on global attractors of multivalued semiprocesses and nonautonomous evolution inclusions

Set-Valued Analysis8: 375–403, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

375

On Global Attractors of Multivalued Semiprocessesand Nonautonomous Evolution Inclusions?

VALERY S. MELNIK 1 and JOSÉ VALERO21Institute of System Applied Analysis, Pr. Pobedy 37, 252056 Kiev, Ukraine2CEU San Pablo-Elche, Comisario 3, 03203-Elche, Alicante, Spain

(Received: 1 February 2000)

Abstract. In this paper we define multivalued semiprocesses and give theorems providing theexistence of global attractors for such systems. This theory generalizes the construction of nonau-tonomous dynamical systems given by V. V. Chepyzhov and M. I. Vishik to the case where the systemis not supposed to have a unique solution for each initial state. Further, we apply these theorems tononautonomous differential inclusions of reaction–diffusion type.

Mathematics Subject Classifications (2000):35B40, 35K55, 35K90.

Key words: global attractor, nonautonomous multivalued dynamical systems, differential inclusions.

1. Introduction

In recent years, several works have been published concerning the existence ofglobal attractors for autonomous multivalued dynamical systems [3, 4, 16–18, 21,22]. In [21, 22, 25] the existence and properties of global attractors were studiedfor autonomous evolution inclusions. In [26], the results of [22] were extended toa class of differential inclusions generated by a difference of subdifferential maps.In all these works, the equations or inclusions generating the dynamical system arenot supposed to have a unique global solution for each initial state.

Some works were also devoted to the approximations of multivalued semiflowsand their attractors [11–13] and in [14] the dependence on a parameter of globalattractors of autonomous evolution inclusions was studied.

Concerning nonautonomous dynamical systems a nice theory was constructedin [7] for abstract processes and semiprocesses with applications to nonautonomousreaction–diffusion equations, Navier–Stokes equations and hyperbolic equations.In [19, 20] this theory was generalized for multivalued semiprocesses and appliedto reaction–diffusion equations in the case where the convergence to the attractoris considered in a weak sense. In [6] was studied a random multivalued systemgenerated by a differential inclusion with random perturbations.

? This work has been supported by PB-2-FS-97 grant (Fundacion Seneca (Comunidad Autonomade Murcia)).

376 VALERY S. MELNIK AND JOSE VALERO

In this paper, we generalize the construction of [7] to multivalued semiprocessesgiving an abstract theory which can be applied to a large class of nonautonomousevolution inclusions. First we define multivalued semiprocesses and study theirω-limit sets and global attractors. Then the abstract theorems are applied to evolutioninclusions of the type

∂u

∂t−

n∑i=1

∂

∂xi

(∣∣∣∣ ∂u∂xi∣∣∣∣p−2

∂u

∂xi

)∈ f (t, u)+ g(t), in �× (τ, T ),

u|∂� = 0, u(τ) = u0,

whereτ > 0,� ⊂ Rn is a bounded open subset,p > 2,g ∈ L∞([0,+∞), L2(�))

andf :R+×R→ 2R is a Lipschitz (in the multivalued sense) multivalued map withnonempty, convex, compact values. Under some additional conditions it is provedin Theorem 4 that this inclusion generates a multivalued semiprocess having auniform compact global attractor.

Inclusions of this type are used for example in some economical models [23].

2. ω-Limit Sets and Global Attractors of Multivalued Semiprocesses

Let X be a complete metric space with the metricρ, 2X (P(X), B(X), C(X),K(X)) be the set of all (nonempty, nonempty bounded, nonempty closed, non-empty compact) subsets of the spaceX and let6 be a compact metric space.DenoteR+ = [0,+∞),R2+ = R+×R+,R+d = {(t, τ ) ∈ R2+ : t > τ }. ForA,B ∈B(X) andx ∈ X set dist(x, B) = infy∈B ρ(x, y), dist(A,B) = supx∈A dist(x, B),Oδ(A) = {y ∈ X : dist(y,A) < δ}, whereδ > 0.

DEFINITION 1. The mapU : R+d × X → P(X) is called a multivalued semi-process (MSP) if:

(1) U(t, t, ·) = Id is the identity map;(2) U(t, τ, x) ⊂ U(t, s, U(s, τ, x)), ∀t > s > τ , ∀x ∈ X, whereU(t, s, U(s,

τ, x)) =⋃y∈U(s,τ,x) U(t, s, y).

Consider the family of MSP{Uσ : σ ∈∑} and define the mapU+: R+d×X→P(X) byU+(t, τ, x) =⋃σ∈6 Uσ(t, τ, x).

Remark 1.It is clear thatU+ is a MSP.

DEFINITION 2. The setA ⊂ X is called uniformly attracting for the family ofMSPUσ if for anyB ∈ B(X) andτ ∈ R+

limt→+∞ dist(U+(t, τ, B),A) = 0. (1)

GLOBAL ATTRACTORS OF MULTIVALUED SEMIPROCESSES 377

ForB ⊂ X defineγ τs,σ (B) =⋃t>s Uσ (t, τ, B) and

ωτ,6(B) =⋂t>τ

⋃σ∈6

γ τt,σ (B).

DEFINITION 3. The family of MSP{Uσ : σ ∈ 6} is called uniformly as-ymptotically upper semicompact if for anyB ∈ B(X) and τ ∈ R+ such thatfor someT = T (B, τ), γ τT ,6(B) =

⋃σ∈6 γ

τT ,σ (B) ∈ B(X), any sequence{ξn},

ξn ∈ Uσn(tn, τ, B), σn ∈ 6, tn→+∞, is precompact inX.

PROPOSITION 1. Let the family of MSP{Uσ : σ ∈ 6} be uniformly asymp-totically upper semicompact and for anyB ∈ B(X) and τ ∈ R+ there existsT = T (B, τ) such thatγ τT ,6(B) =

⋃σ∈6 γ

τT ,σ (B) ∈ B(X).

Thenωτ,6(B) 6= ∅, ∀B ∈ B(X), τ ∈ R+.Moreover, it is compact inX and

dist(U+(t, τ, B), ωτ,6(B)

)→ 0, as t → +∞.If P is another closed set such thatdist(U+(t, τ, B), P ) →

t→+∞ 0, thenωτ,6(B) ⊂ P(minimality property).

Proof.The setωτ,6(B) can be characterized as follows.

LEMMA 1. The following statements are equivalent:

(1) y ∈ ωτ,6(B);(2) There exists a sequence{ξn} such thatξn ∈ Uσn(tn, τ, B) and ξn → y ∈ X,

wheretn→+∞ andσn ∈ 6.

Proof. Consider the sequence of setsγ τtn,6(B), wheretn → +∞ asn→ +∞.The upper and lower limits of these sets are defined by

Lim sup(γ τtn,6(B)) ={y ∈ X : lim inf

n→+∞ dist(y, γ τtn,6(B)) = 0},

Lim inf (γ τtn,6(B)) ={y ∈ X : lim

n→+∞ dist(y, γ τtn,6(B)) = 0}.

It follows from these definitions thaty ∈ Lim inf (γ τtn,6(B)) (Lim sup(γ τtn,6(B)),respectively) if and only if for any neighborhoodO(y) of y there existsn0 suchthatO(y) ∩ γ τtn,6(B) 6= ∅ for n > n0 (respectively fornk > n0, wherenk is somesubsequence). Sinceγ τt2,6(B) ⊂ γ τt1,6(B) if t2 > t1,

Lim sup(γ τtn,6(B)) = Lim inf (γ τtn,6(B)) =⋂tn>τ

γ τtn,6(B)

= ωτ,6(B) (2)

(see [2, p. 18]).Let y ∈ ωτ,6(B). Then in view of (2) for any neighborhoodOδ(y), δ > 0, of y

there existsn0 such thatOδ(y) ∩ γ τtn,6(B) 6= ∅ for n > n0. Let us take a sequence


δn → 0, asn → +∞, and ξn ∈ Oδn(y) ∩ γ τtn,6(B). Henceξn ∈ Uσn(τn, τ, B),whereτn > tn, andξn→ y, asn→+∞.

Let now ξn ∈ Uσn(tn, τ, B) be such thatξn → y ∈ X, wheretn → +∞ andσn ∈ 6. Thenξn ∈ γ τtn,6(B) and for any neighborhoodO(y) there existsn0 forwhich ξn ∈ O(y) if n > n0. Hence,O(y) ∩ γ τtn,6(B) 6= ∅, ∀n > n0, so thaty ∈ ωτ,6(B). 2

Further, let us prove thatωτ,6(B) 6= ∅. If this is not the case, then Lemma 1implies that there cannot be converging sequencesξn ∈ Uσn(tn, τ, B), wheretn →+∞. But being the family of MSP{Uσ : σ ∈ 6} uniformly asymptotically uppersemicompact we can extract a converging subsequence, which is a contradiction.

Let {ξn} ⊂ ωτ,6(B) be an arbitrary sequence. It follows thatξn ∈ γ τt,6(B),∀t > τ . Therefore there exist sequences{tn}, {ζn} such thatζn ∈ Uσn(tn, τ, B),σn ∈ 6, tn → +∞ asn→ +∞, andρ(ζn, ξn) < 1/n, ∀n. Using again Lemma 1and the fact thatUσ is uniformly upper semicompact we obtain the existence of aconverging subsequenceζnk → ζ0 ∈ ωτ,6(B). But thenξnk → ζ0. Henceωτ,6(B)is compact.

Suppose now that there existε > 0 andyn ∈ Uσn(tn, τ, B), σn ∈ 6, tn→+∞,such that

dist(yn, ωτ,6(B)) > ε, ∀n.Then as before we can extract a subsequenceynk → ζ ∈ ωτ,6(B), which is acontradiction. Therefore dist(U+(t, τ, B), ωτ,6(B))→ 0, ast →+∞.

Finally letP ⊂ X be a closed set such that dist(U+(t, τ, B), P ) → 0, ast →+∞. Lemma 1 implies that for anyy ∈ ωτ,6(B) there existsξn ∈ Uσn(tn, τ, B)converging toy astn→+∞. Hencey ∈ P . 2PROPOSITION 2.Let for anyB ∈ B(X) andτ ∈ R+ there exists a compact setK(B, τ) such that

dist(U+(t, τ, B),K(B, τ))→ 0, ast →+∞.Then the conditions of Proposition1 hold.

Proof.It is evident that for anyB ∈ B(X) andτ ∈ R+ there existsT = T (B, τ)such thatγ τT ,6(B) ∈ B(X). Consider an arbitrary sequenceξn ∈ Uσn(tn, τ, B),σn ∈ 6, tn → +∞. Take a sequenceδk → 0, ask → +∞. Then there exist asubsequence{tnk } andζnk ∈ K(B, τ) satisfyingρ(ξnk , ζnk ) 6 δk. BeingK(B, τ)compact we can assume (taking a subsequence if necessary) thatζnk → ζ . Henceξnk → ζ andUσ is uniformly upper semicompact. 2PROPOSITION 3.LetX be an infinite-dimensional Banach space and the condi-tions of Proposition1 (or 2) hold. Then for anyτ ∈ R+ there exists a set2τ 6= Xsuch that∀B ∈ B(X)

dist(U+(t, τ, B),2τ )→ 0, ast →+∞. (3)


Moreover, for any closed setYτ satisfying(3),2τ ⊂ Yτ .Proof. Let {Bi} be a countable set of balls inX centered at 0 and set2τ =⋃∞i=1ωτ,6(Bi). We note that ifB1 ⊂ B2 thenωτ,6(B1) ⊂ ωτ,6(B2) and also that

anyB ∈ B(X) belongs to someBk. Hence, Proposition 1 implies that (3) holds.It follows also from Proposition 1 that ifYτ is a closed set satisfying (3) thenωτ,6(B) ⊂ Yτ , ∀B ∈ B(X), so that2τ ⊂ Yτ . Finally, since a compact set in aninfinite-dimensional Banach space is of first category and2τ is a countable unionof compact sets, the inequality2τ 6= X follows from Baire’s theorem. 2DEFINITION 4. The set26 is a uniform global attractor for the family of MSP{Uσ : σ ∈ 6} if:(1) 26 is a uniformly attracting set;(2) 26 ⊂ U+(t,0,26), ∀t ∈ R+;(3) For any closed uniformly attracting setY ,26 ⊂ Y .

Let Z be a topological space andF(R+, Z) be some space of functions withvalues inZ. We shall further assume that:(L1) 6 ⊂ F(R+, Z) is a compact metric space;(L2) On6 is defined the continuous shift operatorT (h)σ (t) = σ (t+h), h ∈ R+,

andT (h)6 ⊆ 6;(L3) For any(t, τ ) ∈ R+d , σ ∈ 6, h ∈ R+, x ∈ X, we have

Uσ(t + h, τ + h, x) ⊂ UT (h)σ (t, τ, x).

THEOREM 1. LetX be an infinite-dimensional Banach space, the conditions ofProposition1 be satisfied and(L1)–(L3) hold. Suppose that fort > 0 the map6 ×X 3 (σ, x) 7→ Uσ(t,0, x) has closed graph. Then the set

26def=

⋃B∈B(X)

ω0,6(B) =⋃τ>0

⋃B∈B(X)

ωτ,6(B) (4)

is a uniform global attractor and26 6= X. It is σ -compact and Lindelöf inX. It islocally compact in the topology of the disjoint unionτ⊕.

Remark 2.If X is not infinite-dimensional or is a metric space all the statementsremain true except the fact that26 6= X.

Proof.Define the multivalued mapG: R+ × X ×6→ P(X ×6) by

G(t, (x, σ )) = (Uσ(t,0, x), T (t)σ ). (5)

LEMMA 2. The mapG is a multivalued semiflow in the sense of[22] and for anyt > 0 the graph ofG(t, ·) is closed.


Proof.First we have to prove thatG is a multivalued semiflow, that is,G(0, ·) =IdX×6 (the identity map) andG(t1+ t2) ⊂ G(t1,G(t2, ξ )), ∀ξ ∈ X ×6, ∀t1, t2 ∈R+. SinceT (·): 6 → 6 is a semigroup,G(0, (x, σ )) = (x, σ ). For the secondproperty note that in view of Definition 1 and (L3)

G(t1 + t2, (x, σ )) = (Uσ(t1+ t2,0, x), T (t1+ t2)σ )⊂ (Uσ(t1+ t2, t2, Uσ (t2,0, x)), T (t1)T (t2)σ )⊂ (UT (t2)σ (t1,0, Uσ (t2,0, x)), T (t1)T (t2)σ )

= G(t1, (Uσ (t2,0, x), T (t2)σ )) = G(t1,G(t2, (x, σ ))).Finally, the fact thatG(t, ·) has closed graph fort > 0 is a consequence of being

the product of the maps(x, σ ) 7→ Uσ(t,0, x), T (t): 6 → 6, which have closedgraph. 2

ForC ∈ B(X ×6) setγ +t (C) =⋃s>t G(s, C), ω(C) =

⋂t>0 γ

+t (C).

LEMMA 3. The multivalued semiflowG is asymptotically upper semicompact,i.e.∀C ∈ B(X × 6) such thatγ +t1 (C) ∈ B(X ×6) for somet1(C) any sequenceξn ∈ G(tn, C), wheretn→+∞, is precompact inX ×6.

Proof. Consider an arbitrary sequence{ξn} of the type described in the state-ment. Thenξn = (yn, βn), whereyn ∈ Uσn(tn, τ, B), βn = T (tn)σn, B ∈ B(X),σn ∈ 6. The sequenceβn is precompact in6 because the space6 is compact and{yn} is precompact because the familyUσ is asymptotically upper semicompact.Henceξn is precompact inX ×6. 2LEMMA 4. For anyC∈B(X×6) there existsT (C) such thatγ +T (C)∈B(X×6).

Proof. First we note thatC ⊂ B × 6 for someB ∈ B(X). There existsT (B)for whichγ 0

T ,6(B) ∈ B(X). Hence,

γ +T (C) ⊂⋃t>T(U+(t,0, B),6) = (γ 0

T ,6(B),6) ∈ B(X ×6). 2The previous lemmas and Theorem 1 from [22] implies that for anyC ∈ B(X×

6) theω-limit setω(C) is nonempty, compact and negatively invariant. Moreover,it attractsC,

distX×6(G(t, C), ω(C))→ 0, ast →∞,and it is the minimal closed set with this property, i.e., ifP is closed and attractsC, thenω(C) ⊂ P .

Further, Theorem 2 from [22] implies thatG has the global attractor< satisfy-ing:(1) < =⋃C∈B(X×6) ω(C) and< 6= X ×6;(2) < ⊂ G(t,<), ∀t ∈ R+;


(3) ∀C ∈ B(X × 6), distX×6(G(t, C),<) → 0, ast → +∞ (< attracts everybounded setC);

(4) for any closed setP satisfying the previous property< ⊂ P.We note that although in [22, Theorems 1 and 2] the mapG(t, ·) is supposed to beupper semicontinuous this property is only used to prove that its graph is closed.

Further we state that26 = π1<, whereπ1: X×6→ X, π2: X×6 → 6 arethe projection operators, and that it is a uniform global attractor of the familyUσ .

First prove that< = ⋃B∈B(X) ω(B × 6). In fact for anyC ∈ B(X × 6),

ω(C) ⊂ ω(B × 6), whereB ∈ B(X) is such thatC ⊂ B × 6. Hence,< ⊂⋃B∈B(X) ω(B × 6). Conversely,∀B ∈ B(X), B × 6 ∈ B(X × 6) and then⋃B∈B(X) ω(B ×6) ⊂ <.Further, we note that as a consequence of standard theorems for semigroups (see

[10, 15]) the continuous semigroupT (h):6→ 6 has the compact global invariantattractorω(6) = ⋂

t>0

⋃h>t T (h)6 =

⋂t>0 γ

+t (6), which is the minimal set

attracting any bounded set. We shall check thatω(B ×6) ⊂ ω0,6(B)×ω(6) andπ1ω(B ×6) = ω0,6(B), π2ω(B ×6) = ω(6). On the one hand,

ω(B ×6) =⋂t>0

⋃s>tG(s, B ×6) ⊂

⋂t>0

⋃s>t(U+(s,0, B), T (s)6)

⊂⋂t>0

γ 0t,6(B)× γ +t (6) =

⋂t>0

(γ 0t,6(B)× γ +t (6)

)= ω0,6(B)× ω(6). (6)

It follows thatπ1ω(B × 6) ⊂ ω0,6(B), π2ω(B × 6) ⊂ ω(6). On the otherhand, since

distX×6(⋃σ∈6

(Uσ (t,0, B), T (t)σ ), ω(B ×6))

= distX×6(G(t, B ×6),ω(B ×6))→ 0, ast →∞,it is clear that

dist(U+(t,0, B), π1ω(B ×6)

)→ 0,ast →∞. (7)

But in view of Proposition 1 for any closed setP having property (7),ω0,6(B) ⊂P , henceω0,6(B) ⊂ π1ω(B × 6) (we note that sinceω(B × 6) is compact,π1ω(B × 6) is closed). In a similar way using the minimality ofω(6) we provethatω(6) ⊂ π2ω(B ×6).

Therefore,

π1< = π1

( ⋃B∈B(X)

ω(B ×6))=

⋃B∈B(X)

π1ω(B ×6) =⋃

B∈B(X)ω0,6(B),

so thatπ1< = 26. In the same wayπ2< = ω(6).


We have to check the conditions of Definition 4. For arbitraryB ∈ B(X),σ ∈ 6 and(t, τ ) ∈ R+d we have

Uσ(t, τ, B) ⊂ UT (τ)σ (t − τ,0, B) ⊂ π1G(t − τ, B ×6),so that26 is a uniformly attracting set.

On the other hand, since< ⊂ G(t,<), ∀t ∈ R+, we have

26 = π1< ⊂ π1G(t,<) ⊂ π1G(t,26 ×6) ⊂ U+(t,0,26).

Therefore,26 is negatively semi-invariant with respect toU+(t,0, ·).It remains to prove the minimality property. LetP be a closed uniformly at-

tracting set.X ×6 is a metric space with the metric

d((x1, σ1), (x2, σ2)) = ρ(x1, x2)+ r(σ1, σ2),

wherer is the metric of6. Then for anyB ∈ B(X)

distX×6(G(t, B ×6), P ×6)

= supx∈B, σ∈6

distX×6((Uσ (t,0, x), T (t)σ ), P ×6

)6 distX×6

(U+(t,0, B)× T (t)6, P ×6

)= sup

(x,σ )∈(U+(t,0,B)×T (t)6)inf

(y,δ)∈P×6(ρ(x, y) + r(σ, δ))

= dist(U+(t,0, B), P ) →t→+∞ 0.

Then the setP × 6 is attracting forG and in view of property (4) of the globalattractor<,< ⊂ P ×6. Hence26 = π1< ⊂ P .

It is clear that26 is σ -compact, because in view of (4) and the inclusionω0,6(B1) ⊂ ω0,6(B2) if B1 ⊂ B2 the set26 is a countable union of compactsets. Hence, it is Lindelöf.

In the same way as in [22, Theorem 2] we can say that accurate to homeomor-phims26 =⋃∞i=1Di, whereDi = {(x, i) : x ∈ ω0,6(Bi)}, Bi = {x ∈ X : ‖x‖ 6i} andDi are topological spaces with the topologyτi induced byX. We considerthe familyβ⊕ = {U ⊂ 26 | U⋂Di ∈ τi for any i > 1}, which is a sub-base ofa topologyτ⊕ in 26 . The space(26, τ⊕) is called the disjoint union of the spacesω0,6(Bi). In this space the global attractor is locally compact (see [22, proof ofTheorem 2]).

Since26 is a countable union of compact sets, Baire’s theorem implies that26 6= X.

Finally we shall prove the second equality in (4). It is a consequence of thefollowing lemma:

LEMMA 5. For anyτ ∈ R+, B ∈ B(X)

ωτ,6(B) ⊂ ω0,6(B).


Proof.Let σ ∈ 6 and(t, τ ) ∈ R+d . ThenUσ(t, τ, B) ⊂ UT (τ)σ (t − τ,0, B) ⊂U+(t − τ,0, B). Hence∀B ∈ B(X),∀(t, τ ) ∈ R+d

U+(t, τ, B) ⊂ U+(t − τ,0, B).Consequently,

γ 0s,6(B) =

⋃t ′>s

U+(t ′,0, B) =⋃t−τ>s

U+(t − τ,0, B),

γ τs,6(B) =⋃t>sU+(t, τ, B) ⊂ γ 0

s−τ,6(B).

Therefore,

ωτ,6(B) =⋂s>τ

γ τs,6(B) ⊂⋂s−τ>0

γ 0s−τ,6(B) = ω0,6(B). 2

DEFINITION 5. The family of MSP{Uσ } is called pointwise dissipative if thereexistsB0 ∈ B(X) such that∀x ∈ X

dist(U+(t,0, x), B0)→ 0 ast →+∞.DEFINITION 6. LetX,Y be metric spaces. The multivalued mapF : X → 2Y

is said to be w-upper semicontinuous (w-u.s.c.) atx0 if for any ε > 0 there existsδ > 0 such that

F(x) ⊂ Oε(F(x0)), ∀x ∈ Oδ(x0).

The mapF is w-u.s.c. if it is w-u.s.c. at anyx ∈ D(F) = {y ∈ X : F(x) 6= ∅}.If we replace theε-neighborhoodOε by an arbitrary oneO thenF is called

upper semicontinuous.

Remark 3.Any upper semicontinuous map is w-upper semicontinuous, the con-verse being valid ifF has compact values [1, p. 67].

LEMMA 6. A multivalued w-u.s.c. mapF with closed values and closed domainD(F) has closed graph.

Proof.Let (xn, yn) ∈ Graph(F ) andxn → x ∈ D(F), yn → y ∈ Y . For anyε-neighborhoodOε(F(x)) there existsn0 such thatyn ∈ Oε(F(x)), ∀n > n0. Sincethe setF(x) is closed this implies thaty ∈ F(x). 2THEOREM 2. Let the conditions of Proposition1 be satisfied,(L1)–(L3) hold,Uσbe pointwise dissipative and let the map(x, σ ) 7→ Uσ(t,0, x) have closed valuesand be w-u.s.c. for anyt ∈ R+. Then the family of MSPUσ has the global compactuniform attractor26.


Proof. It follows from the definition of the mapG that for any fixedt ∈ R+,G(t, ·) has closed values and is w-upper semicontinuous. Since the semiflowG

is uniformly asymptotically upper semicompact in view of Lemma 3, Theorem 1from [22] implies that the setω(B × 6) is nonempty, compact, negatively semi-invariant and the minimal closed set attractingB×6. We note that in this theoremthe mapG(t, ·) is supposed to be upper semicontinuous instead of w-upper semi-continuous, but this property is only used to prove thatG(t, ·) has closed graph,which is also a consequence of the w-upper semicontinuity (see Lemma 6).

In view of Proposition 1 for anyB ∈ B(X) and τ ∈ R+ the setωτ,6(B) isnonempty, compact and the minimal closed set uniformly attractingB. Moreover,it is shown in the proof of Theorem 1 (see (6)) thatω(B × 6) ⊂ ω0,6(B) ×ω(6) andπ1ω(B × 6) = ω0,6(B). Further, Lemma 5 implies thatω0,6(B) =⋃τ∈R+ ωτ,6(B).Let B0 ∈ B(X) attracts any pointx ∈ X. Then the setB0 × 6 attracts any

ξ ∈ X ×6, i.e.

distX×6(G(t, ξ), B0×6

)→ 0, ast →+∞,andG is pointwise dissipative (see [22, Definition 7]).

SetB1 = Oε1(B0) for some fixedε1 > 0 and< = ω(B1 × 6). We claim thatthis set is a global compact attractor forG. Since it is compact and negatively semi-invariant it is only necessary to show that it attracts any bounded setC ∈ B(X×6).In fact it is sufficient to take sets of the typeB × 6, whereB ∈ B(X). Since thefamily Uσ is pointwise dissipative for anyξ ∈ X × 6 there existsT (ξ) suchthatG(T, ξ) ⊂ B1 × 6. Now by the w-upper semicontinuity of the mapU wecan find a neighborhoodOδ(ξ)(ξ) for whichG(T,Oδ(ξ)(ξ )) ⊂ B1 × 6. The set{Oδ(ξ)(ξ) : ξ ∈ ω(B × 6)} is an open cover of the compact setω(B × 6). Let{Oδ(ξi)(ξi)}ni=1 be a finite subcover. ThenO(ω(B × 6)) = ⋃n

i=1Oδ(ξi)(ξi) is aneighborhood ofω(B ×6). If ε2 > 0 then for anyξi we have

G(t + T (ξi),Oδ(ξi )(ξi)) ⊂ G(t,G(T (ξi),Oδ(ξi)(ξi)))

⊂ G(t, B1 ×6) ⊂ Oε2(ω(B1×6)),for anyt > T (ε2, B1). Hence

G(t,O(ω(B ×6))) ⊂ Oε2(ω(B1×6)),for any t > maxi{T (ξi)} + T (ε2, B1). For anyε > 0 there existsT (ε, B) suchthatG(t, B × 6) ⊂ Oε(ω(B × 6)), ∀t > T (ε, B). The compacity ofω(B × 6)implies thatO(ω(B×6)) contains some neighborhood of the typeOε(ω(B×6)).Therefore,

G(t, B ×6) ⊂ Oε(B)(ω(B ×6)) ⊂ O(ω(B ×6)), ∀t > T (ε, B),so that

G(t, B ×6) ⊂ G(t − T (ε, B),G(T (ε, B), B ×6)) ⊂ Oε2(ω(B1×6)),


for anyt > T (ε, B)+T (ε2, B1)+maxi{T (ξi)}. This means that the setω(B1×6)attracts any bounded setC ∈ X ×6 and then it is a global compact attractor.

On the other hand, it is clear fromπ1ω(B1 × 6) = ω0,6(B1) that for anyB ∈ B(X)

dist(U+(t,0, B), ω0,6(B1))→ 0, ast →+∞. (8)

The uniformly attracting property follows from the following lemma:

LEMMA 7. Property(8) is equivalent to

dist(U+(t, τ, B), ω0,6(B1))→ 0, ast →+∞,for any(t, τ ) ∈ R+d .

Proof. SinceU+(t + τ, τ, B) ⊂ U+(t,0, B),∀(t, τ ) ∈ R+d , ∀B ∈ B(X), wehave

dist(U+(t + τ, τ, B), ω0,6(B1)) 6 dist(U+(t,0, B), ω0,6(B1)). 2We note that the minimality property ofω0,6(B) implies that ω0,6(B) ⊂

ω0,6(B1), ∀B ∈ B(X). Hence by (4) we haveω0,6(B1) = 26. From Lemma6 and Theorem 1 we obtain that it is a uniform global compact attractor. 2

3. Global Attractors for Nonautonomous Differential Inclusions

In this section we shall consider multivalued semiprocesses generated by non-autonomous evolution incusions.

3.1. ABSTRACT SETTING: CONSTRUCTION OF THE MULTIVALUED

SEMIPROCESS

Let X, X∗ be a real separable Banach space and its dual,Z be a complete metricspace andF (R+, Z) some functional space. Consider the evolution inclusion

du(t)

dt∈ A(u(t))+ Fσ(t)(u(t)), t ∈ [τ, T ],

u(τ) = uτ ,(9)

whereT > τ > 0, σ (·) ∈ 6 ⊂ F (R+, Z) andA: D(A) ⊂ X → 2X, Fσ(·)(·):R+ ×X→ 2X, are multivalued maps satisfying:

(A) The operatorA is m-dissipative, i.e.∀y1, y2 ∈ D(A), ∀ξi ∈ A(yi), i =1,2, ∃j (yi, ξi) ∈ J (y1− y2) such that

〈ξ1− ξ2, j〉 6 0,


and Im(A− λI) = X, ∀λ > 0, whereJ : X→ 2X∗

is the duality map definedby

J (y) = {ξ ∈ X∗ | 〈y, ξ 〉 = ‖y‖2X = ‖ξ‖2X∗ }, ∀y ∈ X.(G1) ∀σ (·) ∈ 6, Fσ(·)(·): R+ × X → Cb(X), where Cb(X) is the set of all

nonempty, closed, bounded subsets ofX.(G2) ∀x ∈ D(A), σ ∈ 6, the mapt 7→ Fσ(t)(x) is measurable and for anyσ ∈ 6,

T > τ > 0, there existsk(·) ∈ L1([τ, T ]) such that∀x1, x2 ∈ D(A)distH (Fσ(t)(x1), Fσ(t)(x2)) 6 k(t)‖x1− x2‖X, a.e.t ∈ (τ, T ).

(G3) For anyT > τ > 0, σ ∈ 6, there existsx ∈ D(A) andn: [τ, T ] → R+,n(·) ∈ L1([τ, T ]), such that

‖Fσ(t)(x)‖+ 6 n(t), a.e.t ∈ (τ, T ),where‖K‖+ = supy∈K ‖y‖X.

We assume also that for anyh ∈ R+ the shift operatorT (h): 6 → 6, T (h)

σ (s) = σ (s + h), is defined.

DEFINITION 7. The continuous functionuσ (·) ∈ C([τ, T ], X) is called an inte-gral solution of (9) ifuσ (τ) = uτ and there existsl(·) ∈ L1([τ, T ], X) such thatl(t) ∈ Fσ(t)(uσ (t)), a.e. on(τ, T ), and∀ξ ∈ D(A), ∀v ∈ A(ξ),‖uσ (t)− ξ‖2X6 ‖uσ (s)− ξ‖2X + 2

∫ t

s

〈l(r)+ v, uσ (r)− ξ 〉+dr, t > s, (10)

where〈η, y〉+ = supj∈J (y)〈η, j〉.

Supposing that conditions (A), (G1)–(G3) hold for anyuτ ∈ D(A) there existsat least one integral solution of (9) for anyT > τ > 0 [24, Theorem 3.1]. We shalldenote this solution byuσ (·) = I (uτ )l(·). For a fixedσ ∈ 6 let Dσ,τ (x) be the setof all integral solutions corresponding to the initial conditionu(τ) = x.

For any integral solutionsuσ (·) = I (uτ )l1(·), vσ (·) = I (vτ )l2(·), the followinginequality holds

‖uσ (t)− vσ (t)‖X6 ‖uσ (s)− vσ (s)‖X +

∫ t

s

‖l1(r)− l2(r)‖Xdr, t > s. (11)

In the sequel we shall writeu instead ofuσ for simplicity of notation. We shalldefine the mapUσ : R+d ×D(A)→ P(D(A)) by

Uσ(t, τ, x) = {z : ∃u(·) ∈ Dσ,τ (x), u(t) = z}.


PROPOSITION 4.For eachσ ∈ 6, (t, τ ) ∈ R+d , h ∈ R+, τ 6 s 6 t , x ∈ D(A)Uσ (t, s, Uσ (s, τ, x)) = Uσ(t, τ, x),UT (h)σ (t, τ, x) = Uσ(t + h, τ + h, x).

Hence,Uσ is a multivalued semiprocess for eachσ ∈ 6 and condition(L3) holds.Proof.Givenz ∈ Uσ(t1+t2, τ, x) we have to prove thatz ∈ Uσ(t1+t2, t2, Uσ (t2,

τ, x)). Takey(·) ∈ Dσ,τ (x) such thaty(τ) = x and y(t1 + t2) = z. Clearly,y(t2) ∈ Uσ(t2, τ, x). Then if we definez(t) = y(t) for t > t2 we have thatz(t2) =y(t2) and obviouslyz(·) ∈ Dσ,t2(y(t2)). Consequently,z = z(t1 + t2) ∈ Uσ(t1 +t2, t2, Uσ (t2, τ, x)).

Conversely, givenz ∈ Uσ(t1 + t2, t2, Uσ (t2, τ, x)) we have to prove thatz ∈Uσ(t1 + t2, τ, x). There existy1(·) ∈ Dσ,τ (x), y2(·) ∈ Dσ,t2(y1(t2)), such thatz = y2(t1+ t2). Define

y(t) ={y1(t), if τ 6 t 6 t2,y2(t), if t2 6 t 6 t1 + t2,

l(t) ={l1(t), if τ 6 t 6 t2,l2(t), if t2 < t 6 t1+ t2,

wherey1(·) = I (x)l1(·), y2(·) = I (y1(t2))l2(·). Clearly,l(t) ∈ Fσ(t)(y(t)), a.e. on(τ, t1 + t2). We claim thaty(·) ∈ Dσ,τ (x). Indeed, firstlyy(τ) = x. Secondly, wehave to check that (10) holds. The casess 6 t 6 t2, t2 6 s 6 t are straightforward.If s 6 t2 6 t then

‖y(t) − ξ‖2X = ‖y2(t)− ξ‖2X6 ‖y2(t2)− ξ‖2X + 2

∫ t

t2

〈l2(r)+ v, y2(r)− ξ 〉+dr

6 ‖y1(s)− ξ‖2X + 2∫ t2

s

〈l1(r)+ v, y1(r)− ξ 〉+dr +

+2∫ t

t2

〈l2(r)+ v, y2(r)− ξ 〉+dr

= ‖y(s) − ξ‖2X + 2∫ t

s

〈l(r)+ v, y(r) − ξ 〉+dr.

Hence,z = y2(t1+ t2) = y(t1 + t2) ∈ Uσ(t1 + t2, τ, x).Consider now the second equality. Givenz ∈ Uσ(t1 + h, τ + h, x), whereh ∈

R+, there existsy(·) ∈ Dσ,τ+h(x) such thatz = y(t1 + h). Letw(t) = y(t + h),lw(t) = l(t + h), wherel(t) ∈ Fσ(t)(y(t)), a.e. on(τ + h, t1+ h), y(·) = I (x)l(·),so thatlw(t) ∈ Fσ(t+h)(y(t + h)) = FT (h)σ (t)(w(t)), a.e. on(τ, t1), andw(τ) = x.We can show thatw(·) ∈ DT (h)σ,τ (x) as follows

‖w(t)− ξ‖2X = ‖y(t + h)− ξ‖2X


6 ‖y(s + h)− ξ‖2X + 2∫ t+h

s+h〈l(r)+ v, y(r) − ξ 〉+dr

= ‖w(s)− ξ‖2X + 2∫ t

s

〈lw(r)+ v,w(r)− ξ 〉+dr.

Therefore,z = w(t1) ∈ UT (h)σ (t1, τ, x).Finally, let z ∈ UT (h)σ (t1, τ, x). Thenz = y(t1), wherey(·) ∈ DT (h)σ,τ (x).

Definew(t) = y(t − h), lw(t) = l(t − h), wherel(t) ∈ Fσ(t+h)(y(t)), a.e. on(τ, t1), y(·) = I (x)l(·). Hence,lw(t) ∈ Fσ(t)(y(t − h)) = Fσ(t)(w(t)), a.e. on(τ + h, t1+ h). As before we can prove thatw(·) ∈ Dσ,τ+h(x), hencez = w(t1+h) ∈ Uσ(t1+ h, τ + h, x). 2

Consider the particular case whereX = ϕ(�,2) is a real separable Banachspace of functions defined on the set� and taking values in the normed space2.Let us define the mapFσ0: R+ ×X→ 2X by

Fσ0(t)(u) = Fσ0(t, u)+ g0(t),

whereg0(·) ∈ Lloc2 (R+, X) andFσ0:R+×X→ 2X is generated by the multivalued

mapf0: R+ ×2→ K(2) (that is, it has compact values) as follows

Fσ0(t, u) = {y ∈ X : y(x) ∈ f0(t, u(x)) for x ∈ �}.Moreover,K(2) is endowed with the Hausdorff metric andf0 ∈ C(R+, C(2,K(2))).

LEMMA 8. Let 2 be a locally compact, Lindelöf, Banach space. ThenC(2,

K(2)) is a complete metrizable space.Proof. Under the above conditions there exists a countable cover{Vi : i =

1,2, . . .} of the space2 by open sets such that their closure is compact. For each

i = 1,2, . . . setKi =⋃is=1Vs. The sequence of compact setsKi has the following

properties:

(1) K1 ⊂ K2 ⊂ · · · ⊂ Kn ⊂ · · · ;(2) For any compact setK ⊂ 2 there existsi such thatK ⊂ Ki .Hence inC(2,K(2)) is defined the topology of uniform convergence on{Ki},i.e.,Fn → F if and only if for anyi

ρi(Fn, F ) = supx∈Ki

distH (Fn(x), F (x))→ 0, asn→+∞.

This convergence can be realized by the metric

ρ(F1, F2) =∞∑i=1

αiρi(F1, F2)

1+ ρi(F1, F2),


whereαi > 0 and∑∞

i=1 αi <∞.Since2 is a locally compact Banach space, each bounded set is precompact and

thenK(2) is a complete metric space. Hence,C(2,K(2)) is also complete. 2Let M ⊂ C(2,K(2)) be a closed subset (and then a complete metrizable

space with the metric ofC(2,K(2))) andσ0(t) = (f0(t, ·), g0(t)) ∈ Z =M×X.Assume thatσ0(t +h) = (f0(t +h, ·), g0(t +h)) ∈ Z, ∀h ∈ R+. Let6 be the hullH+(σ0) of the mapσ0(t) in the spaceC(R+,M) × Lloc

2,w(R+, X) = F (R+, Z),that is,

6 = clC(R+,M){f0(t + h) : h ∈ R+} × clLloc2,w(R+,X){g(t + h) : h ∈ R+},

where clY denotes the clausure in the spaceY and Lloc2,w(R+, X) is the space

Lloc2 (R+, X) with the weak topology.

It is obvious that the shift operatorT (h) is continuous inC(R+,M) ×Lloc

2,w(R+, X). Further, we shall give conditions providing6 to be a compact posi-tively invariant set.

LEMMA 9 [8, Proposition 6.1]. The hullH+(f0) = clC(R+,M){f0(t+h) : h ∈ R+}of f0 is compact inC(R+,M) if and only if:

(1) The set{f0(t) : t ∈ R+} is precompact inM;(2) f0(t) is uniformly continuous inR+.

LEMMA 10. Let conditions(1)–(2)of Lemma9 hold and let

supt>0

∫ t+1

t

‖g0(τ )‖2X dτ < +∞. (12)

Then the hull6 is compact.Proof. Condition (12) implies that the hullH+(g0) = clLloc

2,w(R+,X){g0(t + h) :h ∈ R+} of g0 is compact inLloc

2,w(R+, X) (see [8, p. 931]). From this and Lemma9 we obtain that6 is compact. 2LEMMA 11. For anyh ∈ R+, T (h)6 ⊂ 6.

Proof.Let y = (y1, y2) ∈ 6. Then there existhn such thatf0(t + hn)→ y1 inC(R+,M), g0(t + hn)→ y2 in Lloc

2,w(R+, X), asn→∞. It is clear that

T (h)(f0(t + hn), g(t + hn)) = (f0(t + hn + h), g0(t + hn + h)) ∈ 6and in view of the continuity ofT (h),

(f0(t + hn + h), g0(t + hn + h))→ (y1(t + h), y2(t + h)),in C(R+,M)× Lloc

2,w(R+, X), so thatT (h)y ∈ 6. Hence,T (h)6 ⊂ 6. 2


Now the mapsFσ : R+ ×X→ 2X are defined by

Fσ(t)(u) = Fσ (t, u)+ gσ (t),where

Fσ (t, u) = {y ∈ X : y(x) ∈ fσ (t, u(x)) for x ∈ �}and(fσ , gσ ) ∈ 6.

As a consequence of Theorems 1, 2, Proposition 4 and Lemmas 10, 11 we obtainthe following abstract result:

THEOREM 3. Let2 be a locally compact, Lindelöf, Banach space. Suppose thatconditions(A), (G1)–(G3)are satisfied and that for anyt ∈ R+ the map(σ, x) 7→Uσ(t,0, x) has closed graph. Let the family of MSP{Uσ : σ ∈ 6} be uniformlyasymptotically upper semicompact and for anyB ∈ B(X) andτ ∈ R+ there existsT = T (B, τ) such thatγ τT ,6(B) =

⋃σ∈6 γ

τt,σ (B) ∈ B(X). Assume also that the

conditions of Lemma10hold. Then the familyUσ has the uniform global attractor26 defined by(4).

Moreover, if the familyUσ is pointwise dissipative and for anyt ∈ R+ the map(σ, x) 7→ Uσ(t,0, x) has closed values and is w-upper semicontinuous, then26

is compact.

3.2. GLOBAL ATTRACTORS OF NONAUTONOMOUS REACTION–DIFFUSION

INCLUSIONS

LetCv(R) be the set of all nonempty, compact, convex subsets ofR and� ⊂ Rn bea bounded open subset with smooth boundary∂�. Consider the parabolic inclusion

∂u

∂t−

n∑i=1

∂

∂xi

(∣∣∣∣ ∂u∂xi∣∣∣∣p−2

∂u

∂xi

)∈ f (t, u)+ g(t), on�× (τ, T ),

u |∂�= 0,u |t=τ= uτ ,

(13)

wherep > 2, f : R+ × R → Cv(R), g ∈ L∞(R+, L2(�)) and the followingconditions hold:

(F1) There existsC > 0 such that∀t ∈ R+,∀u, v ∈ R,distH (f (t, u), f (t, v)) 6 C|u− v|.

(F2) For anyt, s ∈ R+ andu ∈ R,distH (f (t, u), f (s, u)) 6 l(|u|)α(|t − s|),

whereα is a continuous function such thatα(t) → 0, ast → 0+, and l isa continuous nondecreasing function. Moreover, there existK1,K2 > 0 suchthat

|l(u)| 6 K1+K2|u|, ∀u ∈ R.


(F3) ∃D ∈ R+, v0 ∈ R for which

|f (t, v0)|+ 6 D, ∀t ∈ R+,where|f (t, v0)|+ = supζ∈(t,v0)

|ξ |.(F4) If p = 2 then there existε > 0 andM > 0 such that∀u ∈ R,∀t ∈ R+,∀y ∈

f (t, u)

yu 6 (λ1− ε)u2+M,whereλ1 is the first eigenvalue of−1 in H 1

0 (�).

LEMMA 12. There existD1,D2 > 0 such that∀u ∈ R,∀t ∈ R+,∀y ∈ f (t, u),|y| 6 D1+D2|u|.

Proof. Sincef (t, v0) is compact, for anyu ∈ R, t ∈ R+, y ∈ f (t, u) thereexistsy0 ∈ f (t, v0) such that dist(y, f (t, v0)) = |y − y0|. Using (F1) and (F3) weobtain

|y| 6 D + C|u| + C|v0|. 2Following the notation of the previous section2 = R,X = ϕ(�,2) = L2(�)

andf0(t, ·) = f (t, ·): R→ K(R), g0 = g ∈ L∞(R+, L2(�))⊂ Lloc2 (R+, L2(�)).

Let W be the spaceCv(R) endowed with the Hausdorff metricρ(x, y) =distH (x, y). The spaceW ⊂ K(R) is complete.

For anyψ ∈ W let |ψ |+ = maxy∈ψ |y|. Define also the space

M = {ψ ∈ C(R,W) : |ψ(v)|+ 6 D1+D2|v|}.The constantsD1,D2 are taken from Lemma 12. If in Lemma 8 takeKi =[−Ri,Ri], whereR1 < R2 < · · · < Rn < · · ·, with Rn → ∞, we have thatψm→ ψ if and only if

max|v|6R

distH(ψm(v), ψ(v))→ 0, asm→∞, ∀R > 0.

The spaceM ⊂ C(R,K(R)) is complete. Indeed, ifψm → ψ then it is clearfrom Lemma 8 thatψ ∈ C(R,K(R)). SinceW is complete,ψ(v) is convex andthenψ ∈ C(R,W). On the other hand if we fixv ∈ R we get

|ψm(v)|+ 6 D1+D2|v|, ∀m.Hence,∀ε > 0, ∀y ∈ ψ(v), ∃ym ∈ ψm(v) such that

|y − ym| < εand then

|y| 6 D1+D2|v| + ε.Beingy, ε arbitrary we have|ψ(v)|+ 6 D1+D2|v|, so thatψ ∈M.


LEMMA 13. In the spaceW each bounded set is precompact.Proof. It is clear that a sequenceIn = [an, bn] is bounded inW if and only if

there existsR > 0 such that

|y| 6 R, ∀y ∈ In, ∀n.If In is bounded there exists a subsequenceI ′n such thata′n → a, b′n → b. There-fore, αa′n + (1 − α)b′n → αa + (1 − α)b, ∀α ∈ [0,1]. Hence, ifI = [a, b]then

distH(I′n, I )→ 0, asn→+∞. 2

Let8 ⊂M be the set

8 = {ψ ∈M : distH(ψ(u), ψ(v)) 6 C|u− v|,∀u, v ∈ R},whereC is defined in (F1).

LEMMA 14. The set8 is compact.Proof.Let πR be the projector ofC([−K,K],W) ontoC([−R,R],W), where

R < K. Let {ψn} ⊂ 8 be an arbitrary sequence. It is clear that these functions areequicontinuous. On the other hand, in view of the definition of the spaceM thereexistsD(R) such that

|ψn(v)|+ 6 D(R), ∀v ∈ [−R,R].By Lemma 13 and Ascoli–Arzelá’s theorem{ψn} is precompact inC([−R,R],W).Hence, we can take a converging subsequenceψ1n→ ψ1 in C([−R,R],W). Fur-ther, we take a subsequence{ψ2n} ⊂ {ψ1n} converging toψ2 in C([−2R,2R],W).It is evident thatπRψ2 = ψ1. In the same way we can construct the chain of subse-quences{ψ1n}⊃{ψ2n}⊃· · · ⊃ {ψjn} · · · such thatψjn→ ψj in C([−jR, jR], X)andπ(j−1)Rψj = ψj−1, ∀j > 2. We define a mapψ ∈ C(R,W) such that

πjRψ = ψj , ∀j > 1.

The functionψ belongs to8. Indeed, sinceψ ⊂M it is sufficient to show thatψjsatisfies the Lipschitz property for anyj > 1. Letε > 0 be arbitrary andn be suchthat distH(ψjn(v), ψj (v)) 6 ε, ∀v ∈ [−jR, jR]. For anyu, v ∈ [−jR, jR] wehave

distH(ψj(u), ψj (v)) 6 distH(ψj(u), ψjn(u))++distH(ψjn(u), ψjn(v))+ distH (ψj(v), ψjn(v))

6 2ε + C|u− v|.Being ε arbitrary smallψj satisfies the Lipschitz property. Thenψ ∈ 8 and

finally we can see that the diagonal subsequence{ψjj } converges toψ in M. 2Recall that the hull off ∈ C(R+,M) is defined by

H+(f ) = clC(R+,M){f (t + h) : h > 0}.


DEFINITION 8. The functionf ∈ C(R+,M) is said to be translation-compactif its hull H+(f ) is compact inC(R+,M).

LEMMA 15. The functionf is translation-compact.Proof. In view of (F1) and Lemma 12,f (s, ·) ∈ 8,∀s > 0. Hence, the set

{f (t) : t ∈ R+} is precompact by Lemma 14. On the other hand, (F2) implies that

ρM(f (t), f (s)) =∞∑i=1

αimax|v|6Ri distH (f (t, v), f (s, v))

1+max|v|6Ri distH(f (t, v), f (s, v))6 ε,

if |t − s| 6 δ(ε). Hence,f (s) is uniformly continuous. By Lemma 9,f is transla-tion-compact. 2

Sinceg∈L∞(R+, L2(�)), we have that (12) holds, so that in view of Lemma 10the symbolσ0(t) = (f (t, ·), g(t)) is translation-compact in the spaceC(R+,M)×Lloc

2,w(R+, L2(�)). The hull of this symbol will be denoted as before by6.It is straightforward to check that for anyfσ ∈ H+(f ) conditions (F1)–(F4)

hold. We note that all the constants and functions in (F1)–(F4) do not depend onσ ∈ 6.LEMMA 16. For anygσ ∈ H+(g)

‖gσ‖L∞(R+,L2(�)) 6 C0 = ‖g‖L∞(R+,L2(�)).

Proof.If gσ (t) = g(t+h) for someh ∈ R+ the statement is obvious. Let us sup-pose thatgσ ∈ clLloc

2,w(R+,X){g(t+h) : h > 0}. Then there exists a sequencegn(t) =

g(t + hn) converging togσ in Lloc2,w(R+, X) asn→ ∞. Since‖gn‖L∞(R+,L2(�)) 6

C0, passing to a subsequencegn → g weakly star inL∞(R+, L2(�)). Hencegn→ g weakly inL2([0, T ], L2(�)), ∀T > 0, so thatg = gσ . Finally,

‖gσ‖L∞(R+,L2(�)) 6 lim supn→∞

‖gn‖L∞(R+,L2(�)) 6 C0. 2Let us now define the family of multivalued mapsFσ : R+ × L2(�)→ 2L2(�)

F σ (t, u) = {y ∈ L2(�) : y(x) ∈ fσ (t, u(x)), a.e. on�},wherefσ ∈ H+(f ).

LetCv(L2(�)) denote the set of all nonempty, bounded, closed, convex subsetsof L2(�).

LEMMA 17. The following properties hold:(1) Fσ (t, u) ∈ Cv(L2(�)), ∀(t, u) ∈ R+ × L2(�), ∀σ ∈ 6.(2) distH (F σ (t, u), F σ (t, v)) 6 C ‖u− v‖L2

, ∀u, v ∈ L2(�),∀t ∈ R+, ∀σ ∈ 6.Proof. It is a consequence of Lemmas 11 and 12 from [22]. 2


LEMMA 18. For anyu ∈ L2(�) there existsD(u) not depending onσ such that

supy∈Fσ (t,u)

‖y‖L2 6 D(u), ∀t ∈ R+.

Proof. In view of Lemma 12 for anyy ∈ Fσ (t, u) we have

|y(x)| 6 D1+D2|u(x)|, a.e. on�.

After integration we obtain the required result. 2LEMMA 19. There existR1, R2 > 0 such that∀σ ∈ 6,

distH (Fσ (t, u), F σ (s, u)) 6 (R1+ R2‖u‖L2)α(|t − s|),

∀t, s ∈ R+,∀u ∈ L2(�).

Proof.Let y ∈ Fσ (t, u) be arbitrary. The mapx 7→ fσ (s, u(x)) is measurable,sincez 7→ fσ (s, z) is continuous (see [2, Theorem 8.2.8.]). For eachε > 0 wedefine the measurable functionx 7−→ l(|u(x)|)α(|t − s|)+ ε = ρε(x). LetB(r, ρ)denote a closed ball centered atr with radiusρ. Condition (F2) implies that the set-valued mapD(x) = B(y(x), ρε(x))∩fσ (s, u(x)) has nonempty values. Moreover,it is measurable and there exists a measurable selectionzε(x) ∈ D(x), a.e.x ∈ �(see Theorems 8.1.3, 8.2.4 and Corollary 8.2.13 in [2]). Again by (F2) we get

|y(x) − zε(x)| 6 (K1+K2|u(x)|)α(|t − s|)+ ε a.e. on�.

Hence, after integration we obtain

‖y − zε‖2L26 ε2µ(�)+K2

1α2(|t − s|)µ(�)+K2

2α2(|t − s|)‖u‖2L2

++2K2K1α

2(|t − s|)‖u‖L2(µ(�))1/2+

+2εα(|t − s|)K1µ(�)++2εα(|t − s|)K2‖u‖L2(µ(�))

1/2.

Passing to the limit asε → 0 we have

dist2(y, F σ (s, u))

6 α2(|t − s|)(K21µ(�)+K2

2‖u‖2L2+ 2K2K1‖u‖L2(µ(�))

1/2)

6 α2(|t − s|)(K21µ(�)+K2

2K21µ(�)+ (1+K2

2)‖u‖2L2),

and then there existR1, R2 > 0 such that

dist(F σ (t, u), F σ (s, u)) 6 (R1+ R2‖u‖L2)α(|t − s|).The converse inequality is proved exactly in the same way. 2

COROLLARY 1. For eachσ ∈ 6, u ∈ L2(�) the mapt 7→ Fσ (t, u) is measur-able.


Proof. In view of Lemma 19 the mapt 7→ Fσ (t, u) is continuous in the Haus-dorff metric. This implies that it is lower semicontinuous, that is, for anyt ∈ R+,y ∈ F(t, u) and tn → t, there existyn ∈ F(tn, u) converging toy. Hence, it ismeasurable (see [2, Theorem 8.2.1]). 2

Further we shall define the family of multivalued mapsFσ : R+ × L2(�) →2L2(�)

Fσ(t)(u) = Fσ (t, u)+ gσ (t), ∀t ∈ R+,∀u ∈ L2(�).

It follows from the previous results that∀t ∈ R+,∀u ∈ L2(�) the followingproperties hold:

(S1) Fσ(t)(u) ∈ Cv(L2(�)).

(S2) distH (Fσ(t)(u), Fσ(t)(v)) 6 C‖u− v‖L2.

(S3) For eachu ∈ L2(�), t 7→ Fσ(t)(u) is measurable.(S4) For anyu ∈ L2(�) there existsD(u) such that

supy∈Fσ(t)(u)

‖y‖L2 6 D(u)+ ‖gσ (t)‖L2 6 D(u)+ C0, a.e.t ∈ R+.

Therefore, conditions (G1)–(G3) hold. Moreover, the functionsk(t) = C,n(t) =D(u)+ C0 do not depend on eitherσ or (τ, T ).

On the other hand, the operator

−A(u) = −n∑i=1

∂

∂xi

(∣∣∣∣ ∂u∂xi∣∣∣∣p−2

∂u

∂xi

),

with D(A) = {u ∈ W 1,p0 (�) : A(u) ∈ L2(�)} is the subdifferential of the proper

convex lower semicontinuous function

ϕ(u) =

1

p

n∑i=1

∫�

∣∣∣∣ ∂u∂xi∣∣∣∣pdx, if u ∈ W 1,p

0 (�) ,

+∞, otherwise.

Therefore, inclusion (13) is a particular case of the abstract one

du(t)

dt+ ∂ϕ(u(t)) ∈ Fσ(t)(u(t)), on (τ, T ),

u |t=τ= uτ .(14)

The operatorA is m-dissipative andD(A) = L2(�) (see [22, Section 3.2]).Hence (A) holds and by Proposition 4 and Lemma 11 we obtain the family of semi-processes{Uσ : σ ∈ 6}, where6 is the hull of the symbolσ0(t) = (f (t, ·), g(t)),which is a compact metric space such thatT (h)6 ⊂ 6, ∀h ∈ R+.


Let us now prove the properties needed to provide the existence of a uniformglobal compact attractor. For this purpose we shall use the fact that ifl(·) ∈L2([τ, T ], X) then the integral solutionu(·) of the problem

du(t)

dt+ ∂ϕ(u(t)) 3 l(t), on (τ, T ),

u |t=τ= uτ ,(15)

which is unique, is in fact a strong one, that is,u(·) is absolutely continuous oncompact sets of(τ, T ), a.e. differentiable on(τ, T ) and satisfies (15) a.e. in(τ, T ),and also that the semigroup generated by−∂ϕ is compact (see [22, Section 3.2]).

PROPOSITION 5. For any t ∈ R+ the map(σ, u0) 7→ Uσ(t,0, u0) has closedgraph.

Proof.Let yn ∈ Uσn(t,0, un0) be such that

yn→ y in L2(�),

un0→ u0 in L2(�),

σn = (fn, gn)→ σ = (fσ , gσ ) in C(R+,M)× Lloc2,w(R+, L2(�)).

We have to prove thaty ∈ Uσ(t,0, u0).There exist sequencesun(·) = I (un0)ln(·), ln(s) ∈ Fσn(s)(un(s)), a.e. in(0, T ),

such thatyn = un(t).Let v ∈ L2(�) be fixed. It follows from (S2) and (S4) that for anyy ∈ L2(�),

σ ∈ 6,

supξ∈Fσ(s)(y)

‖ξ‖ 6 D(v)+ C‖v‖L2 + C‖y‖L2 + ‖gσ (s)‖L2, a.e.s ∈ (0, T ).

Therefore, Lemma 16 implies that there existK1,K2 > 0 for which

‖ln(s)‖L2 6 ‖Fσn(s)(un(s))‖+ 6 K1+K2‖un(s)‖L2, a.e. in(0, T ). (16)

We shall show first the existence of a functionm(·) ∈ L∞(0, T ), m(s) > 0,such that‖ln(s)‖L2 6 m(s), a.e. in(0, T ). Let us introduce the sequencevn(·) =I (u0)ln(·) and letz(·) be the unique solution of

dz(t)

dt+ ∂ϕ(z(t)) 3 0, on (0, T ),

z(0) = u0.

Let r0 = max{z(s) : s ∈ [0, T ]} andr2 = r1 + r0, where‖u0 − un0‖L2 6 r1, ∀n.From (11) we have

‖un(s)− z(s)‖L2 6 ‖u0 − un0‖L2 +∫ s

0‖ln(r)‖L2dr


and then by (16)

‖un(s)‖L2 6 ‖z(s)‖L2 + r1+∫ s

0(K1 +K2‖un(s)‖L2)dr

6 r2 +K1s +K2

∫ s

0‖un(s)‖L2dr.

Hence by Gronwall lemma

‖un(s)‖L2 6 −K1

K2+(r2+ K1

K2

)exp(K2s) = r(s), ∀t ∈ [0, T ].

Therefore by (16)

‖ln(s)‖L2 6 K1+K2r(s) = m(s), a.e. in(0, T ).

The sequence{ln} is then integrable bounded inL1(0, T , L2(�)) and since thesemigroup generated by−∂ϕ is compact, this implies that the sequence{vn} isprecompact inC([0, T ], L2(�)) (see [9, Theorem 2.3]). We obtain that there existsubsequences such that

vn → v in C([0, T ], L2(�)),

ln → l weakly inL2(0, T , L2(�)).

Sinceln→ l weakly inL1(0, T , L2(�)), Lemma 1.3 from [24] implies thatv(·) =I (u0)l(·). Using again (11) we have‖un(s)− vn(s)‖L2 6 ‖un0(s)− u0(s)‖L2, ∀s ∈[0, T ], so thatun → v in C([0, T ], L2(�)) andy = v(t). To conclude the proofwe have to check thatl(s) ∈ Fσ(s)(v(s)), a.e. on(0, T ).

Sinceln→ l, gn→ gσ , weakly inL2(0, T , L2(�)), we haveln−gn = dn(·)→l − gσ = dσ (·), weakly inL2(0, T , L2(�)). Then we need to obtain thatdσ (s) ∈Fσ (s, v(s)), a.e. on(0, T ). Fix s ∈ (0, T ).

Note that sinceun(s) → v(s) in L2(�), passing to a subsequence if necessaryun(s, x)→ v(s, x) for a.a.x ∈ �. Hence by (F1)

dist(fσ (s, un(s, x)), fσ (s, v(s, x))) 6 C|un(s, x)− v(s, x)| → 0,

asn→+∞,for a.a.x ∈ �. On the other hand, beingun(s, x) bounded, that is|un(s, x)| 6C(x), ∀n, andfn converging tofσ in C(R+,M), we get

dist(fn(s, un(s, x)), fσ (s, un(s, x)))→ 0, asn→+∞,for a.a.x ∈ �. Then

dist(dn(s, x), fσ (s, v(s, x)))

6 dist(fσ (s, un(s, x)), fσ (s, v(s, x)))++dist(fn(s, un(s, x)), fσ (s, un(s, x)))→ 0, (17)


for a.a.x ∈ �.In view of [24, Proposition 1.1] for a.a.s ∈ (0, T )

d(s) ∈∞⋂n=1

co∞⋃k>n

dk(s) = A(s).

DenoteAn(s) = co⋃∞k>n dk(s). It is easy to see thatz ∈ A(s) if and only if there

exist zn ∈ An(s) such thatzn → z, asn → ∞, in L2(�). Taking a subsequencewe havezn(x)→ z(x), a.e. in�. Sincezn ∈ An(s), we get

zn(s) =N∑i=1

λidki (s),

whereλi ∈ [0,1], ∑Ni=1 λi = 1 andki > n, ∀i. Now (17) implies that for any

ε > 0 and a.a.x ∈ � there existsn(x, ε) such that

dk(s, x) ⊂ [a(x) − ε, b(x) + ε], ∀k > n,where[a(x), b(x)] = fσ (s, v(s, x)). Hence,

zn(x) ⊂ [a(x) − ε, b(x) + ε],as well. Passing to the limit we obtain

z(x) ∈ [a(x), b(x)], a.e. on�.

Therefore,z(s) ∈ Fσ (s, v(s)), so thatd(s) ∈ A(s) ⊂ Fσ (s, v(s)), a.e. on(0, T ).It follows that l(s) ∈ Fσ(s)(v(s)), a.a. on(0, T ), as required. 2COROLLARY 2. For any t ∈ R+ the map(σ, u0) 7→ Uσ(t,0, u0) has closedvalues.

PROPOSITION 6. For any t ∈ R+ the map(σ, u0) 7→ Uσ(t,0, u0) is uppersemicontinuous, hence w-upper semicontinuous.

Proof.Suppose that for some(σ, u0) the map is not upper semicontinuous. Thenthere exists a neighborhoodO of Uσ(t,0, u0) and sequenceszn ∈ Uσn(t,0, un0),σn → σ in C(R+,M)× Lloc

2,w(R+, L2(�)), un0 → u0 in L2(�), such thatzn /∈ O.Repeating the same lines of the proof of Proposition 5 we can prove that for somesubsequenceznk → z ∈ Uσ(t,0, u0), which is a contradiction. 2LEMMA 20. There existsR0 > 0 such that for any bounded setB and τ > 0there is a numberT (B, τ) for which

‖Uσ(t, τ, B)‖+L26 R0, ∀t > T , ∀σ ∈ 6.


Proof. Let first p = 2. Let u(·) = I (uτ )l(·) be an arbitrary integral solu-tion. Multiplying (15) byu (note that∂ϕ = −1) and using condition (F4) andLemma 16 we obtain

1

2

d

dt‖u‖2L2

+ ε2‖u‖2L2

6 K, (18)

whereK = Mµ(�)+ 12εC

20. By Gronwall lemma we get

‖u(t)‖2L26 exp(−ε(t − τ))‖u(τ)‖2L2

+ 2K

ε(1− exp(−ε(t − τ))). (19)

TakingR20 = 2K

ε+ δ, for someδ > 0, the result follows.

Let nowp > 2. Since in view of Poincaré’s inequality(−A(u), u) > γ ‖u‖pLp >D‖u‖pL2

, ∀u ∈ D(A), whereD > 0, and as shown before (see (16)) there existK1,K2 > 0 such that∀u ∈ L2(�), σ ∈ 6,‖Fσ(t)(u)‖+ 6 K1+K2‖u‖L2, ∀t ∈ R+,

we get

1

2

d

dt‖u‖2L2

+D‖u‖pL26 K1‖u‖L2 +K2‖u‖2L2

.

Applying Young inequality we obtain that for someD,K3,K4 > 0

1

2

d

dt‖u‖2L2

+ D2‖u‖2L2

−K4 61

2

d

dt‖u‖2L2

+ D2‖u‖pL2

6 K3, (20)

and we conclude the proof as before using Gronwall lemma. 2Let us denote byB(0, R) a closed ball ofL2(�) centered at 0 with radiusR.

LEMMA 21. For anyR > R0, t, τ ∈ R+, t > τ , σ ∈ 6, we have

Uσ(t, τ, B(0, R)) ⊂ B(0, R).Proof.Let us suppose the opposite, that is, there existR > R0, u(·) = Iσ (uτ )l(·),

‖u(τ)‖2L26 R2 andt > τ such that‖u(t)‖2L2

> R2. Sinceu(·) is continuous thereexists tR such that‖u(tR)‖2L2

= R2, ‖u(s)‖2L2> R2, ∀tR < s 6 t . Then (18)

implies

d

dt‖u(s)‖2L2

6 −δ, ∀tR 6 s 6 t,

whereδ > 0, and after integration‖u(t)‖2L26 R2− δ(t − tR), ∀tR < s 6 t, which

is a contradiction. 2COROLLARY 3. For anyB ∈ B(X), τ ∈ R+, γ τ0,6(B) ∈ B(X).


For any bounded setB andτ, t ∈ R+ let us introduce the set

M(B, τ, t) = {l ∈ L1(τ, t;L2(�)) : uσ (·)= I (uτ )l(·), uσ ∈ Dσ,τ (uτ ), uτ ∈ B, σ ∈ 6

}.

LEMMA 22. For any bounded setB andτ, t ∈ R+ the setM(B, τ, t) is boundedin the spaceL∞(τ, t;L2(�)).

Proof. As shown before (see (16)) there existK1,K2 > 0 such that∀u ∈L2(�), σ ∈ 6,‖Fσ(t)(u)‖+ 6 K1+K2‖u‖L2, ∀t ∈ R+.

Hence, for anyl ∈ M(B, τ, t)‖l(s)‖L2 6 K1+K2‖uσ (s)‖L2, a.e.s ∈ (τ, t),

whereuσ (·) = I (uτ )l(·). But Lemma 21 implies that for anyσ ∈ 6, τ 6 s 6 t ,uσ (s) ∈ Uσ(s, τ, B), ‖uσ (s)‖L2 6 R for someR > R0, so that the statementfollows. 2PROPOSITION 7. There exists a compact setK such that for anyB ∈ B(X),τ ∈ R+, there existsT (B, τ) for which

U+(t, τ, B) ⊂ K, if t > T .

Proof. SetK = U+(1,0, B(0, R0)). We claim thatK is compact. Lety ∈U+(1,0, B(0, R0)) be arbitrary. Then there existsuσ (·) = I (u0)l(·), with σ ∈ 6,u0 ∈ B(0, R0), such thaty = uσ (1). Multiplying the equation

duσdt− A(uσ ) = l (21)

by uσ and using Lemma 22, the inequality(−A(u), u) > γ ‖u‖pW1,p , ∀u ∈ D(A),

whereγ > 0, and Young inequality we have

1

2

d

dt‖uσ (s)‖2L2

+ γ ‖uσ (s)‖pW1,p 6 ‖l(s)‖L2‖uσ (s)‖L2

6 1

2DC + 1

2D‖uσ (s)‖pL2

.

The continuous injectionsW 1,p0 (�) ⊂ Lp(�) ⊂ L2(�) allow us to chooseD > 0

such thatD‖uσ (s)‖pL26 γ ‖uσ (s)‖pW1,p . Hence, integrating over(0,1) we obtain

‖uσ (1)‖2L2+ γ

∫ 1

0‖uσ (s)‖pW1,p 6

1

DC + ‖u0‖2L2

. (22)

Recall thatϕ(u) = 1p

∑ni=1 ‖ ∂

∂xiu‖pLp , if u ∈ W 1,p

0 (�). Consider first the case

whereu0 ∈ D(ϕ) = W 1,p0 (�). In this case sincel(·) ∈ L2(0,1;L2(�)) it is known


(see [5, p. 189]) thatϕ(u(t)) is absolutely continuous in[0,1] and dds ϕ(u(s)) =

(∂ϕ(u(s)), du(s)ds ), a.e. on(0,1). Further, multiplying (21) bys duσ

ds we have

s

∥∥∥∥ d

dtuσ (s)

∥∥∥∥2

L2

+ s d

dsϕ(u(s)) 6 s‖l(s)‖L2

∥∥∥∥ d

dsuσ (s)

∥∥∥∥L2

6 1

2s‖l(s)‖2L2

+ 1

2s

∥∥∥∥ d

dtuσ (s)

∥∥∥∥2

L2

.

Integrating by parts over(0,1) we get∫ 1

0

1

2s

∥∥∥∥ d

dtuσ (s)

∥∥∥∥2

L2

ds + ϕ(uσ (1)) 6∫ 1

0ϕ(u(s))ds + 1

4C.

Using the fact that the norms‖u‖W1,p and(∑n

i=1 ‖ ∂∂xiu‖pLp)1/p are equivalent in

W1,p0 (�) and (22) we have

ϕ(uσ (1)) 6 α(

1

DC + ‖u0‖2L2

)+ 1

4C. (23)

Let now consider the general caseu0 ∈ L2(�). We takeun0 → u0 with un0 ∈B(0, R0). From [24, Theorem 3.1] we obtain the existence of a sequenceun(·) =I (un0)ln(·), ln(s) ∈ Fσ(s)(un(s)), such thatun→ uσ in C([0,1], L2(�)). Hence by(23) and using the lower semicontinuity ofϕ we obtain

ϕ(uσ (1)) 6 lim inf ϕ(un(1)) 6 α(

1

DC + ‖u0‖2L2

)+ 1

4C.

This implies that the setU+(1,0, B(0, R0)) is bounded in the spaceW 1,p(�).Since the injectionW 1,p(�) ⊂ L2(�) is compact, the setK is compact.

Further, letB ∈ B(X) be arbitrary. Lemma 20 implies that for anyτ ∈ R+ thereexists somet1(τ, B) for whichU+(t, τ, B) ⊂ B(0, R0), ∀t > t1. Then∀σ ∈ 6,∀t = s + 1, wheres > t1(τ, B), we have by Proposition 4

Uσ(t, τ, B) = Uσ(1+ s, s, Uσ (s, τ, B))= UT (s)σ (1,0, Uσ (s, τ, B)) ⊂ UT (s)σ (1,0, B(0, R0)) ⊂ K.

(24)

2COROLLARY 4. The family of semiprocessesUσ is uniformly aymptotically up-per semicompact.

We have proved that the family of semiprocesses generated by (13) satisfies allconditions of Theorem 3. We can then state the main result of this paper:

THEOREM 4. If (F1)–(F4)hold for f : R+ × R → Cv(R) and g ∈ L∞(R+,L2(�)), then the family of semiprocessesUσ has the uniform global compactattractor26.


Remark 4.If we consider the inclusion

∂u

∂t−

n∑i=1

∂

∂xi

(∣∣∣∣ ∂u∂xi∣∣∣∣p−2

∂u

∂xi

)∈f (t, u)− f1(u)+ g(t), on�× (τ, T ),

u|∂� = 0,u|t=τ = uτ ,

wheref1: R → 2R is a maximal monotone map andD(f1) = R, then the sameresult can be proved. We have just to replace in (F4) the functionf by f − f1 anddefine the operator−A in the following way:

−A(u) ={y ∈ L2(�) : y(x) ∈ −

n∑i=1

∂

∂xi

(∣∣∣∣ ∂u∂xi∣∣∣∣p−2

∂u

∂xi

)+

+ f1(u(x)),a.e. on�

},

D(−A) = {u ∈ W 1,p0 (�) : −A(u) ∈ L2(�), ∃ξ ∈ L2(�)

such thatξ(x) ∈ f1(u(x)) a.e.},which is the subdifferential of the proper convex lower semicontinuous function

ϕ(u) =

1

p

n∑i=1

∫�

∣∣∣∣ ∂u∂xi∣∣∣∣pdx +

∫�

∫ u

0f1(s)ds,

if u ∈ W 1,p0 (�),

∫ u

0f1(s)ds ∈ L1(�),

+∞, otherwise.

The operator−∂ϕ generates also a compact semigroup (see [25] or [26]). Smallchanges are needed in the proofs of Lemma 20 and Proposition 7 in order to obtain(20) and (22), respectively. We have to use the existence ofα ∈ R, β ∈ L2(�)

such that∀u ∈ D(−A),∀y ∈ L2(�), y(x) ∈ f1(u(x)) a.e.,

(y, u) > ψ(u)− ψ(0) > α + (β, u),whereψ(u) = ∫

�

∫ u0 f1(s)ds, if

∫ u0 f1(s)ds ∈ L1(�), and argue as before. The

last inequality follows from the fact thaty ∈ ∂ψ(u) andψ is bounded below by anaffine function (see [5]).

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on global attractors of multivalued semiprocesses and nonautonomous evolution inclusions

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