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Existence and continuity of global attractors and nonhomogeneous equilibria for a class of evolution equations with non local terms
J. Math. Anal. Appl. 396 (2012) 590–600
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Existence and continuity of global attractors and nonhomogeneous equilibria for a class of evolution equations with non local terms Flank D.M. Bezerra a,∗ , Antônio L. Pereira b , Severino H. da Silva c a
Departamento de Matemática, CCEN/UFPB, Cidade Universitária-Campus I, 58051-900, João Pessoa-PB, Brazil
b
Instituto de Matemática e Estatística-Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo-SP, Brazil Unidade Acadêmica de Matemática e Estatística UAME/CCT/UFCG, Avenida Aprígio Veloso, 882, Bairro Universitário, Caixa Postal: 10.044, 58109-970, Campina Grande-PB, Brazil c
article
info
abstract
Article history: Received 27 May 2012 Available online 29 June 2012 Submitted by Jie Xiao
In this paper, we are concerned with some aspects of the asymptotic behavior of the dynamical system generated by evolution equations with nonlocal terms of the type
Keywords: Global attractors Nonhomogeneous equilibria Continuity Lyapunov functional
where Ω ⊂ RN , β > 0, N ≥ 1 is a bounded smooth domain and K is an integral operator with symmetric kernel
∂ u(x, t ) = −u(x, t ) + g (β(Ku)(x, t )) , ∂t
(Ku)(x) :=
RN
x ∈ Ω, t > 0
J (x, y)u(y)dy.
© 2012 Elsevier Inc. All rights reserved.
1. Introduction A class of problems that have been considered recently are differential equations with non local diffusion terms. Equations of this type appear, for example, in modeling population dynamics, propagation of nerve impulses and problems of phase transition in ferromagnetic models. The mathematical treatment of these equations is generally simpler than similar models based on parabolic partial equations, from the standpoint of the basic theory of existence and well posedness. On the other hand, the qualitative behavior of solutions can be much richer. Thus, many of the issues that arise for models based on parabolic semilinear equations can also be considered in the new context but often require new techniques and ideas. Our goal here is to consider some of these issues for a nonlinear Dirichlet problem with non local terms, namely,
∂ u(x, t ) = −u(x, t ) + g (β(Ku)(x, t )) , ∂t u(x, 0) = u0 (x), u(x, t ) = 0,
x ∈ Ω, t > 0 x∈Ω
(1.1)
x ̸∈ Ω , t > 0
where Ω ⊂ R , N ≥ 1 is a bounded smooth domain, u(x, t ) is a real function on RN × [0, +∞), β > 0 and K is an integral operator with symmetric kernel N
(Ku)(x) := ∗
RN
J (x, y)u(y)dy.
Corresponding author. E-mail addresses: [email protected] (F.D.M. Bezerra), [email protected] (A.L. Pereira), [email protected] (S.H. da Silva).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.06.042
F.D.M. Bezerra et al. / J. Math. Anal. Appl. 396 (2012) 590–600
591
Here, J is an even non negative function of class C 2 with RN J (x, y)dy = 1, and g : R → R is a nonlinear real function of class C 1 with g (0) = 0. Equation (1.1) is called a non local diffusion equation since the diffusion of the density u at a point x and time t depends on the values of u in a neighborhood of x in RN through the term Ku. The asymptotic behavior of solutions of non local diffusion equations has been extensively studied over the past ten years (see [1–8]). Often, the non local term in (1.1)-like problems is given by convolution. We extend some of these results to (1.1), and consider also some different aspects of the asymptotic behavior of the infinite dimensional dynamical system it generates. For N = 1 there are several works in the literature dedicated to the analysis of similar models (see, [9–11], where existence and continuity of global attractors is proved) and, in particular, for the case of (1.1) with g ≡ tanh, and J (x, y) = J (x − y) that is
∂ u(x, t ) = −u(x, t ) + tanh (β(J ∗ u)(x, t )) , ∂t
t >0
(1.2)
(see [12–15,9]). In this case, if β ≤ 1, Eq. (1.2) has only one (stable) equilibrium, (see [15,9]). If β > 1, the equation mβ = tanh(β mβ )
(1.3)
+ 0 has three roots m− β , mβ , mβ , which are then spatially homogeneous equilibria of Eq. (1.3). The existence and uniqueness + (modulo translation) of a traveling front connecting the equilibria m− β and mβ is proved in [13]. The existence and uniqueness
(modulo translation) of a solution tending asymptotically to ±m+ β , referred to as the ‘‘instanton’’ is proved in [14,16].
This paper is organized as follows. In Section 2, we prove that (1.1) generates a flow in L2 (RN ). Section 3 is dedicated to the proof of existence of the global attractor (for the case N ≥ 1), generalizing results of [12,10,11]. In Section 4, we prove a comparison result, generalizing Theorem 2.7 of [14] and Theorem 4.2 of [10]. In Section 5, we exhibit a Lyapunov functional for the flow of (1.1), and used it to prove the existence of non trivial equilibrium. Finally, in Section 6, we prove continuity of the global attractor with respect to the parameter β . We collect here the conditions on g which will be used as hypotheses when needed. (H1) The function g : R → R, is globally Lipschitz, with g (0) = 0. That is, there exists a positive constant k1 such that
|g (x) − g (y)| ≤ k1 |x − y|,
∀ x, y ∈ R .
In particular,
|g (x)| ≤ k1 |x|,
∀ x ∈ R.
(1.4)
(H2) The function g ∈ C (R) and g is Lipschitz with constant k2 . In particular 1
′
|g ′ (x)| ≤ k2 |x| + k3 ∀ x ∈ R, for some k3 > 0. (H3) The function g has a positive derivative. In particular, it is strictly increasing. (H4) There exists a > 0 such that |g (x)| < a < ∞, for all x ∈ R. In this paper, |Ω | denotes the Lebesgue measure of Ω in RN . 2. Estimates and well posedness In this section, in order to obtain well posedness of (1.1), we initially consider the following Cauchy problem in the space L2 (RN )
∂ u(x, t ) = (Fu)(x, t ) ∂t u(x, 0) = u0
(2.5)
where F : L2 (RN ) → L2 (RN ) is given by
[F (u)](x) =
−u(x) + g (β(Ku)(x)), 0,
x∈Ω x ̸∈ Ω .
(2.6)
Proposition 2.1. Suppose that the hypothesis (H1) holds. Then the function F given above is uniformly Lipschitz in L2 = L2 (RN ).
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Proof. From Generalized Young’s Inequality, (see [17]),
∥K (m − u)∥L2 (Ω ) ≤ ∥m − u∥L2 (Ω ) . Therefore,
∥F (m) − F (u)∥L2 = ≤ = ≤
∥ − (m − u) + g (β Km + β h) − g (β Ku)∥L2 (Ω ) ∥m − u∥L2 (Ω ) + k1 ∥β(Km) − β(Ku)∥L2 (Ω ) ∥m − u∥L2 (Ω ) + k1 β∥K (m − u)∥L2 (Ω ) (1 + k1 β) ∥m − u∥L2 ,
which concludes the proof.
From Proposition 2.1, it follows that the Cauchy problem (1.1) is well posed in L2 (RN ) with a unique global solution, (see [18,19]). More precisely, we have the following corollary. Corollary 2.2. The problem (1.1) has a unique solution for any initial condition in L2 = L2 (RN ), which is globally defined. Remark 2.3. Note that, if u ∈ L2 (RN ) then
|(Ku)(x)| ≤ ∥u∥L2 ,
∀x∈Ω
(2.7)
and
|(K ′ u)(x)| ≤ ∥J ′ ∥L1 ∥u∥L2 ,
∀ x ∈ Ω.
(2.8)
The following result is very similar to Proposition 2.6 of [10] and therefore its proof will be omitted. Proposition 2.4. Assume that the hypotheses (H1) and (H2) hold. Then the function
[F (u)](x) =
−u(x) + g (β(Ku)(x)), 0,
x∈Ω x ̸∈ Ω
is continuously Frechet differentiable in L2 (RN ) with derivative given by
−v(x) + g (β(Ku)(x))β K (v)(x), [DF (u)v](x) = 0,
x∈Ω x ̸∈ Ω .
Remark 2.5. Consider the subspace X of L2 (RN ) given by X = u ∈ L2 (RN ) | u(x) = 0, if x ̸∈ Ω .
Since the range of F is X , this is an invariant subspace for the flow generated by (2.5). In the following sections, we always consider the flow restricted to X , which is an abstract way to impose Dirichlet boundary conditions. 3. Existence of a global attractor We prove, in this section, the existence of a global maximal invariant compact set A in X ⊂ L2 (RN ) for the flow of (1.1), which attracts each bounded set of X (the global attractor, see [20,21]). We recall that a set B ⊂ X is an absorbing set for the flow T (t ) (here, T (t ) denotes the global semi-flow generated by (1.1) in X ) if, for any bounded set C ⊂ X , there is a t1 > 0 such that T (t )C ⊂ B for any t ≥ t1 . The following result is proved in [21]. Theorem 3.1. Let X be a Banach space and T (t ) a semigroup on X . Assume that, for every t , T (t ) = T1 (t ) + T2 (t ) where the operators T1 (·) are uniformly compact for t large, that is, for every bounded set B there exists t0 , which may depend on B, such that
T1 (t )B
t ≥ t0
is relatively compact in X and T2 (t ) is a continuous mapping from X into itself such that the following holds: for every bounded set C ⊂ X , rc (t ) = sup ∥T2 (t )ϕ∥X → 0 as t → ∞. ϕ∈C
Assume also that there exists an open set U and a bounded subset B of U such that B is absorbing in U. Then the ω-limit set of B , A = ω(B ), is a compact attractor which attracts the bounded sets of U. It is the maximal bounded attractor in U (for the inclusion relation). Furthermore, if U is convex and connected, then A is connected.
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√
Lemma 3.2. Assume that the hypotheses (H1) and (H4) hold and let R = a |Ω |. Then, for any ε > 0, the ball of radius R + ε in X is an absorbing set for the flow T (t ) generated by (1.1). Proof. Let u(x, t ) be a solution of (1.1) with initial condition u(x, 0). Then, if x ̸∈ Ω , then u(x, t ) = 0 and, if x ∈ Ω we have, by the variation of the constants formula u(x, t ) = e u(x, 0) + −t
t
e−(t −s) g (β((Ku)(x, s)))ds.
0
Using hypothesis (H4) it follows that
|u(x, t )| ≤ e |u(·, 0)| + −t
t
e−(t −s) |g (β((Ku)(x, s)))|ds
0
≤ e−t |u(x, 0)| + a. Hence,
∥u(·, t )∥L2 ≤ ∥e−t |u(·, 0)| + a∥L2 ≤ e−t ∥u(·, 0)∥L2 + a |Ω |. Therefore, u(·, t ) ∈ B(0, R + ε) for t > ln(
∥u(·,0)∥L2 ε
), and the result is proved.
The next result is an extension of Theorem 3.3 of [12]. Theorem 3.3. Suppose that (H1), (H2) and (H4) hold. √ Then there exists a global attractor A for the flow T (t ) generated by (1.1) in X , which is contained in the ball of radius R = a |Ω |. Proof. If u(x, t ) is the solution of (1.1) with initial condition u(x, 0) we have, if x ∈ Ω , by the variation of constants formula u(x, t ) = e u(x, 0) + −t
t
es−t g (β(Ku(x, s)))ds.
(3.9)
0
Write T1 (t )u(x) =
t
es−t g (β((Ku)(x, s)))ds 0
and T2 (t )u(x) = e−t u(x, 0) and suppose u(·, 0) ∈ C , where C is a bounded set in X . We may suppose that C is contained in the ball of radius ρ . Then
∥T2 (t )u∥L2 → 0 as t → ∞, uniformly in u. √
Also, using (3.9), we have that ∥u(·, t )∥L2 ≤ M, for t ≥ 0, where M = max{ρ, a |Ω |}. Therefore, for t ≥ 0 we have
∂ T1 (t )u(x) = ∂x
t
es−t 0
=β
∂ g (β((Ku)(x, s)))ds ∂x
t
es−t g ′ (β((Ku)(x, s)))(K ′ u)(x, s)ds. 0
Thus
t ∂ T1 (t )u(x) ≤β es−t |g ′ (β(Ku)(x, s))| |(K ′ u)(x, s)|ds. ∂x 0
Using (H2) and Remark 2.3, we obtain
|g ′ (β Ku(x, s))| |(K ′ u)(x, s)| ≤ ≤ ≤
[k2 |β(Ku)(x, s)| + k3 ]|(K ′ u)(x, s)| [k |β(Ku)(x, s)| + k3 ]|(K ′ u)(x, s)| 2 k2 β∥u(·, s)∥L2 + k3 ∥J ′ ∥L1 ∥u(·, s)∥L2
≤ k2 β∥J ′ ∥L1 M 2 + (k3 )∥J ′ ∥L1 M .
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Hence
t ∂ T1 (t )u(x) ≤β es−t k2 β∥J ′ ∥L1 M 2 + (k3 )∥J ′ ∥L1 M ds ∂x 0 2 ′ t s−t e ds = k2 β ∥J ∥L1 M 2 + (k3 β)∥J ′ ∥L1 M 0
≤ k2 β 2 ∥J ′ ∥L1 M 2 + (k3 β)∥J ′ ∥L1 M . It follows that, for t > 0 and any u ∈ C , the value of ∥
∂ T1 (t )u ∥ ∂ x |Ω L2 (Ω ) 1,2
is bounded by a constant (independent of t and u).
Thus, for all u ∈ C , we have that T1 (t )u|Ω belongs to a ball of W (Ω ). From Sobolev’s Embedding Theorem, it follows that 2 2 t ≥0 u∈C T1 (t )u|Ω is relatively compact in L (Ω ). Since X and L (Ω ) are isometric spaces, it follows that
T1 (t )C
t ≥0
is also relatively compact in X . Therefore, the result follows from Theorem 3.1, the attractor A being the set ω-limit of any bounded set in X containing the ball B(0, R). 4. Comparison and boundedness results The following comparison result has been proved in [10] for the case N = 1 (see also [14] for particular case of g ≡ tanh). Its extension for N ≥ 1 is straightforward. Theorem 4.1 (Comparison Theorem). Assume hypotheses (H1) and (H3) hold and let v(x, t ), [V (x, t )] be a sub solution [super solution] of the Cauchy problem of (1.1) with initial condition u(·, 0). Then
v(x, t ) ≤ u(x, t ) ≤ V (x, t ), almost everywhere. Theorem 4.2. Assume the hypotheses (H1) and (H4). Then the attractor A belongs to the ball ∥ · ∥∞ ≤ a in L∞ (RN ).
√
Proof. From Theorem 3.3 it follows that the attractor is contained in the ball B[0, a |Ω |] in L2 (RN ). Let u(x, t ) be a solution of (1.1) in A. Then, if x ∈ Ω , by the variation of constants formula u(x, t ) = e
−(t −t0 )
u(x, t0 ) +
t
e−(t −s) g (β(Ku)(x, s))ds.
t0
Letting t0 → −∞ we obtain, for all (x, t ) ∈ Ω × R+ u(x, t ) =
t
e−(t −s) g (β(Ku)(x, s))ds,
−∞
where the equality above is in the sense of L2 (RN ). Thus, using (H4) again, we have
|u(x, t )| ≤
t
e−(t −s) |g (β(Ku)(x, s))|ds
−∞
t
≤
ae−(t −s) ds
−∞
≤a as claimed.
5. Existence of nonhomogeneous equilibria In this section we exhibit a Lyapunov functional that decreases along the solutions of (1.1), and use it to show the existence of nonhomogeneous equilibria for (1.1), via La Salle’s Invariance Principle (see [22]). Remark 5.1. Since g (0) = 0, under the hypothesis (H1) above, it is easy to see that u ≡ 0 is equilibrium solutions of (1.1). Proposition 5.2. Assume that k1 β < 1. Then u ≡ 0 is the unique equilibrium solution of (1.1).
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Proof. Consider the map Ψ : X → X given by
Ψ (u) := g (β(Ku)). Then u is an equilibrium if and only if it is a fixed point of Ψ . Note that, using (H1) and Generalized Young’s Inequality, obtain
∥Ψ (u) − Ψ (v)∥L2 ≤ k1 β∥K ∥L1 ∥(u − v)∥L2 = k1 β∥(u − v)∥L2 . Thus, since k1 β < 1, it follows that Ψ is a contraction. Hence the result follows from Remark 5.1. Suppose now that k1 β > 1. Let f be given by 1 f (m) = − m2 − β −1 i(m), 2 where i(m) = −
m
g −1 (s)ds.
0
¯ = m+ , is the global minimum of f . It is easy to see that f has two local minimum m− < 0 < m+ , one of which, say m Define m∗ : RN → R by ¯, m 0,
m∗ (x) =
x∈Ω x ̸∈ Ω .
Motivated by [13,14,9–11] we define the following functional in L2 (RN )
F(u) =
Ω
[f (u(x)) − f (m)]dx +
1
4
Ω
where f is given in the hypotheses (H5).
Ω
J (x, z )[u(x) − u(z )]2 dxdz
(5.10)
Note that the functional given in (5.10) is lower bounded, namely, F(u) ≥ 0, for all u. Remark 5.3. It is easy to see that, under hypotheses (H1) and (H4), the functional F is continuous with respect to the topology of L2 (RN ). The following result is proved in [10] using a comparison theorem similar to (4.1). Theorem 5.4. Suppose that the hypotheses (H1)–(H4)hold. Let u(·, t ) be a solution of (1.1) with u(·, t ) ≤ a. Then F(u(·, t )) is differentiable with respect to t for t > 0 and d dt
F(u(·, t )) = −I (u(·, t )) ≤ 0,
where, for any u ∈ L2 = L2 (RN ), I (u(·, t )) =
Ω
[(Ku)(x, t ) − β −1 g −1 (u(x, t ))][−u(x, t ) + g (β(Ku)(x, t ))]dx.
Consequently, I (u) = 0 if and only if u is an equilibrium point. Now, consider m(·, t ) the solution of (1.1) with m(·, 0) = m∗ . Since F(0) > F(m∗ ), it follows from Theorem 5.4 above that the Ω -limit of m∗ does not contain the null stationary solution. Furthermore, from the existence of a global compact attractor we have precompacity of the orbits of T (t ). It follows then by La Salle’s Invariance Principle (see [22]) that m(·, t ) tends to a non constant equilibrium. Using the comparison Theorem 4.1, we can show that this equilibrium is positive. 6. Continuity of the global attractors In this section, we study the continuity of the global attractors with respect to the parameter β at β = β0 . By Proposition 2.4, the map F given in (2.6) is continuously Frechet differentiable in L2 (RN ). Therefore, the problem (2.5) generates a C 1 flow in X which depends on the parameter β . From now on we denote this flow by Tβ (t ). As proved in Section 3 the flow T (t ) = Tβ (t ) admits a global compact attractor, which is denoted by Aβ . We will study the dependence of this attractor with respect to the parameter β .
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Let us recall that a family of subsets {Aβ }, is upper semicontinuous at β0 if dist(Aβ , Aβ0 ) −→ 0,
as β → β0 ,
where dist(Aβ , Aβ0 ) = sup dist(x, Aβ0 ) = sup inf ∥x − y∥L2 .
(6.11)
x∈Aβ y∈Aβ0
x∈Aβ
Analogously, {Aβ } is lower semicontinuous at β0 if dist(Aβ0 , Aβ ) −→ 0,
as β → β0 .
6.1. Upper semicontinuity The upper semicontinuity of the global attractors is an immediate consequence of the continuity of the flow. Lemma 6.1. Under the assumptions (H1) and (H4), the flow Tβ (t ) is continuous with respect to β , uniformly for u in bounded sets and t ∈ [0, b] with b < ∞. Proof. The solutions of (1.1), restricted to Ω , satisfy the ‘variations of constants formula’, Tβ ( t ) u = e u + −t
t
e−(t −s) g β K (Tβ (s)u)ds.
0
Let β0 be fixed, b > 0 and C a bounded set in L2 (RN ). Given ε > 0, we want to find δ > 0 such that |β − β0 | < δ implies
∥Tβ (t )u − Tβ0 (t )u∥L2 < ε, for t ∈ [0, b] and u in C . Since g is globally Lipschitz, for any t > 0 and u ∈ C , it follows that
∥Tβ (t )u − Tβ0 (t )u∥L2 ≤
t
e−(t −s) ∥g (β K (Tβ (s)u)) − g (β0 K (Tβ0 (s)u))∥L2 ds
0 t
≤
e−(t −s) k1 ∥β K (Tβ (s)u) − β0 K (Tβ0 (s)u)∥L2 ds.
0
Subtracting and summing the term β0 K (Tβ (s)u) and using Young’s inequality, we obtain
∥Tβ (t )u − Tβ0 (t )u∥L2 ≤
t
e−(t −s) k1 |β − β0 |∥J ∥L1 ∥Tβ (s)u∥L2 ds +
0
t
e−(t −s) k1 β0 ∥J ∥L1 ∥Tβ (s)u − Tβ0 (s)u∥L2 ds.
0
From Theorem 3.3 in [10] it follows that, for all t ∈ [a, b], ∥Tβ (t )u∥L2 is bounded by a positive constant L depending only of C . Thus, since ∥J ∥L1 = 1, we obtain
t ∥Tβ (t )u − Tβ0 (t )u∥L2 ≤ {Lk1 |β − β0 |} + e−(t −s) k1 β0 ∥Tβ (s)u − Tβ0 (s)u∥L2 ds 0 t = C (β) + k1 β0 ∥Tβ (s)u − Tβ0 (s)u∥L2 ds, 0
where C (β) = {Lk1 |β − β0 |}. Therefore, by Gronwall’s Lemma, it follows that
∥Tβ (t )u − Tβ0 (t )u∥L2 ≤ C (β)ek1 β0 t . From this, the results follows immediately.
The following result is an immediate consequence of Lemma 6.1 and therefore its proof will be omitted. Theorem 6.2. Assume the hypotheses (H1) and (H4). Then, the family of global attractors Aβ is upper semicontinuous with respect to β .
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6.2. Existence and continuity of the local unstable manifolds In order to obtain the existence and continuity of the local unstable manifolds we will need the following additional hypotheses. (H5) For each β0 ≥ 1, the set Eβ0 , of the equilibria of Tβ0 (t ), has only hyperbolic equilibria. (H6) The function g ∈ C 2 (R). We need to assume that the equilibrium points of (2.5) with β0 are stable under perturbation. This stability under perturbation will follow from the hyperbolicity of the equilibrium points. Lemma 6.3. Suppose that the hypotheses (H1)–(H5) hold. Then the set Eβ of the equilibria of Tβ (t ) is continuous with respect to β at β = β0 . Proof. The upper semicontinuity of the equilibria is a consequence of the upper semicontinuity of the global attractors. The lower semicontinuity follows from the Implicit Function Theorem since the equilibria are all hyperbolic. Remark 6.4. Any hyperbolic equilibrium point u of (2.5) with β0 is isolated. The result above guarantees the continuity of equilibrium points. In fact, small neighborhoods of equilibrium points of the problem (2.5) with β0 , have a unique equilibria of the problem (2.5). Therefore, there are only a finite number of hyperbolic equilibrium points of (2.5) for β near β0 . Let us return to Eq. (2.5). Recall that the unstable set Wβu = Wβu (uβ ) of an equilibrium uβ is the set of initial conditions ϕ of (2.5), such that Tβ (t )ϕ is defined for all t ≤ 0 and Tβ (t )ϕ → uβ as t → −∞. For a given neighborhood V of uβ , the set Wβu ∩ V is called a local unstable set of uβ . Using results of [23] we now show that the local unstable sets are actually Lipschitz manifolds in a sufficiently small neighborhood and vary continuously with β . More precisely, we have the following lemma. Lemma 6.5. If u0 is a fixed equilibrium of (2.5) for β = β0 then there is a δ > 0 such that, if |β − β0 | + ∥u0 − uβ ∥L2 < δ and Uβδ := {u ∈ Wβu (uβ ) : ∥u − uβ ∥L2 < δ} then Uβδ is a Lipschitz manifold and dist(Uβδ , Uβδ0 ) + dist(Uβδ0 , Uβδ ) → 0,
as |β − β0 | + ∥u0 − uβ ∥L2 → 0,
with dist defined as in (6.11). Proof. As already mentioned, assuming (H1) and (H2), the map F : L2 (RN ) × R → X , F (u, β) = −u + g (β(K (u))), defined by the right-hand side of (2.5) is continuously Frechet differentiable. Let uβ be an equilibrium of (2.5). Writing u = uβ + v , it follows that u is a solution of (2.5) if and only if v satisfies
∂v = L(β)v + r (uβ , v, β), ∂t
(6.12)
where L(β)v = ∂∂u F (uβ , β) = −v + g ′ (β(K (uβ )))β K (v) and r (uβ , v, β) = F (uβ + v, β) − F (uβ , β) − L(β)v. We rewrite Eq. (6.12) in the form
∂v = L(β0 )v + f (v, β), (6.13) ∂t where f (v, β) = [L(β) − L(β0 )]v + r (uβ , v, β) is the ‘‘nonlinear part’’ of (6.13). Observe that now the ‘‘linear part’’ of (6.13) does not depend on the parameter β , as required by Theorems 2.5 and 3.1 from [23]. To obtain the needed estimates we first observe that, by Hölder inequality
|(K (v))(x)| ≤
√
2τ ∥J ∥∞ ∥v∥L2 ,
∀x ∈ RN
(6.14)
for any v ∈ X . Therefore, since g is of class C 2 , g ′ (β K (uβ )(x) + β K (v)(x)) and g ′′ (β K (uβ )(x) + β K (v)(x)) are bounded by a constant M, for any β in a neighborhood of β0 and ∥v∥L2 (RN ) ≤ 1. We then obtain
∥g ′ (β K (uβ ))β(K (v)) − g ′ (β0 K (uβ0 ))β(K (v))∥2L2 = |g ′ (β K (uβ )(x))β − g ′ (β0 K (uβ0 )(z ))|2 β 2 |K (v)(x)|2 dx RN
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≤ RN
≤ RN
≤ RN
M 2 |[|β K (uβ )(x) − β0 K (uβ0 )(x)|]2 β 2 |K (v)(x)|2 dx M 2 [|β K (uβ )(x) − β0 K (uβ0 )(x)|]2 β 2 2τ ∥J ∥2∞ ∥v∥2L2 dx M 2 [|β K (uβ )(x) − β K (uβ0 )(x)| + |β K (uβ0 )(x) − β0 K (uβ0 )(x)|]2 β 2 2τ ∥J ∥2∞ ∥v∥2L2 dx M 2 [β
≤ RN
√
√
2τ ∥J ∥∞ ∥uβ − uβ0 ∥L2 + |β − β0 | 2τ ∥J ∥∞ ∥uβ0 ∥L2 ]2 β 2 2τ ∥J ∥2∞ ∥v∥2L2 dx
= d1 (β)∥v∥2L2 , with d1 (β) → 0, as β → β0 . Analogously
∥g ′ (β0 K (uβ0 ))(β − β0 )K (v)∥2L2 ≤
RN
≤ RN
∥g ′ (β0 K (uβ0 ))∥2∞ |β − β0 |2 |K (v)(x)|2 dx ∥g ′ (β0 K (uβ0 ))∥2∞ |β − β0 |2 2τ ∥J ∥2∞ ∥v∥2L2 dx
= d2 (β)∥v∥2L2 with d2 (β) → 0, as β → β0 . It follows that
∥ (L(β) − L(β0 )) v∥L2 ≤ ∥g ′ (β K (uβ ))β K (v) − g ′ (β0 K (uβ0 ))β K (v)∥L2 + ∥g ′ (β0 K (uβ0 ))(β − β0 )K (v)∥L2 ≤ C1 (β)∥v∥L2 , √
(6.15)
√
with C1 (β) = d1 (β) + d2 (β) → 0, as β → 0. Observe now that, for any x ∈ RN , we get F (uβ (x) + v(x), β) − F (uβ0 (x) + v(x), β0 )
= [g (β0 K (uβ0 )(x)) − g (β0 K (uβ0 )(x) + β0 K (v)(x))] − [g (β K (uβ )(x)) − g (β K (uβ )(x) + β K (v)(x))] = g ′ (β0 K (uβ0 )(x) + β0 K (¯v (x)))β0 K (v)(x) − g ′ (β K (uβ )(x) + β K (v¯¯ )(x))β K (v)(x), for some v¯ in the segment defined by K (uβ0 ) and K (uβ0 + v) and some v¯¯ in the segment defined by K (uβ ) and K (uβ + v). Then
|F (uβ (x) + v(x), β) − F (uβ0 (x) + v(x), β0 )| ≤ [|g ′ (β0 K (uβ0 )(x) + β0 K (¯v )(x))β0 − g ′ (β0 K (uβ0 )(x) + β0 K (¯v )(x))β| + β|g ′ (β0 K (uβ0 )(x) + β0 K (¯v )(x)) − g ′ (β K (uβ )(x) + β K (v¯¯ )(x))|]|β K (v)(x)|
√ ≤ [M |β − β0 | + β M |β0 K (uβ0 )(x) − β K (uβ )(x)| + β M |β0 K (¯v )(x) − β K (v¯¯ )(x)|] 2τ ∥J ∥∞ ∥v∥L2 ≤ [M |β − β0 | + β M |β − β0 | |K (uβ0 )(x) − K (uβ )(x)| √ + β M (|β − β0 | |K (¯v )(x)| + β|K (¯v )(x) − K (v¯¯ )(x)|)] 2τ ∥J ∥∞ ∥v∥L2 √ √ ≤ [M |β − β0 | + β M |β − β0 | 2τ ∥J ∥∞ ∥uβ − uβ0 ∥L2 + β M |β − β0 | 2τ ∥J ∥∞ ∥¯v ∥L2 √ √ + β 2 M 2τ ∥J ∥∞ ∥¯v − v¯¯ ∥L2 ] 2τ ∥J ∥∞ ∥v∥L2 .
Therefore, since ∥¯v − v¯¯ ∥L2 → 0, as β → β0 ,
∥F (uβ + v, β) − F (uβ0 + v, β0 )∥L2 ≤ C2 (β)∥v∥L2 ,
(6.16)
with C2 (β) → 0, as β → 0. Since r (uβ , v, β) = F (uβ + v, β) − L(β)v , we obtain from (6.15) and (6.16) that
∥r (uβ , v, β) − r (uβ0 , v, β0 )∥ ≤ C3 (β)∥v∥L2 . From (6.15) and (6.17), it follows that
∥f (v, β) − f (v, β0 )∥ ≤ C4 (β)∥v∥L2 , where C4 (β) → 0 as β → β0 . We also obtain which, as ∥v∥L2 , ∥w∥L2 ≤ ρ ,
∥r (uβ (x), v(x), β) − r (uβ (x), w(x), β)∥L2 ≤ ν1 (ρ)∥v − w∥L2 ,
(6.17)
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599
with ν(ρ) → 0, as ρ → 0. Furthermore
∥[L(β) − L(β0 )]v − [L(β) − L(β0 )]w∥L2 ≤ C1 (β) ∥ ∥(v − w)∥L2 . Thus
∥f (v, β) − f (w, β)∥L2 ≤ (ν(ρ) + C1 (β))∥v − w∥L2 ,
(6.18)
where ν(ρ) → 0, as ρ → 0 and ∥v∥L2 , ∥w∥L2 ≤ ρ , and C1 (β) → 0, as β → β0 . Therefore, the conditions of Theorems 2.5 and 3.1 from [23] are satisfied and we obtain the existence of locally invariant sets for (6.13) near the origin, given as graphs of Lipschitz functions which depend continuously on the parameter β near β0 . Using uniqueness of solutions, we can easily prove that these sets coincide with the local unstable manifolds of (6.13). Observing now that the translation u → ( u − uβ ) sends an equilibrium uβ of (2.5) into the origin (which is an equilibrium of (6.13)), the results claimed follow immediately. Using the compacity of the set of equilibria, one can obtain a ‘uniform version’ of Lemma 6.5 that will be needed later. Lemma 6.6. Let β = β0 be fixed. Then, there is a δ > 0 such that, for any equilibrium u0 of (P)β0 , if |β − β0 | + ∥u0 − uβ ∥L2 < δ and Uβδ := {u ∈ Uβ (uβ ) : ∥u − uβ ∥L2 (S 1 ) < δ}; then Uβδ is a Lipschitz manifold and sup dist(Uβδ , Uβδ0 ) + dist(Uβδ0 , Uβδ ) → 0 as |β − β0 | + ∥u0 − uβ ∥L2 → 0,
u0 ∈Eβ
0
with dist defined as in (6.11). Proof. From Lemma 6.5, we know that, for any u0 ∈ Eβ0 , there is a δ = δ(u0 ) such that Uβδ is a Lipschitz manifold, if
|β − β0 | + ∥u0 − uβ ∥L2 < 2δ . Thus, in particular, Uβδ is a Lipschitz manifold, if |β − β0 | + ∥˜u0 − uβ ∥L2 < δ , for any u˜ 0 ∈ Eβ0 with ∥˜u0 − u0 ∥L2 < δ . Taking a finite subcovering of the covering of Eβ0 by balls B(u0 , δ(u0 )), with u0 varying in Eβ0 , the first part of the result follows with δ chosen as the minimum of those δ(u0 ). Now, if ε > 0 and u0 ∈ Eβ0 , there exists, by Lemma 6.5, δ = δ(u0 ) such that, if |β − β0 | + ∥u0 − uβ ∥L2 < 2δ , then dist(Uβδ , Uβδ0 ) + dist(Uβδ 0 , Uβδ ) < ε/2. If u˜ 0 ∈ Eβ0 is such that ∥˜u0 − u0 ∥L2 < δ and |β − β0 | + ∥˜u0 − uβ ∥L2 < δ then, since |β − β0 | + ∥u0 − uβ ∥L2 < 2δ dist(Uβδ (uβ ), Uβδ0 (˜u0 )) + dist(Uβδ0 (˜u0 ), Uβδ (uβ )) < dist(Uβδ (uβ ), Uβδ0 (u0 )) + dist(Uβδ0 (u0 ), Uβδ (uβ ))
+ dist(Uβδ0 (˜u0 ), Uβδ0 (u0 )) + dist(Uβδ0 (u0 ), Uβδ0 (˜u0 )) < ε. By the same procedure above of taking a finite subcovering of the covering of Eβ0 by balls B(u0 , δ(u0 )), and δ the minimum of those δ(u0 ), we conclude that dist(Uβδ (uβ ), Uβδ0 (˜u0 )) + dist(Uβδ0 (˜u0 ), Uβδ (uβ )) < ε if |β − β0 | + ∥˜u0 − uβ ∥L2 < δ , for any u˜ 0 ∈ Eβ0 . This proves the result claimed.
6.3. Lower semicontinuity Now we are ready to prove the lower semicontinuity of the global attractors. Theorem 6.7. Under hypotheses (H1)–(H6), the family of attractors {Aβ } is lower semicontinuous at β0 . Proof. Using Theorems 3.3 and 5.4, Lemmas 6.1, 6.3, 6.5 and 6.6, the result follows from Theorem 4.10.8 of [20].
Acknowledgments The first author was partially supported by FAPESP-Brazil grant 11/04166-5. The second author was partially supported by CNPq-Brazil grants 2003/11021-7, and 03/10042-0. The third author was partially supported by CNPq-Brazil grant Casadinho 620150/2008 and INCTMat.
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References [1] F. Andreu, J.M. Mazón, J.D. Rossi, J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equation, J. Evol. Equ. 8 (1) (2008) 189–215. [2] F. Andreu, J.M. Mazón, J.D. Rossi, J. Toledo, A nonlocal p-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl. (9) 90 (2) (2008) 201–227. [3] F. Andreu, J.M. Mazón, J.D. Rossi, J. Toledo, The limit as p → ∞ in a nonlocal p-Laplacian evolution equation. A nonlocal approximation of a model for sandpiles, Calc. Var. Partial Differential Equations 35 (2009) 279–316. [4] E. Chasseigne, M. Chaves, J.D. Rossi, Asumptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. 86 (2006) 271–291. [5] J.D. Rossi, Asymptotic behaviour of solutions to evolution problems with nonlocal diffusion, in: Course, in: CIEM, Castro Urdiales, Cantabra, Spain, July 6–17, 2009. Available in: http://mate.dm.uba.ar/jrossi/. [6] C. Cortazar, M. Elgueta, J.D. Rossi, A non-local diffusion equation whose solutions develop a free boundary, Ann. Henri Poincaré 6 (2) (2005) 269–281. [7] C. Cortazar, M. Elgueta, J.D. Rossi, N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal. 187 (1) (2008) 137–156. [8] L.I. Ignat, J.D. Rossi, A nonlocal convection–diffusion equation, J. Funct. Anal. 251 (2) (2007) 399–437. [9] A.L. Pereira, Global attractor and nonhomogeneous equilibria for a non local evolution equation in an unbounded domain, J. Differential Equations 226 (2006) 352–372. [10] A.L. Pereira, S.H. da Silva, Existence of global attractors and gradient property for a class of non local evolution equations, São Paulo J. Math. Sci. 2 (2008) 1–20. [11] A.L. Pereira, S.H. da Silva, Continuity of global attractors for a class of non local evolution equations, Discrete Contin. Dyn. Syst. 26 (2010) 1073–1100. [12] S.R.M. Barros, A.L. Pereira, C. Possani, A. Simonis, Spatial periodic equilibria for a non local evolution equation, Discrete Contin. Dyn. Syst. 9 (4) (2003) 937–948. [13] A. Masi, T. Gobron, E. Presutti, Traveling fronts in non local evolution equations, Arch. Ration. Mech. Anal. 132 (1995) 143–205. [14] A. Masi, E. Orland, E. Presutti, L. Triolo, Uniqueness and global stability of the instanton in non local evolution equations, Rend. Mat. Appl. (7) 14 (1994) 693–723. [15] A. Masi, E. Oliveri, E. Presutti, Critical droplet for a non local mean field equation, Markov Process. Related Fields 6 (2000) 439–471. [16] A. Masi, E. Orland, E. Presutti, L. Triolo, Stability of the interface in a model of phase separation, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 1013–1022. [17] G. Folland, Introduction to Partial Differential Equations, Princeton Univ. Press, 1976. [18] H. Brezis, Analisis Funcional, Teoria Y Aplicaciones, Alianza, Madrid, 1984. [19] J.L. Daleckii, M.G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, American Mathematical Society, Providence, Rhode Island, 1974. [20] J.K. Hale, Asymptotic Behavior of Dissipative Systems, in: American Surveys and Monographs, vol. 25, 1988. [21] R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1988. [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, vol. 840, Springer-Verlag, 1981. [23] S.H. da Silva, A.L. Pereira, Exponential trichotomies and continuity of invariant manifolds, São Paulo J. Math. Sci. 5 (2) (2011) 124.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Existence and continuity of global attractors and nonhomogeneous equilibria for a class of evolution equations with non local terms Flank D.M. Bezerra a,∗ , Antônio L. Pereira b , Severino H. da Silva c a
Departamento de Matemática, CCEN/UFPB, Cidade Universitária-Campus I, 58051-900, João Pessoa-PB, Brazil
b
Instituto de Matemática e Estatística-Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo-SP, Brazil Unidade Acadêmica de Matemática e Estatística UAME/CCT/UFCG, Avenida Aprígio Veloso, 882, Bairro Universitário, Caixa Postal: 10.044, 58109-970, Campina Grande-PB, Brazil c
article
info
abstract
Article history: Received 27 May 2012 Available online 29 June 2012 Submitted by Jie Xiao
In this paper, we are concerned with some aspects of the asymptotic behavior of the dynamical system generated by evolution equations with nonlocal terms of the type
Keywords: Global attractors Nonhomogeneous equilibria Continuity Lyapunov functional
where Ω ⊂ RN , β > 0, N ≥ 1 is a bounded smooth domain and K is an integral operator with symmetric kernel
∂ u(x, t ) = −u(x, t ) + g (β(Ku)(x, t )) , ∂t
(Ku)(x) :=
RN
x ∈ Ω, t > 0
J (x, y)u(y)dy.
© 2012 Elsevier Inc. All rights reserved.
1. Introduction A class of problems that have been considered recently are differential equations with non local diffusion terms. Equations of this type appear, for example, in modeling population dynamics, propagation of nerve impulses and problems of phase transition in ferromagnetic models. The mathematical treatment of these equations is generally simpler than similar models based on parabolic partial equations, from the standpoint of the basic theory of existence and well posedness. On the other hand, the qualitative behavior of solutions can be much richer. Thus, many of the issues that arise for models based on parabolic semilinear equations can also be considered in the new context but often require new techniques and ideas. Our goal here is to consider some of these issues for a nonlinear Dirichlet problem with non local terms, namely,
∂ u(x, t ) = −u(x, t ) + g (β(Ku)(x, t )) , ∂t u(x, 0) = u0 (x), u(x, t ) = 0,
x ∈ Ω, t > 0 x∈Ω
(1.1)
x ̸∈ Ω , t > 0
where Ω ⊂ R , N ≥ 1 is a bounded smooth domain, u(x, t ) is a real function on RN × [0, +∞), β > 0 and K is an integral operator with symmetric kernel N
(Ku)(x) := ∗
RN
J (x, y)u(y)dy.
Corresponding author. E-mail addresses: [email protected] (F.D.M. Bezerra), [email protected] (A.L. Pereira), [email protected] (S.H. da Silva).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.06.042
F.D.M. Bezerra et al. / J. Math. Anal. Appl. 396 (2012) 590–600
591
Here, J is an even non negative function of class C 2 with RN J (x, y)dy = 1, and g : R → R is a nonlinear real function of class C 1 with g (0) = 0. Equation (1.1) is called a non local diffusion equation since the diffusion of the density u at a point x and time t depends on the values of u in a neighborhood of x in RN through the term Ku. The asymptotic behavior of solutions of non local diffusion equations has been extensively studied over the past ten years (see [1–8]). Often, the non local term in (1.1)-like problems is given by convolution. We extend some of these results to (1.1), and consider also some different aspects of the asymptotic behavior of the infinite dimensional dynamical system it generates. For N = 1 there are several works in the literature dedicated to the analysis of similar models (see, [9–11], where existence and continuity of global attractors is proved) and, in particular, for the case of (1.1) with g ≡ tanh, and J (x, y) = J (x − y) that is
∂ u(x, t ) = −u(x, t ) + tanh (β(J ∗ u)(x, t )) , ∂t
t >0
(1.2)
(see [12–15,9]). In this case, if β ≤ 1, Eq. (1.2) has only one (stable) equilibrium, (see [15,9]). If β > 1, the equation mβ = tanh(β mβ )
(1.3)
+ 0 has three roots m− β , mβ , mβ , which are then spatially homogeneous equilibria of Eq. (1.3). The existence and uniqueness + (modulo translation) of a traveling front connecting the equilibria m− β and mβ is proved in [13]. The existence and uniqueness
(modulo translation) of a solution tending asymptotically to ±m+ β , referred to as the ‘‘instanton’’ is proved in [14,16].
This paper is organized as follows. In Section 2, we prove that (1.1) generates a flow in L2 (RN ). Section 3 is dedicated to the proof of existence of the global attractor (for the case N ≥ 1), generalizing results of [12,10,11]. In Section 4, we prove a comparison result, generalizing Theorem 2.7 of [14] and Theorem 4.2 of [10]. In Section 5, we exhibit a Lyapunov functional for the flow of (1.1), and used it to prove the existence of non trivial equilibrium. Finally, in Section 6, we prove continuity of the global attractor with respect to the parameter β . We collect here the conditions on g which will be used as hypotheses when needed. (H1) The function g : R → R, is globally Lipschitz, with g (0) = 0. That is, there exists a positive constant k1 such that
|g (x) − g (y)| ≤ k1 |x − y|,
∀ x, y ∈ R .
In particular,
|g (x)| ≤ k1 |x|,
∀ x ∈ R.
(1.4)
(H2) The function g ∈ C (R) and g is Lipschitz with constant k2 . In particular 1
′
|g ′ (x)| ≤ k2 |x| + k3 ∀ x ∈ R, for some k3 > 0. (H3) The function g has a positive derivative. In particular, it is strictly increasing. (H4) There exists a > 0 such that |g (x)| < a < ∞, for all x ∈ R. In this paper, |Ω | denotes the Lebesgue measure of Ω in RN . 2. Estimates and well posedness In this section, in order to obtain well posedness of (1.1), we initially consider the following Cauchy problem in the space L2 (RN )
∂ u(x, t ) = (Fu)(x, t ) ∂t u(x, 0) = u0
(2.5)
where F : L2 (RN ) → L2 (RN ) is given by
[F (u)](x) =
−u(x) + g (β(Ku)(x)), 0,
x∈Ω x ̸∈ Ω .
(2.6)
Proposition 2.1. Suppose that the hypothesis (H1) holds. Then the function F given above is uniformly Lipschitz in L2 = L2 (RN ).
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Proof. From Generalized Young’s Inequality, (see [17]),
∥K (m − u)∥L2 (Ω ) ≤ ∥m − u∥L2 (Ω ) . Therefore,
∥F (m) − F (u)∥L2 = ≤ = ≤
∥ − (m − u) + g (β Km + β h) − g (β Ku)∥L2 (Ω ) ∥m − u∥L2 (Ω ) + k1 ∥β(Km) − β(Ku)∥L2 (Ω ) ∥m − u∥L2 (Ω ) + k1 β∥K (m − u)∥L2 (Ω ) (1 + k1 β) ∥m − u∥L2 ,
which concludes the proof.
From Proposition 2.1, it follows that the Cauchy problem (1.1) is well posed in L2 (RN ) with a unique global solution, (see [18,19]). More precisely, we have the following corollary. Corollary 2.2. The problem (1.1) has a unique solution for any initial condition in L2 = L2 (RN ), which is globally defined. Remark 2.3. Note that, if u ∈ L2 (RN ) then
|(Ku)(x)| ≤ ∥u∥L2 ,
∀x∈Ω
(2.7)
and
|(K ′ u)(x)| ≤ ∥J ′ ∥L1 ∥u∥L2 ,
∀ x ∈ Ω.
(2.8)
The following result is very similar to Proposition 2.6 of [10] and therefore its proof will be omitted. Proposition 2.4. Assume that the hypotheses (H1) and (H2) hold. Then the function
[F (u)](x) =
−u(x) + g (β(Ku)(x)), 0,
x∈Ω x ̸∈ Ω
is continuously Frechet differentiable in L2 (RN ) with derivative given by
−v(x) + g (β(Ku)(x))β K (v)(x), [DF (u)v](x) = 0,
x∈Ω x ̸∈ Ω .
Remark 2.5. Consider the subspace X of L2 (RN ) given by X = u ∈ L2 (RN ) | u(x) = 0, if x ̸∈ Ω .
Since the range of F is X , this is an invariant subspace for the flow generated by (2.5). In the following sections, we always consider the flow restricted to X , which is an abstract way to impose Dirichlet boundary conditions. 3. Existence of a global attractor We prove, in this section, the existence of a global maximal invariant compact set A in X ⊂ L2 (RN ) for the flow of (1.1), which attracts each bounded set of X (the global attractor, see [20,21]). We recall that a set B ⊂ X is an absorbing set for the flow T (t ) (here, T (t ) denotes the global semi-flow generated by (1.1) in X ) if, for any bounded set C ⊂ X , there is a t1 > 0 such that T (t )C ⊂ B for any t ≥ t1 . The following result is proved in [21]. Theorem 3.1. Let X be a Banach space and T (t ) a semigroup on X . Assume that, for every t , T (t ) = T1 (t ) + T2 (t ) where the operators T1 (·) are uniformly compact for t large, that is, for every bounded set B there exists t0 , which may depend on B, such that
T1 (t )B
t ≥ t0
is relatively compact in X and T2 (t ) is a continuous mapping from X into itself such that the following holds: for every bounded set C ⊂ X , rc (t ) = sup ∥T2 (t )ϕ∥X → 0 as t → ∞. ϕ∈C
Assume also that there exists an open set U and a bounded subset B of U such that B is absorbing in U. Then the ω-limit set of B , A = ω(B ), is a compact attractor which attracts the bounded sets of U. It is the maximal bounded attractor in U (for the inclusion relation). Furthermore, if U is convex and connected, then A is connected.
F.D.M. Bezerra et al. / J. Math. Anal. Appl. 396 (2012) 590–600
593
√
Lemma 3.2. Assume that the hypotheses (H1) and (H4) hold and let R = a |Ω |. Then, for any ε > 0, the ball of radius R + ε in X is an absorbing set for the flow T (t ) generated by (1.1). Proof. Let u(x, t ) be a solution of (1.1) with initial condition u(x, 0). Then, if x ̸∈ Ω , then u(x, t ) = 0 and, if x ∈ Ω we have, by the variation of the constants formula u(x, t ) = e u(x, 0) + −t
t
e−(t −s) g (β((Ku)(x, s)))ds.
0
Using hypothesis (H4) it follows that
|u(x, t )| ≤ e |u(·, 0)| + −t
t
e−(t −s) |g (β((Ku)(x, s)))|ds
0
≤ e−t |u(x, 0)| + a. Hence,
∥u(·, t )∥L2 ≤ ∥e−t |u(·, 0)| + a∥L2 ≤ e−t ∥u(·, 0)∥L2 + a |Ω |. Therefore, u(·, t ) ∈ B(0, R + ε) for t > ln(
∥u(·,0)∥L2 ε
), and the result is proved.
The next result is an extension of Theorem 3.3 of [12]. Theorem 3.3. Suppose that (H1), (H2) and (H4) hold. √ Then there exists a global attractor A for the flow T (t ) generated by (1.1) in X , which is contained in the ball of radius R = a |Ω |. Proof. If u(x, t ) is the solution of (1.1) with initial condition u(x, 0) we have, if x ∈ Ω , by the variation of constants formula u(x, t ) = e u(x, 0) + −t
t
es−t g (β(Ku(x, s)))ds.
(3.9)
0
Write T1 (t )u(x) =
t
es−t g (β((Ku)(x, s)))ds 0
and T2 (t )u(x) = e−t u(x, 0) and suppose u(·, 0) ∈ C , where C is a bounded set in X . We may suppose that C is contained in the ball of radius ρ . Then
∥T2 (t )u∥L2 → 0 as t → ∞, uniformly in u. √
Also, using (3.9), we have that ∥u(·, t )∥L2 ≤ M, for t ≥ 0, where M = max{ρ, a |Ω |}. Therefore, for t ≥ 0 we have
∂ T1 (t )u(x) = ∂x
t
es−t 0
=β
∂ g (β((Ku)(x, s)))ds ∂x
t
es−t g ′ (β((Ku)(x, s)))(K ′ u)(x, s)ds. 0
Thus
t ∂ T1 (t )u(x) ≤β es−t |g ′ (β(Ku)(x, s))| |(K ′ u)(x, s)|ds. ∂x 0
Using (H2) and Remark 2.3, we obtain
|g ′ (β Ku(x, s))| |(K ′ u)(x, s)| ≤ ≤ ≤
[k2 |β(Ku)(x, s)| + k3 ]|(K ′ u)(x, s)| [k |β(Ku)(x, s)| + k3 ]|(K ′ u)(x, s)| 2 k2 β∥u(·, s)∥L2 + k3 ∥J ′ ∥L1 ∥u(·, s)∥L2
≤ k2 β∥J ′ ∥L1 M 2 + (k3 )∥J ′ ∥L1 M .
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Hence
t ∂ T1 (t )u(x) ≤β es−t k2 β∥J ′ ∥L1 M 2 + (k3 )∥J ′ ∥L1 M ds ∂x 0 2 ′ t s−t e ds = k2 β ∥J ∥L1 M 2 + (k3 β)∥J ′ ∥L1 M 0
≤ k2 β 2 ∥J ′ ∥L1 M 2 + (k3 β)∥J ′ ∥L1 M . It follows that, for t > 0 and any u ∈ C , the value of ∥
∂ T1 (t )u ∥ ∂ x |Ω L2 (Ω ) 1,2
is bounded by a constant (independent of t and u).
Thus, for all u ∈ C , we have that T1 (t )u|Ω belongs to a ball of W (Ω ). From Sobolev’s Embedding Theorem, it follows that 2 2 t ≥0 u∈C T1 (t )u|Ω is relatively compact in L (Ω ). Since X and L (Ω ) are isometric spaces, it follows that
T1 (t )C
t ≥0
is also relatively compact in X . Therefore, the result follows from Theorem 3.1, the attractor A being the set ω-limit of any bounded set in X containing the ball B(0, R). 4. Comparison and boundedness results The following comparison result has been proved in [10] for the case N = 1 (see also [14] for particular case of g ≡ tanh). Its extension for N ≥ 1 is straightforward. Theorem 4.1 (Comparison Theorem). Assume hypotheses (H1) and (H3) hold and let v(x, t ), [V (x, t )] be a sub solution [super solution] of the Cauchy problem of (1.1) with initial condition u(·, 0). Then
v(x, t ) ≤ u(x, t ) ≤ V (x, t ), almost everywhere. Theorem 4.2. Assume the hypotheses (H1) and (H4). Then the attractor A belongs to the ball ∥ · ∥∞ ≤ a in L∞ (RN ).
√
Proof. From Theorem 3.3 it follows that the attractor is contained in the ball B[0, a |Ω |] in L2 (RN ). Let u(x, t ) be a solution of (1.1) in A. Then, if x ∈ Ω , by the variation of constants formula u(x, t ) = e
−(t −t0 )
u(x, t0 ) +
t
e−(t −s) g (β(Ku)(x, s))ds.
t0
Letting t0 → −∞ we obtain, for all (x, t ) ∈ Ω × R+ u(x, t ) =
t
e−(t −s) g (β(Ku)(x, s))ds,
−∞
where the equality above is in the sense of L2 (RN ). Thus, using (H4) again, we have
|u(x, t )| ≤
t
e−(t −s) |g (β(Ku)(x, s))|ds
−∞
t
≤
ae−(t −s) ds
−∞
≤a as claimed.
5. Existence of nonhomogeneous equilibria In this section we exhibit a Lyapunov functional that decreases along the solutions of (1.1), and use it to show the existence of nonhomogeneous equilibria for (1.1), via La Salle’s Invariance Principle (see [22]). Remark 5.1. Since g (0) = 0, under the hypothesis (H1) above, it is easy to see that u ≡ 0 is equilibrium solutions of (1.1). Proposition 5.2. Assume that k1 β < 1. Then u ≡ 0 is the unique equilibrium solution of (1.1).
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Proof. Consider the map Ψ : X → X given by
Ψ (u) := g (β(Ku)). Then u is an equilibrium if and only if it is a fixed point of Ψ . Note that, using (H1) and Generalized Young’s Inequality, obtain
∥Ψ (u) − Ψ (v)∥L2 ≤ k1 β∥K ∥L1 ∥(u − v)∥L2 = k1 β∥(u − v)∥L2 . Thus, since k1 β < 1, it follows that Ψ is a contraction. Hence the result follows from Remark 5.1. Suppose now that k1 β > 1. Let f be given by 1 f (m) = − m2 − β −1 i(m), 2 where i(m) = −
m
g −1 (s)ds.
0
¯ = m+ , is the global minimum of f . It is easy to see that f has two local minimum m− < 0 < m+ , one of which, say m Define m∗ : RN → R by ¯, m 0,
m∗ (x) =
x∈Ω x ̸∈ Ω .
Motivated by [13,14,9–11] we define the following functional in L2 (RN )
F(u) =
Ω
[f (u(x)) − f (m)]dx +
1
4
Ω
where f is given in the hypotheses (H5).
Ω
J (x, z )[u(x) − u(z )]2 dxdz
(5.10)
Note that the functional given in (5.10) is lower bounded, namely, F(u) ≥ 0, for all u. Remark 5.3. It is easy to see that, under hypotheses (H1) and (H4), the functional F is continuous with respect to the topology of L2 (RN ). The following result is proved in [10] using a comparison theorem similar to (4.1). Theorem 5.4. Suppose that the hypotheses (H1)–(H4)hold. Let u(·, t ) be a solution of (1.1) with u(·, t ) ≤ a. Then F(u(·, t )) is differentiable with respect to t for t > 0 and d dt
F(u(·, t )) = −I (u(·, t )) ≤ 0,
where, for any u ∈ L2 = L2 (RN ), I (u(·, t )) =
Ω
[(Ku)(x, t ) − β −1 g −1 (u(x, t ))][−u(x, t ) + g (β(Ku)(x, t ))]dx.
Consequently, I (u) = 0 if and only if u is an equilibrium point. Now, consider m(·, t ) the solution of (1.1) with m(·, 0) = m∗ . Since F(0) > F(m∗ ), it follows from Theorem 5.4 above that the Ω -limit of m∗ does not contain the null stationary solution. Furthermore, from the existence of a global compact attractor we have precompacity of the orbits of T (t ). It follows then by La Salle’s Invariance Principle (see [22]) that m(·, t ) tends to a non constant equilibrium. Using the comparison Theorem 4.1, we can show that this equilibrium is positive. 6. Continuity of the global attractors In this section, we study the continuity of the global attractors with respect to the parameter β at β = β0 . By Proposition 2.4, the map F given in (2.6) is continuously Frechet differentiable in L2 (RN ). Therefore, the problem (2.5) generates a C 1 flow in X which depends on the parameter β . From now on we denote this flow by Tβ (t ). As proved in Section 3 the flow T (t ) = Tβ (t ) admits a global compact attractor, which is denoted by Aβ . We will study the dependence of this attractor with respect to the parameter β .
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Let us recall that a family of subsets {Aβ }, is upper semicontinuous at β0 if dist(Aβ , Aβ0 ) −→ 0,
as β → β0 ,
where dist(Aβ , Aβ0 ) = sup dist(x, Aβ0 ) = sup inf ∥x − y∥L2 .
(6.11)
x∈Aβ y∈Aβ0
x∈Aβ
Analogously, {Aβ } is lower semicontinuous at β0 if dist(Aβ0 , Aβ ) −→ 0,
as β → β0 .
6.1. Upper semicontinuity The upper semicontinuity of the global attractors is an immediate consequence of the continuity of the flow. Lemma 6.1. Under the assumptions (H1) and (H4), the flow Tβ (t ) is continuous with respect to β , uniformly for u in bounded sets and t ∈ [0, b] with b < ∞. Proof. The solutions of (1.1), restricted to Ω , satisfy the ‘variations of constants formula’, Tβ ( t ) u = e u + −t
t
e−(t −s) g β K (Tβ (s)u)ds.
0
Let β0 be fixed, b > 0 and C a bounded set in L2 (RN ). Given ε > 0, we want to find δ > 0 such that |β − β0 | < δ implies
∥Tβ (t )u − Tβ0 (t )u∥L2 < ε, for t ∈ [0, b] and u in C . Since g is globally Lipschitz, for any t > 0 and u ∈ C , it follows that
∥Tβ (t )u − Tβ0 (t )u∥L2 ≤
t
e−(t −s) ∥g (β K (Tβ (s)u)) − g (β0 K (Tβ0 (s)u))∥L2 ds
0 t
≤
e−(t −s) k1 ∥β K (Tβ (s)u) − β0 K (Tβ0 (s)u)∥L2 ds.
0
Subtracting and summing the term β0 K (Tβ (s)u) and using Young’s inequality, we obtain
∥Tβ (t )u − Tβ0 (t )u∥L2 ≤
t
e−(t −s) k1 |β − β0 |∥J ∥L1 ∥Tβ (s)u∥L2 ds +
0
t
e−(t −s) k1 β0 ∥J ∥L1 ∥Tβ (s)u − Tβ0 (s)u∥L2 ds.
0
From Theorem 3.3 in [10] it follows that, for all t ∈ [a, b], ∥Tβ (t )u∥L2 is bounded by a positive constant L depending only of C . Thus, since ∥J ∥L1 = 1, we obtain
t ∥Tβ (t )u − Tβ0 (t )u∥L2 ≤ {Lk1 |β − β0 |} + e−(t −s) k1 β0 ∥Tβ (s)u − Tβ0 (s)u∥L2 ds 0 t = C (β) + k1 β0 ∥Tβ (s)u − Tβ0 (s)u∥L2 ds, 0
where C (β) = {Lk1 |β − β0 |}. Therefore, by Gronwall’s Lemma, it follows that
∥Tβ (t )u − Tβ0 (t )u∥L2 ≤ C (β)ek1 β0 t . From this, the results follows immediately.
The following result is an immediate consequence of Lemma 6.1 and therefore its proof will be omitted. Theorem 6.2. Assume the hypotheses (H1) and (H4). Then, the family of global attractors Aβ is upper semicontinuous with respect to β .
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6.2. Existence and continuity of the local unstable manifolds In order to obtain the existence and continuity of the local unstable manifolds we will need the following additional hypotheses. (H5) For each β0 ≥ 1, the set Eβ0 , of the equilibria of Tβ0 (t ), has only hyperbolic equilibria. (H6) The function g ∈ C 2 (R). We need to assume that the equilibrium points of (2.5) with β0 are stable under perturbation. This stability under perturbation will follow from the hyperbolicity of the equilibrium points. Lemma 6.3. Suppose that the hypotheses (H1)–(H5) hold. Then the set Eβ of the equilibria of Tβ (t ) is continuous with respect to β at β = β0 . Proof. The upper semicontinuity of the equilibria is a consequence of the upper semicontinuity of the global attractors. The lower semicontinuity follows from the Implicit Function Theorem since the equilibria are all hyperbolic. Remark 6.4. Any hyperbolic equilibrium point u of (2.5) with β0 is isolated. The result above guarantees the continuity of equilibrium points. In fact, small neighborhoods of equilibrium points of the problem (2.5) with β0 , have a unique equilibria of the problem (2.5). Therefore, there are only a finite number of hyperbolic equilibrium points of (2.5) for β near β0 . Let us return to Eq. (2.5). Recall that the unstable set Wβu = Wβu (uβ ) of an equilibrium uβ is the set of initial conditions ϕ of (2.5), such that Tβ (t )ϕ is defined for all t ≤ 0 and Tβ (t )ϕ → uβ as t → −∞. For a given neighborhood V of uβ , the set Wβu ∩ V is called a local unstable set of uβ . Using results of [23] we now show that the local unstable sets are actually Lipschitz manifolds in a sufficiently small neighborhood and vary continuously with β . More precisely, we have the following lemma. Lemma 6.5. If u0 is a fixed equilibrium of (2.5) for β = β0 then there is a δ > 0 such that, if |β − β0 | + ∥u0 − uβ ∥L2 < δ and Uβδ := {u ∈ Wβu (uβ ) : ∥u − uβ ∥L2 < δ} then Uβδ is a Lipschitz manifold and dist(Uβδ , Uβδ0 ) + dist(Uβδ0 , Uβδ ) → 0,
as |β − β0 | + ∥u0 − uβ ∥L2 → 0,
with dist defined as in (6.11). Proof. As already mentioned, assuming (H1) and (H2), the map F : L2 (RN ) × R → X , F (u, β) = −u + g (β(K (u))), defined by the right-hand side of (2.5) is continuously Frechet differentiable. Let uβ be an equilibrium of (2.5). Writing u = uβ + v , it follows that u is a solution of (2.5) if and only if v satisfies
∂v = L(β)v + r (uβ , v, β), ∂t
(6.12)
where L(β)v = ∂∂u F (uβ , β) = −v + g ′ (β(K (uβ )))β K (v) and r (uβ , v, β) = F (uβ + v, β) − F (uβ , β) − L(β)v. We rewrite Eq. (6.12) in the form
∂v = L(β0 )v + f (v, β), (6.13) ∂t where f (v, β) = [L(β) − L(β0 )]v + r (uβ , v, β) is the ‘‘nonlinear part’’ of (6.13). Observe that now the ‘‘linear part’’ of (6.13) does not depend on the parameter β , as required by Theorems 2.5 and 3.1 from [23]. To obtain the needed estimates we first observe that, by Hölder inequality
|(K (v))(x)| ≤
√
2τ ∥J ∥∞ ∥v∥L2 ,
∀x ∈ RN
(6.14)
for any v ∈ X . Therefore, since g is of class C 2 , g ′ (β K (uβ )(x) + β K (v)(x)) and g ′′ (β K (uβ )(x) + β K (v)(x)) are bounded by a constant M, for any β in a neighborhood of β0 and ∥v∥L2 (RN ) ≤ 1. We then obtain
∥g ′ (β K (uβ ))β(K (v)) − g ′ (β0 K (uβ0 ))β(K (v))∥2L2 = |g ′ (β K (uβ )(x))β − g ′ (β0 K (uβ0 )(z ))|2 β 2 |K (v)(x)|2 dx RN
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≤ RN
≤ RN
≤ RN
M 2 |[|β K (uβ )(x) − β0 K (uβ0 )(x)|]2 β 2 |K (v)(x)|2 dx M 2 [|β K (uβ )(x) − β0 K (uβ0 )(x)|]2 β 2 2τ ∥J ∥2∞ ∥v∥2L2 dx M 2 [|β K (uβ )(x) − β K (uβ0 )(x)| + |β K (uβ0 )(x) − β0 K (uβ0 )(x)|]2 β 2 2τ ∥J ∥2∞ ∥v∥2L2 dx M 2 [β
≤ RN
√
√
2τ ∥J ∥∞ ∥uβ − uβ0 ∥L2 + |β − β0 | 2τ ∥J ∥∞ ∥uβ0 ∥L2 ]2 β 2 2τ ∥J ∥2∞ ∥v∥2L2 dx
= d1 (β)∥v∥2L2 , with d1 (β) → 0, as β → β0 . Analogously
∥g ′ (β0 K (uβ0 ))(β − β0 )K (v)∥2L2 ≤
RN
≤ RN
∥g ′ (β0 K (uβ0 ))∥2∞ |β − β0 |2 |K (v)(x)|2 dx ∥g ′ (β0 K (uβ0 ))∥2∞ |β − β0 |2 2τ ∥J ∥2∞ ∥v∥2L2 dx
= d2 (β)∥v∥2L2 with d2 (β) → 0, as β → β0 . It follows that
∥ (L(β) − L(β0 )) v∥L2 ≤ ∥g ′ (β K (uβ ))β K (v) − g ′ (β0 K (uβ0 ))β K (v)∥L2 + ∥g ′ (β0 K (uβ0 ))(β − β0 )K (v)∥L2 ≤ C1 (β)∥v∥L2 , √
(6.15)
√
with C1 (β) = d1 (β) + d2 (β) → 0, as β → 0. Observe now that, for any x ∈ RN , we get F (uβ (x) + v(x), β) − F (uβ0 (x) + v(x), β0 )
= [g (β0 K (uβ0 )(x)) − g (β0 K (uβ0 )(x) + β0 K (v)(x))] − [g (β K (uβ )(x)) − g (β K (uβ )(x) + β K (v)(x))] = g ′ (β0 K (uβ0 )(x) + β0 K (¯v (x)))β0 K (v)(x) − g ′ (β K (uβ )(x) + β K (v¯¯ )(x))β K (v)(x), for some v¯ in the segment defined by K (uβ0 ) and K (uβ0 + v) and some v¯¯ in the segment defined by K (uβ ) and K (uβ + v). Then
|F (uβ (x) + v(x), β) − F (uβ0 (x) + v(x), β0 )| ≤ [|g ′ (β0 K (uβ0 )(x) + β0 K (¯v )(x))β0 − g ′ (β0 K (uβ0 )(x) + β0 K (¯v )(x))β| + β|g ′ (β0 K (uβ0 )(x) + β0 K (¯v )(x)) − g ′ (β K (uβ )(x) + β K (v¯¯ )(x))|]|β K (v)(x)|
√ ≤ [M |β − β0 | + β M |β0 K (uβ0 )(x) − β K (uβ )(x)| + β M |β0 K (¯v )(x) − β K (v¯¯ )(x)|] 2τ ∥J ∥∞ ∥v∥L2 ≤ [M |β − β0 | + β M |β − β0 | |K (uβ0 )(x) − K (uβ )(x)| √ + β M (|β − β0 | |K (¯v )(x)| + β|K (¯v )(x) − K (v¯¯ )(x)|)] 2τ ∥J ∥∞ ∥v∥L2 √ √ ≤ [M |β − β0 | + β M |β − β0 | 2τ ∥J ∥∞ ∥uβ − uβ0 ∥L2 + β M |β − β0 | 2τ ∥J ∥∞ ∥¯v ∥L2 √ √ + β 2 M 2τ ∥J ∥∞ ∥¯v − v¯¯ ∥L2 ] 2τ ∥J ∥∞ ∥v∥L2 .
Therefore, since ∥¯v − v¯¯ ∥L2 → 0, as β → β0 ,
∥F (uβ + v, β) − F (uβ0 + v, β0 )∥L2 ≤ C2 (β)∥v∥L2 ,
(6.16)
with C2 (β) → 0, as β → 0. Since r (uβ , v, β) = F (uβ + v, β) − L(β)v , we obtain from (6.15) and (6.16) that
∥r (uβ , v, β) − r (uβ0 , v, β0 )∥ ≤ C3 (β)∥v∥L2 . From (6.15) and (6.17), it follows that
∥f (v, β) − f (v, β0 )∥ ≤ C4 (β)∥v∥L2 , where C4 (β) → 0 as β → β0 . We also obtain which, as ∥v∥L2 , ∥w∥L2 ≤ ρ ,
∥r (uβ (x), v(x), β) − r (uβ (x), w(x), β)∥L2 ≤ ν1 (ρ)∥v − w∥L2 ,
(6.17)
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with ν(ρ) → 0, as ρ → 0. Furthermore
∥[L(β) − L(β0 )]v − [L(β) − L(β0 )]w∥L2 ≤ C1 (β) ∥ ∥(v − w)∥L2 . Thus
∥f (v, β) − f (w, β)∥L2 ≤ (ν(ρ) + C1 (β))∥v − w∥L2 ,
(6.18)
where ν(ρ) → 0, as ρ → 0 and ∥v∥L2 , ∥w∥L2 ≤ ρ , and C1 (β) → 0, as β → β0 . Therefore, the conditions of Theorems 2.5 and 3.1 from [23] are satisfied and we obtain the existence of locally invariant sets for (6.13) near the origin, given as graphs of Lipschitz functions which depend continuously on the parameter β near β0 . Using uniqueness of solutions, we can easily prove that these sets coincide with the local unstable manifolds of (6.13). Observing now that the translation u → ( u − uβ ) sends an equilibrium uβ of (2.5) into the origin (which is an equilibrium of (6.13)), the results claimed follow immediately. Using the compacity of the set of equilibria, one can obtain a ‘uniform version’ of Lemma 6.5 that will be needed later. Lemma 6.6. Let β = β0 be fixed. Then, there is a δ > 0 such that, for any equilibrium u0 of (P)β0 , if |β − β0 | + ∥u0 − uβ ∥L2 < δ and Uβδ := {u ∈ Uβ (uβ ) : ∥u − uβ ∥L2 (S 1 ) < δ}; then Uβδ is a Lipschitz manifold and sup dist(Uβδ , Uβδ0 ) + dist(Uβδ0 , Uβδ ) → 0 as |β − β0 | + ∥u0 − uβ ∥L2 → 0,
u0 ∈Eβ
0
with dist defined as in (6.11). Proof. From Lemma 6.5, we know that, for any u0 ∈ Eβ0 , there is a δ = δ(u0 ) such that Uβδ is a Lipschitz manifold, if
|β − β0 | + ∥u0 − uβ ∥L2 < 2δ . Thus, in particular, Uβδ is a Lipschitz manifold, if |β − β0 | + ∥˜u0 − uβ ∥L2 < δ , for any u˜ 0 ∈ Eβ0 with ∥˜u0 − u0 ∥L2 < δ . Taking a finite subcovering of the covering of Eβ0 by balls B(u0 , δ(u0 )), with u0 varying in Eβ0 , the first part of the result follows with δ chosen as the minimum of those δ(u0 ). Now, if ε > 0 and u0 ∈ Eβ0 , there exists, by Lemma 6.5, δ = δ(u0 ) such that, if |β − β0 | + ∥u0 − uβ ∥L2 < 2δ , then dist(Uβδ , Uβδ0 ) + dist(Uβδ 0 , Uβδ ) < ε/2. If u˜ 0 ∈ Eβ0 is such that ∥˜u0 − u0 ∥L2 < δ and |β − β0 | + ∥˜u0 − uβ ∥L2 < δ then, since |β − β0 | + ∥u0 − uβ ∥L2 < 2δ dist(Uβδ (uβ ), Uβδ0 (˜u0 )) + dist(Uβδ0 (˜u0 ), Uβδ (uβ )) < dist(Uβδ (uβ ), Uβδ0 (u0 )) + dist(Uβδ0 (u0 ), Uβδ (uβ ))
+ dist(Uβδ0 (˜u0 ), Uβδ0 (u0 )) + dist(Uβδ0 (u0 ), Uβδ0 (˜u0 )) < ε. By the same procedure above of taking a finite subcovering of the covering of Eβ0 by balls B(u0 , δ(u0 )), and δ the minimum of those δ(u0 ), we conclude that dist(Uβδ (uβ ), Uβδ0 (˜u0 )) + dist(Uβδ0 (˜u0 ), Uβδ (uβ )) < ε if |β − β0 | + ∥˜u0 − uβ ∥L2 < δ , for any u˜ 0 ∈ Eβ0 . This proves the result claimed.
6.3. Lower semicontinuity Now we are ready to prove the lower semicontinuity of the global attractors. Theorem 6.7. Under hypotheses (H1)–(H6), the family of attractors {Aβ } is lower semicontinuous at β0 . Proof. Using Theorems 3.3 and 5.4, Lemmas 6.1, 6.3, 6.5 and 6.6, the result follows from Theorem 4.10.8 of [20].
Acknowledgments The first author was partially supported by FAPESP-Brazil grant 11/04166-5. The second author was partially supported by CNPq-Brazil grants 2003/11021-7, and 03/10042-0. The third author was partially supported by CNPq-Brazil grant Casadinho 620150/2008 and INCTMat.
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