Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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Dissipativity of some classes of equations 237

long to the set J

, because J

is a maximal compact invariant set in X. But

the inclusion λx

∈ J holds for all λ ∈ R if and only if x

∈ Θ. The obtained

contradiction shows that J

= Θ.

We will show that in (X, S

, π) there is no nontrivial compact motions deﬁned

on S. In fact, let x ∈ X be such point that there exists ϕ : S → X possessing the

following properties: ϕ(S) is relatively compact, π

ϕ(s) = ϕ(t + s) (t ∈ S

, s ∈ S)

and ϕ(0) = x

. Since J

is a maximal compact invariant set, then

ϕ(S) ⊆ J

and,

in particular, x

∈ J

⊆ Θ. Therefore, |x

| = 0. Thus, we proved that 1. implies

2. The converse implication is evident.

We will show that 1. implies 3. Indeed, since under the conditions of Theorem

6.10 from 1. follows that (X, S

, π) is pointwise dissipative, then, according to

Theorem 1.10, it is locally dissipative. As Y is compact and (X, S

, π) is locally

dissipative, then there is δ > 0 such that

lim

t→+∞

sup{|xt| : |x| < δ} = 0. (6.34)

Taking into consideration (6.34), by standard reasoning (see, for example,

[

208

]

[

275

]

,and also Theorem 2.38) we may prove that there are N, ν > 0 such that

|xt| ≤ N e

−νt

|x| for all x ∈ X and t ≥ 0. Finally, 3. evidently implies 1. The

theorem is proved. 

Remark 6.3 a. The condition of local compactness in Theorem 6.10 is essential.

To conﬁrm this statement it is suﬃcient to consider the example 1.8. In this example

all motions tend to zero as t → +∞, but this system is not exponentially stable. It

is not diﬃcult to see that the dynamical system from the example 1.8 is not locally

compact.

b. Theorem 6.10 is also true if the space Y is just pseudo-metric.

A very important class of linear non-autonomous dynamical systems with inﬁnite

dimensional phase space satisfying the condition of local compactness is the class

of linear non-autonomous functional diﬀerential equations

[

175

]

. Let us recall some

notions and denotations from

[

175

]

. Let r > 0, C([a, b], R

) be a Banach space of all

continuous functions ϕ : [a, b] → R

equipped with the norm sup. If [a, b] = [−r, 0],

then we set C := C([−r, 0], R

). Let σ ∈ R, A ≥ 0 and u ∈ C([σ −r, σ +A], R

). We

will deﬁne u

∈ C for all t ∈ [σ, σ + A] by the equality u

(θ) := u(t + θ), −r ≤ θ ≤ 0.

Denote by D = D(C, R

) a Banach space of all linear continuous operators acting

from C into R

and endowed with operational norm. Consider a linear equation

˙u = A(t, u

), (6.35)

where A : R ×C → R

is continuous and linear with respect to the second variable,

i.e. A(t, ·) ∈ D for all t ∈ R. Let us set H(A) :=

: s ∈ R}, where A

(t, ·) =

A(t + s, ·) and by bar we denote a closure in the compact-open topology on R ×C.

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238 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Along with the equation (6.35) let us consider the family of equations

˙v = B(t, v

), (6.36)

where B ∈ H(A). Let ϕ(·, φ, B) be a solution of (6.36) passing through the point

φ ∈ C for t = 0 deﬁned for all t ≥ 0.

Let Y := H(A) and denote by (Y, R, σ) a dynamical system of translations on

H(A). Let X := C × Y , (X, R

, π) be a dynamical system on X deﬁned in the

following way: π((φ, B), τ) := (ϕ(τ, φ, B), B

). It is easy to see that the system



(X, R

, σ), (Y, R, σ), h



is linear. We remark a very important property of this

system.

The following statement holds.

Lemma 6.3 Let H(A) be compact. Then for any point x ∈ X := C ×H(A) there

exist a neighborhood U

of the point x and a positive number l

> 0 such that π

is relatively compact for all t ≥ l

, i.e. the dynamical system (X, R

, π) is locally

compact.

Proof. This assertion follows from Lemmas 2.2.3 and 3.6.1 from

[

175

]

and from the

compactness of Y . 

Applying to the constructed non-autonomous dynamical system Theorem 6.10

(see also Remark 6.3) and taking into consideration Lemma 6.3, we will obtain the

following statement.

Theorem 6.11 Let H(A) be compact. Then the following statements are equiv-

alent:

(1) the trivial solution of (6.35) is uniformly exponentially stable, i.e. there are

positive numbers N and ν such that |ϕ(t, φ, B)| ≤ N e

−νt

|φ| for all B ∈ H(A)

and t ≥ 0;

(2) the trivial solution of (6.36) is uniformly exponentially stable for all B ∈ H(A).

6.5 Convergence of monotone evolutionary equations

Let H be a Hilbert space. Denote by L

loc

(R, H) the space of all functions ϕ : R → H

that are locally integrable with the power p, that is

|t|≤k

|ϕ(t)|

dt < +∞ (p ≥ 1, k = 0, 1, 2 . . .).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6

Dissipativity of some classes of equations 239

Let us deﬁne a topology on L

loc

(R, H) by the family of semi-norms |· |

|ϕ|



|t|≤k

|ϕ(t)|



(k = 0, 1, 2, . . .)

and by (L

loc

(R, H), R, σ) we denote the dynamical system of translations on

loc

(R, H) (see

[

103

]

Deﬁnition 6.3 Recall

[

]

that an operator A : D(A) → H (D(A) ⊆ H) is called:

a. monotone, if RehAu

−Au

, u

−u

i ≥ 0 for all u

, u

∈ D(A);

b. semi-continuous, if the function ϕ : R → R deﬁned by the equality ϕ(λ) :=

hA(u + λv), wi is continuous;

c. uniformly monotone, if there exists α > 0 such that

RehAu − Av, u −vi ≥ α|u −v|

(6.37)

for all u, v ∈ D(A).

Note, that the family of all monotone operators can be partially ordered with

respect to the inclusion of its graphs.

Deﬁnition 6.4 A monotone operator is called maximal, if it is maximal among

monotone operators.

Consider a diﬀerential equation

+ Ax = f(t), (6.38)

where f ∈ L

loc

(R, H) and A is a maximal monotone operator with the domain

D(A). It is known

[

]

that for all x

∈

D(A) there exists a unique weak solution

ϕ(t, x

, f) of the equation (6.38) satisfying the condition ϕ(0, x

, f) = x

and deﬁned

on R

. Let Y := H

(f) and (Y, R

, σ) be a dynamical system of translations on Y .

We denote by X :=

D(A) ×Y and by (X, R

, π) a dynamical system on X, where

π(t, hv, gi) := hϕ(t, v, g), g

i. Then the triple h(X, R

, π), (Y, R

, σ), hi(h = pr

X → Y ) is the non-autonomous dynamical system

[

187

]

generated by the equation

(6.38). Applying the results obtained in Chapters 1-5 to the constructed in such

way non-autonomous dynamical system, we will obtain the following results for the

equation (6.38).

Theorem 6.12 Let f ∈ L

loc

(R, H) and H(f) be compact. If A is a maximal

monotone operator which is semi-continuous and uniformly monotone, then the

equation (6.38) is convergent, i.e. the equation (6.38) admits a unique compact on

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6

240 Global Attractors of Non-autonomous Dissipative Dynamical Systems

R uniformly compatible solution which is globally asymptotically stable. In addition,

there exist positive numbers N and ν such that the equation

+ Ax = g(t) (6.39)

holds for all g ∈ H(f ) and admits a unique compact on R uniformly compatible

solution ϕ(t, v

, g) (g ∈ H(f)) such that

|ϕ(t, v, g) −ϕ(t, v

, g)| ≤ N e

νt

|v − v

| (6.40)

for all v ∈

D(A) and t ≥ 0.

Proof. Modifying the results from

[

187

]

, we obtain that under the conditions of

Theorem 6.12 all the solutions of the equation (6.38) are compact on R

. On the

other hand, by the condition (6.37) we have

|ϕ(t, x

, f) −ϕ(t, x

, f)| ≤ e

−αt

−x

| (6.41)

for all t ≥ 0 and x

, x

∈ H. And, moreover, if g ∈ H(f ), then for solutions of the

equation (6.39) the following estimation

|ϕ(t, y

, g) −ϕ(t, y

, g)| ≤ e

−αt

−y

| (6.42)

takes place for all t ≥ 0 and y

, y

∈ H. The relation (6.42) implies the condi-

tion 4. of Theorem 2.15, if we take as V : X × X → R

the following function

V ((v

, g), (v

, g)) := |v

−v

| (g ∈ H(f), v

, v

∈

D(A)). The theorem is proved.

Corollary 6.4 Let f ∈ L

loc

(R, H) be stationary (τ−periodic, almost periodic,

recurrent). Then the equation (6.38) admits a unique stationary (τ−periodic, almost

periodic, recurrent) solution wich is globally asymptotically stable.

Finally, we will give an example of equations of the type (6.38).

Example 6.1 Consider the equation

∂

∂t

= ∆u − φ(

∂u

∂t

) + f(t) (6.43)

in the open set D ⊂ R

with the boundary condition u|

∂D

= 0 on the boundary

∂D of D. Suppose that the function φ : R → R satisﬁes the conditions : φ(0) = 0

and 0 < c

≤ φ

(ξ) ≤ c

(ξ ∈ R). Then the equation (6.43) may be rewritten in the

following way:

(

∂u

∂t

= v

∂v

∂t

= ∆u − φ(v) + f(t)

. (6.44)

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Dissipativity of some classes of equations 241

We denote by H := W

1,2

(Ω) × L

(Ω) and deﬁne on H a scalar product

h(u, v), (u

∗

, v

∗

)i =

Ω

[vv

∗

+ ∆u∆u

∗

+ λuv

∗

+ λu

∗

v]dx,

where λ is a certain positive constant depending only on c

and c

. It is possible

to verify (see, for example,

[

131

]

) that Theorem 6.12 can be applied to the system

(6.44), if the set H(f) ⊂ L

loc

(R, H) is compact.

Let I ⊆ R , H be a Hilbert space with scalar product h, i, D(I, R) be a space of

all inﬁnitely diﬀerentiable functions ϕ : I → H with compact support and [H] be

an algebra of all linear bounded operators on H.

Consider the equation

[hu(t), ϕ

(t)i + hA(t)u(t), ϕ(t)i + hf (t), ϕ(t)i]dt = 0, (6.45)

where A ∈ C(R, [H]) and f ∈ C(R, H). A function u ∈ C(I, H) is called a solution

of the equation (6.45), if the equality (6.45) takes place for all ϕ ∈ D(I, H).

Let x ∈ H, ϕ(t, x, A, f ) be a solution of the equation (6.45) deﬁned on R

and

satisfying the condition ϕ(0, x, A, f) = x and

[hu(t), ϕ

(t)i + hB(t)u(t), ϕ(t)i + hg(t), ϕ(t)i]dt = 0 (6.46)

be a family of equations, where (B, g) ∈ H(A, f) :=

{(A

, f

) | τ ∈ R}. We will

suppose that the operator-function A ∈ C(R, [H]) is self-adjoint and negatively

deﬁned, i.e. A(t) = −A

(t) + iA

(t) for all t ∈ R, where A

(t) and A

(t) are

self-adjoint and

(t)u, ui ≥ α|u|

(6.47)

for all t ∈ R and u ∈ H, where α > 0.

Lemma 6.4

[

]

We have

|ϕ(t, x, A, f)|

= −hA

(t)ϕ(t, x, A, f), ϕ(t, x, A, f )i

+Rehf(t), ϕ(t, x, A, f)i (6.48)

for all t > 0.

Lemma 6.5 The following inequality

|ϕ(t, x, A, f)| ≤ |x| +

|f(τ )|dτ (6.49)

holds for all t ≥ 0.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6

242 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. In virtue of the equality (6.48) we have

|ϕ(t, x, A, f)|

≤ |f(t)||ϕ(t, x, A, f )|.

Let v(t) := |ϕ(t, x, A, f)|

. Then

≤ 2|f(t)|

v(t) and, consequently,

v(t) −

v(τ) ≤

|f(s)|ds.

From this follows the inequality (6.49). 

Lemma 6.6 Let l, r and β > 0, x

∈ H, A ∈ C(R, [H]) and f ∈ C(R, [H]). Then

there exists M = M (f, l, r, β, x

) > such that

|ϕ(t, x, B, g) − ϕ(t, x

, A, f)| ≤ |x − x

kB(τ) − A(τ )kdτ +

|g(τ) −f(τ)|dτ (6.50)

for all t ∈ [0, l] and x ∈ B[x

, r] := {x | x ∈ H, |x − x

| ≤ r}, if |g(t) − f(t)| ≤ β

and RehB(t)x, xi ≤ 0 for any t ∈ [0, l] and x ∈ H.

Proof. We denote by v(t) = ϕ(t, x, B, g) − ϕ(t, x

, A, f). Then

[hv(t), ϕ

(t)i + hA(t)v(t), ϕ(t)i+

hB(t) −A(t))v(t), ϕ(t)i + hg(t) −f(t), ϕ(t)i]dt = 0

for any ϕ ∈ D(R, H). By virtue of Lemma 6.4

|v(t)|

= RehA(t)v(t), v(t)i

+Re[h(B(t) −A(t))ϕ(t, x, B, f), v(t)i + hg(t) −f(t), v(t)i]

and, according to Lemma 6.5, we have

|v(t)| ≤ |v(0)| +

|(B(τ) − A(τ))ϕ(τ, x, B, g) + g(τ) − f (τ)|dτ

≤ |v(0)| +

kB(τ) − A(τ )k|ϕ(τ, x, B, g)|dτ +

|g(τ) −f(τ)|dτ. (6.51)

On the other hand, according to Lemma 6.5 for ϕ(t, x, B, g) we have

|ϕ(t, x, B, g)| ≤ |x| +

|g(τ)|dτ ≤ |x

|+ r + βl

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6

Dissipativity of some classes of equations 243

+l max

0≤t≤l

|f(t)| := M(f, l, r, β, x

). (6.52)

Taking into account the inequalities (6.51) and (6.52), we obtain (6.50). The lemma

is proved. 

Let

X = H ×H

(A, f) and denote by X the set of all hu, (B, g)i ∈

X such that

through the point u ∈ H there passes a solution ϕ(t, u, B, g) of the equation (6.46)

deﬁned on R

Lemma 6.7 The set X ⊆ H × H(A, f ) is closed in H × H(A, f ).

Proof. Let hx, (B, g)i ∈

X. Then there exists a sequence hx

, (B

, g

)i ∈ X such

that x

→ x in the space H, B

→ B in C(R, [H]) and g

→ g in C(R, H). Let l

and ε > 0 be such that

−x

| < ε, |g

(t) − g

(t)| < ε and kB

(t) − B

(t)khε (6.53)

for all t ∈ [0, l] and k, l ≥ k

. Denote by r := sup{|x

| : k ∈ N}. Then according

to Lemma 6.6

|ϕ(t, x

, B

, g

) − ϕ(t, x

, B

, g

)| ≤ |x

−x

| + M

(τ) − B

(τ)kdτ

(τ) − g

(τ)|dτ ≤ ε + M εl + εl (6.54)

for all t ∈ [0, l] and k, m ≥ k

, where M is a positive constant which is not

dependent on r, l and g. Taking into account that the space C(R

, H) is com-

plete and the inequality (6.54), we conclude that the sequence {ϕ(t, x

, B

, g

)}

is convergent in C(R

, H) and, according to the inequality (6.54), ϕ(t, x, B, g) =

lim

k→+∞

ϕ(t, x

, B

, g

). The lemma is proved.



Lemma 6.8 The mapping ϕ : R

× X → H(ϕ : (t, hu, B, gi) → ϕ(t, u, B, g)) is

continuous.

Proof. Let t

→ t, x

→ x, B

→ B and g

→ g. Then

|ϕ(t, x

, B

, g

) − ϕ(t, x, B, g)| ≤ |ϕ(t, x

, B

, g

) − ϕ(t

, x, B, g)|

+|ϕ(t

, x, B, g) −ϕ(t, x, B, g)| ≤ max

0≤t≤l

|ϕ(t, x

, B

, g

) − ϕ(t, x, B, g)|

+|ϕ(t

, x, B, g) − ϕ(t, x, B, g)|. (6.55)

By virtue of the inequality (6.55) and Lemma 6.6 we obtain the necessary assertion.

The lemma is proved. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6

244 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 6.9 For all (B, g) ∈ H

(A, f) and x

, x

∈ H we have

|ϕ(t, x

, B, g) −ϕ(t, x

, B, g)| ≤ e

−αt

−x

for any t ∈ R

Proof. If the operator-function A(t) is negatively deﬁned, then every operator-

function B ∈ H

(A) is negatively deﬁned too and RehB(t)u, ui ≥ α|u|

(t ∈

R, u ∈ H), where α > 0 is the same constant as the one ﬁguring in (6.47) for the

operator-function A(t). Let ω(t) := ϕ(t, x

, B, g) − ϕ(t, x

, B, g). Then, according

to Lemma 6.4, we have

|ω(t)|

= RehB(t)ω(t), ω(t)i ≤ −α|ω(t)|

and, consequently, |ω(t)| ≤ |ω(0)|e

−αt

for all t ∈ R

. 

Let us deﬁne on X a dynamical system in the following way: π(t, x) :=

π(hu, (b, g)i, t) = hϕ(t, u, B, g), (B

, g

)i for all hu, (B, g)i ∈ X and R

. Let

Y = H(A, f) and (Y, R, σ) be a dynamical system of translations on Y and

h = pr

: X → Y . Then the triple h(X, R

, π), (Y, R, σ), hi is the non-autonomous

dynamical system generated by equation (6.45).

Deﬁnition 6.5 We will call the equation (6.45) convergent, if it admits a compact

solution on R that is globally asymptotically stable.

According to the results of Chapter 2 (see section 2.5), the equation (6.45) will

be convergent if and only if the non-autonomous dynamical system h(X, R

, π), (Y,

, σ), hi generated by the equation (6.45) is convergent.

Theorem 6.13 Let A ∈ C(R, [H]), f ∈ C(R, H) and H(A, f ) be compact. Then

the equation (6.45) is convergent.

Proof. According to the results from

[

]

[

187

]

, all the solutions of the equation

(6.45) are compact on R

and by virtue of Lemma 6.9 every solution of the equation

(6.45) is globally asymptotically stable. 

6.6 Global attractors of non-autonomous Lorenz systems

This section is devoted to the study of non-autonomous Lorenz systems. The

problems are formulated and solved in the context of non-autonomous dynamical

systems. First, we prove that such systems admit a compact global attractor and

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6

Dissipativity of some classes of equations 245

characterize its structure. Then, we obtain conditions of convergence of the non-

autonomous Lorenz systems, under which all solutions approach a point attrac-

tor. Third, we derive a criterion for existence of almost periodic (periodic, quasi-

periodic) and recurrent solutions of the systems. Finally, we prove a global averaging

principle for non-autonomous Lorenz systems.

The following n-dimensional systems of diﬀerential equations are called systems

of hydrodynamic type or autonomous Lorenz systems(

[

306

]

= Σ

j,k

ijk

+ Σ

+ f

, i = 1, 2, ..., n, (6.56)

where Σb

ijk

is identically equal to zero, Σa

is negative deﬁnite, and

are constants. The well known three-dimensional Lorenz system for geophysical

ﬂows or climate modelling

[

242

]

is a special case of this type of systems.

It is known that solutions of (6.56) imbed in some ellipsoid and do not leave it

later, i.e. the autonomous system (6.56) is dissipative, and hence admits a compact

global attractor. In the vector-matrix form the system (6.56) may be written as:

= Au + B(u, u) + f, (6.57)

where A is a positive deﬁnite matrix and B : H × H → H (H is a n-dimensional

real or complex Euclidean space) is a bilinear form satisfying the identity

RehB(u, v), wi = −RehB(u, w), vi

for every u, v, w ∈ H.

When f is not a constant vector but a bounded function of time t, it is known

that the equation (6.57) also admits a compact global attractor

[

218

]

Our aim is to study the non-autonomous version of the equation (6.57). Namely,

in this case, the matrix A, the bilinear form B, and the function f all depend on

time t. We will consider issues like compact global attractors, convergence, almost

periodic (including periodic and quasi-periodic) solutions and recurrent solutions,

and averaging principles.

6.6.1 Non-autonomous Lorenz systems

Let Ω be a compact metric space, R = (−∞, +∞), (Ω, R, σ) be a dynamical system

on Ω and H be a real or complex Hilbert space. We denote by L(H) (L

(H))

the space of all linear (bilinear) operators on H. When W is some metric space,

C(Ω, W ) denotes the space of all continuous functions f : Ω → W , endowed with

the topology of uniform convergence.

Let us consider the non-autonomous Lorenz system

= A(ωt)u + B(ωt)(u, u) + f(ωt), ω ∈ Ω, (6.58)

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246 Global Attractors of Non-autonomous Dissipative Dynamical Systems

where ωt := σ(t, ω), A ∈ C(Ω, L(H)), B ∈ C(Ω, L

(H)) and f ∈ C(Ω, H). Note

that when the autonomous Lorenz system (6.57) is perturbed by periodic, quasi-

periodic, almost periodic or recurrent forces, it can then be written as (6.58). More-

over, we assume that the following conditions are fulﬁlled:

(1) There exists α > 0 such that

RehA(ω)u, ui ≤ −α|u|

(6.59)

for all ω ∈ Ω and u ∈ H, where |· | is a norm in H ;

(2)

RehB(ω)(u, v), wi = −RehB(ω)(u, w), vi (6.60)

for every u, v, w ∈ H and ω ∈ Ω .

Remark 6.4 a. It follows from (6.60) that

RehB(ω)(u, v), v)i = 0 (6.61)

for every u, v ∈ H and ω ∈ Ω.

b. From bilinearity and continuity, we obtain

|B(ω)(u, v)| ≤ C

|u||v| (6.62)

for all u, v ∈ H and ω ∈ Ω, where C

:= sup{|B(ω)(u, v)| : ω ∈ Ω, u, v ∈ H, |u| ≤ 1

and |v| ≤ 1}.

Deﬁnition 6.6 We will call the system (6.58) with conditions (6.59) and (6.60)

a non-autonomous Lorenz system or a non-autonomous system of hydrodynamic

type.

We note that from the conditions (6.60) -(6.62) it follows that

|B(ω)(x

, x

) − B(ω)(x

, x

)| ≤ C

(|x

| + |x

|)|x

−x

| (6.63)

for all x

, x

∈ H and ω ∈ Ω.

Since the coeﬃcients of (6.58) are locally Lipschitzian with respect to u ∈ H,

through every point x ∈ H passes a unique solution ϕ(t, x, ω) of equation (6.58) at

the initial moment t = 0. And this solution is deﬁned on some interval [0, t

(x,ω)

Let us note that

(t) = 2Rehϕ

(t, x, ω), ϕ(t, x, ω)i = 2RehA(ωt)ϕ(t, x, ω), ϕ(t, x, ω)i

+2RehB(ωt)(ϕ(t, x, ω), ϕ(t, x, ω)), ϕ(t, x, y)i + 2Rehf(ωt), ϕ(t, x, ω)i

= 2RehA(ωt)ϕ(t, x, ω), ϕ(t, x, ω)i + 2Rehf(ωt), ϕ(t, x, ω)i

≤ −2α|ϕ(t, x, ω)|

+ 2kfk|ϕ(t, x, ω)|, (6.64)