半線形熱方程式の解の挙動について

東京工業大大学院 理工学研究科 数学専攻
修士課程 解析系中間発表（2015.02.05）

Transcript

1. ൒ઢܗ೤ํఔࣜͷղͷڍಈʹ͍ͭͯ ౦ژ޻ۀେֶେֶӃ ཧ޻ֶݚڀՊ ਺ֶઐ߈ म࢜՝ఔ ໦ଜҰً
2. ࿦จ঺հ T. Cazenave, F. Dickstein, F. B. Weissler, Structural properties of the set of global solutions of the nonlinear heat equation (2010) ࿩͢͜ͱ 2
3. ಋೖ ೤ํఔࣜʢ֦ࢄํఔࣜʣ ൒ઢܗ೤ํఔࣜʢ൓Ԡ֦ࢄํఔࣜʣ ut = u ut = u + f(u) u = u ( x, t ) ut = u + |u|p 1u p > 1 3
4. ಋೖ ⌦ ⇢ RN ༗քͰͳΊΒ͔ͳྖҬ ͜ͷܗͷ൒ઢܗ೤ํఔࣜʹ͍ͭͯߟ͑Δɻ ۭؒ࣍ݩ ඇઢܗ߲ u = u ( x, t ) x 2 ⌦ , t 0 f 2 C1(R) N 1 8 > < > : ut u = f(u) in ⌦ ⇥ (0, T) u = 0 on @⌦ ⇥ (0, T) u(x, 0) = u0(x) in ⌦ 4
5. ಋೖ X = C0(⌦) 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 X 5
6. ಋೖ ͷͱ͖ɺ࣌ؒେҬղͱ͍͏ɻ ͷͱ͖ɺʢ༗ݶ࣌ؒʣരൃղͱ͍͍ɺ limt!Tu0 ku(·, t)kX = 1 Tu0 = 1 Tu0 < 1 Tu0 Λരൃ࣌ࠁͱ͍͏ɻ ͕੒ཱ͢Δɻ ͜ͷͱ͖ɺॳظ৚݅ʹରͯ͠Ұҙʹ ࣌ؒہॴతͳղ͕ଘࡏ͢Δɻ ղͷ࠷େଘࡏ࣌ؒ Tu0 6
7. ໰୊ G := {u0 2 X | Tu0 = 1} B := {u0 2 X | Tu0 < 1} X = G [ B ࣌ؒେҬղ രൃղ Λɺೋͭͷू߹ʹΘ͚Δɻ G \ B = ; ղۭؒ ͱͳ͍ͬͯΔɻ X = C0(⌦) 7
8. ໰୊ G ͸ ʹ͓͍ͯހঢ়࿈͔݁ʁ X ࣌ؒେҬղͷू߹ 8
9. ໰୊ ͕ತू߹ͱ͸ɺ Ұൠʹɺತू߹ͳΒ͹ɺހঢ়࿈݁Ͱ͋Δɻ ͕ހঢ়࿈͔݁Ͳ͏͔ʁ ʹ େҬղΛ΋ͨΒ͢ॳظؔ਺Λ ࿈ଓతʹมԽͤ͞Δͱരൃ͢Δղ͕Ͱͯ͘Δ͔ʁ G G v, w 2 G ✓ 2 [0, 1] ೚ҙͷ ʹର͠ɺ ͕੒ཱ͢Δ͜ͱɻ u := ✓v + (1 ✓)w 2 G 9
10. ໰୊ G ͸ ʹ͓͍ͯತू߹͔ʁ X ࣌ؒେҬղͷू߹ G ͸ ʹ͓͍ͯހঢ়࿈͔݁ʁ X ࣌ؒେҬղͷू߹ 10
11. ྫ f = 0 ʢ೤ํఔࣜʣͳΒ͹ɺ X = G G 6= ;, B 6= ; G ͸ತ͔Ͳ͏͔ʁ ͳΒ͹ɺ G ͸ತͰ͋Δɻ ͱͳΔɻ ͱͳΔɻ 11 f(s) = |s|p 1s, p > 1
12. ઌߦ݁Ռ ͕ತؔ਺ͳΒ͹ɺ P.L.Lions (1982) f G ͸ತू߹ɻ 12
13. ఆཧ ͸ತͰ͸ͳ͍ɻ G T. Cazenave, F. Dickstein, F. B. Weissler(2010) 1 < p < ps ΛؚΉ৚݅Լʹ͓͍ͯɺ ತʹͳΒͳ͍ྫ͕ଘࡏ͢Δɻ f(s) = |s|p 1s 13
14. ఆཧ ⌦ ⇢ RN ༗քͰͳΊΒ͔ͳྖҬ ͸ತͰ͸ͳ͍ɻ G ͱ͢Δɻ ͜ͷͱ͖ɺ |f(s)|  C(1 + |s|p) 8s 2 R sf ( s ) > 0 for s large ", C > 0 ps := ( 1 N = 1, 2 (N + 2)/(N 2) N 3 f 2 C1(R) T. Cazenave, F. Dickstein, F. B. Weissler (2010) 1 < p < ps sf0(s) (1 + ")f(s) 8s 2 R 14
15. ূ໌ ✓v + (1 ✓)w 62 G v, w 2 G ͋Δ Ͱ Λຬͨ͢΋ͷΛݟ͚͍ͭͨɻ ූ߸มԽ͢Δఆৗղ Ͱ͋Δɻ ͕ଘࡏ͢Δ͜ͱ͕ࣔͤΔɻ w 2 G w ( w = f ( w ) in ⌦ w = 0 on @ ⌦ ✓ 2 (0, 1) 15
16. ূ໌ ͷୈҰݻ༗஋ʹର͢Δݻ༗ؔ਺ͱ͢Δɻ L ූ߸มԽ͢Δఆৗղͱ͢Δɻ f ʹର͢ΔԾఆ͔Βɺ ͸઴ۙ҆ఆͳղɻ ͱ͢Δɻ u = 0 " ͕ͨͬͯ͠ɺ Λे෼খ͘͞औΕ͹ɺ w L ઢܗ࡞༻ૉ Lu := u f0(w)u Λ ͱఆΊΔɻ v1 v := "v1 v 2 G Λ 16
17. ূ໌ v, w 2 G Ͱ͋Δ͜ͱ͸͔֬Ίͨɻ ූ߸มԽ͢Δఆৗղ w ͷୈҰݻ༗஋ʹର͢Δݻ༗ؔ਺ͷఆ਺ഒ L v ✓v + (1 ✓)w 62 G Λຬͨ͢͜ͱΛ͍ࣔͨ͠ɻ 17
18. ূ໌ ิ୊ ූ߸มԽ͢Δఆৗղͱ͢Δɻ ·ͨ͸ ͸ Λຬͨ͢ͱ͢Δɻ ͜ͷͱ͖ɺ u0 ʹରԠ͢Δํఔࣜͷղ͸രൃ͢Δɻ ͭ·Γɺ f ͷԾఆΛ༻͍ͯɺ࣍ͷ͜ͱ͕ࣔͤΔɻ w u0 2 X, u0 6= w u0 w u0  w u0 2 B 18
19. ূ໌ ͕ࣔͤΔɻ ௐ΂Δ͜ͱͰɺ ✓ > 0 ʹରͯ͠ɺ ූ߸มԽ͢Δఆৗղ w ͷୈҰݻ༗஋ʹର͢Δݻ༗ؔ਺ͷఆ਺ഒ L v w ʹ͍ͭͯઢܗԽͨ͠ํఔࣜΛ ݩͷํఔࣜΛ u✓ 0 := ✓v + (1 ✓)w u✓ 0 > w ͱஔ͘ɻ 19 े෼খ͞ͳ
20. ূ໌ ͕ͨͬͯ͠ɺิ୊ΑΓɺ ✓v + (1 ✓)w 62 G G ͕ತू߹Ͱͳ͍͜ͱ͕ࣔͤͨɻ ͕ࣔͤͨɻ ͸ූ߸มԽ͢ΔఆৗղͰ͋Γɺ w u✓ 0 > w Ҏ্ʹΑΓɺ 20
21. ࢒͞Εͨ໰୊ ಉ͡ԾఆͷݩͰɺހঢ়࿈͔݁ʁ ತͰͳ͍͔Βͱ͍ͬͯɺހঢ়࿈݁Ͱͳ͍ͱ͸ݶΒͳ͍ɻ ͜ΕΒͷ໰୊͸ɺະղܾ໰୊ͱͯ͠঺հ͞Ε͍ͯΔɻ p ps ͷͱ͖ɺತ͔ʁ ඇઢܗ߲ͷ΂͖ͷڧ͞ p ͸੍ݶ͞Ε͍ͯͨɻ 21
22. ·ͱΊ G := {u0 2 X | Tu0 = 1} B := {u0 2 X | Tu0 < 1} ࣌ؒେҬղ രൃղ X = C0(⌦) ղۭؒ ͸ತͰ͸ͳ͍ɻ G 1 < p < ps f(s) = |s|p 1s 8 > < > : ut u = f(u) in ⌦ ⇥ (0, T) u = 0 on @⌦ ⇥ (0, T) u(x, 0) = u0(x) in ⌦ 22