Алексеев Г.В. Классические методы математической физики. Часть 2
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u(x
0
, t
0
) = M
v(x, t) = u(x, t) +
M − m
6d
2
(x − x
0
)
2
+ (y − y
0
)
2
+ (z − z
0
)
2
,
u d Ω
Σ
T
Q
T
Ω
0
v(x, t) ≤ m +
M − m
6
=
M
6
+
5m
6
< M.
v(x
0
, t
0
) = u(x
0
, t
0
) = M v
u Σ
T
Q
T
t = 0
v
(x
1
, t
1
) = (x
1
, y
1
, z
1
, t
1
) ∈ Q
T
= Ω × (0, T ] x
1
Ω 0 < t
1
≤ T (x
1
, t
1
)
v
gradv = 0,
∂
2
v
∂x
2
≤ 0,
∂
2
v
∂y
2
≤ 0,
∂
2
v
∂z
2
≤ 0,
∂v
∂t
≥ 0.
t
1
< T ∂v/∂t = 0
(x
1
, t
1
)
∂v
∂t
− a
2
∆v ≥ 0.
(x, t) ∈ Ω × (0, T ]
∂v
∂t
− a
2
∆v =
∂u
∂t
− a
2
∆u − a
2
M − m
d
2
= −a
2
M − m
d
2
< 0.
u ∈ C
2,1
(Q
T
) ∩ C(Q
T
)
(x, t) ∈ Q
T
(x, t) ∈ Γ × (0, T ]
x ∈ Ω
u
u
g = 0 ϕ = 0
Q
T
u Σ
T
Ω
0
u ≡ 0
u
1
u
2
u = u
1
− u
2
t = 0 Γ ×(0, T ]
u ≡ 0 Q
T
u
1
= u
2
C(Q
T
) g
ϕ
g
1
− g
2
g
1
g
2
ϕ
1
− ϕ
2
ϕ
1
ϕ
2
ε > 0 |g
1
−g
2
| ≤ ε Γ ×[0, T ] |ϕ
1
−ϕ
2
| ≤ ε
Ω t = 0 u
1
u
2
(g
1
, ϕ
1
) (g
2
, ϕ
2
) u = u
1
− u
2
g
1
−g
2
ϕ
1
− ϕ
2
|u| ≤ ε
T = ∞
R
n
R
n
u
u ∈ C(Q
T
) u
ϕ ∈ C(Ω) g ∈ C(Γ × [0, T ]) ϕ(x) = g(x, 0) ∀x ∈ Γ
u
n = 1
Ω ⊂ R
n
R
n
Ω
R
n
Γ Q
∞
= Ω×(0, ∞ )
Q
∞
= Ω × [0, ∞) Σ
∞
= Γ × (0, ∞) Σ
∞
= Γ × [0, ∞) Q
T
= Ω × (0, T ]
T < ∞ Q
∞
ρ
∂u
∂t
− div(pgradu) + qu = f(x, t).
ρ p q f
ρ ∈ C(Ω) p ∈ C
1
(Ω) q ∈ C(Ω) ρ > 0 p > 0 q ≥ 0 Ω f ∈ C(Q
∞
)
q = 0
p
ρ = 1
p
q ≥ 0
u ∈ C
2,1
(Q
∞
) ∩C(Q
∞
)
Q
∞
T > 0 f(x, t) ≤ 0 Q
T
u ≤ 0 Q
T
u
Q
T
Ω
0
Q
T
Γ×(0, T ]
u(x, t) ≤ M ≡ max[0, ma x
x∈
Ω
u(x, 0), max
(x,t)∈Γ×[0,T ]
u(x, t)] ∀(x, t) ∈
Q
T
;
f(x, t) ≥ 0 Q
T
u(x, t) ≥ m ≡ m in [0, min
x∈
Ω
u(x, 0), min
(x,t)∈Γ×[0,T ]
u(x, t)],
u
u
Q
T
Ω
0
Σ
T
(x
0
, t
0
), x
0
∈
Ω, 0 < t
0
≤ T
u(x
0
, t
0
) > M > 0.
ε = u(x
0
, t
0
) − M > 0,
v u
v(x, t) = u(x, t) +
ε
2
(2T − t)
T
, (x, t) ∈
Q
T
.
v(x, t) ≤ u(x, t)+ ε ∀(x, t) ∈ Q
T
(x, t) ∈ Σ
T
t = 0
v(x
0
, t
0
) ≥ u(x
0
, t
0
) = ε + M ≥ ε + u(x, t) ≥ ε + v( x, t) − ε = v(x, t).
v
Q
T
(x
1
, t
1
) ∈ Ω × (0, T ]
v(x
1
, t
1
) ≥ v(x
0
, t
0
) ≥ ε + M > ε.
v (x
1
, t
1
)
gradv = 0, ∆v ≤ 0,
∂v
∂t
≥ 0.
f ≤ 0 Q
T
(x
1
, t
1
)
ρ
∂u
∂t
− div(pgradu) + qu − f ( x, t) = ρ
∂v
∂t
− p∆v − (g radp, gradv) + qv − f+
+
ε
2
(
ρ
T
− q
2T − t
1
T
) ≥ qv +
ε
2
(
ρ
T
− q
2T − t
1
T
) ≥ qε
1 −
2T − t
1
2T
+
ερ
2T
> 0.
u Q
∞
T > 0
M
0
= kϕk
C(
Ω)
, M
1
= kgk
C(Γ×[0,T ])
, M = kfk
C(Q
T
)
.
u : Q
∞
→ R
v : Q
T
→ R
v(x, t) = u(x, t) −
M
ρ
0
t, ρ
0
= min
x∈
Ω
ρ(x) > 0.
v
f g
˜
f = f −
ρ
ρ
0
M −
q
ρ
0
Mt
˜g = g −
M
ρ
0
t
˜
f(x, t) ≤ 0 ∀(x, t) ∈
Q
T
, ˜g(x, t) ≤ M
1
∀(x, t) ∈ Γ × [0, T ].
v
v(x, t) ≤ max(M
0
, M
1
) ∀(x, t) ∈ Q
T
.
u
u(x, t) ≤ max(M
0
, M
1
) +
M
ρ
0
T ∀(x, t) ∈
Q
T
.
w(x, t) = u(x, t) + (M/ρ
0
)t
u
u(x, t) ≥ −max(M
0
, M
1
) −
M
ρ
0
T ∀(x, t) ∈
Q
T
.
u
kuk
C(
Q
T
)
≤ max
n
kϕk
C(Ω)
, k gk
C(Γ×[0,T ])
o
+
T
ρ
0
kfk
C(Q
T
)
.
ϕ = 0 g = 0 f = 0
C(Q)
C(Q
T
)
ϕ g f u
(ϕ, g, f) ˜u
( ˜ϕ, ˜g,
˜
f) ˜u−u
k˜u − uk
C(
Q
T
)
≤ max
k˜ϕ − ϕk
C(Ω)
, k˜g −gk
C(Γ×[0,T ])
+
T
ρ
0
k
˜
f − f k
C(Q
T
)
.
C(Q
T
) ϕ
C(Ω) g C(Γ×[0, T ]) f
C(Q
T
)
αu + β
∂u
∂n
Γ
= g
(0, T ].
α β Γ
u Γ
ρ
0
= 1
ρ
∂u
∂t
− div(pgradu) + a · gr adu + qu = f.
a ∈ C
1
(Ω)
diva = 0
R
u ∈
C
2,1
(Q
T
) ∩C(Q
T
) Q
T
= (0, l) × (0, T ]
∂u
∂t
= a
2
∂
2
u
∂x
2
+ f(x, t)
Q
T
u|
x=0
= g
1
(t), u|
x=l
= g
2
(t) (0, T ]
u|
t=0
= ϕ(x) (0, l).
f, g
1
, g
2
ϕ
f
g
1
, g
2
ϕ
n = 1 u
C(Q
T
) C
ϕ g
1
g
2
f
u
u
∂u
∂t
= a
2
∂
2
u
∂x
2
Q
T
u|
x=0
= 0, u|
x=l
= 0 (0, T ]
l
ϕ
u
u ∂u/∂t
u(x, t) = X(x)T (t).
T
T
′
(t) + a
2
λT (t) = 0,
X
′′
+ λX = 0 (0, l), X(0) = X(l) = 0
X
λ
k
X
k
λ
k
=
kπ
l
2
, X
k
(x) = sin
kπ
l
x, k = 1 , 2, ... .
λ
k
λ
T
k
(t) = a
k
e
−λ
k
a
2
t
≡ a
k
exp
−(
kπa
l
)
2
t
, k = 1, 2, ... .
a
k
, k = 1, 2, ...
u
k
(x, t) = T
k
(t)X
k
(x) = a
k
e
−λ
k
a
2
t
sin
kπx
l
, k = 1, 2... .
u
k
k
a
k
u(x, t) =
∞
X
k=1
a
k
e
−λ
k
a
2
t
X
k
(x)
Q
T
= [0, l]×
[0, T ]
x
t Q
T
(x, t) ∈ Q
T
a
k
ϕ(x) =
∞
X
k=1
a
k
sin
kπx
l
.
ϕ
(0, l)
C[0, l] a
k
ϕ
a
k
=
2
l
l
Z
0
ϕ(x) sin
kπx
l
dx, k = 1, 2... .
u
a
k
Q
T
x t Q
T
ϕ
ϕ ∈ C[0, l] ϕ
′
[0, l] ϕ(0) = ϕ(l) = 0
a
k
ϕ [0, l] t ≥ 0
0 < e
−λ
k
a
2
t
≤ 1
a
k
0 ≤ x ≤ l t ≥ 0
u Q
T
u Q
T
−a
2
∞
X
k=1
a
k
kπ
l
2
e
−λ
k
a
2
t
sin
kπx
l
−
∞
X
k=1
a
k
kπ
l
2
e
−λ
k
a
2
t
sin
kπx
l
,
t
x Q
T
t > 0
0 <
k
2
π
2
a
2
l
2
e
−λ
k
a
2
t
< 1, 0 <
k
2
π
2
l
2
e
−λ
k
a
2
t
< 1,
k
C
2,1
(Q
T
) ∩ C(Q
T
)
u
ϕ
t ≥ 0
t = 0
t
u [−T, 0]
t = 0 x = 0
x = 1
u
−
t u
−
t
ϕ
u
−
u
−
k
(x, t) = α
k
e
−λ
k
a
2
t
sin
kπx
l
,
{α
k
}
∞
k=1
u
−
k
(x, 0 ) ≡ α
k
sin
kπx
l
k → ∞
t < 0 u
−
k
(x, t)
k → ∞ u
−
t
t = 0