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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)

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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

  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 1ɽ ࣗ໌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