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）

Transcript

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

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

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

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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
   

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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
   

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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
   

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

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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
   

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

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

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

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12. ઌߦ݁Ռ
    ̎ɽ͋Δ ɹɹɹɹ
    ⌘, " > 0
    f(s) ⌘s1+✏
    s 2 R
    f(0) = 0 ͱ͍͏৚݅ΛՃ͑Ε͹ɺ
    1ɽ
    ࣗ໌ղͱ͍͏େҬղ͕ଘࡏ͢Δɻ
    രൃղ͕ଘࡏ͢Δɻ
    ͕ଘࡏͯ͠ɺे෼େ͖ͳ
    ʹରͯ͠ɺ ͱ͍͏৚݅ΛՃ͑Δͱɺ
    G 6= ;, B 6= ; ͱͳΔඇઢܗ߲ͷ৚݅ʹ͍ͭͯ
    12
    

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

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

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

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16. ओఆཧ
    ओఆཧ͸ɺۭؒ̍࣍ݩʹ੍ݶͨ͠ͱ͖ʹɺେҬղͷू
    ߹͕ހঢ়࿈݁ੑΛ໌Β͔ʹͨ͠ɻ
    16
    

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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
    

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

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19. ূ໌
    C0(⌦)
    2 S
    v
    0
    19
    

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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
    

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21. ূ໌
    C0(⌦)
    2 S
    v
    0
    u
    21
    

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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
    

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23. ূ໌
    C0(⌦)
    2 S
    v
    0
    2 G
    u0
    u
    23
    

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

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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32. ূ໌
    C0(⌦)
    2 S
    v
    0
    2 G
    u0
    u
    32
    

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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
    

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