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非線形熱方程式の大域解の集合の連結性

 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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  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

    ࣗ໌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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