Asymptotic behavior for a class of non autonomous non local problems

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ASYMPTOTIC BEHAVIOR FOR A CLASS OF NON AUTONOMOUS

NON LOCAL PROBLEMS

F. D. M. BEZERRA1, S. H. DA SILVA2, AND A. L. PEREIRA3

Abstract. In this paper we consider the non local non autonomous evolution problem

∂tu = −u+ g (β(t)(Ku)) in Ω,

u = 0 in RN\Ω,

where Ω is a smooth bounded domain in RN , β denotes the functional parameter given by

a continuous bounded function on R, and K is an integral operator with symmetric kernel.We prove existence and some regularity properties of the pullback attractor.

Mathematical Subject Classification 2010: 35B40, 35B41, 37B55.Key words and phrases: Pullback attractors; non local diffusion equations; non-autonomousequations; evolution process.

1. Introduction

The understanding of the dynamics of non-autonomous evolution equations has attractedthe attention of many researchers in recent years; see for instance [5, 6, 7, 11, 19] and [21].Unlike autonomous problems where we already have a pretty good knowledge of the dynamics(at least, in the context of global attractors for problems with gradient structure), issues suchas the existence, regularity, characterization, and continuity (with rate) of attractors for non-autonomous problems are still the object of intense investigation; see for instance ([1] and[3]) for problems with ‘local character’, and [2, 16, 17] and [18] for problems with ‘non localcharacter’.In this paper, we seek to clarify some aspects of the asymptotic behavior for a class of

non-autonomous evolution problem with non local terms, which arise as a continuum limitof Ising spin systems with Glauber dynamics and Kac potentials, when the temperatureexplicitly depends on the time.More precisely, we consider here the non local non-autonomous evolution problem

(1.1)

∂tu = −u+ g(β(t)(Ku)) in Ω,

u(τ) = uτ in Ω,

u = 0 in RN\Ω,

Date: December 23, 2013.2Research partially supported by CAPES/CNPq.

1

http://arxiv.org/abs/1312.6045v1

2 F. D. M. BEZERRA, S. H. DA SILVA, AND A. L. PEREIRA

where Ω is a bounded smooth domain in RN (N ≥ 1), g : R → R, β is a real function and

K is an integral operator with symmetric kernel

(Ku)(x) :=

∫

RN

J(x, y)u(y)dy.

We will suppose, without loss of generality, that∫

RN J(x, y) d y =∫

RN J(x, y) d x = 1.

In [10, 13, 14, 15] and [16] the authors considered the autonomous evolution equation

∂tu = −u + tanh(β(J ∗ u) + βh) in R,

where u = u(x, t) represents the magnetization density at the point x ∈ R and time t ∈[0,+∞), β > 0 is the inverse of the temperature, J defined on R is an even, non negativefunction which gives the strength of the spin-spin interaction, h > 0 is an external magneticfield, and J ∗ u is the convolution between J and u.Autonomous versions of (1.1) have been considered in [17] and [18] (in the one-dimensional

case) and in [2] and [9] (in the n-dimensional case).

The paper is organized as follows. After recalling some concepts and results in Section 2,we prove well posedness of (1.1) in Section 3 and existence and some regularity properties forthe pullback attractor on Section 4. In Section 5 we prove some comparison results. Section6 is dedicated to the proof of upper semi-continuity of the family of global attractors withrespect to the parameter β.

2. Some preliminary definitions and results

We start by recalling the concepts of evolution processes and pullback attractors whichproved to be a useful tool to study the asymptotic dynamics of infinite dimensional non-autonomous dynamical system. We refer to [4, 5, 6, 7, 11] and [19] for more details andproofs.

Definition 2.1 (Nonlinear Evolution Processes). A nonlinear evolution processes in a Ba-nach space X is a family of maps S(t, τ); t ≥ τ ∈ R (not necessarily linear) from X intoitself 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 τ ≤ s ≤ t,(3) The map (t, τ) ∈ R

2; t ≥ τ × X ∋ (t, τ, x) 7→ S(t, τ)x ∈ X is continuous.

In the particular case where each S(t, τ) is linear, t ≥ τ ∈ R, we say that S(t, τ); t ≥ τ ∈R is a linear process.If S(t, τ) = S(t− τ, 0) for all t ≥ τ ∈ R, we say that the process is autonomous, and the

family S(t) = S(t, 0); t ≥ 0 is then called a semigroup or a dynamical system.

Definition 2.2. The pullback (resp. forward) orbit of a subset B of X , at time t ∈ R, isthe set γp(B, t) := ∪s≤tS(t, s)B (resp. γf(B, t) := ∪s≥tS(s, t)B).

Definition 2.3. A globally-defined solution (or simply a global solution) of the evolutionprocesses S(t, τ); t ≥ τ ∈ R through ψ0 ∈ X at time τ0 is a function ψ : R → X such thatψ(τ0) = ψ0, and S(t, τ)ψ(τ) = ψ(t), for all t ≥ τ . A global solution through ψ0 ∈ X is a Aglobal solution through ψ0 ∈ X at some time.

NON-AUTONOMOUS NONLOCAL PROBLEMS 3

Definition 2.4 (Pullback Attraction). A family of sets K(t); t ∈ R pullback attractsbounded subsets of X under S(t, τ); t ≥ τ ∈ R if K(t) pullback attracts all boundedsubsets at t under S(t, τ); t ≥ τ ∈ R, for each t ∈ R, i.e., for each bounded set C ⊂ X ,

limτ→−∞

dist(S(t, τ)C,K(t)) = 0,

where dist(·, ·) denotes the Hausdorff semi-distance in X ,

(2.1) dist(A,B) = supa∈A

infb∈B

|a− b|X .

Definition 2.5. A family of bounded subset B(t); t ∈ R of X is said to be strongly pullbackabsorbing if, for all bounded set C ⊂ X and τ ≤ t, there exists τ0(τ, C) ∈ R such that

S(τ, τ − r)C ⊂ B(t), for all r ≥ τ0(τ, C).

Definition 2.6. An evolution process S(t, τ); t ≥ τ ∈ R is said to be pullback stronglybounded dissipative if it has a family of bounded strongly pullback absorbing subsets.

Next, we present the notion of a set of pullback asymptotic states.

Definition 2.7 (Pullback Attractor). A family A(t); t ∈ R of compact subsets of X is saidto be a pullback attractor for the evolution process S(t, τ); t ≥ τ ∈ R if it is invariant, i.e.,S(t, τ)A(τ) = A(t) for all τ ≤ t, pullback attracts bounded subsets of X , and is minimal,that is, if there is another family of closed sets C(t); t ∈ R which pullback attracts boundedsubsets of X, then A(t) ⊂ C(t), for all t ∈ R.

Remark 2.8. The minimality requirement in the Definition 2.7 is an addition with respectto the theory of attractors for semigroups and is necessary to ensure uniqueness (see [5]. Itcan be dropped if we require that

⋃

τ≤t A(τ) is bounded for any t ∈ R. In this case, we alsohave

A(t) =

ξ(t) : ξ : R → X is a global backwards bounded solution of S(t, τ)

,

for all t ∈ R.

Definition 2.9. An evolution process S(t, τ); t ≥ τ ∈ R in a Banach space X is pullbackasymptotically compact if, for each t ∈ R, each sequence τkk∈N in (−∞, t] such that τk →−∞, and each bounded sequence zkk∈N in X with S(t, τk)zkk∈N bounded, the sequenceS(t, τk)zkk∈N possesses a convergent subsequence.

Definition 2.10. An evolution process S(t, τ); t ≥ τ ∈ R in a Banach space X is pullbackstrongly bounded if, for each t ∈ R and each bounded subset B of X ,

⋃

s≤t γp(B, s) is bounded.

The following two results are proved in [5] (see also [6]) and will be used to prove theexistence of the pullback attractor for the evolution process generated by (1.1), (see Section4).

Theorem 2.11. If an evolution process S(t, τ); t ≥ τ ∈ R satisfies the properties

S(t, τ) = T (t, τ) + U(t, τ), t ≥ τ,

where U(t, τ) is a compact operator and there exists a non-increasing function k : [0,+∞)×[0,+∞) → R with k(σ, r) → 0 as σ → +∞, and for all τ ≤ t and z ∈ X with |z|X ≤ r,


|T (t, τ)|X ≤ k(t − τ, r), then the process S(t, τ); t ≥ τ ∈ R is pullback asymptoticallycompact.

Theorem 2.12. If an evolution process S(t, τ); t ≥ τ ∈ R in a Banach space X is pullbackstrongly bounded dissipative and pullback asymptotically compact, then S(t, τ); t ≥ τ ∈ Rpossesses a pullback attractor A(t); t ∈ R. Moreover, the union

⋃

τ≤tA(τ) is bounded foreach t ∈ R, and the global pullback attractor is given by

A(t) = Ω℘(B(t), t) :=⋂

s≤t

⋃

τ≤s

S(t, τ)B, for each t ∈ R,

where B(t); t ∈ R is a family of bounded subsets of X which is strongly pullback absorbing.

This theorem extends the analogous result for semigroups (cf. Theorem 1.1, Chapter 1 in[22]). The pullback attractor of strongly bounded dissipative processes however, is alwaysbounded in the past. To be more precise, for every t ∈ R the union

⋃

τ≤tA(τ) is boundedin X .

3. Well posedness

In this section we show that the problem (1.1) is well posed in a suitable phase space X .

Definition 3.1. A function u : [τ − s, τ)× RN → L2(RN) (s > 0) is a solution for (1.1) if

it is continuous, u is continuously differentiable in (τ − s, τ)× RN and (1.1) is satisfied for

all (t, x) ∈ (τ − s, τ)× RN .

Consider the subspace X of L2(RN) given by

X =

u ∈ L2(RN); u(x) = 0, if x ∈ RN\Ω

with the induced norm. The space X is canonically isomorphic to L2(Ω) and we sometimesidentify the two spaces, without further comment. We also use the same notation for afunction in R

n and its restriction to Ω for simplicity wherever we believe the intention isclear from the context.In order to obtain well posedness of (1.1), we consider the Cauchy problem

(3.1)

∂tu = −u+ F (t, u),

u(τ) = uτ ,

where the map F : R×X → X is defined by

F (t, u)(x) =

g(β(t)(Ku)(x)), x ∈ Ω,

0, x ∈ RN\Ω.

The map given by

(3.2) Ku(x) :=

∫

RN

J(x, y)u(y) d y

is well defined as a bounded linear operator in various function spaces, depending on theproperties assumed for J . We collect here some estimates for this map which will be usedin the sequel.


Lemma 3.2. Let K be the map defined by (3.2), ‖J‖1:= supx∈Ω ‖J(x, ·)‖L1(Ω), ‖J‖2:=supx∈Ω ‖J(x, ·)‖L2(Ω) and ‖J‖∞:= supx∈Ω ‖J(x, ·)‖L∞(Ω). Then, for all u ∈ X, we have

(3.3) ‖Ku‖L2(Ω) ≤ ‖J‖∞‖u‖L2(Ω),

(3.4) ‖Ku‖L∞(Ω) ≤ ‖J‖1‖u‖L∞(Ω),

(3.5) ‖Ku‖L∞(Ω) ≤ ‖J‖∞‖u‖L∞(Ω)|Ω|,

(3.6) ‖Ku‖L2(Ω) ≤ ‖J‖1‖u‖L2(Ω) = ‖u‖L2(Ω),

(3.7) |Ku(x)| ≤ ‖J‖2‖u‖L2(Ω), for all x ∈ Ω.

Proof. The estimates (3.3), (3.4) and (3.5) are immediate, (3.6) follows from Young’sinequality and (3.7) follows from Holder inequality.

Proposition 3.3. Suppose, in addition to the hypotheses of Lemma 3.2, that the function gis locally Lipschitz continuous and β is a bounded function on R. Then the function F (t, ·)is Lipschitz continuous on bounded subsets of X, uniformly for t ∈ R.

Proof. Suppose u belongs to a ball BR about the origin of radius R in X . Then, writing||β||∞ := ||β||L∞(R), it follows from (3.7), that

(3.8) |β(t)Ku(x)| ≤ ||β||∞‖J‖2‖u‖L2(Ω),

for any x ∈ Ω and t ∈ R.Therefore, if u and v belong to a ball BR about the origin of radius R in X and kO is the

Lipschitz constant of g in the interval [−||β||∞‖J‖2R, ||β||∞‖J‖2R], we obtain

|g(β(t)(Ku(x)))− g(β(t)(Kv(x)))| ≤ kO|β(t)(Ku(x))− β(t)(Kv(x))|≤ kO‖β‖∞|K(u− v)(x)|≤ kO‖β‖∞‖J‖2‖u− v‖L2(Ω)

Thus,

‖g(β(t)(Ku))− g(β(t)(Kv))‖L2(Ω) ≤∫

Ω

(kO‖β‖∞‖J‖2‖u− v‖L2(Ω))2dx

1/2

= kO‖β‖∞‖J‖2|Ω|1/2‖u− v‖L2(Ω),

and the map

u ∈ X 7→ g(β(t)(K(u(·)) ∈ X

is Lipschitz in BR, with Lipschitz constant independent of t and, therefore, so is the mapF (t, ·).


From the result above, it follows from well known results that the problem (3.1) has alocal solution for any initial condition in X . For the global existence, we need the followingresult ([12] - Theorem 5.6.1)

Theorem 3.4. Let X be a Banach space, and suppose that f : [t0,∞]×X → X is continuousand satisfies

(1) ‖f(t, u)‖ ≤ g(t, ‖u‖); for all (t, u) ∈ [t0,∞]×X ;(2) g : [t0,∞]× R

+ → R+ is continuous and g(t, r) is non decreasing in r ≥ 0, for each

t ∈ [t0,∞] and the maximal solution r(t, t0, r0) of the scalar initial value problem

r′ = g(t, r), r(t0) = r0,

exists throughout [t0,∞]

Then the maximal interval of existence of any solution u(t, t0, y0) of the initial value problem

du

dt= f(t, u), t ≥ t0, u(t0) = u0,

is [t0,∞].

Corollary 3.5. Suppose in addition to the hypotheses of Proposition 3.3, that β and J arecontinuous and g satisfies:

(3.9) lim sup|x|→∞

|f(x)||x| < k1, for some constant k1 ∈ R.

Then the problem (3.1) has a unique globally defined solution for any initial condition in X.which is given for t ≥ τ by the “variation of constants formula”

u(t, τ, x; uτ) = e−(t−τ)uτ (x) +

∫ t

τ

e−(t−s)F (s, u(s, τ, x, uτ))(x)ds x ∈ RN

=

e−(t−τ)uτ (x) +

∫ t

τ

e−(t−s)g(β(s)(Ku)(s, τ, x; uτ))ds, x ∈ Ω,

0, x ∈ RN\Ω.

(3.10)

Proof. From Proposition 3.3, it follows that the right-hand-side of (3.1) is Lipschitzcontinuous in bounded sets of X and, therefore, the Cauchy problem (3.1) is well posed inX with a unique local solution u(t, τ, x; uτ), given by (3.10) (see [8]). From condition (3.9),it follows that there is a constant k2 such that

(3.11) ‖f(x)‖ ≤ k2 + k1‖x‖, for any x in R.

From (3.7), it follows that |g(β(t)Ku(x))| ≤ k2+ k1‖β‖∞‖J‖2‖u‖L2(Ω), for all u ∈ X , and

x ∈ Ω. Therefore, ‖g(β(t)Ku(·))‖L2(Ω) ≤ k2|Ω|1/2+k1‖β‖∞‖J‖2|Ω|1/2‖u‖L2(Ω), for all u ∈ X.

and then ‖F (t, u)‖L2(Ω) ≤ k2|Ω|1/2 + (k1‖β‖∞‖J‖∞|Ω|+ 1)‖u‖L2(Ω), for all u ∈ X.

Defining h : [t0,∞] × R+ → R

+, by h(t, r) = k2|Ω|1/2 + (k1‖β‖∞‖J‖2|Ω|1/2 + 1)r, itfollows that F satisfies the hypothesis of Theorem 3.4 and the desired conclusion followsimmediately.

The result below has been proven in [20].


Proposition 3.6. Let Y and Z be normed linear spaces, F : Y → Z a map and supposethat the Gateaux derivative of F , DF : Y → L(Y, Z) exists and is continuous at y ∈ Y .Then the Frechet derivative F ′ of F exists and is continuous at y.

Proposition 3.7. Suppose, in addition to the hypotheses of Corollary 3.5 that the function gis continuously differentiable on R. Then, for each t ∈ R the function F (t, ·) is continuouslyFrechet differentiable on X with derivative given by

DF (t, u)v(x) :=

g′(β(t)(Ku)(x))β(t)(Kv)(x), x ∈ Ω,

0, x ∈ RN\Ω.

Proof. From a simple computation, using the fact g is continuously differentiable on R, itfollows that the Gateaux’s derivative of F (t, ·) is given by

DF (t, u)v(x) :=

g′(β(t)(Ku)(x))β(t)(Kv)(x), x ∈ Ω,

0, x ∈ RN\Ω.

The operator DF (t, u) is clearly a linear operator in X . Using estimate (3.8) and thecontinuity of g′ it follows that g′(β(t)(Ku)(·)) is bounded by a constant k3 for any t ∈ R.Using (3.6) and the fact that ‖J(x, ·)‖L1 = 1, we obtain

‖DF (t, u)v‖X ≤ k3‖β‖∞‖Kv‖L2(Ω)

≤ k3‖β‖∞‖v‖X,proving that DF (t, ·) is a bounded operator.We now prove that DF (t, ·) : X → L(X) is continuous.If w and u in X , it follows from (3.7) that

‖DF (t, u)v −DF (t, w)v‖X ≤∫

Ω

|g′(β(t)(Ku)(x))− g′(β(t)(Kw)(x))|2|β(t)|2|(Kv)(x)|2dx1/2

≤

‖β‖L∞‖‖J‖2∫

Ω

|g′(β(t)(Ku)(x))− g′(β(t)(Kw)(x))|2 d x1/2

‖v‖L2(Ω).

Now, from (3.8), it follows that β(t)(Kw)(x) is in a bounded set of RN for w in a neigh-borhood of u and any t ∈ R, x ∈ Ω . Therefore, |g′(β(t)(Ku)(x)) − g′(β(t)(Kw)(x))| → 0uniformly in Ω, as w → u in L2(Ω) by estimate (3.7), and the continuity of g′. From thisthe continuity of DF (t, ·) at u follows immediately.Hence, it follows from Proposition 3.6 that F (t, ·) is Frechet differentiable with continuous

derivative in X .

Remark 3.8. Since the right-hand side of (3.1) is a C1 function, the flow generated by (3.1)in X is C1 with respect to initial conditions.

From the results above, we have, for each τ ∈ R and u(τ, ·) ∈ X , the unique solutionof (3.1) with initial condition u(τ, ·) exists for all t ≥ τ , this solution (t, x) 7→ u(t, τ, x; uτ)(defined by (3.10)) gives rise to a family of nonlinear C1 evolution processes on X given by


S(t, τ)uτ (x) := u(t, τ, x; uτ), t ≥ τ ∈ R.

4. Existence and regularity of the pullback attractor

We first prove the existence of a pullback attractor A(t); t ∈ R in X for the evolutionprocess S(t, τ); t ≥ τ ∈ R.

Lemma 4.1. Suppose that the hypotheses of Proposition 3.7 hold with the constant k1 in

(3.9) satisfying k1‖β‖L∞(R) < 1 Then the ball centered at the origin with radius2k2

√|Ω|

1−k1‖β‖L∞(R),

where k1 and k2 are the constants appearing in (3.11) is strongly pullback absorbing for theevolution process S(t, τ); t ≥ τ ∈ R generated by (3.1).

Proof. If u(t, τ, x; uτ) is a solution of (3.1) with initial condition uτ , then

d

dt

∫

RN

|u(t, τ, x; uτ)|2dx =d

dt

∫

Ω

|u(t, τ, x; uτ)|2dx

= −2

∫

Ω

u2(t, τ, x; uτ)dx+ 2

∫

Ω

u(t, τ, x; uτ)g(β(t)Ku(t, τ, x; uτ))dx.

But, by Holder’s inequality

∫

Ω

u(t, τ, x; uτ)g(β(t)Ku(t, τ, x; uτ))dx ≤ ‖u(t, τ, ·; uτ)‖L2(Ω)

(∫

Ω

(g(β(t)Ku(t, τ, x; uτ)))2dx

)12

.

Using Holder’s inequality, condition (3.9), estimate (3.6) and the fact that ‖J(x, ·)‖L1 = 1,we obtain

∫

Ω

u(t, τ, ·; uτ)g(β(t)Ku(t, τ, ·; uτ))dx

≤ ‖u(t, τ, ·; uτ)‖L2(Ω)

(∫

Ω

(k1‖β‖L∞(R)|Ku(t, τ, x; uτ)|+ k2)2dx

)12

≤ ‖u(t, τ, ·; uτ)‖L2(Ω)

[

k1‖β‖L∞(R)‖J‖L1‖u(t, τ, ·; uτ)‖L2(Ω) + k2√

|Ω|]

= ‖u(t, τ, ·; uτ)‖L2(Ω)

[

k1‖β‖L∞(R)‖u(t, τ ; ·, uτ)‖L2(Ω) + k2√

|Ω|]

.

Hence

d

dt‖u(t, τ, ·; uτ)‖2L2(Ω) ≤ −2‖u(t, τ, ·; uτ)‖2L2(Ω) + 2k1‖β‖L∞(R)‖u(t, τ, ·; uτ)‖2L2(Ω)

+ 2k2√

|Ω|‖u(t, τ, ·; uτ)‖L2(Ω)

= 2‖u(t, τ, ·; uτ)‖2L2(Ω)

[

−1 + k1‖β‖L∞(R) +k2√

|Ω|‖u(t, τ, ·; uτ)‖L2(Ω)

]

.


Since k1‖β‖L∞ < 1, ε = 1 − k1‖β‖L∞ > 0. Then, while ‖u(t, τ, ·; uτ)‖L2(Ω) ≥ 2k2√

|Ω|

ε, we

have

d

dt‖u(t, τ, ·; uτ)‖2L2(Ω) ≤ 2‖u(t, τ, ·; uτ)‖2L2(Ω)(−ε+

ε

2)

= −ε‖u(t, τ, ·; uτ)‖2L2(Ω).

Therefore while ‖u(t, τ, ·; uτ)‖L2(Ω) ≥2k2

√|Ω|

1−k1‖β‖L∞

, we have

‖u(t, τ, ·; uτ)‖L2(Ω) ≤ e−ε2t‖uτ‖L2(Ω)

= e−12(1−k1‖β‖L∞(R))t‖uτ‖L2(Ω),(4.1)

which concludes the result.

Theorem 4.2. In addition to the conditions of Lemma 4.1, suppose that ‖Jx(x, ·)‖L2(Ω)

is bounded. Then there exists a pullback attractor A(t); t ∈ R for the evolution processS(t, τ); t ≥ τ ∈ R generated by (3.1) in X and the set

⋃

τ≤tA(τ) is contained in the ball

of radius2k2

√|Ω|

1−k1‖β‖L∞

in L2(Ω) for any t ∈ R.

Proof. From (3.10), it follows that

S(t, τ)uτ = T (t, τ)uτ + U(t, τ)uτ , t ≥ τ, x ∈ Ω

where

T (t, τ)uτ (x) := e−(t−τ)uτ(x)

and

U(t, τ)uτ (x) :=

∫ t

τ

e−(t−s)g(β(s)(Ku)(s, τ, x; uτ))ds.

Suppose uτ ∈ B, where B is a bounded subset of X . We may suppose that B is containedin the ball centered at the origin of radius r > 0. Then

‖T (t, τ)uτ‖L2(Ω) ≤ re−(t−τ), t ≥ τ ∈ R.

Also, from (4.1), we have that ‖u(t, τ, ·; uτ)‖L2(Ω) ≤M , for t ≥ τ , whereM = r+2k2

√|Ω|

1−k1‖β‖L∞

.

Hence, for all t ≥ τ , and x ∈ Ω, we obtain

∂xU(t, τ)u(τ, x) =

∫ t

τ

e−(t−s)∂xg(β(s)(Ku)(s, τ, x; uτ))ds

=

∫ t

τ

e−(t−s)β(s)g′(β(s)(Ku)(s, τ, x; uτ))∂x(Ku)(s, τ, x; uτ)ds.

Thus,

|∂xU(t, τ)u(τ, x)| ≤∫ t

τ

e−(t−s)|β(s)||g′(β(s)(Ku)(s, τ, x; uτ))||∂x(Ku)(s, τ, x; uτ)|ds.


By Holder’s inequality we have that

|∂x(Ku)(s, τ, x; uτ)| ≤∫

Ω

|∂xJ(x, y)||u(s, τ, y; uτ)| d y

≤ ‖∂xJ(x, ·)‖L2(Ω)‖u(s, τ, ·; uτ)‖L2(Ω)

≤ C‖u(s, τ, ·; uτ)‖L2(Ω)

(4.2)

for some C > 0 such that supx∈Ω ‖∂xJ(x, ·)‖L2(Ω) ≤ C. Using estimate (3.7) we obtain, as inthe proof of Theorem 3.3 that g′(β(t)(Ku)(·)) is bounded by a constant k3 for any u in theball of radius M and t ∈ R. Therefore

|∂xU(t, τ)uτ (x)| ≤ k3C‖β‖∞M∫ t

τ

e−(t−s)ds

≤ k3C‖β‖∞M.

Therefore, for t ≥ τ and any uτ ∈ B, the value of ‖∂xU(t, τ)uτ‖L2(Ω) is bounded by aconstant (independent of τ and uτ ∈ B). Thus, for all uτ ∈ B, we have that U(t, τ)uτbelongs to a ball of H1(Ω). From Sobolev’s Embedding Theorem, it follows that for eachs ∈ R and each bounded subset B of X ,

⋃

τ≤t

⋃

s≤τ U(τ, s)B|Ω is relatively compact in L2(Ω).Then

⋃

t≥τ

U(t, τ)B,

is also relatively compact in X .Therefore, it follows from Theorem 2.12 that the pullback attractor A(t); t ∈ R ex-

ists and is the pullback Ω-limit set of any bounded subset of X containing B, where B =

B

(

0,2k2

√|Ω|

1−k1‖β‖L∞(R)

)

, i.e., for instance

A(t) = Ω℘(B, t) :=⋂

s≤t

⋃

τ≤s

S(t, τ)B, for each t ∈ R.

From this, it also follows that⋃

τ≤tA(τ) is contained in the ball centered at the origin of

radius2k2

√|Ω|

1−k1‖β‖L∞(R)in L2(RN ).

Theorem 4.3. Assume the same conditions as in Lemma 4.1. Then the union⋃

τ≤tA(τ)

is in a bounded set of H1(Ω).

Proof. Proceeding as in the proof of the Lemma 4.1, we obtain that⋃

τ≤tA(τ) is contained

in the ball centered at the origin and radius2k2

√|Ω|

1−k1‖β‖L∞

in L2(Ω). Furthermore, from Theorem

4.2, we get that the pullback attractor can be written as the set of all global boundedsolutions. Hence, if u(t, τ, x; uτ) is a solution of (3.1) such that u(t, τ, x; uτ) ∈ A(t) for allt ∈ R, then

(4.3) u(t, τ, x; uτ) =

∫ t

−∞

e−(t−s)g(β(s)(Ku)(s, τ, x; uτ))ds,

where the equality above is in the sense of L2(RN ).


Proceeding as in the proof of the Theorem 4.2 (see the estimate (4.2) above), we have that

|∂xu(t, τ, x; uτ)| ≤∫ t

−∞

e−(t−s)|β(s)||g′(β(s)(Ku)(s, τ, x; uτ))||∂x(Ku)(s, τ, x; uτ)|ds

≤ k3C‖β‖∞M.

for some C > 0 such that supx∈Ω ‖∂xJ(x, ·)‖L2(Ω) ≤ C, and k3 is a bound of |g′(β(t)(Ku)(·))|for u in the ball of radius M =

2k2√

|Ω|

1−k1‖β‖L∞

in L2(Ω). It follows that A(t) = S(t, τ)A(τ) is in

a bounded set of H1(Ω) uniformly for t ∈ R.

5. A comparison result

In this section we prove a comparison result in L∞(RN ) and use it to prove the invarianceof some sets under S(t, τ).

Definition 5.1. A function v = v(t, τ, x; vτ) is a sub solution of the Cauchy problem (3.1)with initial condition vτ in L∞(RN) if v(τ, x) = vτ (x) for almost everywhere (a.e.) x ∈ Ω, vis continuously differentiable with respect to t and satisfies

∂tv ≤ −v + g(β(t)(Kv))

a. e. in [τ,+∞]× Ω.

Analogously, the function V (t, τ, x; uτ) is a super solution of (3.1) with initial conditionVτ if V (τ, x) = Vτ (x), for almost everywhere (a.e.) x ∈ Ω, V is continuously differentiablewith respect to t and satisfies

−V + g(β(t)(KV )) ≤ ∂tV.

Theorem 5.2. Assume the same conditions of Proposition 3.7 and additionally, that thefunction g is increasing. Let v(t, τ, x; uτ ), (V (t, τ, x; uτ)) be a sub solution (super solution) ofthe Cauchy problem (3.1) with initial condition vτ (Vτ ) ∈ L∞(RN) and u(t, τ, x; uτ) a solutionof (3.1), with initial condition uτ ∈ L∞(RN) . Then, if v(τ, ·) ≤ u(τ, ·) ≤ V (τ, ·)a.e. in Ω.,then

v(t, τ, x; vτ ) ≤ u(t, τ, x; uτ) ≤ V (t, τ, x;Vτ ) a.e. in [τ,+∞]× Ω.

Proof. Consider the operator G on L∞([τ, T ] × Ω), for some T ∈ R (to be fixed later),defined by

(Gφ)(t, x) =

e−(t−τ)φ(τ, x) +

∫ t

τ

e−(t−s)g(β(s)(Kφ)(s, x))ds, x ∈ Ω,

0, x ∈ RN\Ω.

From (3.4), it follows that, for all x ∈ Ω and t ∈ R

|β(t)Kφ)(t, x)| ≤ ‖β‖L∞(R)‖φ‖L∞([τ,T ]×Ω)

and, therefore, |g(β(t)(Kφ)(t, x))| is bounded by a constant B.


Thus, for any (t, x) ∈ [τ, T ]× Ω

|(Gφ)(t, x)| ≤ e−(t−τ)|φ(τ, x)|+∫ t

τ

e−(t−s)|g(β(s)(Kφ)(s, x))|ds

≤ e−(t−τ)|φ(τ, x)|+B

∫ t

τ

e−(t−s) d s

≤ ‖φ(τ, ·)‖L∞(Ω) +B(T − τ)

Therefore G is well defined as an operator on L∞([τ, T ]× Ω).Furthermore (Gφ)(τ, x) = φ(τ, x) and, from the monotonicity properties of g, it follows

that G is monotonic, i.e., for any φ1, φ2 ∈ L∞([τ, T ] × Ω) with φ1 ≥ φ2 a.e. in [τ, T ] × Ω,G(φ1) ≥ G(φ2) a.e. in [τ, T ]× Ω.Let uτ ∈ L∞(Ω), M ∈ R a positive number, and consider the set Muτ in L∞([τ, T ] × Ω)

given by

Muτ= φ; φ(τ, x) = uτ(x) and |φ(τ, x)− uτ (x)| ≤ M

.if φ ∈ Muτ

, then from (3.4), it follows that, for all (t, x) ∈ [τ, T ]× Ω and φ ∈ Muτ

|β(t)Kφ)(t, x)| ≤ ‖β‖L∞(R)‖φ‖L∞([τ,T ]×Ω) ≤ ‖β‖L∞(R)(M + ‖uτ‖L∞(Ω))

and, therefore, |g(|β(t)(Kφ)(t, x)))| is bounded by a constant B for (t, x) ∈ [τ, T ] × Ω andφ ∈ Muτ

.Thus, for any (t, x) ∈ [τ, T ]× Ω

|(Gφ)(t, x)− uτ(x) ≤ (1− e−(t−τ))|uτ (x)|+∫ t

τ

e−(t−s)|g(β(s)(Kφ)(s, x))|ds

≤ (1− e−(t−τ))|uτ (x)|+B

∫ t

τ

e−(t−s) d s

≤ (1− e−(t−τ))‖uτ‖L∞(Ω) +B(T − τ).

and (Gφ)(t, x) belongs to Muτ, for any φ ∈ Muτ

if (T − τ) is small enough.We now prove that G is a contraction in Muτ

. First, using again (3.4), we find that, for(t, x) ∈ [τ, T ]× Ω and φ ∈ Muτ

|β(t)Kφ)(t, x)| ≤ ‖β‖L∞(R)‖φ‖L∞([τ,T ]×Ω) ≤ ‖β‖L∞(R)(M + ‖uτ‖L∞(Ω))

and, since g is locally Lipschitz, there is a constant k3, such that, for φ1, φ2 ∈ Muτ

|g(β(t)(Kφ1)(t, x)))− g(β(t)(Kφ2)(t, x)))| ≤ k2|β(t)|(Kφ1)(t, x)− (Kφ2)(t, x)|≤ k2‖β‖L∞(R)|φ1(t, x)− φ2(t, x)|


Therefore

|(Gφ1)(t, x)− (Gφ2)(t, x)|

≤∫ t

τ

e−(t−s)|g(β(s)(Kφ1)(s, x))− g(β(s)(Kφ2)(s, x))|ds

≤ k2‖β‖L∞(R)

∫ t

τ

e−(t−s)|(φ1)(s, x)− (φ2)(s, x)|ds

≤ k2‖β‖L∞(R)‖(φ1)− (φ2)‖L∞([τ,T ]×Ω)

∫ t

τ

e−(t−s)ds

≤ k2‖β‖L∞(R)(T − τ)‖(φ1)− (φ2)‖L∞([τ,T ]×Ω).

Hence G is a contraction in Muτfor T − τ small enough.

If φ(t, x) belongs to Muτand z(t, x) = limn→∞(Gnφ)(t, x) then, from the continuity of G,

we obtain that

(Gz)(t, x) = G(

limn→∞

(Gnφ)(t, x))

= limn→∞

(Gn+1φ)(t, x) = z(t, x).

Therefore z(t, x) is (the unique) fixed point of G in Muτand it follows that z(t, x) =

u(t, τ, x; uτ), the unique solution of (3.1) in [τ, T ]× Ω with initial condition z(τ, x) = uτ(x).In particular

u(t, τ, x; uτ) = limn→∞

(Gnuτ)(t, x) in L∞([τ, T ]× Ω).

Now, if v is a sub solution of (3.1), with initial condition vτ ≤ uτ a.e., we obtain

v(t, τ, x; vτ ) ≤ e−(t−τ)v(τ, x) +

∫ t

τ

e−(t−s)g(β(s)(Kv)(s, τ, x; vτ))ds a.e. in [τ, T ]× Ω.

Hencev(t, τ, x; vτ) ≤ (Gv)(t, τ, x; uτ) a.e. in [τ, T ]× Ω.

and it follows that v(t, τ, x; vτ ) ≤ (Gnv)(t, τ, x; vτ) a.e. in [τ, T ] × Ω. Thus, passing to thelimit as above, with an eventually smaller T to ensure that G is a contraction in Mvτ weobtain

(5.1) v(t, τ, x; vτ ) ≤ u(t, τ, x; vτ ), a.e. in [τ, T ]× Ω

where u(t, τ, x; vτ) is the unique solution of (3.1) in [τ, T ]× Ω with initial condition vτ .Since v(τ, x) ≤ u(τ, x) a.e. in Ω, it follows by the monotonicity of G in L∞([τ, T ]×Ω) that

(Gnvτ )(t, x) ≤ (Gnuτ )(t, x) a.e. in [τ, T ]× Ω.

Passing to the limit again, we obtain

(5.2) u(t, τ, x; vτ ) ≤ u(t, τ, x; uτ), a.e. in [τ, T ]× Ω.

From (5.1) and (5.2) it follows that

v(t, τ, x; vτ ) ≤ u(t, τ, x; uτ).

If V (t, τ, x; uτ ) is a super solution of the Cauchy problem of (3.1) with initial conditionVτ , uτ ≤ Vτ we obtain, by analogous arguments


u(t, τ, x; uτ) ≤ V (t, τ, x;Vτ ), a.e. in [τ, T ]× Ω.

Since the estimates above are uniform in a bounded set containing the initial conditions,we may extend the result to the interval [T, (2T − τ)] and, by iteration, we can complete theproof of the theorem.

From Theorem 5.2, we can obtain invariance of some subsets of C0(Ω) (or the functionsin C0(Rn) vanishing outside Ω, with the obvious identification) under the process S(t, τ).

Corollary 5.3. Suppose that βm ≤ β(t) ≤ βM , for any t ∈ R. Then, any non negative (nonpositive) solution of (3.1) with β(t) replaced by βM (βm) is a super (sub) solution of (3.1).

Proof. Let v be a non negative solution of (3.1) with β(t) replaced by βM . Then, bythe monotonicity of g, it follows that vt(t, x) = −v(t, x) + g(βMKv(t, x)) ≥ −v(t, x) +g(β(t)Kv(t, x)). Thus v is a super solution. The proof for sub solutions is analogous.

Corollary 5.4. Suppose V (v) is a non negative (non positive) equilibrium of (3.1) with βM(βm) substituted for β(t). Then the set

U = u ∈ C0(RN ); 0 ≤ u ≤ V (U = u ∈ C0(RN); v ≤ u ≤ 0)is positively invariant under the process S(t, τ), that is S(t, τ)U ⊂ U for all t ≥ τ .

Proof. Let uτ ≥ 0 be a initial condition in U . Since u ≡ 0 is a non negative equilibriumsolution of (3.1) it follows by Theorem 5.2 and Corollary 5.3, that

u(t, τ, ·; 0) ≡ 0 ≤ u(t, τ, ·; uτ) ≤ u(t, τ, ·;V ) ≡ V, for all t ≥ τ.

The proof for v is completely similar.

6. Upper semi-continuity of the pullback attractors

A natural question to examine is the dependence of the pullback attractors of (3.1) on thefunction β. From now on, we will denote as Sβ(t, τ)uβ(τ, x) = uβ(t, τ, x; uτ) the evolutionprocess associated with the problem (3.1) with respect to β. We prove the upper semi-continuity of the pullback attractors as β → β0 in L∞(R), i.e., we show that

dist(Aβ(t),Aβ0(t)) → 0 as β → β0 in L∞(R)

where Aβ(t); t ∈ R denotes the pullback attractor of Sβ(t, τ) in X , and dist(Aβ(t),Aβ0(t))is the Hausdorff semi-distance in L2(RN) defined by (2.1), for each t ∈ R.

Theorem 6.1. Assume the same hypotheses of Lemma 4.1, for β in a neighborhood of β0.Then the evolution process Sβ(t, τ); t ≥ τ ∈ R is continuous with respect to variations ofβ, in the L∞(R)−norm at β0, uniformly for t ∈ [τ, b] with b <∞ and uτ in a bounded subsetof X.

Proof. Let uβ(t, τ, x; uτ ) = Sβ(t, τ)uβ(τ, x) and uβ0(t, τ, x; uτ ) = Sβ0(t, τ)uβ(τ, x) be thesolutions of (3.1) with initial condition uτ , and parameters β and β0, respectively, Then

(uβ − uβ0)(t, τ, x; uτ )

=

∫ t

τ

e−(t−s)[g(β(s)(Kuβ)(s, τ, x; uτ ))− g(β0(s)(Kuβ0)(s, τ, x; uτ))]ds.


Summing and subtracting the term∫ t

τe−(t−s)g(β(s)(Kuβ0)(s, τ, x; uτ))ds, we get

(uβ − uβ0)(t, τ, x; uτ )

=

∫ t

τ

e−(t−s)[g(β(s)(Kuβ)(s, τ, x; uτ ))− g(β(s)(Kuβ0)(s, τ, x; uτ))]ds

+

∫ t

τ

e−(t−s)[g(β(s)(Kuβ0)(s, τ, x; uτ))− g(β0(s)(Kuβ0)(s, τ, x; uτ))]ds

for any t ≥ τ and x ∈ Ω.From estimates (4.1) and (3.8) it follows that β(s)(Kuβ)(s, τ, x; uτ) and β(s)(Kuβ0)(s, τ, x; uτ)

remain in a bounded set B ⊂ R. If k3 denotes the Lipschitz constant of g in B, we obtain

|(uβ − uβ0)(t, τ, x; uτ)|

≤∫ t

τ

e−(t−s)|g(β(s)(Kuβ)(s, τ, x; uτ))− g(β(s)(Kuβ0)(s, τ, x; uτ))|ds

+

∫ t

τ

e−(t−s)|g(β(s)(Kuβ0)(s, τ, x; uτ))− g(β0(s)(Kuβ0)(s, τ, x; uτ))|ds

≤ k3‖β‖L∞(R)

∫ t

τ

e−(t−s)|(Kuβ)(s, τ, x; uτ)− (Kuβ0)(s, τ, x; uτ)|ds

+ k3

∫ t

τ

e−(t−s)|(β − β0)(s)||(Kuβ0)(s, τ, x; uτ )|ds

for any t ≥ τ and x ∈ Ω.By (3.7) we then obtain

|(uβ − uβ0)(t, τ, x; uτ)|

≤ k3‖β‖L∞(R)‖J‖L2

∫ t

τ

e−(t−s)‖(uβ − uβ0)(s, τ, ·; uτ)‖Xds

+ k3‖J‖L2‖β − β0‖L∞(R)

∫ t

τ

e−(t−s)‖uβ0(s, τ, ·; uτ)‖Xds

for any t ≥ τ and x ∈ Ω. By (4.1), there exists a constant R such that ‖uβ0(t, τ, ·; uτ)‖X ≤ Rfor any t > τ and uτ in a bounded set of X .Hence, by Holder inequality

‖(uβ − uβ0)(t, τ, ·; uτ)‖X ≤ Rk3‖J‖L2

√

|Ω|‖βǫ − β0‖L∞(R)

+ k3‖J‖L2

√

|Ω|‖β‖L∞(R)

∫ t

τ

e−(t−s)‖(uβ − uβ0)(s, τ, ·; uτ)‖Xds.(6.1)

Finally, setting ϕ(t) = et‖(uβ−uβ0)(t, τ, ·; uτ)‖X for any t ∈ Rτ , we can write the inequalityin (6.1) of the following form

ϕ(t) ≤ k3R‖J‖L2

√

|Ω|‖β − β0‖L∞(R)et + k3‖J‖L2

√

|Ω|‖β‖L∞(R)

∫ t

τ

ϕ(s)ds,


and using Gronwall’s inequality

‖(uβ − uβ0)(t, τ, ·; uτ)‖X ≤ k3R‖J‖L2

√

|Ω|‖β − β0‖L∞(R)ek3‖J‖L∞‖β‖L∞(R)(t−τ)

concluding the proof.

Theorem 6.2. Under the same conditions of Theorem 6.1, the family of pullback attractorsAβ(t); t ∈ R is upper semicontinuous at β = β0.

Proof. Denote by M the radius of an absorbing ball for the process Sβ for β closeto β0, whose existence is guaranteed by Lemma 4.1. Given δ > 0, let τ ∈ R such thatdist(Sβ0(t, τ)B (0;M) ,Aβ0(t)) < δ/2. By Theorem 6.1

‖(uβ − uβ0)(t, τ, ·; uτ)‖X ≤ k3R‖J‖L∞‖β − β0‖L∞(R)

√

|Ω|ek3‖J‖L∞‖β‖L∞(R)

√|Ω|(t−τ) → 0

as β → β0 uniformly for uτ in bounded subsets of X . Since⋃

s∈RAβ(s) ⊂ B(0;M), for β ina neighborhood of β0 , there exists r0 > 0 such that

supaβ∈Aβ(τ)

‖Sβ(t, τ)aβ − Sβ0(t, τ)aβ‖X < δ/2

for all β ∈ B(β0, r0), the ball centered at β0 of radius r0 > 0 in L∞(R).Then

dist(Aβ(t),Aβ0(t)) ≤ dist(Sβ(t, τ)Aβ(τ), Sβ0(t, τ)Aβ(τ)) + dist(Sβ0(t, τ)Aβ(τ), Sβ0(t, τ)Aβ0(τ))

= supaβ∈Aβ(τ)

dist(Sβ(t, τ)aβ, Sβ0(t, τ)aβ) + dist(Sβ0(t, τ)Aβ(τ),Aβ0(t))

<δ

2+δ

2and the upper semicontinuity is proved.

References

[1] J. M. Arrieta, F. D. M. Bezerra and A. N. Carvalho, Rate of convergence of global attractors of some

perturbed reaction-diffusion problems. Topological Methods in Nonlinear Analysis 41 (2) 229-2532013.

[2] F. D. M. Bezerra, A. L. Pereira and S. H. Da Silva, Existence and continuity of global attractors and

nonhomogeneous equilibria for a class of evolution equation with non local terms. J. Math. Anal.Appl., 396 (2012) 590-600.

[3] T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped

wave equation: Existence and continuity of the pullback attractor. Nonlinear Analysis 74 (2011)2272–2283.

[4] A. N. Carvalho, J. A. Langa, J. C. Robinson, On the continuity of pullback attractors for evolution

processes. Nonlinear Analysis, 71 (2009) 1812-1824.[5] A. N. Carvalho, J. A. Langa, J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous

Dynamical Systems. Applied Mathematical Sciences 182, Springer-Verlag, 2012.[6] A. N. Carvalho, S. Sonner, Pullback exponential attractors for evolution process in Banach spaces:

theoretical results. Communications on Pure and Applied Analysis, 12, 6, 2013.[7] V. V. Chepyzhov, M. I. Vishik, Attractors for Equations of Mathematical Physics, in: Colloquium

Publications, vol. 49, American Mathematical Society, 2002.[8] J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces.

American Mathematical Society, Providence, Rhode Island, 1974.


[9] S. H. Da Silva, F. D. M. Bezerra, Finite fractal dimensionality of attractors for nonlocal evolution

equations. Electronic Journal of Differential Equations, 2013, No. 221, (2013) 1-9.[10] A. De Masi, E. Oliveri and E. Presutti, Critical droplet for a non local mean field equation. Markov

Process. Related Fields, 6 (2000) 439-471.[11] P. E. Kloeden, Pullback attractors in nonautonomous difference equations. J. Differ. Equations

Appl. 6 (1) (2000) 33-52.[12] G.E. Ladas and V. Lakshmikantham, Differential equations in abstract spaces , Academic Press,

N.Y., 1972.[13] A. De Masi, E. Orland, E. Presutti and L. Triolo, Stability of the interface in a model of phase

separation. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 1013-1022.[14] A. De Masi, T. Gobron, E. Presutti, Traveling fronts in non local evolution equations. Arch. Rational

Mech. Anal., 132 (1995) 143-205.[15] A. De Masi, E. Orland, E. Presutti and L. Triolo, Uniqueness and global stability of the instanton

in non local evolution equations, Rend. Mat. Appl., 14 (7) (1994) 693-723.[16] 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.[17] A. L. Pereira and S. H. Da Silva, Existence of global attractors and gradient property for a class of

non local evolution equations, Sao Paulo J. Math. Sci. 2 (2008) 1-20.[18] A. L. Pereira and S. H. Da Silva, Continuity of global attractors for a class of non local evolution

equations, Discrete Cont. Dyn. Syst. 26 (2010) 1073-1100.[19] B. Schmalfuss, Attractors for the non-autonomous dynamical systems. International Conference on

Differential Equations, Vol. 1, 2 (Berlin, 1999), 684–689, World Sci. Publishing, River Edge, NJ,2000.

[20] Rall, L.B., Nonlinear Functional Analysis and Applications. Academic Press, New York- London,1971.

[21] G. R. Sell, Non-autonomous differential equations and dynamical systems. Trans. Amer. Math.Soc., 127 (1967) 241-283.

[22] Teman, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, 1988.

(F. D. M. Bezerra) Departamento de Matematica, Universidade Federal da Paraıba, Cidade

Universitaria-Campus I, 58051-900 Joao Pessoa PB, Brazil.

E-mail address : [email protected]

(S. H. Da Silva) Unidade Academica de Matematica, Universidade Federal de Campina

Grande, 58051-900 Campina Grande PB, Brazil.


(A. L. Pereira)Universidade de Sao Paulo, Instituto de Matematica e Estatıstica, 13565-905

Sao Paulo SP, Brazil.


Asymptotic behavior for a class of non autonomous non local problems

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