March 19, 2016
530

# 非線形熱方程式の大域解の集合の連結性

March 19, 2016

## Transcript

1. ඇઢܗ೤ํఔࣜͷେҬղͷ࿈݁ੑ
౦ژ޻ۀେֶେֶӃ ཧ޻ֶݚڀՊ
਺ֶઐ߈ म࢜՝ఔ
໦ଜҰً
1

2. ໨࣍
1. ಋೖ
2. ໰୊ઃఆ
3. ઌߦݚڀ
4. ఆཧ
5. ূ໌
6. ·ͱΊ
2

3. ಋೖ
೤ํఔࣜʢ֦ࢄํఔࣜʣ
ඇઢܗ೤ํఔࣜʢ൓Ԡ֦ࢄํఔࣜʣ
ut = u
ut = u + f(u)
u
=
u
(
x, t
)
ut = u + |u|p 1u p > 1
3

4. ಋೖ
ut = u + |u|p 1u
p > 1 pF := (N + 2)/N
Fujita(1966) ઌۦతݚڀʢྟքࢦ਺ͷൃݟʣ
౻ాܕํఔࣜ
ͳΒɺਖ਼஋େҬղ͕ଘࡏ͢Δɻ
ͳΒɺ͢΂ͯͷਖ਼஋ղ͕༗ݶ࣌ؒͷ͏ͪʹ
p < pF
p > pF
in RN
ൃࢄ͢Δʢղͷരൃʣ
4

5. ಋೖ
⌦ ⇢ RN ༗քͰͳΊΒ͔ͳྖҬ
ۭؒ࣍ݩ
u
=
u
(
x, t
)
x
2 ⌦
, t
0
N 1
8
>
<
>
:
ut u = f(u) in ⌦

(0, T)
u = 0 on @⌦

(0, T)
u(x, 0) = u0(x) in ⌦
͜ͷܗͷ൒ઢܗ೤ํఔࣜʹ͍ͭͯߟ͑Δɻ
5

6. ಋೖ
ղۭؒ ʢsup ϊϧϜʣ
8
>
<
>
:
ut u = f(u) in ⌦

(0, T)
u = 0 on @⌦

(0, T)
u(x, 0) = u0(x) in ⌦
ॳظ৚݅
u0
2 C0(⌦)
ඇઢܗ߲
f 2 C1(R)
C0(⌦)
6

7. ಋೖ
ͷͱ͖ɺ࣌ؒେҬղͱ͍͏ɻ
ͷͱ͖ɺʢ༗ݶ࣌ؒʣരൃղͱ͍͍ɺ
Tu0
= 1
Tu0
< 1
Tu0
Λരൃ࣌ࠁͱ͍͏ɻ
͜ͷͱ͖ɺॳظ৚݅ʹରͯ͠Ұҙʹ
࣌ؒہॴతͳղ͕ଘࡏ͢Δɻ
ղͷ࠷େଘࡏ࣌ؒ
Tu0
7

8. ໰୊ઃఆ
G := {u0
2 X | Tu0
= 1}
B := {u0
2 X | Tu0
< 1}
࣌ؒେҬղ
രൃղ
Λɺೋͭͷू߹ʹΘ͚Δɻ
G \ B = ;
ղۭؒ
ͱͳ͍ͬͯΔɻ
C0(⌦)
C0(⌦) = G [ B
8

9. ໰୊ઃఆ
G ͸ ʹ͓͍ͯހঢ়࿈͔݁ʁ
࣌ؒେҬղͷू߹
G ͸ ʹ͓͍ͯ࿈͔݁ʁ
࣌ؒେҬղͷू߹
େҬղͱരൃղ͕ࠞࡏ͢Δํఔࣜʹ͓͍ͯɺେҬղ
ͷू߹͕࿈݁Ͱ͋Δ͔Ͳ͏͔͸ɺํఔࣜͷղͷߏ଄
Λཧղ͢ΔͨΊͷॏཁͳಛ௃Ͱ͋Δͱߟ͑ΒΕΔɻ
C0(⌦)
C0(⌦)
9

10. ઌߦ݁Ռ
P. L. Lions (1982)
͕ತؔ਺ͳΒ͹ɺ
f G ͸ತू߹ɻ
→ G ͸࿈݁ɺހঢ়࿈݁Ͱ͋Δɻ
10

11. ઌߦ݁Ռ
T. Cazenave, F. Dickstein, F. B. Weissler (2010)
|f| ͕1ΑΓେ͖͘1ʹे෼͍ۙႈؔ਺Ͱ
্͔Β͓͑͞ΒΕɺ͞Βʹ͍͔ͭ͘ͷ৚݅
Λຬͨ͢ͳΒɺG ͸ತͰ͸ͳ͍ɻ
11

12. ઌߦ݁Ռ
̎ɽ͋Δ ɹɹɹɹ
⌘, " > 0
f(s) ⌘s1+✏
s 2 R
f(0) = 0 ͱ͍͏৚݅ΛՃ͑Ε͹ɺ

ࣗ໌ղͱ͍͏େҬղ͕ଘࡏ͢Δɻ
രൃղ͕ଘࡏ͢Δɻ
͕ଘࡏͯ͠ɺे෼େ͖ͳ
ʹରͯ͠ɺ ͱ͍͏৚݅ΛՃ͑Δͱɺ
G 6= ;, B 6= ; ͱͳΔඇઢܗ߲ͷ৚݅ʹ͍ͭͯ
12

13. ઌߦ݁Ռ
ͱ͢Δͱɺ
f(s) = |s|p 1s, p > 1
G 6= ;, B 6= ; ͱͳΔɻ
ઌ΄Ͳͷ̍ɽ̎ɽͷ৚݅Λຬͨ͠ɺ
13

14. ઌߦ݁Ռ
T. Cazenave, F. Dickstein, F. B. Weissler (2010)
ɹɹɹ͕ɺ1ʹे෼͍ۙͳΒ͹ɺ
ٿରশͳେҬղͷू߹͸࿈݁Ͱ͋Δɻ
p > 1
14

15. ઌߦ݁Ռ
T. Cazenave, F. Dickstein, F. B. Weissler (2010)
→ހঢ়࿈݁ੑ͸ෆ໌
ɹɹɹ͕ɺ1ʹे෼͍ۙͳΒ͹ɺ
ٿରশͳେҬղͷू߹͸࿈݁Ͱ͋Δɻ
p > 1
15

16. ओఆཧ
ओఆཧ͸ɺۭؒ̍࣍ݩʹ੍ݶͨ͠ͱ͖ʹɺେҬղͷू
߹͕ހঢ়࿈݁ੑΛ໌Β͔ʹͨ͠ɻ
16

17. ओఆཧ
͸ހঢ়࿈݁Ͱ͋Δɻ
G
͜ͷͱ͖ɺ
ͱ͢Δɻ
8
>
<
>
:
ut = uxx +
|
u
|p
1
u in ( 1, 1)

(0, T)
u = 0 on @
{
( 1, 1)
} ⇥
(0, T)
u(x, 0) = u0(x) in [ 1, 1]
N = 1, ⌦ = ( 1, 1)
17

18. ূ໌
• ൃදʹ͓͚Δূ໌ͷྲྀΕ
• ໋୊Λ༻ҙ
• ໋୊Λ࢖ͬͯఆཧΛূ໌
• ໋୊Λূ໌
18

19. ূ໌
C0(⌦)
2 S
v
0
19

20. ূ໌
ఆཧΛূ໌͢ΔͨΊʹɺ࣍ͷ໋୊Λࣔ͢ɻ
໋୊
Λͭͳ͙ϔςϩΫϦχοΫيಓ͕ଘࡏ͢Δɻ
S
v 0
v 2 S
Λఆৗղͷू߹ͱ͠ɺ
೚ҙʹඇࣗ໌ͳఆৗղ ΛͱΔɻ
ͱ
u ! v (t ! 1), u ! 0 (t ! 1)
u 2 G
ͱͳΔΑ͏ͳ ͷ͜ͱʣ
20

21. ূ໌
C0(⌦)
2 S
v
0
u
21

22. ূ໌
͕੒ཱ͢Δ͜ͱ͕஌ΒΕ͍ͯΔɻ
ఆཧͷূ໌
!(u0) ⇢ S
!(u0) ͷਖ਼ͷۃݶू߹ʢω-ۃݶू߹ʣ
͜ͷํఔࣜͰ͸ɺ೚ҙͷ ʹର͠ɺ
u0
u0
2 G
ରԠ͢Δղ ͕ t ! 1 ͰҰ༷ʹ༗քͰ͋Γɺ
ʢํఔࣜʹରԠ͢ΔΤωϧΪʔ൚ؔ਺Λௐ΂Δʣ
22

23. ূ໌
C0(⌦)
2 S
v
0
2 G
u0
u
23

24. ূ໌
͕੒ཱ͢Δɻ
!(u0) ⇢ S
໋୊ʹΑΓɺ೚ҙͷඇࣗ໌ͳఆৗղ v 2 S
͸ɺࣗ໌ղ0΁ͱͭͳ͛ΒΕΔɻ
ΑͬͯɺେҬղͷू߹ ͸ހঢ়࿈݁Ͱ͋Δɻ
G
ఆཧͷূ໌ऴΘΓ
24

25. ূ໌
ิ୊
໋୊ͷূ໌
(
'
xx +
p|v|p
1'
=
'
in ( 1
,
1)
'
= 0 on
@
( 1
,
1)
ͷ·ΘΓͰͷઢܗԽݻ༗஋໰୊Λߟ͑Δɻ
v 2 S
࠷େͷݻ༗஋Λ
͕ෆ҆ఆͰ͋Δ͜ͱΛҙຯ͢Δɻ
ɺରԠ͢Δݻ༗ؔ਺Λ
1 '1 ͱ͢Δɻ
͜ͷͱ͖ɺ 1 > 0 Ͱ͋Δɻ
͜Ε͸ v
25

26. ূ໌
1 = sup
U2H1
0
(⌦),U6⌘0
R

{ |r
U
|2 +
p'
p 1
1 U
2}
dx
R
⌦ U
2
dx
ม෼ݪཧʹΑΓ࠷େݻ༗஋͸ϨΠϦʔ঎ͱͯ͠දͤΔɻ
U = '1 ͱͯ͠෼ࢠΛܭࢉ͢Δɻ
'1 ͕ఆৗղͰ͋Δ͜ͱɺάϦʔϯͷఆཧ
Λ࢖͍ܭࢉ͢Δͱ 1 > 0
26

27. ূ໌
Λͭͳ͙ϔςϩΫϦχοΫيಓ͕ଘࡏ͢Δɻ
S
v 0
v 2 S
Λఆৗղͷू߹ͱ͠ɺ
೚ҙʹඇࣗ໌ͳఆৗղ ΛͱΔɻ
ͱ
͜ͷ໋୊Λࣔͨ͢Ίʹ͸ɺ
u ! v (t ! 1), u ! 0 (t ! 1)
u 2 G
ͱͳΔΑ͏ͳ ͕ଘࡏ͢Δ͜ͱΛࣔͤ͹ྑ͍ɻ
27

28. ূ໌
༏ղɾྼղͷํ๏Ͱ u ! v (t ! 1), u ! 0 (t ! 1)
u 2 G
ͱͳΔΑ͏ͳ Λߏ੒͢Δɻ
Ͱද͠ɺ৔߹෼͚͍ͯࣔͯ͘͠͠ɻ
u := v "e (t)'1(t < 0)
u := v "e 1t'1(t < 0)
৔߹̍ɽ ͷͱ͖
Λద౰ʹܾΊɺ
ͱ͓͘ɻ
z[v]
v
z[v] = 0
ͷྵ఺਺Λ
",
28

29. ূ໌
u, u ! v (t ! 1)
u := v "e (t)'1(t < 0)
u := v "e 1t'1(t < 0)
Λద౰ʹܾΊΔ͜ͱͰ
͜ΕΒ͕༏ղɾྼղͱͳΔɻ
΋੒ཱɻ
(
t
) = 1t 1
p
1
log(1 +
1
1
e 1(p 1)t
)
",
29

30. ূ໌
Y. Fukao, Y. Morita, H. Ninomiya(2004) Λࢀߟʹ
͕ͨͬͯ͠ɺ༏ղɾྼղͷํ๏ʹΑΓɺ
ui(
x, i
) :=
u
(|
x
|
, i
)
{ui
}i2N
ͱ͓͖ɺ
ui
! u (i ! 1) ͕ࣔͤΔɻ
ΞείϦɾΞϧπΣϥͷఆཧΛ༻͍ͯ
Λߏ੒͢Δɻ
ղͷྻ
t < 0
Ͱఆٛ͞Εͨղ u ͕ଘࡏ͢Δɻ
͜͜Ͱ
30

31. ূ໌
v u
u ! 0 (t ! 1)
ۭؒ1࣍ݩͷ໰୊Ͱྵ఺਺͕ඇ૿ՃͰ͋Δ͜ͱɺ
͕ࣔͤͨɻ
৔߹2. z[v] = k(k 2 N)
͜ͷ৔߹͸ɺ z[v] = 0
v Λ ͱͳΔղͷͭͳ͗߹Θͤ
ͱͯ͠ߟ͑Δ͜ͱͰɺz[v] = 0 ͷ৔߹ʹؼண͢Δɻ
ূ໌ऴΘΓ
͕ෆ҆ఆͰ͋Δ͜ͱʢิ୊1ʣΑΓɺ
v
31
ܭࢉʹΑΓ

32. ূ໌
C0(⌦)
2 S
v
0
2 G
u0
u
32

33. ·ͱΊ
͸ހঢ়࿈݁Ͱ͋Δɻ
G
͜ͷͱ͖ɺ
ͱ͢Δɻ
8
>
<
>
:
ut = uxx +
|
u
|p
1
u in ( 1, 1)

(0, T)
u = 0 on @
{
( 1, 1)
} ⇥
(0, T)
u(x, 0) = u0(x) in [ 1, 1]
G ͸ ʹ͓͍ͯހঢ়࿈͔݁ʁ
X
࣌ؒେҬղͷू߹
N = 1, ⌦ = ( 1, 1)
33