Tính đặt đúng trong lớp Gevrey đối với các toán tử hyperbolic yếu với các hệ số không liên tục Lipschitz

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■

▼❆■ ❚❍➚ ❚❯❨➌❚ ▼❆■

❚➑◆❍ ✣➄❚ ✣Ó◆● ❚❘❖◆● ▲❰P ●❊❱❘❊❨ ✣➮■ ❱❰■ ❈⑩❈
❚❖⑩◆ ❚Û ❍❨P❊❘❇❖▲■❈ ❨➌❯ ❱❰■ ❈⑩❈ ❍➏ ❙➮
❑❍➷◆● ▲■➊◆ ❚Ö❈ ▲■P❙❈❍■❚❩
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿ ❚❙✳ P❍❸▼ ❚❘■➋❯ ❉×❒◆●

❍⑨ ◆❐■✱ ✷✵✶✼


▼ö❝ ❧ö❝
P❤➛♥ ♠ð ✤➛✉

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

✶✳✷

✶✳✸

❈ì sð ❣✐↔✐ t➼❝❤ ❋♦✉r✐❡r ❝ê ✤✐➸♥✳ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✶✳✶

❈ì sð ❣✐↔✐ t➼❝❤ ❋♦✉r✐❡r ❝ê ✤✐➸♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✶✳✷

❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣✳ P❤÷ì♥❣ ♣❤→♣ ♥➠♥❣ ❧÷ñ♥❣ ✳ ✳ ✳

✶✵

✶✳✷✳✶

P❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣ ❝ê ✤✐➸♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✷✳✷

P❤÷ì♥❣ ♣❤→♣ ♥➠♥❣ ❧÷ñ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✶✳✷✳✸

P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣✳ ✳ ✳ ✳ ✳ ✳

✶✹

▲î♣ ●❡r✈❡② ✈➔ ❝→❝ ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ ✤÷ì♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✸✳✶

✣à♥❤ ♥❣❤➽❛ ❧î♣ ●❡r✈❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✸✳✷

❈→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❧î♣ ●❡✈r❡②✳

✶✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ❚➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✤è✐ ✈î✐ t♦→♥ tû ❤②♣❡r❜♦❧✐❝ ②➳✉
✈î✐ ❤➺ sè ♣❤ö t❤✉ë❝ t❤í✐ ❣✐❛♥ ❦❤æ♥❣ ❧✐➯♥ tö❝ t❤❡♦ ▲✐♣s❝❤✐t③
✶✽
✷✳✶

❑➳t q✉↔ ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ tr♦♥❣ ❧î♣ ●❡r✈❡② γ (s) ✤è✐ ✈î✐ 1

q<3 ✳ ✳ ✳

✶✽

✷✳✷

❑➳t q✉↔ ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ tr♦♥❣ ❧î♣ ●❡✈r❡② ✈î✐ sè ♠ô q ♥❤ä✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷✳✸

❑➳t ❧✉➟♥ ❝❤÷ì♥❣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✶


❑➳t ❧✉➟♥ ❝❤✉♥❣

✸✶

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✷



ữủ t ữợ sỹ ữợ ừ P r ữỡ ỳ
tr ờ tú ú ừ t ú tổ ồ ọ ữủ rt
ổ tự ỡ tr ỏ tự ụ ữ t ở
ừ t ỹ ở ỏ t tữ ừ t ổ ú tổ ố
ự tổ tọ ỏ ỡ t ố ợ t
ổ ụ ỷ ớ ỡ t ổ tr trữớ
ồ ữ ở õ t ổ ở ổ t õ r
t tổ tr sốt tớ q ố ũ tổ ữủ ỡ ữớ
ỗ tt ồ ữớ q t ở ú ù
tổ õ t t ử ừ
ở t
ồ

t


P
ố tữủ ử ự
r t ú tổ ự t ợ t tỷ r
ự số ử tở tớ
n

utt

aij (t)uxi xj = 0,

(t, x) [0, T ] ì Rn

(1)

i,j=1

u(0, x) = u0 (x);

u (0, x) = u1 (x).

tr aij ữủ tt tỹ ố ự tọ r
n

aij (t)i j /||2

a(t, ) =

0

i,j=1

ợ ồ (t, ) ([0, T ] ì (Rn \ {0})

(2)

sỷ (s) ợ r ợ số s õ t tr t ú ợ
ồ ỳ u0 , u1 (s) tỗ t t ừ s

u W 2,1 ([0, T ]; (s) )
ử ự ừ t t s tr t q t
ữủ tr ổ r ừ t ợ trữớ ủ ổ tử
st ừ số trữớ ủ tố ở t s ừ
số s ữủ t ởt r t

sỷ t
r ự trữợ r r số ừ



❈❍×❒◆● ✵✳ P❍❺◆ ▼Ð ✣❺❯

✷

tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ t❤➻ ❜➔✐ t♦→♥ ✭✶✮ ❧➔ ✤➦t ✤ó♥❣ tr♦♥❣ γ (s) ✈î✐ ♠å✐

1

s < 3/2✳ ❈á♥ ♥➳✉ aij ❧➔ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r ✈î✐ sè ♠ô α ✭t❛ ❦➼ ❤✐➺✉ ❧➔ C 0,α ([0, T ])✮ t❤➻

✭✶✮ ❧➔ γ (s) − ✤➦t ✤ó♥❣ ✈î✐ ♠å✐ 1

s < 1 + α/2 ✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ❤②♣❡r❜♦❧✐❝ ♠↕♥❤

♥➳✉ ❧➔♠ ❣✐↔♠ ♥❤➭ t➼♥❤ ❝❤➼♥❤ q✉② ❝õ❛ ❤➺ sè aij t❛ ✈➝♥ ♥❤➟♥ ✤÷ñ❝ ♠ët sè ❦➳t q✉↔ ❧➼ t❤ó✳
❱➼ ❞ö ♥❤÷ ❝â t❤➸ ❣✐↔♠ ♥❤➭ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ ❝õ❛ ❝→❝ ❤➺ sè t❤❡♦ ♠ët ❝→❝❤ ❦❤→❝✿ ✈➼
❞ö ❝❤♦ aij ❧➔ ❝→❝ ❤➔♠ C 1 tr➯♥ [0, T ]✴{t} ✈➔ →♣ ✤➦t ✤✐➲✉ ❦✐➺♥ ✈➲ ✤↕♦ ❤➔♠ ❝õ❛ aij ð ❞↕♥❣

|aij (t)|

C|t − t|−q

(3)

✈î✐ ♠å✐ t ∈ [0, T ]/{t} ✭ð ✤➙② t ❝â t❤➸ ❜➡♥❣ ✵✮✳
❚r♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥➔② t❛ s➩ ❝❤➾ ❣✐↔ t❤✐➳t r➡♥❣ ✈î✐ ♠ët t ∈ [0, T ] ❝è ✤à♥❤✱ ai,j (t) ∈

C 1 ([0, T ]/{t}) ∩ L1 (0, T ) ✈î✐ ♠å✐ i, j = 1, 2, ..., n ✈➔ ✤✐➲✉ ❦✐➺♥ ✭✷✮ ❝❤➾ ❝➛♥ t❤ä❛ ♠➣♥ ✈î✐
♠å✐ (t, ξ) ∈ ([0, T ], t) × (Rn \ {0}). ❚✉② ♥❤✐➯♥ ✤➸ ❝â t➼♥❤ ✤➦t ✤ó♥❣ tr♦♥❣ ❧î♣ ●❡r✈❡②✱
♠ët ♣❤↔♥ ✈➼ ❞ö ✤÷ñ❝ ①➙② ❞ü♥❣ tr♦♥❣ ✭✷✮ ❝ô♥❣ ✤➣ ❝❤➾ r❛ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❞↕♥❣ ✭✸✮ ❧➔ ❝➛♥
t❤✐➳t ♣❤↔✐ ❜ê s✉♥❣ t❤➯♠✳

✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✿
⑩♣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ♥➠♥❣ ❧÷ñ♥❣ ✈➔ ❤✐➺✉ ❝❤➾♥❤ ❝❤➼♥❤ q✉② ✤è✐ ✈î✐ ❤➺
sè aij (t) ✤➸ ❝↔✐ t✐➳♥ ✤ë trì♥✳
✣→♥❤ ❣✐→ ♥➠♥❣ ❧÷ñ♥❣ tr✉②➲♥ t❤è♥❣ ✈î✐ ❜➔✐ t♦→♥ ♠î✐ ✤➣ ✤÷ñ❝ ❤✐➺✉ ❝❤➾♥❤ ✈➔ ❦✐➸♠
s♦→t tè❝ ✤ë s❛✐ sè✳
❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✳

✹✳ ❈➜✉ tró❝ ❝õ❛ ❧✉➟♥ ✈➠♥✿
▲✉➟♥ ✈➠♥ ✤÷ñ❝ tr➻♥❤ ❜➔② ✈î✐ 2 ❝❤÷ì♥❣ tr♦♥❣ ✤â
❈❤÷ì♥❣ 1 ✤÷ñ❝ ❞➔♥❤ ❝❤♦ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❜ê trñ ❧✐➯♥ q✉❛♥ tî✐ ❝ì sð ✈➲
♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣✱ ♣❤÷ì♥❣ ♣❤→♣ ♥➠♥❣ ❧÷ñ♥❣✱ ❈ì sð ❣✐↔✐ t➼❝❤ ❋♦✉r✐❡r ❝ê ✤✐➸♥✱
❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❧î♣ ●❡✈r❡② ✈➔ ❝→❝ ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ ✤÷ì♥❣✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ ✤è✐ ✈î✐ t♦→♥ tû ❤②♣❡r❜♦❧✐❝ ②➳✉ ✈î✐ ❤➺ sè ♣❤ö


❈❍×❒◆● ✵✳ P❍❺◆ ▼Ð ✣❺❯

✸

t❤✉ë❝ t❤í✐ ❣✐❛♥ ❦❤æ♥❣ ❧✐➯♥ tö❝ t❤❡♦ ▲✐♣s❝❤✐t③✿ ❝→❝ ❧î♣ ❞❛♦ ✤ë♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❤➺ sè✱
❦➳t q✉↔ ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ tr♦♥❣ C ∞ ✤è✐ ✈î✐ ❝→❝ tr÷í♥❣ ❤ñ♣ t➛♠ t❤÷í♥❣ q❂✶✱ ❦➳t q✉↔
✈➲ t➼♥❤ ✤➦t ✤ó♥❣ tr♦♥❣ ❧î♣ ●❡✈r❡② ✈î✐ sè ♠ô q ♥❤ä✱ t➼♥❤ ✤➦t ✤ó♥❣ ●❡✈r❡② ✈î✐ ❝→❝ sè
♠ô q ✤õ ❧î♥ ❦➳t ❤ñ♣ ✈î✐ t➼♥❤ ❍☎
♦❧❞❡r ❝õ❛ ❤➺ sè✳
❉♦ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳
✈➔ t❤✐➳✉ sât✱ tæ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥
❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ✈➲ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ ù♥❣ ❞ö♥❣ ❝õ❛ ♣❤➨♣
❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈➔ ✤è✐ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣❀ ♣❤÷ì♥❣
tr➻♥❤ tr✉②➲♥ sâ♥❣❀ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈❀ ♣❤÷ì♥❣ ♣❤→♣ ♥➠♥❣ ❧÷ñ♥❣ ✈➔ ❧î♣ ●❡r✈❡② ✈➔ ❝→❝
✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t q✉❛♥ tr♦♥❣ ❝õ❛ ❧î♣ ●❡r✈❡②✳

✶✳✶✳ ❈ì sð ❣✐↔✐ t➼❝❤ ❋♦✉r✐❡r ❝ê ✤✐➸♥✳ ❑❤æ♥❣ ❣✐❛♥
❙♦❜♦❧❡✈✳
✶✳✶✳✶✳ ❈ì sð ❣✐↔✐ t➼❝❤ ❋♦✉r✐❡r ❝ê ✤✐➸♥✳
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝â þ ♥❣❤➽❛ q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦
❤➔♠ r✐➯♥❣✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❧➼ ❞♦ ✤â ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ tr♦♥❣ ♥❤✐➲✉ tr÷í♥❣ ❤ñ♣✱ ❝❤♦
♣❤➨♣ ✤÷❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ♠➔ t❛ ❝â
t❤➸ ❣✐↔✐ ✤÷ñ❝ ♠ët ❝→❝❤ ❝❤➼♥❤ ①→❝✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❧➔ ♠ët ❜✐➳♥ ✤ê✐ t➼❝❤ ♣❤➙♥ ✤➦❝ ❜✐➺t✱
♥â t❤÷í♥❣ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❝→❝❤ ❝ê ✤✐➸♥ ♥❤÷ s❛✉

F (f )(ξ) := √

1
n
2π

✹

e−ix·ξ f (x)dx
Rn


ì



ợ t ổ ữợ tr Rn ữủ x ã =

n

xk k ố ự ợ

k=1

ờ rr t õ ờ rr ữủ ữủ ổ tự s

F 1 (g)(x) :=

1
n
2

eixã g()d.
Rn

ử tợ ổ ừ ổ
t ổ C (Rn ) r õ f õ t ổ
4

|x| ử f (x) = e|x| r trữớ ủ tờ qt ổ
ừ f g t ổ ũ õ t s r sỹ ổ ở tử ừ t F (f ) F 1 (g)
õ t t ổ C0 (Rn ) ỗ tt tở C0 õ
t õ t tr F (f ) F 1 (g) tỗ t f, g C0
f g tở C0 (Rn ) t ữ r F (f ) F 1 (g) ụ tở

C0 (Rn ) tỗ t ởt ổ tr ỳ C (Rn ) C0 (Rn ) õ
t t f g tở ổ t s r F (f ) F 1 (g) ụ tở
ổ õ ổ õ ổ rt S (Rn ) ổ ừ


S (R ) ổ ừ C
n



(Rn ) ỗ tt

f tọ s

p, (f ) = sup x x f (x) <
xRn

tt số ổ ổ tr S (Rn ) ữủ t t ỷ

{p, (f ) }, .
ổ rt ổ ợ t ừ L1 (Rn ) t ố ợ
t tỷ x ợ x

ờ rr ờ rr ữủ tử tứ ổ
rt õ ờ rr ừ xk f ik F (f )

F (xk f )() = ik F (f )()
ờ rr ừ xk f ik F (f )

F (xk f )() = ik F (f )().


ì



ởt t tr ổ t Rnx s tữỡ ự ợ ởt
ợ ởt ổ tr ổ Rn ữủ
trỹ t ủ õ s ố ợ ổ tự rr ữủ

F 1 (F (f )) (x) = f (x) .
ổ tự ữủ ú tt f S (Rn ) õ t ự
ữỡ q õ s õ q ỵ tữ ừ ự
trữợ ởt g C0 (Rn ) g(x) 0

n g

x


g(x)dx = 1 t g (x) =

Rn

f Lp (Rn ) ợ p [1, ) õ t t tỷ q õ

J (f ) ừ f ớ t
g (x y)f (y)dy.

J (f )(x) := (g f )(x) :=
Rn

ợ ồ t tỷ q õ J (f ) tở C (Rn ) ởt t tỷ q õ
tọ t t ú ỵ ữ s

lim J (f ) f

0

Lp (Rn )

= lim g f f
0

Lp (Rn )

=0

ử tợ ổ L

p

ử ử ờ rr

F (f )() :=

1
(2)

eixã f (x)dx

n
2

Rn

ổ Lp (Rn )
rữợ t t f L1 (Rn ) õ F (f ) tở L (Rn ) ỡ ỳ F (f ) tử tr

Rn lim F (f )() = 0 t s ữủ tọ
||

n

F (f g)() = (2) 2 F (f )()F (g)().
é t sỷ ử t t ừ L1 (Rn ) ữ ởt số ợ t tỷ ừ t

ớ t f L2 (Rn ) õ ờ F (f ) ổ ử
ữủ ữ r ởt ũ ủ ỡ F (f ) f tở L2 (Rn ) ữ s


ì


n

ồ f, g S(Rn ) õ ờ F 1 (f ) = (2) 2
n

F (f ) = (2) 2

Rn

Rn

eix. f (x)dx

eix. f (x)dx ử ữủ ữủ ố q s
f ()F (g)()d



f ()F 1 (g)()d



F 1 (f )(x)g (x)dx =

Rn

Rn

F (f )(x)g (x)dx =
Rn

Rn

tự t ữủ s r ổ tự s

1

F 1 (f )(x)g (x)dx =
Rn

Rn

=

(2)
f ()

Rn

f ()

=
Rn

=

eix. f ()d g(x)dx

n
2

Rn

1
(2)
1
(2)

n
2

n
2

eix g(x) d
eix g(x)dxd
Rn

f ()F (g)()d
Rn

ú t sỷ ử ờ ợ F 1 (f )(x) ố q õ t ữủ
t (F 1 (f ), g) = (f, F (g)) ợ tt f, g S(Rn ) ớ t ồ f L2 (Rn )
õ t ổ ữợ (f, F (g)) ữủ ồ g S(Rn ) t trũ

t ừ C0 (Rn ) S(Rn ) tr L2 (Rn ) tự (, g) = (f, F (g))
ởt tr L2 (Rn ) ỗ t t ởt L2 (Rn ) s tự

(, g) = (f, F (g)) tọ ợ ồ f L2 (Rn ) ữ ờ
rr ữủ ừ f L2 (Rn ) ỷ ử ố q ởt tữỡ tỹ t
õ t ờ rr F (f ) L2 (Rn ) ợ ởt trữợ f L2 (Rn )
ờ qt õ t t F (f ) F 1 (f ) ợ ởt trữợ f L2 (Rn )
ỹ ố q

(F 1 (f ), g) = (f, F (g)) ợ ồ g S(Rn )
(F (f ), g) = (f, F 1 (g)) ợ ồ g S(Rn ).

ờ tự Psr ợ ồ f L (R ) õ t õ
2

|f (x)|2 dx =
Rn

n

|F (f )()|2 d.
Rn


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✽

❈→❝ ❝æ♥❣ t❤ù❝ ❋♦✉r✐❡r ♥❣÷ñ❝
❈❤♦ ✤➳♥ ✤➙② t❛ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝æ♥❣ t❤ù❝ ❋♦✉r✐❡r ♥❣÷ñ❝ ❝❤♦ ❝→❝ ❤➔♠ t❤✉ë❝ S(Rn )

✈➔ L2 (Rn )✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ s➩ ✤÷❛ r❛ ❝→❝ ❝æ♥❣ t❤ù❝ ❋♦✉r✐❡r ♥❣÷ñ❝ ❝❤♦ ❝→❝ ❦❤æ♥❣
❣✐❛♥ ❤➔♠ ❦❤→❝✳

• ▲➜② f ∈ L1 (R) ❝õ❛ ❤➔♠ ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ tr➯♥ ♠å✐ ✤♦↕♥ ❝♦♠♣❛❝t [a, b]✱ ♥❣❤➽❛
❧➔

b

|f (x)|dx < +∞ ✈î✐ ♠å✐ ✤♦↕♥ [a, b]
a

✈➔ f ❧✐➯♥ tö❝✳ ❑❤✐ ✤â ♥â t❤ä❛ ♠➣♥

1
f (x) = √ (p.v)
2π

eixξ F (f )(ξ)dξ.
R

◆➳✉ ❦❤æ♥❣ ❝â ❣✐↔ sû f ❧✐➯♥ tö❝✱ t❛ s➩ ❝â t❤❛② ✈➔♦ ✤â

f (x + 0) + f (x − 0)
1
= √ p.v.
2
2π

eixξ F (f )(ξ)dξ.
R

• ▲➜② f ∈ Lp (R) ✈î✐ p ∈ [1, 2]✳ ❑❤✐ ✤â ♥â t❤ä❛ ♠➣♥
1
f (x) = √ lim
2π ε→0

eixξ F (f )(ξ)χ(εξ)dξ
R

✈î✐ ♠ët ❤➔♠ χ ∈ C0∞ (Rn )✱ ð ✤â χ(η) =



1 ♥➳✉ |η| ≤ 1,

0 ♥➳✉ |η| ≥ 2.

• ▲➜② f ∈ Lp (R) ✈î✐ p ∈ (1, 2]✳ ❑❤✐ ✤â
1
f (x) = √ p.v.
2π

eixξ F (f )(ξ)dξ.
R

Ù♥❣ ❞ö♥❣ tî✐ ❤➔♠ ♣❤➙♥ ❜è t➠♥❣ ❝❤➟♠
❑❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ S(Rn ) ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ S (Rn )✳ ❈→❝ ♣❤✐➳♠ ❤➔♠ tø S (Rn )
✤÷ñ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ♣❤➙♥ ❜è t➠♥❣ ❝❤➟♠✳ ❚❛ ✤➣ ù♥❣ ❞ö♥❣ ❧þ t❤✉②➳t ❝õ❛ ❝→❝ ♣❤✐➳♠ ❤➔♠
✤➸ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✈➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ ❝õ❛ ♠ët ❤➔♠ tø L2 (Rn )✳
P❤÷ì♥❣ ♣❤→♣ ♥➔② ❦❤æ♥❣ ❞ò♥❣ ✤÷ñ❝ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ✈➻ tr♦♥❣ S (Rn ) t❛ ❦❤æ♥❣

❝â t➼❝❤ ✈æ ❤÷î♥❣✳ ◆❤÷♥❣ t❛ ❝â t❤➸ sû ❞ö♥❣ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❝→❝ ♣❤✐➳♠ ❤➔♠ f (g) ❝❤♦


ì



f S (Rn ) g tũ ỵ tở S(Rn ) ờ rr ờ rr
ữủ ừ ởt ố t

F (f )(g) = f (F 1 (g)) F 1 (f )(g) = f (F (g)).
ổ tự ờ rr ữủ ố t

F 1 (F (f ))(g) = F (f )(F (g)) = f (F 1 (F (g))) = f (g)



F 1 (F (f )) = f

õ tr t tọ ố t G S (Rn )
n
T E (Rn ) : F (G T ) = 2 F (G)F (T ) é t E (Rn ) ổ
ố t õ t

ổ
ữ ổ H m (Rn ), m N ổ t số
1/2
m

n

H (R ) =

n

u S (R ) : u

H m (Rn )

2

2 m

|F (u)()| (1 + || ) d

=

<

Rn

t t r x u L2 (Rn ) ợ ồ ||

m

u H m (Rn ), m N t ử ổ tự PrsPr q t ừ
ờ rr t õ F (x u)
số tọ ố q

Rn

L2 (Rn )

= F (u)

L2 (Rn )

ợ ồ ||

m

|F (u)()|2 (1 + ||2 )m d < ữủ ởt

số tọ q ố ũ t x u L2 (Rn ) ợ ồ ||

m

ừ H m ữủ sỷ ử ũ ợ ờ rr ủ t õ õ t ữủ tờ
qt tứ ồ số tỹ s R

ứ H (R ), s R t ởt t ỗ ố t
s

n

tr strts
1/2

H s (Rn ) =

u S (Rn ) : u

H s (Rn )

|F (u)()|2 (1 + ||2 )s d

=

<

Rn

ử ỵ s ổ H s (Rn ) ú tr ổ CB2 (Rn )
s >

n
2

+ 2 é CB2 (Rn ) ổ ừ số tử

ợ ợ s 0 t ồ tỷ tr H m (Rn ) tở

L2 (Rn )


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✵

✶✳✷✳ ✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣✳ P❤÷ì♥❣

♣❤→♣ ♥➠♥❣ ❧÷ñ♥❣
✶✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣ ❝ê ✤✐➸♥✳
❳➨t ❜➔✐ t♦→♥ ❈❛✉❝❤②✿

utt − uxx = 0; u(0, x) = ϕ(x); ut (0, x) = ψ(x)
✣➦t ξ = x − t; η = x + t ⇒ −4uξη = 0
❑❤✐ ✤â✱ ❜➔✐ t♦→♥ ❝â ♥❣❤✐➺♠ u = u(ξ, η) = u1 (ξ) + u2 (η) ✈î✐ u1 , u2 ❜➜t ❦➻✳ ❚❤❛② ❧↕✐ t❤❡♦
❝→❝❤ ✤➦t t❛ ✤÷ñ❝ u = u(t, x) = u1 (x − t) + u2 (x + t)✳ ◆❣❤✐➺♠ ✉ ♥➔② ❧➔ sü ❝❤ç♥❣ ❝❤➜t
❝õ❛ ❤❛✐ sâ♥❣✿

• ❙â♥❣ u1 (x − t) ❧➔ ✶ ❝❤✉②➸♥ ✤ë♥❣ ♥❤✐➵✉ ❧♦↕♥ ✈î✐ ✈➟♥ tè❝ ✶ ✈➲ ❜➯♥ ♣❤↔✐✳
• ❙â♥❣ u2 (x + t) ❧➔ ✶ ❝❤✉②➸♥ ✤ë♥❣ ♥❤✐➵✉ ❧♦↕♥ ✈î✐ ✈➟♥ tè❝ ✶ ✈➲ ❜➯♥ tr→✐✳
❈↔ ❤❛✐ ♥❣❤✐➺♠ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣ ✳ ⑩♣ ❞ö♥❣ ❞ú
❦✐➺♥ ❜❛♥ ✤➛✉ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② t❛ ✤÷ñ❝✿

u(0, x) = ϕ(x) = u1 (x) + u2 (x)
ut (0, x) = ψ(x) = −u1 (x) + u2 (x)
x

❚➼❝❤ ♣❤➙♥ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ t❛ ✤÷ñ❝ −u1 (x) + u2 (x) =

ψ(r)dr ✈î✐ x0 ❧➔ ♠ët
x0

✤✐➸♠ ❜➜t ❦➻ ❦❤æ♥❣ ✤ê✐✳
❍ì♥ ♥ú❛✱

x

1

1
u1 (x) = ϕ(x) +
2
2

ψ(r)dr
x0
x

1
1
u2 (x) = ϕ(x) −
2
2

ψ(r)dr
x0


ì



õ t t ữủ
x+t

1
1
u(t, x) = ((x t) + (x + t)) +
2

2

(r)dr
xt

ữủ ồ ổ tự rts

t ừ ữỡ tr tr sõ
t t utt uxx = 0; u(0, x) = (x); ut (0, x) = (x) ợ ỳ

C k (R1 ); C k1 (R1 ) õ t õ ởt ởt
u C k ([0, ) ì R1 ) ử tở tử ỳ é
t t ờ ởt ữủ q trú ừ C k (R1 ) C k1 (R1 ) t
u ụ t ờ ởt ữủ q trú t ừ C k ([0, ) ì R1 )
t sỹ tỗ t ữủ ổ tự rts t
ữủ s r tứ ừ ữỡ tr utt uxx = 0 ổ tự u(t, x) =

u1 (x t) + u2 (x + t) ứ ổ tự rts t t ữủ sỹ ử tở
tử ừ ỳ t t ợ ỳ C 2 (R1 )
C 1 (R1 ) t ờ ỳ s C 1 (R1 ) tr [a, b]
t t utt uxx = 0; u(0, x) = s (x); ut (0, x) = s (x) ợ s = s =

0 \ [a, b]
ừ t
x+t

1
1
us (t, x) = (s (x t) + s (x + t)) +
2

2

s (r)dr
xt

õ s tớ ỳ T = dist(x0 , [a, b]) t t ữủ
sỹ ừ ởt x0 R1 [a, b] ợ t ừ ọ t õ

u(t, x0 ) = 0 t x0 t ữủ ồ tố ở tr ỳ sỹ
sỹ tỗ t ừ ởt sõ ừ ữỡ tr tr sõ ử tở
ừ ỳ u(t0 , x0 ) t tớ (t0 , x0 )
t t [x0 t0 ; x0 + t0 ] ợ ỳ t tớ x0 t0 x0 + t0

[x0 t0 ; x0 + t0 ] ữủ ồ ử tở ừ t tớ (t0 , x0 )

ổ tr sõ ợ ỗ tt


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✷

❳➨t ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣✿

 u − u = F (t, x)
tt
xx
 u(0, x) = ϕ(x); u (0, x) = ψ(x)
t

●✐↔ sû ♥❣✉ç♥ ❋ ❧➔ ❦❤↔ t➼❝❤ ✈➔ F ∈ L1loc ([0, ∞) × R1 ))✳ ❳➨t ♥❣❤✐➺♠ ❦❤æ♥❣ ❝ê ✤✐➸♥

u = v + ω tr♦♥❣ ✤â v ✈➔ ω ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤②
vtt − vxx = F (t, x); v(0, x) = 0; vt (0, x) = 0
ωtt − ωxx = 0); ω(0, x) = ϕ(x); ωt (0, x) = ψ(x)
❚❤❡♦ ❝æ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥ ♥❣❤✐➺♠ ❉✬❆❧❡♠❜❡rt✬s t❛ ✤➣ ❝â ❜✐➸✉ ❞✐➵♥ ❝õ❛ ♥❣❤✐➺♠
x+t

1
1
ω(t, x) = (ϕ(x − t) + ϕ(x + t)) +
2
2

ψ(r)dr
x−t

t x+(t−t )

1
❝á♥ ♥❣❤✐➺♠ v(t, x) =
2

F (x , t )dx dt
0 x−(t−t )
t x+(t−t )

1
⇒ u = v +ω =
2

x+t

1
1
F (x , t )dx dt + (ϕ(x − t) + ϕ(x + t))+
2
2
0 x−(t−t )

ψ(r)dr
x−t

✶✳✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ ♥➠♥❣ ❧÷ñ♥❣
❑❤→✐ ♥✐➺♠ ✈➲ ♥➠♥❣ ❧÷ñ♥❣ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣ ❧➔ ❝æ♥❣ ❝ö ❝ì ❜↔♥
✤➸ s✉② r❛ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ♠æ ❤➻♥❤ tr✉②➲♥ sâ♥❣✳ ▲➜② u ❧➔ ♠ët ❤➔♠ sè t❤✉ë❝ ✈➔♦

C([0, T ], H 1 (Rn )) ∩ C 1 ([0, T ], L2 (Rn ))
❳➨t tr➯♥ C([0, T ], H 1 (Rn )) ∩ C 1 ([0, T ], L2 (Rn ))✳ ❚❛ ❦➼ ❤✐➺✉✿

E(u)(t) :=

1
2

|ut (t, x)|2 + |∇x u(t, x)|2 dx
Rn

=

1
ut (t, .)
2

2
L2

+

1
∇x u(t, .)
2

2
L2

❧➔ ♥➠♥❣ ❧÷ñ♥❣ ❤❛② tê♥❣ ♥➠♥❣ ❧÷ñ♥❣✱ ❝❤ó þ r➡♥❣ ð ✤➙② ❊ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❜✐➳♥ t❤í✐
❣✐❛♥ t


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
•

1
2

ut (t, .)

•

1
2

∇x u(t, .)

2
L2

✶✸

❜✐➸✉ t❤à ✤ë♥❣ ♥➠♥❣✳
2
L2

❜✐➸✉ t❤à ♥➠♥❣ ❧÷ñ♥❣ ✤➔♥ ❤ç✐✳

◆➳✉ ❦❤æ♥❣ q✉❛♥ t➙♠ tî✐ tê♥❣ ♥➠♥❣ ❧÷ñ♥❣ ✱ t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ♠ët t➟♣ K ⊂ Rn ✭❑ ❧➔
♠ët ♠✐➲♥ ✤â♥❣✮ ✈➔ ❦❤✐ ✤â t❛ ✤à♥❤ ♥❣❤➽❛ ✤÷ñ❝ ♥➠♥❣ ❧÷ñ♥❣ ✤à❛ ♣❤÷ì♥❣✿

E(u, K)(t) :=

1
2

(|ut (t, x)|2 + |∇x u(t, x)|2 )dx
K

▲➜② (t0 , x0 ) ❧➔ ♠ët ✤✐➸♠ ❜➜t ❜✐➳♥ tr♦♥❣ Rn+1 ; t0 > 0
❚➟♣ {(t, x) : |x − x0 | = |t − t0 |} ♠æ t↔ ♠➦t ♥❣♦➔✐ ❝õ❛ ♠ët ❤➻♥❤ ♥â♥ ✈î✐ ✤➾♥❤ t↕✐ (t0 , x0 )
▲➜② T

t0 P❤➛♥ ♠➦t ♣❤➥♥❣ ✤÷ñ❝ ❝❤✐❛ r❛ t↕✐ t = T ♥➡♠ ❜➯♥ tr♦♥❣ ❤➻♥❤ ♥â♥ ✤➦❝ tr÷♥❣

♣❤➼❛ s❛✉ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ K(x0 , t0 − T ). P❤➛♥ ♥➔② ❧➔ ♠ët ❤➻♥❤ ❝➛✉ ✤â♥❣ ❝â t➙♠

x = x0 ✈➔ ❜→♥ ❦➼♥❤ t0 − T

✣à♥❤ ❧➼ ✶✳✸✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ✈➲ ♠✐➲♥ ♣❤ö t❤✉ë❝✮ ▲➜② (t , x ) ∈ R
0

n+1

0

✈î✐ t0 > 0✳ ❚❛

✤à♥❤ ♥❣❤➽❛ Ω ❧➔ ♠ët ♠✐➲♥ ❤➻♥❤ ♥â♥ ❜à ❝❤➦♥ ❜ð✐ ❤➻♥❤ ♥â♥ ✤➠❝ tr÷♥❣ ♣❤➼❛ s❛✉ ✈î✐ ✤➾♥❤
t↕✐ (t0 , x0 ) ✈➔ ♠➦t ♣❤➥♥❣ t = 0✳ ▲➜② u ∈ C 2 (Ω) ❧➔ ♠ët ♥❣❤✐➺♠ ❝ê ✤✐➸♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
tr✉②➲♥ sâ♥❣ utt − ∆u = 0✳ ❑❤✐ ✤â t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝✿

E(u, K(x0 , t0 − t))

✈î✐

E(u, K(x0 , t0 ))

t ∈ [0, t0 ]

✣à♥❤ ❧➼ ✶✳✹✳ ✭✣à♥❤ ❧➼ ✈➲ ❜↔♦ t♦➔♥ ♥➠♥❣ ❧÷ñ♥❣✮
▲➜② u ∈ C([0, T ], H 1 (Rn )) ∩ C 1 ([0, T ], L2 (Rn )) ❧➔ ♠ët ♥❣❤✐➺♠ ❙♦❜♦❧❡✈ ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ tr✉②➲♥ sâ♥❣ utt − ∆u = 0,

ut (0, x) = ψ(x) ✈î✐ ❞ú ❦✐➺♥ ϕ ∈

u(0, x) = ϕ(x);

H 1 (Rn ) ✈➔ ψ ∈ L2 (Rn )✳ ❑❤✐ ✤â t❛ ❝â✿
E(u)(t) = E(u)(0) =

1
2

ψ

2
L2

+ ∇ϕ

2
L2

✈î✐

∀t

0.

✣à♥❤ ❧➼ ✶✳✺✳ ❳➨t ❜➔✐ t♦→♥ ❈❛✉❝❤② ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ❑❧❡✐♥✲●♦r❞♦♥✿
utt − ∆u + m2 u = 0; u(0, x) = ϕ(x); ut (0, x) = ψ(x), m2 > 0 ❦❤æ♥❣ ✤ê✐✱ ✈î✐ ❞ú

❦✐➺♥ ϕ ∈ H 1 (Rn ) ✈➔ ψ ∈ L2 (Rn )✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ tê♥❣ ♥➠♥❣ ❧÷ñ♥❣ ❝õ❛ ♥❣❤✐➺♠ ✉ ♥❤÷
s❛✉✿

E(u)(t) :=

1
2

(|ut (t, .)|2 + |∇x u(t, .)|2 + m2 |u(t, x)|2 )dx
Rn


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✹

✣à♥❤ ❧➼ ✶✳✻✳ ▲➜② u ∈ C([0, T ], H (R )) ∩ C ([0, T ], L (R )) ❧➔ ♥❣❤✐➺♠ ❙♦❜♦❧❡✈ ❝õ❛ ❜➔✐
1

n

1

2

n

t♦→♥ tr➯♥✳ ❑❤✐ ✤â✱ ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② ✤÷ñ❝ t❤ä❛ ♠➣♥✿

E(u)(t) = E(u)(0) =

1
2

ψ

2
L2

+ ∆ϕ

2
L2

+ m2 ϕ

2
L2

✈î✐

∀t ≥ 0.

✶✳✷✳✸✳ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ sâ♥❣✳
❳➨t ❜➔✐ t♦→♥ ❈❛✉❝❤②✿

utt − ∆u = 0, u(0, x) = ϕ(x), ut (0, x) = ψ(x), x ∈ Rn , n ≥ 1
❙❛✉ ❦❤✐ →♣ ❞ö♥❣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ (v(t, ξ) = Fx→ξ (u(t, x))) t❛ t❤✉ ✤÷ñ❝ ❜➔✐
t♦→♥ ❈❛✉❝❤② ♣❤ö s❛✉✿

vtt + |ξ|2 v = 0, v(0, ξ) = F (ϕ)(ξ), v(0, ξ) = F (ψ)(ξ)

✤è✐ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ♣❤ö t❤✉ë❝ ❤➺ sè ξ ∈ Rn ✳ ❱î✐ ξ = 0 t❛ ❝â ♥❣❤✐➺♠

v(t, ξ) = c1 (ξ)e−i|ξ|t + c2 (ξ)ei|ξ|t
❚ø ✤✐➲✉ ❦✐➺♥ ❈❛✉❝❤② t❛ s✉② r❛ c1 (ξ) + c2 (ξ) = F (ϕ)(ξ), −i|ξ|c1 (ξ) + i|ξ|c2 (ξ) = F (ψ)(ξ)
1
1
❞➝♥ ✤➳♥ c1 (ξ) = 21 F (ϕ)(ξ) −
F (ψ)(ξ), c2 (ξ) = 12 F (ϕ)(ξ) +
F (ψ)(ξ)
2i|ξ|
2i|ξ|
❚â♠ ❧↕✐✱ t❛ t❤✉ ✤÷ñ❝✿

v(t, ξ) = cos(|ξ|t)F (ϕ)(ξ) +

sin(|ξ|t)
F (ψ)(ξ)
|ξ|

●✐↔ sû r➡♥❣ t↕✐ ♠ët t❤í✐ ✤✐➸♠ ❝â ❤✐➺✉ ❧ü❝ ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ u(t, x) =
−1
(Fx→ξ (u(t, x))) t❛ ✤✐ ✤➳♥ ♠ët ✤↕✐ ❞✐➺♥ ❝õ❛ ✉ ❧➔✿
Fξ→x
−1
−1
u(t, x) = Fξ→x
(cos(|ξ|t)F (ϕ)(ξ)) + Fξ→x

sin(|ξ|t)
F (ψ)(ξ)
|ξ|

❚❛ ❝ô♥❣ ❝â t❤➸ sû ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣✿

1
1
−1
−1
e−i|ξ|t F (ϕ)(ξ) − Fξ→x
e−i|ξ|t
u(t, x) = Fξ→x
F (ψ)(ξ)
2
2i|ξ|


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✺

−1
−1
1
ei|ξ|t 2i|ξ|
F (ψ)(ξ)
+Fξ→x
ei|ξ|t 12 F (ϕ)(ξ) − Fξ→x

❇✐➸✉ ❞✐➵♥ ❝❤ù❛ ♥❤➙♥ tû ❋♦✉r✐❡r
−1
Fξ→x
(eiφ(t,ξ) a(t, ξ)F (u0 )(ξ)).

Ð ✤➙②✱ φ = φ(t, ξ) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ sè ♣❤❛ ✈➔ a = a(t, ξ) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ sè ❜✐➯♥ ✤ë✳
❚ø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t❛ ✤➣ ❝â ϕ ∈ H s (Rn ) ✈➔ ψ ∈ H s−1 (Rn ) ✈î✐ s ≤ 1✳
❙❛✉ ✤➙② t❛ ♣❤→t ❜✐➸✉ ✤à♥❤ ❧➼ ✈➲ s÷ tç♥ t↕✐ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥
sâ♥❣ ✈➔ ❤➺ q✉↔ ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ H s ✳

✣à♥❤ ❧➼ ✶✳✼✳ ▲➜② ϕ ∈ H (R ) ✈➔ ψ ∈ H
s

n

s−1

(Rn ) ✈î✐ s ≤ 1, n ≤ 1 tr♦♥❣ ❜➔✐ t♦→♥

❈❛✉❝❤②✿

utt − ∆u = 0, u(0, x) = ϕ(x), ut (0, x) = ψ(x)
❑❤✐ ✤â tç♥ t↕✐ ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠ ♥➠♥❣ ❧÷ñ♥❣ u ∈ C ([0, T ], H s (Rn ))∩C 1 ([0, T ], H s−1 (Rn ))

❍➺ q✉↔ ✶✳✶✳ ❳➨t ❜➔✐ t♦→♥ ❈❛✉❝❤②✿
utt − ∆u = 0, u(0, x) = ϕ(x), ut (0, x) = ψ(x), x ∈ Rn , n ≤ 1
❧➔ H s ✤➦t ✤ó♥❣✱ ð ✤➙② ❞ú ❦✐➺♥ ❜❛♥ ✤➛✉ ✤➣ ❝❤♦ ϕ ∈ H s (Rn ), ψ ∈ H s−1 (Rn ) ❦❤✐ ✤â tç♥
t↕✐ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ①→❝ ✤à♥❤ u ∈ C ([0, T ], H s (Rn )) ∩ C 1 ([0, T ], H s−1 (Rn ))✳ ◆❣❤✐➺♠
♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✈➔♦ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✱ tr♦♥❣ ✤â ✈î✐ ♠é✐ ε > 0✱ tç♥ t↕✐ aδ(ε) s❛♦ ❝❤♦

ϕ1 − ϕ2

Hs

+ ψ1 − ψ2

H s−1

< δ t❤➻ u1 − u2

C([0,T ],H s )∩C 1 ([0,T ],H s−1

< ε.

✶✳✸✳ ▲î♣ ●❡r✈❡② ✈➔ ❝→❝ ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ ✤÷ì♥❣✳
✶✳✸✳✶✳ ✣à♥❤ ♥❣❤➽❛ ❧î♣ ●❡r✈❡②
❳➨t ❜➔✐ t♦→♥ ❈❛✉❝❤② ✈î✐ ❝→❝ t♦→♥ tû ❤②♣❡r❜♦❧✐❝ ②➳✉ ❝❤ù❛ ❤➺ sè ♣❤ö t❤✉ë❝ ✈➔♦ t❤í✐
❣✐❛♥✿

n

utt −

aij (t)uxj xj = 0,

(t, x) ∈ [0, T ] × Rn

i,j=1

u(0, x) = u0 (x);

u (0, x) = u1 (x).

(1)


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✻

❚❛ ①➨t ❝→❝ ❤➔♠ sè t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿

•

a(t) > 0

•

∃λ(t) ∈ C 1 ([0, ∞), λ (t) ≥ 0, ∀t ≥ 0; c1 λ(t)

•

∃Θ(t)❧➔ ❤➔♠ sè ❞÷ì♥❣✱ t➠♥❣ ♠↕♥❤

a(t)

c2 λ(t)

t

Θ(t) = o(Λ(t))(t → ∞)

ð ✤➙②

λ(s)ds, Λ−1 (t) ∈ L1 ([1, ∞)) ❦❤✐

✤â tç♥ t↕✐ 0 < ε << 1 :

Θ(t + ε)
❜à ❝❤➦♥ ❜ð✐ ❤➡♥❣ sè ❝
Θ(t)

Λ(t) :=
0

t

|a(s) − λ(s)|ds

c Θ(t)∀t.

∃Γ(t) ❞÷ì♥❣✱ ❧✐➯♥ tö❝✱ ✤ì♥ ✤✐➺✉ d1 Θ(t)

λ(t)Γ(t)

0

•

d2 Λ(t)

∞

1

•

λm−1 (s)Γm (s)

ds

C
Θm (t

m≥0

t
(k) a(t)
|
λ(t)

• |dt

C(k!)(Γ(t))−k ,

∀k > 0.

❙❛✉ ❦❤✐ →♣ ❞ö♥❣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❛ ✤✐ ✤➳♥ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ❦❤→❝ ❝â ❞↕♥❣ ♥❤÷ s❛✉✿

vtt + a2 (t)|ξ|2 v = 0

v(ξ, 0) = v0 (ξ)
vt (ξ, 0) = v1 (ξ).
✈î✐ v = v(ξ, t) = Fx→ξu ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ u(x, t).
✣✐➲✉ ❦✐➺♥ ❝✉è✐ ❝ò♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧î♣ ●❡✈r❡② γρv (I) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ♠ët t➟♣
❝♦♠♣❛❝t I ⊂ R :

γρv (I) = {f (t) ∈ C ∞ (I) : ∃C > 0 : |f (k) (t)|

C(k!)v ρ−k ,

k = 0, 1, 2, ...}

(∗)

✈î✐ ❤➡♥❣ sè v > 1 ✈➔ ♠é✐ ❤➔♠ sè ρ > 0 ✈➔ t❛ ✈✐➳t f ∈ γ v (I) ♥➳✉ tç♥ t↕✐ ρ(t) > 0, f ∈

γρv (I).


❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✼

✶✳✸✳✷✳ ❈→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❧î♣ ●❡✈r❡②✳
• ◆➳✉ fi (t) ∈ γρv (I) ✈î✐ ❤➡♥❣ sè Ci , i = 1, 2 tr♦♥❣ ✭✯✮ t❤➻ tç♥ t↕✐ ♠ët ❤➡♥❣ sè ♣❤ê
K > 0 ♣❤ö t❤✉ë❝ ✈➔♦ Ci , v, ❦❤✐ ✤âf = f1 f2 ❝ô♥❣ t❤✉ë❝ ✈➔♦ γρv (I). ✈î✐ ❤➡♥❣ sè
KC1 C2
• ❚ç♥ t↕✐ ♠ët ❤➡♥❣ sè L > 1, s❛♦ ❝❤♦ ✈î✐ f ∈ γρv (I), f > f0 >, 0✈➔ ❤➡♥❣ sè C0 t❤➻
❤➔♠ ♥❣❤à❝❤ ✤↔♦

1
f

t❤✉ë❝ ✈➔♦ f0 ✈➔ v.

❝ô♥❣ t❤✉ë❝ ✈➔♦ γρv (I). ✈î✐ ❤➡♥❣ sè L.C0 ✱ tr♦♥❣ ✤â L ❝❤➾ ♣❤ö


❈❤÷ì♥❣ ✷
❚➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤②
✤è✐ ✈î✐ t♦→♥ tû ❤②♣❡r❜♦❧✐❝ ②➳✉ ✈î✐
❤➺ sè ♣❤ö t❤✉ë❝ t❤í✐ ❣✐❛♥ ❦❤æ♥❣
❧✐➯♥ tö❝ t❤❡♦ ▲✐♣s❝❤✐t③
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ♠ët sè ✤à♥❤ ❧þ✱ ❦➳t q✉↔ ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛
❜➔✐ t♦→♥ ❈❛✉❝❤② tr♦♥❣ ❧î♣ ●❡r✈❡② γ (s) tò② t❤✉ë❝ ✈➔♦ sè ♠ô q

✷✳✶✳ ❑➳t q✉↔ ✈➲ t➼♥❤ ✤➦t ✤ó♥❣ tr♦♥❣ ❧î♣ ●❡r✈❡② γ (s)
✤è✐ ✈î✐ 1 q < 3
✣à♥❤ ❧➼ ✷✳✶✳ ▲➜② 1

q < 3 ✳ ●✐↔ sû tç♥ t↕✐ C > 0 s❛♦ ❝❤♦✿
|a (t, ξ)|

C|t − t|−q

✭✷✳✶✮

✈î✐ ♠å✐ (t, ξ) ∈ ([0, T ] \ {t}) × (Rn \ {0}) t❤➻ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✭✶✳✶✮ ✈î✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥
✤➛✉ ✭✶✳✷✮ ❧➔ γ (s) − ✤➦t ✤ó♥❣ ✈î✐ ♠å✐ 1

s < 3/2.

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❚r÷í♥❣ ❤ñ♣ ✶✿ t = T ✳ ●✐↔ sû v ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ u ♣❤ö t❤✉ë❝
✶✽


❈❍×❒◆● ✷✳ ❚➑◆❍ ✣➄❚ ✣Ó◆● ❈Õ❆ ❇⑨■ ❚❖⑩◆ ❈❆❯❈❍❨ ✣➮■ ❱❰■ ❚❖⑩◆ ❚Û ❍❨P❊❘❇❖▲■❈
✈➔♦ x✳
❚ø (1.1) ⇒ v (t, ξ) = −a(t, ξ)|ξ|2 v(t, ξ)
❉♦ u0 , u1 ∈ γ (s) ∩ C0∞ ;
⑩♣ ❞ö♥❣ ✤à♥❤ ❧þ P❛❧❡②✲❲✐❡♥❡r → ∃M ; δ > 0 s❛♦ ❝❤♦
1

|v(0, ξ)|2 + |v (0, ξ)|2

✭✷✳✷✮

∀ξ ∈ Rn ; |ξ| ≥ 1.

M exp(−δ|ξ| s )

✣➸ ❝❤ù♥❣ ♠✐♥❤ u ∈ W 2,1 ([0, T ]; γ (s) ) t❛ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ∃M , δ > 0 s❛♦ ❝❤♦✿
1

|v(t, ξ)|2 + |v (t, ξ)|2

M exp(−δ |ξ| s )

∀t ∈ [0, T ]; ∀ξ ∈ Rn ; |ξ| ≥ 1.

✭✷✳✸✮

▲➜② ε ∈ (0, T ]✳ ✣➦t✿


 ε3−q (T − t)−2 + a(t, ξ)
aε (t, ξ) =
 ε1−q + a(T − ε, ξ)

✈î✐ t ∈ [0, T − ε]

✭✷✳✹✮

✈î✐ t ∈ [T − ε, T ]

✈➔
✭✷✳✺✮

Eε (t, ξ) = aε (t, ξ)|ξ|2 |v(t, ξ)|2 + |v (t, ξ)|2

✈î✐ Eε ❧➔ ❤➔♠ ♥➠♥❣ ❧÷ñ♥❣ ❝õ❛ ♥❣❤✐➺♠ u✳
❚r♦♥❣ ✤â✱ Eε ♣❤ö t❤✉ë❝ ✈➔♦ t ✈➔ tø ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ s✉② r❛

Eε (t, ξ) = aε (t, ξ)|ξ|2 |v(t, ξ)|2 + 2aε (t, ξ)|ξ|2 Re(v (t, ξ)v(t, ξ)) + 2Re(v ” (t, ξ)v (t, ξ))
1
aε (t, ξ) aε (t, ξ) − a(t, ξ)|ξ| 2
⇔ Eε (t, ξ)
+
Eε (t, ξ)
aε (t, ξ)

aε (t, ξ)
❍ì♥ ♥ú❛✱ tø ❜ê ✤➲ ●r♦♥✇❛❧❧ t❛ s✉② r❛✿
 T
aε (t, ξ)
Eε (t, ξ) Eε (0, ξ)exp 
dt + |ξ|
aε (t, ξ)
0

✈î✐ ∀t ∈ [0, T ]

∀ξ ∈ Rn ;

0

T−

aε (t, ξ)
dt
aε (t, ξ)
0

0



|aε (t, ξ) − a(t, ξ)| 
dt
1
aε (t, ξ) 2

|ξ| ≥ 1.

T

❚ø ✭✷✳✶✮ ✈➔ ✭✷✳✹✮ t❛ ❝â✿

T

2ε3−q (T − t)−3 + |a (t, ξ)|
dt
ε3−q (T − t)−2

✭✷✳✻✮


❈❍×❒◆● ✷✳ ❚➑◆❍ ✣➄❚ ✣Ó◆● ❈Õ❆ ❇⑨■ ❚❖⑩◆ ❈❆❯❈❍❨ ✣➮■ ❱❰■ ❚❖⑩◆ ❚Û ❍❨P❊❘❇❖▲■❈
T −ε

((T − t)−1 + εq−3 (T − t)2−q )dt

C

C εq−3 (1 + |logε|)

0

▼➦t ❦❤→❝✱ t❛ ❝❤ó þ r➡♥❣✿

 C(|log(T − t)| + 1)

a(t, ξ)
 C(T − t)1−q
✈➔
T

a(t, ξ)dt
T −ε




Cε(|log(ε)| + 1)


Cε2−q



 ||a|| 1

♥➳✉

1
♥➳✉

q=1
1
♥➳✉

2

T −ε

1

|aε (t, ξ) − a(t, ξ)| 2
dt
aε (t, ξ)

❍ì♥ ♥ú❛✱t❛ ❝â

q=1

♥➳✉

L (0,T )

T

♥➳✉

ε

0

3−q
2

T
−1

(T −t) dt+

q<3
ε1−q + a(T − ε, ξ) + a(t, ξ)
dt
ε(1−q)/2

T −ε

0
T

Cε(3−q)/2 (|logε|+1)+ε(q+1)/2 a(T −ε, ξ)+ε(q−1)/2

a(t, ξ)dt C ε(3−q)/2 (|logε|+1) ≤

T −ε

C ε(3−q)/2 (|logε| + 1).
❉♦ ✤â✱

Eε (t, ξ)

Eε (0, ξ)exp(C (εq−3 + |ξ|ε(3−q)/2 )(1 + |logε|))

✈î✐ ♠å✐ t ∈ [0, T ] ✈➔ ♠å✐ ξ ∈ Rn , |ξ|

1.

❚ø ✭✷✳✹✮✈➔ ✭✷✳✺✮ t❛ ❝â✿

Eε (0, ξ)

(T 1−q + a(0, ξ)|ξ|2 |v(0, ξ)|2 + |v (0, ξ)|2 )

✈➔

Eε (t, ξ)

T −2 ε3−q |ξ|2 |v(t, ξ)|2 + |v (t, ξ)|2 .

▲➜② ε = |ξ|−2/(3(3−q) t❛ s✉② r❛✿

T −2 |ξ|4/3 |v(t, ξ)|2 + |v (t, ξ)|2

(C|ξ|2 |v(0, ξ)|2 + |v (0, ξ)|2 )exp(C |ξ|2/3 (1 + |log|ξ||))

✭✷✳✼✮


❈❍×❒◆● ✷✳ ❚➑◆❍ ✣➄❚ ✣Ó◆● ❈Õ❆ ❇⑨■ ❚❖⑩◆ ❈❆❯❈❍❨ ✣➮■ ❱❰■ ❚❖⑩◆ ❚Û ❍❨P❊❘❇❖▲■❈
❑➳t ❤ñ♣ ✈î✐ ✭✷✳✷✮ ✈➔ ✭✷✳✸✮ ✈î✐ s < 3/2 ✈➔ ❦➳t q✉↔ ❝õ❛ ✤à♥❤ ❧➼ ✷✳✶ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤
tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✳

❚r÷í♥❣ ❤ñ♣ ✷✿ t = 0✳ ✣➦t✿

 ε1−q + a(ε, ξ)

aε (t, ξ) =
 ε3−q t−2 + a(t, ξ)

✈î✐ t ∈ [0, ε]
✈î✐ t ∈ [ε, T ]

❇✐➺♥ ❧✉➟♥ ♥❤÷ tr÷î❝ ✤â t❛ s✉② r❛ ✤÷ñ❝ ✭✷✳✼✮✳ ❈❤ó þ r➡♥❣ tr♦♥❣ tr÷í♥❣ ❤ñ♣

a(ε, ξ)

C(|logε| + 1) ♥➳✉ q = 1 ✈➔ a(ε, ξ)

Cε1−q ♥➳✉ 1 < q < 3✳

❚❛ ❝â✿

Eε (0, ξ)

C(|logε| + 1)|ξ|2 |v(0, ξ)|2 + |v (0, ξ)|2

✈➔Eε (0, ξ)

Cε1−q |ξ|2 |v(0, ξ)|2 + |v (0, ξ)|2 ,

❍ì♥ ♥ú❛✱ ❧➜② ε = |ξ|−2/(3(3−q)) t❛ ✤÷ñ❝✿

T −2 |ξ|4/3 |v(t, ξ)|2 + |v (t, ξ)|2

(C(|logε| + 1)|ξ|2 |v(0, ξ)|2 + |v (0, ξ)|2 )exp(C |ξ|2/3 (1 + |log|ξ||)) ♥➳✉ q = 1
✈➔ T −2 |ξ|4/3 |v(t, ξ)|2 + |v (t, ξ)|2

(C|ξ|(4(4−q))/(3(3−q)) |v(0, ξ)|2 +|v (0, ξ)|2 )exp(C |ξ|2/3 (1+|log|ξ||)) ♥➳✉ 1 < q < 3.
t÷ì♥❣ tü ♥❤÷ ✭✷✳✸✮ t❛ s✉② r❛ ✤à♥❤ ❧➼ ✤ó♥❣ ❝❤♦ s < 3/2.

❚r÷í♥❣ ❤ñ♣ ✸✿ ❱î✐ t ∈ (0, T ) ❧➔ ✤õ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❜➔✐ t♦→♥ ❈❛✉❝❤② tr♦♥❣ [0, t]✱
s❛✉ ✤â ✤➸ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ tr♦♥❣ [t, T ] ✈î✐ ❞ú ❦✐➺♥ ❜❛♥ ✤➛✉ ✤➣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤
tr÷î❝ ✤â✳ ❈✉è✐ ❝ò♥❣✱ ❦➳t ❤ñ♣ ✷ tr÷í♥❣ ❤ñ♣ tr➯♥ t❛ t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠✳