,33], whereas the existence of a Lyapunov function was established in [21]. Related results concerning attractors for nonautonomous differential equations and inclusions can be found in [22,28,60].
The paper is organized as follows. In Section 2 we prove the existence of the global attractor for problem (1) in the case of a Lipschitz nonlinearity f (in the multivalued sense) with g (t) ≡ 0 and the upper semicontinuity with respect to the parameter λ, as well. In Section 3 we extend the results of the previous section by proving the existence of a pullback attractor when g (t) ≡ 0. In Section 4 we consider a more general nonlinear term.
For a space X denote by P(X) the set of all non-empty subsets of X, and by C v (X) the set of all nonempty, bounded, closed, convex subsets of X. If X is a metric space with metric ρ, we define the Hausdorff semidistance between two sets by dist (A, B) = sup y∈A inf x∈B ρ(y, x), and the Hausdorff distance by Let H be the space L 2 (R n ) with norm · and scalar product (·, ·) . Also, we will denote the norms in the spaces L p (R n ), 2 < p ≤ ∞ by · p .
Let f : R → C v (R) be a multivalued map which is Lipschitz, that is, there exists Moreover, suppose there exists D > 0 such that As a particular example, we could consider f : Then we can define the associated map F : D(F ) ⊂ H → P(H), given by The author in [46] proved that the operator is maximal monotone in H and is the subdifferential of a proper, convex and lower semicontinuos function

+∞, otherwise
where E := u ∈ W 1,p (R n ); R n a(x)|u(x)| p dx < +∞ is a reflexive Banach space with the norm given by . Moreover, the author proved that there are constants w 1 = w 1 (σ) > 0, w 2 = w 2 (p, M ) > 0 such that for all u ∈ E the following two conditions hold: As a consequence, D(A D λ ) = H and the operator A D λ : D(A D λ ) ⊂ H → H generates a compact semigroup S D λ [14]. We consider our equation (2) in the abstract form It can be proved, with some adjustments, in an analogous way as in [39,Lemmas 11,12], that the operator F has values in C v (H) (in particular, D(F ) = H) and that it is Lipschitz (in the multivalued sense) with the same constant C from (3), that is, Remark 1. In order to consider at once the multivalued and the single-valued cases we could consider the operator U : H → H, which is a globally Lipschiz map with Lipschitz constant L ≥ 0, and the problem It is easy to show then that the mapF : H → P(H) defined byF (u) := F (u)+U (u) has values in C v (H) and that it is Lipschitz.
The main trouble to deal with an unbounded domain problem is the fact that p > q does not imply L p (R n ) ⊂ L q (R n ), and also that the usual compact embeddings for Sobolev spaces in bounded domains fail. However, the author in [46] solved this problem in a very simple way considering the space E and a compact embedding theorem for this space. We extend the result in [46] in the following lemma.
Proof. The first statement is proved in [46].
Let n ≤ p, 2 ≤ s < ∞ and choose some q > max{p, s}. We note that E ⊂ W 1,p (R n ) and from the proof of Lemma 1 in [46] we know that E ⊂ L 2 (R n ). For n = p it follows (from the embedding theorems of Sobolev spaces) that W 1,p (R n ) ⊂ L q (R n ) and then by the interpolation inequality W 1,p (R n ) ⊂ L s (R n ). When n < p we obtain as a consequence of Morrey's inequality that W 1,p (R n ) ⊂ L ∞ (R n ), and then (again using the interpolation inequality) W 1,p (R n ) ⊂ L s (R n ).
Let {u n } be a bounded sequence in E. Then up to a subsequence u n → v weakly in L 2 (R n ). We can show that v n := u n − v → 0 in L 2 (R n ). Indeed, denote by B R a ball of radius R centered at 0 in L 2 (R n ), and by B c R its complementary. Condition (C) and v ∈ L 2 (R n ) imply that for any > 0 there exists R = R( ) such that Then by the Hölder inequality we have Hence, Combining this with the compact embedding W 1,p (B R ) ⊂ L 2 (B R ), it is easy to see that v n → 0 in L 2 (R n ).
If n = p, as {u n } is a bounded sequence in E, then u n → v weakly in L q (R n ). Thus, using the interpolation inequality we can get where α = 2 s . Therefore, u n → v strongly in L s (R n ) and the embedding E ⊂ L s (R n ) is compact.

JACSON SIMSEN AND JOSÉ VALERO
Any integral solution u λ (·) of problem (8) is the unique integral solution of (12) with ϕ = ϕ D λ and f = f λ . The properties of the map F imply that for every integral solution of (8) the selection f λ (·) belongs to L 2 (0, T ; H). Then, Proposition 1 implies that it is in fact the unique strong solution of problem (12), as a strong solution of (12) is also an integral one [8]. Therefore, u λ (·) is also a strong solution of (8) and the sets of integral and strong solutions of (8) coincide.
Lemma 2.2. The following property is satisfied: Proof. Since E ⊂⊂ H by Lemma 2.1 and it is sufficient to show that for each K > 0, M K is a bounded set in E. Let K > 0 and u ∈ M K . Then, p . So, the condition (H) is satisfied.
We recall that A λ is a global attractor for the multivalued semiflow G λ if dist(G λ (t, B), A λ ) → 0, as t → +∞, for all bounded set B, and A λ ⊂ G λ (t, A λ ) for all t ≥ 0 (negatively semi-invariance). It is invariant if A λ = G λ (t, A λ ) for all t ≥ 0. Now, we prove the existence of the global attractor A λ for problem (8). Proof. We know from Lemma 2.2 that the condition (H) is satisfied. Then the result follows from [39,Theorem 9] if we prove that there exist δ > 0, M > 0 such that for any u ∈ D(A D λ ) such that u ≥ M and for all y ∈ −A D λ (u) + F (u), Indeed, let u ∈ D(A D λ ), ξ ∈ F (u). Using the embedding E ⊂ H in Lemma 2.1 we have that u H ≤ γ u E for some γ > 0. Using (5), (6), the Cauchy-Schwarz and the Young inequalities we get where C 1 is a positive constant. Considering M := 2γ p w1 (1 + C 1 ) condition (14) is satisfied.
2.1. Uniform estimates. In this section we obtain estimates in the spaces H and E for the solutions u λ of problem (8) which are uniformly on λ ∈ [0, ∞). As a consequence of (5), there exists D > 0 such that As commented before, each integral solution u λ (·) of problem (8) is a strong solution of this problem.
, working with selections we can repeat the same arguments used in [46,48,49] to obtain the desired estimates. What essentially changes is the control on the right hand side, i.e., if u λ is a solution of (8), then there exists Multiplying the equation by u λ (t) we control the right hand side using (16): for all λ ∈ [0, ∞). Thus, we obtain the following results, which are proved for example as in [ Remark 3. We observe that the constants r 0 , t 0 in Lemma 2.4 depend neither on the initial data nor on λ.
, where u λ is any integral solution of (8) with initial data u 0,λ .
where t 0 is as in Lemma 2.4 and u λ is any integral solution of (8) in [0, ∞).

Remark 5. For any bounded set
, where u λ is any integral solution of (8) with initial condition u 0,λ .
As an important consequence of Lemma 2.5 it follows that λ∈[0,∞) A λ is a bounded subset of E and once E ⊂⊂ H, we can conclude:

Upper semicontinuity of the global attractors. A multivalued map
denotes an ε-neighborhood of the set B. It is said to be upper semicontinuous if for any x 0 ∈ D (U ) and any neighborhood O (U (x 0 )) there exists δ > 0 such that . Obviously, any upper semicontinuous map is w-upper semicontinuous, the converse being valid if U has compact values [3, p.45].
In this section we prove that {A λ } λ∈[0,∞) is upper semicontinuous at any λ 1 . Since λ → A λ has compact values, this is equivalent to the property We first prove the w-upper semicontinuity of the map λ −→ G λ (t, A) with respect to λ.
Proof. For simplicity, we consider λ 1 = 0. Suppose, on the contrary, that there exists , and so we obtain a contradiction. Indeed, we have that u λn is an integral solution of (8) with u λn (0) ∈ A. So, there exists f λn ∈ L 1 (0, T ; H), with f λn (t) ∈ F (u λn (t)), a.e. in (0, T ), and such that u λn is an integral solution over (0, T ) of the problem (17): denoted by u λn (·) = I(u 0,λn )f λn (·). We can suppose that t 0 ∈ (0, T ). As A is compact, u λn (0) → u 0 ∈ A. Let z λn (·) = I(u 0 )f λn (·) be the integral solution of the problem By (16) and Remark 4, there exists In the same way as in Statement 1 in [48], but now using the compact embedding of E in H (see Let v ∈ E be arbitrary. Since z λn (·) are strong solutions of problem (18) we can obtain that 1 2 Hence, after integration we obtain for all 0 < s ≤ t. In fact, the continuity of s → z λn (s) implies that the inequality holds for 0 ≤ s ≤ t.
As f λn (τ ) ≤ L, for a.a. 0 ≤ τ ≤ T and for all n ∈ N, we conclude that there exists a positive constant L such that f λn L 2 (0,T ;H) ≤ L for all n ∈ N.
where we have used the inequality [8]: In view of [55, Proposition 1.1] for a.a. τ ∈ (0, t) , where co denotes the closure of the convex hull in H. Fix τ ∈ (0, t). Since F is Lipschitz, we obtain that for any δ > 0 there exists n > 0 such that for any λ k ≥ n, As F (z (τ )) is convex and closed, this implies that co ∪ , Thus, passing to the limit in (19) and taking into account that D(A D λ ) ⊂ E, we obtain that z (·) is an integral solution of problem (8) with λ = 0 and initial data u 0 .
which is a contradiction, and so we conclude that the map is w-upper semicontinuous on λ 1 for each t > 0.
Therefore, using Theorem 2.6 and Corollary 2, we obtain immediately from Theorem 1.2 in [30] the following result. (8) is upper semicontinuous at any λ.
3. The nonautonomous Lipschitz case. Let us consider now the following nonautonomous problem: where f : R → C v (R) satisfies as before (3) and (4) and We define then the map F : R × H → P(H), given by It follows from the results in Section 2 that sup y∈F (t,u) for all u, v ∈ H, t ∈ R. Also, it is obvious that for any u ∈ H the map t → F (t, u) is measurable, which means that for any open set U the inverse image for u fixed given by We then rewrite our problem in the abstract form where A D λ is the same as before. Integral and strong solutions of (24) on an interval [s, T ] are defined in the same way as in (10), (11). It is well know [55, Theorem 3.1] that for any T > s and u s,λ ∈ D(A D λ ) = H there exists at least one integral solution u λ (·) to problem (24). Also, the set of all integral and strong solutions coincide. We define then the multivalued map U λ : R 2 d × H → P(H) by U λ (t, s, u s,λ ) = {u λ (t) : u λ is an integral solution of (24)}.
This map is a strict multivalued process, that is: 1. U λ (t, t, ·) is the identity map for any t ∈ R; 2. U λ (t, s, x) = U λ (t, r, U λ (r, s, x)) for all s ≤ r ≤ t, x ∈ H.
This fact can be proved in the same way as in [61,Proposition 4.6]. We note that it also follows from this proof that the concatenation of two integral solutions is a new integral solution, which implies that every integral solution is global, that is, it can be continued for any forward time till +∞.
In order to study the asymptotic behaviour of solutions of problem (24) we will consider the pullback attraction of parametrized families of sets rather than the forward attraction of bounded sets as in the autonomous case. Namely, let D β be the class of all families D = {D (t) : t ∈ R} such that D (t) are bounded sets and We recall that a family A = {A(t) : t ∈ R} is called a global pullback D β -attractor if: 1. A(t) is compact for any t ∈ R; 2. A is pullback D β -attracting, that is, for any D ∈ D β one has lim s→−∞ dist(U λ (t, s, D(s)), A(t)) = 0 for any t ∈ R,

3.
A is negatively semi-invariant, which means that If in the third property we have an equality, then A is said to be invariant. We shall establish some previous statements.
where R > 0 is an universal constant, is pullback D β -absorbing for U λ . This means that for any t ∈ R and D ∈ D β there exists T t, D such that U λ (t, s, D (s)) ⊂ B 0 (t) for any s ≤ T.
Proof. Let D ∈ D β , u s,λ ∈ D (s) and u λ (·) be an arbitrary integral solution with u λ (s) = u s,λ . Since u λ is a strong solution as well, we have where f λ ∈ L 2 loc (s, +∞; H) and f λ (t) ∈ F (u λ (t)) for a.a. t > s. Multiplying (25) by u λ and using (15) we obtain where R 1 is a positive constant. Young's inequality gives By Gronwall's lemma we obtain which implies by choosing R = 1 + R3 β that B 0 is a pullback D β -absorbing family. Also, it is clear that B 0 ∈ D β . Remark 6. The constants in inequality (26) are independent of λ, so that the estimate is uniform with respect to this parameter.
Lemma 3.2. For any t ∈ R, r > 0 the map U λ (t + r, t, ·) maps bounded set of H into precompact ones.
Corollary 3. U λ is pullback D β -asymptotically compact, which means that for any D ∈ D β , t ∈ R and every sequence of times s n → −∞, any sequence y n ∈ U λ (t, s n , D (s n )) is precompact.
Proof. In view of Lemma 3.1 there exists T t − 1, D such that Then and the result is a consequence of Lemma 3.2.
Further, we shall prove the upper semicontinuity of the map x → U λ (t, s, x) .
where C is the constant from (9).
Since the operator A D λ generates a compact semigroup S D λ , it follows from [55, Theorem 3.4] that the set of all integral solutions of problem (24) with an initial data u λ,s in H is compact in the space C ([s, t], H) for any t > s. Thus, the set U λ (t, s, x) is compact in H for any x ∈ H, t ≥ s. Proof. As in view of (27) x → U λ (t, s, x) is continuous with respect to the Hausdorff metric, it is w-upper semicontinuous. Therefore, since U λ has compact values, it follows that x → U λ (t, s, x) is upper semicontinuous (see the beginning of Section 2.2). Now we are ready to state the main result in this section.
where B 0 is the absorbing familiy given in Lemma 3.1. Moreover, A ∈ D β , it is invariant and unique.
Proof. Putting together the results of this section we have obtain that: 1. There exists a pullback D β -absorbing family B 0 ∈ D β such that the sets B 0 (t) are closed; 2. U λ is pullback D β -asymptotically compact; 3. U λ has compact values; 4. The map x → U λ (t, s, x) is upper semicontinuous; 5. U λ is a strict multivalued process. Then the statement follows from [10, Theorem 3.3].

The case of a non-Lipschitz nonlinearity. Let us consider now the problem
where the (possibly) multivalued map f : R n × R →2 R satisfies the following assumptions: (F 1) f (x, u) ∈ C v (R) for a.a. x ∈ R n and any u ∈ R. (F 2) For some α > 0, c (·) ∈ H = L 2 (R n ) we have sup y∈f (x,u) |y| ≤ α |u| q + c (x) , for a.a. x ∈ R n and any u ∈ R, where 1 ≤ q < p 2 . (F 3) f is Caratheodory, that is, measurable in x and continuous in u (in the setvalued sense). We recall briefly the definition of continuity of set-valued maps. The definition of upper semicontinuity is given in Section 2.2. The map g : R →2 R is lower semicontinuous if for any x ∈ D(g), y ∈ g(x) and any sequence x n ∈ D(g) such that x n → x, there exists a sequence y n ∈ g(x n ) such that y n → y. It is continuous if it is upper and lower semicontinuous.

Denote by D(B) the domain of the operator B.
We consider then problem (28) in the abstract form We shall obtain some properties of the operator B.
Lemma 4.1. Let f satisfy (F 1), (F 3) and the following: , for a.a. x ∈ R n and any u ∈ R, Moreover, the constants C, γ do not depend on λ.

JACSON SIMSEN AND JOSÉ VALERO
Now, let n = p. In the same way, using the embedding E ⊂ L q (R n ) ∩ L 2 (R n ), for all q ≥ p, we prove ξ(·) ∈ H and (33).
We prove also some additional properties of the operator B. Proof. It follows from Lemma 4.1 that the set B(u) is non-empty and bounded if u ∈ D(ϕ D λ ). Let y n → y in H and y n ∈ B(u). Then y n (x) → y(x) for a.a. x ∈ R n . Since the set f (x, u(x)) is closed by (F1), we obtain that y(x) ∈ f (x, u(x)) for a.a. x ∈ Ω. Hence, y ∈ B(u) and B(u) is closed.
(B2) B is demiclosed in the following sense: If u n → u in L 2 (a, b; H) and b n → b weakly in L 2 (a, b; H) with b n (t) ∈ B(u n (t)) for a.a. t ∈ (a, b), then b(t) ∈ B(u(t)) for a.a. t ∈ (a, b).
Proof. First, we prove (B1). Arguing as in the proof of Lemma 4.1 we obtain that the composition (t, x) → f (x, u(t, x)) is measurable, the existence of a measurable map b(·,·) such that b(t, x) ∈ f (x, u(t, x)), for a. a. (t, x) ∈ (a, b) × Ω, and the inequality For a fixed t the map x → b(t, x) is measurable. Hence, since for a.a. t we have that u(t) ∈ E ⊂ L 2q (R n ) (see again the proof of Lemma 4.1), we obtain that b(t) = b(t,·) ∈ H for a.a. t ∈ (a, b). Now we define the sequence of measurable sets Ω t n by Ω t n = {x ∈ R n : |x| > n or |b(t, x)| > n} and the sequence of measurable approximations Since for any ε > 0 there exists M (ε) such that it follows easily that b n (t) → b(t) in L 2 (R n ). Then t → b(t) ∈ H is the pointwise limit of a sequence of measurable functions, so that it is measurable. Let us check (B2). Since u n → u in L 2 ((a, b) × R n ), we have that u n (t, x) → u (t, x) for a.a. (t, x). On the other hand, the continuity of the map u → f (x, u) implies that for a.a. (t, x) we have dist(b n (t, x), f (x, u(t, x))) ≤ dist(f (x, u n (t, x), f (x, u(t, x))) → 0 as n → ∞. (35) Using [55, Proposition 1.1] for a.a. t ∈ (a, b) we obtain Fix t and denote A n (t) = co ∞ ∪ k≥n b k (t) . Clearly, z ∈ A (t) if and only if there exist z n ∈ A n (t) such that z n → z in H. Thus, up to a subsequence we have z n (x) → z (x), a.e. in R n . On the other hand, z n ∈ A n (t) implies that where λ i ∈ [0, 1], N i=1 λ i = 1 and k i ≥ n, for any i. Note that f (x, u(t, x)) = [a 1 (t, x) , a 2 (t, x)] as f has convex, bounded, closed values. Then (35) implies that for any ε > 0 and a.a. x ∈ Ω there exists n = n (t, x, ε) such that and passing to the limit we have on (a, b).
Using general results proved in [44] about the existence of solutions for abstract differential inclusions we will obtain the existence of strong solutions of problem (29) in the following sense.
In view of Lemmas 4.1, 4.2, 4.3 and Remark 7 the conditions of Theorems III and IV in [44] are satisfied, from which the result follows.
Remark 8. Every function u λ satisfying properties 1-6 can be continued globally, that is, to a function defined on [0, +∞) satisfying the same properties in every interval [0, T ]. This follows also from Theorem IV in [44]. Proof. Let u λ (·) be the function from Proposition 3. It remains to prove only that b λ (·) ∈ L 2 (0, T ; H) for any T > 0. From (30) ).
We have obtained the existence of a strong solution with good regularity properties. We need to prove further that every strong solution also satisfies good properties.
Remark 9. In view of Remark 8 and Lemma 4.6 every strong solution of (29) can be extended to the whole semiline [0, ∞) and satisfies the properties given in this lemma for every T > 0.
We prove now that the concatenation of two global strong solutions is a new global strong solution.
Proof. It is easy to see that the translation v λ (·) = u λ (· + s) of a strong solution is again a strong solution. Hence, the inclusion G λ (t + s, u 0,λ ) ⊂ G λ (t, G λ (s, u 0,λ )) follows. The converse inequality is a consequence of Lemma 4.7.
We shall obtain now some estimates for strong solutions. In the following two lemmas the constants are independent on λ and the initial data.
Also, there exist positive constants δ 1 , δ 2 > 0 such that Proof. In view of (31) we have for some κ j > 0, we have Then by [52, p.164] and Gronwall's lemma we have
Then the uniform Gronwall's lemma [52] gives for all t ≥ 0. Integrating in (44) over (r, T ) and using (45) we obtain Thus, the result follows.
As a consequence of Lemma 4.10 we obtain that the map G λ is compact for positive times.
We note that by a diagonal argument we can choose a common subsequence for all N.
The result is proved for an arbitrary T > 0 using a diagonal argument.
Proof. If not, then there exists a neighborhood O of G λ (t, u 0,λ ) and sequences u n 0,λ → u 0,λ , y n ∈ G λ (t, u n 0,λ ) such that y n ∈ O. But then Lemma 4.11 implies the existence of a subsequence y nj such that y nj → y ∈ G λ (t, u 0,λ ), which is a contradiction. Proof. The case t = 0 is obvious. If t > 0, it follows from Corollary 5 that the set G λ (t, u 0,λ ) is relatively compact, and from Lemma 4.11 that it is closed.
Further, we establish the existence of a bounded absorbing set for G λ , that is, a set B 0 such that for any bounded set B ⊂ H there exists a time T (B, λ) such that G λ (t, B) ⊂ B 0 for all t ≥ T . Proof. In view of (40) the set B 0 = {v ∈ H : v 2 ≤ δ 2 + 1} is absorbing. We note that the time T (B) does not depend on λ. Now we are ready to prove the existence of a global compact attractor. Proof. It follows easily from Lemma 4.12 and Corollary 5 the existence of a compact set K λ such that dist(G λ (t, B), K λ ) → 0, as t → +∞, for any bounded set B. Also, by Corollaries 6, 7 the map u 0,λ → G λ (t, u 0,λ ) is upper semicontinuous and has compact values. The result follows then from [39,Theorem 4 and Remark 8]. From the definition of global attractor it follows easily that A λ ⊂ B 0 .
As an important consequence of Lemma 4.10 it follows that λ∈[0,∞) A λ is a bounded subset of E and once E ⊂⊂ H, we can conclude: Finally, we shall prove the upper semicontinuity of the attractors with respect to λ.