Differ Equ Dyn Syst DOI 10.1007/s12591-016-0302-1 ORIGINAL RESEARCH

Pullback Attractors for a Nonlocal Nonautonomous Evolution Model in R N Flank D. M. Bezerra1 · Miriam da S. Pereira1 · Severino H. da Silva2

© Foundation for Scientific Research and Technological Innovation 2016

Abstract In this work we consider the nonlocal evolution equation with time-dependent terms which arises in models of phase separation in R N ∂t u = −u + g (β(J ∗ u) + βh(t)) under some restrictions on h, growth restrictions on the nonlinear term g and β > 1. We prove the existence, regularity and upper-semicontinuity of pullback attractors with respect to functional parameter h(t) in some weighted spaces. Keywords Nonlocal evolution equation · Pullback attractors · Weighted spaces · Upper semicontinuity Mathematics Subject Classification 35B40 · 35B41 · 37B55

Introduction In this paper we study the following nonlocal nonautonomous evolution equation  ∂t u(t, x) = −u(t, x)+ g (β(J ∗ u)(t, x) + βh(t)) , (J ∗u)(t, x) = J (x − y)u(t, y)dy, RN

B

(1.1)

Miriam da S. Pereira [email protected] Flank D. M. Bezerra [email protected] Severino H. da Silva [email protected]; [email protected]

1

Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB 58051-900, Brazil

2

Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB 58429-900, Brazil

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Differ Equ Dyn Syst

with initial condition at t = τ , τ ∈ R u(τ, x) = u τ (x).

(1.2)

Here u(t, x) is a real function on R × R N , β

is a non negative constant, g is a real function on R, J is a non negative function on R N and h is a non negative function on R. We also assume that g is a globally Lipschitz continuous function of class C 1 on R with g(0) = 0; the non negative function h is such that there exists a constant h ∗ > 0, such that 0  h(t) < h ∗ , t ∈ R, and the non negative even function J is continuously differentiable in  the ball centred at the origin, of radius 1 and R N J (x)d x = 1. Moreover, it follows from hypothesis on J that  max ∂xi J (x − y)dy  S i∈{1,...,N } R N

(1.3) RN

with support in

(1.4)

for some constant 0 < S < ∞, and for p ∈ (1, +∞) max

i∈{1,...,N }

∂xi J (x − ·) L p (R N )  σ.

(1.5)

for some 0 < σ < ∞. It is interesting to note that if we take g(t) ≡ t, β = 1 and h(t) ≡ 0, then the linear map Au = −u + J ∗ u shares some properties with the Laplace operator, such as a form of maximum principle (see Theorems 2.1 and 2.2 in [6]). One can also  see that A is a nonpositive operator on L 2 (R N ) by taking Fourier transforms since J(ξ ) = R N eiξ ·x J (x)d x is real and bounded by 1. The Eq. (1.1) also generalize the non local evolution equation with convolution, (1.6), below ∂t u(x, t) = −u(x, t) + tanh (β(J ∗ u)(x, t) + βh) , (1.6) which arise as continuous limit of Ising systems with Glauber dynamics and Kac potentials. In (1.6), u(t, x) represents the magnetization density in x ∈ R at time t ∈ [τ, +∞); β > 0 the inverse temperature of the Ising system; J ∈ C 1 (R) a non-negative even function which gives the strength of the spin interaction; h a constant external magnetic field, cf. [9,10], (see also Appendix). Nonlocal equations like (1.6) are not only studied in phase separation and interface dynamics, see for instance [13–15], but also in many other fields as biology, population dynamics, and epidemiology, see for instance [11,12]. In the theory of dynamical systems in infinite dimensional spaces these equations also have been studied widely in different contexts. For instance, in [9] the authors proved the existence of travelling fronts and their uniqueness modulo translation for bounded values of h, they study the case of an external magnetic constant field h, characterizing the travelling-front solutions of (1.6) for small values of h, and proving that their shape is globally stable. When in (1.1), h is a constant function, as [1,7,20,21], the analysis of the asymptotic behavior of solution is performed under the point of view of the theory of compact global attractors. In [19] the author proves the existence of a global attractor for this equation in some weighted spaces in the context one dimensional, the author also proves the existence of nonhomogeneous equilibria. In this paper we are concerned with the study of the asymptotic behavior of solutions to initial value problems associated with nonautonomous equations motivated by (1.6), see

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Differ Equ Dyn Syst

(1.1) above. These equations can be seen as a nonautonomous ODEs in Banach spaces, and therefore the properties of (local) existence and uniqueness follow from standard results of the classical theory. Our interest comes from the fact that the solutions of these problems act as the solutions of semilinear parabolic (or hyperbolic) nonautonomous problems (see [2–4]). However, the investigation of qualitative properties of the evolution process given by these equations is a much harder topic. The essential difference between the results here and the ones mentioned before is that our goal is to prove, under some hypotheses about the function g on R, the existence, regularity and upper semicontinuity of pullback attractors for the nonlocal nonautonomous model (1.1). While the notion of attractors in the autonomous setting is well established and well understood, its counterpart for nonautonomous evolution equations is not. The outline of the paper is as follows. In Sect. 2, we define the functional spaces and we recall some definitions of the theory of pullback attractors; in Sect. 3 we show the well posedness of (1.1)–(1.2) on weighted spaces L p (R N , ρ), p ∈ [1, +∞) [see (2.1) below]. For this, we assume the same technic hypothesis on weight function, ρ, considered in [7,19]. In Sect. 4, proceeding again as in [7,19], that is, using the weight function to overcome the lack of compactness, we prove the existence of pullback attractors for the nonlinear evolution process S(t, τ )u(τ, x):=u(t, x), where u(t, x) is given by (3.6) and we study its properties. In Sect. 5, using similar arguments to [2,3], we show that the pullback attractor is a bounded set in W 1, p (R N , ρ) and C 1 (R N ). In Sect. 6 we prove the upper semicontinuity of the attractors with respect to functional parameter h(t) using standard techniques based on the continuity of the processes. A concrete example to illustre our results is given in Sect. 7. Finally, in Appendix, we give a brief description of the model (1.6).

Preliminaries The Spaces L p (R N , ρ) Let ρ be a positive continuous function on R N with L 1 (R N )−norm equal to 1. Given p ∈ [1, +∞), the Banach space L p (R N , ρ) is defined by   1 (R N ); L p (R N , ρ):= u ∈ L loc

RN

with the norm

 ρ(x)|u(x)| p d x < +∞ ,

 u L p (R N ,ρ) :=

RN

(2.1)

1/ p ρ(x)|u(x)| p d x

.

We notice that the constant functions are on L p (R N , ρ) and u ≡ 1 has norm 1. The corresponding higher-order weighted Sobolev space W m, p (R N , ρ), m ∈ N, is the space of functions u ∈ L p (R N , ρ) whose distributional derivatives up to order m are also in L p (R N , ρ), with norm ⎛ uW m , p (R N ,ρ) := ⎝



|α|m

⎞1/ p p D α u L p (R N ,ρ) ⎠

where D α u = ∂ |α| u/∂ x1α1 ∂ x2α2 . . . ∂ x Nα N , α ∈ N N .

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Differ Equ Dyn Syst

Pullback Attractors for Nonautonomous Dynamical Systems We remember now the definition of nonlinear evolution process (or nonautonomous dynamical systems) generated by problem of the type (1.1)–(1.2) and pullback attractors, see [4,5,16,17,22]. Definition 2.1 An evolution process in a Banach space Z is a family of maps {S(t, τ ); t ∈ Rτ , τ ∈ R} from Z into itself, where Rτ = [τ, ∞), with the following properties: (1) S(t, t) = I , for all t ∈ R, (2) S(t, τ ) = S(t, s)S(s, τ ), for all t  s  τ , (3) The map {(t, τ ) ∈ R2 ; t ∈ Rτ } × Z  (t, τ, x)  → S(t, τ )x ∈ Z is continuous. Definition 2.2 A globally-defined solution (or simply a global solution) of the nonlinear evolution process {S(t, τ ); t ∈ Rτ , τ ∈ R} through ξτ ∈ Z is a function ξ : R → Z such that ξ(τ ) = ξτ and for all t  s we have S(t, s)ξ(s) = ξ(t). Definition 2.3 A subset B of Z pullback absorbs bounded subsets of Z under {S(t, τ ); t ∈ Rτ , τ ∈ R} if B pullback absorbs all bounded subsets at time t ∈ R under the process {S(t, τ ); t ∈ Rτ , τ ∈ R}, for each t ∈ R, i.e., for each bounded subset D ⊂ Z there exists τ0 = τ0 (t, B) with S(t, τ )D ⊂ B for any τ  τ0 . Definition 2.4 A family of sets {K (t); t ∈ R} pullback attracts bounded subsets of Z under {S(t, τ ); t ∈ Rτ , τ ∈ R} if K (t) pullback attracts all bounded subsets at t under the process {S(t, τ ); t ∈ Rτ , τ ∈ R}, for each t ∈ R, i.e., for each bounded subset C ⊂ Z , lim dist(S(t, τ )C, K (t)) = 0,

τ →−∞

where dist(·, ·) denotes the Hausdorff semi-distance, dist(A, B) = sup inf |a − b|, a∈A b∈B

and | · | is the norm in the Banach space Z . Definition 2.5 The pullback omega-limit set at time t of a subset B of Z is defined by  S(t, τ )B. (2.2) ω℘ (B, t):= s t τ s

In the sequel we introduce the concept of pullback attractor (see [16,17] for more details). Definition 2.6 A family {A(t); t ∈ R} of compact subsets of Z is said to be the pullback attractor for the evolution process {S(t, τ ); t ∈ Rτ , τ ∈ R} if it is invariant, i.e., S(t, τ )A(τ ) = A(t) for all t ∈ Rτ , for all τ ∈ R, pullback attracts bounded subsets of Z , and is minimal, that is, if there is another family of closed sets {C(t); t ∈ R} which pullback attracts bounded subsets of Z , then A(t) ⊂ C(t), for all t ∈ R. In [5] the sets A(t) are referred to as kernel sections.

Estimates and Well-Posedness in L p (R N , ρ) In order to obtain well posedness of (1.1)–(1.2) on L p (R N , ρ), we initially consider the following nonautonomous ODE on L p (R N , ρ), for 1  p < ∞, ∂t u = f (t, u)

123

(3.1)

Differ Equ Dyn Syst

with initial condition at t = τ , τ ∈ R u(τ, x) = u τ (x)

(3.2)

f (t, u) = −u + g(β(J ∗ u) + βh(t))

(3.3)

where the map is defined on Rτ × L p (R N , ρ). Lemma 3.1 Assume that there exists a constant K > 0 such that sup{ρ(x); |x − y|  1}  Kρ(y) for all y ∈ R N . Then, for any u ∈ L p (R N , ρ) J ∗ u L p (R N ,ρ)  K 1/ p u L p (R N ,ρ) . Proof By Hölder’s inequality, for 1 < p < ∞, we have  |(J ∗ u)(x)|  |J (x − y)||u(y)|dy RN  |J (x − y)|( p−1)/ p |J (x − y)|1/ p |u(y)|dy = RN  ( p−1)/ p   1/ p  |J (x − y)|dy |J (x − y)||u(y)| p dy RN RN  1/ p p  |J (x − y)||u(y)| dy . RN

By Fubini’s theorem p ∗ u L p (R N ,ρ)

J

 =  

RN

R

=

N

RN

ρ(x)|(J ∗ u)(x)| p d x   ρ(x) |J (x − y)||u(y)| p dy d x RN   ρ(x)|J (x − y)|d x |u(y)| p dy RN

and using the characteristic function, χ B[y;1] , of the ball B[y; 1] in R N , we get p



J ∗ u L p (R N ,ρ) =



RN

 K

RN

 RN

 ρ(x)|J (x − y)|χ B[y;1] (x)d x |u(y)| p dy

ρ(y)|u(y)| p dy.

Hence p

p

J ∗ u L p (R N ,ρ)  K u L p (R N ,ρ) . The case p = 1 is treated as follows. From Fatou’s Lemma   ρ(x)|(J ∗ u)(x)|d x  lim inf ρ(x)|(J ∗ u)(x)| p d x. RN

p→1+

RN

(3.4)

(3.5)

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Differ Equ Dyn Syst

Thus, using (3.5), we obtain 

p

lim sup J ∗ u L p (R N ,ρ) = lim sup p→1+

p→1+



 lim inf p→1+

RN

RN

ρ(x)|J ∗ u(x)| p d x ρ(x)|(J ∗ u)(x)| p d x p

= lim inf J ∗ u L p (R N ,ρ) . p→1+

Since p

p

lim inf J ∗ u L p (R N ,ρ) ≤ lim sup J ∗ u L p (R N ,ρ) , p→1+

p→1+

p

it follows that there exists lim p→1+ J ∗u L p (R N ,ρ) . Then, since |(J ∗u)(x)| p → |(J ∗u)(x)|, as p → 1+ , by Lebesgue’s Dominated Convergence Theorem, we have p

lim J ∗ u L p (R N ,ρ) = J ∗ u L 1 (R N ,ρ) .

p→1+

Again using Lebesgue’s Dominated Convergence Theorem, we obtain p

lim u L p (R N ,ρ) = u L 1 (R N ,ρ) .

p→1+

Therefore, applying the limit in (3.4), as p → 1+ , we conclude the proof of the lemma. We will show that f is a globally Lipschitz continuous function on L p (R N , ρ) with respect to the second variable. Proposition 3.2 Assume that g is globally Lipschitz continuous in R with constant g > 0. For each t ∈ R, the map f (t, ·) given in (3.3) is globally Lipschitz continuous in L p (R N , ρ) with  f (t, u) − f (t, v) L p (R N ,ρ)  (1 + g β K 1/ p )u − v L p (R N ,ρ) where K > 0 is such that sup{ρ(x); |x − y|  1}  Kρ(y) for all y ∈ R N (as well as in Lemma 3.1). Proof Since J is bounded and compactly supported, (J ∗ u)(x) is well defined for u ∈ 1 (R N ). From hypotheses under h and g it follows that f (t, u) ∈ L p (R N , ρ) if (t, u) ∈ L loc R × L p (R N , ρ). Now, from Lemma 3.1, we have  f (t, u) − f (t, v) L p (R N ,ρ)  u − v L p (R N ,ρ) + g(β(J ∗ u) + βh(t)) − g(β(J ∗ v) + βh(t)) L p (R N ,ρ)  u − v L p (R N ,ρ) + g βJ ∗ (u − v) L p (R N ,ρ)  (1 + g β K 1/ p )u − v L p (R N ,ρ)

as claimed.



From Proposition 3.2 and basic theory of ODE’s in Banach spaces it follows that, for any u τ ∈ L p (R N , ρ), the Cauchy problem (3.1)–(3.2) has a unique local solution in

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Differ Equ Dyn Syst

C ([τ, s(u τ )], L p (R N , ρ)) ∩ C 1 ((τ, s(u τ )], L p (R N , ρ)) for some s(u τ ) > 0 which is con-

tinuous with respect to u τ . By standard arguments (Gronwall’s Lemma), for each u τ the solution is actually globally defined, i.e., s(u τ ) = ∞ and, by variation of constants formula,  t e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds, (t, x) ∈ Rτ × R N , u(t, x) = e−(t−τ ) u τ (x) + τ

(3.6)

since the equation (1.1) is equivalent to ∂t [et u(t, x)] = et g (β(J ∗ u)(t, x) + βh(t)) . The natural notation for the global solution of the Cauchy problem (3.1)–(3.2) is u(t, τ, x; u τ ). In this paper for simplicity of notation, we use u(t, x) to denote the global solution.

Existence of Pullback Attractors In this section, we will prove that S(t, τ )u τ (x):=u(t, x) (t ∈ Rτ , τ ∈ R), where u(t, x) denote the global solution of the Cauchy problem (3.1)–(3.2), provides an infinite-dimensional nonautonomous dynamical system in L p (R N , ρ) that has a pullback attractor {A(t); t ∈ R}. The next result is a extension of Lemma 3 of [19]. Lemma 4.1 Assume the same hypotheses of Proposition 3.2. If g is globally bounded by a constant a > 0, then the ball B(0; a + ε) is a pullback absorbing for the evolution process S(t, τ ) generated by (3.1)–(3.2) in L p (R N , ρ), for any ε > 0. Proof Let u(t, x) be the solution of (3.1)–(3.2) with initial condition u(τ, x) ∈ B, where B is a bounded subset of L p (R N , ρ), namely  t e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds. (4.1) u(t, x) = e−(t−τ ) u(τ, x) + τ

We observe that p  t    e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds   p N  τ L (R ,ρ)  t p  −(t−s)  ρ(x) e |g(β(J ∗ u)(s, x) + βh(s))|ds d x. RN

τ

Using the boundedness of g and the fact that the norm of ρ is equal to 1, it follows that p  t    e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds   p N  τ L (R ,ρ)  t p   ap ρ(x) e−(t−s) ds d x  a p . RN

Hence

τ

  t    e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds    τ

L p (R N ,ρ)

 a.

(4.2)

Now, we notice that from (4.1), (4.2) and Minkowski’s inequality u(t, ·) L p (R N ,ρ)  e−(t−τ ) u τ  L p (R N ,ρ) + a.

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Differ Equ Dyn Syst

Therefore, if u τ  L p (R N ,ρ)  = 0 then u(t, ·) ∈ B(0; a + ε) for all τ  τ0 (t, B), where ε + t, where L = sup u τ  L p (R N ,ρ) . τ0 (t, B) = ln L u τ ∈B On other hand, if u τ  L p (R N ,ρ) = 0 then u(t, ·) L p (R N ,ρ)  a, i.e., u(t, ·) ∈ B(0; a) for all τ ∈ R. Therefore, for each t ∈ R there exists τ0 = τ (t, B) ∈ R such that, for any τ  τ0 , S(t, τ )B ⊂ B(0; a + ε).



Then, the result follows.

Our next goal is to prove that the pullback attractor is the family of pullback omega-limit {ω℘ (B(0; a + ε), t); t ∈ R} [see (2.2)]. Next, we established a result that is an extension of Lemma 4 of [19]. It will be used to prove the compactness of the sets ω℘ (B(0; a + ε), t). Lemma 4.2 Assume the same hypotheses of Lemma 4.1. If g  is globally Lipschitz continuous with constant k1 > 0 and k2 = |g  (0)| > 0, then for any η > 0 and t ∈ R there exists τη  t such that S(t, τη )B(0; a + ε) has a finite covering by balls of L p (R N , ρ) with radius smaller than η. Proof From Lemma 4.1 it follows that S(t, τ )B(0; a + ε) ⊂ B(0; 2a + ε), for any t ∈ Rτ , τ  τ0 (B(0, a + ε)). Given u τ ∈ B(0; a + ε), we consider the nonautonomous coupled system  ∂t v(t, x) = −v(t, x) (4.3) ∂t w(t, x) = −w(t, x) + g(β(J ∗ (v + w)(t, x)) + βh(t)) with initial conditions at t = τ , τ ∈ R 

v(τ, x) = u τ (x), w(τ, x) = 0

(4.4)

which is a way to rewrite the problem (1.1)–(1.2) on L p (R N , ρ). Note that if (v, w) is the solution of (4.3)–(4.4) in L p (R N , ρ)× L p (R N , ρ) then u = v +w is a solution of (1.1) in L p (R N , ρ) with u(τ ) = u τ . In fact, ∂t u(t, x) = ∂t v(t, x) + ∂t w(t, x) = −v(t, x) − w(t, x) + g(β(J ∗ (v + w)(t, x)) + βh(t)) = −(v + w)(t, x) + g(β(J ∗ (v + w)(t, x)) + βh(t)) = −u(t, x) + g(β(J ∗ u(t, x)) + βh(t)). Conversely, any solution u of (1.1)–(1.2) in L p (R N , ρ) can be written as u = v + w, with (v, w) the solution of (4.3)–(4.4) in L p (R N , ρ) × L p (R N , ρ). Since v(t, x) = e−(t−τ ) u τ (x), given η > 0, we may find τη ∈ R such that if t  τη  τ then v(t, ·) L p (R N ,ρ)  η/2, for any u τ ∈ B(0; a + ε). By variation of constants formula  t w(t, x) = e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds τ

and therefore, by boundedness of g, we obtain |w(t, x)|  a, ∀ (t, x) ∈ Rτ × R N .

123

(4.5)

Differ Equ Dyn Syst

From now on, we write the solution w of the following way w(t, ·) = w(t, ·)χ B(0;R) + w(t, ·)(1 − χ B(0;R) ) where χ B(0;R) denotes the characteristic function of the ball B(0; R) and R > 0 is a constant to be chosen. Since J has support in the ball B(0, 1), if q is such that 1p + q1 = 1, using Hölder inequality, we have  |J (x − y)||u(s, y)|dy |(J ∗ u)(s, x)|  B(x,1)

 1 



|J (x − y)| dy q

 B(x,1)

|u(s, y)| dy p

p

B(x,1)

1



 J  L q (R N )

1

q

|u(s, y)| p dy

p

.

B(x,1)

Now, let u ∈ B(0; a + ε), R > 0 and x ∈ B(0; R), then  |(J ∗ u)(s, x)|  p

p J  L q (R N )

|u(s, y)| dy

p

B(0,R+1)



p

 1 p p

 J  L q (R N ) |u(s, y)| p dy B(0,R+1)



p

 J  L q (R N )

RN

|u(s, y)| p χ B(0;R+1) ρ(y)

1 ρ R+1

dy,

where ρ R+1 = inf{ρ(y) : |y|  R + 1}. Thus |(J ∗ u)(s, x)| p 

p J  L q (R N ) 

ρ R+1

RN

|u(s, y)| p ρ(y)dy

p



J  L q (R N ) ρ R+1

p

u(s, ·) L p (R N ,ρ) .

Since S(t, τ )B(0; a + ε) ⊂ B(0; 2a + ε), for any t  Rτ , τ  τ0 (B(0, a + ε)), we have p

|(J ∗ u)(s, x)| p 

J  L q (R N )

ρ R+1  C1 (ε),

2 p J 

(2a + ε) p (4.6)

p q

L (R ) where C1 (ε) = (2 p a p + ε p ) ρ R+1 Using similar arguments and the hypothesis (1.5), we get N

|∂xi (J ∗ u)(x, s)| p  C2 (ε), where C2 (ε) =

2pσ p p p ρ R+1 (2 a

(4.7)

+ ε p ) > 0 and σ = maxi∈{1,...,N } ∂xi J (x − ·) L p (R N ) .

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Differ Equ Dyn Syst

Since g  is globally Lipschitz continuous on R with constant k1 > 0, using (1.3) we find  t   ∂x w(t, x)  β e−(t−s) |g  (β(J ∗ u)(s, x) + βh(s))||∂xi (J ∗ u)(s, x)|ds i τ  t  β 2 k1 e−(t−s) |(J ∗ u)(x, s) + βh(s)||∂xi (J ∗ u)(s, x)|ds τ  t e−(t−s) |∂xi (J ∗ u)(s, x)|ds +βk2 τ  t 2  β k1 e−(t−s) |(J ∗ u)(s, x)||∂xi (J ∗ u)(s, x)|ds τ  t e−(t−s) |∂xi (J ∗ u)(s, x)|ds +β(k2 + h ∗ ) τ

for any t ∈ Rτ , τ ∈ R, x ∈ B(0; R). Therefore, by (4.6) and (4.7) we obtain   ∂x w(t, x)  β 2 k1 C 1/ p (ε)C 1/ p (ε) + β(k2 + h ∗ )C 1/ p (ε). i 1 2 2 Let R > 0 be chosen such that  ηp ρ(x)(1 − χ B(0;R) (x))d x  p p . 4 a RN Then, by (4.5) we get p

w(t, ·)(1 − χ B(0;R) ) L p (R N ,ρ) =

 RN

ρ(x)(1 − χ B(0;R) (x))|w(t, x)| p d x 

ηp . 4p

Moreover, by (4.5) the function w(t, ·)χ B(0;R) is bounded in W 1, p (R N , ρ) (by a constant independent of u τ ∈ B(0; a + ε)) and, therefore the set {w(t, ·)}, with u τ ∈ B(0; a + ε) is a compact subset of L p (R N , ρ) for any t ∈ Rτ and, thus, it can be covered by a finite number of balls with radius smaller than η/4. Therefore, since u(t, ·) is the solution of the system (4.3)–(4.4) in L p (R N , ρ) we can be writer as u(t, ·) = v(t, ·) + w(t, ·)χ B(0;R) + w(t, ·)(1 − χ B(0;R) ) it follows that S(t, τη )B(0; a + ε) has a finite covering by balls of L p (R N , ρ) with radius smaller than η, where τη is such that if t  τη  τ then v(t, ·) L p (R N ,ρ)  η/2.

By the Lemmas 4.1 and 4.2 we have a family of compact sets that attracts bounded subsets of L p (R N , ρ) under S(·, ·). Then the following result is a immediate consequence of the Theorem 2.12 in [4]. Theorem 4.3 Assume the same hypotheses of Lemma 4.2. The family of sets A(t) = {ω℘ (B, t); B ⊂ L p (R N , ρ) bounded}

is a pullback attractor for the process S(t, τ ) generated by (3.1)–(3.2) in L p (R N , ρ), and A(t) is minimal in the sense that, if there exists another family of closed bounded sets Aˆ (t) that pullback attractors bounded subsets of L p (R N , ρ) under S(t, τ ), then A(t) ⊂ Aˆ (t) for all t ∈ R.

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Regularity of the Attractors In this section, we will show that the pullback attractor is contained in a fixed bounded subset of the Banach spaces W 1, p (R N , ρ) and C 1 (R N ). First, we will prove that the attractor is a bounded subset of W 1, p (R N , ρ). Since the attractor can be written as the set of all global bounded solutions, if u(t, x) is a solution of (3.1)–(3.2) in A(t) for all t ∈ R, then we obtain, letting τ → −∞  u(t, x) =

t

−∞

e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds.

(5.1)

Due to Lemma 4.2, u(t, ·) ∈ C(R × R N , W 1, p (R N , ρ)) and ∂t u(t, x) ∈ C(R × R N , L p (R N , ρ)). Therefore A(t) = S(t, τ )A(τ ) ∈ W 1, p (R N , ρ) for all t ∈ R. Furthermore, we have the following result: Theorem 5.1 Assume the same assumptions of Theorem 4.3. Then, for each t ∈ R, the set A(t) is bounded in C 1 (R N ). Proof If u(t, x) is the solution of (3.1)–(3.2) in A(t), we have by (5.1)  u(t, x) =

t −∞

e−(t−s) g(β(J ∗ u)(s, x) + βh(s))ds.

The equality above is in the sense of L p (R N , ρ) but, since the right-hand side is as regular as J we have  |u(t, x)|  a

t

−∞

e−(t−s)  a, ∀(t, x) ∈ Rτ × R N .

(5.2)

From (5.2) we obtain |(J ∗ u)(t, x)|  aJ  L 1 (R N ) and |∂xi (J ∗ u)(t, x)|  a∂xi J  L 1 (R N )

(5.3)

for any t ∈ Rτ , x ∈ R N . Deriving (5.1) with respect to x, we obtain for t  τ  ∂xi u(t, x) = β

t

e−(t−s) g  (β(J ∗ u)(s, x) + βh(s))∂xi (J ∗ u)(s, x)ds

−∞

which is well defined by arguments entirely similar to the ones used in the proof of Lemma 4.2. Since g  is globally Lipschitz continuous on R with constant k1 > 0, we find  |∂xi u(t, x)|  β

t

−∞



 β k1 2

e−(t−s) |g  (β(J ∗ u)(s, x) + βh(s))||∂xi (J ∗ u)(s, x)|ds t

−∞

e−(t−s) |(J ∗ u)(s, x)||∂xi (J ∗ u)(s, x)|ds

+ β 2 (k2 + h ∗ )



t −∞

e−(t−s) |∂xi (J ∗ u)(s, x)|ds

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Differ Equ Dyn Syst

and from (1.4) and (5.3), we obtain  ∂xi u(t, ·) L p (R N ,ρ)  β 2 k1

t

−∞

e−(t−s) a 2 ∂xi J  L 1 (R N ) J  L 1 (R N ) ds ∗

+ β (k2 + h ) 2



t −∞

e−(t−s) a∂xi J  L 1 (R N ) ds

 aβ 2 ∂xi J  L 1 (R N ) [ak1 J  L 1 (R N ) + k2 + h ∗ ]  aβ 2 S[ak1 J  L 1 (R N ) + k2 + h ∗ ]



concluding the proof.

Remark 5.2 Assuming that g is a function of class C k+1 , for any integer k  0, and its derivatives up to order k + 1 are bounded we can use the estimates in L p (R N , ρ) have been obtained for solution in the pullback attractor to obtain C k estimates.

Upper Semicontinuity of the Attractors We suppose that there exist functions h ε : R → R satisfying (1.3), for any ε ∈ [0, 1], and we assume the convergence h ε (t) → h 0 (t), as ε → 0+ , uniformly on R. From now on, we will denote as {Sε (t, τ ); t ∈ Rτ , τ ∈ R} the process associated with the problem (3.1)–(3.2) with h = h ε . We prove the upper semicontinuity of the pullback attractors for (1.1)–(1.2) as ε → 0+ , i.e., we show that lim dist(Aε (t), A0 (t)) = 0

ε→0+

where {Aε (t); t ∈ R} denotes the pullback attractor of Sε (t, τ ) on L p (R N , ρ), for any ε ∈ [0, 1]. Theorem 6.1 Let {Sε (t, τ ); t ∈ Rτ , τ ∈ R} as above. For each u 0 ∈ L p (R N , ρ), we have p

Sε (t, τ )u 0 − S0 (t, τ )u 0  L p (R N ,ρ)  M1 h ε − h 0  L ∞ (R) e M1 K

1 p

(t−τ )

where M1 = g β. Proof Let u 0 ∈ L p (R N , ρ) and u ε = Sε (t, τ )u 0 , for any ε ∈ [0, 1]. Then (u ε − u 0 )(t, x) =

 τ

t

e−(t−s) [g(β(J ∗ u ε )(s, x) + βh ε (s))

−g(β(J ∗ u 0 )(s, x) + βh 0 (s))]ds for any (t, x) ∈ Rτ × R N .

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Differ Equ Dyn Syst

Firstly, we notice that if g is globally Lipschitz continuous on R with constant g > 0, then (u ε − u 0 )(t, ·) L p (R N ,ρ)  t  e−(t−s) g(β(J ∗ u ε )(s, ·) + βh ε (s)) − g(β(J ∗ u 0 )(s, ·) + βh 0 (s)) L p (R N ,ρ) τ  t  e−(t−s) g β(J ∗ u ε )(s, ·) + βh ε (s) − β(J ∗ u 0 )(s, ·) − βh 0 (s) L p (R N ,ρ) τ  t    e−(t−s) g β J ∗ (u ε − u 0 )(s, ·) L p (R N ,ρ) + h ε (s) − h 0 (s) L p (R N ,ρ) ds τ  t   e−(t−s) g β J ∗ (u ε − u 0 )(s, ·) L p (R N ,ρ) + |h ε (s) − h 0 (s)| ds = τ  t    e−(t−s) g β J ∗ (u ε − u 0 )(s, ·) L p (R N ,ρ) + h ε − h 0  L ∞ (R) ds. τ

Using Lemma 3.1, we obtain (u ε − u 0 )(t, ·) L p (R N ,ρ)  t   1  e−(t−s) g β K p (u ε − u 0 )(s, ·) L p (R N ,ρ) + h ε − h 0  L ∞ (R) ds. τ  t  t 1 = e−(t−s) g β K p (u ε − u 0 )(s, ·) L p (R N ,ρ) ds + e−(t−s) g βh ε − h 0  L ∞ (R) ds. τ τ  t 1  e−(t−s) g β K p (u ε − u 0 )(s, ·) L p (R N ,ρ) ds + g βh ε − h 0  L ∞ (R) ds. τ

Hence et (u ε − u 0 )(t, ·) L p (R N ,ρ)  t 1  g β K p es (u ε − u 0 )(s, ·) L p (R N ,ρ) ds + et g βh ε − h 0  L ∞ (R) ds. τ

Then,using Gronwall’s inequality, we have et (u ε − u 0 )(t, ·) L p (R N ,ρ)  M1 et h ε − h 0  L ∞ (R) e M1 K

1 p

(t−τ )

where M1 = g β > 0 and K is the constant given in the Lemma 3.1. Thus (u ε − u 0 )(t, ·) L p (R N ,ρ)  M1 h ε − h 0  L ∞ (R) e M1 K

1 p

(t−τ )

,



concluding the proof. Theorem 6.2 The pullback attractor {Aε (t); t ∈ R} is upper semicontinuous in ε = 0.

Proof Fix δ > 0 and t ∈ R. Thenchoose τ ∈ R, with τ  t, such that dist(S0 (t, τ )B(0; a), A0 (t)) < δ/2, where s∈R A(s) ⊂ B(0; a). By Theorem 6.1 Sε (t, τ )u 0 − S0 (t, τ )u 0  L p (R N ,ρ)  M1 h ε − h 0  L ∞ (R) e M1 K

1 p

(t−τ )

→0

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Differ Equ Dyn Syst

as ε → 0+ in compact subsets of R uniformly for u 0 ∈ L p (R N , ρ). Hence, there exists ε0 > 0 such that sup

aε ∈Aε (τ )

Sε (t, τ )aε − S0 (t, τ )aε  L p (R N ,ρ) < δ/2, ∀ ε ∈ [0, ε0 ].

Then dist(Aε (t), A0 (t))  dist(Sε (t, τ )Aε (τ ), S0 (t, τ )Aε (τ )) + dist(S0 (t, τ )Aε (τ ), S0 (t, τ )A0 (τ ))

=

sup

aε ∈Aε (τ )

dist(Sε (t, τ )aε , S0 (t, τ )aε ) + dist(S0 (t, τ )Aε (τ ), A0 (t)) <

δ δ + =δ 2 2



and the upper semicontinuity is proved.

Remark 6.3 As well as in the autonomous case, h(t) = h 0 for all t ∈ R, it follows from Theorem 5.1 that the pullback attractor is contained in the space of bounded continuous differentiable functions.

An Example Motivated by the example given in [8], we consider the following particular case of (1.6) −

1

[consequently of (1.1)] where g ≡ tanh, J (x) = e 1−|x|2 if |x| < 1 and J (x) = 0 if |x|  1, β > 1, h : R → R is a nonnegative and bounded function; that is, we consider the following equation    1 − ∂t u = −u + tanh β e 1−|x−y|2 u(y)dy + βh(t) , (7.1) R

in the phase space L p (R, ρ), where the weight function ρ is given by ρ(x) = π −1 (1+ x 2 )−1 . Note that the function g ≡ tanh satisfies the following conditions: (a) g is a globally Lipschitz continuous function in R, with constant of Lipschitz g = 1, that is, | tanh(x) − tanh(y)|  |x − y|. (b) It is well known that tanh(0) = 0 and, for a = 1, we have | tanh(x)|  a, for all x ∈ R. (c) Since g  (x) = sech2 (x) and g  (x) = 2sech2 (x) tanh(x), we have |g  (x)|  2. Hence, with k1 = 2, we obtain |g  (x) − g(y)|  k1 |x − y|. Furthermore, k2 = |g  (0)| = 1 > 0.

−

1

On the other hand, it is easy to see that J given by J (x) = e 1−|x|2 if |x| < 1 and J (x) = 0 if |x|  1 is an even continuous function with compact support. Furthermore, J satisfies the following conditions: (d) Since 0  J (x)  e−1 follows that,  1  1 1 2 − 1 e 1−x 2 d x  dx = . J  L1 = e −1 e −1

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Differ Equ Dyn Syst

(e) The function J is differentiable and it has derivative, given by J  (x) =

2x − 1 2 1−|x| i f |x| < 1 and J  (x) = 0 i f |x|  1, e (1 − x 2 )2

which it is continuous and satisfies |J  (x)|  3. Hence  1  |J  (x)| p d x  3 p d x = 2(3 p ). −1

R

Finally, note that ρ(x) =

π −1 (1 +



x 2 )−1 R

is a non negative even function such that

ρ(x)d x = 1.

Furthermore 1 ρ(y) 2y + . =1+ 2 ρ(y + 1) 1+y 1 + y2 y Then, since | 1+y 2| 

1 2

1 and | 1+y 2 |  1, it follows that

ρ(y) ρ(y+1)

 3. Hence

ρ(y)  3ρ(y + 1). Thus the hypothesis sup{ρ(x); |x − y|  1}  Kρ(y) required in the Lemma 3.1 can be verified, for example, with K = 3.

−

1

Therefore the functions ρ(x) = π −1 (1 + x 2 )−1 , g ≡ tanh and J (x) = e 1−|x|2 if |x| < 1 and J (x) = 0 if |x|  1 meet all the hypotheses assumed in the results of Sects. 3–6 and then the results presented in these sections are valid for the evolution process generated by (7.1). Acknowledgments The authors would like to thank the anonymous referees for his careful reading of the manuscript and his helpful suggestions.

Appendix In this section we give a brief description of as arises the model (1.6). For this we faithfully follow Section 2.1 of [10]. A spin configuration can be described as follows: at all lattice sites there is a spin variable with two different values in {−1, 1}.The value of the spin at x is flipped at rates which depend on the value of the others spin. More precisely, we have the following definition. Definition 8.1 A spin configuration is a map σ : Zd → {−1, 1}, that is an element of {−1, 1}Z . The value σ (x) of the spin at x is a function of the configd uration σ , thus a random variable on the space of all the spin configurations {−1, 1}Z . The d restriction to  ⊂ Z of a configuration σ , is denoted by σ , which is therefore a function on  with values {−1, 1}. d

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Differ Equ Dyn Syst

Definition 8.2 A Kac potential is a function Jγ : Zd × Zd → R, which depends on a (scaling) parameter γ and has the form Jγ (x, y) = γ d J (γ [x − y]),

(8.1)

where γ varies in the set {2−n , n ∈ Z+ }. As in [10], we assume that J is a smooth symmetric probability density (thus J  0) with compact support. Given (a magnetic field) h ∈ R, we define the energy of the spin configuration σ as Hγ (σ ) = −h



−

x∈

1  Jγ (x, y)σ (x)σ (y), 2

(8.2)

x= y∈

while its energy inclusive of the interaction with the spins in the complement, c, of , is  Hγ (σ − σc ) = Hγ (σ ) − Jγ (x, y)σ (x)σ (y), (8.3) x∈,y ∈ /

Definition 8.3 Given (the “the inverse temperature”) β > 0 and γ > 0, we denote by Glauber d dynamics the unique Markov process on {−1, 1}Z , (given in [18]) whose pregenerator is the operator L γ with domain the set of all cylinder functions f on which it acts as  L γ f (σ ) = cγ (x, σ )[ f (σ x) − f (σ )]. (8.4) x∈Zd

In (8.4),

σx

is the configuration obtained from σ by flipping the spin at x, namely, σ x (y) = σ (y) i f y  = x, σ x (y) = −σ (x) i f y = x.

(8.5)

The “flip rate” cγ (x, σ ) of the spin at x in the configuration σ is Cγ (x, σ ) =

e−βh γ (x)σ (x) . e−βh γ (x) − eβh γ (x)

where h γ (x) = h + (Jγ ◦ σ )(x) and (Jγ ◦ σ )(x) =



(8.6)

Jγ (x, y)σ (y).

(8.7)

y=x

The space of realizations of Glauber dynamics is D(R+ , {−1, 1}Z ), the Skorohod space of cadlag trajectories, (continuous from the right and with limits from the left). The value of the d spin in x at time t is σ (x, t) which is thus a random variable on D(R+ , {−1, 1}Z ). d

Note that −β

cγ (x, σ ) = Z γ (σx c )−1 e 2x Hγ (σ ) where x Hγ (σ ) is the change of energy due to the spin flip at x, that is x Hγ (σ ) = Hγ ((σ x ) ) − Hγ ((σ ) where  is any set which contains x and such that the spin at x does not interact with those in c . Here Z γ (σx c )−1 is the denominator in (8.6), but, for what we say below, it may be

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Differ Equ Dyn Syst

any other function, provided it is independent of σ (x), as implied by the notation. In fact, the important point about the rates is that they verify the “detailed balance” condition e−βx Hγ (σ ) =

cγ (x, σ x ) . cγ (x, σ )

(8.8)

Thus the Glauber dynamics is intimately related to the notion of Gibbs measures. Definition 8.4 The Gibbs measure μβ,h,γ is any probability on {−1, 1}Z such that, for x ∈ Zd , satisfies the equation d

μβ,h,γ (σ (x) = ±1 | {σ (y), y  = x}) =

e±βh γ (x) , + eβh γ (x)

(8.9)

e−βh γ (x)

where the left hand side is the probability that σ (x) = ±1 conditioned on the σ -algebra generated by all the spin σ (y), y  = x. In definition above, β has the physical meaning of an inverse temperature and h of an external magnetic field, Jγ of the spin-spin interaction strength. Notice that the left hand side of (8.9) is a function of σ (x) and all σ (y), y  = x, it is thus a function of the whole spin configuration σ . Then, from (8.8) and (8.9), it follows that   μβ,h,γ (σ (x) | {σ (y), y  = x}) cγ (x, σ ) = μβ,h,γ σ x (x) | {σ (y), y  = x} cγ (x, σ x ) (x)

so that the operator L γ defined by (8.4) after setting cγ (y, σ ) = 0 for all y  = x, is self d adjoint in L 2 ({−1, 1}Z , μβ,h,γ ). It then follows that also the full generator of the Glauber dynamics is self adjoint and μβ,h,γ is stationary, the Glauber dynamics then being a reversible process. Finally we mention that (8.8) does not depend on the choice of Z γ , which appears in the definition of cγ , thus different choice of Z γ define other, equally acceptable, reversible evolutions. The choice (8.6) gives rise to a simpler limiting mesoscopic equation. The “scale separation” between the two levels is specified by γ in the transition micromesoscopic (x, t) → (r, t) = (γ x, t), thus, time is unchanged while space is shrunk by γ . The microscopic points x ∈ Zd are represented in the mesoscopic space Rd by the lattice γ Zd . It is thus convenient to partition Rd into the “elementary squares” {r : [r ]γ = γ x}, with x ∈

Rd

and, denoting by r = (r1 , . . . , rd ), x = (x1 , . . . , xn ),

[r ]γ = γ x

if x ∈ Zd and γ xi ≤ ri < γ (xi + 1)

for all i = 1, . . . , d.

(8.10)

Definition 8.5 Let X be a measurable space. Denoting by M(X ) the space of all the real valued, measurable functions on X . We define γ : M(Zd ) → M(Rd ) by (γ ( f ))(r ) = f (x), x = γ −1 [r ]γ and f ∈ M(Zd ),

(8.11)

where [r ]γ is defined in (8.10). In particular, we denote by σγ = γ (σ ), σγ ,t = γ (σt ),

(8.12)

σγ (r ) is thus the image of the spin configuration σ in the mesoscopic representation.

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Differ Equ Dyn Syst

Definition 8.6 We define, for any 0 < α < 1 and γ as in (8.1), the block spin transformation f → f (α,γ ) , f and f (α,γ ) both in M(Rd ), as  f (α,γ ) (r ) = Nγ−1 1({|[r ]γ − [r  ]γ |  γ 1−α }) f (r  )dr  (8.13)  (8.14) Nγ = 1({|[r ]γ − [r  ]γ |  γ 1−α })dr  As well as in [10], we use the shorthand notation (α)

σγ(α) :=(σγ )(α,γ ) ; σγ ,t :=(σγ ,t )(α,γ )

(8.15)

to avoid redundancy in the formulas, and in general we may omit the superscript γ in (α, γ ), when γ already appears as a subscript. The more familiar form of the block spin transformation is recovered when we apply the transformation to a function g = γ ( f ), f ∈ M(Zd ). In that case g α,γ (r ), [r ]γ :=γ x, is given by  1 g α,γ (r ) = Aγ −α ,x ( f ):= f (y) (8.16) |Bγ −α | y∈Bγ −α ,x

where

Bγ −α ,x = {|x − y|  γ −α }, |Bγ −α | = cardinality of Bγ −α ,x .

(8.17)

We end this section with the following theorem, that it has been proved in [10] (see Sect. 2.1, Theorem 2.1.6). Theorem 8.7 For any α ∈ (0, 1) there are ζ , a and b all positive and for any n and any d k ∗  2, there is c so that the following holds: Given γ small enough, for all σ ∈ {−1, 1}Z d and m ∈ [−1, 1]Z , m∞  1, for which (see (8.12)–(8.15) for notation) sup |r |≤k ∗ γ −1

we have that Pγσ

|σγ(α) (r ) − m (α,γ ) (r )|  γ ζ

 sup t≤a log γ −1 |r |≤(k ∗ −1)γ −1

    (α) ) (r, t)  γ b σγ ,t (r ) − m (α,γ γ

  cγ n

γ

where Pσ is the law of the Glauber dynamics when the process starts at time 0 from σ ; ) m (α,γ (·, t) = (m γ (·, t))(α,γ ) γ

and m γ (r, t) is the unique solution of the Cauchy problem (1.6) with initial datum m γ (r, 0) = (γ )(r ).

References 1. Bezerra, F.D.M., Pereira, A.L., da Silva, S.H.: Existence and continuity of global attractors and nonhomogeneous equilibria for a class of evolution equations with nonlocal terms, J. Math. Anal. Appl. 396, 590–600 (2012) 2. Caraballo, T., Carvalho, A.N., Langa, J.A., Rivero, F.: A non-autonomous strongly damped wave equation: existence and continuity of the pullback attractor. Nonlinear Anal. 74(6), 2272–2283 (2011) 3. Caraballo, T., Carvalho, A.N., Langa, J.A., Rivero, F.: A gradient-like nonautonomous evolution processes. Internat. J. Bifur. Chaos 20(9), 2751–2760 (2010)

123

Differ Equ Dyn Syst 4. Carvalho, A.N., Langa, J.A., Robinson, J.C.: Attractors for infinite-dimensional nonautonomous dynamical systems. Applied Mathematical Sciences, vol. 182. Springer, New York (2012) 5. Chepyzhov, V.V., Vishik, M.I.: Attractors for equations of mathematical physics. In: Colloquium Publications, vol. 49. America Mathematical Society, Providence, RI (2002) 6. Coville, J., Dupaigne, L.: Propagation speed of travelling fronts in nonlocal reaction-diffusion equations. Nonlinear Anal. 60, 797–819 (2005) 7. da Silva, S.H.: Existence and upper semicontinuity of global attractors for nueral fields in an unbounded domain. Eletron. J. Differ. Equ. 138, 1–12 (2010) 8. da Silva, S.H.: Properties of an equation for neural fields in a bounded domain. Electron. J. Differ. Equ. 2012(42), 1–9 (2012) 9. De Masi, A., Gobron, T., Presutti, E.: Travelling fronts in non local evolution equations. Arch. Ration. Mech. Anal. 132, 143–205 (1995) 10. De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with Kac potentials: I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity 7, 633–696 (1994) 11. Diekmann, O., Kaper, H.G.: On the bounded solutions of a non linear convolution equation. Non Linear Anal. Theory Methods Appl. 2, 721–737 (1978) 12. Fife, P., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977) 13. Kac, M., Uhlenbeck, G., Hemmer, P.C.: On the Van der Waals theory of vapor-liquid equilibrium. I. Discussion of a one dimensional model. J. Math. Phys. 4, 216–228 (1963) 14. Kac, M., Uhlenbeck, G., Hemmer, P.C.: On the Van der Waals theory of vapor-liquid equilibrium. II. Discussion of the distribution functions. J. Math. Phys. 4, 229–247 (1963) 15. Kac, M., Uhlenbeck, G., Hemmer, P.C.: On the Van der Waals theory of vapor-liquid equilibrium. III. Discussion of the critical region. J. Math. Phys. 5, 60–74 (1964) 16. Kloeden, P.E.: Pullback attractors in nonautonomous difference equations. J. Differ. Equ. Appl. 6(1), 33–52 (2000) 17. Kloeden, P.E., Schmalfuß, B.: Asymptotic behaviour of non-autonomous difference inclusions. Syst. Control Lett. 33, 275–280 (1998) 18. Ligget, T.M.: Interacting particle systems. Springer-Verlag, New York (1985) 19. Pereira, A.L.: Global attractor and nonhomogeneous equilibria for a nonlocal evolution equation in an unbounded domain. J. Differ. Equ. 226, 352–372 (2006) 20. Pereira, A.L., da Silva, S.H.: Existence of global attractors and gradient property for a class of nonlocal evolution equations. São Paulo J. Math. Sci. 2, 1–20 (2008) 21. Pereira, A.L., da Silva, S.H.: Continuity of global attractors for a class of nonlocal evolution equations. Discrete Continuous Dyn. Syst. 26, 1073–1100 (2010) 22. Sell, G.R.: Nonautonomous differential equations and dynamical systems. Trans. Am. Math. Soc. 127, 241–283 (1967)

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