International Journal of Differential Equations
Volume 2014, Issue 1 625271
Research Article
Open Access

Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

Severino Horácio da Silva, 

Corresponding Author

Severino Horácio da Silva

Unidade Acadêmica de Matemática UAMAT/CCT/UFCG, Bairro Universitário, Rua Aprígio Veloso 882, 58429-900 Campina Grande, PB, Brazil ufcg.edu.br

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Jocirei Dias Ferreira, 

Jocirei Dias Ferreira

Insituto de Ciencias Exatas e da Terra, Universidade Federal de Mato Grosso, Campus Universitário, Rodovia MT-100 Km 3,5, 78.698-000 Barra do Garças, MT, Brazil ufmt.br

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Flank David Morais Bezerra, 

Flank David Morais Bezerra

Departamento de Matemática UFPB/CCEN, Cidade Universitária, Campus I, 58051-900 João Pessoa, PB, Brazil ufpb.br

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First published: 24 April 2014
https://doi.org/10.1155/2014/625271

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

Academic Editor: Tuncay Candan

Abstract

We show the normal hyperbolicity property for the equilibria of the evolution equation ∂m(r, t)/∂t = −m(r, t) + g(βJ*m(r, t) + βh),    h,  β ≥ 0, 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 J.

1. Introduction

We consider the nonlocal evolution equation
(1)
where m(r, t) is a real function on ℝ × ℝ+, h and β are nonnegative constants, and J ∈ C1(ℝ) is a nonnegative even function supported in the interval [−1,1] with integral equal to 1. The * denotes the convolution product, namely,
(2)

There are several works in the literature dedicated to the analysis of (1) or its particular case when g ≡ tanh (see [1–8]).

In the particular case when g ≡ tanh, the existence of a compact global attractor for the flow of (1) was proved in [1] for bounded domain and h = 0 and in [9] for unbounded domain.

If g is globally Lipschitz, the Cauchy problem for (1) is well posed, for instance, in the space of continuous and bounded functions Cb(ℝ), 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 ℙ2τ of the 2τ periodic functions is invariant for the flow of (1). We consider here (1) restricted to ℙ2τ, with τ > 1. As shown in the previous work [7], this leads naturally to the consideration of the flow generated by (1) in L2(S1) where S1 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 g.
  • (H1)

    The function g : ℝ → ℝ is globally Lipschitz; that is, there exists a positive constant k1 such that

    (3)

  •  

    and there exist nonnegative constants k2 and k3, with k2 ≤ k1, such that

    (4)

  •  

    If g is globally Lipschitz with constant k1 it follows that (4) also holds with k2 = k1 and k3 = |g(0)|. However, we are most interested in the case where k2 < k1 because k1β < 1 can leave the attractor to the trivial case of only point.

  • (H2)

    The function g ∈ C1(ℝ) and g′ is Lipschitz with Lipschitz constant k4. In particular, there exists a nonnegative constant k5, such that

    (5)

  • (H3)

    The function g has positive derivative.

  • (H4)

    There exists a > 0 such that, for all x ∈ ℝ, |g(x)| < a. In particular, when a < ∞ inequality (4) holds with k2 = 0 and k3 = a.

  • (H5)

    The function g−1 is continuous in (−a, a) and the function

    (6)

  •  

    where i defined by

    (7)

  •  

    has a global minimum in (−a, a).

Under hypothesis (H1) it was proved in [7] that the problem (1) is well posed in L2(S1) and its flow is C1 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 L∞ 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 J. 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., [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 [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 (h, β) is studied and here we study the continuity with respect to a functional parameter, namely, the function J. 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 J, 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
(8)
is continuously Frechet differentiable in L2(S1), with * being now the convolution product in L2(S1); that is
(9)
Hence, the problem
generates a C1 flow in L2(S1) which depends on the function J, which is given by the variation of constants formula
(10)
From now on we denote this flow for problem (P)J by TJ(t) or T(J, t). It was proved in [7] that, in a certain range of the parameters, TJ(t) admits a compact global attractor. Furthermore, assuming the hypotheses (H1)–(H5) we see that TJ(t) has a gradient structure with Lyapunov functional F : L2(S1) → ℝ given by
(11)
where f and are given in the hypothesis (H5).

A natural question to examine is the dependence of the compact global attractor of TJ(t) on the parameter J. We denote by 𝒜J the global attractor of  (P)J whose existence was proved in [7].

Let us recall that a family of subsets {𝒜J} is upper semicontinuous at J0 if
(12)
where
(13)
Analogously, {𝒜J} is lower semicontinuous at J0 if
(14)

3. Upper Semicontinuity of the Attractors

In this section, we prove that the family of attractors AJ is upper semicontinuous with respect to parameter J at J0, with J, J0 ∈ 𝒥, where
(15)

Lemma 1. Assume that assumptions (H1) and (H2) hold and that k2β < 1. Then, the flow TJ(t) is continuous with respect to J in the L1-norm at J0, uniformly for u in bounded sets and t ∈ [0, b] with b < ∞.

Proof. As shown in [7] the solutions of (P)J satisfy the “variations of constants formula”:

(16)
Let J0 ∈ 𝒥. Given ɛ > 0, we want to find δ > 0 such that implies
(17)
for t ∈ [0, b] and u in C, where C is a bounded set in L2(S1). Since g is globally Lipschitz, for any t > 0 and u ∈ C, it follows that
(18)

Adding and subtracting the term J0*TJ(s)u inside the norm we get

(19)
Using Young′s inequality, we obtain
(20)
From Theorem 3.3 of   [7] it follows, for all nonnegative J ∈ 𝒥, that if k2β < 1 and (H1) and (H2) hold then is bounded by a positive constant K depending only on C. Thus, since we obtain
(21)
Therefore, by Gronwall′s lemma, it follows that
(22)
where . This last assertion completes the proof.

Remark 2. Under hypotheses (H1) and (H2) and k2β < 1, from Theorem 3.3 of [7] it follows that, for all nonnegative J ∈ 𝒥, there exists a global attractor AJ in L2(S1), which is contained in the ball with center at the origin of L2(S1) 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 k2β < 1. Then the family of attractors 𝒜J is upper semicontinuous with respect to J ∈ 𝒥 at J0.

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 F 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 J0 ∈ 𝒥, the set E of the equilibria of is such that E = E1 ∪ E2, where

    • (a)

      the equilibria in E1 are constant hyperbolic equilibria;

    • (b)

      the equilibria in E2 are nonconstant (consequently, nonhyperbolic).

  • (H7)

    The function g ∈ C2(ℝ).

From hypotheses (H2) and (H7) it follows that g′ is bounded; that is, there exists b > 0 such that |g′(x)| ≤ b.

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

Remark 4. A straightforward calculation shows that if u0 is nonconstant equilibria of then zero is always an eigenvalue of the operator

(23)
with eigenfunction .

Remark 5. Let u0 ∈ E2. It is easy to show that DFu(u0, J0) is a self-adjoint operator with respect to the inner product

(24)
where dν(w) = dw/g′(β(J0*u0)(w) + βh) is equivalent to the Lebesgue measure.

Remark 6. In [8] in the hypothesis (H6)-(b) it was also assumed as hypothesis that, for each u0 ∈ E2, zero is simple eigenvalue of the operator DFu(u0, J0). 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 J0 ∈ 𝒥, (H1), (H6), and (H7) hold. Given u ∈ E2 and α ∈ S1, define γ(α; u) ∈ L2(S1) by

(25)
Then Γ = γ(S1; u) is a closed, simple C2 curve of equilibria of which is isolated in the set of equilibria; that is, no point of Γ is an accumulation point of E∖Γ.

Corollary 8. Let M be a closed connected curve of equilibria in E2 and u0 ∈ M. Then M = Γ, where Γ = γ(S1, u0).

Proof. Suppose that Γ⊄M. Then there exist equilibria in M∖Γ accumulating at u0 contradicting Lemma 7. Therefore Γ⊆M. Since Γ is a simple closed curve, it follows that M = Γ.

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

4.1. Normal Hyperbolicity of the Equilibria

Recall that if T(t) : X → X is a semigroup a set M ⊂ X is invariant under T(t) if T(t)M = M, for any t > 0.

Definition 9. Suppose that T(t) is a C1 semigroup in a Banach space X and that M ⊂ X is an invariant manifold for T(t). We say that M is normally hyperbolic under T(t) if

  • (i)

    for each m ∈ M there is a decomposition

    (26)

  •  

    by closed subspaces with being the tangent space to M at m.

  • (ii)

    For each m ∈ M and t ≥ 0, if m1 = T(t)(m)

    (27)

  •  

    and is an isomorphism from onto .

  • (iii)

    There is t0 ≥ 0 and μ < 1 such that for all t ≥ t0

    (28)
    (29)

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

The following result has been proved in [18].

Theorem 10 (normal hyperbolicity). Suppose that T(t) is a C1 semigroup on a Banach space X and M is a C2 compact connected invariant manifold which is normally hyperbolic under T(t) (i.e., (i) and (ii) of Definition 9 hold and there exists 0 ≤ t0 < ∞ such that (iii) holds for all t ≥ t0). Let be a C1 semigroup on X and t1 > t0. Consider N(ɛ), the ɛ-neighborhood of M, given by

(30)
Then, there exists ɛ* > 0 such that, for each ɛ < ɛ*, there exists σ > 0 such that if
(31)
there is a unique compact connected invariant manifold of class C1, , in N(ɛ). Furthermore, is normally hyperbolic under and, for each t ≥ 0, is a C1-diffeomorphism from to .

Remark 11. For u ∈ L2(S1) we have

(32)
where we have used Hölder′s inequality in the last estimate.

Motivated by [21] we prove below that, for each u0 ∈ E2, zero is simple eigenvalue of DFu(u0, J). But specifically we have the following result.

Proposition 12. Assume that bβ2τ∥J∥∞ < 1. Then, for each u0 ∈ E2, zero is simple eigenvalue of DFu(u0, J) with eigenfunction .

Proof. From Remark 5, DFu(u0, J) is self-adjoint operator. Then, to prove that zero is simple eigenvalue, it is enough to show that if v ∈ ker⁡(DFu(u0, J)) then , for some λ0 ∈ ℝ. For this, let v ∈ L2(S1) be such that DFu(u0, J)v = 0. Then

(33)
Suppose that, in L2(S1), for all λ ∈ ℝ; that is,
(34)
But, using Remark 11, for any λ ∈ ℝ and almost every point of S1, we have
(35)
Hence
(36)
Since bβ2τ∥J∥∞ < 1, and , we obtain a contradiction. Therefore, there exists λ0 ∈ ℝ such that .

Remark 13. Since

(37)
is a compact operator in L2(S1), it follows from (H6) that
(38)
contains only real eigenvalues of finite multiplicity with −1 as the unique possible accumulation point.

Proposition 14. Assume that the hypotheses (H1), (H2), and (H6) and that bβ∥J∥∞2τ < 1 holds. Then, for each J ∈ 𝒥, any curve of equilibria of TJ(t) is a normally hyperbolic invariant manifold under TJ(t).

Proof. Here we follow closely a proof given in Pereira and Silva [8]. Let M be a curve of equilibria of TJ(t) and m ∈ M. From Proposition 12 it follows that

(39)
Let Y = ℛ(DFu(m, J)) be the range of DFu(m, J). Since DFu(m, J) is self-adjoint and Fredholm of index zero, it follows that
(40)
where σu and σs correspond to the positive and negative eigenvalues, respectively.

From (H1) and (H2), it follows that TJ(t) is a C1 semigroup. Consider the linear autonomous equation

(41)
Then DTJ(t)v0 is the solution of (41) with initial condition v0; that is, . In particular .

Let Pu and Ps be the spectral projections corresponding to σu and σs. Thus, the subspaces and are invariant under DTJ(t) and the following estimates hold (see [11, pages 73, 81] or [22, page 37]):

(42)
for some positive constant ν and some constant N > 1.

It is clear that DTJ(t) ≡ 0 when restricted to . Therefore, we have the decomposition

(43)
Since DFu(m, J)|Y is an isomorphism, then
(44)
is an isomorphism. Consequently, the linear flow
(45)
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 DTJ(t)(u) be the linear flow generated by the equation

(46)
Then, for a fixed J0 ∈ 𝒥, we have
(47)
when , uniformly for u in bounded sets of L2(S1) and t ∈ [0, b], b < ∞.

Proof. From Lemma 1 it follows that

(48)
for u in bounded sets of L2(S1) and t ∈ [0, b].

By the variation of constants formula, we have

(49)
Thus
(50)
Subtracting and adding the term g′(βJ*u + βh)β(J0*v), we have
(51)
Now, using hypothesis (H2) and Remark 11, we obtain
(52)
Thus, by Young′s inequality and from the fact that u belongs to a bounded set (e.g., the ball in L2 with radius K), it follows that
(53)
From Remark 11 we obtain that
(54)
Assuming (H2), Young′s inequality, and the fact that we get
(55)
Thus
(56)
Hence, from (53) and (56) it follows that
(57)

Therefore

(58)
That is,
(59)
where
(60)
tends to zero when .

4.2. Lower Semicontinuity of the Equilibria

Theorem 16. Suppose that the hypotheses (H1), (H2), and (H5) with a < ∞ and (H6) and (H7) hold. Then, if bβ2τ∥J∥∞ < 1, the set EJ of the equilibria of TJ(t) is lower semicontinuous with respect to J at J0.

Proof. The continuity of the constant equilibria follows from the Implicit Function Theorem and the hypothesis of normal hyperbolicity.

Suppose now that m is a nonconstant equilibrium of (P)J and let Γ = γ(α; m) be the isolated curve of equilibria containing m given in Lemma 7. We wish to show that, for every ɛ > 0, there exists δ > 0 so that if J ∈ 𝒥 there exists ΓJ ∈ EJ such that where is the ɛ-neighborhood of ΓJ.

From Lemma 7 and Propositions 14 and 15, the assumptions of the normal hyperbolicity theorem are satisfied. Thus, given ɛ > 0, there is δ > 0 such that if there is a unique C1 compact connected invariant manifold ΓJ normally hyperbolic under TJ(t), such that ΓJ is ɛ-close and C1-diffeomorphic to Γ.

Since TJ(t) is gradient and ΓJ is compact, there exists at least one equilibrium mJ ∈ ΓJ. In fact, the ω limit of any u ∈ ΓJ is nonempty and belongs to ΓJ by invariance. From Lemma 3.8.2 of [12], it must contain an equilibrium. Since ΓJ is ɛ-close to Γ, there exists m ∈ Γ such that .

Let be the curve of equilibria given by which is a normally hyperbolic invariant manifold under TJ(t) by Proposition 14. Then, for each α ∈ S1, we have

(61)
Thus
(62)
And Γ is ɛ-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
(63)
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

(64)

Note that (64) has, besides the origin, the curve of equilibria given by

(65)
which is generated, in polar coordinate, by the rotation of a fixed equilibrium.

However, for any ɛ ≠ 0, the perturbed system

(66)
has no nontrivial equilibrium, although the system is still invariant under the action of S1.

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 (P)J. Recall that the unstable set of an equilibrium uJ is the set of initial conditions φ of (P)J, such that TJ(t)φ is defined for all t ≤ 0 and TJ(t)φ → uJ as t → −∞. For a given neighborhood V of uJ, the set is called a local unstable set of uJ.

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 J. More precisely, we have the following.

Lemma 18. If u0 is a fixed equilibrium of  (P)J for J = J0 then there is a δ > 0 such that if and

(67)
then is a Lipschitz manifold and
(68)
with dist⁡ defined as in (13).

Proof. As already mentioned, assuming (H1) and (H2), the map F : L2(S1) × 𝒥 → L2(S1),

(69)
defined by the right hand side of  (P)J is continuously Frechet differentiable. Let uJ be an equilibrium of (P)J. Writing u = uJ + v, it follows that u is a solution of (P)J if and only if v satisfies
(70)
where L(J)v = (∂/∂u)F(uJ, J) = −v + g′(β(J*uJ) + βh)β(J*v) and r(uJ, v, J) = F(uJ + v, J) − F(uJ, J) −L(J)v. We rewrite (70) in the form
(71)
where f(v, J) = [L(J) − L(J0)]v + r(uJ, v, J) is the “nonlinear part” of (71). Observe that now the “linear part” of (71) does not depend on the parameter J, as required by Theorems 2.5 and 3.3 from [19].

Note that

(72)
So, using (H2) and Young′s inequality we obtain
(73)
and consequently,
(74)
On the other hand, since by hypothesis (H7) g is C2, the functions g′(β(J*uJ) + β(J*v) + βh) and g′′(β(J*uJ) + β(J*v) + βh) are bounded by a constant M; for any J in a neighborhood of J0 with , we have
(75)
From (74) and (75) it follows that
(76)

Therefore,

(77)
where
(78)

Now, note that

(79)
Thus
(80)
for some in the segment defined by J*uJ and J*(uJ + v) and for some in the segment defined by and . Then, using (H2) and the fact that g′(β(J*uJ) + β(J*v) + βh) is bounded by a constant M, for any J in a neighborhood of J0 with , we have
(81)

With this

(82)

Once the following estimates hold

(83)
it follows that
(84)
Therefore, as provided that , it follows that
(85)
with C2(J) → 0 when .

Since r(uJ, v, J) = F(uJ + v, J) − L(J)v, we obtain from (77) and (85) that

(86)

From (77) and (86), it follows that

(87)
where C4(J) → 0 as .

In a similar way, we obtain that

(88)
for any v, w with and smaller than 1, with in the segment defined by βJ*v + βh and βJ*w + βh and in the segment defined by 0 and . As , it follows that
(89)
with ν(ρ) → 0 when ρ → 0 and . Furthermore
(90)

Thus

(91)
with ν(ρ) → 0 when ρ → 0, and are less than or equal to ρ, and C1(J) → 0 when J → J0.

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 J near J0. Using uniqueness of solutions, we can easily prove that these sets coincide with the local unstable manifolds of (71).

Observing now that the translation

(92)
sends an equilibrium uJ of (P)J 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 J = J0 be fixed. Then, there exists a δ > 0 such that, for any equilibrium u0 of  ,  if and

(93)
then is a Lipschitz manifold and
(94)
with dist⁡ defined as in  (13).

Proof. From Lemma 18 we know that, for any , there exists a δ = δ(u0) 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  B(u0, δ(u0)),   with u0 varying in  ,  the first part of the result follows with  δ  chosen as the minimum of those  δ(u0).

Now, if ɛ > 0 and there exists, by Lemma 18, δ = δ(u0) such that if then

(95)
If is such that and then, since ,
(96)

By the same procedure given above, taking a finite subcovering of the covering of    by balls  B(u0, δ(u0))  and  δ  the minimum of those  δ(u0),  we conclude that

(97)
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 (P)J 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
(98)
where B is a bounded linear operator on a Banach space X and g : X → X is a C2 function. We may write (98) in the form
(99)
where A = B − g′(x0) and f(x) = g(x0) + r(x), with r differentiable and r(0) = 0.

The following result has been proved in [20].

Theorem 20. Suppose that the spectrum σ(A) contains 0 as a simple eigenvalue, while the remainder of the spectrum has real part outside some neighborhood of zero. Let γ be a C2 curve of equilibria of the flow generated by (99). Then there exists a neighborhood U of γ such that, for any x0 ∈ U whose positive orbit is precompact and whose ω-limit set ω(x0) belongs to γ, there exists a unique point y(x0) ∈ γ with ω(x0) = y(x0). Similarly, for any x0 ∈ U with bounded negative orbit and α-limit set α(x0) in γ, there exists a unique point y(x0) ∈ γ such that α(x0) = y(x0).

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

(100)

Proof. From Theorem 5.5 of [7] we have

(101)
There exist only a finite number {u1, …, uk} of constant equilibria since they are all hyperbolic. For each nonconstant equilibrium u ∈ EJ, there is a curve Mu ⊂ EJ ⊂ 𝒜J. From Lemma 7 these curves Mu are all isolated and, since 𝒜J is compact, it follows that there exist only a finite number of them, namely, M1, …, Mn. Thus
(102)
By Theorem 20, it follows that
(103)
Therefore
(104)

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 ɛ > 0 there exists T > 0 such that, for all ,

(105)
for some t ∈ [0, T], where is the ɛ-neighborhood of . Furthermore, when ɛ is sufficiently small,
(106)
for some , where is the local unstable manifold of .

Proof. Let ɛ > 0 be given and . From Proposition 21, it follows that

(107)
for some . Thus, there exists tu = tu(ɛ) < ∞ such that . By the continuity of the operator , there exists ηu > 0 such that , where B(u, ηu) is the ball of center u and radius ηu. By compactness, there are such that
(108)
with , for j = 1, …, n. Let . Then, for any , for some t ∈ [0, T]. Since for some and then, to conclude that when ɛ is sufficiently small, it is enough to show that there exists δ > 0 such that , for all . But this result follows immediately from Lemma 18.

Theorem 23. Assume the hypotheses (H1), (H2), and (H5) with a < ∞ and (H6) and (H7) hold. Then the family of attractors 𝒜J is lower semicontinuous with respect to the parameter J at J0 ∈ 𝒥.

Proof. Let ɛ > 0 be given. From Lemma 22 there is a T > 0 such that, for all , there exists tu ∈ [0, T] such that

(109)
for some . Since is a continuous family of bounded operators there exists η > 0 such that, for all t ∈ [0, T],
(110)

Now, by the uniform continuity of the equilibria and the local unstable manifolds with respect to the parameter J guaranteed by Theorem 16 and Lemma 19, there exists δ* > 0 independent of u such that implies the existence of uJ ∈ EJ and some with

(111)
where UJ(uJ) denotes the local unstable manifold of the equilibrium uJ of TJ(t). Hence, when we obtain, from (110) and (111),
(112)

On the other hand, from the continuity of the flow with respect to parameter J, there exists such that implies

(113)
for any and t ∈ [0, T], and in particular for and t = tu.

Consider and let . It is clear that vJ ∈ 𝒜J since .

Using (112) and (113) we obtain

(114)
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 g(x) = tanh(x).

In this case, we can rewrite (P)J as follows:
(115)
In this case, if β ≤ 1 (115) has only one (stable) equilibrium (see [9]). If β > 1, there is h*, implicity defined by (116), such that, for 0 ≤ h ≤ h*, (115) has three equilibria, , , and , each of which is identically equal to one of the roots of the equations:
(116)
The Lyapunov functional for (115) is given by
(117)
where f (the free energy density) is given by
(118)
and i is the entropy density, given by
(119)

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

Note that g satisfies (H1) and (H4) with k1 = a = 1. Moreover, it is easy to see that (4) is satisfied with k2 = 1/3 and k3 = 3.

Now, we observe that g ∈ C2(ℝ), g′(x) = sech2(x), and g′′(x) = −2sech2(x)tanh(x). Thus (H2), (H3), and (H7) hold. In particular
(120)
where k4 = 2 and k5 = 0 and g′(x) ≤ b with b = 1.

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.

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