§ 5-1 Main Results Chapter 5 C -evolution System and the Conditional Stability for the Solutions of Abstract Semilinear Differential Equations

(1)

Chapter 5

C

₀-evolution System and the Conditional Stability for the Solutions of Abstract Semilinear Differential Equations

§ 5-1 Main Results

In the preceding chapter, we get sufficient conditions to ensure that the zero solution to the abstract semilinear equation (4.1) is conditionally stable and conditionally asymptotically stable. In that chapter, we considered the linear parts and forcing term function of the equation together. However, if the equation can be linearized, we may approach the desired conclusion by some perturbation theory of linear operators. We consider the asymptotic behavior of some solutions of the abstract semilinear initial value problem:

( ) ( ) ( ) ( ( ) )

( )

0

, 0

d u t A t u t f t u t dt

u ξ

 = +



 =



(5.1)

where the family of operators

{

^{A t}

( )

^:^t^≥⁰

}

generates a non-trivial C -evolution ₀ system

{

^{U t s}

( )

^{, : 0}≤ ≤ < ∞ on a Banach space X , and the forcing term function ^s ^t

}

[ )

: 0,

f ∞ × →X X satisfies the following conditions:

( )

^F1 f t x is continuous in

( )

^, ^t^∈

[

^0,∞ for each fixed x X

)

∈ .

( )

^F2 f is locally Lipschitz continuous respect to x in X with Lipschitz constant γ , that is,

( )

^,

( )

^,

f t x − f t y ≤γ x− for y all t≥ and x , y0 ≤ . α

( )

^F3 ^{f t}

( )

^{, 0} = for all ⁰ t≥ . 0

Here we assume that there exists none trivial supplementary projections P , ₁ P ₂ and P on the Banach space X such that ₃ P X_i = X_i for i =1, 2, 3. Where the dimensions of X₁ and X₂ are finite and the C₀ -evolution system

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ satisfies the following conditions: ^s ^t

}

(2)

( )

^A1

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ restricted on ^s ^t

}

X and ₁ X are total evolution ₂ system (here U t s x is defined by

( )

^, ^{U t s x}

( )

^, ⁼^{U s t}

( )

^, ⁻¹^x for all t< , s x∈X₁ and x∈X₂).

( )

^A2 U t s P

( )

^, j =PU t sj

( )

^, for all 0≤ ≤ < ∞ and j =1, 2, 3. s t

( )

^A3 0^t

( )

, 3

( )

, 1

U tτ P dτ+ t^∞ U t τ P dτ ≤K

∫ ∫

for all 0≤ < ∞ . t

( )

^A4 U t s P

( )

, 2 ≤L2 for all 0≤s t, < ∞ .

Furthermore, the forcing term function f satisfies the condition

( )

^{F4 :}

( )

^F4

∫

0^∞ P f2

(

τ ϕ τ,

( ) )

−P f2

(

τ φ τ,

( ) )

dτ γ ϕ φ≤ 2 − _∞ for all ϕ , φ ∈ , where D

^D=

{

^ϕ∈^C

( [

^0,∞

)

^;^X

)

^: ^ϕ ∞ ≤^α

}

, 0α > and ⋅ _∞ denotes supremum norm on ^C

( [

^0,^∞

)

^;^X

)

^.

Under the above notations and assumptions, we have following results:

Lemma 5.1 Suppose that the C -evolution system ₀

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ ^s ^t

}

satisfies conditions (A1)~(A4) and the function ^f ^{: 0,}

[

∞ × →

)

^X ^X satisfies conditions (F1)~(F4). Let ξ ∈₃ X₃ and the operator ^{G D}^: ^→^C

( [

^0,^∞

)

^;^X

)

^is

defined by

( )( )

Gϕ t =U t

( )

, 0 ξ3+

∫

0^tU t

( )

,τ P f3

(

τ ϕ τ,

( ) )

dτ

( )

, 2

(

,

( ) )

t^∞U tτ P f τ ϕ τ dτ

−

∫

( )

^, ¹

(

^,

( ) )

t^∞U tτ P f τ ϕ τ dτ

−

∫

for all ϕ ∈ , then G is well-defined and D

2 2

( )

Gϕ−Gφ _∞ ≤ γK+γ L ϕ φ− _∞ for any φ , ϕ ∈ , D where γ , K , γ , ₂ L are the constants in (A3), (A4), (F2) and (F4). ₂

Proof. From the conditions (A1), (A3), (A4), (F3) and (F4), for any ε > , 0

1 2 0

t > ≥ and t ϕ ∈ , we have D

(3)

( )

1

( )

2

Gϕ t −Gϕ t

( ) ( )

{

U t1, 0 ξ3 U t2, 0 ξ3

}

0^t¹U t

^{( )}

1,τ P f3

(

τ ϕ τ,

^{( )} )

dτ

= − +

∫

( ) ( ( ) ) ( ) ( ( ) )

1 1

1, 2 , 1, 1 ,

t^∞U t τ P f τ ϕ τ dτ t^∞U t τ P f τ ϕ τ dτ

−

∫

−

∫

( ) ( ( ) )

2

2 3

0^tU t ,τ P f τ ϕ τ, dτ

−

∫

( ) ( ( ) ) ⁽ ⁾ ( ^{( )} )

2 2

2, 2 , 2, 1 ,

t^∞U t τ P f τ ϕ τ dτ t^∞U t τ P f τ ϕ τ dτ

+

∫

+

∫

( )

1, 0 3

(

2, 0

)

3 0^t²

{ ( )

1,

(

2,

) }

3

(

,

( ) )

U t ξ U t ξ U t τ U t τ P f τ ϕ τ dτ

≤ − +

∫

−

( ) ( ( ) ) ( ) ( ( ) )

1 1

2 2

1, 3 , 2, 2 ,

t t

t U t τ P f τ ϕ τ dτ t U t τ P f τ ϕ τ dτ

+

∫

+

∫

( ) ( ( ) )

1 2

2, 1 ,

t

t U t τ P f τ ϕ τ dτ +

∫

( ) ( )

{ } ( ) ( ( ) )

1

1, 2, 1 2 ,

t^∞ U t τ U t τ P P f τ ϕ τ dτ

+

∫

− + ^.

Since ^f

(

^{τ ϕ τ}^,

( ) )

^≤^{γ ϕ τ}

( )

^≤^γα on the interval

[

^0,^{∞ ,}

) ( )

1

1, 1

t^∞ U t τ P dτ ≤K

∫

^and

∫

t^∞₂ U t

⁽

²^,τ

⁾

P d¹ τ ≤K ^, there exists a constant T₁> such that t₁

( )

1, 1

(

,

( ) ) (

2,

)

1

(

,

( ) )

T^∞U t τ P f τ ϕ τ dτ+ T^∞U t τ P f τ ϕ τ dτ

∫ ∫

( )

1, 1

(

,

( ) ) (

2,

)

1

(

,

( ) )

T^∞ U t τ P f τ ϕ τ dτ T^∞ U t τ P f τ ϕ τ dτ

≤

∫

+

∫

( ) ( )

(

^T ^{U t}¹^, ^{P d}¹ ^T ^{U t}²^, ^{P d}¹

)

γα ^∞ τ τ ^∞ τ τ

≤

∫

+

∫

ε

<

for any T ≥ . From the condition (A4) and (F4), we obtain that T₁ U t

( )

,τ P2 ≤L2

for all 0≤τ, t< ∞ and

∫

0^∞ P f2

(

τ ϕ τ,

( ) )

dτ γ ϕ≤ 2 _∞ ≤γ α2 < ∞ . Thus there exists a constant T₂ > such that t₁

( )

1, 2

(

,

( ) ) (

2,

)

2

(

,

( ) )

T^∞U t τ P f τ ϕ τ dτ + T^∞U t τ P f τ ϕ τ dτ

∫ ∫

( )

1, 2 2

(

,

( ) ) (

2,

)

2 2

(

,

( ) )

T^∞ U t τ P P f τ ϕ τ dτ T^∞ U t τ P P f τ ϕ τ dτ

≤

∫

+

∫

(4)

( ( ) )

2 2

2 ,

L T^∞ P f τ ϕ τ dτ

≤

∫

ε

<

for any T ≥ . Let T₂ T0 =max

{

T T1, 2

}

, then

( ) ( ( ) ) ( ) ( ( ) )

0 0

1, 1 , 2, 1 ,

T^∞U t τ P f τ ϕ τ dτ+ T^∞U t τ P f τ ϕ τ dτ

∫ ∫

( ) ( ( ) )

0

1, 2 ,

T^∞U t τ P f τ ϕ τ dτ

+

∫

+

∫

T^∞₀U t

⁽

²^,τ

⁾

P f²

⁽

τ ϕ τ^,

^{( )} ⁾

dτ ^<^2ε and

( )

1

( )

2

Gϕ t −Gϕ t

( )

1, 0 3

(

2, 0

)

3 0^t²

{ ( )

1,

(

2,

) }

3

(

,

( ) )

U t ξ U t ξ U t τ U t τ P f τ ϕ τ dτ

≤ − +

∫

−

( ) ( ( ) ) ( ) ( ( ) )

1 1

2 2

1, 3 , 2, 2 ,

t t

t U t τ P f τ ϕ τ dτ t U t τ P f τ ϕ τ dτ

+

∫

+

∫

( ) ( ( ) )

1 2

2, 1 ,

t

t U t τ P f τ ϕ τ dτ +

∫

( ) ( )

{ } ( ) ( ( ) )

0 1

1, 2, 1 2 ,

T

t U t τ U t τ P P f τ ϕ τ dτ

+

∫

− + ^{+ 2ε .}

Since the function taU t s

( )

, ξ3 is continuous on 0≤ ≤ < ∞ , there exists a s t constant δ > such that ₁ 0

( )

1, 0 3

(

2, 0

)

3

U t ξ −U t ξ < for ε all t₁− < . t₂ δ₁

From the facts that functions τ aU t

( )

1,τ P f3

(

τ ϕ τ,

( ) )

, τ aU t

(

2,τ

)

P f_i

(

τ ϕ τ,

( ) )

are continuous on the compact interval

[ ]

t t for each 2,1 i=1, 2, one obtain that they are bounded on

[ ]

t t and there exists a constant 2,1 δ > such that ₂ 0

( ) ( ( ) ) ( ) ( ( ) )

1 1

2 2

1, 3 , 2, 2 ,

t t

t U t τ P f τ ϕ τ dτ + t U t τ P f τ ϕ τ dτ

∫ ∫

( ) ( ( ) )

1 2

2, 1 ,

t

t U t τ P f τ ϕ τ dτ

+

∫

^<^ε

for all t₁− <t₂ δ₂. On the other hand, since the mappings

( )

t,τ aU t

( )

,τ P f3

(

τ ϕ τ,

( ) )

and

( )

t,τ aU t

( )(

,τ P1+P2

)

f

(

τ ϕ τ,

( ) )

are uniformly continuous on compact sets

{ ( )

t,τ : 0≤ ≤ ≤τ t T0

}

and

(5)

{ ( )

t,τ : 0≤ ≤ ≤t τ T0

}

respectively, there exists a constant δ > such that ₃ 0

( )

1, 3

(

,

( ) ) (

2,

)

3

(

,

( ) )

U t τ P f τ ϕ τ −U t τ P f τ ϕ τ <εT₀⁻¹ for all t₁− < , t₂ δ₃ 0≤ ≤ ≤ and τ t₂ T₀

( ) ( )

{

U t1,τ −U t2,τ

} (

P1+P2

)

f

(

τ ϕ τ,

( ) )

<εT₀⁻¹ for all t₁− < , t₂ δ₃ 0≤ ≤ ≤ . Let t₁ τ T₀ δ =min

{

δ δ δ1, 2, 3

}

, then

( )

1

( )

2

Gϕ t −Gϕ t ≤ +ε

(

t2−s

)

εT0⁻¹+

(

T0−t1

)

εT0⁻¹+2ε ≤6ε for all 0≤ ≤ ≤ + . t₂ t₁ t₂ δ

Hence, G is well-defined and ^G^{ϕ ∈}^C

( [

^0,^∞

)

^;^X

)

^{for all}ϕ ∈ . Moreover, ^D Gϕ−Gφ _∞

( ) ( ( ( ) ) ( ^{( )} ) )

{

⁰ ³

0

sup ^t , , ,

t

U tτ P f τ ϕ τ f τ φ τ dτ

≤ ≥

∫

−

( )

^, ²

(

²

(

^,

( ) )

²

(

^,

( ) ) )

t^∞U tτ P P f τ ϕ τ P f τ φ τ dτ

+

∫

−

( )

^, ¹

( (

^,

( ) ) (

^,

^{( )} ) ) }

t^∞U tτ P f τ ϕ τ f τ φ τ dτ

+

∫

−

( ) ( ) ( ) ( ) ( ) ( )

{

⁰ ³ ¹

}

0

sup ^t , ,

t t

U tτ P γ ϕ τ φ τ dτ ^∞ U tτ P γ ϕ τ φ τ dτ

≥

≤

∫

− +

∫

−

( )

²

(

²

( ( ) )

²

( ^{( )} ) )

0

sup , , ,

t t

U t τ P P f τ ϕ τ P f τ φ τ dτ

∞

≥

+

∫

−

( ) ( )

{

⁰ ³ ¹

}

0

sup ^t , ,

t t

U t P d U t P d

γ ϕ φ _∞ τ τ ^∞ τ τ

≥

≤ −

∫

+

∫

( ( ) ) ( ^{( )} )

2 2 2

0 0

sup , ,

t

L ^∞ P f τ ϕ τ P f τ φ τ dτ

+ ≥

∫

−

2 2

K L

γ ϕ φ _∞ γ ϕ φ _∞

≤ − + −

(

γK γ2L2

)

ϕ φ _∞

≤ + −

for all φ , ϕ ∈ . This lemma is proved now. D

Lemma 5.2. Suppose that the C -evolution system ₀

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ ^s ^t

}

satisfies conditions (A1)~(A4). Then lim

( )

, 3 0

t U t s P

→∞ = for all s≥ , and there is 0 a constant L₃> such that 0 U t

( )

, 0 P3 ≤L3 for all t≥ . 0

(6)

Furthermore, if the function ^f ^{: 0,}

[

^{∞ × →}

)

^X ^X satisfies conditions (F1)~(F4) and the constants K , L , ₂ γ and γ in (A3), (A4), (F2) and (F4) satisfy ₂

2 2 1

K L

γ +γ < , then for any ξ ∈₃ P X₃ with ξ3 < −

(

1 γK−γ2L2

)

αL3⁻¹, the operator G is a contraction mapping from D into itself.

Proof. From the condition (A3), we obtain that

( )

3

0^t U t,τ P dτ ≤K

∫

^{for all}

0

t≥ . For any fixed s≥ , let 0

( )

t U t s P

( )

, 3 ¹

ϕ = ⁻ for all t≥ . s Then for any fixed ξ ∈ and X t≥ ≥ , s 0

( ^∫

^s^t^{ϕ τ τ}

( )

^d

)

^{U t s P}

^{( )}

^, ³^ξ ⁼

^∫

^s^t^{ϕ τ}

^{( ) ( )}

^{U t s P d}^, ³^{ξ τ}

( ) ( )

, 3

( )

, 3 t

sϕ τ U tτ PU τ s P dξ τ

=

∫

( ) ( )

, 3

( )

, 3 t

sϕ τ U tτ PU τ s Pξ τd

≤

∫

( ) ( )

, 3

( )

, 3 t

sϕ τ U t τ P U τ s P ξ τd

≤

∫

( ) ( )

, 3

( )

, 3 t

s U s P U t P d

ξ ϕ τ τ τ τ

=

∫

( )

, 3 t

s U t P d

ξ τ τ

=

∫

( )

3 0^t U t, P d

ξ τ τ

≤

∫

K ξ

= .

This implies that

( )

^, ³

(

^s^t

( ) ) (

^s^t

^{( )} ) ^{( )}

^, ³

U t s P

∫

ϕ τ τd =

∫

ϕ τ τd U t s P ≤K

for all t≥ ≥ and s 0

( )

¹ ^t

( )

t s d K

ϕ ⁻

∫

ϕ τ τ ≤ ^for^all^t^{≥ ≥ .}^s ⁰ Let

( )

^t

( )

t sϕ τ τd

Ψ =

∫

^{for all}^t≥ ≥ . Then ^s ⁰

( )

^t

( )

^t ¹ _s^t

( )

^d ¹

( )

^t

K K

ϕ ϕ τ τ

Ψ′ = ≥

∫

= Ψ ^,

(7)

and hence ^Ψ^′

( ) ( )

^t ^Ψ ^t ⁻¹^≥^K⁻¹^{for all}^t≥ ≥ . This shows that for any fixed ^s ⁰

0 0

t > ≥ , s

( )

^t

( )

^t⁰ ^exp

{

^K⁻¹

(

^t ^t⁰

) }

Ψ ≥ Ψ − for all t≥ . t₀ and

( )

, 3

( )

¹

U t s P =ϕ t ⁻

( )

¹

K t ⁻

≤ Ψ

( )

⁰ ¹^exp

{

¹

(

⁰

) }

K t ⁻ K⁻ t t

≤ Ψ − −

( ) ( )

{

^K ^t⁰ ⁻¹^exp ^{K t}⁻¹⁰

}

^exp

⁽

^{K t}⁻¹

⁾

≤ Ψ −

for all t≥ . Therefore, t₀ lim

( )

, 3 0

t U t s P

→∞ = for all s≥ and 0

( )

^{, 0} ³

{ ( )

¹ ¹^exp

( )

¹

}

^exp

⁽

¹

⁾

U t P ≤ KΨ ⁻ K⁻ −K t⁻

( )

¹ ¹^exp

( )

¹

K ⁻ K⁻

≤ Ψ

for all t≥ . With a similar proof as that for Lemma 2.1, there exists a constant 1

1 0

M > such that U t

( )

, 0 P3 ≤M1 for all ^t^∈

[ ]

^0,1 ^{. Let}

( ) ( )

{

¹ ¹

}

3 max 1, 1 exp

L = M KΨ ⁻ K⁻ .

Then U t

( )

, 0 P3 ≤L3 for all t≥ . 0

If γK+γ₂L₂ < and 1 ξ ∈₃ P X₃ with ξ3 < −

(

1 γK−γ2L2

)

αL3⁻¹, then for any ϕ ∈ , D

Cϕ _∞

( )

3

( )

3

( ( ) )

0 0

sup ,0 ^t , ,

t

U t ξ U tτ P f τ ϕ τ dτ

≥

= +

∫

( )

, 2

(

,

( ) ) ^{( )}

, 1

(

,

^{( )} )

t^∞U t τ P f τ ϕ τ dτ t^∞U tτ P f τ ϕ τ dτ

−

∫

−

∫

( )

3 3

( )

2 2

( ( ) )

0 0

sup , 0 sup , ,

t t t

U t Pξ ^∞ U t τ P P f τ ϕ τ dτ

≥ ≥

≤ +

∫

( ) ( ( ) )

(

⁰ ³

sup ^t , ,

t s

U tτ P f τ ϕ τ dτ

≥

+

∫

( )

^, ¹

(

^,

( ) ) )

t^∞ U tτ P f τ ϕ τ dτ +

∫

(8)

( )

3 3

( )

2 2

( ( ) )

0 0

sup , 0 sup , ,

t t t

U t P ξ ^∞ U t τ P P f τ ϕ τ dτ

≥ ≥

≤ +

∫

( ) ( ) ( ) ( )

(

⁰^t ^{U t}^,^τ ^P³ ^{γ ϕ τ} ^d^τ ^t^∞ ^{U t}^,^τ ^P¹ ^{γ ϕ τ} ^d^τ

)

+

∫

+

∫

( ( ) )

3 3 2 0 2 ,

L ξ L ^∞ P f τ ϕ τ dτ γK ϕ _∞

≤ +

∫

+

( )

3 3 2 2

L ξ γ L γK ϕ _∞

≤ + +

(

² ²

) ( )

3 2 2

3

1 L K

L L K

L

γ γ α γ γ α

− −

≤ + +

α

= .

Hence, Gϕ ∈ for all D ϕ ∈ and D ^{G D}

( )

^{⊂ .}^D

Moreover, from Lemma 5.1,

2 2

( )

Gϕ−Gφ _∞ ≤ γK+γ L ϕ φ− _∞ for any φ , ϕ ∈ . D

Hence, :G D→ is a contraction mapping on D with a contraction constant D

2 2

K L

γ +γ . The assertion of this lemma is established now.

Theorem 5.3. Suppose that the C -evolution system ₀

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ ^s ^t

}

[

^{∞ × →}

)

^X ^X satisfies conditions (F1)~(F4). If the constants K , L , ₂ γ and γ in (A3), (A4), (F2) and ₂ (F4) satisfy γK+γ₂L₂< , then for any 1 ξ ∈₃ X₃ with ξ3 < −

(

1 γK−γ2L2

)

αL3⁻¹, there exists ξ ∈ such that ₀ X Pξ_{3 0} = and the corresponding unique mild solution ξ₃

( )

u t to the abstract semilinear initial value problem (5.1) is bounded on

[

^0,^{∞ .}

)

Furthermore, ^lim

( )

⁰

t u t

→∞ = .

Proof. From Lemma 5.2, G D: → is a contraction mapping on D with a D contraction constant γK+γ₂L₂ . Then there exists u∈D such that Gu= . u Hence, u t is bounded on

( ) [

^0,^{∞ , and}

)

( ) ( )

3

( )

3

( ( ) )

0

, 0 ^t , ,

u t =U t ξ +

∫

U tτ P f τ u τ dτ

( )

, 2

(

,

( ) ) ( )

, 1

(

,

( ) )

t^∞U t τ P f τ u τ dτ t^∞U tτ P f τ u τ dτ

−

∫

−

∫

^.

Thus

(9)

( )

0 3 0

( )

0, 2

(

,

( ) )

0

^{( )}

0, 1

(

,

^{( )} )

u =ξ −

∫

^∞U τ P f τ u τ dτ −

∫

^∞U τ P f τ u τ dτ ^. Let ξ =0 u

( )

0 ∈X . Following from the facts PU t s3

( )

, =U t s P

( )

, 3 for all

0

t≥ ≥ and s P P_j ₃ = for each j =1, 2, we have 0 P3 0ξ =P u3

( )

0 = . On the other ξ3

hand,

( )

u t

( )

,0 3 0^t

( )

, 3

(

,

( ) ) ( )

, 2

(

,

( ) )

U t ξ U tτ P f τ u τ dτ t^∞U tτ P f τ u τ dτ

= +

∫

−

∫

( )

, 1

(

,

( ) )

t^∞U t τ P f τ u τ dτ

−

∫

( )

^{, 0} ⁰

( )

^{, 0} ₀

( )

^0, ²

(

^,

( ) )

U t ξ U t ^∞U τ P f τ u τ dτ

= +

∫

( )

, 0

( )

0, 1

(

,

( ) )

U t s ^∞U τ P f τ u τ dτ

+

∫ ^{( )}

3

( ^{( )} )

0^tU t,τ P f τ,u τ dτ +

∫

( )

, 2

(

,

( ) ) ( )

, 1

(

,

( ) )

t^∞U t τ P f τ u τ dτ t^∞U tτ P f τ u τ dτ

−

∫

−

∫

( )

, 0 0 0^t

( )(

, 1 2 3

) (

,

( ) )

U t ξ U t τ P P P f τ u τ dτ

= +

∫

+ +

( )

, 0 0 0^t

( )

,

(

,

( ) )

U t ξ U t τ f τ u τ dτ

= +

∫

^for^any^t^{≥ .}⁰

This shows that ^{u t}

( )

is a bounded mild solution to the abstract semilinear initial value problem (5.1) with initial value ξ on ₀

[

^0,∞ which satisfies

)

Pξ_{3 0} =ξ₃. From Theorem 2.14, the solution ^{u t}

( )

is unique on

[

^0,^{∞ .}

)

Since ^{u t}

( )

≤α for all t≥ , there exists a constant 0 µ ∈ ∞ such that

[

^0,

)

t

( )

lim u t

µ = →∞ . If µ > , then there is a constant 0 ^{θ ∈}

( )

^0,1 ^andt1≥ such that 0

2 2

K L

θ γ> +γ and ^{u t}

( )

≤θ µ⁻¹ for all t≥ . From Lemma 5.2, one may have t₁

( )

3

( )

1 3

lim , 0 lim , 0

t U t P t U t t P

→∞ = →∞ = . For any t≥ ≥ with t large enough, t₁ 0

( )

u t

( )

, 0 3 0^t

( )

, 3

(

,

( ) )

U t ξ U tτ P f τ u τ dτ

= +

∫

( )

, 2

(

,

( ) ) ^{( )}

, 1

(

,

^{( )} )

t^∞U tτ P f τ u τ dτ t^∞U tτ P f τ u τ dτ

−

∫

−

∫

(10)

( )

, 0 3 3

( )

, 1 3 0^t¹

( )

1, 3

(

,

( ) )

U t P ξ U t t P U t τ P f τ u τ dτ

≤ +

∫

( ) ( ( ) ) ( ) ( ( ) )

1

3 1

, , , ,

t

t U t τ P f τ u τ dτ t^∞U tτ P f τ u τ dτ

+

∫

+

∫

( )

, 2

(

,

( ) )

t^∞U tτ P f τ u τ dτ

+

∫

( )

3 3

( )

1 3 ¹

( )

1 3

( ( ) )

, 0 , 0^t , ,

≤ +

∫

( ) ( )

1

, 3 t

t U tτ P γ u τ dτ

+

∫

+

∫

t^∞ U t

^{( )}

^,τ P¹ γ u

^{( )}

τ dτ

( )

, 2 2

(

,

( ) )

t^∞ U tτ P P f τ u τ dτ +

∫

( )

, 0 3 3

( )

, 1 3 0^t¹

( )

1, 3

(

,

( ) )

≤ +

∫

( ) ( )

(

1

)

1

3 1

, ,

t

t U t τ P dτ t^∞ U tτ P dτ γθ µ⁻

+

∫

+

∫

( ( ) )

2 2 ,

L t^∞ P f τ u τ dτ +

∫

( )

^{, 0} ³ ³

( )

^, ¹ ³ ₀^t¹

( )

¹^, ³

(

^,

( ) )

¹

U t P ξ U t t P U t τ P f τ u τ dτ Kγθ µ⁻

≤ +

∫

+

( ( ) )

2 2 ,

L t^∞ P f τ u τ dτ

+

∫

^.

Thus lim

( ) (

2 2

)

¹

t u t K L

µ γ γ θ µ µ⁻

= →∞ ≤ + < . This is impossible, and hence µ = . 0 This shows that ^lim

( )

⁰

t u t

→∞ = , and this theorem is completely proved now.

With the same processes as in the proofs of Lemma 5.1 and Lemma 5.2, one may easily obtain following Lemma 5.4 and Lemma 5.5.

Lemma 5.4. Suppose the C -evolution system ₀

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ satisfies ^s ^t

}

conditions (A1)~(A4) and the function ^f ^{: 0,}

[

^{∞ × →}

)

^X ^X satisfies conditions (F1)~(F4). For any fixed ξ ∈₂ X₂, ξ ∈₃ X₃, let the operator ^{B D}^: ^→^C

( [

^0,^∞

)

^;^X

)

be defined by

( )( ) ( )

2

( )

3

( )

2

( ( ) )

, 0 , 0 0^t , ,

Bϕ t =U t ξ +U t ξ +

∫

U tτ P f τ ϕ τ dτ

( )

3

( ( ) )

0^tU t,τ P f τ ϕ τ, dτ

+

∫

−

∫

t^∞U t

^{( )}

^,τ P f¹

⁽

τ ϕ τ^,

^{( )} ⁾

dτ

(11)

for all ϕ ∈ , then B is well-defined and D

2 2

( )

Bϕ−Bφ _∞ ≤ γK+γ L ϕ φ− _∞ for any φ , ϕ ∈ . D Where γ , K , γ , ₂ L are the constants in (A3), (A4), (F2) and (F4). ₂

Lemma 5.5. Suppose that the C -evolution system ₀

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ ^s ^t

}

[

^{∞ × →}

)

^X ^X satisfies conditions (F1)~(F4). If the constants K , L , ₂ γ and γ in (A3), (A4), (F2) and ₂ (F4) satisfy γK+γ₂L₂ < , then for any 1 ξ ∈₂ X₂, ξ ∈₃ X₃ with both ξ and ₂ ξ ₃ strictly less than

(

1−γK−γ2L2

) (

α L2+L3

)

⁻¹, where L is as in Lemma 5.2, the ₃ operator B is a contraction mapping from D into itself.

Theorem 5.6. Suppose that the C -evolution system ₀

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ ^s ^t

}

[

^{∞ × →}

)

^X ^X satisfies conditions (F1)~(F4). If the constants K , L , ₂ γ and γ in (A3), (A4), (F2) and ₂ (F4) satisfy γK+γ₂L₂ < . Then for any 1 ξ ∈₂ X₂ , ξ ∈₃ X₃ with ξ , ₂

( ) ( )

¹

3 1 K 2L2 L2 L3

ξ < −γ −γ α + ⁻ , there exists ξ ∈₀ X such that Pξ_{3 0} =ξ₃ ,

2 0 2

Pξ =ξ and the corresponding unique mild solution u t to the abstract

( )

semilinear initial value problem (5.1) is bounded on

[

^0,∞ . More precisely,

)

2 3

2 2 2 2

1 1

L L

u K L ξ K L ξ

γ γ γ γ

∞ ≤ +

− − − − .

Proof. From Lemma 5.5, B D: → is a contraction mapping on D with a D contraction constant γK+γ₂L₂ . Hence there exists u∈ such that Bu uD = ,

( )

u t is bounded on

[

^0,^{∞ ,}

)

( ) ( )(

, 0 2 3

)

0^t

( )

, 2

(

,

( ) )

u t =U t ξ ξ+ +

∫

U tτ P f τ u τ dτ

( )

3

( ( ) ) ( )

1

( ( ) )

0^t , , , ,

U tτ P f τ u τ dτ t^∞U t τ P f τ u τ dτ

+

∫

−

∫

and

( )

⁰ ² ³ ₀

( )

^, ¹

(

^,

( ) )

u =ξ + −ξ

∫

^∞U sτ P f τ u τ dτ^.

(12)

Let ξ =0 u

( )

0 ∈X . Since P U t sj

( )

^, =U t s P

( )

^, j and P P_j _i = for 0 i j, 1, 2, 3∈

{ }

and i≠ , this implies j P2 0ξ =P u2

( )

0 = , ξ2 P3 0ξ =P u3

( )

0 = . On the other hand, ξ3

( )

u t ⁼^{U t}

( )(

^,0 ^{ξ ξ}²⁺ ³

)

⁺

∫

₀^t^{U t}

( )

^,^τ ^{P f}²

(

^τ^,^u

( )

^τ

)

^d^τ

( )

3

( ( ) ) ( )

1

( ( ) )

0^t , , , ,

U tτ P f τ u τ dτ t^∞U t τ P f τ u τ dτ

+

∫

−

∫

( )

, 0 0

( )

, 0 0

( )

, 1

(

,

( ) )

U t ξ U t ^∞U sτ P f τ u τ dτ

= +

∫

( )

2

( ( ) ) ( )

3

( ( ) )

0^tU t,τ P f τ,u τ dτ 0^tU t,τ P f τ,u τ dτ

+

∫

+

∫

( )

, 1

(

,

( ) )

t^∞U t τ P f τ u τ dτ

−

∫

( )

0

( )(

1 2 3

) ( ( ) )

, 0 0^t , ,

U t ξ U t τ P P P f τ u τ dτ

= +

∫

+ +

( )

, 0 0 0^t

( )

,

(

,

( ) )

U t ξ U t τ f τ u τ dτ

= +

∫

^for^any^t^{≥ .}⁰

Thus ^{u t}

( )

is a bounded mild solution to the abstract semilinear initial value problem (5.1) on

[

^0,∞ with initial value

)

ξ which satisfies ₀ P_{2 0}ξ = , ξ₂ Pξ_{3 0}= . The ξ₃ uniqueness of the solution can be obtained as in the proof of Theorem 2.14 immediately. Furthermore,

( )(

2 3

)

0

( )

2

( ( ) )

0

sup , 0 ^t , ,

t

u _∞ U t ξ ξ U tτ P f τ u τ dτ

= ≥ + +

∫

( )

, 3

(

,

( ) ) ^{( )}

, 1

(

,

^{( )} )

t

sU tτ P f τ u τ dτ t^∞U tτ P f τ u τ dτ

+

∫

−

∫

( )

2 2

( )

3 3

0 0

sup , 0 sup , 0

t t

U t P ξ U t P ξ

≥ ≥

≤ +

( )

2 2

( ( ) )

0 0

sup ^t , ,

t

U tτ P P f τ u τ dτ

≥

+

∫

( ) ( ( ) )

(

⁰ ³

0

sup ^t , ,

t

U tτ P f τ u τ dτ

≥

+

∫

( )

^, ¹

(

^,

( ) ) )

t^∞ U tτ P f τ u τ dτ

+

∫

( )

2 2

( )

3 3

0 0

sup , 0 sup , 0

t t

U t P ξ U t P ξ

≥ ≥

≤ +

( )

2 2

( ( ) )

0 0

sup ^t , ,

t

U tτ P P f τ u τ dτ + ≥

∫

(13)

( ) ( )

(

⁰ ³

0

sup ^t ,

t

U tτ P γ u τ τd

≥

+

∫

( )

^, ¹

( ) )

t^∞ U tτ P γ u τ τd

+

∫ ( ( ) )

2 2 3 3 2 0 2 ,

L ξ L ξ L ^∞ P f τ u τ dτ γK u _∞

≤ + +

∫

+

( )

2 2 3 3 2 2

L ξ L ξ γ L γK u _∞

≤ + + + .

Thus

(

1−γK−γ2L2

)

u _∞ ≤L2ξ2 +L3ξ3 and

2 3

2 2 2 2

1 1

L L

u K L ξ K L ξ

γ γ γ γ

∞ ≤ +

− − − − .

The proof of this theorem is completed now.

§ 5-2 Applications

Example 5.1 We will consider the semilinear differential equations:

( ) ( ) ( ) ( ) ( )

( ) [ )

( )

0

( )

, , , , , on 0,

, 0 on 0, 0, on u t x u t x u t x f t x u t

u t x

u x x

β

ξ

∂ = ∆ + + ∞ × Ω

∂

= ∞ × ∂Ω

 = Ω



(5.2)

where Ω ⊂Rⁿ is a bounded domain with smooth boundary, β > is a constant, the 0 function f satisfy conditions (F1)~(F4), and ξ ⋅ is in 0

( )

^L²

( )

Ω . Let X be the Hilbert space ^L²

( )

Ω , and let the operator ^{A D A}^:

( )

^→^X be defined by

Aϕ = ∆ +ϕ βϕ for all ϕ ∈^{D A}

( )

,

where ^{D A}

( )

⁼

{

^ϕ^∈^C¹

( )

^Ω ^I^C²

^{( ) ( )}

^Ω ^:^ϕ ^x ⁼^{0 on}^∂Ω

}

. Then the semilinear differential equation (5.2) can be replaced by the semilinear initial value problem:

( ) ( ) ( ) ( )

( )

0

, on 0, 0

d u t Au t f t u dt

u ξ X

 = + ∞



 = ∈



(5.3)

for all ^u

( )

⋅ ∈^{D A}

( )

. From [13, P.205], it can be shown that there exists a sequence

(14)

of eigenfunctions

{

ϕn ^:n∈N

}

corresponding to the sequence of eigenvalues

{

λn^:n∈N

}

for A and

{

ϕn^:n∈N

}

is an orthonormal basis for the Hilbert space X.

This implies that

1 , _k _k

ϕ =

∑

k^∞₌ ϕ ϕ ϕ ^for^all^{ϕ ∈ ,}^X

and the C -semigroup ₀

{

^{S t}

( )

^:^{t ≥}⁰

}

generated by A on X is given by

(

^{S t}

( )

^ϕ

) ^{( )}

^x ⁼

∑

^∞_k₌₁^exp

^{( )}

^λ^k^t ^{ϕ ϕ ϕ}^, ^k ^k

^{( )}

^x ^for^allϕ ∈ , x ∈Ω . ^X Suppose that β > be a constant such that the eigenvalues of A satisfies 0

1 2

Reλ ≥Reλ ≥L≥Reλ_n >0,

1 2

Reλ_n₊ =Reλ_n₊ =L=Reλ_m =0 and

1 2

0>Reλ_m₊ ≥Reλ_m₊ ≥ LL.

We may define linear operators P , ₁ P and ₂ P on X by ₃

1 ⁿ 1 , _k _k

Pϕ =

∑

k₌ ϕ ϕ ϕ for all ϕ ∈ , X

2 ^m_{k n} 1 , _k _k

Pϕ =

∑

_{= +} ϕ ϕ ϕ ^for^all^{ϕ ∈ ,}^X and

3 _{k m}1 , _k _k

Pϕ =

∑

^∞_{= +} ϕ ϕ ϕ ^for^all^{ϕ ∈ .}^X

Then operators P , ₁ P and ₂ P are projections on the Hilbert space X. Let ₃ X be _i the range of a projection P for each i =1, 2, 3. Thus dimensions of _i X and ₁ X ₂ are n and m− , respectively. Let n

( )

^,

( )

U t s =S t− for s all t≥ ≥ , s 0

then

{

^{U t s}

( )

^, ^{: 0}≤ ≤ < ∞ is a ^s ^t

}

C -evolution system with the infinitesimal ₀ generator ^{A t}

( )

≡ . Since ^A

( ) ( )

ⁿk ₁ ^, k k

S t ϕ =S t

∑

₌ ϕ ϕ ϕ

1 ,

( )

n

k k

k₌ ϕ ϕ S t ϕ

=

∑

( ( ) )

1 , 1exp ,

n

k j k j j

k₌ ϕ ϕ ^∞j₌ λt ϕ ϕ ϕ

=

∑ ∑

1exp

( )

,

n

k k k

k₌ λt ϕ ϕ ϕ

=

∑

^∈^X¹ for any t≥ and 0 ϕ ∈X₁,

(15)

and similarly, S t

( )

ϕ =

∑

^mk n_{= +}₁^exp

( )

λkt ϕ ϕ ϕ^, k k ^{for all}^t^{≥ and}⁰ ^{ϕ ∈}^X²^{. This}

implies that

{

S t

( )

| :_X1 t∈R

}

and

{

S t

( )

| :_X2 t∈R

}

are C -groups on ₀ X and ₁ X , ₂ respectively. Therefore, the condition (A1) holds. On the other hand, since

( ) ( )

1 1 1exp _k , _k _k

P S t ϕ =P

∑

k^∞₌ λt ϕ ϕ ϕ 1exp

( )

_k , _k 1 _k

k λt ϕ ϕ Pϕ

∞

=

∑

=

⁼

^∑

^∞^k⁼¹^exp

( )

^λ^k^t ^{ϕ ϕ}^, ^k

( ^∑

ⁿ^j⁼¹ ^{ϕ ϕ ϕ}^k^, ^j ^j

)

=

∑

ⁿk₌₁^exp

( )

λkt ϕ ϕ ϕ^, k k

=

∑

ⁿk₌₁ ϕ ϕ^, k S t

( )

ϕk

=S t Pϕ

( )

1 for all t≥ and 0 ϕ ∈ , X and

( ) ( )

2 2 _k 1exp _k , _k _k

P S t ϕ =P

∑

^∞₌ λt ϕ ϕ ϕ 1exp

( )

_k , _k 2 _k

k λt ϕ ϕ Pϕ

∞

=

∑

=

⁼

^∑

^∞^k⁼¹^exp

( )

^λ^k^t ^{ϕ ϕ}^, ^k

( ^∑

^m^{j n}^{= +}¹ ^{ϕ ϕ ϕ}^k^, ^j ^j

)

=

∑

^mk n_{= +}₁^exp

( )

λkt ϕ ϕ ϕ^, k k

=

∑

^mk n_{= +}₁ ϕ ϕ^, k S t

( )

ϕk

=S t P

( )

2ϕ for all t≥ and 0 ϕ ∈ . X

This shows that P S t1

( )

=S t P

( )

1 and P S t2

( )

=S t P

( )

2 on X for all t≥ . Thus, 0

( ) ( ) ( ) ( ) ( ) ( ) ( )

3 1 2 1 2 3

P S t = I − −P P S t =S t −P S t −P S t =S t P on X for all t≥ . Hence, the condition (A2) holds. 0

For all t∈ and R ϕ ∈ , X

( )

1 ⁿ 1exp

( )

_k , _k _k S t Pϕ =

∑

k₌ λt ϕ ϕ ϕ

1exp

( )

,

n

k k k

k₌ λt ϕ ϕ ϕ

≤

∑

≤

∑

ⁿk₌₁^exp

( )

λkt ϕ ϕk²

^≤

( ^∑

ⁿ^k⁼¹^{exp Re}

(

^t ^λ^k

) )

^ϕ ^,

and hence