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Estimativas de Strichartz e a
Equação Não Linear de Schrödinger
em Espaços Euclidianos.
Alex Santana dos Santos
Maceió
Fevereiro de 2009
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Catalogação na fonte
Universidade Federal de Alagoas
Biblioteca Central
Divisão de Tratamento Técnico
Bibliotecária Responsável: Helena Cristina Pimentel do Vale
S237e Santos, Alex Santana dos.
Estimativas de Strichartz e a equação não linear de Schrödinger em espaços
euclidianos / Alex Santana dos Santos. – Maceió, 2009.
83f.
Orientador: Adán José Corcho Fernández.
Dissertação (mestrado em Matemática) – Universidade Federal de Alagoas.
Instituto de Matemática. Maceió, 2009.
Bibliografia: f. 83.
1. Análise harmônica. 2. Estimativas de Strichartz. 3. Schrödinger,Equação de.
I. Título.
CDU: 517.955
•
•
•
•
•
•
•
•
•
L
2
(R
N
)
iu
t
(t, x) + ∆
x
u(t, x) = γ |u(t, x)|
α
u(t, x)
u(x, 0) = ϕ(x) ∈ L
2
(R
N
), x ∈ R
N
, t ∈ R.
u α <
4
N
.
L
2
(R
N
)
iu
t
(t, x) + ∆
x
u(t, x) = γ |u(t, x)|
α
u(t, x)
u(x, 0) = ϕ(x) ∈ L
2
(R
N
), x ∈ R
N
, t ∈ R.
u 0 < α <
4
N
.
L
p
R
N
L
1
(R
N
)
L
2
(R
N
)
L
2
(R
N
)
L
2
H
1
(R
N
)
L
2
(R
N
)
L
2
(R
N
)
(1)
iu
t
(t, x) + ∆
x
u(t, x) = γ |u(t, x)|
α
u(t, x)
u(x, 0) = ϕ(x) ∈ L
2
(R
N
), x ∈ R
N
, t ∈ R,
u α <
4
N
[3] [7]
α =
4
N
[3] [7]
(1)
(1)
Φ(u)(t) = S(t)ϕ + iγ
T
0
S(t − s) |u(s)|
α
uds. (2)
|u(t, x)|
2
u(t, x)
(1)
(1)
[−T, T ] (1)
(3)
iu
t
(t, x) + ∆
x
u(t, x) = 0
u(x, 0) = ϕ(x) ∈ L
2
(R
N
), x ∈ R
N
, t ∈ R.
(3)
S(t)ϕ
L
2
S(t)ϕ
L
p
≤ (4iπ |t|)
−
N
2
(
1−
1
p
)
ϕ
L
p
,
L
2
(R
N
)
(1)
(2)
L
2
(R
N
)
L
2
(R
N
) H
1
(R
N
)
L
2
(R
N
)
L
p
1 ≤ p ≤ ∞ X ⊂ R
N
L
p
(X)
f : X −→ C
f
L
p
=
X
|f(x)|
p
dx
1
p
< ∞, 1 ≤ p < ∞
inf {λ > 0 ; µ(A
λ
) = 0} < ∞, p = ∞,
A
λ
= {x ∈ X ; |f(x)| > λ} µ R
N
p, q ∈ [1, ∞]
1
p
+
1
q
= 1.
p
p 1 e ∞
1 ≤ p ≤ ∞ X ⊂ R
N
f ∈ L
p
(X) g ∈ L
p
fg ∈ L
1
fg
L
1
≤ f
L
p
g
L
p
.
[8] .
1 ≤ p ≤ r ≤ q ≤ ∞
1
r
=
θ
p
+
1 − θ
q
.
f ∈ L
p
(X) ∩ L
q
(X) f ∈ L
r
(X)
f
L
r
≤ f
θ
L
p
f
1−θ
L
q
.
θr
p
+
(1−θ)r
q
= 1
1
p
θr
+
1
q
(1−θ)r
= 1
X
|f|
r
dx =
X
|f|
θr
|f|
(1−θ)r
dx
≤
X
|f|
θr
p
θr
dx
θr
p
X
|f|
(1−θ)r
q
(1−θ)r
dx
(1−θ)r
q
=
X
|f|
p
dx
θr
p
X
|f|
q
dx
(1−θ)r
q
.
1
r
X
|f|
r
dx
1
r
≤
X
|f|
p
dx
θ
p
X
|f|
q
dx
1−θ
q
.
L
p
(X) 1 ≤ p ≤ ∞ X ⊂ R
N
[8] .
L
2
(X)
[8] .
1 ≤ p ≤ ∞ X ⊂ R
N
f ∈ L
p
(X)
f
L
p
= sup
X
f(x)g(x)dx; g(x)
L
p
= 1
.
[2] .
X, Y ⊂ R
N
f : X × Y −→ C
1 ≤ p < ∞
X
Y
|f(x, y)| dy
p
dx
1
p
≤
Y
X
|f(x, y)|
p
dx
1
p
dy.
p = ∞ p < ∞ F (x) =
Y
|f(x, y)| dy
F
L
p
x
= sup
g
L
p
x
=1
X
g(x)
Y
|f(x, y)| dy
dx
= sup
g
L
p
x
=1
Y
X
|f(x, y)| g(x)dxdy
≤ sup
g
L
p
x
=1
Y
f
L
p
x
g
L
p
x
dy
=
Y
f
L
p
x
dy.
X
Y
|f(x, y)| dy
p
dx
1
p
≤
Y
X
|f(x, y)|
p
dx
1
p
dy.
C
0
(R
N
)
1 ≤ p < ∞ C
0
(R
N
) L
p
(R
N
)
[8] .
I ⊂ R p, q ≥ 1 L
p
t
(I; L
q
x
(R
N
))
f : R
N
× I −→ C
f
L
p
t
(I;L
q
x
(R
N
))
=
I
f(·, t)
p
L
q
x
dt
1
p
< ∞.
I = [0, T ] T > 0 I = R
L
p
t
(I; L
q
x
(R
N
)) L
p
T
L
q
x
L
p
t
L
q
x
f : R
N
× I −→ C
f
L
p
t
(R;L
q
x
)
= sup
+∞
−∞
R
N
f(x, t)g(x, t)dtdx; g
L
p
t
(R;L
q
x
)
= 1
.
1.4
f, g ∈ L
1
R
N
f g
(f ∗ g)(x) =
R
N
f(x − y)g(y)dy.
(1.2)
y −→
R
N
f(x − y)g(y)dy L
1
(R
N
) x
f ∗ g
L
1
=
R
N
R
N
f(x − y)g(y)dy
dx
≤
R
N
dx
R
N
|f(x − y)g(y)| dy
=
R
N
dy |g(y)|
R
N
|f(x − y)| dx
= f
L
1
g
L
1
.
f ∗ g = g ∗ f
(f ∗ g) ∗ h = f ∗ (g ∗ h)
(f + g) ∗ h = f ∗ h + g ∗ h
[5]
f ∈ L
1
(R
N
) g ∈ L
q
(R
N
) f ∗ g ∈ L
q
(R
N
)
f ∗ g
L
q
≤ f
L
1
g
L
q
.
q ∈ [1, ∞) p
|f(x − y)g(y)| = |f(x − y)|
1
p
|f(x − y)|
1
q
|g(y)| .
(1.4) y
|f ∗ g| =
R
N
f(x − y)g(y)dy
≤
R
N
|f(x − y)|
1
p
|f(x − y)|
1
q
|g(y)| dy
≤
R
N
|f(x − y)| dy
1
p
R
N
|f(x − y)| |g(y)|
q
dy
1
q
,
x q
R
N
|f ∗ g|
q
dx ≤
R
N
|f(x − y)| dy
q
p
R
N
R
N
|f(x − y)| |g(y)|
q
dydx
= f
q
p
L
1
R
N
|g(y)|
q
dy
R
N
|f(x − y)| dx
= f
q
p
L
1
f
L
1
g
q
L
q
= f
q
L
1
g
q
L
q
.
f ∗ g
L
q
≤ f
L
1
g
L
q
,
f ∈ L
p
(R
N
) g ∈ L
q
(R
N
) f ∗ g ∈ L
r
(R
N
)
1
r
=
1
q
+
1
p
− 1
p, q ∈ [1, ∞)
H A H B(A, H)
T : A −→ H
T = inf {C > 0; T f ≤ C f , ∀f ∈ A}
H A H
(B(A, H), ·)
T = sup
f=0
T f
f
= sup
f=1
T f ∀ T ∈ B(A, H)
T ∈ B(A, H) T ∈ B(A, H) A
A H
T =
T
.
[6]
F
S = {z = x + iy /0 ≤ x ≤ 1} ,
F S y ∈ R
|F (iy)| ≤ M
0
e |F(1 + iy)| ≤ M
1
,
z = x + iy ∈ S
|F (x + iy)| ≤ M
1−x
0
M
x
1
.
F S
|F (z)| ≤ 1 z ∈ ∂S z ∈ S |F (z)| ≤ 1
[4]
M
0
, M
1
> 0 F (z)/M
1−z
0
M
z
1
M
0
= M
1
= 1
|F (iy)| ≤ 1 e |F (1 + iy)| ≤ 1.
|F (z)| ≤ 1 z ∈ S
lim
|y|→∞
F (x + iy) = 0,
0 ≤ x ≤ 1
y
0
> 0 |F (x + iy)| ≤ 1 |y| ≥ y
0
|F (x + iy)| ≤ 1
iy
0
, 1 + iy
0
, −iy
0
, 1 − iy
0
.
F
n
(z) = F (z)e
(z
2
−1)/n
∀ n ∈ N
F
n
0 ≤ x ≤ 1
|F
n
(z)| = |F (x + iy)| e
−y
2
/n
e
(x
2
−1)/n
≤ |F (x + iy)| e
−y
2
/n
−→ 0,
|y| −→ ∞ 0 ≤ x ≤ 1 |F (iy)| ≤ 1 e |F (1 + iy)| ≤ 1
|F
n
(z)| ≤ 1 n −→ ∞
|F (z)| ≤ 1 ∀ z ∈ S.
X ⊂ R
N
Y ⊂ R
N
p
0
= p
1
q
0
= q
1
T ∈ B(L
p
0
(X), L
q
0
(Y )) ∩ B(L
p
1
(X), L
q
1
(Y ))
M
0
= sup
f=0
T f(x)
L
q
0
f
L
p
0
e M
1
= sup
f=0
T f(x)
L
q
1
f
L
p
1
T ∈ B(L
p
θ
(X), L
q
θ
(Y )) M
θ
M
θ
≤ M
1−θ
0
M
θ
1
,
1
p
θ
=
1 − θ
p
0
+
1
p
1
;
1
q
θ
=
1 − θ
q
0
+
1
q
1
θ ∈ (0, 1).
h, g =
Y
h(y)g(y)dν(y).
h
L
q
= sup {|h, g| : g
L
q
= 1}
T
p,q
= sup {|T f, g| : g
L
q
= f
L
p
= 1} ,
1
p
+
1
p
= 1 =
1
q
+
1
q
p, q = ∞ p < ∞ q
< ∞
f, g
f(x) =
j
a
j
X
A
j
(x) e g(x) =
k
b
j
X
B
k
(x),
A
j
j B
k
k
f
p
θ
= g
q
θ
= 1.
0 ≤ Rez ≤ 1
1
p(z)
=
1 − z
p
0
+
z
p
1
e
1
q(z)
=
1 − z
q
0
+
z
q
1
ϕ(z) = ϕ(x, z) =
j
|a
j
|
p
θ
p(z)
e
iarg(a
j
)
p
θ
p(z)
X
A
j
(x)
ψ(z) = ψ(x, z) =
k
|b
k
|
q
θ
q(z)
e
iarg(b
k
)
p
θ
p(z)
X
B
k
(x).
ϕ(z) = ϕ(x, z) ∈ L
p
j
j = 0, 1
|ϕ(z)|
p
=
j
|a
j
|
p
θ
p(z)
e
iarg(a
j
)
X
A
j
(x)
p
=
j
|a
j
|
p
θ
p(z)
e
iarg(a
j
)
X
A
j
(x)
p
=
j
|a
j
|
p
θ
p(Rez)
p
X
A
j
(x),
A
j
X
|ϕ(z)|
p
dx
1
p
=
j
|a
j
|
p
θ
p(z)
p
X
X
A
j
(x)dx
=
j
|a
j
|
p
θ
p(Rez)
p
µ(A
j
) < ∞.
ϕ(z) = ϕ(x, z) ∈ L
p
j
j = 0, 1
ψ(z) = ψ(x, z) ∈ L
q
j
, j = 0, 1 T ϕ(z) ∈ L
q
j
j = 0, 1
ϕ
(z) ∈ L
p
j
ψ
(z) ∈ L
q
j
(T ϕ)
(z) ∈ L
q
j
0 ≤ Rez ≤ 1
F (z) = T ϕ(z); ψ(z) .
F 0 ≤ Rez ≤ 1
F
|ϕ(it)|
p
0
=
j
|a
j
|
p
θ
p(it)
e
iarg(a
j
)
X
A
j
(x)
p
0
=
j
|a
j
|
p
θ
p(it)
p
0
X
A
j
(x).
p
θ
p(it)
p
0
= p
θ
p
0
1 − it
p
0
+ p
θ
p
0
it
p
1
= p
θ
− itp
θ
+
p
θ
p
0
it
p
1
= p
θ
+ itp
θ
−1 +
p
0
p
1
.
|ϕ(it)|
p
0
=
j
|a
j
|
p
θ
p(it)
p
0
X
A
j
(x)
=
j
|a
j
|
p
θ
+itp
θ
−1+
p
0
p
1
X
A
j
(x)
=
j
|a
j
| X
A
j
(x)
p
θ
.
ϕ(it)
p
0
=
X
|ϕ(it)|
p
0
dx
1
p
0
=
X
j
|a
j
| X
A
j
(x)
p
θ
dx
1
p
0
=
X
j
|a
j
| X
A
j
(x)
p
θ
p
0
p
0
dx
1
p
0
.
ϕ(it)
p
0
=
X
|f|
p
θ
p
0
p
0
dx
1
p
0
=
|f|
p
θ
p
0
p
0
= f
p
θ
/p
0
p
θ
= 1.
ϕ(1 + it)
p
1
=
|f|
p
θ
p
1
p
1
= f
p
θ
/p
1
p
θ
= 1
ψ(it)
q
0
= ψ(1 + it)
q
1
= 1.
|F (it)| = |T ϕ(it); ψ(it)|
=
Y
T ϕ(it)ψ(it)dν(y)
≤
Y
|T ϕ(it)ψ(it)| dν(y)
= T ϕ(it)ψ(it)|
L
1
≤ T ϕ(it)
L
q
ψ(it)
L
q
≤ M
0
|F (1 + it)| = |T ϕ(1 + it); ψ(1 + it)|
≤ T ϕ(1 + it)
L
q
1
ψ(1 + it)
L
q
1
≤ M
1
.
ϕ(θ) = f ψ(θ) = g F (θ) = T f, g
|T f, g| ≤ M
1−θ
0
M
θ
1
.
(1.5)
M
θ
≤ M
1−θ
0
M
θ
1
,
1
p
θ
=
1 − θ
p
0
+
1
p
1
;
1
q
θ
=
1 − θ
q
0
+
1
q
1
θ ∈ (0, 1).
p, q ∈ [1, ∞)
1
q
+
1
p
≥ 1 f ∈ L
p
(R
N
)
g ∈ L
q
(R
N
) f ∗ g ∈ L
r
(R
N
)
1
r
=
1
q
+
1
p
− 1
f ∗ g|
L
r
≤ f
L
p
g
L
q
.
g ∈ L
q
(R
N
)
T f(x) = (f ∗ g)(x) =
R
N
f(x − y)g(y)dy
1.3 T : L
1
−→ L
q
T f
L
q
≤ f
L
1
g
L
q
.
T
0
= sup
f=0
T f
L
q
f
L
1
≤ g
L
q
.
|T f(x)| =
R
N
f(x − y)g(y)dy
≤
R
N
|f(x − y)g(y)| dy
≤
R
N
|f(x − y)|
q
dy
1
q
R
N
|g(y)|
q
dy
1
q
= f
L
q
g
L
q
.
T f(x)
L
∞
≤ f
L
q
g
L
q
.
T
1
= sup
f=0
T f(x)
L
∞
f
L
q
≤ g
L
q
.
T : L
q
−→ L
∞
T : L
p
(R
N
) −→ L
r
(R
N
)
T
θ
≤ T
1−θ
0
T
θ
1
≤ g
1−θ
L
q
g
θ
L
q
= g
L
q
,
1
p
=
1 − θ
1
+
θ
q
= 1 −
θ
q
1
r
=
1 − θ
q
+ 0 =
1
q
+
1 −
θ
q
− 1 =
1
q
+
1
p
− 1.
T f
L
r
= f ∗ g
L
r
≤ f
L
p
g
L
q
.
0 < α < N α I
α
I
α
f(x) = C
α
R
N
f(y)
|x − y|
N−α
dy
C
α
= π
−
N
2
2
−α
Γ(N/2 − α/2)/Γ(α/2)
0 < α < N 1 ≤ p < q < ∞
1
q
=
1
p
−
α
N
.
f ∈ L
P
(R
N
) (1.7) x ∈ R
N
p > 1 I
α
f(x)
L
q
≤ C f
L
p
[7]
R
N
L
1
(R
N
)
f : R
N
−→ C L
1
(R
N
) f
ˆ
f = F(f) : R
N
−→ C
ˆ
f (ξ) = (2π)
−N/2
R
N
f (x) e
−iξ.x
dx,
x = (x
1
, x
2
, ..., x
n
) ξ = (ξ
1
, ξ
2
, ..., ξ
n
) ∈ R
N
x.ξ =
n
i=1
x
i
ξ
i
.
ˆ
f f ∈ L
1
(R
N
)
f −→
ˆ
f
ˆ
f
L
∞
≤ (2π)
−n
2
f
L
1
.
ˆ
f
lim
|ξ|→∞
ˆ
f(ξ) = 0.
f
h
(x) = f(x + h)
(f
h
)(ξ) =
ˆ
f(ξ)e
ih.ξ
e (
e
ih.ξ
f) =
f(ξ − h).
[5]
f g ∈ L
1
(f ∗ g)(ξ) = (2π)
N
2
ˆ
f(ξ)ˆg(ξ).
f, g ∈ L
1
f ∗ g ∈ L
1
(R
N
)
(f ∗ g)(ξ) = (2π)
−N/2
R
N
(f ∗ g) (x) e
−iξ.x
dx
= (2π)
−N/2
R
N
e
−iξ.x
dx
R
N
f(x − y)g(y)dy.
(f ∗ g)(ξ) = (2π)
−N/2
R
N
g(y)dy
R
N
e
−iξ.x
f(x − y)dx
= (2π)
−N/2
R
N
g(y)dy
R
N
e
−iξ.(x+y−y)
f(x − y)dx
= (2π)
−N/2
R
N
g(y)e
−iξ.y
dy
R
N
e
−iξ.(x−y)
f(x − y)dx
= (2π)
−N/2
(2π)
N/2
ˆ
f(ξ) (2π)
N/2
ˆg(ξ)
= (2π)
N/2
ˆ
f(ξ)ˆg(ξ).
f : R −→ R
f(x) =
1, x ∈ [−1, 1]
0, x /∈ [−1, 1]
f ∈ L
1
(R)
ˆ
f(ξ) =
(2π)
−N/2
sin ξ
ξ
, ξ = 0
(2π)
−N/2
, ξ = 0
ˆ
f /∈ L
1
(R)
f
L
1
(R
N
)
L
1
(R
N
)
[5]
f ∈ L
1
(R
N
)
ˆ
f ∈ L
1
(R
N
)
[2] [5]
N N
N
=
N × N × ... × N

N
α ∈ N
N
α = (α
1
, α
2
, ..., α
N
)
x ∈ R
N
α
|α| =
N
j=1
α
j
; x
α
= x
α
1
1
· x
α
2
2
· .... · x
α
N
N
.
∂
α
=
∂
∂x
1
α
1
·
∂
∂x
2
α
2
· ... ·
∂
∂x
N
α
N
.
S(R
N
)
f : R
N
−→ C f ∈ C
∞
(R
N
)
f
α,β
= sup
x∈R
N
x
α
∂
β
f(x)
< ∞.
S(R
N
)
(1.13)
f ∈ C
∞
(R
N
) f ∈ S(R
N
)
lim
|x|→∞
x
α
∂
β
f(x) = 0, ∀ α, β ∈ N
N
.
[2]
S(R
N
) L
p
(R
N
)
[2]
{f
k
} S(R
N
) f ∈ S(R
N
)
lim
k→∞
f
k
− f
α,β
= 0 ∀ α, β ∈ N
N
.
f
k
S
→ f.
p ∈ [1, ∞] f
k
S
→ f f
k
L
p
→ f
[5]
f ∈ S(R
N
)
(1.8)
f ∈ S(R
N
)
ˆ
f ∈ S(R
N
)
a) (−i)
|α|
(∂
α
f)ˆ(ξ) = ξ
α
ˆ
f(ξ)
b) (−i)
|α|
(x
α
ˆ
f )(ξ) = (∂
α
ˆ
f )(ξ).
[2]
a ∈ C/ {0} Re(a) ≥ 0 g(x) = e
−a|x|
2
ˆg(ξ) = e
−|ξ|
2
4a
(2a)
−
N
2
.
[2]
f, g ∈ S(R
N
)
f ∈ S(R
N
)
f(x) = (2π)
−N/2
R
N
ˆ
f(x)e
iξ.x
dξ
R
N
f ˆg =
R
N
ˆ
fg.
[2]
ˇ
f = F
−1
: S(R
N
) −→ S(R
N
)
ˇ
f = (2π)
−N/2
R
N
f(ξ)e
iξ.x
dξ.
ˇ
f S(R
N
) ⊂ L
1
(R
N
) e
iξ.x
f(x) = (
ˆ
f)ˇ(ξ).
f(x) = (
ˇ
f)ˆ(x).
F : S(R
N
) −→ S(R
N
)
S(R
N
)
f, g ∈ S(R
N
)
f, g =
R
N
f(x)g(x)dx =
R
N
ˆ
f(ξ)ˆg(ξ)dξ =
ˆ
f, ˆg
.
f
L
2
=
ˆ
f
L
2
.
[5]
f ∈ L
2
(1.8)
f(x) =
0 x ∈ (−∞, 1]
x
−1
, (1, +∞).
f ∈ L
2
(R
N
) L
1
(R
N
)
L
2
(R
N
) 1.1
S(R
N
) L
2
(R
N
)
1.10
L
2
(R
N
)
f ∈ L
2
(R
N
) {f
n
} S(R
N
) f
n
−→ f
L
2
(R
N
)
lim
n→∞
ˆ
f
n
=
ˆ
f e lim
n→∞
ˇ
f
n
=
ˇ
f,
L
2
(R
N
)
L
2
(R
N
)
S(R
N
) L
2
(R
N
)
[2]
T : S(R
N
) −→ C
T
T ϕ
k
S
→ 0 T ϕ
k
−→ 0 k −→ ∞
S
(R
N
)
S
(R
N
)
L
p
1 ≤ p ≤ ∞
T
f
(ϕ) =
R
N
f(x)ϕ(x)dx, f ∈ L
p
(R
N
), ϕ ∈ S(R
N
).
|T
f
(ϕ)| =
R
N
f(x)ϕ(x)dx
≤
R
N
|f(x)ϕ(x)| dx
≤ f
L
p
ϕ
L
p
,
1
p
+
1
q
= 1 1.2
δ x ∈ R
N
δ
x
(ϕ) = ϕ(x), ϕ ∈ S(R
N
).
δ x ∈ R
N
|δ
x
(ϕ)| ≤ ϕ
L
∞
.
δ δ
{T
k
} S
(R
N
) T ∈ S
(R
N
) {T
k
}
T
lim
k→∞
T
k
(ϕ) = T (ϕ) ∀ ϕ ∈ S(R
N
).
T
k
S
→ T f
k
∈ L
p
(R
N
) f
k
L
p
→ f
|T
f
k
(ϕ) − T
f
(ϕ)| =
R
N
(f
k
− f)(x)ϕ(x)dx
≤
R
N
|(f
k
− f)(x)| |ϕ(x)| dx
≤ f
k
− f
L
p
ϕ(x)
L
q
,
1
p
+
1
q
= 1 k −→ ∞ T
f
k
−→ T
f
ϕ(x) ∈ S(R
N
)
f ∈ L
p
(R
N
) −→ T
f
∈ S
(R
N
)
L
p
(R
N
) S
(R
N
)
S
(R
N
) f, g
f(ϕ) = f, ϕ , f ∈ S
(R
N
) ϕ ∈ S(R
N
).
f ∈ L
p
(R
N
) 1 ≤ p < ∞
f, ϕ =
R
N
f(x)ϕ(x)dx ϕ ∈ S(R
N
).
7 9 [5] 4
[2]
α ∈ N
N
f ∈ S
(R
N
) ∂
α
f
∂
α
f : S(R
N
) −→ C
ϕ −→ (−1)
|α|
f(∂
α
ϕ).
ϕ ∈ S(R
N
) T ∈ S
(R
N
) T ϕ
T ∗ ϕ : S(R
N
) −→ C
T ∗ ϕ(φ) = T (ϕ ∗ φ), ∀ φ ∈ S(R
N
).
f ∈ S
(R
N
) f
F(f) =
ˆ
f
ˆ
f(ϕ) = f( ˆϕ) ∀ ϕ ∈ S(R
N
)
f F
−1
(f) =
ˇ
f
ˇ
f(ϕ) = f( ˇϕ).
F : S
(R
N
) −→ S
(R
N
)
2.8 [2]
S
(R
N
)
Q(R
N
) C
∞
(R
N
)
φ ∈ Q(R
N
) φ ∈ C
∞
(R
N
) α ∈ N
N
C(α)
n(α)
|∂
α
φ(x)| ≤ C(α)(1 + |x|
2
)
n(α)
∀ x ∈ R
N
.
T ∈ S
(R
N
) φ ∈ Q(R
N
) φT
T φ
φT (ϕ) = T (φϕ), ∀ ϕ ∈ S(R
N
).
f ∈ S
(R
N
) x
α
f φ(x) = x
α
f
(∂
α
f)= i
|α|
ξ
α
ˆ
f
∂
α
(
ˆ
f) = (−i)
|α|
(x
α
f)ˆ.
[2]
L
2
(R
N
)
L
2
(R
N
)
s ∈ R S
(R
N
)
H
s
(R
N
)
s ∈ R s S
(R
N
)
H
s
(R
N
) =
f ∈ S
(R
N
) : ∧
s
f(ξ) = (1 + |ξ|
2
)
s
2
ˆ
f ∈ L
2
(R
N
)
·
s
f
H
s
= ∧
s
f
L
2
=
R
N
(1 + |ξ|
2
)
s
ˆ
f(ξ)
2
dξ
1
2
.
s ≥ s
H
s
(R
N
) ⊆ H
s
(R
N
) f ∈ H
s
(R
N
)
f ∈ H
s
(R
N
) f
H
s
< ∞ s ≥ s
(1 + |ξ|
2
)
s
2
≤ (1 + |ξ|
2
)
s
2
f
2
H
s
=
R
N
(1 + |ξ|
2
)
s
ˆ
f(ξ)
2
dξ
≤
R
N
(1 + |ξ|
2
)
s
ˆ
f(ξ)
2
dξ
= f
2
H
s
< ∞.
H
s
(R
N
) ⊆ H
s
(R
N
)
H
s
(R
N
)
f, g
H
s
=
R
N
(1 + |ξ|
2
)
s
ˆ
f(ξ)ˆg(ξ)dξ.
s, s
∈ R
H
s
(R
N
)
= L
2
(R
N
, (1 + |ξ|
2
)
s
dx)
H
s
(R
N
) H
−s
(R
N
)
[5]
m ∈ N f ∈ H
m
(R
N
) ∂
α
f ∈ L
2
(R
N
)
|α| ≤ m
f ∈ H
m
(R
N
) ∂
α
f ∈ L
2
(R
N
) ∂
α
f
L
2
< ∞
(1.23)
(∂
α
f)= i
|α|
ξ
α
ˆ
f.
|ξ|
α
= |ξ
α
1
1
. . . ξ
α
n
n
|
≤ (1 + |ξ|
2
)
α
1
2
. . . (1 + |ξ|
2
)
α
n
2
≤ (1 + |ξ|
2
)
|α|
2
.
|α| ≤ m
∂
α
f
2
L
2
= (∂
α
f)
2
L
2
=
R
N
|(∂
α
f(x))|
2
dx
=
R
N
ξ
α
ˆ
f(ξ)
2
dξ
=
R
N
|ξ|
2α
ˆ
f(ξ)
2
dξ.
∂
α
f
2
L
2
≤
R
N
(1 + |ξ|
2
)
|α|
ˆ
f(ξ)
2
dξ
≤
R
N
(1 + |ξ|
2
)
m
ˆ
f(ξ)
2
dξ
= f
H
m
< ∞.
∂
α
f ∈ L
2
(R
N
)
∂
α
f ∈ L
2
(R
N
) α ∈ N |α| ≤ m
(1 + |ξ|
2
)
m
ˆ
f(ξ)
2
=
m
j
c
j
|ξ|
2j
ˆ
f(ξ)
2
.
m
j
c
j
|ξ|
2j
ˆ
f(ξ)
2
ξ
α
ˆ
f(ξ)
2
(1.23)
f
H
m
=
R
N
(1 + |ξ|
2
)
m
ˆ
f(ξ)
2
dξ
=
R
N
m
j
c
j
|ξ|
2j
ˆ
f(ξ)
2
2 < ∞.
f ∈ H
m
(R
N
)
s >
n
2
H
m
(R
N
)
C
∞
(R
N
) R
N
C
|x| → ∞
f
L
∞
≤ (2π)
−
n
2
f
H
s
R
N
(1 + |ξ|
2
)
−s
1
2
.
f ∈ H
s
(R
N
)
ˆ
f ∈ L
1
f
L
1
=
R
N
ˆ
f(ξ)
dξ
=
R
N
(1 + |ξ|
2
)
−s/2
(1 + |ξ|
2
)
s/2
ˆ
f(ξ)
dξ
≤
R
N
(1 + |ξ|
2
)
−s
dξ
1
2
R
N
(1 + |ξ|
2
)
s
ˆ
f(ξ)
2
dξ
1
2
= f
H
s
R
N
(1 + |ξ|
2
)
−s
1
2
< ∞.
(1.22)
(1) (1.15)
f
L
∞
=
(
ˆ
f)ˇ
L
∞
≤ (2π)
−
n
2
ˆ
f
L
1
≤ (2π)
−
n
2
f
H
s
R
N
(1 + |ξ|
2
)
−s
1
2
.
[3]
W
m,p
u : R
N
−→ C D
α
u ∈ L
p
(R
N
)
α |α| ≤ m
u
W
m,p
=
|α|≤m
D
α
u
L
p
.
1 ≤ p, q, r ≤ ∞ j, m
0 ≤ j ≤ m
1
p
=
j
n
+ θ
1
r
−
m
n
+
1 − θ
q
θ ∈
j
m
, 1
C = C(n, m, j, θ, q, r)
|α|=j
D
α
u ≤ C
|α|=m
D
α
u
θ
u
1−θ
L
q
u ∈ S(R
N
)
[1]
u ∈ H
m
(R
N
) v ∈ L
2
u, v
H
m
,H
−m
=
R
N
u(x)v(x)dx.
∆ : H
1
−→ H
−1
∆u ∈ H
−1
H
1
∆u, v
H
−1
,H
1
=
R
N
∇u(x)∇v(x)dx.
> 0 J
H
−1
(R
N
)
J
u = (I − ∆)
−1
u.
u ∈ H
−1
(R
N
) u
= J
u ∈ H
1
(R
N
)
u
− ∆u
= u.
λ > 0 u ∈ H
1
(R
N
) f ∈ H
−1
(R
N
) −∆u + λu = f
f ∈ L
p
(R
N
) p ∈ [1, ∞) u ∈ L
p
(R
N
)
λ u
L
p
≤ f
L
p
.
(1.5.1) [3]
J
f
X
≤ f
X
X = H
1
(R
N
) H
−1
(R
N
) L
p
(R
N
) p ∈ [1, ∞)
J
B(X)
J
f
X
≤ 1
X H
1
(R
N
) H
−1
(R
N
) L
p
(R
N
)
p ∈ [1, ∞)
J
f, g
X,X
∗
= f, J
g
X,X
∗
, ∀ f ∈ X, g ∈ X
∗
J
f −→ f X −→ 0 f ∈ X
f
X −→ 0 (J
f
− f) −→ 0 −→ 0
(1.5.2) [3]
iu
t
+ ∆
x
u = 0
u(x, 0) = ϕ(x).
L
2
(R
N
)
ϕ ∈ S(R
N
) u ∈ C
∞
(R; S(R
N
)) (2.1)
i ˆu
t
− 4π
2
|ξ|
2
ˆu = 0
ˆu(ξ, 0) = ˆϕ(ξ).
(2.2)
ˆu(t) = e
−i4π
2
|ξ|
2
t
C.
(2.3)
ˆu(t) = e
−i4π
2
|ξ|
2
t
ˆϕ(·).
(2.1)
u(t) = (e
−i4π
2
|ξ|
2
t
ˆϕ)ˇ(·)
u(t) = (e
−it∆
)ϕ(·).
S(t) = e
−it∆
(2.6)
u(t) = S(t)ϕ(·).
S(t)
F(S(t)ϕ)(ξ) = e
−i4π
2
|ξ|
2
t
ˆϕ(ξ)
{S(t)} L
2
(R
N
)
∀ t ∈ R S(t) : L
2
(R
N
) −→ L
2
(R
N
)
S(t)f
L
2
(R
N
)
= f
L
2
(R
N
)
∀f ∈ L
2
(R
N
).
S(t)S(s) = S(t + s) S(t)
−1
= S(−t) = S(t)
∗
S(0) = I
f ∈ L
2
(R
N
) φ
f
: R −→ L
2
(R
N
) φ
f
= S(t)ϕ
φ
f
L
2
(R
N
)
f f ∈ C
2k
φ
f
∈ C
k−1
(baseado em ∂
t
φ
f
= ∆φ
f
)
f ∈ L
2
(R
N
)
S(t)f
2
L
2
=
(e
−i4π
2
|ξ|
2
t
ˆ
f)ˇ
L
2
=
e
−i4π
2
|ξ|
2
t
ˆ
f
L
2
=
ˆ
f
L
2
= f
L
2
.
S(t) : L
2
(R
N
) −→ L
2
(R
N
)
S(t + s)f =
e
−i4π
2
|ξ|
2
(t+s)
ˆ
f
ˇ
=
e
−i4π
2
|ξ|
2
t
e
−i4π
2
|ξ|
2
s
ˆ
f
ˇ
=
e
−i4π
2
|ξ|
2
t

e
−i4π
2
|ξ|
2
s
ˆ
f
ˇ
ˆ
ˇ.
S(t + s)f =
e
−i4π
2
|ξ|
2
t
[S(s)f ] ˆ
ˇ
= S(t) ◦ S(s)(f).
S(t + s) = S(t)S(s)
S(t)f, g
L
2
=
R
N
(e
−i4π
2
|ξ|
2
t
ˆ
f)ˇg(x)dx
=
R
N
e
−i4π
2
|ξ|
2
t
ˆ
f
g(x)
ˇdx
=
R
N
e
−i4π
2
|ξ|
2
t
ˆ
f
ˆg(x)dx
=
R
N
ˆ
f
e
i4π
2
|ξ|
2
t
ˆg(x)
dx
=
R
N
f
e
i4π
2
|ξ|
2
t
ˆg(x)
ˆdx
=
R
N
f
e
i4π
2
|ξ|
2
t
ˆg(x)
ˇdx
= f, S(−t)g
L
2
.
S(t)
∗
= S(−t).
S(0) = I
S(0)f =
e
−i4π
2
|ξ|
2
.0
ˆ
f
ˇ
=
ˆ
f
ˇ= f.
φ
f
(t) L
2
(R
N
)
lim
t→τ
φ
f
(t) − φ
f
(τ)
L
2
= lim
t→τ
S(t)f − S(τ)f
L
2
= lim
t→τ
e
−i4π
2
|ξ|
2
(t)
ˆ
f
ˇ−
e
−i4π
2
|ξ|
2
(τ)
ˆ
f
ˇ
L
2
= lim
t→τ
e
−i4π
2
|ξ|
2
(t)
ˆ
f − e
−i4π
2
|ξ|
2
(τ)
ˆ
f
ˇ
L
2
= lim
t→τ

e
−i4π
2
|ξ|
2
(t)
− e
−i4π
2
|ξ|
2
(τ)
ˆ
f
ˇ
L
2
= 0,
(2.8)
|F(S(t)ϕ)(x)| =
e
−i4π
2
|ξ|
2
t
ϕ(x)
=
e
−i4π
2
|ξ|
2
t
|ϕ(x)|
= | ϕ(x)| ,
t ∈ R x ∈ R
N
∀ s, t ∈ R
S(t)ϕ
H
s
=
R
N
1 + |ξ|
2
s
2
|F(S(t)ϕ)(ξ)| dξ
=
R
N
1 + |ξ|
2
s
2
|ϕ(ξ)| dξ = ϕ
H
s
.
S(R
N
) H
s
(R
N
) s ∈ R
{S(t)} t ∈ R H
s
(R
N
)
(M.H Stone) {T (t)} t ∈ R
H A
H
T (t) = e
itA
.
D(A) A H
f ∈ D(A)
lim
t→o
T (t)f − f
t
= iAf.
[7]
t = 0 K
t
K
t
(x) =
1
4itπ
N
2
e
i|x|
2
4t
.
S(t)ϕ = K
t
∗ ϕ
S(t)ϕ =
1
4itπ
N
2
R
N
e
i|x−y|
2
4t
ϕ(y)dy
∀t = 0 ∀ ϕ ∈ S(R
N
)
[7]
p ∈ [2; ∞] t = 0 S(t) : L
p
(R
N
) −→ L
p
(R
N
)
S(t)ϕ
L
p
(R
N
)
≤ (4π |t|)
−N
(
1
2
−
1
p
)
ϕ
L
p
(R
N
)
.
∀ ϕ ∈ L
p
(R
N
)
ϕ ∈ S(R
N
)
S(t)ϕ
L
∞
= K
t
∗ ϕ
L
∞
≤ K
t
(x)
L
∞
ϕ(x)
L
1
=
(4itπ)
−
N
2
e
i|x|
2
4t
L
∞
ϕ(x)
L
1
=
(4itπ)
−
N
2
ϕ(x)
L
1
.
S(t)ϕ
L
∞
≤
(4itπ)
−
N
2
ϕ
L
1
.
S(t) : L
1
(R
N
) −→ L
∞
(R
N
)
S(t)
0
= sup
S(t)ϕ
L
∞
ϕ
L
1
≤
(4itπ)
−
N
2
.
2.1 S(t) : L
2
(R
N
) −→ L
2
(R
N
)
S(t)
1
= sup
S(t)ϕ
L
2
ϕ
L
2
= 1.
S(t) :
L
p
(R
N
) −→ L
p
(R
N
)
S(t)
p
≤ S(t)
1−θ
0
S(t)
θ
1
= 1
1−θ
(4π |t|)
−
N
2
θ
= (4π |t|)
−
N
2
θ
,
1
p
=
1−θ
2
+ θ
1
p
=
1−θ
2
∀ θ ∈ (0, 1)
1
p
=
1
p
+ θ ⇒
1
p
−
1
p
= θ
(2.20)
S(t)ϕ
L
p
≤ (4iπ |t|)
−
N
2
θ
ϕ
L
p
= (4iπ |t|)
−
N
2
1
p
−
1
p
ϕ
L
p
.
1
p
+
1
p
= 1 ⇒
1
p
= 1 −
1
p
S(t)ϕ
L
p
≤ (4iπ |t|)
−
N
2
(
1−
1
p
)
ϕ
L
p
.
(q, r)
2
q
= N
1
2
−
1
r
2 ≤ r ≤
2N
N − 2
.
N ≥ 3
(3.2) N = 1 2 ≤ r ≤ ∞ N = 2
(3.2) 2 ≤ r < ∞
(q, r) 2 ≤ q ≤ ∞ (∞, 2)
r =
2N
N−2
r =
2N
N−2
[3]
1) ϕ ∈ L
2
(R
N
) t −→ S(t)ϕ
L
q
(R, L
r
(R
N
)) ∩ C(R, L
2
(R
N
))
C
S(t)ϕ
L
q
(R,L
r
(R
N
))
≤ C ϕ
L
2
∀ϕ ∈ L
2
(R
N
).
2) I R J =
¯
I t
0
∈ J (q, r) f ∈
L
q
(I, L
r
(R
N
)) t −→ θ
f
(t) ∀ t ∈ I
θ
f
(t) =
t
t
o
S(t − s)f(s)ds.
L
q
(I, L
r
(R
N
)) C I
θ
f
(t)
L
q
T
L
r
x
≤ C f
L
q
T
L
r
x
∀ϕ ∈ L
2
(R
N
), q, q
= 2.
3) (q, r) f ∈ L
q
(I, L
r
(R
N
)) θ
f
(t)
C(I, L
2
(R
N
))
θ
f
L
∞
T
L
2
x
≤ C f
L
q
T
L
r
x
.
2 e 3 I = [0, T )
T ∈ (0, ∞) t
0
= 0
φ
f
( onde t ∈ (0, T ))
φf(s) =
t
0
S(t − τ)f(τ )dτ ∀s ∈ [0, T ).
2) (q, r) φ
f
e θ
f
L
q
(I, L
r
(R
N
)) −→ L
q
(I, L
r
(R
N
)) θ
f
φ
f
f ∈ L
q
(I, L
r
(R
N
))
θ
f
L
r
x
=
R
N
t
0
S(t − s)f(s)ds
r
dx
1
r
≤
t
0
R
N
|S(t − s)f(s)|
r
dx
1
r
dt
=
t
0
S(t − s)f(s)
L
r
x
ds
≤
t
0
(|t − s|)
−N
(
1
2
−
1
r
)
f
L
r
ds
=
t
0
(|t − s|)
−
2
q
f
L
r
ds
=
R
X
[0,T )
(|t − s|)
−
2
q
f
L
r
ds ≤ I
α
(f(s)
L
r
).
(3.6) (3.7)
2.1 α = −
2
q
+ 1
(3.9)
α = −
2
q
+1 (3.2) 2 ≤ q ≤ ∞
0 ≤ α ≤ 1
1
q
+
1
q
= 1 = α +
2
q
.
1
q
−
2
q
= −
1
q
+ α =⇒
−
1
q
= −
1
q
+ α =⇒
1
q
=
1
q
− α.
I
α
(f(s))
N = 1 α = −
2
q
+ 1
θ
f
L
q
T
L
r
x
≤ I
α
(f(s)
L
r
)
L
q
T
≤ C f
L
q
T
L
r
x
.
θ
f
L
q
T
L
r
x
≤ C f
L
q
T
L
r
x
.
(q, r)
3) (q, r) θ
f
L
q
(I, L
r
(R
N
)) −→ C(I, L
2
(R
N
)) f ∈ L
q
(I, L
r
(R
N
)
θ
f
(t)
2
L
2
=
t
0
S(t − s)f(s)ds,
t
0
S(t − τ)f(τ )dτ
L
2
.
=
R
N
t
0
S(t − s)f(s)ds
t
0
S(t − τ)f(τ )dτdx
=
R
N
t
0
t
0
S(t − s)f(s)S(t − τ)f(τ)dτdsdx
=
t
0
t
0
R
N
S(t − s)f(s)S(t − τ)f(τ)dxdτds
=
t
0
t
0
(S(t − s)f(s), S(t − τ)f(τ))
L
2
dτds.
θ
f
(t)
2
L
2
=
t
0
t
0
(f(s), S
∗
(t − s) [S(t − τ)f(τ )])
L
2
dτds
=
t
0
t
0
(f(s), S(−t + s) [S(t − τ)f(τ)])
L
2
dτds.
θ
f
(t)
2
L
2
=
t
0
t
0
(f(s), S(s − τ)f(τ ))
L
2
dτds
=
t
0
f(s),
t
0
S(s − τ)f(τ )dτ
L
2
ds
=
t
0
(f(s), φ
f
(s))
L
2
ds
=
t
0
R
N
f(s)φ
f
(s)dxds.
θ
f
(t)
2
L
2
≤
t
0
f(s)
L
r
x
φ
f
(s)
L
r
x
ds
≤ f
L
q
T
L
r
x
φ
f
(s)
L
q
T
L
r
x
≤ f
L
q
T
L
r
x
C f
L
q
T
L
r
x
= C f
2
L
q
T
L
r
x
.
(3.13)
(3.14)
θ
f
L
∞
T
L
2
x
≤ C f
L
q
T
L
r
x
.
1) (q, r)
S(t)ϕ
L
q
(R,L
r
(R
N
))
≤ C ϕ
L
2
∀ϕ ∈ L
2
(R
N
).
δ
f
(t) =
R
S(t − s)f(s)ds ∀t ∈ [0, T ).
Γ
f
(t) =
R
S(−t)f(t)dt ∀t ∈ [0, T ).
δ
f
(t) : L
q
(I, L
r
(R
N
)) −→ L
q
(I, L
r
(R
N
))
δ
f
(t)
L
q
t
L
r
x
≤ C f
L
q
t
L
r
x
,
(q, r)
(3.18)
Γ
f
(t) : L
q
(R, L
r
(R
N
)) −→ C(R, L
2
(R
N
))
Γ
f
(t)
L
2
≤ C f
L
q
t
L
r
x
.
R
S(t)ϕ(·), ψ(t)
L
2
dt
=
R
ϕ(·), S(−t)ψ(t)
L
2
dt
=
R
R
N
ϕ(·)S(−t)ψ(t)dxdt
=
R
N
R
ϕ(·)
S(−t)ψ(t)dtdx
=
R
N
ϕ(·)Γ
ψ
(t)dx
.
(3.19)
R
S(t)ϕ(·), ψ(t)
L
2
dt
=
ϕ(·), Γ
ψ
(t)
L
2
≤ ϕ
L
2
Γ
ψ
(t)
L
2
≤ ϕ
L
2
ψ
L
q
t
L
r
x
.
S(t)ϕ
L
q
T
L
r
x
= sup
t
0
(S(t)ϕ, ψ)
L
2
dt
; ψ ∈ C
∞
c
(R, R
N
); ψ
L
q
T
L
r
x
= 1
.
S(t)ϕ
L
q
T
L
r
x
≤ ϕ
L
2
.
(q
0
, p
0
) (q
1
, p
1
) ∀ T > 0
φ
f
(t)
L
q
1
(I,L
p
1
(R
N
))
≤ C f
L
q
0
(I,L
p
0
(R
N
))
.
(q
0
, p
0
) (q
1
, p
1
)
P = (0, 1/2) Q =
N
4
−
N
2p(N)
;
1
p(N)
p(N) = ∞ N = 1, 2 p(N) =
2N
N−2
N ≥ 3 p
0
∈ [2, p
1
)
φ
f
(t)
L
q
0
T
L
p
0
x
=
T
0
φ
f
(t)
q
0
L
p
0
dt
1
q
0
≤
T
0
φ
f
(t)
q
0
(1−θ)
L
2
φ
f
(t)
q
0
θ
L
p
1
dt
1
q
0
≤ sup
[0,T ]
φ
f
(t)
(1−θ)
L
2
T
0
φ
f
(t)
q
0
θ
L
p
1
dt
1
q
0
, (∗)
1
p
0
=
θ
p
1
+
1 − θ
2
,
θ =
p
1
(2 − p
0
)
p
0
(2 − p
1
)
.
(q
0
, p
0
) (q
1
, p
1
)
2
q
0
=
N
2
−
N
p
0
e
2
q
1
=
N
2
−
N
p
1
.
N
4
=
p
0
q
0
(p
0
− 2)
=
p
1
q
1
(p
1
− 2)
.
q
0
q
1
=
p
0
(2 − p
1
)
p
1
(2 − p
0
)
=
1
θ
.
q
1
= q
0
θ.
(∗)
φ
f
(t)
L
q
0
T
L
p
0
x
≤ φ
f
(t)
(1−θ)
L
∞
T
L
2
x
T
0
φ
f
(t)
q
1
L
p
1
dt
θ
q
1
.
=
φ
f
(t)
L
∞
T
L
2
x
(1−θ)
φ
f
(t)
L
q
1
T
L
p
1
x
θ
.
φ
f
(t)
L
q
0
T
L
p
0
x
≤ C
f
L
q
1
T
L
p
1
x
(1−θ)
f
L
q
1
T
L
p
1
x
θ
= f
L
q
1
T
L
p
1
x
.
φ
f
(t)
L
q
1
(I,L
p
1
(R
N
))
≤ C f
L
q
0
(I,L
p
0
(R
N
))
.
(p
0
, q
0
) (p
1
, q
1
) m ≥ 0
S(t)ϕ
L
q
T
W
m,r
≤ C ϕ
H
m
φ
f
(t)
L
q
0
(I,W
m,p
0
)
≤ C f
L
q
1
(I,W
m,p
1
)
.
L
2
(R
N
)
L
2
(R
N
)
iu
t
+ ∆
x
u = γ |u|
α
u γ ∈ R
u(x, 0) = ϕ(x) x ∈ R
N
, t ∈ R,
S(t)
u (4.1) u
u(t) = S(t)ϕ + iγ
t
0
S(t − s) |u(s)|
α
u(s)ds.
f(t) = γ |u(t)|
α
u(t)
w(s) = S(t − s)u(s)
w(s + h) − w(s) = S(t − s − h)u(s + h) − S(t − s)u(s).
w(s + h) − w(s)
h
=
S(t − s − h)u(s + h) − S(t − s − h + h)u(s)
h
=
S(t − s − h)u(s + h) − S(t − s − h)S(h)u(s)
h
=
S(t − s − h)u(s + h) − S(t − s − h)u(s)
h
+
S(t − s − h)u(s) − S(t − s − h)S(h)u(s)
h
= S(t − s − h)
u(s + h) − u(s)
h
−
S(h)u(s) − u(s)
h

.
h −→ 0
u(s + h) − u(s)
h
−→ ∂
s
u(s).
2.2
S(h)u(s) − u(s)
h
−→ i∆u.
w
(s) = S(t − s) (∂
s
u(s) − i∆u)
= iS(t − s)f(s).
τ τ ∈ [0, t)
τ
0
w
(s)ds = i
τ
0
S(t − s)f(s)ds.
S(t − τ)u(τ ) − S(t)u(0) = i
τ
0
S(t − s)f(s)ds.
τ −→ t
S(0)u(t) − S(t)u(0) = i
t
0
S(t − s)f(s)ds.
u(t) = S(t)ϕ + iγ
t
0
S(t − s) |u(s)|
α
u(s)ds.
α <
4
N
(q, r) r = α + 2 q =
4(α+2)
αN
|u|
α
u
L
r
x
= u
α+1
L
r
x
.
|u|
α
u
L
r
x
=
R
N
(|u|
α
|u|)
r
dx
1
r
=
R
N
|u|
(α+1)r
dx
1
r
=
R
N
|u|
α+1
r−1
r
dx
r−1
r
=
R
N
|u|
r−1
r−1
r
dx
α+1
r
=
R
N
|u|
r
dx
α+1
r
= u
α+1
L
r
x
.
α, q e r
|u|
α
u
L
q
T
L
r
x
≤ T
(1−
Nα
4
)
u
α+1
L
q
T
L
r
x
.
|u|
α
u
L
q
T
L
r
x
=
T
0
|u|
α
u
q
L
r
x
dt
1
q
=
T
0
1. u
(α+1)q
L
r
x
dt
1
q
.
p =
q
q−(1+α)q
p
=
q
(1+α)q
|u|
α
u
L
q
T
L
r
x
≤
T
0
1dt
1−
α+1
q
q
T
0
u
(α+1)q
q
(α+1)q
L
r
x
dt
α+1
q
q
1
q
=
T
0
1dt
1
q
−
α+1
q
T
0
u
q
L
r
x
dt
α+1
q
= T
1
q
−
α+1
q
u
α+1
L
q
T
L
r
x
= T
(
1−
1
q
−
α+1
q
)
u
α+1
L
q
T
L
r
x
= T
(
1−
1
q
−
α+1
q
)
u
α+1
L
q
T
L
r
x
= T
(
1−
α+2
q
)
u
α+1
L
q
T
L
r
x
= T
(
1−
Nα
4
)
u
α+1
L
q
T
L
r
x
.
α, q e r
|u|
α
u − |v|
α
v
L
q
T
L
r
x
≤ CT
(1−
Nα
4
)
u
α
L
q
T
L
r
x
+ v
α
L
q
T
L
r
x
u − v
L
q
T
L
r
x
.
g : C −→ C
g(z) = |z|
α
z.
| |z
1
|
α
z
1
− |z
2
|
α
z
2
| ≤ C (|z
1
|
α
+ |z
2
|
α
) |z
1
− z
2
|
α − 1 < 0
|z
1
| < |z
2
|
| |z
1
|
α
z
1
− |z
2
|
α
z
2
| = | |z
1
|
α
z
1
− |z
2
|
α
z
1
+ |z
2
|
α
|z
1
− z
2
||
≤ |z
2
|
α
|z
1
− z
2
| + |z
1
| | |z
1
|
α
− |z
2
|
α
|
θ ∈ (0, 1)
| |z
1
|
α
z
1
− |z
2
|
α
z
2
| ≤ |z
2
|
α
|z
1
− z
2
| + |z
1
| α ((1 − θ) |z
1
| + θ |z
2
|)
α−1
| |z
1
| − |z
2
| |
≤ |z
2
|
α
|z
1
− z
2
| + |z
1
| α |z
1
|
α−1
|z
1
− z
2
|
≤ |z
1
− z
2
| (|z
1
|
α
+ |z
2
|
α
)
(4.3)
|u|
α
u − |v|
α
v
L
r
x
≤ C
|u|
α
|u − v|
L
r
x
+ |v|
α
|u − v|
L
r
x
. (∗)
r = α + 2 r
=
α+2
α+1
1
r
=
1
r
+
1
r
1
r
1
=
r
α
|u|
α
|u − v|
L
r
x
≤ |u|
α
L
r
1
x
u − v
L
r
x
= u
α
L
r
x
u − v
L
r
x
.
|v|
α
|u − v|
L
r
x
≤ v
α
L
r
x
u − v
L
r
x
.
|u|
α
|u − v|
L
q
T
L
r
x
=
T
0
|u|
α
|u − v|
L
r
x
q
dt
1
q
≤
T
0
|u|
αq
L
r
x
u − v
q
L
r
x
dt
1
q
.
p =
q
(α+1)q
p
=
q
q−(1+α)q
|u|
α
|u − v|
L
q
T
L
r
x
≤
T
0
1.
|u|
αq
L
r
x
u − v
q
L
r
x
dt
1
q
≤
T
0
1dt
1
q
−
α+1
q
T
0
u
αq
α+1
L
r
x
u − v
q
(α+1)
L
r
x
dt
(α+1)
q
= T
1−
Nα
4
T
0
u
αq
α+1
L
r
x
u − v
q
(α+1)
L
r
x
dt
(α+1)
q
.
p =
α+1
α
p
= α + 1
|u|
α
|u − v|
L
q
T
L
r
x
≤ T
1−Nα/4
T
0
u
αq
α+1
.
α+1
α
L
r
x
dt
α
q
T
0
u − v
q
α+1
.α+1
L
r
x
dt
1
q
= T
1−
Nα
4
T
0
u
q
L
r
x
dt
α
q
T
0
u − v
q
L
r
x
dt
1
q
= T
1−
Nα
4
u
α
L
q
T
L
r
x
u − v
L
q
T
L
r
x
.
|u|
α
|u − v|
L
q
T
L
r
x
≤ T
1−
Nα
4
u
α
L
q
T
L
r
x
u − v
L
q
T
L
r
x
.
|v|
α
|u − v|
L
q
T
L
r
x
≤ v
α
L
q
T
L
r
x
T
1−
Nα
4
u − v
L
q
T
L
r
x
.
(4.7) (4.8) (∗)
|u|
α
u − |v|
α
v
L
q
T
L
r
x
≤ CT
(1−
Nα
4
)
u
α
L
q
T
L
r
x
+ v
α
L
q
T
L
r
x
u − v
L
q
T
L
r
x
.
L
2
(4.1)
L
2
(R
N
)
γ ∈ C α <
4
N
r = α + 2 (q, r)
ϕ ∈ L
2
(R
N
) T = T (ϕ
L
2
) > 0 M = M(ϕ
L
2
) > 0 u
(4.1)
u
L
q
T
L
r
x
≤ M.
(4.1) L
2
(R
N
) Φ
(4.1)
Φ(u) = S(t)ϕ + iγ
t
0
S(t − s) |u(s)|
α
u(s)ds.
Φ
Φ(u) = u.
T > 0 M > 0
E =
u ∈ L
q
[0, T ]; L
r
(R
N
)
∩ C
[0, T ], L
2
(R
N
)
: u
L
q
T
L
r
x
≤ M
.
(E, D
E
)
D
E
(u, v) = u − v
L
q
T
L
r
x
.
Φ(u) ∈ E u ∈ E Φ
u ∈ E
φ(u)
L
q
T
L
r
x
=
S(t)ϕ + iγ
t
0
S(t − s) |u(s)|
α
u(s)ds
L
q
T
L
r
x
.
φ(u)
L
q
T
L
r
x
≤ |S(t)ϕ
L
q
T
L
r
x
+ |γ|
t
0
S(t − s) |u(s)|
α
u(s)ds
L
q
T
L
r
x
≤ C
0
ϕ
L
2
+ C
1
|γ| |u|
α
u
L
q
T
L
r
x
≤ C
0
ϕ
L
2
+ C
1
T
(1−
Nα
4
)
u
α+1
L
q
T
L
r
x
≤ C
0
ϕ
L
2
+ C
1
T
(1−
Nα
4
)
M
α+1
.
M = 2C
0
ϕ
L
2
T > 0
C
1
T
(1−
Nα
4
)
M
α+1
≤
M
2
,
φ(u)
L
q
T
L
r
x
≤
M
2
+
M
2
= M.
φ(u) ∈ E
u, v ∈ E
Φ(u) − Φ(v)
L
q
T
L
r
x
= |γ|
T
0
S(t − s) (|u|
α
u − |v|
α
v) (s)ds
L
q
T
L
r
x
≤ C |u|
α
u − |v|
α
v
L
q
T
L
r
x
≤ CT
(1−
Nα
4
)
u
α
L
q
T
L
r
x
+ v
α
L
q
T
L
r
x
u − v
L
q
T
L
r
x
≤ CT
(1−
Nα
4
)
M
α
u − v
L
q
T
L
r
x
.
T (4.13)
CT
(1−
Nα
4
)
M
α
≤
1
2
.
Φ(u) − Φ(v)
L
q
T
L
r
x
≤
1
2
u − v
L
q
T
L
r
x
,
Φ(u)
u ∈ L
q
T
L
r
x
Φ(u) = u E
(4.1).
u, v ∈ L
q
T
L
r
x
(4.1) [0, T ) u ≡ v
θ(τ) = u(τ) − v(τ )
L
q
τ
L
r
x
0 ≤ τ ≤ T
θ(0) = u(0) − v(0)
L
q
τ
L
r
x
= ϕ(x) − ϕ(x)
L
q
τ
L
r
x
= 0.
θ(t) > 0 ∀ t = 0
t
0
= inf {t ∈ [0, T ] / θ(t) > 0} .
θ θ
t
n
−→ t
0
θ(t
n
) =
u − v
L
r
x
X
[0,t
n
]
(t
n
)
L
q
T
≤
u − v
L
r
x
L
q
T
.
lim
n→∞
θ(t
n
) = lim
n→∞
u − v
L
r
x
X
[0,t
n
]
(t
n
)
L
q
T
=
u − v
L
r
x
lim
n→∞
X
[0,t
n
]
(t
n
)
L
q
T
=
u − v
L
r
x
X
[0,t
0
]
(t
0
)
L
q
T
= θ(t
0
).
θ
u(t
0
, x) = v(t
0
, x) = ψ(x) ˜u(t) = u(t + t
0
)
˜v(t) = v(t + t
0
) ˜u, ˜v
i˜u + ∆
x
˜u = γ |˜u|
α
˜u
˜u(x, t
0
) = ψ(x)
i˜v + ∆
x
˜v = γ |˜v|
α
˜u
˜v(x, t
0
) = ψ(x).
(4.14)
˜u − ˜v
L
q
δ
L
r
x
≤ Cδ
(1−
Nα
4
)
˜u
α
L
q
δ
L
r
x
+ ˜v
α
L
q
δ
L
r
x
˜u − ˜v
L
q
δ
L
r
x
,
δ ∈ [0, T − t
0
]
˜
θ(δ) = ˜u
α
L
q
δ
L
r
x
+ ˜v
α
L
q
δ
L
r
x
.
˜
θ θ δ > 0
˜
θ(δ) < 1
˜
θ(0) = 0
˜u − ˜v
L
q
δ
L
r
x
≤ Cδ
(1−
Nα
4
)
˜u − ˜v
L
q
δ
L
r
x
.
δ > 0
Cδ
(1−
Nα
4
)
≤
1
2
,
˜u − ˜v
L
q
δ
L
r
x
≤
1
2
˜u − ˜v
L
q
δ
L
r
x
.
˜u − ˜v
L
q
δ
L
r
x
= 0.
0 = ˜u − ˜v
L
q
δ
L
r
x
= u − v
L
q
t
0
+δ
L
r
x
= θ(t
0
+ δ).
θ(t
0
+ δ) = 0 t
0
t
θ(t) > 0 θ(t) = 0 t ∈ [0, T ) u ≡ v
γ ∈ C α <
4
N
r = α + 2 (q, r)
ϕ ∈ L
2
(R
N
)
∃ T
∗
(ϕ) > 0 u [0, T
∗
) u ∈ L
q
T
L
r
x
∀ T < T
∗
• T
∗
= ∞
• T
∗
< ∞ lim
t→T
∗
u(t)
L
2
= ∞
( ) T < T
∗
δ > 0 C > 0
ϕ − ψ
L
2
< δ T
∗
(ψ) > T
u − v
L
∞
T
L
2
x
+ u − v
L
q
T
L
r
x
≤ C ϕ − ψ
L
2
.
T
∗
= sup {T > 0 / ∃ u ∈ L
q
T
L
r
x
}
T
∗
< ∞ ∃ t
n
−→ T
∗
M > 0
u(t
n
)
L
2
≤ M
ψ
n
= u(t
n
) ψ
n
L
2
≤ M v
n
(t) (4.1)
ψ
n
v
n
(t) [0, T (M)) (4.1)
˜u
n
(t) =
u(t), t < t
n
v
n
(t − t
n
), t
n
≤ t < t
n
+ T (M).
˜u
n
(t) (4.1) [0, t
n
+ T (M)) t
n
−→ T
∗
n
0
> 1 t
n
0
+ T (M) > T
∗
˜u
n
0
(t) [0, t
n
0
+ T (M))
[0, T
∗
)
T
∗
= ∞ ou
T
∗
< ∞ e lim
t→T
∗
u(t)
L
2
= ∞.
T < T
∗
M = 2 sup
0≤t≤T
u(t)
L
2
ψ ∈ L
2
ψ
L
2
≤ M 4.1 K(M) > 0
T (M) > 0 v (4.1)
v(0, x) = ψ(x) e v
L
q
T (M )
L
r
x
≤ K(M).
ψ
1
e ψ
2
ψ
j
L
2
≤ M j = 1, 2 v
1
, v
2
(4.1) [0, T (M)] ψ
1
e ψ
2
(4.14)
v
1
− v
2
L
∞
T (M )
L
2
x
≤ S(t)(ψ
1
− ψ
2
)
L
∞
T (M )
L
2
x
+ |γ|
T
0
S(t − s) (|v
1
|
α
v
1
− |v
2
|
α
v
2
) (s)ds
L
∞
T (M )
L
2
x
≤ C
0
ψ
1
− ψ
2
L
2
x
+ P v
1
− v
2
L
∞
T (M )
L
2
x
P = C
1
T (M)
(
1−
Nα
4
)
K(M)
α
v
1
− v
2
L
q
T (M )
L
r
x
≤ CT (M)
1−
Nα
4
v
1
− v
2
L
q
T (M )
L
r
x
.
v
1
− v
2
L
∞
T (M )
L
2
x
+ v
1
− v
2
L
q
T (M )
L
r
x
≤ C
0
ψ
1
− ψ
2
L
2
x
+
+ P
v
1
− v
2
L
∞
T (M )
L
2
x
+ v
1
− v
2
L
q
T (M )
L
r
x
.
P
= C
α,M
T (M)
1−
Nα
4
T (M) C
α,M
T (M)
1−
Nα
4
≤
1
2
v
1
− v
2
L
∞
T (M )
L
2
x
+ v
1
− v
2
L
q
T (M )
L
r
x
≤ 2C
0
ψ
1
− ψ
2
L
2
x
(∗)
[0, T (M)]
(4.1)
u (4.1) [0, T ] A = max {1, 2C
0
} n ∈ N
T ≤ nT (M) δ =
M
2A
n
u [0, T ]
u(0)
L
2
= ϕ
L
2
≤
M
2
.
ψ ψ − ϕ
L
2
≤ δ ψ
L
2
≤ ϕ
L
2
+ δ
ψ
L
2
≤
M
2
+ δ ≤ M.
(4.1) v ψ [0, T (M)]
[0,
T
n
] u v (4.1) ϕ ψ
(∗)
u − v
L
∞
T /n
L
2
x
+ u − v
L
q
T /n
L
r
x
≤ A ψ − ϕ
L
2
x
.
u
T
n
L
2
≤
M
2
M
v
T
n
L
2
=
v
T
n
− u
T
n
L
2
+
u
T
n
L
2
≤ A ψ − ϕ
L
2
+
M
2
≤ Aδ +
M
2
≤ M.
˜ϕ = u
T
n
,
˜
ψ = v
T
n
˜u(t) = u
t +
T
n
, ˜v(t) = v
t +
T
n
˜u(t) ˜v(t) ˜ϕ
˜
ψ [0, T /n]
u, v [T/n, 2T/n]
u − v
L
∞
2T /n
L
2
x
+ u − v
L
q
2T /n
L
r
x
≤ A
2
ψ − ϕ
L
2
x
.
n u
i
[(i − 1)
T
n
, i
t
n
]
i = 1, 2, 3, ..., n
u
i
− v
L
∞
iT /n
L
2
x
+ u
i
− v
L
q
iT /n
L
r
x
≤ A
i
ψ − ϕ
L
2
x
.
u
u |
[(i−1)
T
n
,i
t
n
]
= u
i
.
u − v
L
∞
T
L
2
x
+ u − v
L
q
T
L
r
x
≤ C ψ − ϕ
L
2
x
(4.1)
H
1
(R
N
)
(4.1)
H
1
(R
N
) L
2
(R
N
)
H
1
(R
N
)
L
2
(R
N
) H
1
(R
N
)
α <
4
N−2
ϕ ∈ H
1
(R
N
) T = T (ϕ
H
1
) > 0
u (4.2) [0, T ]
u ∈ C([0, T ], H
1
(R
N
)) ⊂ L
q
([0, T ], W
1,r
(R
N
))
r =
N(α+2)
N+α
q =
4(α+2)
α(N−2)
W
1,r
(R
N
) f ∈ L
r
(R
N
)
u
0
−→ u(t)
E =
u ∈ C([0, T ], H
1
(R
N
)) ⊂ L
r
([0, T ], W
1,r
(R
N
) : u
T
< R
u
T
= sup
[0,T ]
u
H
1
+
T
0
u(t)
q
L
r
x
+ ∇u(t)
q
L
r
x
dt
1
q
Φ : E −→ E
Φ(u) = S(t)ϕ + iγ
t
0
S(t − s) |u(s)|
α
u(s)ds.
|u|
α
∇u
L
r
x
≤ C |u|
α
L
l
x
∇u
L
r
x
= C u
α
L
αl
x
∇u
L
r
x
,
1
r
=
1
l
+
1
r
p = αl, θ = 1, m = 1, j = 0
u
α
L
αl
x
≤ ∇u
L
r
x
|u|
α
∇u
L
r
x
≤ C ∇u
α+1
L
r
x
,
1
αl
=
1
r
−
1
N
1
l
=
α
r
−
α
N
1
l
=
1
r
−
1
r
= 1 −
1
r
−
1
r
= 1 −
2
r
α
r
−
α
N
= 1 −
2
r
=⇒
α
r
=
N + α
N(α + 2)
|u|
α
u
W
1,r
≤ C u
α+1
W
1,r
(∗)
Φ(u) ∈ E
Φ(u)
T
= sup
[0,T ]
Φ(u)
H
1
+
T
0
Φ(u)
q
L
r
x
+ ∇Φ(u)
q
L
r
x
dt
1
q
≤ sup
[0,T ]
Φ(u)
H
1
+
T
0
(Φ(u)
L
r
x
+ ∇Φ(u)
L
r
x
q
dt
1
q
= Φ(u)
L
∞
T
H
1
+ Φ(u)
L
q
T
W
1,r
Φ(u)
L
∞
T
H
1
≤ S(t)ϕ
L
∞
T
H
1
+
T
0
S(t − s).γ |u|
α
uds
L
∞
T
H
1
≤ C ϕ
H
1
+ C |u|
α
u
L
q
T
W
1,r
.
Φ(u)
L
q
T
W
1,r
≤ S(t)ϕ
L
q
T
W
1,r
+
T
0
S(t − s).γ |u|
α
uds
L
q
T
W
1,r
≤ C ϕ
H
1
+ C |u|
α
u
L
q
T
W
1,r
.
Φ(u)
T
≤ C ϕ
H
1
+ C |u|
α
u
L
q
T
W
1,r
= C ϕ
H
1
+ C
T
0
|u|
α
u
q
W
1,r
dt
1
q
(∗)
Φ(u)
T
≤ C ϕ
H
1
+
T
0
|u
(α+1)q
W
1,r
dt
1
q
4.2
Φ(u)
T
≤ ϕ
H
1
+ T
δ
u
α+1
L
q
T
W
1,r
δ = 1 −
α+2
q
= 1 −
α(N−2)
4
R = 2C ϕ
H
1
Φ(u)
T
≤ ϕ
H
1
+ T
δ
u
α+1
T
≤
R
2
+ CT
δ
R
α+1
(2C)
α+1
≤ R,
CT
δ
R
α
(2C)
α+1
≤
1
2
Φ : E −→ E
Φ
u Φ(u) = u
L
2
(R
N
)
L
2
(R
N
)
(4.1)
L
2
(R
N
)
γ ∈ R (4.1) (4.2)
u(t, x)
L
2
= ϕ(x)
L
2
∀ t ∈ [0, T
∗
)
(4.1)
H
1
(R
N
)
> 0 J
H
−1
(R
N
)
J
u = (I − ∆)
−1
u
H
−1
(R
N
) g(u) = γ |u|
α
u g
(u
) = J
g(J
u)
4.1 T = T (ϕ
L
2
) u
= J
u
[0, T ]
i (u
)
t
+ ∆
x
u
= g
(u
)
u
(x, 0) = ϕ(x).
J
u
∈ L
q
T
L
r
x
(q, r)
H
1
(R
N
) → L
r
(R
N
) ⇒ L
r
(R
N
) → H
−1
(R
N
)
g(J
u
) ∈ L
q
T
L
r
x
→ L
q
T
H
−1
x
g
= J
g(J
u
) ∈ L
q
T
H
1
x
.
i (u
)
t
+ ∆
x
u
= g
(u
) = f ∈ L
q
T
H
1
x
u
(x, 0) = ϕ(x) ∈ H
1
(R
N
).
u
iu
t
+ ∆
x
u = f(t, x)
u(x, 0) = 0.
u ∈ C(R, L
2
(R
N
))
u
L
∞
t
L
2
x
≤ C f
L
q
t
L
r
x
,
(q, r) f ∈ L
q
t
L
r
x
f
n
∈ C
∞
c
(R, R
N
) f
n
−→ f L
q
t
L
r
x
u
n
= i
t
0
S(t − s)f
n
(s, x)ds
u
n
∈ C(R, L
2
(R
N
))
u − u
n
L
∞
t
L
2
x
=
i
t
0
S(t − s)(f
n
− f)(s, x)ds
L
∞
t
L
2
x
≤ C f
n
− f
L
q
t
L
r
x
−→ 0.
u
n
−→ u u ∈ C(R, L
2
(R
N
))
4.4
u
(4.21)
u
(4.21)
i (u
)
t
+ ∆
x
u
= g
(u
)
iu
i (u
)
t
iu
+ iu
∆
x
u
= iu
g
(u
)
R
N
i (u
)
t
iu
dx +
R
N
∆
x
u
iu
dx =
R
N
g
(u
)iu
dx
R
N
i (ˆu
)
t
iu
dx +
R
N
∆
x
u
iu
dx =
R
N
ˆg
(u
)
iu
dx
R
N
1
1 + |ξ|
2
i (ˆu
)
t
(1 + |ξ|
2
)
iu
dx +
R
N
1
1 + |ξ|
2
∆
x
u
(1 + |ξ|
2
)
iu
dx
=
R
N
1
1 + |ξ|
2
ˆg
(u
)(1 + |ξ|
2
)
iu
dx
i (u
)
t
, iu
H
−1
,H
1
+ ∆
x
u
, iu
H
−1
,H
1
= g
(u
), iu
H
−1
,H
1
∆
x
u
, iu
H
−1
,H
1
= −i ∇u
, ∇u
L
2
= i ∇u
2
L
2
g
(u
), iu
H
−1
,H
1
= J
g(J
u
), iu
H
−1
,H
1
= g(J
u
), iJ
(u
)
H
−1
,H
1
=
R
N
|γ| |J
(u
)|
α
J
(u
)(−i)J
(u
)dx
= −i |γ|
R
N
|J
(u
)|
α+2
dx.
1
2
d
dt
u
2
L
2
=
1
2
R
N
d
dt
|u
|
2
dx
=
R
N
|u
| (u
)
t
dx
= −
R
N
iu
i(u
)
t
dx
= − i (u
)
t
, iu
H
−1
,H
1
= i∆
x
u
, iu
H
−1
,H
1
− g
(u
), iu
H
−1
,H
1
= −i ∇u
2
L
2
+ i |γ|
R
N
|J
(u
)|
α+2
dx.
1
2
d
dt
u
2
L
2
+ i
∇u
2
L
2
− |γ|
R
N
|J
(u
)|
α+2
dx
= 0
1
2
d
dt
u
(t, x)
2
L
2
= 0
u
(t, x)
L
2
= u
(0, x)
L
2
= ϕ(x)
L
2
u
−→ u −→ 0 L
∞
(I, L
2
(R
N
))
u
= S(t)ϕ + i
t
0
S(t − s)g
(u
)ds.
u − u
= i
t
0
S(t − s) (g(u) − g
(u
)) ds.
u − u
L
∞
T
L
2
x
≤ C g(u) − g
(u
)
L
q
T
L
r
x
u − u
L
q
T
L
r
x
≤ C g(u) − g
(u
)
L
q
T
L
r
x
u − u
L
∞
T
L
2
x
+ u − u
L
q
T
L
r
x
≤ C g(u) − g
(u
)
L
q
T
L
r
x
g(u) − g
(u
)
L
q
T
L
r
x
= g
(u
) − g
(u) + g
(u) − J
(g(u)) + J
(g(u)) − g(u)
L
q
T
L
r
x
≤ g
(u
) − g
(u)|
L
q
T
L
r
x
+ g
(u) − J
(g(u))
L
q
T
L
r
x
+ J
(g(u)) − g(u)
L
q
T
L
r
x
= I + II + III.
(4.3)
I = g
(u
) − g
(u)|
L
q
T
L
r
x
≤ CT
(1−
Nα
4
)
u
α
L
q
T
L
r
x
+ u
α
L
q
T
L
r
x
u
− u
L
q
T
L
r
x
T
CT
(1−
Nα
4
)
u
α
L
q
T
L
r
x
+ u
α
L
q
T
L
r
x
≤
1
2
,
I = g
(u
) − g
(u)
L
q
T
L
r
x
≤
1
2
u
− u
L
q
T
L
r
x
J
(u) −→ u −→ 0 1.4
II = g
(u) − J
(g(u))
L
q
T
L
r
x
= J
(g(J
u) − J
(g(u))
L
q
T
L
r
x
= J
(g(J
u) − g(u))
L
q
T
L
r
x
→0
−→ 0
III = J
(g(u)) − g(u)
L
q
T
L
r
x
→0
−→ 0
a
= II + III
g(u) − g
(u
)
L
q
T
L
r
x
≤
1
2
u
− u
L
q
T
L
r
x
+ a
T
u − u
L
∞
T
L
2
x
+ u − u
L
q
T
L
r
x
≤
1
2
u
− u
L
q
T
L
r
x
+ a
−→ 0 u
−→ u
(4.27)
u(t, x)
L
2
= ϕ(x)
L
2
T
ϕ ∈ L
2
(R
N
) ϕ ∈ L
2
(R
N
)
ϕ
n
∈ H
1
(R
N
) ϕ
n
L
2
−→ ϕ H
1
(R
N
) L
2
(R
N
)
u
n
(4.1) ϕ
n
u
n
L
2
−→ u
u
n
(t, x)
L
2
=
n−→∞
ϕ
n
(x)
L
2
n−→∞
u(t, x)
L
2
=
ϕ
(
x)
L
2
T
u(t, x)
L
2
= ϕ(x)
L
2
∀ t ∈ [0, T
∗
).
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