Qualitative Theory of Differential, Difference, and Dynamic EquationsView this Special Issue
Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations
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.
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 [1–8]).
In the particular case when , the existence of a compact global attractor for the flow of (1) was proved in  for bounded domain and and in  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 , 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 . 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  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  was also proved in . A comparison result under the hypotheses (H1) and (H3) was also proved. Assuming (H1), (H2), (H3), and (H4), the authors in  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 .
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 , 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., [13–17]). 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 , 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  is that in  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  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  on the permanence of normally hyperbolic invariant manifolds and one result given in  concerning the continuity properties of the local unstable manifolds of the (nonnecessarily isolated) equilibria with respect to the parameter , together with some results of  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 , 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  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  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 .
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  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  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  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 , 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 ) 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  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 .
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 .
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  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 . 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 , 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 ).
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  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  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 .
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
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
Once the following estimates hold