¨o



∂u
∂t
= a(t)
∂
2
u
∂x
2
, (x, t) ∈ (−∞; +∞) × (0; 1),
u(x, 1) = ϕ(x).
1.
¨o
(1.1)



∂u
∂t
= a(t)
∂
2
u
∂x
2
, (x, t) ∈ (−∞; +∞) × (0; 1),
u(x, 1) = ϕ(x).
ϕ(x)

1
(R) fg
1
 f
p
g
q
.
2 ( L
1
(R) f ∈ L
1
(R)
f

f(ξ) :=
1
√
2π

+∞
−∞
e
−ix.ξ
f(x)dx (y ∈ R).
2 ( f ∈ L
1
(R) ∩L
2
(R)



f
k
−

f
j
 = 

f
k
− f
j
 = f
k
− f
j
 {

f
k
}
∞
k=1
L
2
(R)

f

f ˆgdξ

D
α
f = (iξ)
α

f α D
α
f ∈ L
2
(R)
3.
1 ( u(x, t)
(3.1)







∂u
∂t
= a(t)
∂
2
u
∂x
2

u(x, t)
∂u
∂t
= a(t)
∂
2
u
∂x
2
∂u
∂t
(ξ, t) = −ξ
2
a(t)u(ξ, t).(3.2)
u(ξ, t) = e
A(1)(1−µ(t))ξ
2
u(ξ, 1), (ξ, t) ∈ R × [0, 1].(3.3)
ˆu(·, t) ∈ L
2
(R), t ∈ [0, 1]
|u(ξ, t)|
µ(t)
= e
A(1)µ(t)(1−µ(t))ξ
2
|u(ξ, 1)|
µ(t)
, (ξ, t) ∈ R ×[0, 1].(3.4)
t = 0

t = 0 t = 1
t ∈ (0, 1)
f = |u(·, 1)|
2µ(t)
, g = |u(·, 0)|
2(1−µ(t))
, p =
1
µ(t)
, q =
1
1 − µ(t)
u(·, t)
2
= u(·, t)
2
=

+∞
−∞
|u(ξ, 1)|
2µ(t)
|u(ξ, 0)|
2(1−µ(t))
dξ
=

+∞
−∞
f(ξ)g(ξ)dξ =



+∞
−∞
u
2
(ξ, 1)dξ

µ(t)
.


+∞
−∞
u
2
(ξ, 0)dξ

(1−µ(t))
= u(·, 1)
2µ(t)
.u(·, 0)
2(1−µ(t))
= u(·, 1)
2µ(t)
.u(·, 0)
2(1−µ(t))
 
2µ(t)
E