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

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Linear almost periodic dynamical systems 437

Let r := diam A and t

be a positive number such that m(t

, r) < 1. We note

that

ϕ(t

, A, Y ) ⊆ ψ(t

, A, Y ) + γ(t

, A, Y )

= ∪{ψ(t

, A

, Y

)|i = 1, 2, . . . , n; j = 1, 2, .., m}+ γ(t

, A, Y ). (13.68)

According to the conditions of Theorem 13.21, α(γ(t

, A, Y )) = 0 and

diam ψ(t

, A

, y) ≤ m(t

, r) diam A

for all y ∈ Y . Thus we have

|ψ(t

, u

, y

) − ψ(t

, u

, y

)| ≤ |ψ(t

, u

, y

) − ψ(t

, u

, y

+|ψ(t

, u

, y

) − ψ(t

, u

, y

)| (13.69)

and, consequently,

diam ψ(t

, A

, y) ≤ m(t

, r) diam A

+ diam ψ(t

, u

, Y

) for all y ∈ Y

. (13.70)

Since Y is compact, from (13.69)-(13.70) follows the inequality

diam ψ(t

, A

, Y

) ≤ m(t

, r) diam A

≤ m(t

, r)(α(A) + ε)

and, consequently, α(ϕ(t

, A, Y )) ≤ m(t

, r)α(A). The theorem is proved. 

13.6 Exponential stable systems

Theorem 13.22 Let h(X, S

, π), (Y, S, σ), hi be a linear non-autonomous

dynamical system, Y be a compact set. Then the following conditions are equiv-

alent:

1. The non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is uniformly ex-

ponentially stable, i.e. there exist two positive constants N and ν such that

|π(t, x)| ≤ Ne

−νt

|x| for all t ∈ S

and x ∈ X.

2. kπ

k → 0 as t → +∞, where kπ

k = sup{|π

x| : x ∈ X, |x| ≤ 1}.

3. The non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is locally dissi-

pative.

Proof. According to Theorem 2.38, conditions 1. and 3. are equivalent. Now we

will prove that the conditions 1. and 2. are equivalent. It is clear that from 1.

follows 2. According to condition 2. there exists L > 0 such that

kπ

k ≤ 1 (13.71)

for all t ≥ L. We claim that the family of operators {π

: t ∈ [0, L]} is uniformly

continuous, that is, for any ε > 0 there is a δ(ε) > 0 such that |x| ≤ δ implies

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438 Global Attractors of Non-autonomous Dissipative Dynamical Systems

|xt| ≤ ε for all t ∈ [0, L]. On the contrary, assume that there are ε

> 0, δ

> 0

with δ

→ 0, |x

| < δ

and t

∈ [0, L] such that

| ≥ ε

. (13.72)

Since (X, h, Y ) is a locally trivial Banach ﬁber bundle and Y is compact, the zero

section Θ := {θ

: y ∈ Y, θ

∈ X

, |θ

| = 0} of (X, h, Y ) is compact and, con-

sequently, we can assume that the sequences {x

} and {t

} are convergent. Put

= lim

n→+∞

and t

= lim

n→+∞

, then x

= θ

= h(x

)). Passing to the

limit in (13.72) as n → +∞, we obtain 0 = |x

| ≥ ε

. This last inequality contra-

dicts the choice of ε

, and hence proves the above assertion. If γ > 0 is such that

|π

x| ≤ 1 for all |x| ≤ γ and t ∈ [0, L], then

|xt| ≤

|x| (13.73)

for all t ∈ [0, L] and x ∈ X. We put M := max{γ

−1

, 1}, then from (13.71) and

(13.73) follows

kπ

k ≤ M (13.74)

for all t ≥ 0 and x ∈ X. Consider the function m(t) := kπ

k. We note that

m(t + τ) ≤ m(t)m(τ) for all t, τ ∈ S

and m(t) ≤ M for all t ∈ S

and m(t) → 0 as

t → +∞. According to Lemma 2.19 there exist positive numbers N and ν such that

m(t) ≤ Ne

−νt

for all t ∈ S

. Thus |π(t, x)| ≤ kπ

k|x| ≤ Ne

−νt

|x| for all t ∈ S

and

x ∈ X. The theorem is proved. 

Let B := {x ∈ X : ∃γ ∈ Φ

such that sup

t∈S

|γ(t)| < +∞}.

Theorem 13.23 Let h(X, S

, π), (Y, S, σ), hi be a linear non-autonomous

dynamical system, Y be compact and (X, S

, π) be conditionally α-condensing.

Then the following assertions are equivalent:

(1) The non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is point dissi-

pative and this system doesn’t admit non-trivial bounded trajectories on S, i.e.

B ⊆ Θ = {θ

: y ∈ Y, θ

∈ X

, |θ

| = 0}.

(2) The non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is uniformly ex-

ponentially stable.

Proof. Denote by Θ = {θ

: y ∈ Y, θ

∈ X

, |θ

| = 0} the zero section of the vector

ﬁbering (X, h, Y ). Since (X, h, Y ) is locally trivial and Y is compact, the zero section

Θ is compact and an invariant set of the dynamical system (X, S

, π). Taking

into account that the dynamical system (X, S

, π) is conditionally α-condensing,

according to Theorem 2.4.8

[

179

]

the set Θ is orbitally stable and in particular there

exists a positive constant N such that |xt| ≤ N|x| for all t ∈ S

and x ∈ X. By

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Linear almost periodic dynamical systems 439

virtue of Theorem 2.38 the dynamical system (X, S

, π) is compact dissipative and

according to Theorem 2.38 (X, S

, π) is local dissipative. It follows from Theorem

13.22 that (X, S

, π) is uniformly exponentially stable.

Let now the non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi be uni-

formly exponentially stable, then according to Theorem 13.22 it is locally dissipa-

tive. Let J be its Levinson’s centre (i.e. maximal compact invariant set of dynamical

system (X, S

, π)) . We note that according to the linearity of non-autonomous

dynamical system h(X, S

, π), (Y, S, σ), hi we have J = Θ. Let ϕ be a entire

bounded trajectory of dynamical system (X, S

, π) . Since the non-autonomous

dynamical system h(X, S

, π), (Y, S, σ), hi is conditionally α-condensing, in partic-

ularly, it is asymptotically compact and the set M = ϕ(S) is relatively compact. In

fact, the set M is invariant Ω(M) =

M and in view of Lemma 3.3

[

]

the set M is

relatively compact. We note that ϕ(S) ⊆ J = Θ because J is the maximal compact

invariant set of (X, S

, π). The theorem is proved. 

Remark 13.7 Theorem A in

[

277

]

implies a version of Theorem 13.23 under

slightly stronger assumptions.

Theorem 13.24 Let h(X, S

, π), (Y, S, σ), hi be a linear non-autonomous

dynamical system, Y be compact and (X, S

, π) be completely continuous, i.e. for

any bounded set A ⊆ X there exists a positive number ` such that π

(A) is relatively

compact. Then the following assertions are equivalent:

1. The non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is uniformly ex-

ponentially stable.

2. lim

t→+∞

|π

x| = 0 for all x ∈ X.

Proof. It is clear that condition 1 implies 2. Now we will show that condition 1

follows from 2. According to Theorem 2.38 the non-autonomous dynamical system

h(X, S

, π), (Y, S, σ), hi is point dissipative. Since the dynamical system (X, S

, π)

is completely continuous, by virtue of Theorem 2.38 the dynamical system (X, S

, π)

is locally dissipative. To prove the theorem it is suﬃcient to refer to Theorem 13.22.

13.7 Linear system with a minimal base

In this section we study a linear system h(X,S

,π), (Y,S,σ),hi with compact minimal

base (Y, S, σ).

Theorem 13.25 Let h(X, S

, π), (Y, S, σ), hi be a linear non-autonomous

dynamical system and the following conditions hold:

(1) Y is compact and minimal.

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440 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) The dynamical system (X, S

, π) is asymptotically compact.

(3) The mapping y 7→ kπ

k is continuous, where kπ

k is the norm of the linear

operator π

:= π

, for every t ∈ S

or (X, S

, π) is a skew-product dynamical

system.

Then the non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is uni-

formly exponentially stable if and only if

lim

t→+∞

|π

x| = 0 (13.75)

for all x ∈ X.

Proof. It is clear that from uniform exponential stability of the non-autonomous

dynamical system h(X, S

, π), (Y, S, σ), hi follows the equality (13.75).

From condition (13.75) and minimality of (Y, S, σ) by virtue of Theorem 13.14 it

follows that for the non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi the

inequality

|π(t, x)| ≤ M|x| (13.76)

holds for all t ∈ S

and x ∈ X. According to Theorem 2.38 this system is com-

pact dissipative. Since the dynamical system (X, S

, π) is asymptotically com-

pact, to ﬁnish the proof of the Theorem it is suﬃcient to remark that accord-

ing to Theorem 2.13

[

]

every compact dissipative and asymptotically compact

dynamical system is local dissipative. Thus a linear non-autonomous dynamical

system h(X, S

, π), (Y, S, σ), hi is local dissipative and, consequently, it is uniform

exponentially stable. The theorem is proved. 

Theorem 13.26 Suppose that the following conditions are satisﬁed:

(1) A dynamical system (Y, S, σ)is compact and minimal.

(2) A linear non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is genera-

ted by cocycle ϕ (i.e. X = E × Y , π = (ϕ, σ) and h = pr

: X 7→ Y ).

(3) The dynamical system (X, S

, π) is conditionally α-condensing.

(4) There exists a positive number M such that |ϕ(t, u, y)| ≤ M|u| for all t ∈ Y

and t ∈ S

Then there are two vectorial positively invariant sub-ﬁberings (X

, h, Y ) and

, h, Y ) of (X, h, Y ) such that:

a. X

= X

+ X

and X

∩X

= θ

for all y ∈ Y , where θ

= (0, y) ∈ X = E ×Y

and 0 is the zero in the Banach space E.

b. The vector sub-ﬁbering (X

, h, Y ) is ﬁnite dimensional, invariant (i.e. π

for all t ∈ S

) and every trajectory of the dynamical system (X, S

, π)

belonging to X

is recurrent.

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Linear almost periodic dynamical systems 441

c. There exist two positive numbers N and ν such that |ϕ(t, u, y)| ≤ N e

−νt

|u| for

all (u, y) ∈ X

and t ∈ S

Proof. Let X

:= B, then according to Theorem 13.2, statement b. holds. Denote

by P

the projection of X

:= h

−1

(y) to B

:= B∩h

−1

(y), then P

(u, y) = (P(y)u, y)

for all u ∈ E, P

(y) = P(y) and the mapping P : Y → [E] ( y 7→ P(y)) is continuous,

where by [E] denotes the set of all linear continuous operators acting on E. Now we

set X

:= Q(y)X

and X

:= ∪{X

: y ∈ Y }, where Q(y) := Id

− P(y). We will

show that X

is closed in X. In fact, let {x

} = {(u

, y

)} ⊆ X

and x

= (u

, y

) =

lim

n→∞

. Note that P

) = (P(y

, y

) = ( lim

n→∞

P(y

, y

) = (0, y

) = θ

and, consequently, x

∈ X

⊆ X

Let (X

, S

, π) be the dynamical system induced by (X, S

, π). It is clear

that under the conditions of Theorem 13.26 the dynamical system (X

, S

, π) is

asymptotically compact and every positive semi-trajectory is relatively compact

and, consequently, lim

t→∞

|π(t, x)| = 0 for all x ∈ X

because the dynamical system

, S

, π) doesn’t have a non-trivial entire trajectory bounded on S. In fact, if

we suppose that it is not true, then there exist x

= (u

, y

) and t

→ +∞ such

that: |u

| 6= 0, lim

n→+∞

π(t

, x) = x

and through point x

pass a non-trivial entire

trajectory bounded on S. This contradiction proves the necessary assertion. Thus

we can apply Theorem 2.38 according which there exist two positive constants N

and ν such that |ϕ(t, u, y)| ≤ N e

−νt

|u| for all (u, y) ∈ X

and t ∈ S

. The theorem

is proved. 

13.8 Some classes of uniformly exponentially stable equations

Let Λ be a complete metric space of linear operators that act on a Banach space E

and C(R, Λ) be the space of all continuous operator-functions A : R → Λ equipped

with the open-compact topology and (C(R, Λ), R, σ) be the dynamical system of

shifts on C(R, Λ).

Linear ordinary diﬀerential equations. Let Λ = [E] and consider the linear

diﬀerential equation

= A(t)u , (13.77)

where A ∈ C(R, Λ). Along with equation (13.77), we shall also consider its H-class,

that is, the family of equations

= B(t)v , (13.78)

where B ∈ H(A) :=

: τ ∈ R}, A

(t) = A(t + τ) (t ∈ R), and the bar denotes

closure in C(R, Λ). Let ϕ(t, u, B) be the solution of equation (13.78) that satis-

ﬁes the condition ϕ(0, u, B) = u. We put Y := H(A) and denote the dynamical

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442 Global Attractors of Non-autonomous Dissipative Dynamical Systems

system of shifts on H(A) by (Y, R, σ). Then the triple h(X, R

, π), (Y, R, σ), hi is

a linear non-autonomous dynamical system, where X := E × Y , π := (ϕ, σ); i.e.,

π((v, B), τ ) := (ϕ(τ, v, B), B

) and h := pr

: X → Y ).

Lemma 13.13

[

51, 60

]

(i) The mapping (t, u, A) 7→ ϕ(t, u, A) of R × E × C(R, [E]) to E is continuous,

and

(ii) the mapping A 7→ U (·, A) of C(R, [E]) to C(R, [E]) is continuous, where U(·, A)

is the Cauchy’s operator

[

132

]

of equation (13.77).

Theorem 13.27 Let A ∈ C(R, Λ) be compact (i.e. H(A) is a compact set of

(C(R, Λ), R, σ) ), then the following conditions are equivalent:

1. The trivial solution of equation (13.77) is uniformly exponentially stable, i.e.

there exist positive numbers N and ν such that kU(t, A)U(τ, A)

−1

k ≤ Ne

−(t−τ)

for all t ≥ τ.

2. There exist positive numbers N and ν such that kU(t, B)U(τ, B)

−1

k ≤ Ne

−(t−τ)

for all t ≥ τ and B ∈ H(A).

3. lim sup

t→+∞

{kU(t, B)k : B ∈ H(A)} = 0.

Proof. Applying Theorem 13.22 to the non-autonomous system h(X,R

,π),

(Y,R,σ), hi, generated by equation (13.77), we obtain the equivalence of conditions

2. and 3. According to Lemma 3

[

]

conditions 1. and 2. are equivalent. The

theorem is proved. 

Theorem 13.28 Let A ∈ C(R, Λ) be recurrent with respect to t ∈ S (i.e. H(A)

is a compact and minimal set of (C(R, Λ), R, σ) ), the non-autonomous system

h(X, R

, π), (Y, R, σ), hi generated by equation (13.77) is asymptotically compact.

Then the following conditions are equivalent:

1. The trivial solution of equation (13.77) is uniformly exponentially stable, i.e.

there exist positive numbers N and ν such that kU(t, A)U(τ, A)

−1

k ≤ Ne

−(t−τ)

for all t ≥ τ.

2. lim sup

t→+∞

|ϕ(t, u, B)| = 0 for every u ∈ E and B ∈ H(A).

Proof. According to Lemma 13.13 the mapping U(t, ·) : [E] → [E] is continuous

and, consequently, the mapping B 7→ kU(t, B)k is also continuous for every t ∈ S.

Now applying Theorem 13.22 to non-autonomous system h(X, R

, π), (Y, R, σ), hi

generated by equation (13.77), we obtain the equivalence of conditions 1. and 2.

The theorem is proved. 

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Linear almost periodic dynamical systems 443

We now formulate some suﬃcient conditions for the α− condensedness (in par-

ticular, asymptotic compactness) of the linear non-autonomous dynamical system

generated by equation (13.77).

Theorem 13.29 Let A ∈ C(R, [E]), A(t) = A

(t) + A

(t) for all t ∈ R, and

assume that H(A

) (i = 1, 2) are compact and the following conditions hold:

(i) The zero solution of the equation

= A

(t)u (13.79)

is uniformly asymptotically stable, that is, there are positive numbers N and ν

such that

kU(t, A

−1

(τ, A

)k ≤ Ne

−ν(t−τ)

(13.80)

for all t ≥ τ (t, τ ∈ R), where U(t, A

) is the Cauchy’s operator of equation

(13.79).

(ii) The family of operators {A

(t) : t > 0} is uniformly completely continuous, that

is, for any bounded set A ⊂ E the set {A

(t)A : t > 0} is relatively compact.

Then the linear non-autonomous dynamical system generated by equation

(13.77) is an α-contraction.

Proof. First of all we note that the set ϕ(t, A, Y ) is bounded for every t > 0

and bounded set A ⊆ E. Let B ∈ H(A). Then there are {t

} ⊂ S such that

B(t) = B

(t) + B

(t) and B

(t) = lim

t→+∞

(t + t

). Note that

ϕ(t, v, B) = U(t, B

)v +

U(t, B

−1

(τ, B

(τ)ϕ(τ, v, B)dτ.

By Lemma 13.13,

U(t, B

) = lim

t→+∞

U(t, A

), A

(t) = A

(t + t

and the equality

U(t, A

−1

(τ, A

) = U(t + t

, A

−1

(τ + t

, A

)

and inequality (13.80) imply that

kU(t, B

−1

(τ, B

)k ≤ Ne

−ν(t−τ)

(13.81)

for all t ≥ τ and B

∈ H(A

). Theorem 13.29 will be proved if we can prove that

the set

{

U(t, B

−1

(τ, B

(τ)ϕ(τ, v, B)dτ : (v, B) ∈ A}

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444 Global Attractors of Non-autonomous Dissipative Dynamical Systems

is relatively compact for every t > 0 and every bounded positively invariant set

A ⊆ E × Y . We put

(τ)ϕ(τ, v, B) : τ ∈ [0, t], (v, B) ∈ A}

and we note that the set K

is compact. Really, the set ϕ([0, t], A) = ∪{ϕ(τ, v, B) :

τ ∈ [0, t], (v, B) ∈ A} is bounded because |ϕ(τ, v, B)| ≤ e

r for all τ ∈ [0, t]

and (v, B) ∈ A, where r = sup{|v| : ∃B ∈ H(A), such that (v, B) ∈ A} and

M = sup{kA(t)k : t ∈ S}. Then

U(t, B

−1

(τ, B

(τ)ϕ(τ, v, B)dτ (13.82)

∈ t

conv{U(t, B

−1

(τ, B

)w : 0 ≤ τ ≤ t, B

∈ H(A

), w ∈ K

Since H(A

), H(A), and K

are compact sets, then formula (13.82), condition (ii)

of Theorem 13.29, and Lemma 13.13 imply that {U(t, B

−1

(τ, B

)w : 0 ≤ τ ≤

t, B

∈ H(A

), w ∈ K

} is compact, which completes the proof of the theorem. 

Theorem 13.30 Let H(A) be compact and assume that there is a ﬁnite-

dimensional projection P ∈ [E] such that

(i) the family of projections {P (t) : t ∈ R}, where P (t) := U (t, A)P U

−1

(t, A), is

relatively compact in [E], and

(ii) there are positive numbers N and ν such that

kU(t, A)QU

−1

(τ, A)k ≤ N e

−ν(t−τ)

for all t ≥ τ, where Q := I − P .

Then the linear non-autonomous dynamical system generated by equation

(13.77) is an α-contraction.

Proof. Since the family of projections P (t) = U(t, A)P U

−1

(t, A) is relatively

compact in [E], the family H =

{P (t) : t ∈ R} is uniformly completely continuous,

where the bar denotes closure in [E]. Indeed, let A be a bounded subset of E,

} ⊆ {QA : Q ∈ H}, and ε

↓ 0. Then there are t

∈ R and v

∈ A such

that |x

−P (t

| ≤ ε

. Since the sequence {P (t

)} is relatively compact, we can

assume that it converges. Let L := lim

n→+∞

P (t

). Then L is completely continuous,

which implies that the sequence {x

} = {Lv

} is relatively compact. Note that

−x

| ≤ |x

−P (t

| + |P (t

−Lv

| ≤ ε

+ kP (t

) − Lk|v

which implies that |x

−x

| → 0 as n → +∞. Hence, {x

} is relatively compact.

Assume that B ∈ H(A) and {t

} ⊂ R are such that

B = lim

n→+∞

, P (B) = lim

n→+∞

P (A

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

Linear almost periodic dynamical systems 445

where P (A

) = U(t

, A)P U

−1

, A). The assertions proved above imply that the

family {P (B) : B ∈ H(A)} is uniformly completely continuous. Note that Q(B) =

lim

n→+∞

Q(A

), where Q(B) := I − P (B) and Q(A

) = I − P (A

). Moreover,

condition (ii) of Theorem 13.30 implies that

kU(t, A

)QU

−1

(τ, A

)k ≤ Ne

−ν(t−τ)

(13.83)

for all t ≥ τ. Passing to the limit in (13.83) as n → +∞ and taking into account

Lemma 13.13, we obtain that

kU(t, B)QU

−1

(τ, B)k ≤ N e

−ν(t−τ)

for all t ≥ τ and B ∈ H(A). We complete the proof of the theorem by observing

that U(t, B)Q(B) + U (t, B)P (B) = U(t, B) and applying Theorem 13.21. 

Theorem 13.31 Suppose that the following conditions are satisﬁed:

(1) The operator-function A ∈ C(R, [E]) is recurrent with respect to t ∈ R.

(2) The linear non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi gener-

ated by equation (13.77) is conditionally α-condensing.

(3) the trivial solution of equation (13.77) is uniformly stable in the positive direc-

tion, i.e. there exist a positive number M such that

kU(t, A)U

−1

(τ, A)k ≤ M (13.84)

for all t ≥ τ.

Then there are two vectorial positively invariant sub-ﬁbers (X

, h, Y ) and

, h, Y ) of (X, h, Y ) such that:

a. X

= X

+ X

and X

∩X

= θ

for all y ∈ Y , where θ

= (0, y) ∈ X = E ×Y

and 0 is the zero in the Banach space E.

b. The vectorial sub-ﬁber (X

, h, Y ) is ﬁnite dimensional, invariant (i.e. π

for all t ∈ S

) and every trajectory of a dynamical system (X, S

, π) be-

longing to X

is recurrent.

c. There exist two positive numbers N and ν such that |ϕ(t, u, B)| ≤ N e

−νt

|u| for

all (u, B) ∈ X

and t ∈ S

, where ϕ(t, u, B) := U(t, B)u.

Proof. Assume that B ∈ H(A) and {t

} ⊂ R are such that B = lim

n→+∞

, then

condition (3) of Theorem 13.31 implies that

kU(t, A

−1

(τ, A

)k ≤ N (13.85)

for all t ≥ τ. Passing to the limit in (13.85) as n → +∞ and taking into account

Lemma 13.13, we obtain that

kU(t, B)U

−1

(τ, B)k ≤ N

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

446 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for all t ≥ τ and B ∈ H(A) and, consequently,

kU(t, B)k ≤ N

for all t ≥ 0 and B ∈ H(A). Now to ﬁnish the proof of Theorem 13.31 it is

suﬃciently to refer on Theorem 13.26. 

Linear partial diﬀerential equations. Let Λ be some complete metric space

of linear closed operators acting into Banach space E ( for example Λ = {A

C|C ∈ [E]}, where A

is a closed operator that acts on E). We assume that the

following conditions are fulﬁlled for equation (13.77) and its H− class (13.78):

a. for any v ∈ E and B ∈ H(A) equation (13.78) has exactly one mild solution

ϕ(t, v, B) (i.e. ϕ(·, v, B) is continuous, deﬁned on R

and satisﬁes of equation

ϕ(t, v, B) = U(t, B)v +

U(t − τ, B)ϕ(τ, v, B)dτ. (13.86)

and the condition ϕ(0, v, B) = v;

b. the mapping ϕ : (t, v, B) → ϕ(t, v, B) is continuous in the topology of R

E × C(R; Λ);

c. for every t ∈ R

the mapping U(t, ·) : H(A) → [E] is continuous, where U(t, ·)

is the Cauchy’s operator of equation (13.78), i.e. U(t, B)v := ϕ(t, v, B) (t ∈

, v ∈ E and B ∈ H(A) ).

Under the above assumptions the equation (13.77) generates a linear non-

autonomous dynamical system h(X, R

, π), (Y, R, σ), hi, where X := E × Y, π =

(ϕ, σ) and h := pr

: X → Y. Applying the results from the sections 13.6 and 13.7

to this system, we will obtain the analogous assertions for diﬀerent classes of partial

diﬀerential equations.

We will consider examples of partial diﬀerential equations which satisfy the

above conditions a.-c.

Example 13.4 Consider the diﬀerential equation

= (A

+ A

(t))u, (13.87)

where A

is a sectorial operator that does not depend on t ∈ R, and A

∈ C(R, [E]).

The results of

[

188

]

[

238

]

imply that equation (13.87) satisﬁes conditions a.-c.

Example 13.5 Let H be a Hilbert space with a scalar product h·, ·i = | · |

D(R

, H) be the set of all inﬁnite diﬀerentiable, bounded functions on R

with

values into H.