ROBUSTNESS OF DYNAMICALLY GRADIENT MULTIVALUED DYNAMICAL SYSTEMS

. In this paper we study the robustness of dynamically gradient multivalued semiﬂows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a diﬀerential inclusion studied in [3], proving that the weak solutions of these problems generate a dynamically gradient multivalued semiﬂow with respect to suitable Morse sets.

1. Introduction. One of the main goals of the theory of dynamical systems is to characterize the structure of global attractors. It is possible to find a wide literature about this problem for semigroups; however, it has been recently when new results in this direction for multivalued dynamical systems have been proved [3], [13], [14].
In this sense, the theory of Morse decomposition plays an important role. In fact, the existence of a Lyapunov function, the property of being a dynamically gradient semiflow and the existence of a Morse decomposition are shown to be equivalent for multivalued dynamical systems in [9].
In this work we show under suitable assumptions that a dynamically gradient multivalued semiflow is stable under perturbations, that is, the family of perturbed multivalued semiflows remains dynamically gradient.
For a fixed dynamically gradient multivalued semiflow with a global attractor we also analyze the rearrangement of a pairwise disjoint finite family of isolated weakly invariant sets, included in the attractor, in such a way that the dynamically gradient property is satisfied in the stronger sense of [16].
These results extend previous ones in the single-valued framework in [7,1,2] to the case where uniqueness of solution does not hold. Additionally, it is worth saying that the multivalued semiflows here are not supposed to be general dynamical systems as in [16], where a robustness theorem for Morse decompositions of multivalued dynamical systems is also proved under a suitable continuity assumption.
We also apply this general robustness theorem in order to show that a family of Chafee-Infante problems approximating a differential inclusion is dynamically gradient if it is close enough to the original problem.
This paper is organized as follows.
Firstly, we introduce in Section 2 basic concepts and properties related to fixed points, complete trajectories and global attractors. In this way, we are able to present in Section 3 the main result about robustness of dynamically gradient multivalued semiflows. Further, in Section 4 we prove a theorem which allows us to reorder the family of weakly invariants sets, thus establishing an equivalent definition of dynamically gradient families.
Afterwards, we consider a Chafee-Infante problem in Section 5, where the equivalence of weak and strong solutions is established. Once the set of fixed points is analyzed, we consider a family of Chafee-Infante equations, approximating the differential inclusion tackled in [3]. We check that this family of Chafee-Infante equations verifies the hypotheses of the robustness theorem in order to obtain, therefore, that the multivalued semiflows generated by the solutions of the approximating problems are dynamically gradient if this family is close enough to the original one.
2. Preliminaries. Consider a metric space (X, d) and a family of functions R ⊂ C(R + ; X). Denote by P (X) the class of nonempty subsets of X. Then, define the multivalued map G : R + × X → P (X) associated with the family R as follows In this abstract setting, the multivalued map G is expected to satisfy some properties that fit in the framework of multivalued dynamical systems. The first concept is given now, although a more axiomatic construction will be provided below. Definition 1. A multivalued map G : R + × X → P (X) is a multivalued semiflow (or m-semiflow) if G(0, x) = x for all x ∈ X and G(t + s, x) ⊂ G(t, G(s, x)) for all t, s ≥ 0 and x ∈ X. If the above is not only an inclusion, but an equality, it is said that the m-semiflow is strict.
In order to obtain a detailed characterization of the internal structure of a global attractor, we introduce an axiomatic set of properties on the set R (see [4] and [13]).
The set of axiomatic properties that we will deal with is the following.
From now on (K1)-(K2) are always satisfied and G will be the multivalued semiflow associated to R.
Once a multivalued semiflow is defined, we recall the concepts of invariance and global attractor, with evident differences with respect to the single-valued case.

Definition 3.
A set B ⊂ X is said to be negatively invariant if B ⊂ G(t, B) for all t ≥ 0, and strictly invariant (or, simply, invariant) if the above relation is not only an inclusion but an equality.
The set B is said to be weakly invariant if for any x ∈ B there exists a complete trajectory γ of R contained in B such that γ(0) = x. We observe that weak invariance implies negative invariance. Definition 4. A set A ⊂ X is called a global attractor for an m-semiflow if it is negatively semi-invariant and it attracts all attainable sets through the msemiflow starting in bounded subsets, i.e., dist X (G(t, B), A) → 0 as t → ∞, where dist X (A, B) = sup a∈A inf b∈B d(a, b).

Remark 1.
A global attractor for an m-semiflow does not have to be unique, nor a bounded set (see [24] for a non-trivial example of an unbounded non-locally compact attractor). However, if a global attractor is bounded and closed, it is minimal among all closed sets that attract bounded sets [19]. In particular, a bounded and closed global attractor is unique.
Several properties concerning fixed points, complete trajectories and global attractors are summarized in the following results [13]. Lemma 1. Let (K1)-(K2) be satisfied. Then every fixed point (resp. complete trajectory) of R is also a fixed point (resp. complete trajectory) of G.
If R fulfills (K1)-(K4), then the fixed points of R and G coincide. Besides, a map γ : R → X is a complete trajectory of R if and only if it is continuous and a complete trajectory of G.
The standard well-known result in the single-valued case for describing the attractor as the union of bounded complete trajectories reads in the multivalued case as follows.
Theorem 1. Consider R satisfying (K1) and (K2), and either (K3) or (K4). Assume also that G possesses a compact global attractor A. Then where K denotes the set of all bounded complete trajectories in R.
Now we recall the definitions of some important sets in the literature of dynamical systems. Let B ⊂ X and let ϕ ∈ R. We define the ω−limit sets ω(B) and ω(ϕ) as follows: ω(B) ={y ∈ X : there are sequences t n → ∞, y n ∈ G(t n , B) such that y n → y}, ω(ϕ) ={y ∈ X : there is a sequence t n → ∞ such that ϕ(t n ) → y}.
If γ is a complete trajectory of R, then the α−limit set is defined by Some useful properties of these sets [4,Lemma 3.4] are summarized in the following lemma.
In order to give a more detailed description of the internal structure of the attractor under special cases, additional concepts are required.
Definition 5. Consider the m-semiflow G associated with R.
1. We say that S = {Ξ 1 , . . . , Ξ n } is a family of isolated weakly invariant sets if there exists For an m-semiflow G on (X, d) with a global attractor A and a finite number of weakly invariant sets S, a homoclinic orbit in A is a collection {Ξ p(1) , . . . , Ξ p(k) } ⊂ S and a collection of complete trajectories {γ i } 1≤i≤k of R in A such that (putting p(k + 1) := p(1)) 3. We say that the m-semiflow G on (X, d) with the global attractor A is dynamically gradient if the following two properties hold: (G1) there exists a finite family S = {Ξ 1 , . . . , Ξ n } of isolated weakly invariant sets in A with the property that any bounded complete trajectory γ of R in A satisfies for some 1 ≤ i, j ≤ n; (G2) S does not contain homoclinic orbits.
Remark 2. The last definition generalizes the concept of dynamically gradient semigroups (see [7], where they are called gradient-like semigroups) to the multivalued case. Observe that the above definitions are concerned with weakly invariant families, which need not to be unitary sets. This is to deal with the more general concept of generalized gradient-like semigroups [7], in contrast with gradient-like semigroups (when the invariant sets are unitary). Now, we introduce the concept of unstable manifold, that will allow us to describe more precisely the structure of a global attractor of a dynamically gradient msemiflow.
Now the following result, relating the global attractor with unstable manifolds, is standard. The first statement is straightforward to see. The second one, supposing that the global attractor is compact, follows directly from the structure described in Theorem 1 and the definition of dynamically gradient semiflows.
Moreover, assume that R satisfies either (K3) or (K4), and that the global attractor A is compact. Suppose also that the associated m-semiflow G defined in (1) is dynamically gradient. Then 3. Robustness of dynamically gradient m-semiflows. Our first main goal is to prove that a dynamically gradient multivalued semiflow is stable under suitable perturbations, that is, a family of perturbed multivalued semiflows remains dynamically gradient if it is close enough to the original semiflow, generalizing the corresponding result in the single-valued case [7]. This is rigorously formulated in the following theorem.
Proof. Observe that assumption (H5) concerning certain neighborhood V i of Ξ 0 i involves a hyperbolicity condition of G 0 w.r.t. each Ξ 0 i , and as far as (H3) is also assumed, there exist . . , n. By Theorem 1, we have that A η is composed by all the orbits of bounded complete trajectories of R η , K η .
Since G 0 is dynamically gradient, there exists i ∈ {1, . . . , n} such that dist X (γ 0 (t), Ξ 0 i ) → 0 as t → ∞. Therefore, for all r ∈ N, there exist t r and k r such that dist X (ξ k (t r ), Ξ 0 i ) < 1/r for all k ≥ k r . Indeed, this is done as follows: dist X (γ 0 (s), Ξ 0 i ) < 1/r for all s ≥ t r (for some t r , w.l.o.g. t r ≥ r > 1/δ); now, combining this with the uniform convergence on [0, t r ] of ξ k toward γ 0 , the existence of k r follows. However, from (4), there exists t r > t r such that dist X (ξ kr (t), Ξ 0 i ) < δ for all t ∈ [t r , t r ) and dist X (ξ kr (t r ), Ξ 0 i ) = δ. Now we distinguish two cases and we will arrive to the same conclusion in both of them.
Case (1a). Suppose that t r −t r → ∞ as r → ∞ (at least for a certain subsequence).
Since {ξ kr (t r )} is also relatively compact (by (H1), again), and ξ 1 kr (·) = ξ kr (t r +·) is a bounded complete trajectory of R kr , from (H4) we deduce that a subsequence (relabeled the same) is converging on bounded time-intervals of [0, ∞), i.e. γ 1 (t) := lim r→∞ ξ kr (t + t r ) holds for certain γ 1 ∈ R 0 . Moreover, as before, a diagonal argument, using not t r above, but t r − 1, t r − 2, . . . implies that γ 1 can be extended to the whole real line (the function will still be denoted the same; and the convergence holds in bounded time-intervals of R), in particular, by (H1) and (H4), γ 1 ∈ K 0 .
As long as Ξ 0 i is the biggest weakly invariant set contained in . From (H4), there exist a subsequence {ξ 1 kr } and ξ 1 ∈ R 0 with ξ 1 (0) = y such that ξ 1 kr converge towards ξ 1 uniformly in bounded intervals of [0, ∞). In particular, ξ 1 This is exactly the same conclusion we arrived in Case (1a).
Reasoning now with the subsequence {ξ 1 kr }, and proceeding as above, we obtain the existence of Thus, in a finite number of steps we arrive to a contradiction, since G 0 satisfies (G2). Therefore, (4) is absurd, and Step 1 is proved.
Step 2. There exists η 1 > 0 such that for all η < η 1 , any bounded complete trajectory ξ η of R η satisfies that there exist j ∈ {1, . . . , n} and t 1 such that dist X (ξ η (t), Ξ 0 j ) ≤ δ for all t ≤ t 1 . The above claim can be proved analogously as before, and since for any bounded complete trajectory ξ η ∈ K η , by Lemma 2, α(ξ η ) is weakly invariant for G η , and contained in some V j , the 'backward part' of property (G1) of a dynamically gradient m-semiflow will follow immediately.
If not, there exist a sequence η k → 0, with G η k having an homoclinic structure. We may suppose that the number of elements of weakly invariant subsets connected on each homoclinic chain in S η k is the same. Moreover, by assumption (H3) each Ξ η k j is contained in V j for η k small enough and w.l.o.g. the order in the route of the homoclinics visiting the V j sets is the same.
Therefore, for k ≥ k 0 there exist a sequence of subsets If we argue now as in the proof of (G1), we may construct a homoclinic structure of G 0 , getting a contradiction with the fact that the m-semiflow G 0 is dynamically gradient. 4. An equivalent definition of dynamically gradient families. We will give an equivalent definition of dynamically gradient families. For proving the main result in this section we will need a stronger condition than (K4). Namely, we shall consider the following stronger condition: (K4) For any sequence {ϕ n } ⊂ R such that ϕ n (0) → x 0 in X, there exists a subsequence {ϕ n } and ϕ ∈ R such that ϕ n converges to ϕ uniformly in bounded subsets of [0, ∞). As before, let A be the global attractor of the m-semiflow G associated with R.
Remark 4. We have seen that the property of being dynamically gradient for a disjoint family of isolated weakly invariant sets S = {Ξ 1 , . . . , Ξ n } ⊂ A is stable under perturbations. We observe that in the paper [16] a slightly different definition was used for dynamically gradients families. Namely, instead of conditions (G1)-(G2) it is assumed that any bounded complete trajectory γ(·) of R in A satisfies one of the following properties: These assumptions are clearly stronger than (G1)-(G2) and imply that the sets Ξ j are ordered. Our aim is to show that when S is a disjoint family of isolated weakly invariant sets, these conditions are equivalent. For this we will need to introduce the concept of local attractor and its repeller and study their properties.
We say that A ⊂ A is a local attractor in A if for some ε > 0 we have that . Some properties about local attractors and its repeller as well as the proof of the following three lemmas can be found in [9]. Remark 5. Although in [9] the stronger assumption (K4) is assumed, the proof is valid for just (K4).
Lemma 5. Assume that (K1)-(K3), (K4) hold and that a global compact attractor A exists. Then the repeller A * of a local attractor A ⊂ A is weakly invariant and compact.
Lemma 6. Assume that (K1)-(K3), (K4) hold and that a global compact attractor A exists. Let us consider the sequences x k ∈ A, t k → +∞ and ϕ k (·) ∈ R such that ϕ k (0) = x k . Then from the sequence of maps ξ k (·) : one can extract a subsequence converging to some ψ(·) ∈ K uniformly on bounded subsets of R.
In order to prove the equivalent definition of dynamically gradient families, we have to ensure the existence of one local attractor in a family of isolated weakly invariant sets.
Lemma 7. Assume that (K1)-(K3), (K4) hold and that a global compact attractor A exists. Let S = {Ξ 1 , . . . , Ξ n } ⊂ A be a disjoint family of isolated weakly invariant sets. If G is dynamically gradient with respect to S, then one of the sets Ξ j is a local attractor in A.
It is clear that G 1 possesses a global compact attractor, which is the union of all bounded complete trajectories of R 1 , and that G 1 is dynamically gradient with respect to {Ξ 2 , . . . , Ξ n }. Then, again by Lemma 7 we can reorder the sets in such a way that Ξ 2 is a local attractor in Ξ * 1 . Let Ξ * 2,1 be the repeller of Ξ 2 in Ξ * 1 . Then we restrict as before the dynamics to the set Ξ * 2,1 and so on. Hence, we have reordered the sets Ξ j in such a way that Ξ 1 is a local attractor and Ξ j is a local attractor for the dynamics restricted to the repeller of the previous local attractor Ξ * j−1,j−2 for j ≥ 2, and Ξ then we shall prove that i ≤ j. Moreover, if γ(·) is not completely contained in some Ξ k , then i < j.
To finish this section, we recall that the disjoint family of isolated weakly invariant sets S = {Ξ 1 , . . . , Ξ n } ⊂ A is a Morse decomposition of the global compact attractor A if there is a sequence of local attractors ∅ = A 0 ⊂ A 1 ⊂ . . . ⊂ A n = A such that for every k ∈ {1, . . . , n} it holds It is well known [16] that for general dynamical systems conditions 1-2 in Theorem 3 are equivalent to the fact that S generates a Morse decomposition. This fact can be proved also under conditions (K1)-(K3), (K4) [9].
Thus, Theorem 3 implies that under conditions (K1)-(K3),(K4) the family S generates a Morse decomposition if and only if G is dynamically gradient.
where the equality is understood in the sense of distributions.
Finally, if p > 2 by condition (A6)(a) we have Hence, u(·) is a weak solution as well.

Stationary points.
We now focus on the properties of the stationary points. To this end, we have followed the classic procedure from [11] and [12]. Moreover, we have also taken some ideas from [18].
If follows from conditions (A2) and (A3) of f that −∞ ≤ a − < 0 < a + ≤ +∞. Since f is positive on (0, a + ) and negative on (a − , 0), we have that F is strictly increasing on [0, a + ), strictly decreasing on (a − , 0] and F (0) = 0. We consider Then, F has the inverse functions U + : We also define the following functions with domains (0, E + ) and (0, E − ), respectively, with values on [0, ∞): Let us consider v 0 ∈ R and a solution u of Note that the solution of the problem (8) is unique, since f is convex for u < 0 and concave for u > 0, so it is Lipschitz on compact intervals (see [28, p.4] or [10, p.8]).
If we define E = v 2 0 /2, then: On the other hand, the functions τ + , τ − evaluated in E = v 2 0 /2 give us √ 2 the x-time necessary to go from the initial condition u(0) = 0, with initial velocity v 0 , −v 0 respectively, to the point where u (T + (E)) = 0. Indeed, u(x) satisfies . Since u (T + (E)) = 0 for u = U + (E), By symmetry with respect to the u axis, the x−time it takes for u(x) to go from (U + (E), 0) to (0, −v 0 ) is T + (E). By this way, if 2T + (E) = 1, that is, τ + (E) = 1 √ 2 , then u(·) is a solution satisfying the boundary conditions u(0) = u(1) = 0. Applying a similar reasoning for τ − (E), we obtain that u satisfies the boundary conditions if, and only if, E satisfies for some k ∈ N only one of the following conditions: Remark 7. Note that if E satisfies (9) or (10) for a certain k, then u has 2k zeros and if E satisfies (11), then u has 2k + 1 zeros. Our goal is to solve these equations for E as a function of f (0). To this end, we study the properties of τ ± .
In order to obtain solutions of the equations (9), (10) and (11) it is necessary to make a change of variable for the functions τ ± . Given E ∈ (0, E ± ), we put and Hence, du = (2yE/f (u))dy and E − F (u) = E(1 − y 2 ). By this change, we obtain The next results show some properties of these functions.
From [10, p.8] we have that the function f is differentiable almost everywhere in is differentiable as well. Hence, Ey 2 )) .
Recall the change of variable F (u) = Ey 2 . Consider the numerator of α , that is, Then we obtain Since f is strictly concave, if s < u, then f (s) > f (u) (cf. [28, p.5]). As a result, β(u) > 0. In order to finish the proof rigorously, we have to justify the previous calculations. Indeed, from [10, p.5], we have that the function f is absolutely continuous and from [5, p.16], f ∈ L 1 loc . Therefore, α ∈ L 1 loc and α > 0 a.e., which implies that α(E) is strictly increasing and the proof is finished.
Our aim now is to prove that for ε sufficiently small the multivalued semiflow G ε generated by the weak solutions of problem (13) is dynamically gradient. Problem (13) is an approximation of the following problem, governed by a differential inclusion We say that the function u ∈ C([0, T ], L 2 (Ω)) is a strong solution of (15) if 1. u(0) = u 0 ; 2. u(·) is absolutely continuous on (0, T ) and u(t) ∈ H 2 (Ω) ∩ H 1 0 (Ω) for a.e. t ∈ (0, T ); 3. There exists a function g(·) such that g(t) ∈ L 2 (Ω), a.e. on (0, T ), g(t, x) ∈ H 0 (u(t, x)), for a.e. (t, x) ∈ (0, T ) × Ω, and du dt − ∂ 2 u ∂x 2 − g(t) = 0, a.e. t ∈ (0, T ). In this case we put R as the set of all strong solutions such that the map g belongs to L 2 (0, T ; L 2 (Ω)). Conditions (K1)-(K4) are satisfied (cf. [9]) and the map G : R + × L 2 (Ω) → P (L 2 (Ω)) defined by (1) is a strict multivalued semiflow possessing a global compact attractor A 0 (cf. [25]) in L 2 (Ω), which is connected (cf. [26]). The structure of this attractor is studied in [3]. It is shown that there exists an infinite (but countable) number of fixed points . . , and that A 0 consists of these fixed points and all bounded complete trajectories ψ(·), which always connect two fixed points, that is, where z i = 0, z i = v + n or z i = v − n for some n ≥ 1. Moreover, if ψ is not a fixed point, then either z 2 = 0 and z 1 = v ± n , for some n ≥ 1, or We fix some N 0 ∈ N. Denote (16) holds with z j ∈ Z N0 , j = 1, 2 and y = ψ(t) for some t ∈ R , where K stands for the set of all bounded complete trajectories. We note that set Ξ 0 N0 contains the fixed points in Z N0 and all bounded complete trajectories connecting them. Remark 8. It is known [9] that the family M = {Ξ 0 1 , . . . , Ξ 0 N0 } is a disjoint family of isolated weakly invariant sets and that G 0 is dynamically gradient with respect to M in the sense of Remark 4. Hence, G 0 is dynamically gradient with respect to M in the sense of Definition 5. Now our purpose is to adapt some lemmas from [3, p.2979] to problem (13). In view of Theorems 7 and 8 and the third condition on f ε , there exists a sequence ε k → 0, as k → ∞, such that for every ε ∈ (ε k , ε k+1 ] and any k ≥ 1 problem (13) has exactly 2k + 1 fixed points {v ε 0 = 0, {v + ε,j } k j=1 } such that for each 1 ≤ n ≤ k v ± ε,n has n + 1 zeros in [0, 1]. Let us consider a sequence {ε m } converging to zero.
Proof. It is easy to see that v + εmk is bounded in H 2 (Ω)∩H 1 0 (Ω), so v + εmk → v strongly in H 1 0 (Ω) and C([0, 1]) up to a subsequence. The proof will be finished if we prove that v = v + k . We observe that since in such a case every subsequence would have the same limit, the whole sequence would converge to v + k It is clear that the functions g εm = f εn (v + εmk ) are bounded in L ∞ (0, 1). Passing to a subsequence we can then assume that g εn converges to some g weakly in L 2 (0, 1). It is clear that −(∂ 2 v/∂x 2 ) = g and v is a fixed point if we prove the inclusion g(x) ∈ H 0 (v(x)) for a.e. x ∈ (0, 1). By Mazur's theorem [29, p.120] there exist z m ∈ V m = conv(∪ ∞ k≥m g ε k ) such that z m → g, as m → ∞, strongly in L 2 (0, 1). Taking a subsequence we have z m (x) → g(x), a.e. in (0, 1). Since z m ∈ V m , we get If v(x) < 0, we apply a similar argument.
To conclude the proof, we have to prove that Since v + n has n + 1 zeros, the convergence v + εmk → v + n implies that v + εmk has n + 1 zeros for m ≥ N . But v + εmk possesses k + 1 zeros. Thus, k = n. For the sequence v − εmk the proof is analogous.
Proof. In the same way as in the proof of Lemma 9 we obtain that up to a subsequence v + εm,km → v in H 1 0 (Ω) and C([0, 1]), where v is a fixed point of problem (15). We will prove that v = 0 by contradiction. If not, then v = v ± n for some n ∈ N. However, since v ± n has exactly n + 1 zeros and v + εm,km → v in C([0, 1]), we have that v + εm,km has n + 1 zeros for any m ≥ M with M big enough. This contradicts the fact that v + εm,km possesses k m + 1 zeros and k m → ∞. As the limit is 0 for every converging subsequence, the whole sequence converges to 0.
For the sequence v − εmk the proof is analogous.
Once we have described the preliminary properties, we are now ready to check that (13) satisfies the conditions given in Theorem 2 for certain families M ε . We recall that [27,Theorem 10] guarantees the existence of the global compact invariant attractors A ε , where each A ε is the union of all bounded complete trajectories.
As we have seen before, condition (H2) follows from Remark 8. Therefore, we prove now condition (H1).
In order to establish that (13) satisfies the rest of conditions given in Theorem 2, we need to prove two previous results related to the convergence of solutions of the approximations and the connections between fixed points. Theorem 9. If u εn0 → u 0 in L 2 (Ω) as ε n → 0, then for any sequence of solutions of (13) u εn (·) with u εn (0) = u εn0 there exists a subsequence of ε n such that u εn converges to some strong solution u of (15) in the space C([0, T ], L 2 (Ω)), for any T > 0.
Hence, u n → u in C([0, T ], L 2 (Ω)). By a diagonal argument we obtain that the result is true for every T > 0.
As a consequence of the last theorem, condition (H4) follows.