về một phương pháp chỉnh hóa cho phương trình parabolic tuyến tính ngược thời gian với hệ số phụ thuộc thời gian trong không gian banach

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH





NGUYỄN VĂN TÁM






VỀ MỘT PHƯƠNG PHÁP CHỈNH HÓA
CHO PHƯƠNG TRÌNH PARABOLIC TUYẾN TÍNH
NGƯỢC THỜI GIAN VỚI HỆ SỐ PHỤ THUỘC
THỜI GIAN TRONG KHÔNG GIAN BANACH




LUẬN VĂN THẠC SĨ TOÁN HỌC







NGHỆ AN - 2014

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH






NGUYỄN VĂN TÁM




VỀ MỘT PHƯƠNG PHÁP CHỈNH HÓA
CHO PHƯƠNG TRÌNH PARABOLIC TUYẾN TÍNH
NGƯỢC THỜI GIAN VỚI HỆ SỐ PHỤ THUỘC
THỜI GIAN TRONG KHÔNG GIAN BANACH


CHUYÊN NGÀNH: TOÁN GIẢI TÍCH
MÃ SỐ: 60.46.01.02


LUẬN VĂN THẠC SĨ TOÁN HỌC



Người hướng dẫn khoa học:
TS. NGUYỄN VĂN ĐỨC

NGHỆ AN - 2014

···

du
dt
= a(t)Au(t) 0  s  t < T
u(s) = χ
X −A
θ ∈

0,
π
2

X a ∈ C([0, T ] : R
+
)
C
0
θ
X
X X T (t), 0  t < ∞
X
T (0) = I, ( )
T (t + s) = T (t)T (s) t, s  0
T (t)

lim
t↓0
T (t) − I = 0
A
D(A) = {x ∈ X : lim
t↓0
T (t)x − x
t
}
Ax = lim
t↓0
T (t)x − x
t
x ∈ D(A)
T (t) D(A)
A
X
T (t), 0  t < ∞ lim
t↓0
T (t)x = x
x ∈ X
X C
0
C
0
A
A
A X
T (t) = e
tA

=
∞

n=0
(tA)
n
n!
t  0
T (t) T (0) = I
T (t + s) = T (t)T (s)
T (t) − I  tAe
tA




T (t) − I
t
− A




 A max
0st
T (s) − I.
T (t)
X A
T (t)
X

ρ > 0 I − ρ
−1

ρ
0
T (s)ds < 1
ρ
−1

ρ
0
T (s)ds

ρ
0
T (s)ds
h
−1
(T (h) − I)

ρ
0
T (s)ds = h
−1


ρ
0
T (s + h)ds −


ρ
0
T (s)ds

= h
−1


ρ+h
ρ
T (s)ds −

h
0
T (s)ds

.
h
−1
(T (h) − I) =

h
−1

ρ+h
0
T (s)ds − h
−1

h

0
T (s)ds



ρ
0
T (s)ds

−1
.
h
−1
(T (h)−I)
(T (ρ) −I)


ρ
0
T (s)ds

−1
h ↓ 0
(T (ρ) −I)


ρ
0
T (s)ds


−1
T (t)
T (t) C
0
A
x ∈ X lim
h→0
1
h

t+h
t
T (s)xds = T(t)x.
x ∈ X

t
0
T (s)xds ∈ D(A)
A


t
0
T (s)xds

= T (t)x − x.
x ∈ D(A) T (t)x ∈ D(A)
d
dt
T (t)x = AT(t)x = T (t)Ax.

x ∈ D(A),
T (t)x − T (s)x =

t
s
T (τ)Axdτ =

t
s
AT (τ)xdτ.
A C
0
D(A) A X A
x ∈ X x
t
=
1
t

t
0
T (s)xds
x
t
∈ D(A) t > 0
t ↓ 0 D(A) ≡ X D(A) D(A)
A
x
n
∈ D(A) x

n
→ x Ax
n
→ y n → ∞
T (t)x
n
− x
n
=

t
0
T (s)Ax
n
ds.
T (s)y
n → ∞
T (t)x − x =

t
0
T (s)yds.
t > 0 t ↓ 0
x ∈ D(A) Ax = y
T (t) C
0
ω  0 M  1
T (t)  Me
ωt
0  t < ∞.

ω = 0 T (t) C
0
ω = 0
M = 1 T (t) C
0
η > 0 T (t) 0  t  η
{t
n
} t
n
 0, lim
n→∞
t
n
= 0
T (t
n
)x  n
x ∈ X T (t
n
)x
C
0
, T(t) M > 0
T (t)  M 0  t  η T (0) = 1 M  1
ω = η
−1
ln M  0 t  0 t = nη + δ n ∈ N
0  δ < η
T (t) = T(δ)T (nη)

n
  M
n+1
 MM
t/n
= Me
ωt
.
A C
0
T (t), t  0
A D(A) = X
ρ(A) A R
+
λ > 0
R(λ : A) 
1
λ
,
R(λ : A) = (λI −A)
−1
I X
θ ∈ (0;
π
2
]
T (t) t > 0 X
θ
T (t)
T (z)

S
θ
= {re
iθ

: r > 0, | θ

|< θ}
T (z + ω) = T (z)T (ω) z, ω ∈ S
θ
θ
1
< θ T(z)x → x z → 0 S
θ1
x ∈ X
θ
1
< θ T(z) S
θ1
T (t) X
θ T(t)
θ
A
X −A θ
θ
1
< θ M
1
> 0
ω /∈ S

π
/
2
−
θ
1
ω ∈ ρ(A)
(ω − A)
−1
 ≤
M
1
dist(ω, S
π/
2
−θ
1
)
.
−A
θ 0 ∈ ρ(A) σ > 0
A
A
−σ
=
1
2πi

Γ
ω

−σ
(ω − A)
−1
dω,
Γ ∞e
i∅
∞e
−i∅
π > φ > π/
2
−θ
A
σ
= (A
−σ)
−1
A
o
= I
−A
θ 0 ∈ ρ(A)
A
σ
σ > 0
A
σ
(A
σ
) ⊆ (A
σ


) σ > σ

> 0
(A
σ
) X σ ≥ 0
A
σ
1
+σ
2
x = A
σ
1
A
σ
2
x σ
1
, σ
2
∈ R x ∈ (A
σ
)
σ = max{σ
1
, σ
2
, σ

1
+ σ
2
}
L
2

u
t
= Au(t) 0  t < T,
u(0) = x,
−A
X
u
t
= −Au(t), t ∈ [0, ∞), u(0) = x
0
,
E A D(A)
E ρ(−A) = ∅
−A
α (0 < α <
π
2
) X
u
t
= Au(t), u(0) = x, (0  t < T ), −A
A
A

A
u
t
= Au(t) + h(t), u(0) = x, (0  t < T ).
u
t
+ Au(t) = 0, (0  t < T), u(T ) = x, A
−A
X x ∈ X

du
dt
= A(t, D)u(t), 0  s  t < T,
u(s) = χ
X −D
θ ∈ (
π
4
,
π
2
] χ ∈ X
u(t) u(t)
L
p
(R), 1  p < ∞
X

du
dt

= a(t)Au(t) 0  s  t < T
u(s) = χ
−A θ ∈ (0,
π
2
] X a ∈
C([0, T] : R
+
)
{R
β
(t) : β > 0, t ∈ [s, T ]}
X
u(t)
χ ∈ X δ > 0 β(δ) > 0
β(δ) −→ 0 δ −→ 0
u(t) −R
β(δ)
(t)χ
δ
 → 0 δ → 0 s ≤ t ≤ T χ −χ
δ
  δ

dv
dt
= f
β
(t, A)v(t) 0 ≤ s ≤ t < T
v(s) = χ

β > 0 f
β
(t; A), 0 ≤ t ≤ T a(t)A
θ ∈ (0,
π
2
]
f
β
(t, A) =

a(t)A − βA
σ
θ ∈ (0,
π
4
]
a(t)A(I + βA)
−1
θ ∈ (
π
4
,
π
2
]
σ > 1 θ ∈ (0;
π
4
]

β → 0 f
β
(t; A)
a(t)A u(t) v
β
(t)
u(t) − v
β
(t) → 0 β → 0
t ∈ [s; T] {V
β
(t; s) : β > 0
t ∈ [s; T]
V
β
(t; s)χ = v
β
(t)
χ − χ
δ
 ≤ δ β > 0 β → 0
δ → 0 u(t) − V
β
(t; s)χ
δ
 → 0 δ → 0 s ≤ t ≤ T
A A − βA
σ
σ > 1 σ(
π

2
− θ) <
π
2
A(I + βA)
−1
θ ∈ (
π
4
;
π
2
]
−A
θ ∈ (0; π/
4
] 0 ∈ ρ(A) 0 < β < 1 σ > 1
σ(π/
2
− θ) < π/
2
f
β
(t; A); 0 ≤ t ≤ T
f
β
(t, A) = a(t)A −βA
σ
.
v

β
(t) = V
β
(t; s)χ, χ ∈ X
V
β
(t, s) =

1
2πi

Γ
φ
e

t
s
f
β
(τ,ω)dτ
(ω − A)
−1
dω 0 ≤ s < t ≤ T,
I t = s,
Γ
φ
ρ(A) π/
2σ
> φ > π/
2

−θ
φ
σ(π/
2
− θ) < π/
2
.
V
β
(t; s) 0 ≤ s ≤ t ≤ T
0 ≤ s < t ≤ T 0 ∈ ρ(A)
d ∈ (0; 1)
ρ(A)
(t − s)
−1/
σ
≤ d
Γ
φ
ρ(A)
Γ
1
= {re
iφ
: r ≥ (t −s)
−1/
σ
},
Γ
2

= {(t −s)
−1
σ
e
−iθ

: −φ ≤ θ

≤ φ},
Γ
3
= {re
−iφ
: r ≥ (t −s)
−1/
σ
}.
ω ∈ Γ
1
∪Γ
3
θ
1
< θ φ > π/
2
−θ
1
> π/
2
−θ

(ω, S
π/
2
−θ
1
) = |ω|sin(φ −(π/
2
−θ
1
))


(ω − A)
−1


≤
M
1
|ω|sin(φ − (π/
2
− θ
1
))
.
M

1
= M
1

/
sin(φ−(π/
2
−θ
1
))
B = max
t∈[0,T ]
|a(t)|





Γ
1
∪Γ
3




≤ M

1

Γ
1
Γ
3

|e

t
s
(a(τ)ω−βω
σ
)dτ
||ω|
−1
dω
= 2M

1

∞
(t−s)
−1/
σ
e

t
s
(a(τ)τ cos φ−βτ
σ
cos σφ)dτ
τ
−1
dτ
≤ 2M


1

∞
(t−s)
−1/
σ
e
B(t−s)τ cos φ−β(t−s)τ
σ
cos σφ
τ
−1
dτ
= 2M

1

∞
1
e
B(t−s)
1−1/σ
x cos φ−βx
σ
cos σφ
x
−1
dx
≤ 2M


1

∞
1
e
BT
1−1/σ
x cos φ−βx
σ
cos σφ
dx ≤ K,
K t s σ > 1 π/
2
σ >
φ > π/
2
− θ 0 < φ < σφ < π/
2
cos φ > 0
cos(σφ) > 0 ω ∈ Γ
2





Γ
2





≤ M
d

Γ
2
|e

t
s
(a(τ)ω−βω
σ
)dτ
|dω
= M
d

φ
−φ
e

t
s
(a(τ)(t−s)
1−/
σ
cos θ

−β(t−s)

−1
cos σθ

)dτ
(t − s)
−1/
σ
dθ

≤ dM
d

φ
−φ
e
B(t−s)
1−1/σ
cos θ

−β cos σθ

dθ

≤ dM
d

φ
−φ
e
BT

1−1/σ
cos θ

dθ

≤ dM
d
e
BT
1−1/σ
2φ
M
d
= max
|ω|≤d
(ω − A)
−1
 ω → (ω −A)
−1
ρ(A) V
β
(t; s) 0 ≤ s ≤ t ≤ T
(t − s)
−1/σ
> d
Γ
φ
Γ
1
= {re

iφ
: r ≥ (t −s)
−1/σ
},
Γ
2
= {(t −s)
−1/σ
e
iθ

: −φ ≤ θ

≤ π},
Γ
3
= {re
iπ
: d ≤ r ≤ (t −s)
−1/σ
},
Γ
4
= {de
−iθ

: −π ≤ θ

≤ π},
Γ

5
= {re
−iπ
: d ≤ r ≤ (t −s)
−1/σ
},
Γ
6
= {(t −s)
−1/σ
e
iθ

: −π ≤ θ

≤ −φ},
Γ
7
= {re
−iφ
: r ≥ (t −s)
−1/σ
}.
Γ
1
= Γ
1
Γ
7
= Γ

3


Γ
1
∪Γ
7
 = 

Γ
1
∪Γ
3
 ≤ K
ω ∈ Γ
2
ω ∈ Γ
2
(ω, S
π/2−θ
1
) ≥ ((t −s)
−1/σ
e
iφ
, S
π/2−θ
1
).






Γ
2




≤ M

1

Γ
2
|e

t
s
(a(τ)ω−βω
σ
)dτ
||ω|
−1
dω
= M

1


π
φ
e

t
s
(a(τ)(t−s)
−1/σ
cos θ

−β(t−s)
−1
cos σθ

)dτ
dθ

≤ M

1

π
φ
e
BT
1−1/σ
cos φ−βcosσθ

dθ


≤ M

1

π
φ
e
1+BT
1−1/σ
cos φ
dθ

= M

1
e
1+BT
1−1/σ
cos φ
(π − φ)
0 < β < 1 

Γ
6



Γ
3
+


Γ
5

= 

(t−s)
−1/σ
d
(e

t
s
(−a(τ)τ−βr
σ
e
−iπσ
)dτ
− e

t
s
(−a(τ)τ−βτ
σ
e
iπσ
)dτ
)(−τ − A)
−1
dτ

≤ M

1

(t−s)
−1/σ
d
|e
−(

t
s
(a(τ)dτ)τ
(e
−β(t−s)τ
σ
e
−iπσ
− e
−β(t−s)τ
σ
e
iπσ
) | τ
−1
dτ
= M

1


(t−s)
−1/σ
d
e
−(

t
s
(a(τ)dτ)τ
|(e
−β(t−s)τ
σ
cos σπ
2i sin(β(t −s)τ
σ
sin σπ)|τ
−1
dτ
≤ M

1

(t−s)
−1/σ
d
e
−β(t−s)τ
σ
cos σπ
2|sin(β(t −s)τ

σ
sin σπ)|τ
−1
dτ
= M

1

1
(t−s)
−1/σ
d
e
−βx
σ
cos σπ
2|sin(βx
σ
sin σπ)|x
−1
dx
= M

1

1
(t−s)
−1/σ
d
x

−1/2
e
−βx
σ
cos σπ
{4x
−1
sin
2
(βx
σ
sin σπ)}
1/2
dx
= M

1

1
(t−s)
−1/σ
d
x
−1/2
e
−βx
σ
cos σπ
{2x
−1

(1 − cos(2βx
σ
sin σπ))}
1/2
dx.
2x
−1
(1 − cos(2βx
σ
sin σπ)) → 0 x → 0.
M

1
β


Γ
3
+

Γ
5
 ≤ M

1

1
0
x
−1/2

e
−βx
σ
cosσπ
dx ≤ M

1
e

1
0
x
−1/2
dx = M

1
2e
0 < β < 1





Γ
4




≤ M

d

Γ
4
|e

t
s
(a(τ)ω−βω
σ
)dτ
|dω
= dM
d

π
−π
e

t
s
(a(τ)d cos θ

−βd
σ
cos σθ

)dτ
dθ


≤ dM
d

π
−π
e
BTd
e
−β(t−s)d
σ
cos σθ

dθ

≤ dM
d
e
BTd
(1 + e
T d
σ
)2π,
M
d
= max
|ω|≤d
(ω − A)
−1

V (t; s) 0 ≤ s ≤ t ≤ T

(t; s) → V
β
(t; s)
0 ≤ s ≤ t ≤ T
V
β
(t, s)A
−σ
=
1
2πi

Γ
φ
ω
−σ
e

t
s
f
β
(τ,ω)dτ
(ω − A)
−1
dω.
t → f
β
(t; w)
V

β
(t; s)A
−σ
− V
β
(t
0
, s
0
)A
−σ
 → 0 (t, s) → (t
0
, s
0
).
x ∈ (A
σ
)
V
β
(t; s)x − V
β
(t
0
, s
0
)x
≤ V
β

(t; s)A
−σ
− V
β
(t
0
, s
0
)A
−σ
A
σ
x → 0 (t, s) → (t
0
, s
0
).
V
β
(t; s)
(A
σ
) X
V
β
(t; s)
[s; T ] → X
t → V
β
(t; s)χ χ ∈ X

t → V
β
(t; s)χ [s; T ]
∂
∂t
V
β
(t, s)χ = f
β
(t, A)V
β
(t, s)χ t ∈ (s, T )
∂
∂t
V
β
(t, s)χ =
1
2πi

Γ
φ
(
∂
∂t
e

t
s
f

β
(τ,ω)dτ
)(ω − A)
−1
χdω
=
1
2πi

Γ
φ
e

t
s
f
β
(τ,ω)dτ
)f
β
(t, ω)(ω − A)
−1
χdω
=
1
2πi

Γ
φ
e


t
s
f
β
(τ,ω)dτ
)a(t)ω(ω − A)
−1
χdω
+
1
2πi

Γ
φ
e

t
s
f
β
(τ,ω)dτ
)(−βω
σ
)(ω − A)
−1
χdω.
1
2πi


Γ
φ
e

t
s
f
β
(τ,ω)dτ
)a(t)ω(ω − A)
−1
χdω
= a(t)
1
2πi

Γ
φ
e

t
s
f
β
(τ,ω)dτ
)(ω − A) + A)(ω −A)
−1
χdω
= (a(t)
1

2πi

Γ
φ
e

t
s
f
β
(τ,ω)dτ
dω)χ
+ a(t)
1
2πi

Γ
φ
e

t
s
f
β
(τ,ω)dτ
A(ω − A)
−1
χdω
= a(t)AV
β

(t, s)χ
ω → e

t
s
f
β
(τ,ω)dτ
A
t ∈ (s; T)
G =
1
2πi

Γ
φ
ω
σ
e

t
s
f
β
(τ,ω)dτ
(ω − A)
−1
dω.
G X
A

−σ
G = V
β
(t; s) A
σ
= (A
σ
)
−1
(V
β
(t; s)) ⊆ (A
−σ
) = (A
σ
) G = A
σ
V
β
(t; s)
1
2πi

Γ
φ
e

t
s
f

β
(τ,ω)dτ
)(−βω
σ
)(ω − A)
−1
χdω = −βGχ = −βA
σ
V
β
(t, s)χ.
∂
∂t
V
β
(t, s) = a(t)AV
β
(t, s)χ − βA
σ
V
β
(t, s)χ
= f
β
(t, A)V
β
(t, s)χ t ∈ (s, T ).
V
β
(s, s)χ = χ t → V

β
(t; s)χ
t → f
β
(t, A)V
β
(t, s)χ
(s; T ) t → e

t
s
f
β
(τ,ω)dτ
f
β
(t, ω)
t → V
β
(t; s)χ (s; T ) t → V
β
(t; s)χ
t → V
β
(t; s)χ
0 < β < 1 f
β
(t, A) 0 ≤ t ≤
T V
β

(t; s) 0 ≤ s ≤ t ≤ T
β
V
β
(t, s) ≤ K

e
Kβ
−1/(σ−1)
0 ≤ s ≤ t ≤ T K

K β
t s
0 ≤ s < t ≤ T V
β
(t; s)
β


Γ
1
∪Γ
3
 = 

Γ
1
∪Γ
7


V
β
(t, s) ≤ K
1
+
M

1
π

∞
1
e
BT
1−1/σ
x cos φ−βx
σ
cos σφ
dx,
K
1
β
β q(x) = 2BT
1−1/σ
x cos φ − βx
σ
cos(σφ)
[1; ∞) x
0
= (

BT
1−1/σ
cos φ
βσ cos(σφ)
)
1/(σ−1)
x ∈ [1; ∞)
2BT
1−1/σ
x cos φ − βx
σ
cos(σφ) ≤ q(x
0
)
= β
−1/(σ−1)
(BT
1−1/σ
cos φ)
σ/(σ−1)
σ
σ/(σ−1)
cos
1/(σ−1)
(σφ)
(σ − 1)
:= K
2
β
1/(σ−1)

.

∞
1
e
BT
1−1/σ
x cos φ−βx
σ
cos σφ
dx ≤ e
K
2
β
1/(σ−1)

∞
1
e
BT
1−1/σ
x cos φ
dx
=
K
2
β
1/(σ−1)
BT
1−1/σ

cos φe
BT
1−1/σ
cos φ
.
V
β
(t, s) ≤ K
1
+K
3
e
K
2
β
1/(σ−1)
0 ≤ s ≤ t ≤ T
K
1
K
2
K
3
β t s
β V
β
(t, s) ≤ K

3
e

K
2
β
−1/(σ−1)
0 ≤ s ≤ t ≤ T
K

3
K
3
−A
θ ∈ (π/4; π/2] 0 ∈ ρ(A) β ∈ (0, 1)
f
β
(t; A) 0 ≤ t ≤ T
f
β
(t, A) = a(t)A(I + βA)
−1
.
v
β
(t) =
V
β
(t; s)χ, χ ∈ X V
β
(t; s) 0 ≤ s ≤ t ≤ T
V
β

(t, s) ≤ e
CT/β
, 0 ≤ s ≤ t ≤ T
C β t s
−A 1/β ∈
ρ(−A)
(I + βA)
−1
= (1/β)(1/β)I − (−A))
−1
 ≤ 1/β ×Cβ = C
C β f
β
(t; A)
X t ∈ [0; T]
f
β
(t, A) = a(t)A(I + βA)
−1

= a(t)
1
β
(I − (I + βA)
−1

≤
B
β
(I+ (I + βA)

−1
)
≤
B(1 + C)
β
,
B = max
t∈[0;T ]
|a(t)| t → f
β
(t; A)
A(I + βA) a(t)
v
β
(t) χ ∈ X v
β
(t)
v
β
(t) = V
β
(t; s)χ V
β
(t; s)
V
β
(t, s) ≤ e

t
s

f
β
(τ,A)dτ
−A
θ 0 ∈ ρ(A) β ∈ (0, 1) f
β
(t; A)
0 ≤ t ≤ T
V
β
(t; s) 0 ≤ s ≤ t ≤ T
f
β
(t; A) 0 ≤ t ≤ T ∈ X v
β
(t) = V
β
(t; s)χ