class="bi x0 y0 w1 h1"

class="bi x0 y2 w2 h3"
A(u) = f,
A : E → F E
F f F
A(u) = f
A(u) = f f ∈ F
u f u = R(f)
(E, F ) ε > 0
δ(ε) > 0 f
1
− f
2
 ≤ δ(ε) u
1
− u
2
 ≤ ε
u
i
= R(f
i
), u
i
∈ E, f
i
∈ F, i = 1, 2.
A(u) = f
f
f

|
y=0
= ϕ(x), −∞ < x < +∞,
f(x) ϕ(x)
f(x) = f
1
(x) ≡ 0 ϕ(x) = ϕ
1
(x) =
1
a
sin(ax) (a > 0)
u
1
(x, y) =
1
a
2
sin(ax) (ay).
f(x) = f
2
(x) = ϕ(x) = ϕ
2
(x) ≡ 0
u
2
(x, y) ≡ 0
sup
f
1

2
 = sup
x∈R
|u
1
(x, y) − u
2
(x, y)|
= sup
x∈R
|
1
a
2
sin(ax) (ay)|
=
1
a
2
(ay),
y > 0

u
t
+ Au = 0, 0 < t < T,
u(T ) − f   ε
{φ
i
}
i1

t
+ Av = 0, 0 < t < T,
v(T ) =
1
λ
n
φ
n
v(t) = 0, 0  t  T

v
t
+ Av = 0, 0 < t < T,
v(T ) = 0.
v
n
(T ) − v(T ) =



1
λ
n
φ
n



=
1

n
e
λ
n
(T −t)
→ +∞ n → +∞
t = T
Au = f
A X
Y f Y
f
f
M X
X
Y  · 
X
 · 
Y
M u ∈ M
f ω
ω(0) = 0
u
X
 ω(f 
Y
).
M
ω(t) = ct
α
α > 0

1
) − F (t, w
2
) ≤ kw
1
− w
2
,
F
2
(t, w)  k
1
w
2
+ k
2
w
2
x
.
M M
1

|a
t
|  M
a, |b|, |b
x
|  M
1

d
dt
h
2
= 2

d
c
hh
t
dx = 2

d
c
h ((ah
x
)
x
− (bh)
x
+ G(h)) dx
= 2

d
c
h (ah
x
)
x
dx − 2

bh
2



d
c
−

d
c
bhh
x
dx

+ 2

d
c
hG(h)dx
= −2

d
c
ah
2
x
dx −

d

dx + 2hG(h)  (M
1
+ 2k)h
2
,
d
dt

h(t)
2
e
−(M
1
+2k)t

 0.
ε t
h(t)
2
 h(ε)
2
e
(M
1
+2k)(t−ε)
.
h(ε)
u(x, ε) = g(x) +

ε


2
dx.
h(ε)
2
 3

d
c


ε
0
u
t
(x, t)dt

2
dx +
3
2

d
c


ε
εf
v
t

t
(x, t)dtdx
= 3ε

d
c

ε
0
u
2
t
(x, t)dtdx +
3
2
ε(1 − f)

d
c

ε
εf
v
2
t
(x, t)dtdx
 3ε


ε

=

ε
0

d
c
u
2
t
dxdt,
J
2
=

d
c

ε
εf
v
2
t
dtdx.
J
1
J
2
J
1


d
c
au
x
u
xt
dxdt +

ε
0

d
c
u
t
(−(bu)
x
+ F (t, u)) dxdt
= −
1
2

ε
0

d
c

au

+ F (t, u)) dxdt
= −
1
2

d
c
au
2
x
|
t=ε
dx +
1
2

d
c
ag
2
x
dx +
1
2

ε
0

d
c


ε
0

d
c
a
t
u
2
x
dxdt +
1
2

ε
0

d
c
u
2
t
dxdt
+
1
2

ε
0

u
2
x
dxdt +
1
2

ε
0

d
c
u
2
t
dxdt
+
3
2

ε
0

d
c

b
2
u
2

d
c
u
2
x
dxdt +
1
2
J
1
+
3
2

ε
0

d
c

M
2
1
u
2
x
+ M
2
1
u

2


ε
0

d
c
u
2
x
dxdt +
1
2
J
1
+
3
2

M
2
1
+ k
1


ε
0


d
c
au
2
x
dxdt
+ 3(M
2
1
+ k
1
)

ε
0

d
c
u
2
dxdt.
d
dt

d
c
u
2
dx = 2


dx + 2

d
c
uF (t, u)dx
 −2

d
c
au
2
x
dx −

d
c
b
x
u
2
dx +
k
2
+ 1
a
0

d
c
u

0
k
2
k
2
+ 1

d
c
u
2
x
dx +

M
1
+
k
2
+ 1
a
0
+
a
0
k
1
k
2
+ 1

0
+ a
0
k
1


d
c
u
2
dx
= −

d
c
au
2
x
dx +

M
1
+
k
2
+ 1
a
0
+ a

g
2
dx + C
0

ε
0

d
c
u
2
dxdt,
C
0
= M
1
+
k
2
+ 1
a
0
+ a
0
k
1
.

ε

c
g
2
x
dx +
1
a
0

M + 3M
2
1
+ 3k
2


d
c
g
2
dx + C
1

ε
0

d
c
u
2

u
2
dx  C
0

d
c
u
2
dx.
0 t

d
c
u
2
(t)dx 

d
c
g
2
dx + C
0

t
0

d
c

t



d
c
g
2
dxe
−C
0
t


d
c
g
2
dx.
0 ε
ψ(ε)  ε


d
c
g
2
dx

e


d
c
g
2
x
dx +

1
a
0

M + 3M
2
1
+ 3k
2

+ C
1
εe
C
0
ε


d
c
g
2

)
x
− (bv)
x
+ F (t, v)) dtdx
= −

d
c

ε
εf
av
tx
v
x
dtdx +

Σ
av
t
v
x
n
x
dσ +

d
c


a
t
v
2
x
dtdx +

Σ
av
t
v
x
n
x
dσ
+

d
c

ε
εf
v
t
(−(bv)
x
+ F (t, v)) dtdx
 −
1
2

dσ
+

d
c

ε
εf
v
t
(−bv
x
− b
x
v + F(t, v)) dtdx


Σ
av
t
v
x
n
x
dσ −
1
2

Σ
av

t
dtdx +
1
2

d
c

ε
εf
(−bv
x
− b
x
v + F(t, v))
2
dtdx


Σ
av
t
v
x
n
x
dσ −
1
2


v
2
t
dtdx +
3
2

d
c

ε
εf

b
2
v
2
x
+ b
2
x
v
2
+ F
2
(t, v)

dtdx
J
2

d
c

ε
εf
v
2
x
dtdx
+
3
2

M
2
1
+ k
1


d
c

ε
εf
v
2
dtdx +
1
2

+ 3k
2


d
c

ε
εf
av
2
x
dtdx
+ 3

M
2
1
+ k
1


d
c

ε
εf
v
2
dtdx.

d
c

ε
εf
v((v
t
+ (bv)
x
− F (t, v)) dtdx +

Σ
av
x
gn
x
dσ
= −

d
c

ε
εf
vv
t
dtdx −

d
c

1
2

d
c

ε
εf
b
x
v
2
dtdx −
1
2

Σ
bg
2
n
x
dσ +

Σ
av
x
gn
x
dσ
+

c

ε
εf
av
2
x
dtdx
 −
1
2

Σ
g
2
n
t
dσ −
1
2

Σ
bg
2
n
x
dσ +

Σ
av

εf
v
2
dtdx +
a
0
k
2
2(k
2
+ 1)

d
c

ε
εf
v
2
x
dtdx


Σ
av
x
gn
x
dσ −
1

dtdx +
1
2

d
c

ε
εf
av
2
x
dtdx.

d
c

ε
εf
av
2
x
dtdx  2

Σ
av
x
gn
x
dσ −

t
v
x
n
x
dσ −

Σ
av
2
x
n
t
dσ + C
1

d
c

ε
εf
v
2
dtdx
+
1
a
0

M + 3M


d
c

ε
εf
v
2
dtdx =

d
c

ε
εf


t
εf
v
s
ds + g

2
dtdx
≤ 2

d
c


t
εf
v
2
s
dsdtdx + 2ε

d
c
(1 − f)g
2
dx
 2

d
c
(ε − εf)
2

ε
εf
v
2
t
dtdx + 4ε

d
c
g
2

J
2
 2

Σ
av
t
v
x
n
x
dσ −

Σ
av
2
x
n
t
dσ + 8C
1
ε
2
J
2
+ 4C
1
ε

d

dσ −

Σ
bg
2
n
x
dσ

.
ε > 0
1 − 8C
1
ε
2
 α
1
> 0.
α
1
J
2
 2

Σ
av
x
(v
t
n

2
)

2

Σ
av
x
gn
x
dσ −

Σ
g
2
n
t
dσ −

Σ
bg
2
n
x
dσ


1
α
2

c
g
2
dx
+
1
a
0
(M + 3M
2
1
+ 3k
2
)

α
3

Σ
av
2
x
dσ +
1
α
3

Σ
ag
2

x
n
t
)
2
dσ +

α
2
+
α
3
a
0
(M + 3M
2
1
+ 3k
2
) − δ


Σ
av
2
x
dσ
+
1
a

n
x
dσ

+ 4C
1
ε

d
c
g
2
dx,
α
2
, α
3
α
2
, α
3
α
2
+
α
3
a
0
(M + 3M
2

a
0
(M + 3M
2
1
+ 3k
2
)

1
α
3

Σ
ag
2
n
2
x
dσ −

Σ
g
2
n
t
dσ −

Σ
bg

+
1
α
1
Q
2

.
h(t)
2
 3ε

Q
1
+
1
α
1
Q
2

e
(M
1
+2k)(t−ε)
.
∂u
∂t
= − (a(x, t)u
x

2
,
F
2
(t, w)  k
1
w
2
+ k
2
w
2
x
.
M M
1
M
2
M
3

|a
t
|  M
a, |b|, |b
x
|  M
1

−

2
(t)
ε > 0
h(t)
2
 ε
δ
1
(t)
F
2
(t),
δ
1
(t) = 1 −
e
−λ
1
ε
− e
−λ
1
t
e
−λ
1
ε
− e
−λ
1

hG(h)dx

d
c
h
2
dx
.
|l(t)|  M
1
+ 2k.
ω = h exp

1
2

T
t
l(s)ds

.
∂ω
∂t
= −(aω
x
)
x
+ (bω)
x
−

c
ω

−(aω
x
)
x
+ (bω)
x
−
1
2
l(t)ω +

G(ω)

dx
= 2

d
c
aω
2
x
dx +

d
c
b
x

d
c
hG(h)dx

d
c
h
2
dx

d
c
h
2
exp


T
t
l(s)ds

dx
=

d
c
b
x
ω
2


d
c
ω
2
dx
.
ϕ = (bω)
x
−
1
2
l(t)ω +

G(ω)
1
2


d
c
ω
2
dx

2
d
dt

Φ

d
c
aω
2
x
dx

d
c
ωω
t
dx
=


d
c
a
t
ω
2
x
dx − 2

d
c
(aω
x
)
x

c
a
t
ω
2
x
dx + 2

d
c
(aω
x
)
x
((aω
x
)
x
− ϕ) dx


d
c
ω
2
dx
− 2

d
c

ω
2
dx + 2

d
c

(aω
x
)
x
−
1
2
ϕ

2
dx

d
c
ω
2
dx
−
1
2

d
c


d
c
ϕωdx

2
.
2

d
c

(aω
x
)
x
−
1
2
ϕ

2
dx

d
c
ω
2
dx − 2



(t)
Φ(t)



2

d
c
a
t
ω
2
x
dx −

d
c
ϕ
2
dx


d
c
ω
2
dx,


2
=

(bω)
x
−
1
2
l(t)ω +

G(ω)

2
=

bω
x
+ b
x
ω −
1
2
l(t)ω +

G(ω)

2

3
2

1
2
l(t)

2
ω
2
+ 6

G
2
(ω)

3
2
M
2
1
ω
2
x
+ 6

3
2
M
1
+ k

2

1


d
c
ω
2
x
dx − 6

3
2
M
1
+ k

2

d
c
ω
2
dx
− 6

d
c

G
2

l(s)ds

= k
2

d
c
ω
2
dx,


d
c
ω
2
dx

d
dt

Φ

(t)
Φ(t)

 −

2M +
3


(t)
Φ(t)


−
2
a
0

M +
3
4
M
2
1


d
c
aω
2
x
dx

d
c
ω
2
dx

1
a
0

M +
3
4
M
2
1


λ
1
= 6

3
2
M
1
+ 2k

2
Φ(t)  e
−

λ
1
t/λ
1

− e
−λ
1
t
e
−λ
1
ε
− e
−λ
1
T
.
Φ(t) = ω
2
= h
2
exp


T
t
l(s)ds

l(t)  M
1
+2k
h(t)
2


2
 3ε(J
1
+ J
2
),
J
1
=

ε
0

d
c
u
2
t
dxdt J
2
=

d
c

ε
εf
v
2
t

)
x
+ (bu)
x
+ F (t, u)) dxdt
=

T
0

d
c
aρu
xt
u
x
dxdt +

T
0

d
c
ρu
t
((bu)
x
+ F (t, u)) dxdt
=
1

c
ρu
t
((bu)
x
+ F (t, u)) dxdt
=
1
2

T
0

d
c

aρu
2
x

t
dxdt −
1
2

T
0

d
c


T
0

d
c
(aρu
2
x
)
t
dxdt = −

d
c
aρu
2
x
|
t=0
dx  0

T
0

d
c
ρu
2
t

2

T
0

d
c
ρu
2
t
dxdt +
1
2

T
0

d
c
ρ (bu
x
+ b
x
u + F (t, u))
2
dxdt
 −
1
2


d
c
ρu
2
t
dxdt +
3
2

T
0

d
c
ρ

b
2
u
2
x
+ b
2
x
u
2
+ F
2
(t, u)



T
0

d
c
au
2
x
dxdt
+ 3

T
0

d
c
ρ

b
2
u
2
x
+ b
2
x
u
2
+ F

+
3T
T − ε

T
0

d
c

M
2
1
u
2
x
+ M
2
1
u
2
+ k
1
u
2
+ k
2
u
2
x

au
2
x
dxdt
+
3T
T − ε

M
2
1
+ k
1


T
0

d
c
u
2
dxdt

T
a
0
(T − ε)

M + 3M

T
0

d
c
u
2
dxdt.

T
0

d
c
au
2
x
dxdt = −

T
0

d
c
u(au
x
)
x
dxdt =


u(bu)
x
dxdt −

T
0

d
c
uF (t, u)dxdt
=
1
2

d
c
(u
2
)


T
0
dx −
1
2

T
0


u
2
dxdt +
k
2
+ 1
2a
0

T
0

d
c
u
2
dxdt
+
a
0
2(k
2
+ 1)

T
0

d
c
F

2
+ 1
2a
0

T
0

d
c
u
2
dxdt
+
a
0
2(k
2
+ 1)

T
0

d
c
(k
1
u
2
+ k



T
0

d
c
u
2
dxdt
+
a
0
k
2
2(k
2
+ 1)

T
0

d
c
u
2
x
dxdt

1

T
0

d
c
au
2
x
dxdt  u(T )
2
+ C
0

T
0

d
c
u
2
dxdt.
J
1

T
(T − ε)a
0

M + 3M
2

1
+ 3k
2
+
a
0
T

C
0
+ 3(M
2
1
+ k
1
)
T
T − ε
.
d
dt

d
c
u
2
dx = 2

d
c

c
au
2
x
dx − M
1

d
c
u
2
dx −
k
2
+ 1
a
0

d
c
u
2
dx −
a
0
k
2
+ 1

d


d
c
u
2
dx −
a
0
k
2
k
2
+ 1

d
c
u
2
x
dx


2 −
k
2
k
2
+ 1



c
u
2
dx.
d
dt


d
c
u
2
dx

e
C
0
t

 0.
t T

d
c
u
2
dx 


d