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非線形熱方程式の大域解の集合の連結性
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木村すらいむ
March 19, 2016
Science
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非線形熱方程式の大域解の集合の連結性
東京工業大大学院 理工学研究科 数学専攻
修士前期課程 修論発表(2016.02.16)
木村すらいむ
March 19, 2016
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Transcript
ඇઢܗํఔࣜͷେҬղͷ࿈݁ੑ ౦ژۀେֶେֶӃ ཧֶݚڀՊ ֶઐ߈ म࢜՝ఔ ଜҰً 1
࣍ 1. ಋೖ 2. ઃఆ 3. ઌߦݚڀ 4. ఆཧ 5.
ূ໌ 6. ·ͱΊ 2
ಋೖ ํఔࣜʢ֦ࢄํఔࣜʣ ඇઢܗํఔࣜʢԠ֦ࢄํఔࣜʣ ut = u ut = u +
f(u) u = u ( x, t ) ut = u + |u|p 1u p > 1 3
ಋೖ ut = u + |u|p 1u p > 1
pF := (N + 2)/N Fujita(1966) ઌۦతݚڀʢྟքࢦͷൃݟʣ ౻ాܕํఔࣜ ͳΒɺਖ਼େҬղ͕ଘࡏ͢Δɻ ͳΒɺͯ͢ͷਖ਼ղ͕༗ݶ࣌ؒͷ͏ͪʹ p < pF p > pF in RN ൃࢄ͢Δʢղͷരൃʣ 4
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 ⌦ ͜ͷܗͷઢܗํఔࣜʹ͍ͭͯߟ͑Δɻ 5
ಋೖ ղۭؒ ʢsup ϊϧϜʣ 8 > < > : ut
u = f(u) in ⌦ ⇥ (0, T) u = 0 on @⌦ ⇥ (0, T) u(x, 0) = u0(x) in ⌦ ॳظ݅ u0 2 C0(⌦) ඇઢܗ߲ f 2 C1(R) C0(⌦) 6
ಋೖ ͷͱ͖ɺ࣌ؒେҬղͱ͍͏ɻ ͷͱ͖ɺʢ༗ݶ࣌ؒʣരൃղͱ͍͍ɺ Tu0 = 1 Tu0 < 1 Tu0
Λരൃ࣌ࠁͱ͍͏ɻ ͜ͷͱ͖ɺॳظ݅ʹରͯ͠Ұҙʹ ࣌ؒہॴతͳղ͕ଘࡏ͢Δɻ ղͷ࠷େଘࡏ࣌ؒ Tu0 7
ઃ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
ઃఆ G ʹ͓͍ͯހঢ়࿈͔݁ʁ ࣌ؒେҬղͷू߹ G ʹ͓͍ͯ࿈͔݁ʁ ࣌ؒେҬղͷू߹ େҬղͱരൃղ͕ࠞࡏ͢Δํఔࣜʹ͓͍ͯɺେҬղ
ͷू߹͕࿈݁Ͱ͋Δ͔Ͳ͏͔ɺํఔࣜͷղͷߏ Λཧղ͢ΔͨΊͷॏཁͳಛͰ͋Δͱߟ͑ΒΕΔɻ C0(⌦) C0(⌦) 9
ઌߦ݁Ռ P. L. Lions (1982) ͕ತؔͳΒɺ f G ತू߹ɻ →
G ࿈݁ɺހঢ়࿈݁Ͱ͋Δɻ 10
ઌߦ݁Ռ T. Cazenave, F. Dickstein, F. B. Weissler (2010) |f|
͕1ΑΓେ͖͘1ʹे͍ۙႈؔͰ ্͔Β͓͑͞ΒΕɺ͞Βʹ͍͔ͭ͘ͷ݅ Λຬͨ͢ͳΒɺG ತͰͳ͍ɻ 11
ઌߦ݁Ռ ̎ɽ͋Δ ɹɹɹɹ ⌘, " > 0 f(s) ⌘s1+✏ s
2 R f(0) = 0 ͱ͍͏݅ΛՃ͑Εɺ 1ɽ ࣗ໌ղͱ͍͏େҬղ͕ଘࡏ͢Δɻ രൃղ͕ଘࡏ͢Δɻ ͕ଘࡏͯ͠ɺेେ͖ͳ ʹରͯ͠ɺ ͱ͍͏݅ΛՃ͑Δͱɺ G 6= ;, B 6= ; ͱͳΔඇઢܗ߲ͷ݅ʹ͍ͭͯ 12
ઌߦ݁Ռ ͱ͢Δͱɺ f(s) = |s|p 1s, p > 1 G
6= ;, B 6= ; ͱͳΔɻ ઌ΄Ͳͷ̍ɽ̎ɽͷ݅Λຬͨ͠ɺ 13
ઌߦ݁Ռ T. Cazenave, F. Dickstein, F. B. Weissler (2010) ɹɹɹ͕ɺ1ʹे͍ۙͳΒɺ
ٿରশͳେҬղͷू߹࿈݁Ͱ͋Δɻ p > 1 14
ઌߦ݁Ռ T. Cazenave, F. Dickstein, F. B. Weissler (2010) →ހঢ়࿈݁ੑෆ໌
ɹɹɹ͕ɺ1ʹे͍ۙͳΒɺ ٿରশͳେҬղͷू߹࿈݁Ͱ͋Δɻ p > 1 15
ओఆཧ ओఆཧɺۭؒ̍࣍ݩʹ੍ݶͨ͠ͱ͖ʹɺେҬղͷू ߹͕ހঢ়࿈݁ੑΛ໌Β͔ʹͨ͠ɻ 16
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
ূ໌ • ൃදʹ͓͚Δূ໌ͷྲྀΕ • ໋Λ༻ҙ • ໋ΛͬͯఆཧΛূ໌ • ໋Λূ໌ 18
ূ໌ C0(⌦) 2 S v 0 19
ূ໌ ఆཧΛূ໌͢ΔͨΊʹɺ࣍ͷ໋Λࣔ͢ɻ ໋ 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
ূ໌ C0(⌦) 2 S v 0 u 21
ূ໌ ཱ͕͢Δ͜ͱ͕ΒΕ͍ͯΔɻ ఆཧͷূ໌ !(u0) ⇢ S !(u0) ͷਖ਼ͷۃݶू߹ʢω-ۃݶू߹ʣ ͜ͷํఔࣜͰɺҙͷ ʹର͠ɺ
u0 u0 2 G ରԠ͢Δղ ͕ t ! 1 ͰҰ༷ʹ༗քͰ͋Γɺ ʢํఔࣜʹରԠ͢ΔΤωϧΪʔ൚ؔΛௐΔʣ 22
ূ໌ C0(⌦) 2 S v 0 2 G u0 u
23
ূ໌ ཱ͕͢Δɻ !(u0) ⇢ S ໋ʹΑΓɺҙͷඇࣗ໌ͳఆৗղ v 2 S ɺࣗ໌ղ0ͱͭͳ͛ΒΕΔɻ
ΑͬͯɺେҬղͷू߹ ހঢ়࿈݁Ͱ͋Δɻ G ఆཧͷূ໌ऴΘΓ 24
ূ໌ ิ ໋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
ূ໌ 1 = sup U2H1 0 (⌦),U6⌘0 R ⌦ {
|r U |2 + p' p 1 1 U 2} dx R ⌦ U 2 dx มݪཧʹΑΓ࠷େݻ༗ϨΠϦʔͱͯ͠දͤΔɻ U = '1 ͱͯ͠ࢠΛܭࢉ͢Δɻ '1 ͕ఆৗղͰ͋Δ͜ͱɺάϦʔϯͷఆཧ Λ͍ܭࢉ͢Δͱ 1 > 0 26
ূ໌ Λͭͳ͙ϔςϩΫϦχοΫيಓ͕ଘࡏ͢Δɻ S v 0 v 2 S Λఆৗղͷू߹ͱ͠ɺ ҙʹඇࣗ໌ͳఆৗղ
ΛͱΔɻ ͱ ͜ͷ໋Λࣔͨ͢Ίʹɺ u ! v (t ! 1), u ! 0 (t ! 1) u 2 G ͱͳΔΑ͏ͳ ͕ଘࡏ͢Δ͜ͱΛࣔͤྑ͍ɻ 27
ূ໌ ༏ղɾྼղͷํ๏Ͱ u ! v (t ! 1), u !
0 (t ! 1) u 2 G ͱͳΔΑ͏ͳ Λߏ͢Δɻ Ͱද͠ɺ߹͚͍ͯࣔͯ͘͠͠ɻ u := v "e (t)'1(t < 0) u := v "e 1t'1(t < 0) ߹̍ɽ ͷͱ͖ ΛదʹܾΊɺ ͱ͓͘ɻ z[v] v z[v] = 0 ͷྵΛ ", 28
ূ໌ u, u ! v (t ! 1) u :=
v "e (t)'1(t < 0) u := v "e 1t'1(t < 0) ΛదʹܾΊΔ͜ͱͰ ͜ΕΒ͕༏ղɾྼղͱͳΔɻ ཱɻ ( t ) = 1t 1 p 1 log(1 + 1 1 e 1(p 1)t ) ", 29
ূ໌ 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༻͍ͯ Λߏ͢Δɻ ղͷྻ t < 0 Ͱఆٛ͞Εͨղ u ͕ଘࡏ͢Δɻ ͜͜Ͱ 30
ূ໌ v u u ! 0 (t ! 1) ۭؒ1࣍ݩͷͰྵ͕ඇ૿ՃͰ͋Δ͜ͱɺ
͕ࣔͤͨɻ ߹2. z[v] = k(k 2 N) ͜ͷ߹ɺ z[v] = 0 v Λ ͱͳΔղͷͭͳ͗߹Θͤ ͱͯ͠ߟ͑Δ͜ͱͰɺz[v] = 0 ͷ߹ʹؼண͢Δɻ ূ໌ऴΘΓ ͕ෆ҆ఆͰ͋Δ͜ͱʢิ1ʣΑΓɺ v 31 ܭࢉʹΑΓ
ূ໌ C0(⌦) 2 S v 0 2 G u0 u
32
·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