Proceedings of 46th Sapporo Symposium on Partial Differential Equations

[1] G. Akagi and M. Efendiev, Allen-Cahn equation with strong irreversibility, European

J. Appl. Math. 30 (2019), 707–755.

[2] G. Akagi, C. Kuehn and K.-I. Nakamura, Traveling wave dynamics for Allen-Cahn

equations with strong irreversibility, arXiv:2004.12386 [math.AP].

[3] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997), 125–160.

[4] P.C. Fife and J.B. McLeod, The approach of solutions of nonlinear diffusion equations

to travelling front solutions, Arch. Ration. Mech. Anal. 65 (1977), 335–361.

[5] R.A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7 (1937),

353–369.

[6] I.Ya. Kanel, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.) 59 (1962), 245–288.

[7] A. Kolmogorov, I. Petrovskii, N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem,

Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh. 1 (1937) 1–26.

Goro Akagi: Mathematical Institute and Graduate School of Science, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai 980-8578 Japan

Email address: goro.akagi@tohoku.ac.jp

−101−

Necessary and sufficient condition for global existence

of L2 solutions for 1D periodic NLS

with non-gauge invariant quadratic nonlinearity

౻‫ ݪ‬࿨ক

1 ֓ཁ

ຊߨԋͰ͸, ࣍ͷप‫ڥظ‬ք৚݅ʹԙ͚Δඇઢ‫ ܕ‬Schr¨

odinger ํఔࣜͷॳ‫ظ‬஋໰୊ͷ࣌ؒେҬՄղੑΛѻ͏.

i∂t u + Δu = |u|2 , t ∈ (0, T ),

u(0, x) = φ(x),

x ∈ T.

x ∈ T,

(1)

i ͸‫਺ڏ‬୯ҐͰ͋Γ, T > 0, Δ = ∂x2 , T = R/2πZ ͱ͢Δ. ຊߨԋͷ໨త͸, (1) ͷ L2 (T) ʹԙ͚Δ࣌ؒେҬՄ

ղੑʹର͢Δ φ ∈ L2 (T) ͷඞཁॆ෼৚݅Λ༩͑ΔࣄͰ͋Δ.

(1) ͷղͷ‫ڍ‬ಈΛߟ͑Δʹઌཱͬͯ, प‫ڥظ‬ք৚݅Ͱͷࣗ༝ Schr¨odinger ํఔࣜͷॳ‫ظ‬஋໰୊ͷղͷ‫ڍ‬ಈ͸,

odinger ํఔࣜͷॳ‫ظ‬஋໰୊ͷղͷ‫ڍ‬ಈͱ͸ҟͳΔࣄʹ஫ҙ͢Δ. R ্ͷࣗ༝ Schr¨odinger ํ

R ্ͷࣗ༝ Schr¨

ఔࣜͷղ͸, ೚ҙͷ t > 0 ʹରͯ͠ධՁࣜ

eitΔ φLp (R) ≤ (4πt)−(1/2−1/p) φLp (R)

(2)

Λຬͨ͢ (ྫ͑͹ [5] Λࢀর). ୠ͠, p ≥ 2 Ͱ͋Γ p ͸ p ͷ H¨

older ‫ڞ‬໾ࢦ਺ͱͨ͠. (2) ͸‫ج‬ຊղͷ۩ମ‫ܗ‬

odinger ํఔࣜͷղ͸, ࣌ؒൃలʹࡍͯ͠ॳ‫ͱࠁ࣌ظ‬

Λ༻͍ͯࣔ͢ࣄ͕Ͱ͖Δ. (2) ʹΑΓ, R ্ͷࣗ༝ Schr¨

͸ҟͳΔՄੵ෼ੑΛ༗͢Δࣄ͕෼͔Δ. ಛʹ, (2) ʹΑΔॳ‫ʹࠁ࣌ظ‬ԙ͚ΔಛҟੑͷධՁΛ༻͍ΔࣄͰ, (1) ͷ

༷ͳඇઢ‫ظॳܕ‬஋໰୊ͷ࣌ؒ‫ॴہ‬ղ͕ߏ੒Ͱ͖Δࣄ͸Α͘஌ΒΕ͍ͯΔ. ҰํͰ, प‫ڥظ‬ք৚݅ʹԙ͚Δࣗ༝

Schr¨

odinger ํఔࣜͷղʹରͯ͠͸, (2) ͷ༷ͳධՁࣜ͸੒ཱ͠ͳ͍. ྫ͑͹, f ∈ L2 (T) Λ೚ҙͷ p > 2 ʹର

/ Lp (T) ͱͳΔ༷ʹऔΔ. ࣍ʹ φ = e−iΔ f ͱ͢Ε͹,

ͯ͠, f ∈

sup φLq (T) ≤

q∈[1,2]

2πφL2 (T) =

Ͱ͋Δ͕,

eiΔ φ = f ∈

2πf L2 (T)

Lp (T)

p>2

Ͱ͋Δ͔Β, (2) ͷ༷ʹ࣌ؒൃలʹࡍͯ͠ॳ‫ظ‬஋ͷಛҟੑ͕؇࿨͞ΕΔͱ͸‫ݶ‬Βͳ͍. ैͬͯ, L∞ (T) ʹຒଂ

͞Εͳ͍ Sobolev ۭؒ H s (s ≤ 1/2) ʹର͢Δ (1) ͷ࣌ؒ‫ॴہ‬ղͷߏ੒ʹ͸, (2) ͱ͸ҟͳΔධՁ͕ඞཁͱͳ

Δ. Bourgain[1] ͸ (t, x) ∈ T × T ʹԙ͚Δࣗ༝ Schr¨

odinger ํఔࣜͷղ͕ධՁࣜ

eitΔ φL4 (T×T) ≤

−103−

2φL2 (T)

(3)

Λຬͨ͢ࣄΛࣔͨ͠. (3) ͸࣌ؒप‫ظ‬ղʹର͢ΔධՁͰ͋Δ͕, (3) ʹΑΓ (1) ͸ L2 (T) ʹԙ͍ͯ࣌ؒ‫ʹॴہ‬ద

੾Ͱ͋Δࣄ͕ࣔ͞ΕΔ. (3) ͸ؔ܎ࣜ

|eitΔ φ(x)|2 =



k∈Z ...