Tập hút toàn cục đối với một lớp phương trình Parabolic phi tuyến chứa toán tử Caffarelli-Kohn-Nirenberg

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

∈ L
2
(Ω) p, γ
f : R × R → R
|f(t, u)| ≤ C
1
|u|
q−1
+ k
1
(1.2)
uf(t, u) ≥ C
2
|u|
q
− k
2
(1.3)
q ≥ 2, C
1
, C
2
, k
1
, k
2
g ∈ L
2
c
(R; L

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

loc
(R; L
2
(Ω))
D
1,p
0,Ω
(Ω)
γ = 0, p = 2
p = 2 γ = 0, p = 2
g = 0 p = 2
f

u
(t, u) ≥ −C
3
t > τ u ∈ R
(f(t, u) − f(t, v)).(u − v) ≥ −C |u − v|
2
t > τ u, v ∈ R
•
•
•
•
• L
p
(Ω)
Ω
||u||

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

L
p
γ
(Ω) =

u : Ω → R |x|
−γ
u(x) ∈ L
p
(Ω)

||u||
L
p
γ
(Ω)
:=


Ω
|x|
−pγ
|u(x)|
p
dx

1/p
.

L

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

1,p
0,γ
(Ω)
= ||∇u||
L
p
γ
(Ω)
=


Ω
|x|
−pγ
|∇u(x)|
p
dx

1
p
.
1 < p < ∞ D
1,p
0,γ
(Ω)
D
1,p
0,γ
(Ω) D
−1,p


−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


1/p
< +∞.
X
X
S(t) : X → X, t ≥ 0
S(0) = I, I
S(t)S(s) = S(s)S(t) = S(t + s),
S(t)u
0
(t, u
0
) ∈ [0; +∞) × X
Y ⊂ X S(t)Y ⊂
Y, ∀t ≥ 0.
Y ⊂ X S(t)Y ⊃ Y, ∀t ≥ 0.
Y ⊂ X S(t)Y = Y, ∀t ≥ 0.
S(t)
B
0
⊂ X
S(t) B
0
⊂ X
B ⊂ X T = T(B) ≥ 0 S(t)B ⊂ B
0
, ∀t ≥ T
B
0
(X, S(t)).

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] =



t≥t
0
S
(2)
tB

X [γ] γ.
S
(1)
(t) ≡ 0 S(t) 1.3.4
1.3.4
K X
B ⊂ X t
0
(B) S
(2)
(t)B ⊂ K, ∀t ≥ t
0
(B).
S(t)
K
lim
t→+∞
dist(S(t)B, K) = 0,
B X.
K t > 0 u ∈ X
v := S
(2)
(t)u ∈ K

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

A S(t)A = A t > 0;
A B X
lim
t→∞
dist(S(t)A, B) = 0,
dist(E, F ) = sup
a∈E
inf
b∈F
d(a, b)
E F X
S(t) A
B X B ⊂ A
B X A ⊂ B
A
(X, S(t))
A
A S(t)
A A
(X, S(t))
A u(t) = S(t)u
0
 > 0
T > 0 τ = τ(, T )
v
0
∈ A
||u(τ + t) − S(t)v
0
|| ≤  0 ≤ t ≤ T.

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

||u(t) − S(t − t
n
)v
n
|| ≤ 
n
t
n
≤ t ≤ t
n+1
.
||v
n+1
− S(t
n+1
− t
n
)v
n
|| n → ∞.
S(t)
X S(t)
B S(t) A = ω(B)
S(t)
A X.
S(t) B
S(t)
A = ω(B)
{S(t)}
t≥0

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
}

∞
n=1
⊂ X, x
n
→ x, t
n
≥ 0, t
n
→ t, S(t
n
)x
n
 S(t)x
X.
X Y X
∗
Y
∗
X
Y i : X → Y
i
∗
: Y
∗
→ X
∗
{S(t)}
t≥0
X Y S(t)
Y {S(t)}

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

r
(Ω) {S(t)}
t≥0
L
q
(Ω)
{S(t)}
t≥0
L
q
(Ω)
 > 0 B L
q
(Ω)
M = M(, B) T = T (, B)

Ω(|S(t)u
0
|≥M)
|S(t)u
0
|
q
< ,
u
0
∈ B t ≥ T.
X {S(t)}
t≥0
X {S(t)}

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

E
ds < ∞.
ϕ ∈ L
2
loc
(R; E)
{ϕ(. + h)/h ∈ R} L
2
loc
(R; E).
ϕ ∈ L
2
loc
(R; E)
ε > 0 η > 0
sup
t∈R
t+η

t
||ϕ||
2
E
ds < ε.
L
2
b
(R; E), L
2
c

(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||

2
L
2
b
≤ ||g||
2
L
2
b
;
{T (h)} H
ω
(g);
T (h)H
ω
(g) = H
ω
(g) h ∈ R;
H
ω
(g)
g ∈ L
2
n
(R; E) τ ∈ R;
lim
γ→+∞
sup
t≥τ
t


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

σ
(τ, τ) = Id, τ ∈ R.

σ ∈

B(X) X
B
0
∈ B(Y ) (X, Y )
U
σ
(t, τ)
σ∈

τ ∈ R
B ∈ B(X) t
0
= t
0
(τ, B) ≥ τ

σ∈

U
σ
(t, τ)B ⊂ B
0
t ≥ t
0
P ⊂ Y (X, Y ) τ ∈ R

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

σ
(t, τ)}
σ∈

(X ×

, Y )
t ≥ τ τ ∈ R (u, r) → U
σ
(t, τ)u
X ×

Y
{U
σ
(t, τ)}
σ∈

(X, Y )
(X, Y ) {U
σ
(t, τ)}
σ∈

(X, Y )
A

Y X
Y
A


= 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

(Ω)) =

u(., .) : Ω × (τ, T ) → R u(., t) ∈ D
1,p
0,γ
(Ω)
t ∈ (τ, T ), ||u(., t)||
D
1,p
0,γ
(Ω)
∈ L
p
(τ, T )

||u||
L
p
(τ,T;D
1,p
0,γ
(Ω))
=

T

τ
||u(., t)||
p
D

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


(τ, T ; D
−1,p

−γ
(Ω)).
−∆
p,γ
u = −div(|x|
−pγ
|∇u|
p−2
∇u), u ∈ D
1,p
0,γ
(Ω).
•
−∆
p,γ
.
−∆
p,γ
D
1,p
0,γ
(Ω)
D
−1,p

−γ
(Ω).

−∆
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||

L
2
b
(R;ε)
= sup
t∈R
t+1

t
||ϕ||
2
ε
ds < ∞.
ϕ ∈ L
2
loc
(R; ε)
{φ(. + h)/h ∈ R} L
2
loc
(R; ε).
L
2
b
(R; ε) L
2
c
(R; ε)
L
2

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
τ,T
= Ω × (τ, T )
V = L
p
(τ, T ; D
1,p
0,γ
(Ω)) ∩ L
q
(τ, T ; L
q
(Ω)),
V

= L
p

(τ, T ; D
−1,p

−γ
(Ω)) + L
q

(τ, T ; L
q

(Ω),
p


, 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

+
1
q
= 1
ab ≤
a
p
p
+
b
q
q
, (a, b > 0).
ab ≤
a
2
2
+
b
2
2
.
x(t)
dx
dt
≤ g(t)x + h(t), ,
g(t) h(t)
x(t) ≤ x(0)e
G(t)
+

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


t
x(s)ds ≤ X,
t+r

t
a(s)ds ≤ A
t+r

t
b(s)ds ≤ B
r > 0 t ≥ t
0
x(t) ≤

X
r
+ B

e
A
t ≥ t
0
+ r.
Ω
R
N
, u|
∂Ω
= 0, t > τ.

||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

N
|x|
−pγ
|∇u(x)|
p
dx