Abstract

We show the normal hyperbolicity property for the equilibria of the evolution equation and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameter .

The first author dedicates this work to his daughter Luana Barros.

1. Introduction

We consider the nonlocal evolution equation where is a real function on , and are nonnegative constants, and is a nonnegative even function supported in the interval with integral equal to 1. The denotes the convolution product, namely,

There are several works in the literature dedicated to the analysis of (1) or its particular case when (see [18]).

In the particular case when , the existence of a compact global attractor for the flow of (1) was proved in [1] for bounded domain and and in [9] for unbounded domain.

If is globally Lipschitz, the Cauchy problem for (1) is well posed, for instance, in the space of continuous and bounded functions , with the sup norm since the function given by the right hand side of (1) is uniformly Lipschitz in this space (see [10, 11]).

It is an easy consequence of the uniqueness theorem that the subspace of the periodic functions is invariant for the flow of (1). We consider here (1) restricted to , with . As shown in the previous work [7], this leads naturally to the consideration of the flow generated by (1) in where is the unit sphere and the convolution product in it. In what follows, we summarize the assumptions and results of [7]. For sake of clarity, it is convenient to start with a list of hypotheses satisfied by the function .(H1)The function is globally Lipschitz; that is, there exists a positive constant such that and there exist nonnegative constants and , with , such that If is globally Lipschitz with constant it follows that (4) also holds with and . However, we are most interested in the case where because can leave the attractor to the trivial case of only point.(H2)The function and is Lipschitz with Lipschitz constant . In particular, there exists a nonnegative constant , such that (H3)The function has positive derivative.(H4)There exists such that, for all , . In particular, when inequality (4) holds with and .(H5)The function is continuous in and the function where defined by has a global minimum in .

Under hypothesis (H1) it was proved in [7] that the problem (1) is well posed in and its flow is if we assume hypothesis (H2). Furthermore, assuming (H1) and (H2) the existence of a global compact attractor for the flow of (1) in the sense of [12] was also proved in [7]. A comparison result under the hypotheses (H1) and (H3) was also proved. Assuming (H1), (H2), (H3), and (H4), the authors in [7] showed an estimate for the attractors; finally, assuming (H5), they exhibited a continuous Lyapunov functional for the flow of (1) and proved under hypotheses (H1), (H2), (H3), (H4), and (H5) that its flow is gradient in the sense of [12].

The main purposes of this paper are showing normal hyperbolicity property of curves of equilibria and proving the continuity of global attractors for the flow of (1) with respect to the function . To the extent of our knowledge, with the exception of [8], the proofs available in the literature concerning the continuity of global attractors assume that the equilibrium points of (1) are all hyperbolic and therefore isolated (see, e.g., [1317]). However, this property cannot hold true in our case, due to the symmetries present in the equation. In fact, it is a consequence of these symmetries that the nonconstant equilibria arise in families and therefore it cannot be hyperbolic. To overcome this difficulty, in [8], the hypothesis of hyperbolicity of equilibria has been replaced by normal hyperbolicity of curves of equilibria.

The difference between our proof and the proof given in [8] is that in [8] the continuity with respect to scalar parameters is studied and here we study the continuity with respect to a functional parameter, namely, the function . Moreover, in [8] it is assumed that the zero is a simple eigenvalue of the Frechét derivative of (8) which implies in normal hyperbolicity of curves of equilibria, and in this paper this property is also proven (see Propositions 12 and 14). To prove our results, we use some results given in [18] on the permanence of normally hyperbolic invariant manifolds and one result given in [19] concerning the continuity properties of the local unstable manifolds of the (nonnecessarily isolated) equilibria with respect to the parameter , together with some results of [20] regarding the limiting behavior of the trajectories.

This paper is organized as follows. In Section 2, we show some preliminary results. Section 3 is devoted to the proof of the upper semicontinuity of the attractors. In Section 4, we show that families of equilibria are normally hyperbolic and we use this property to show the continuity of the families of equilibria with respect to the parameter. In Section 5, using the same techniques given in [8], we prove the lower semicontinuity of attractors. Finally, in Section 6, we illustrate our results with a concrete example.

2. Background Results

It is well known from [7] that under hypotheses (H1) and (H2) the map is continuously Frechet differentiable in , with being now the convolution product in ; that is Hence, the problem generates a flow in which depends on the function , which is given by the variation of constants formula

From now on we denote this flow for problem by or . It was proved in [7] that, in a certain range of the parameters, admits a compact global attractor. Furthermore, assuming the hypotheses (H1)–(H5) we see that has a gradient structure with Lyapunov functional given by where and are given in the hypothesis (H5).

A natural question to examine is the dependence of the compact global attractor of on the parameter . We denote by the global attractor of   whose existence was proved in [7].

Let us recall that a family of subsets is upper semicontinuous at if where

Analogously, is lower semicontinuous at if

3. Upper Semicontinuity of the Attractors

In this section, we prove that the family of attractors is upper semicontinuous with respect to parameter at , with , where

Lemma 1. Assume that assumptions (H1) and (H2) hold and that . Then, the flow is continuous with respect to in the at , uniformly for in bounded sets and with .

Proof. As shown in [7] the solutions of satisfy the “variations of constants formula”: Let . Given , we want to find such that implies for and in , where is a bounded set in . Since is globally Lipschitz, for any and , it follows that
Adding and subtracting the term inside the norm we get Using Young's inequality, we obtain From Theorem 3.3 of   [7] it follows, for all nonnegative , that if and (H1) and (H2) hold then is bounded by a positive constant depending only on . Thus, since we obtain Therefore, by Gronwall's lemma, it follows that where . This last assertion completes the proof.

Remark 2. Under hypotheses (H1) and (H2) and , from Theorem 3.3 of [7] it follows that, for all nonnegative , there exists a global attractor in , which is contained in the ball with center at the origin of and radius .

Now, using Remark 2 and proceeding as in [8], we obtain the following result.

Theorem 3. Assume that hypotheses (H1) and (H2) hold and that . Then the family of attractors is upper semicontinuous with respect to at .

4. Normal Hyperbolicity and Lower Semicontinuity of the Attractors

Due to the symmetries present in our model the nonconstant equilibria are nonisolated. In fact, as we will see shortly, the equivariance property of the map defined in (8) implies that the nonconstant equilibria appear in curves. Therefore, it cannot be hyperbolic preventing the use of tools like the Implicit Function Theorem to obtain their continuity with respect to the parameters. To overcome this difficulty, we need the concept of normal hyperbolicity (see [18]) and we also will need to assume the following additional hypotheses.(H6)For each , the set of the equilibria of is such that , where(a)the equilibria in are constant hyperbolic equilibria;(b)the equilibria in are nonconstant (consequently, nonhyperbolic).(H7)The function .

From hypotheses (H2) and (H7) it follows that is bounded; that is, there exists such that .

We start with some remarks on the spectrum of the linearization for around equilibria.

Remark 4. A straightforward calculation shows that if is nonconstant equilibria of then zero is always an eigenvalue of the operator with eigenfunction .

Remark 5. Let . It is easy to show that is a self-adjoint operator with respect to the inner product where is equivalent to the Lebesgue measure.

Remark 6. In [8] in the hypothesis (H6)-(b) it was also assumed as hypothesis that, for each , zero is simple eigenvalue of the operator . However, in this paper, this property is proved (see Proposition 12).

In what follows we enunciated a result on the structure of the sets of nonconstant equilibria. The proof will be omitted since it is very similar to Lemma 3.3 in [8].

Lemma 7. Suppose that for some , (H1), (H6), and (H7) hold. Given and , define by Then is a closed, simple curve of equilibria of which is isolated in the set of equilibria; that is, no point of is an accumulation point of .

Corollary 8. Let be a closed connected curve of equilibria in and . Then , where .

Proof. Suppose that . Then there exist equilibria in accumulating at contradicting Lemma 7. Therefore . Since is a simple closed curve, it follows that .

The main results of this section will be presented in the next two subsections.

4.1. Normal Hyperbolicity of the Equilibria

Recall that if is a semigroup a set is invariant under if , for any .

Definition 9. Suppose that is a semigroup in a Banach space and that is an invariant manifold for . We say that is normally hyperbolic under if (i)for each there is a decomposition by closed subspaces with being the tangent space to at .(ii)For each and , if and is an isomorphism from onto .(iii)There is and such that for all

Condition (28) suggests that, near , is expansive in the direction of and at rate greater than that on , while (29) suggests that is contractive in the direction of and at a rate greater than that on .

The following result has been proved in [18].

Theorem 10 (normal hyperbolicity). Suppose that is a semigroup on a Banach space and is a compact connected invariant manifold which is normally hyperbolic under (i.e., (i) and (ii) of Definition 9 hold and there exists such that (iii) holds for all ). Let be a semigroup on and . Consider , the -neighborhood of , given by Then, there exists such that, for each , there exists such that if there is a unique compact connected invariant manifold of class , , in . Furthermore, is normally hyperbolic under and, for each , is a -diffeomorphism from to .

Remark 11. For we have where we have used Hölder's inequality in the last estimate.

Motivated by [21] we prove below that, for each , zero is simple eigenvalue of . But specifically we have the following result.

Proposition 12. Assume that . Then, for each , zero is simple eigenvalue of with eigenfunction .

Proof. From Remark 5, is self-adjoint operator. Then, to prove that zero is simple eigenvalue, it is enough to show that if then , for some . For this, let be such that . Then Suppose that, in , for all ; that is, But, using Remark 11, for any and almost every point of , we have Hence Since , and , we obtain a contradiction. Therefore, there exists such that .

Remark 13. Since is a compact operator in , it follows from (H6) that contains only real eigenvalues of finite multiplicity with as the unique possible accumulation point.

Proposition 14. Assume that the hypotheses (H1), (H2), and (H6) and that holds. Then, for each , any curve of equilibria of is a normally hyperbolic invariant manifold under .

Proof. Here we follow closely a proof given in Pereira and Silva [8]. Let be a curve of equilibria of and . From Proposition 12 it follows that Let be the range of . Since is self-adjoint and Fredholm of index zero, it follows that where and correspond to the positive and negative eigenvalues, respectively.
From (H1) and (H2), it follows that is a semigroup. Consider the linear autonomous equation Then is the solution of (41) with initial condition ; that is, . In particular .
Let and be the spectral projections corresponding to and . Thus, the subspaces and are invariant under and the following estimates hold (see [11, pages 73, 81] or [22, page 37]): for some positive constant and some constant .
It is clear that when restricted to . Therefore, we have the decomposition Since is an isomorphism, then is an isomorphism. Consequently, the linear flow is also an isomorphism.
Finally, the estimates (28) and (29) follow from estimate (42).

Proposition 15. Suppose that the hypotheses (H1) and (H2) hold. Let be the linear flow generated by the equation Then, for a fixed , we have when , uniformly for in bounded sets of and , .

Proof. From Lemma 1 it follows that for in bounded sets of and .
By the variation of constants formula, we have Thus Subtracting and adding the term , we have Now, using hypothesis (H2) and Remark 11, we obtain Thus, by Young's inequality and from the fact that belongs to a bounded set (e.g., the ball in with radius ), it follows that From Remark 11 we obtain that Assuming (H2), Young's inequality, and the fact that we get Thus Hence, from (53) and (56) it follows that
Therefore That is, where tends to zero when .

4.2. Lower Semicontinuity of the Equilibria

Theorem 16. Suppose that the hypotheses (H1), (H2), and (H5) with and (H6) and (H7) hold. Then, if , the set of the equilibria of is lower semicontinuous with respect to at .

Proof. The continuity of the constant equilibria follows from the Implicit Function Theorem and the hypothesis of normal hyperbolicity.
Suppose now that is a nonconstant equilibrium of and let be the isolated curve of equilibria containing given in Lemma 7. We wish to show that, for every , there exists so that if there exists such that where is the -neighborhood of .
From Lemma 7 and Propositions 14 and 15, the assumptions of the normal hyperbolicity theorem are satisfied. Thus, given , there is such that if there is a unique compact connected invariant manifold normally hyperbolic under , such that is -close and -diffeomorphic to .
Since is gradient and is compact, there exists at least one equilibrium . In fact, the limit of any is nonempty and belongs to by invariance. From Lemma of [12], it must contain an equilibrium. Since is -close to , there exists such that .
Let be the curve of equilibria given by which is a normally hyperbolic invariant manifold under by Proposition 14. Then, for each , we have Thus And is -close to . Since there are only a finite number of curves of equilibria the result follows immediately.

The example given below shows that the curves of equilibria of the equation generated by the action of a group, may disappear even when the symmetry is preserved. In other words, we are unable to obtain a result by using the Implicit Function Theorem without additional hypotheses of normal hyperbolicity (see [23]).

Example 17 (an example with symmetry, see [8, 23]). Consider the planar system
Note that (64) has, besides the origin, the curve of equilibria given by which is generated, in polar coordinate, by the rotation of a fixed equilibrium.
However, for any , the perturbed system has no nontrivial equilibrium, although the system is still invariant under the action of .

5. Lower Semicontinuity of the Attractors

In this section, using the same techniques of [8] we present the proof of the lower semicontinuity of the attractors in the next two subsections below.

5.1. Existence and Continuity of the Local Unstable Manifolds

Let us return to . Recall that the unstable set of an equilibrium is the set of initial conditions of , such that is defined for all and as . For a given neighborhood of , the set is called a local unstable set of .

In the following, using results of [19] we show that the local unstable sets are actually Lipschitz manifolds in a sufficiently small neighborhood and vary continuously with . More precisely, we have the following.

Lemma 18. If is a fixed equilibrium of   for then there is a such that if and then is a Lipschitz manifold and with defined as in (13).

Proof. As already mentioned, assuming (H1) and (H2), the map , defined by the right hand side of   is continuously Frechet differentiable. Let be an equilibrium of . Writing , it follows that is a solution of if and only if satisfies where and . We rewrite (70) in the form where is the “nonlinear part” of (71). Observe that now the “linear part” of (71) does not depend on the parameter , as required by Theorems 2.5 and 3.3 from [19].
Note that So, using (H2) and Young's inequality we obtain and consequently, On the other hand, since by hypothesis (H7) is , the functions and are bounded by a constant ; for any in a neighborhood of with , we have From (74) and (75) it follows that
Therefore, where
Now, note that Thus for some in the segment defined by and and for some in the segment defined by and . Then, using (H2) and the fact that is bounded by a constant , for any in a neighborhood of with , we have
With this
Once the following estimates hold it follows that Therefore, as provided that , it follows that with when .
Since , we obtain from (77) and (85) that
From (77) and (86), it follows that where as .
In a similar way, we obtain that for any with and smaller than , with in the segment defined by and and in the segment defined by and . As , it follows that with when and . Furthermore
Thus with when , and are less than or equal to , and when .
Therefore, the conditions of Theorems 2.5 and 3.3 from [19] are satisfied and we obtain the existence of locally invariant sets for (71) near the origin, given as graphics of Lipschitz functions which depend continuously on the parameter near . Using uniqueness of solutions, we can easily prove that these sets coincide with the local unstable manifolds of (71).
Observing now that the translation sends an equilibrium of into the origin (which is an equilibrium of (71)), the results follow immediately.

Using the compactness of the set of equilibria, one can obtain a “uniform version” of Lemma 18 that will be needed later.

Lemma 19. Let be fixed. Then, there exists a such that, for any equilibrium of  ,  if and then is a Lipschitz manifold and with defined as in  (13).

Proof. From Lemma 18 we know that, for any , there exists a such that is a Lipschitz manifold if . In particular, is a Lipschitz manifold if for any with . Taking a finite subcovering of the covering of    by balls  ,   with varying in  ,  the first part of the result follows with    chosen as the minimum of those  .
Now, if and there exists, by Lemma 18, such that if then If is such that and then, since ,
By the same procedure given above, taking a finite subcovering of the covering of    by balls    and    the minimum of those  ,  we conclude that if for any .

5.2. Characterization and Proof of Lower Semicontinuity of the Attractors

As a consequence of its gradient structure (see [7]), the attractor of the flow generated by is given by unstable set of the set of equilibria (see [12]). Using results of [20], we obtain below a more precise characterization of the attractors.

Consider an equation of the form where is a bounded linear operator on a Banach space and is a function. We may write (98) in the form where and , with differentiable and .

The following result has been proved in [20].

Theorem 20. Suppose that the spectrum contains as a simple eigenvalue, while the remainder of the spectrum has real part outside some neighborhood of zero. Let be a curve of equilibria of the flow generated by (99). Then there exists a neighborhood of such that, for any whose positive orbit is precompact and whose -limit set belongs to , there exists a unique point with . Similarly, for any with bounded negative orbit and -limit set in , there exists a unique point such that .

Proposition 21. Assume that the hypotheses (H1), (H2), and (H5) with and (H6) and (H7) hold. Let be the set of equilibria for . For , let be the unstable set of . Then the attractor of the flow is given by

Proof. From Theorem 5.5 of [7] we have There exist only a finite number of constant equilibria since they are all hyperbolic. For each nonconstant equilibrium , there is a curve . From Lemma 7 these curves are all isolated and, since is compact, it follows that there exist only a finite number of them, namely, . Thus By Theorem 20, it follows that Therefore

5.2.1. Proof of the Lower Semicontinuity

We now turn back to the proof of our main result, starting with some auxiliary results.

Lemma 22. Assume the same hypotheses of Proposition 21. Thus, given there exists such that, for all , for some , where is the -neighborhood of . Furthermore, when is sufficiently small, for some , where is the local unstable manifold of .

Proof. Let be given and . From Proposition 21, it follows that for some . Thus, there exists such that . By the continuity of the operator , there exists such that , where is the ball of center and radius . By compactness, there are such that with , for . Let . Then, for any , for some . Since for some and then, to conclude that when is sufficiently small, it is enough to show that there exists such that , for all . But this result follows immediately from Lemma 18.

Theorem 23. Assume the hypotheses (H1), (H2), and (H5) with and (H6) and (H7) hold. Then the family of attractors is lower semicontinuous with respect to the parameter at .

Proof. Let be given. From Lemma 22 there is a such that, for all , there exists such that for some . Since is a continuous family of bounded operators there exists such that, for all ,
Now, by the uniform continuity of the equilibria and the local unstable manifolds with respect to the parameter guaranteed by Theorem 16 and Lemma 19, there exists independent of such that implies the existence of and some with where denotes the local unstable manifold of the equilibrium of . Hence, when we obtain, from (110) and (111),
On the other hand, from the continuity of the flow with respect to parameter , there exists such that implies for any and , and in particular for and .
Consider and let . It is clear that since .
Using (112) and (113) we obtain provided that .
When this conclusion follows straightforwardly from the continuity of equilibria. Thus the lower semicontinuity of attractors follows.

6. A Concrete Example

In this section we illustrate the results of the previous sections to the particular case of (1) where .

In this case, we can rewrite as follows: In this case, if (115) has only one (stable) equilibrium (see [9]). If , there is , implicity defined by (116), such that, for , (115) has three equilibria, , , and , each of which is identically equal to one of the roots of the equations:

The Lyapunov functional for (115) is given by where (the free energy density) is given by and is the entropy density, given by

As was observed in [4, 8, 9], the functional given in (117) has minimum value at .

Note that satisfies (H1) and (H4) with . Moreover, it is easy to see that (4) is satisfied with and .

Now, we observe that , , and . Thus (H2), (H3), and (H7) hold. In particular where and and with .

Therefore all results of the previous sections are valid for the particular case of the flow generated by (115).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the anonymous referee for his/her careful reading of the paper and valuable suggestions. The first author is supported by CAPES/CNPq-Brazil, the second author is supported by INCTMat, and the third author is supported by FAPESP.