class="bi x0 y0 w1 h1"
class="bi x0 y1 w1 h2"
class="bi x1 y2 w2 h3"
class="bi x1 y3 w3 h4"
class="bi x1 y4 w2 h5"
class="bi x1 y5 w3 h6"
T
T
Ω R
N
, N ≥ 2
∂
Ω
∂u
∂t
− div(|x|
−pγ
|∇u|
p−2
∇u) + f(t, u) = g(x, t), x ∈ Ω, t > τ
u|
t=τ
= u
τ
(x), x ∈ Ω, (1.1)
u|
∂
Ω
= 0,
τ ∈ R, u
τ

2
(Ω)) L
2
c
(R; L
2
(Ω))
L
2
loc
(R; L
2
(Ω))
2N
N + 2
≤ p ≤ 2
N
p
−
N
2
≤ γ + 1 <
N
p
.
f(t, u) = |u|
q−2
u.arctant, q ≥ 2
L
2

L
p
(Ω)
=


Ω
|u|
p
dx

1/p
.
L
p
(Ω) 1 < p < +∞.
• L
∞
(Ω)
Ω
||u||
L
∞
(Ω)
:= ess sup
x∈Ω
|u(x)|.
• 1 < p < ∞ γ <
N − p
p

p
γ
(Ω)


L
p
γ
(Ω)
L
p

−γ
(Ω)
1
p
+
1
p

= 1
• D
1,p
0,γ
(Ω) C
∞
0
(Ω)
||u||
D


−γ
(Ω).
X
• C([a, b]; X)
u : [a, b] → X [a, b] X
||u||
C([a,b];X)
= sup
t∈[0,T ]
||u(t)||
X
.
• L
p
(a, b; X)
u : (a, b) → X
||u||
L
p
(a,b;X)
:=

b

a
||u(t)||
p
X
dt

X
S(t) t > 0, S(t)
S(t) = S
(1)
(t) + S
(2)
(t),
S
(1)
(t) S
(2)
(t)
B ⊂ X
r
B
(t) = sup
y∈B
||S
1
(t)y||
X
→ 0 khi t → +∞;
B X t
0
[γ
(2)
(t
0
)B] =


dist(S(t)u, K) = ||S(t)u − S
(2)
(t)u||.
S
(1)
(t)u = S(t)u − S
(2)
(t)u S(t)
(1.3.4)
X S(t)
B
S(t)
S(t) AK {x
k
}
X t
k
→ ∞, {S(t
k
)x
k
}
∞
k=1
X
K ⊂ X
dist(S(t)B, K) → 0 khi t → ∞.
A X
S(t)
A

u(t)
A
1.3.9
u(t)
{
n
}
∞
n=1

n
→ 0,
{t
n
}
∞
n=1
t
n+1
− t
n
→ ∞ khi n → ∞,
{v
n
}
∞
n=1
v
n
∈ A

L
r
(Ω)
{S(t)}
t≥0
L
r
(Ω)
 > 0 B ⊂ L
r
(Ω)
T = T (B) M = M()
mes(Ω(|S(t)u
0
| ≥ M)) ≤ ,
u
0
∈ B t ≥ T mes(e)
e ⊂ Ω Ω(|S(t)u
0
| ≥ M) := {x ∈ Ω/|S(t)u
0
(x)| ≥ M}.
X
{S(t)}
t≥0
X X
{x
n
}

t≥0
X
{S(t)}
t≥0
X × R
+
X
{S(t)}
t≥0
(C) X B X
 > 0 t
B
X
1
X {P S(t)x/x ∈ B, t ≥ t
B
}
|(I − P)S(t)x| ≤  t ≥ t
B
x ∈ B,
P : X → X
1
{S(t)}
t≥0
L
q
(Ω) L
r
(Ω)
r ≤ q L

t≥0
X
{S(t)}
t≥0
X
{S(t)}
t≥0
(C) X
E
ϕ ∈ L
2
loc
(R; E)
||ϕ||
2
L
2
b
= ||ϕ||
L
2
b
(R;E)
= sup
t∈R
t+1

t
||ϕ||
2

(R; E) L
2
n
(R; E)
L
2
loc
(R; E)
L
2
c
(R; E) ⊂ L
2
n
(R; E) ⊂ L
2
b
(R; E).
H
ω
(g) {g(. + h)/h ∈ R} L
2
b
(R; L
2
(Ω))
[28, 4.2 ]
σ ∈ H
ω
(g), ||σ||


τ
e
−γ(t−τ)
||ϕ||
2
E
ds = 0 ϕ ∈ H
ω
(g).

X, Y
Y X {U
σ
(t, τ)/t ≥ τ, τ ∈ R}, σ ∈

X σ ∈

, {U
σ
(t, τ)/t ≥ τ, τ ∈ R}
X X
U
σ
(t, s)U
σ
(s, τ) = U
σ
(t, τ) t ≥ s ≥ τ, τ ∈ R,
U

B ∈ B(X), lim
t→+∞
sup
σ∈

dist
Y
(U
σ
(t, τ)B, P ) = 0
{U
σ
(t, τ)}
σ∈

X
U
σ
(t + h, τ + h) = U
T (h)σ
(t, τ) {T (h)/h ≥ 0}

T (h)

=

h ∈ R
+

{U


= ω
τ,σ
(B
0
) =
 
σ∈


s≥t
U
σ
(s, τ)B
0
,
B
0
(X, Y ) {U
σ
(t, τ)}
σ∈

.
•
L
p
(τ, T ; D
1,p
0,γ

1,p
0,γ
(Ω)
dt

1
p
=

T

τ

Ω
|x|
−pγ
|∇u|
p
dxdt

1
p
L
p
(τ, T ; D
1,p
0,γ
(Ω)) L
p


−∆
p,γ
∀u, v, w ∈ D
1,p
0,γ
(Ω),
λ →< −∆
p,γ
(u + λv), w > R → R
−∆
p,γ
< −∆
p,γ
u + ∆
p,γ
v, u − v >≥ 0, ∀u, v ∈ D
1,p
0,γ
(Ω).
ε
ϕ ∈ L
2
loc
(R; ε)
||ϕ||
2
L
2
b
= ||ϕ||

loc
(R; ε).
L
2
c
(R; ε) ⊂ L
2
b
(R; ε)
• H(g) {g(. + h)/h ∈ R} L
2
b
(R; L
2
(Ω)).
H(g)
σ ∈ H(g), ||σ||
2
L
2
b
≤ ||g||
2
L
2
b
;
{T (h)}
T (h)σ(s) = σ(h + s), s, h ∈ R H(g)
T (h)H(g) = H(g) h ≥ 0.


, q

p, q
1 ≤ p, q ≤ ∞,
1
p
+
1
q
= 1
u ∈ L
p
(Ω), v ∈ L
q
(Ω)

Ω
|uv|dx ≤ ||u||
L
p
(Ω)
.||v||
L
q
(Ω)
.
1 < p, q < ∞,
1
p

t

0
e
G(t)−G(s)
h(s)ds,
0 ≤ t ≤ T,
G(t) =
t

0
g(r)dr.
a b
dx
dt
≤ ax + b,
x(t) ≤ (x(0) +
b
a
)e
at
−
b
a
.
x, a
dx
dt
≤ ax + b
t+r

||u||
p
≤ C||∇u||
p
.
1 < p < N
C
N,p,γ,q
∀u ∈ C
∞
0
(R
N
),


R
N
|x|
−δq
|u(x)|
q
dx

p
q
≤ C
N,p,γ,q

R