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

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Global attractors of V -monotone dynamical systems 397

According to the conditions 1. and 3. of Theorem 12.8 we have

a|u

(t) − u

(t)|

≤ 0

for all t ∈ [0, t

) and, consequently, u

(t) = u

(t) for all t ∈ [0, t

Now we will prove that every solution of the equation (12.17) is deﬁned on R

Let u ∈ R

and ϕ(t, u, ω) be the unique solution of the equation (12.17) deﬁned

on [0, t

(u,ω)

). To prove that t

(u,ω)

= +∞ it is suﬃcient to show that the solution

ϕ(t, u, ω) is bounded on [0, t

(u,ω)

). We denote by b := max

ω∈Ω

kW (ω)k and T

(u, ω) =

{t ∈ [0, t

(u,ω)

) | |ϕ(t, u, ω)| ≤ r } and T

(u, ω) = [0, t

(u,ω)

) \T

(u, ω). It is clear that

the set T

(u, ω) is open and, consequently, T

(u, ω) =

{(t

, t

) | β = β(α) }. For

all t ∈ T

(u, ω) there exists α such that t ∈ (t

, t

) , |ϕ(t

, u, ω)| = |ϕ(t

, u, ω)| and

|ϕ(t, u, ω)| > r. Consequently,

a|ϕ(t, u, ω)|

≤ hW (σ

ω)ϕ(t, u, ω), ϕ(t, u, ω)i ≤ (12.18)

hW (ωt

)ϕ(t

, u, ω), ϕ(t

, u, ω)i ≤ br

From the inequality (12.18) follows that |ϕ(t, u, ω)| ≤

r and, consequently,

we obtain sup{|ϕ(t, u, ω)| | t ∈ [0, t

(u,ω)

) } ≤ r

:= max{r,

r}. Thus, the

equation (12.17) deﬁnes a cocycle ϕ on R

Let X = R

× Ω, (X, R

, π) be a skew-product dynamical system and

h(X, R

, π), (Ω, R, σ), hi be the non-autonomous dynamical system generated by

the equation (12.17). Denote by V : X → R

the function deﬁned by the equal-

ity V(u, ω) := hW (ω)u, ui for all (u, ω) ∈ X := R

× Ω. If |ϕ(t, u, ω)| > r for all

t ∈ (t

, t

) ⊂ R

, then

V(σ

ω, ϕ(t, u, ω)) = h

W (σ

ω)ϕ(t, u, ω), ϕ(t, u, ω)i+

hW (σ

)F (σ

ω, ϕ(t, u, ω)), ϕ(t, u, ω)i ≤ −c(|ϕ(t, u, ω)|) < 0.

In view of Theorem 5.3 the non-autonomous dynamical system h(X, R

, π),

(Ω, R, σ), hi admits a compact global attractor.

Let V : X

×X → R

be the function deﬁned by the equality V (ω, u

, u

) :=

hW (ω)(u

−u

), u

−u

i. Then

V (σ

ω, ϕ(t, u

, ω), ϕ(t, u

, ω)) = h

W (σ

ω)(ϕ(t, u

, ω) −

ϕ(t, u

, ω)), ϕ(t, u

, ω) −ϕ(t, u

, ω)i+

hW (σ

ω)(F (σω, ϕ(t, u

, ω)) −

F (σω, ϕ(t, u

, ω)))

ϕ(t, u

, ω) −ϕ(t, u

, ω)i ≤ 0

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

398 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and, consequently, V (σ

ω, ϕ(t, u

, ω), ϕ(t, u

, ω)) ≤ V (ω, u

, u

) for all ω ∈

Ω, u

, u

∈ R

and t ∈ R

. It is easy to verify that the function V satisﬁes

all the conditions of Theorems 12.4, 12.5 and the statements a.-h. follow from

Corollary 12.1, Theorems 12.4, 12.5 and Lemma 12.4. The theorem is proved. 

Example 12.1 As an example that illustrates this theorem we can consider the

following equation

= g(u) + f(σ

ω),

where f ∈ C(Ω, R) and

g(u) =











(u + 1)

, u < −1

0 , |u| ≤ 1

−(u − 1)

) , u > 1.

Example 12.2 Let us consider the equation

+ p(x)x

+ ax = f (σ

ω),

where p ∈ C(R, R), f ∈ C(Ω, R) and a is a positive number. Denote by y :=

+ F (x), where F (x) :=

p(s)ds. Then we obtain the system







= y − F (x)

= −ax + f (σ

ω).

(12.19)

Theorem 12.9 Suppose the following conditions hold:

1. p(x) ≥ 0 for all x ∈ R.

2. there exist positive numbers r and k such that p(x) ≥ k for all |x| ≥ r.

Then the non-autonomous dynamical system generated by (12.19) is compactly

dissipative and V −monotone.

Proof. Let X := Ω × R and h(X, R

, π), (Ω, R, σ), hi be the non-autonomous

dynamical system generated by (12.19). We deﬁne the function V : X → R

the equality

V (ω, x, y) := y

−yF (x) +

(x) + ax

Then

V (π

(ω, x, y))|

t=0

= −p(x)[y − F (x)]

−axF (x) + (2y − F (x))f(ω).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

Global attractors of V -monotone dynamical systems 399

According to

[

270

]

(see the proof of Theorem 12.1.2), there exists R > 0 such

that

V (π

(ω, x, y))|

t=0

< 0 for all x

+ y

≥ R

and V (ω, x, y) → +∞ as

+ y

→ +∞. In view of Theorem 5.3 the non-autonomous dynamical system

h(X, R

, π), (Ω, R, σ), hi is compactly dissipative.

Let V : X

×X → R

be the function deﬁned by the equality

V ((u

, ω), (u

, ω)) := hu

−u

), u

−u

i. Then

V (σ

ω, ϕ(t, u

, ω), ϕ(t, u

, ω)) = −(x

(t) − x

(t))[F (x

(t)) − F (x

(t))] ≤ 0

for all t ∈ R, where x

(t) = pr

ϕ(t, u

, ω) (i = 1, 2), and, consequently,

V (π

, ω), π

, ω)) ≤ V ((u

, ω), (u

, ω)) for all t ∈ R

. To ﬁnish the proof

it is suﬃcient to refer to Theorem 12.8. The theorem is proved. 

12.5.2 Caratheodory’s diﬀerential equations

Let us consider now the equation (12.17) with the right hand side f satisfying the

conditions of Caratheodory (see, for example,

[

292

]

). The space of all Caratheodory‘s

functions we denote by C(R × R

, R

). Topology on this space is deﬁned by the

family of semi-norms (see

[

292

]

)

k,m

(f) :=

−k

max

|x|≤m

|f(t, x)|dt.

This space is metrizable, and on C(R × R

, R

) there can be deﬁned a dynamical

system of translations (C(R × R

, R

), R, σ).

We consider the equation

= f(t, x), (12.20)

where f ∈ C(R × R

, R

), and the family of equations

= g(t, x), (12.21)

where g ∈ H(f ) :=

| τ ∈ R }, and f

is a τ−translation of the function f w.r.t.

the variable t, i.e. f

(t, x) := f(t + τ, x) for all t ∈ R and x ∈ R

and by bar is

denoted the closure in the space C(R ×R

, R

). Denote by ϕ(t, x, g) the solution of

equation the (12.21) with the initial condition ϕ(0, g, x) = x. Then ϕ is a cocycle on

(see, for example

[

292

]

) with the base H(f ). Hence, we may apply the general

results from section 1-4 to the cocycle ϕ generated by the equation (12.20) with a

Caratheodory‘s right hand side and obtain some results for this type of equations.

For instance, the following assertion holds.

Theorem 12.10 Let f ∈ C(R × R

, R

) be an almost periodic function in t ∈ R

(in the sense of Stepanoﬀ

[

238

]

) uniformly w.r.t. x on compacts from R

, i.e. for

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

400 Global Attractors of Non-autonomous Dissipative Dynamical Systems

every ε > 0 and compact K ⊂ R

the set

T(ε, f, K) := {τ ∈ R |

max

x∈K

|f(t + τ + s, x) − f (t + s, x)|ds < ε }

is relatively dense on R. Suppose that

1. hf (t, x

) − f (t, x

), x

−x

i ≤ 0 for all t ∈ R and x

, x

∈ R

2. there exists a positive constant r and a function c : [r, +∞) → (0, +∞) such

that hf(t, u), ui ≤ −c(|u|) for all |u| > r.

Then on R

the equation (12.20) generates a cocycle ϕ which admits a compact

global attractor I = {I

| g ∈ H(f) } with the following properties:

a. I

is a nonempty, compact and convex subset of R

for every g ∈ H(f).

b. I =

| g ∈ H(f) } is connected.

c. the mapping g → I

is continuous with respect to the metric of Hausdorﬀ.

d. I = {I

| g ∈ H(f )} is invariant, i.e. ϕ(t, g, I

) = I

for all g ∈ H(f) and

t ∈ R

e. lim

t→+∞

β(ϕ(t, σ

−t

g)M, I

) = 0 for all M ∈ C(R

) and g ∈ H(f).

f. lim

t→+∞

sup{β(ϕ(t, σ

−t

g)M, I) | g ∈ H(f) } = 0 for any M ∈ C(R

), where

I =

| g ∈ H(f) }.

g. I = {I

| g ∈ H(f) } is a uniform forward attractor, i.e.

lim

t→+∞

sup

g∈H(f )

β(ϕ(t, g)M, I

) = 0

for any M ∈ C(R

h. the equation (12.20) admits at least one stationary (τ−periodic, quasi-periodic,

almost periodic) solution, if the function f ∈ C(R × R

, R

) is stationary

(τ−periodic, quasi-periodic, almost periodic) in t ∈ R uniformly w.r.t. x on

compacts from R

Proof. This theorem is proved using the same arguments that we used in the proof

of Theorem 12.8. 

12.5.3 ODEs with impulse

Let {t

}

k∈Z

be a two-sided sequence of real numbers, p : R → R

be a continuously

diﬀerentiable on every interval (t

, t

k+1

) function, continuous to the right in every

point t = t

, bounded on R, almost periodic in the sense of Stepanoﬀ and

(t) =

k∈Z

where s

:= p(t

+ 0) − p(t

−0).

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Global attractors of V -monotone dynamical systems 401

Consider the equation with impulse

= f(t, x) +

k∈Z

or, what is equivalent,

= f(t, x) + p

(t) (12.22)

and paralelly consider the family of equations

= g(t, x) + q

(t), (12.23)

where (g, q) ∈ H(f, p) :=

{(f

, p

)|τ ∈ R} and by bar we denote the closure in the

product-space C(R ×R

, R

) × C(R, R

Denote by ϕ(t, x, g, q) the unique solution of the equation (12.23) (see

[

170

]

and

[

279

]

) satisfying the initial condition ϕ(0, x, g, q) = x. This solution is continuous

on every interval (t

, t

k+1

) and continuous to the right in every point t = t

(see

[

170

]

and

[

279

]

By the transformation

x := y + q(t) (12.24)

we can bring the equation (12.23) to the equation

= g(t, y + q(t)). (12.25)

Theorem 12.11 Let f ∈ C(R ×R

, R

) be a Bohr’s almost periodic function in

t ∈ R uniformly with respect to x on every compact from R

and p ∈ C(R, R

) be a

Stepanoﬀ’s almost periodic function. Suppose that hf(t, x

) −f (t, x

), x

−x

i ≤ 0

for all t ∈ R and that there exist positive numbers α, L

, L

and r such that

hf(t, x), xi ≤ −L

|x|

α+1

and |f(t, x)| ≤ L

|x|

for all t ∈ R and |x| > r (x ∈ R

Then the equation (12.22) admits a compact global attractor {I

(g,q)

| (g, q) ∈

H(f, p) } with the following properties:

a. I

(g,q)

is a nonempty, compact and convex subset of R

for every (g, q) ∈ H(f, p).

b. I =

(g,q)

| (g, q) ∈ H(f, p) } is disconnected.

c. I = {I

(g,q)

| (g, q) ∈ H(f, p) } is invariant, i.e. ϕ(t, I

(g,q)

, g, q) = I

(g,q)

for all

(g, q) ∈ H(f, p) and t ∈ R

d. lim

t→+∞

β(ϕ(t, σ

(g, q))M, I

(g,q)

) = 0 for all M ∈ C(R

) and (g, q) ∈ H(f, g).

e. lim

t→+∞

sup{β(ϕ(t, σ

−t

(g, q))M, I) | (g, q) ∈ H(f, p) } = 0 for any M ∈ C(R

where I =

(g,q)

| (g, q) ∈ H(f, p) }.

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402 Global Attractors of Non-autonomous Dissipative Dynamical Systems

f. I = {I

| g ∈ H(f) } is a uniform forward attractor ,i.e.

lim

t→+∞

sup

(g,q)∈H(f,p)

β(ϕ(t, g, q)M, I

(g,q)

) = 0

for any M ∈ C(R

g. the equation (12.22) admits at least one stationary (τ - periodic, quasi-periodic,

almost periodic in the sense of Stepanoﬀ ) solution, if the function (f, p) ∈

C(R × R

, R

) × C(R, R

) is stationary (τ - periodic, quasi-periodic, almost

periodic in t ∈ R) uniformly w.r.t. x on compacts of R

Proof. Let ϕ(t, x, g, q) be the cocycle generated by the family of equations (12.23)

and ˜ϕ(t, y, g, q) be the cocycle generated by the family of equations (12.25). Then

we have the following equality

ϕ(t, x, g, q) = q(t) + ˜ϕ(t, x − q(0), g, q)). (12.26)

We will show that it is possible to apply Theorem12.10 to the equation

= f(t, y + p(t)).

Really,

hf(t, y

+ p(t)) − f (t, y

+ p(t)), y

−y

for all t ∈ R and y

, y

∈ R

, and

hf(t, y + p(t)), yi = hf (t, y + p(t)), y + p(t)i − hf (t, y + p(t)), p(t)i ≤ (12.27)

−L

|y + p(t)|

α+1

+ L

kpk|y + p(t)|

for all t ∈ R and |y + p(t)| > r, where kpk := sup{|p(t)| | t ∈ R }. Taking into

account the fact that the function p is bounded on R, from (12.27) we obtain the

existence of positive numbers R (that suﬃciently large) and L

, L

such that

hf(t, y + p(t)), yi ≤ −L

|y|

α+1

and hf(t, y + p(t))i ≤ L

|y|

for all t ∈ R and |y| > R. To ﬁnish the proof of the theorem it is suﬃcient to apply

Theorem 12.10 and take into consideration the relations (12.24) and (12.26). The

theorem is proved. 

12.5.4 Evolution equations with monotone operators

Let H be a real Hilbert space with the inner product h, i, | · | =

h, i and E be a

reﬂexive Banach space contained in H algebraically and topologically. Furthermore,

let E be dense in H, and here H can be identiﬁed with a subspace of the dual E

of E and h, i can be extended by continuity to E

×E.

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Global attractors of V -monotone dynamical systems 403

We consider the initial value problem

(t) + Au(t) = f(σ

ω) (12.28)

u(0) = u, (12.29)

where A : E → E

is bounded (generally nonlinear),

|Au|

≤ C|u|

p−1

+ K, u ∈ E, p > 1,

coercive,

hAu, ui ≥ a|u|

, u ∈ E, a > 0,

monotone,

hAu

−Au

, u

−u

i ≥ 0, u

, u

∈ E,

and semi-continuous (see

[

240

]

A nonlinear ”elliptic” operator

Au = −

i=1

∂

∂x

φ(

∂u

∂x

) in D ⊂ R

u = 0 on ∂D,

where D is a bounded domain in R

, φ(·), is an increasing function satisfying

φ|

[−1,1]

= 0, c|ξ|

≤

i=1

φ(ξ

) ≤ C|ξ|

( for all |ξ| ≥ 2 ),

and provides an example with H = L

(D), E = W

1,p

(D), E

= W

−1,p

(D), p

p−1

The following result is established in

[

240

]

(Ch.2 and Ch.4). If x ∈ H and

f ∈ C(Ω, E

), p

p−1

, then there exists a unique solution ϕ ∈ C(R

, H) of

(12.28) – (12.29).

We denote by ϕ(·, u, ω) the unique solution of (12.28) and (12.29). According

[

187

]

, ϕ(·, u, ω) is a continuous cocycle on H.

Theorem 12.12 Suppose that the operator A satisﬁes the conditions above and

the cocycle ϕ generated by the equation (12.28) is asymptotically compact. Then the

equation (12.28) admits a compact global attractor I = {I

| ω ∈ Ω } possessing the

following properties:

a. I

is a nonempty, compact and convex subset of H for every ω ∈ Ω.

b. I =

| ω ∈ Ω } is connected.

c. the mapping ω → I

is continuous with respect to the metric of Hausdorﬀ.

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404 Global Attractors of Non-autonomous Dissipative Dynamical Systems

d. I = {I

| ω ∈ Ω } is invariant, i.e. ϕ(t, I

, ω) = I

for all ω ∈ Ω and t ∈ R

e. lim

t→+∞

β(ϕ(t, σ

−t

ω)M, I

) = 0 for all M ∈ C(H) and ω ∈ Ω.

f. lim

t→+∞

sup{β(ϕ(t, σ

ω)M, I)| ω ∈ Ω } = 0 for any M ∈ C(H), where I =

| ω ∈ Ω }.

g. I = {I

| ω ∈ Ω } is a uniform forward attractor ,i.e.

lim

t→+∞

sup

ω∈Ω

β(ϕ(t, ω)M, I

) = 0

for any M ∈ C(H).

h. the equation (12.28) admits at least one stationary (τ - periodic, quasi-periodic,

almost periodic) solution, if the point ω ∈ Ω is stationary (τ - periodic, quasi-

periodic, almost periodic).

Proof. Let X := H × Ω, (X, R

, π) be a skew-product dynamical system and

h(X, R

, π), (Ω, R, σ), hi be a non-autonomous dynamical system generated by the

equation (12.28). Denote by V → R

the function deﬁned by the equality V(ω, u) :=

hu, ui for all (u, ω) ∈ X = H × Ω. If |ϕ(t, u, ω)| > r := (

kfk

)

p−1

(where kfk :=

max{|f(ω)| | ω ∈ Ω }) for all t ∈ (t

, t

) ⊂ R

, then

V(σ

ω, ϕ(t, u, ω)) = hA(ϕ(t, u, ω)), ϕ(t, u, ω)i

≤ −a|ϕ(t, u, ω)|

+ kfk|ϕ(t, u, ω)| < 0.

In view of Theorem 5.3 the non-autonomous dynamical system h(X, R

, π),

(Ω, R, σ), hi admits a compact global attractor.

Let V : X

×X → R

be a function deﬁned by the equality V (ω, u

, u

) :=

−u

, u

−u

i. Then

V (σ

ω, ϕ(t, u

, ω), ϕ(t, u

, ω))

= 2h

(ϕ(t, u

, ω) −ϕ(t, u

, ω)), ϕ(t, u

, ω) −ϕ(t, u

, ω)i

= 2hA(ϕ(t, ω, u

)) − A(ϕ(t, u

, ω)), ϕ(t, u

, ω) −ϕ(t, u

, ω)i ≤ 0

and, consequently, V (σ

ω, ϕ(t, u

, ω), ϕ(t, u

, ω)) ≤ V (ω, u

, u

) for all ω ∈

Ω, u

, u

∈ H and t ∈ R

. It is easy to verify that the function V satisﬁes all the

conditions of Theorems 12.4, 12.5 and the statements a.-h. follow from Theorems

12.4, 12.5, 12.6 and Corollary 12.5. The theorem is proved. 

Remark 12.2 If the injection of E into H is compact, then the cocycle ϕ gener-

ated by the equation (12.28), evidently, is asymptotically compact.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

Global attractors of V -monotone dynamical systems 405

Example 12.3 A typical example of the equation of the type (12.28) is the

equation

∂

∂t

u =

i=1

∂

∂x

φ(

∂u

∂x

) + f(σ

ω), u|

∂D

= 0 (12.30)

with ”nonlinear Laplacian” Au =

i=1

∂

∂x

φ(

∂u

∂x

), where φ(·) is an increasing func-

tion satisfying the condition

c|ξ|

≤

i=1

φ(ξ

) ≤ C|ξ|

for all |ξ| ≥ 2 and φ|

[−1,1]

= 0, provides an example with H = L

(D), E =

1,p

(D), E

= W

−1,p

(D), p

p−1

. It is possible to verify (see, for example,

[

240

]

[

]

and

[

]

) that the ”nonlinear Laplacian” satisﬁes all the conditions of

Theorem 12.12 and, consequently, (12.30) admits a compact global attractors with

the properties a.-h. We note that the attractor of the equation (12.30) is not trivial,

i.e. the set I

is not single point set, at least for certain ω ∈ Ω.

Remark 12.3 If the operator A :=

i=1

∂

∂x

φ(

∂u

∂x

) is uniformly elliptic, i.e.

c|ξ|

≤

i=1

φ(ξ

) ≤ C|ξ|

(for all ξ ∈ R

), then the set I

is the single point

set for all ω ∈ Ω (for autonomous systems see

[

314

]

, Ch.III), because in this case

the non-autonomous dynamical system generated by the equation (12.30) is strictly

monotone.

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406 Global Attractors of Non-autonomous Dissipative Dynamical Systems