Structure and regularity of the global attractor of a reaction-di{\S}usion equation with non-smooth nonlinear term

In this paper we study the structure of the global attractor for a reaction- di{\S}usion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.


Introduction
In this paper we study the structure of the global attractor of a reaction-diffusion equation in which the nonlinear term satisfy suitable growth and dissipative conditions, but there is no condition ensuring uniqueness of the Cauchy problem (like e.g. a monotonicity assumption). Such equation generates in the general case a multivalued semiflow having a global compact attractor (see [7], [14]). Also, it is known [12] that the attractor is the union of all bounded complete trajectories of the semiflow.
If we study the global attractor in more detail we can get a better understanding of the dynamics of the semiflow by restricting our attention inside the attractor. In particular, it is important to establish the relationship between the attractor and the stable and unstable manifolds of the set of stationary points. In the single-valued case, when for example the nonlinear term is a polynomial or its derivative satisfies some assumptions, it is well known [3], [4], [22] that the attractor is the unstable manifold of the set of stationary points. Moreover, if the set of stationary points is discrete, then it is the union of all heteroclinic orbits connecting the stationary points. In more particular parabolic equations the structure of the attractor has been completely understood by obtaining a list of which stationary points are joined to each other. This the case of the famous Chafee-Infante equation [11] or general scalar parabolic equations under suitable restrictions [8], [9], [19], [20]. Also, in [10] similar results are obtained for retarded differential equations.
In [2] the structure of the global attractor of a scalar parabolic differential inclusion generating a multivalued semiflow is studied, obtaining a partial description about which pairs of stationary points are joined. As far as we know this is the only published result about the heteroclinic connections between stationary points in the multivalued case.
We prove in this paper that the global attractor of a rather general reaction-diffusion equation without uniqueness can be described in terms of either M + (R) or M − (R), that is, the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.
In Section 4 it is proved that the attractor of the multivalued semiflow generated by weak solutions in the phase space L 2 (Ω) is the closure of M − (R). Also, M + (R) is contained in the attractor, and coincides with it when uniqueness takes place for regular initial data.
In Section 5 we consider the multivalued semiflow generated by regular solutions, which are the weak solutions which become strong after an arbitrary small time. We prove first the existence of a global attractor in the phase space L 2 (Ω) which is, moreover, compact in H 1 0 (Ω). After that we establish that it coincides with the unstable manifold of the set of stationary points, and also with the stable one when we consider only bounded complete solutions.
In Section 6 we consider the multivalued semiflow generated by strong solutions. We prove first the existence of a global attractor in the phase space H 1 0 (Ω) and that the attractors of the regular and strong cases coincide. Finally, the same result about the structure of the attractor as in the case of regular solutions is given.
We denote by A the operator −∆ with Dirichlet boundary conditions, so that D (A) = H 2 (Ω) ∩ H 1 0 (Ω) . As usual, denote the first eigenvalue of A by λ 1 .
Denote F (u) = u 0 f (s)ds. From (2) we have that lim inf |u|→∞ f (u) u = ∞, and for some The function u ∈ L 2 loc (0, +∞; H 1 0 (Ω)) L 4 loc (0, +∞; L 4 (Ω)) is called a weak solution of (1) on (0, +∞) if for all where · , (·, ·) are the norm and the scalar product in L 2 (Ω). We denote by · X the norm in the abstract Banach space X, whereas (·, ·) H will be the scalar product in the abstract Hilbert space H. Also, P (X) will be the set of all non-empty subsets of X.
It is well known [1,Theorem 2] or [6, p.284] that for any u 0 ∈ L 2 (Ω) there exists at least one weak solution of (1) with u(0) = u 0 (and it may be non unique) and that any weak solution of (1) belongs to C ([0, +∞); L 2 (Ω)). Moreover, the function t → u(t) 2 is absolutely continuous and We define Definition 1 Let X be a complete metric space. The multivalued map G : R + × X → P (X) is a multivalued semiflow (m-semiflow) if: Definition 2 The set Θ ⊂ X is called a global attractor of G, if: where 3. It is mininal, gthat is, for any closed set C satisfying (7) it holds Θ ⊂ C.
The map G defined by (6) is a strict multivalued semiflow which possesses a global compact invariant connected attractor [7], [14], [15]. Our aim is to give a characterization of the attractor. First we shall define complete trajectories for problem (1).
In the following section (see Theorem 13) it will be shown that this is equivalent to the following: Let K be the set of all bounded (in the L 2 (Ω) norm) complete trajectories. It is known that the global attractor of G is the union of all bounded complete trajectories. We recall that the global attractor Θ is called stable if for any ǫ > 0 there exists δ > 0 such that Theorem 4 [12, Theorem 3.18] Under conditions (2) the m-semiflow (6) has a global compact invariant attractor Θ ⊂ L 2 (Ω) which is connected, stable and Let R be the set of all stationary points of (1), i.e., the points u ∈ H 1 0 (Ω) such that and It is known [22], [4, p.106], [3] that under additional conditions on f , like G is a single-valued semigroup, the set Θ is bounded in H 2 (Ω) H 1 0 (Ω) and Moreover, in [4, p.106] it is proved that M + (R) is the unstable set of R. We note that under conditions (11) attraction takes place in H 1 0 (Ω). We observe that in this case an equivalent definition of the set M + (R) is the following as for every complete trajectory γ (·) ∈ F as in (10) we have that the set ∪ t∈R γ (t) is bounded, so that the inclusion γ (·) ∈ K follows.
The aim of our paper is to obtain something like (12) for K + under the general conditions (2). Moreover, taking a more regular set of solutions we will show that the equality (12) holds.

About some properties of complete trajectories and fixed points of m-semiflows
We will prove in this section some useful properties of fixed points and complete trajectories for abstract multivalued semiflows. Consider a complete metric space X and let Let R ⊂ W + be some set of functions such that the following conditions hold: (K1) For any x ∈ X there exists ϕ ∈ R such that ϕ (0) = x.
We define now the concept of fixed point and complete trajectory for R.
The set of all fixed points will be denoted by R R . The map γ : R → X is called a complete trajectory of R if We will show that the fixed points of R coincide with the stationary points of G under assumptions (K1) − (K4).
By conditions (K3) − (K4) we have that u n belongs to R and the existence of u ∈ R and a subsequence u n k such that We will show further the relation between complete trajectories of R and G.
Lemma 8 If (K1) − (K4) hold, then the map γ : R → X is a complete trajectory of R if and only if γ (t + s) ∈ G (t, γ (s)) for all s ∈ R and t ≥ 0.
Let K be the set of all bounded complete trajectories of R. Now we will establish equality (8) in the abstract setting.
Proof. Let γ (·) ∈ K. We note that , where u n ∈ R and u n (0) ∈ Θ. Consider the functions v 0 n (·) = u n (· + t n ), which belong to R. In view of (K4) there exist v 0 (·) ∈ R with v 0 (0) = z and a subsequence (denoted again by u n ) such that v 0 , we obtain that v 0 (t) ∈ Θ for all t ≥ 0. Let us take a sequence t j → +∞ such that t 0 = 0 < t j < t j+1 for any j ∈ N. Consider now the sequence of functions v 1 n (·) = u n (· + t n − t 1 ), which belong to R. By (7) it is clear that (up to a subsequence) v 1 n (0) is convergent in X. As before there exist then v 1 (·) ∈ R and a subsequence (denoted again by v n ) such that v 1 n (t) → v 1 (t) in X for all t ≥ 0. Also, v 1 (t) ∈ Θ and v 1 (t + t 1 ) = v 0 (t) for all t ≥ 0. In this way we can define inductively a sequence v j (·) ∈ R such that v j (t) ∈ Θ and v j (t + t j − t j−1 ) = v j−1 (t) for all t ≥ 0 and j ∈ N. We define the function v (t) by taking for all t ∈ R the commom value at t of the functions v j (·). Namely, for any j such that t ≥ −t j we put The first equality is proved. The second one is obvious from the definition of a complete trajectory.
The last theorem is also true if we replace (K4) by (K3) .
Theorem 10 Assume that (K1) − (K3) hold and that G possesses a compact global attractor Θ. Then Proof. As in the proof of Theorem 9 we obtain that B γ = ∪ t∈R γ (t) ⊂ Θ for any γ (·) ∈ K.
The second one is obvious from the definition of a complete trajectory.

Remark 11
The map γ : R → X is a complete trajectory of G if (13) holds. Let K G be the set of all bounded complete trajectories of G. If (K1) − (K4) hold, then by Lemma 8 we have K G = K and equality (14) is the same as If either (K3) or (K4) fails to be true, then we can say only that K ⊂ K G . Nevertheless, if either (K3) or (K4) holds, then (15) is still true. Indeed, Theorems 9, 10 imply that and as in the proof of Theorem 9 we obtain that B γ = ∪ t∈R γ (t) ⊂ Θ for any γ (·) ∈ K G , so that (15) holds.
If both (K3) and (K4) fail, then equality (15) can be obtained under some assumptions on G. Namely, in [12, Lemmas 2.25 and 2.27] it is shown that (15) holds if either G is strict or the following condition is true: for any sequence ϕ n : R + → X satisfying (13), for s, t ≥ 0, and ϕ n (0) → ϕ 0 in X, there exists a subsequence and ϕ : R + → X satisfying (13) for s, t ≥ 0 such that We shall apply these results to the set K + generated by the weak solutions of (1). We note that in view of Lemmas 3 and 15 in [14] (see also [12, Theorems 3.11 and 3.18]) assumptions (K1) − (K4) are satisfied for K + . Moreover, the sets of stationary points R of problem (1) coincides with the set R R = R K + given in Definition 6.
Proof. Let u 0 ∈ R K + . Then u (t) ≡ u 0 belongs to K + . Therefore, u (·) satisfies (4), so that (9) holds. Conversely, let v ∈ R. Then it is obvious that v (t) ≡ v 0 is a weak solution, so that is belongs to K + . Then Lemmas 7, 8 and Theorem 9 imply the following result.
Theorem 13 Let (2) hold. Then the set of weak solutions K + of (1) satisfies:

Structure of the global attractor for weak solutions
In this section we will study the structure of the global attractor generated by weak solutions of equation (1).
First, let us prove some regularity properties of the stationary points.

Lemma 14
Under conditions (2) the set R of solutions of the problem is nonempty, compact in L 2 (Ω), and bounded in H 1 0 (Ω) ∩ H 2 (Ω). Proof is monotone and continuous. Also, from (2) we can obtain that f : . Hence, L is pseudomonotone, coercive and bounded, so that a classical theorem of Brezis (see [23]) implies that R = ∅. It is clear also that it is weakly compact in H 1 0 (Ω) and therefore compact in L 2 (Ω). We remark that R is bounded in H 1 0 (Ω), as L : and (2), together with the continuous imbedding H 1 0 (Ω) ⊂ L 6 (Ω) , imply For initial data in H 1 0 (Ω) we shall obtain the existence of more regular solutions for (1).
Proof. We take as in [6, p.281] the Galerkin approximations using the basis of eigenfunctions {w j (x), j ∈ N}, of the Laplace operator with Dirichlet boundary conditions. Let X m = {w 1 , ..., w m } and let P m be the orthogonal projector from L 2 (Ω) onto X m . Then u m (t, x) = m j=i a j,m (t) w j (x) will be a solution of the system of ordinary differential equations It is proved in [6, p.281] that passing to a subsequence u m converges to a weak solution u of (1) weakly star in L ∞ (0, T ; L 2 (Ω)), weakly in L 4 (0, T ; L 4 (Ω)) and weakly in So from the Poincaré inequality we obtain where R j > 0. By the choise of the special basis we have that so that (18) holds and u mt → u t weakly in L 2 (0, T ; L 2 (Ω)). Thus from the Ascoli-Arzelà On the other hand, for any t ≥ 0 up to a subsequence u mn (t) → a weakly in H 1 0 (Ω). But u mn (t) → u(t) in L 2 (Ω), so that a = u(t) and with the weak topology. Moreover, the equality ∆u = u t + f (u)−h and (2), (17), (18) imply that u ∈ L 2 loc (0, +∞; D (A)). Thus, by standard results [21, p.102], we obtain that u ∈ C([0, +∞), H 1 0 (Ω)).
Now we are ready to prove the main result of this section about the structure of the global attractor. From now on for any A ⊂ L 2 (Ω) we will denote by A its closure in L 2 (Ω) .

Remark 17
Even in the case of uniqueness we cannot use the Lyapunov function method as in [22], because we know nothing about the boundedness of Θ in H 2 (Ω) H 1 0 (Ω). However, it is possible to use the Lyapunov function if the attractor is compact in H 1 0 (Ω), as we will see in the next section.

So, τ <0
Γ τ is connected and compact. As for all ε > 0 there exists T < 0 such that For M − (R) the proof is similar.
We finish this section with a reularity result of the global attractor in the space L ∞ (Ω).

Remark 21
The set M + (R) can be used in order to study properties of the global attractor as the fractal dimension. Let us consider an example which shows that a finite estimate of the fractal dimension of the global attractor for problem (1) is not preserved under small, but unregular perturbations (even in the single-valued case). Let h(x) ≡ 0, f k (u) = u 3 − k − 1 2 sin (k · u). Then for any k ∈ Z f k satisfies (2) with constants which do not depend on k. In this case z = 0 ∈ R, G k (t, u 0 ) = S k (t)u 0 is a single-valued semigroup and due to [22, p.496] z = 0 is a hyperbolic point if λ i = k Thus, if k → ∞, then for the attractors Θ k we obtain So, under conditions (2), we can have arbitrary large dimension of the global attractor, although f k (u) is a small perturbation of f 0 (u) = u 3 , for which it is easy to see that R ≡ {0}, so dim Θ 0 = 0.

Existence and structure of the global attractor for regular solutions
We shall prove in this section that the equality holds if we consider more regular solutions than in the previous section. The function u ∈ L 2 loc (0, +∞; H 1 0 (Ω)) L 4 loc (0, +∞; L 4 (Ω)) is called a regular solution of (1) on (0, +∞) if for all T > 0, v ∈ H 1 0 (Ω) and η ∈ C ∞ 0 (0, T ) we have and On the other hand, from (2) we get Then the equality ∆u = u t + f (u) − h and (27)-(28) imply that for any regular solution u.

Lemma 23
Let g, w, y be three positive integrable functions on (t 0 , +∞) such that y ′ is locally integrable on (t 0 , +∞) and such that where a i > 0. Then y (t + r) ≤ a 3 r + a 2 e a 1 .
We note also that the compact embedding H 1 0 (Ω) ⊂ L 2 (Ω) implies that for any t > 0 the sequence u n (t) is precompact in L 2 (Ω). Hence, applying the Ascoli-Arzelà theorem we obtain, passing to a subsequence and using a diagonal argument, that there exists a function u : [0, +∞) → L 2 (Ω) such that for all 0 < r < T, u n → u weakly star in L ∞ r, T ; u n → u in C([r, T ], L 2 (Ω)), u n → u weakly in L 2 (r, T ; D (A)) , u n t → u t weakly in L 2 r, T ; L 2 (Ω) .
(t, x), and then f (u n ) → f (u) weakly in L (Ω) (see [17, p.12]). In a standard way we can check then that u (·) is a weak solution of (1). Moreover, by the previous arguments it is clear that u is a regular solution.
The set K + r satisfies conditions (K1) − (K2), so that by Lemma 5 G r is a multivalued semiflow.
Remark 24 In this case we are not able to prove that the semiflow G r is strict. The reason is that if we take u 1 , u 2 ∈ K + r with u 2 (0) = u 1 (s) and concatenate them, that is, then u is a weak solution of (1), but we cannot state that it is regular, as properties (27)-(28) can fail now. Hence, condition (K3) is not known to be true.
We shall obtain further some properties of the semiflow G r .
Lemma 25 Assume that (2) holds. Let {u n } ⊂ K + r be a sequence such that u n (0) → u 0 weakly in L 2 (Ω). Then there exists a subsequence (denoted again by u n ), and a regular solution of (1) u ∈ K + r satisfying u(0) = u 0 , such that for any sequence of times t n ≥ 0 such that t n → t 0 we have u n (t n ) → u(t 0 ) weakly in L 2 (Ω). Also, if t 0 > 0, then u n (t n ) → u(t 0 ) strongly in H 1 0 (Ω). Moreover, if u n (0) → u 0 strongly in L 2 (Ω), then for t n ց 0 we get u n (t n ) → u 0 strongly in L 2 (Ω).
Proof. Arguing as in the proof of Theorem 22 we obtain the existence of a subsequence of u n and a weak solution u of (1) with u (0) = u 0 such that the convergences (52), (53), (54), (55) hold. Hence, u ∈ K + r . It follows that if t 0 > 0 and t n → t 0 , then u n (t n ) → u(t 0 ) strongly in L 2 (Ω) and weakly in H 1 0 (Ω). We shall prove that u n (t n ) → u(t 0 ) strongly in H 1 0 (Ω). Let t n , t 0 ∈ (r, T ). Multiplying (1) by u n t and using (40) we obtain 1 2 Theorem [17] imply that J n (t) → J (t) for a.a. t ∈ (r, T ). Take r < t m < t 0 such that t m → t 0 and J n (t m ) → J n (t m ) for all m. Then if t n ≥ t m , so that for any ε > 0 there exist m (ε) and N (m) such that J n (t 0 )−J (t 0 ) ≤ ε if n ≥ N. Then lim sup J (t n ) ≤ lim sup J (t 0 ), so that lim sup u n (t) 2 H 1 0 (Ω) , so that u n (t n ) → u(t 0 ) strongly in H 1 0 (Ω). Finally, let u n (0) → u 0 strongly in L 2 (Ω). In view of (43) we have for some C > 0, so that the functions J n (t) = u n (s) 2 + Ct, J(t) = u (s) 2 + Ct, are continuous and non-increasing for t ≥ 0. Hence, so that lim sup J (t n ) ≤ lim sup J (0) and by the same argument as before we obtain that u n (t n ) → u 0 strongly in L 2 (Ω).
By standar arguments from Lemma 25 the following result follows.
Corollary 27 Let (2) hold. Then the multivalued semiflow G r has compact values and the map u 0 → G r (t, u 0 ) is upper semicontinuous for all t ≥ 0, that is, for any neighborhood Lemma 28 Let (2) hold. Then the ball , is a compact absorbing for G r , that is, for any set B bounded in L 2 (Ω) there exists T (B) such that Proof. The fact that B 0 is absorbing follows from (30) by taking r = 1. Since B 0 is closed in L 2 (Ω) and bounded in H 1 0 (Ω), the compacity in L 2 (Ω) follows.
Now we are ready to prove the existence of a global compact attractor.
Theorem 29 Let (2) hold. Then the multivalued semiflow G r posseses a global compact attractor Θ r . Moreover, for any set B bounded in L 2 (Ω) we have and also that Θ r is compact in H 1 0 (Ω) .
Proof. The existence of a global compact attractor Θ r follows from Corollary 27, Lemma 28 and Theorem 4 in [18]. Let now suppose that (56) is not true. Then there exists ε > 0 and a sequence y n ∈ G r (t n , B) with t n → +∞ such that dist H 1 0 (Ω) (y n , Θ r ) > ε for all n. There exists y ∈ Θ r and a subsequence y n k such that y n k → y in L 2 (Ω). In view of Lemma 25 the set G r (t, B) is precompact in H 1 0 (Ω) for any t > 0 and any bounded set in L 2 (Ω). Therefore, from y n k ∈ G r (1, G r (t n − 1, B)) ⊂ G r (1, B 0 ) for n great enough, we obtain that y n k → y in H 1 0 (Ω), which is a contradiction. Finally, Θ r ⊂ G r (1, Θ r ) implies, by the same reason, that Θ r is precompact in H 1 0 (Ω). Moreover, since Θ r is closed in L 2 (Ω), it is closed in H 1 0 (Ω), as well, so that Θ r is compact in H 1 0 (Ω) .
The map γ : R → L 2 (Ω) is called a complete trajectory of K + r if γ (· + h) | [0,+∞) ∈ K + r for any h ∈ R. The set of all complete trajectories of K + r will be denoted by F r . Let K r be the set of all complete trajectories which are bounded in L 2 (Ω), and let K 1 r be the set of all complete trajectories which are bounded in H 1 0 (Ω) .
Further, we shall prove that Θ r is the union of all points lying in a bounded complete trajectory, that is, Theorem 32 Let (2) hold. Then equalities (57) hold true.
Theorem 37 Under conditions (2) it holds Moreover, Proof. We can prove this theorem arguing in a rather similar way as in Theorem 16. However, we shall prove it using the method of the Lyapunov function, as in [4], [22]. Let z ∈ Θ r and let u ∈ K r be such that u(0) = z. We note that the energy function E (u (t)), t > 0, given in (36) is nonincreasing and bounded below (which follows easily from (3)) for any u ∈ K + r . Hence, E (u (t)) → l, as t → +∞, for some l ∈ R. Suppose that there exist ε > 0 and a sequence u (t n ), t n → +∞, such that In view of Theorem 29 we have that Θ r is compact in H 1 0 (Ω), and then we can take a converging subsequence (denoted again u (t n )) for which u (t n ) → y in H 1 0 (Ω), where t n → +∞. Since the function E : H 1 0 (Ω) → R is continuous, we obtain that E (y) = l. We shall prove that y ∈ R. Fix t > 0. In view of Lemma 25 there exists v ∈ K + r and a subsequence of v n (·) = u (· + t n ) (denoted v n again) such that v (0) = y and v n (t) → v (t) = z in H 1 0 (Ω). Hence, E (v n (t)) → E(z) implies that E (z) = l. We note that v (·) satisfies (36) for all 0 ≤ s ≤ t, so that This implies that v r (r) = 0 for a.a. r, and therefore y ∈ R. Hence, we obtain a contradiction. Thus, Θ r ⊂ M − r (R). The converse inclusion is obvious from Theorem 32, so that Θ r = M − r (R). On the other hand, we observe that for any u ∈ F r equality (36) is satisfied for all −∞ < s ≤ t. Let z ∈ Θ r and let u ∈ K r = K 1 r be such that u(0) = z. In view of (3) the function (F (u (t)) , 1) is bounded above. Hence, E (u (t)) → l, as t → −∞, for some l ∈ R. As before, suppose that there exist ε > 0 and a sequence u (t n ), t n → ∞, such that dist L 2 (Ω) (u(−t n ), R) > ε, and we have that u (−t n ) → y in H 1 0 (Ω), E (y) = l. Also, for a fixed t > 0 there exists v ∈ K + r and a subsequence of v n (·) = u (· − t n ) (denoted v n again) such that v (0) = y and v n (t) → v (t) = z in H 1 0 (Ω). Hence, E (v n (t)) → E(z) implies that E (z) = l and therefore, arguing as before, y ∈ R, which is a contradiction. As before, we obtain that Θ r = M + r (R). Finally, let us prove that the convergence takes place in H 1 0 (Ω). Let us suppose that there exist ε > 0 and a sequence u (t n ), t n → +∞, such that dist H 1 0 (Ω) (u(t n ), R) > ε.
In view of dist L 2 (Ω) (u(t n ), R) → 0 and the compacity of R, we can extract a subsequence u(t n k ) such that u(t n k ) → u ∈ R in L 2 (Ω). It follows from the compacity of Θ r in H 1 0 (Ω) that in fact u(t n k ) → u ∈ R in H 1 0 (Ω), which is a contradiction. Hence, the first part of (61) holds. The second one is proved in the same way. 6 An attractor in H 1 0 (Ω). Existence and structure of the global attractor for strong solutions In this section we shall define a semiflow in the phase space H 1 0 (Ω). For this aim we introduce now a stronger concept of solution for (1).
Lemma 40 Assume that (2) holds. Let {u n } ⊂ K + s be a sequence such that u n (0) → u 0 weakly in H 1 0 (Ω). Then there exists a subsequence (denoted again by u n ), and a strong solution of (1) u ∈ K + s satisfying u(0) = u 0 , such that for any sequence of times t n ≥ 0 such that t n → t 0 we have u n (t n ) → u(t 0 ) weakly in H 1 0 (Ω). Also, if t 0 > 0, then u n (t n ) → u(t 0 ) strongly in H 1 0 (Ω). Moreover, if u n (0) → u 0 strongly in H 1 0 (Ω), then for t n ց 0 we get u n (t n ) → u 0 strongly in H 1 0 (Ω).
Corollary 42 Let (2) hold. Then the multivalued semiflow G s has compact values and the map u 0 → G s (t, u 0 ) is upper semicontinuous for all t ≥ 0.
We prove further the existence of a global compact attractor.
Theorem 43 Let (2) hold. Then the multivalued semiflow G s posseses a global compact invariant attractor Θ s .
Proof. Since G s (t, u 0 ) ⊂ G r (t, u 0 ), the ball B 0 defined in Lemma 28 is absorbing for G s .
Also, the operator G s (t, ·) is compact for t > 0 by Lemma 40, so that K = G s (1, B 0 ) is a compact absorbing set. Then using Corollary 42 the existence of a global compact minimal attractor Θ s follows from Theorem 4 in [18]. As G s is strict, it follows from Remark 8 in [18] that G s (t, Θ s ) = Θ s for all t ≥ 0.
We will prove further that in fact the attractors Θ s and Θ r coincide.