dXRX AMi`Q/m+iBQM

h?2 ai`B+?`ix 2biBKi2b `2 7KBHv Q7 /BbT2`bBp2 2biBKi2b 7Q` i?2 a+?`ƺ/BM;2` ~Qr QM _B2KMMBM JMB7QH/ M,

∥e−it∆u₀_∥_Lp_{(0,T )L}q_{(M )}≤ C_T∥u₀∥_L2,

r?2`2 i?2 2tTQM2Mib (p, q) ?p2 iQ 7QHHQr i?2 /KBbbB#BHBiv +QM/BiBQM ;Bp2M #v i?2 b+HBM; Q7 i?2 2[miBQM, p, q_{≥ 2,} 2 p+ n q = n 2, (p, q, n)̸= (2, ∞, 2),

M/ n Bb i?2 /BK2MbBQM Q7 MX A7 M Bb KMB7QH/ rBi? #QmM/`B2b- r2 mM/2`biM/ ∆ iQ #2 i?2 .B`B+?H2i GTH+BMX

GQ+HHv BM iBK2- i?2b2 2biBKi2b /2b+`B#2 `2;mH`BxBM; 2z2+i- `2~2+i2/ #v ;BM Q7 BMi2;`#BHBivX :HQ#HHv- i?2v /2b+`B#2 /2+v 2z2+i, i?2 Lq

x MQ`K ?b iQ /2+v- i H2bi BM

Lp t b2Mb2X

q2 bv i?i HQbb Q++m`b BM i?2 ai`B+?`ix 2biBKi2b- B7 i?2 2biBKi2b /Q MQi ?QH/ #mi r2 `2 #H2 iQ b?Qr i?i

∥e−it∆u0∥Lp_{(0,T )L}q_{(M )} ≤ C_T∥u₀∥_Hs,

rBi? s > 0X

h?2 biQ`v Q7 ai`B+?`ix 2biBKi2b BM i?2 +b2 Q7 i?2 ~i GTH+BM QM M = Rn_{#2;BMb rBi?}

i?2 rQ`F Q7 (ai`dd) 7Q` p = q- 7Q` i?2 rp2 2[miBQM- i?2M ;2M2`HBx2/ 7Q` Mv /KBbbB#H2 +QmTH2 M/ i?2 a+?`ƺ/BM;2` 2[miBQM #v (:o38#)- M/ #v (EhN3) 7Q` i?2 2M/TQBMi Up = 2) +b2X AM i?2 p`B#H2 +Q2}+B2Mib +b2- i?2 bBimiBQM Bb KQ`2 /B{+mHiX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb 7Q` i?2 a+?`ƺ/BM;2` 2[miBQM r2`2 Q#iBM2/ BM b2p2`H ;2QK2i`B+H bBimiBQMb r?2`2 HH i?2 ;2Q/2bB+b ;Q iQ BM}MBiv @ r2 bv i?i i?2 KMB7QH/ Bb MQMi`TTBM; @ , ("hyd)- ("QmRR)- (>hqye)- (ahyk)- BM i?2 +b2 Q7 KMB7QH/ rBi?Qmi #QmM/`vc M/ BM i?2 +b2 Q7 KMB7QH/ rBi? #QmM/`v- #v (ApRy) r?Q Q#iBM2/ i?2 2biBKi2b QmibB/2 QM2 +QMp2t Q#bi+H2X

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

MQi?2` iQQH 7Q` MHvxBM; i?2 #2?pBQ` Q7 i?2 a+?`ƺ/BM;2` ~Qr Bb i?2 HQ+H bKQQi?BM;

2z2+i

∥χe−it∆D_u

0∥L2_(R,H1/2_{(M ))}≤ C∥u0∥L2

r?2`2 χ Bb bKQQi? +QKT+iHv bmTTQ`i2/ 7mM+iBQMX h?2 BM~m2M+2 Q7 i?2 ;2QK2i`v QM i?2 HQ+H bKQQi?BM; 2z2+i Bb MQr 7mHHv mM/2`biQQ/, ("m`y9) b?Qr2/ i?i M2+2bb`v M/ bm{+B2Mi +QM/BiBQM 7Q` i?Bb 2biBKi2 iQ ?QH/ Bb i?2 MQM@i`TTBM; +QM/BiBQMX q?2M i`TT2/ ;2Q/2bB+b 2tBbi- HQbb BM i?2 bKQQi?BM; 2z2+i ?b iQ Q++m`X

"mi BM i?2 +b2 Q7 i?2 ai`B+?`ix 2biBKi2b- i?2 bBimiBQM Bb MQi 7mHHv mM/2`biQQ/ v2i, (":>Ry) b?Qr2/ ai`B+?`ix 2biBKi2b rBi?Qmi HQbb 7Q` bvKTiQiB+HHv 2m+HB/BM KMB7QH/b rBi?Qmi #QmM/`v 7Q` r?B+? i?2 i`TT2/ b2i Bb bKHH 2MQm;? M/ 2t?B#Bi M ?vT2`#QHB+ /vMKB+bX

q2 ;Q BM i?2 bK2 /B`2+iBQM M/ b?Qr ;HQ#H ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ +QMp2t Q#bi+H2b- T`QpB/BM; }`bi 2tKTH2 Q7 ai`B+?`ix 2biBKi2b rBi?Qmi HQbb BM i`TT2/ b2iiBM; 7Q` i?2 T`Q#H2K rBi? #QmM/`B2b,

h?2Q`2K dXRX G2i Θ1 M/ Θ2 #2 irQ bKQQi?- +QKT+i- bi`B+iHv +QMp2t bm#b2ib Q7 R3- M/

Ω_{= R}3_\(Θ₁_{∪ Θ}₂₎X A7 (p, q) p2`B}2b

p >2, q≥ 2, 2_p +3 q =

3 2,

i?2M- i?2 7QHHQrBM; 2biBKi2 ?QH/b,

∥e−it∆D_u

0∥Lp_(R,Lq_(Ω))≤ C∥u₀∥_L2. UdXRXRV

ai`B+iHv +QMp2t Bb ?2`2 mM/2`biQQ/ BM i?2 b2Mb2 Q7 (ApRy)- i?i Bb rBi? b2+QM/ 7QM/@ K2MiH 7Q`K #QmM/2/ #2HQrX

h?2 #2?pBQ` Q7 i?2 rp2b 2[miBQM QmibB/2 b2p2`H +QMp2t Q#bi+H2b rb }`bi BMp2biB@ ;i2/ #v (AF33- AF3k)- r?Q b?Qr2/ HQ+H 2M2`;v /2+vX "m`[ mb2/ BM ("m`Nj) bQK2 Q7 ?Bb `2bmHib iQ b?Qr +QMi`QH T`QT2`iv QM M 2[miBQM +HQb2Hv `2Hi2/ iQ i?2 a+?`ƺ/BM;2` 2[miBQM QmibB/2 b2p2`H +QMp2t #Q/B2bX

6Q` /i Q7 7`2[m2M+B2b ∼ h−1_{- B7 r2 `2 #H2 iQ T`Qp2 ai`B+?`ix 2biBKi2b BM iBK2b h}

∥e−it∆D_u

0∥Lp_(0,h)Lq_(Ω) ≤ C∥u₀∥_L2,

i?2M i?2 bKQQi?BM; 2biBKi2 T`QpB/2b i?2 ai`B+?`ix 2biBKi2bX "mi BM Qm` 7`K2rQ`F- HQ;`Bi?KB+ HQbb Q++m` BM i?2 2biBKi2- /m2 iQ i?2 T`2b2M+2 Q7 i`TT2/ `v,

∥χe−it∆D_u

0∥L2_(R,H1/2_{(M ))}≤ C| HM h| 1

2∥u₀∥_L₂_.

>Qr2p2`- (":>Ry) `2K`F2/ i?i i?Bb HQ;`Bi?KB+ HQbb +M #2 +QKT2Mbi2/ B7 r2 b?Qr ai`B+?`ix 2biBKi2b mT iQ iBK2b h| HQ; h|X 6QHHQrBM; i?2B` B/2 M/ `;mK2Mib Q7 (ApRy)- r2 }`bi b?Qr i?i r2 +M `2bi`B+i Qm`b2Hp2b iQ T`Qp2 i?2 2biBKi2 QM M2B;?#Q`?QQ/ Q7 i?2 T2`BQ/B+ `v M/ BM iBK2b h HQ; h- 7Q` /ib Q7 7`2[m2M+B2b ∼ h−1_{X h?2M- r2 `2/m+2 ;BM}

i?2 T`Q#H2K- iQ /i r?B+? KB+`Q@HQ+HHv +QMiBM QMHv TQBMib Q7 i?2 iM;2Mi bT+2 r?B+? /Q MQi 2b+T2 ;Bp2M M2B;?#Q`?QQ/ Q7 i?2 T2`BQ/B+ `v 7i2` HQ;`Bi?KB+ iBK2bX 6BMHHv- 7QHHQrBM; B/2b Q7 (AF33-AF3k)- ("m`Nj)- r2 +QMbi`m+i M TT`QtBKi2 bQHmiBQM 7Q` bm+? /i- M/ r2 b?Qr i?i i?Bb TT`QtBKiBQM ;Bp2b i?2 /2bB`2/ 2biBKi2X

dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v

_2K`F dXkX q2 `2bi`B+i ?2`2 Qm`b2Hp2b iQ i?2 i?`22 /BK2MbBQMH +b2 #2+mb2 r2 mb2 `2bmHib

Q7 (AF33- AF3k) M/ ("m`Nj) r?Q `2 bii2/ BM /BK2MbBQM i?`22X h?2 Qi?2` T`ib Q7 i?2 T`QQ7 ?QH/ BM Mv /BK2MbBQM rBi?Qmi KQ/B}+iBQMX "2+mb2 i?2 #Qp2 K2MiBQM2/ `2bmHib b?QmH/ #2 2ti2M/2/ rBi?Qmi /B{+mHiv iQ i?2 `#Bi``v /BK2MbBQMH +b2- Qm` T`QQ7 b?QmH/ rQ`F BM Mv /BK2MbBQMX

_2K`F dXjX q2 /Q MQi Q#iBM i?2 2M/TQBMi +b2 (p, q) = (2, 6) #2+mb2 r2 KF2 mb2 Q7

i?2 *?`Bbi@EBb2H2p H2KKX >Qr2p2`- i?Bb `2bi`B+iBQM b?QmH/ #2 pQB/2/ rBi? KQ`2 +`27mH MHvbBbX

dXRXRX LQiiBQMb

q2 rBHH mb2 i?2 7QHHQrBM; MQiiBQMb M/ +QMp2MiBQMb,

Θ₁_{, Θ}₂ rBHH #2 irQ bKQQi?- +QKT+i- bi`B+iHv +QMp2t bm#b2ib Q7 R3- +HH2/ i?2 Q#bi+H2b- Xi_{(x, p)} i?2 iǶi? TQBMi Q7 BMi2`b2+iBQM Q7 i?2 `2~2+i2/ `v bi`iBM; i x BM i?2 /B`2+iBQM

p rBi? i?2 Q#bi+H2b-

Ξi_{(x, p)} i?2 /B`2+iBQM Q7 i?2 `2~2+i2/ `v bi`iBM; i x BM i?2 /B`2+iBQM p 7i2` i `2~2+@ iBQMb-

(Xt(x, p), Ξt(x, p)) i?2 TQBMi M/ i?2 /B`2+iBQM iiBM2/ #v 7QHHQrBM; i?2 `2~2+i2/ `v

bi`iBM; i x BM i?2 /B`2+iBQM p /m`BM; iBK2 t i bT22/ |p|- Φt(x, p) i?2 #BHHB`/ ~Qr- Φt(x, p) = (Xt(x, p), Ξt(x, p))X

JQ`2Qp2`- r2 `272` iQ i?2 T2`BQ/B+ `v- /2MQi2/ R- b i?2 b2;K2Mi DQBMBM; Θ1 M/ Θ2 i?i

Bb i?2 mMB[m2 T2`BQ/B+ i`D2+iQ`v Q7 R3_{\ (Θ}

1∪ Θ2) .

G2i h > 0 #2 }t2/ bKHH T`K2i2`X q2 }t α0, β0>0M/ bKQQi? 7mMiBQM ψ ∈ C0∞

bmTTQ`i2/ BM [α0, β0]X 6Q` h > 0 M/ u ∈ L2- ψ(−h2∆)u Bb i?2 HQ+HBbiBQM Q7 u BM

7`2[m2M+B2b Q7 bBx2b [α0h−1, β0h−1]X q2 `272` iQ (ApRy) 7Q` i?2 /2}MBiBQM Q7 ψ(−h2∆)X

6BMHHv- 7Q` a ∈ C∞_(Rn_{× R}n)- r2 rBHH /2bB;M #v PT(a) i?2 KB+`QHQ+H QT2`iQ` rBi?

bvK#QH a QT2`iBM; i b+H2 h- /2}M2/ 7Q` u ∈ L2_(Rn₎ #v

(PT(a)u)(x) = 1 (2πh)n

" "

e−i(x−y)·ξ/ha(x, ξ)u(y)dξdy.

q2 `272` iQ (wrQRk) 7Q` +HbbB+H T`QT2`iB2b Q7 i?2b2 QT2`iQ`bX

dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v

h?2 BK Q7 i?Bb b2+iBQM Bb iQ b?Qr i?i i?2 7QHHQrBM; T`QTQbBiBQM BKTHB2b i?2Q`2K dXR,

S`QTQbBiBQM dX9 Ua2KB+HbbB+H ai`B+?`ix 2biBKi2b QM HQ;`Bi?KB+ BMi2`pH M2` i?2

T2`BQ/B+ `vVX h?2`2 2tBbib . > 0 bm+? i?i 7Q` Mv χ ∈ C∞ 0 pMBb?BM; QmibB/2 bKHH 2MQm;? M2B;?#Q`?QQ/ Q7 i?2 i`TT2/ `v- r2 ?p2 ∥χe−it∆D_ψ(_−h2_∆_)u 0∥Lp_(0,%h|HQ; h|)Lq_(Ω)≤ C∥u₀∥_L2. UdXkXRV Nj

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

6B;m`2 dXRXRX, LQiiBQMb

AM i?2 `2bi Q7 i?Bb b2+iBQM- r2 rBHH bbmK2 i?i dX9 ?QH/b rBi? . = 1 M/ T`Qp2 i?2Q`2K dXRX h?2 T`QQ7 7Q` Qi?2` pHm2b Q7 . > 0 Bb i?2 bK2X

G2i χQ#bi, χ_`v _{∈ C}₀∞ #2 bm+? i?i χ_Q#bi = 1 BM M2B;?#Q`?QQ/ Q7 Θ1∪ Θ2 ∪ R- M/

χ`v∈ C0∞ bm+? i?i χ`v= 1 BM M2B;?#Q`?QQ/ Q7 R- M/ H2i ˜ψ bm+? i?i ˜ψ= 1QM i?2

bmTTQ`i Q7 ψX q2 /2+QKTQb2 ψ(−h2_∆_)eit∆D_u 0 BM bmK Q7 i?`22 i2`Kb ψ(−h2∆_)eit∆D_u 0 = ˜ψ(−h2∆)(1− χQ#bi)ψ(−h2∆)eit∆Du0 + ˜ψ(_−h2∆_)χ_Q#bi₍₁_{− χ}_`v_)ψ(_−h2∆_)eit∆D_u 0 + ˜ψ(_−h2∆_)χ_Q#bi_χ_`v_ψ(_−h2∆_)eit∆D_u 0. UdXkXkV

q2 +M /2H rBi? i?2 }`bi irQ i2`Kb HBF2 BM (ApRy) B7 r2 b?Qr bKQQi?BM; 2z2+i rBi?Qmi

HQbb 7Q` χ 2[mH iQ 0 M2` i?2 i`TT2/ `vX h?2 i?B`/ i2`K +M #2 ?M/H2/ rBi? i?2 K2i?Q/

Q7 (":>Ry), i?2 bKQQi?BM; 2z2+i rBi? HQ;`Bi?KB+ HQbb Q#iBM2/ BM ("m`y9) +QK#BM2/ rBi? S`QTQbBiBQM dX9 M/ i?2 bKQQi?BM; 2z2+i BM i?2 MQM i`TTBM; `2;BQM Bb bm{+B2Mi iQ b?Qr ai`B+?`ix 2biBKi2b rBi?Qmi HQbbX

G2i mb }`bi b?Qr

S`QTQbBiBQM dX8 UGQ+H bKQQi?BM; rBi?Qmi HQbb BM i?2 MQM i`TTBM; `2;BQMVX 6Q` Mv

χ_{∈ C}₀∞ pMBb?BM; BM M2B;?#Q`?QQ/ Q7 i?2 T2`BQ/B+ `v- r2 ?p2 ∥χe−it∆D_u

0∥L2_(R,H1/2_(Ω))!∥u0∥L2. UdXkXjV

S`QQ7X G2i K #2 bKQQi?- +QMM2+i2/- MQM i`TTBM; Q#bi+H2- +QBM+B/BM; rBi? Θ1 ∪ Θ2

QmibB/2 M2B;?#Q`?QQ/ Q7 i?2 T2`BQ/B+ `v QM r?B+? χ Bb pMBb?BM; U};m`2 dXkXRVX LQi2 i?i- BM T`iB+mH`- K +QBM+B/2b rBi? Θ1∪ Θ2 QM i?2 bmTTQ`i Q7 χX q2 b2i ˜Ω = Rn\KX AM

T`iB+mH`- ∆_{D, ˜}Ω M/ ∆D,Ω +QBM+B/2b QM i?2 bmTTQ`i Q7 χX b K Bb MQM i`TTBM;- r2 ?p2

dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v

6B;m`2 dXkXRX,

#v i?2 rQ`F Q7 (owyy) M/ (Jad3-Ja3k) 7Q` i?2 ?B;? 7`2[m2M+B2b T`i- ("m`N3) 7Q` i?2 HQr 7`2[m2M+B2b

∥χ(−∆D, ˜Ω− (λ ± .)2)−1χ∥L2_{( ˜}Ω₎_→L2_{( ˜}Ω₎ ≤ C|λ|−1.

LQr- B7 u ∈ L2(Ω)- r2 ?p2

∥χ(−∆D,Ω− (λ ± .)2)−1χu∥L2_(Ω)=∥χ(−∆_{D, ˜}_Ω− (λ ± .)2)−1χu∥_L2_{( ˜}Ω₎

≤ C∥u∥L2_{( ˜}Ω₎|λ|−1≤ C∥u∥L2_(Ω)|λ|−1.

aQ- i?2 7QHHQrBM; `2bQHp2Mi 2biBKi2 ?QH/b

∥χ(−∆D,Ω− (λ ± .)2)−1χ∥L2_(Ω)_→L2_(Ω) ≤ C|λ|−1.

M/ r2 +M /2/m+2 i?2 bKQQi?BM; 2biBKi2 UdXkXjV 2t+iHv HBF2 BM (":hy9)X dXkXRX h?2 }`bi irQ i2`Kb, rv 7`QK i?2 T2`BQ/B+ `v

AM i?Bb b2+iBQM- r2 /2H rBi? i?2 }`bi irQ i2`Kb Q7 UdXkXkV- 7QHHQrBM; (ApRy)X

dXkXRXRX h?2 }`bi i2`K

q2 b2i wh(x, t) = (1− χQ#bi)ψ(−h2∆)eit∆Du0- r?Q biBb}2b

i∂twh+ ∆Dwh = −[χQ#bi, ∆D]ψ(−h2∆)eit∆Du0

wh(t = 0) = (1− χQ#bi)ψ(−h2∆)u0

"v i?2 `;mK2Mib Q7 (ApRy)- b χ = 1 M2` ∂Ω- wh bQHp2 T`Q#H2K BM i?2 7mHH bT+2- M/

i?2 .m?K2H 7Q`KmH +QK#BM2/ rBi? i?2 ai`B+?`ix 2biBKi2b 7Q` i?2 mbmH GTH+BM QM Rn M/ i?2 *?`Bbi@EBb2H2p H2KK ;Bp2b i?2 2biBKi2

∥wh∥Lp_Lq !∥(1 − χQ#bi)ψ(−h2∆)eit∆Du₀∥_L2_(Rn₎+∥[χQ#bi, ∆_D]ψ(−h2∆)eit∆Du₀∥_L2_H−1/2.

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

LQi2 i?i [χ, ∆D]Bb bmTTQ`i2/ BM +QKT+i b2i M/ QmibB/2 bQK2 M2B;?#Q`?QQ/ Q7 Θ1∪

Θ₂ _{∪ RX aQ- i?2`2 2tBbib ˜}_χ _{∈ C}∞

0 bm+? i?i ˜χ = 1 QM i?2 bmTTQ`i Q7 [χQ#bi, ∆D] M/

pMBb?BM; BM M2B;?#Q`?QQ/ Q7 RX h?2M- #v i?2 bKQQi?BM; 2biBKi2 rBi?Qmi HQbb BM i?2 MQM i`TTBM; `2;BQM UdXkXjV- r2 ;2i

∥[χQ#bi, ∆D]ψ(−h2∆)eit∆Du0∥L2_H−1/2 = ∥[χQ#bi, ∆D] ˜χψ(−h2∆)eit∆Du0∥L2_H−1/2

≤ ∥ ˜χψ(_−h2∆_)eit∆D_u

0∥L2_H1/2

! ∥ψ(−h2∆_)u₀_∥_L2.

6BMHHv- i?2 2biBKi2

∥ ˜ψ(_−h2∆_)w_h_∥_Lr ≤ ∥w_h∥_Lr

Q7 (AS) T2`KBib b BM (ApRy) iQ +QM+Hm/2 i?i ∥ ˜ψ(−h2∆₎₍₁_{− χ}_Q#bi_)ψ(_−h2∆_)eit∆D_u

0∥Lp_Lq ≤ ∥ψ(−h2∆)eit∆Du₀∥_L2.

dXkXRXkX h?2 b2+QM/ i2`K

6QHHQrBM; ;BM (ApRy)- r2 HQ+HBb2 BM bT+2 BM i?2 7QHHQrBM; rv, H2i ϕ ∈ C∞

0 (−1, 2) 2[mH iQ QM2 QM [0, 1]- M/ 7Q` l ∈ Z vh,l = ϕ(t/h− l)χQ#bi(1− χ`v)ψ(−h2∆)eit∆Du0 Vh,l = + ϕ(t/h_{− l)A + i}ϕ′(t/h− l) h χQ#bi(1− χ`v) , ψ(_−h2∆_)eit∆D_u 0 r?2`2

A= ∆χQ#bi(1− χ`v)− χQ#bi∆χ_`v_{− ∇χ}_`v·∇χQ#bi+ (1− χ`v)∇χQ#bi·∇ − χQ#bi∇χ`v·∇. h?2b2 [mMiBiB2b p2`B7v

i∂tvh,l+ ∆Dvh,l = Vh,l

v_{h,l|hl+2h>t>hl−h}= 0

LQi2 i?i i?2 bmTTQ`i Q7 A Bb BM+Hm/2/ BM i?i Q7 χQ#bi(1− χ`v)X G2i Q ⊂ R3 #2 +m#2 bm{+B2MiHv H`;2 iQ +QMiBM Θ1∪ Θ2X q2 /2MQi2 #v S i?2 TmM+im`2/ iQ`mb Q#iBM2/ #v

`2KQpBM; Θ1∪Θ2 Q7 QX LQr- H2i ˜χ ∈ C0∞#2 2[mH iQ QM2 QM i?2 bmTTQ`i Q7 χQ#bi(1−χ`v)-

bmTTQ`i2/ QmibB/2 M2B;?#Q`?QQ/ Q7 R M/ BM M2B;?#Q`?QQ/ Q7 ∂Ω bm+? i?i ∆D M/

∆_S +QBM+B/2b QM Bib bmTTQ`i U};m`2 dXkXkVX q`BiBM; v_h,l _{= ˜}_χv_h,l M/ V_h,l _{= ˜}_χV_h,l- i?2 `2bi Q7 i?2 T`QQ7 7QHHQr b BM (ApRy) +QMbB/2`BM; i?2 T`Q#H2K BM S- mbBM; Qm` bKQQi?BM; 2biBKi2 rBi?Qmi HQbb QmibB/2 i?2 i`TTBM; `2;BQM UdXkXjV BMbi2/ Q7 i?2 bKQQi?BM; 2biBKi2 Q7 ("m`y9)- M/ r2 Q#iBM

∥ ˜ψ(−h2∆_)χ_Q#bi₍₁_{− χ}_`v_)ψ(_−h2∆_)eit∆D_u

0∥Lp_Lq ≤ ∥ψ(−h2∆)eit∆Du₀∥_L2.

dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v

6B;m`2 dXkXkX, dXkXkX h?2 i?B`/ i2`K, M2` i?2 T2`BQ/B+ `v

q2 `2+HH i?2 bKQQi?BM; 2z2+i rBi? HQ;`Bi?KB+ HQbb Q#iBM2/ BM ("m`y9) BM i?2 +b2 Q7 i?2 2ti2`BQ` Q7 KMv +QMp2t Q#bi+H2b p2`B7vBM; i?2 AFrǶb +QM/BiBQM @ r?B+? Bb Hrvb p2`B}2/ BM i?2 T`2b2Mi 7`K2rQ`F Q7 i?2 2ti2`BQ` Q7 irQ #HHb,

S`QTQbBiBQM dXeX 6Q` Mv χ ∈ C∞ 0 (Rd) M/ Mv u0 ∈ L2(Ω)bm+? i?i u0 = ψ(−h2∆)u0 - r2 ?p2 ∥χeit∆D_u 0∥L2_(R,L2₎!(h|HQ; h|) 1 2∥u₀∥_L2 UdXkX9V

G2i mb /2MQi2- BM i?Bb bm#b2+iBQM- χ = χQb#iχ_`vM/ u = ψ(−h2∆_)eit∆D_u

0X q2 rBHH ?2`2

bbmK2 KQ`2Qp2` i?i S`QTQbBiBQM dX9 ?QH/b M/ 7QHHQr i?2 K2i?Q/ Q7 (":>Ry) iQ +QMi`QH i?2 i?B`/ i2`K Q7 UdXkXkVX

AM i?2 bTB`Bi Q7 (":>Ry)- r2 rBHH HQ+HBx2 BM iBK2 BMi2`pHb Q7 H2M;i? h| HQ; h|- QM r?B+? r2 +M TTHv i?2 b2KB@+HbbB+H ai`B+?`ix 2biBKi2b Q7 S`QTQbBiBQM dX9X *QMbB/2` ϕ ∈ C₀∞((_{−1, 1)) biBb7vBM; ϕ ≥ 0- ϕ(0) = 1 M/}(_j_∈Zϕ(s_{− j) = 1X q2 /2+QKTQb2} χu=) j_∈Z ϕ( t h|HQ; h| −j)χu =: ) j_∈Z uj.

h?2 uj biBb7v i?2 2[miBQM

(i∂t− ∆D)uj = Fj+ Gj

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b rBi? Fj = (h|HQ; h|)−1ϕ′( t h|HQ; h| −j)χu, Gj = 2ϕ( t h|HQ; h| −j) (∇χ · ∇u − ∆Dχu) . G2i mb /2MQi2 vj(t) = " t (j−1)h| HQ; h| ei(t−s)∆D_F j(s)ds, wj(t) = " t (j−1)h| HQ; h| ei(t−s)∆D_G j(s)ds. *H2`Hv- uj = vj+ wjX A7 r2 /2}M2 ˜ vj(t) = eit∆D " (j+1)h|HQ; h| (j−1)h| HQ; h| e−is∆D_F j(s)ds, ˜ wj(t) = eit∆D " (j+1)h|HQ; h| (j−1)h| HQ; h| e−is∆D_G j(s)ds,

i?2 *?`Bbi@EBb2H2p H2KK HHQrb mb iQ 2biBKi2 i?2 Lp_Lq MQ`Kb Q7 ˜v

j M/ ˜wj BMbi2/ Q7 vj

M/ wjX

q2 +M mb2 i?2 b2KB@+HbbB+H ai`B+?`ix 2biBKi2 QM HQ;`Bi?KB+ BMi2`pH Q7 S`QTQbBiBQM dX9 iQ 2biBKi2 ∥˜vj∥Lp_Lq, ∥˜vj∥Lp_Lq !∥ " (j+1)h|HQ; h| (j−1)h| HQ; h| eis∆D_F j(s)ds∥L2 q2 iF2 ˜χ ∈ C∞

0 2[mH iQ QM2 QM i?2 bmTTQ`i Q7 χ M/ mb2 i?2 /mH p2`bBQM Q7 UdXkX9V iQ

;2i ∥˜vj∥Lp_Lq ! ∥ " (j+1)h|HQ; h| (j−1)h| HQ; h| eis∆D_F j(s)ds∥L2 = 1 h|HQ; h|∥ " (j+1)h|HQ; h| (j−1)h| HQ; h| eis∆D_χϕ_˜ ′₍ t h|HQ; h| −j)χuds∥L2 ! 1 h|HQ; h| ×(h|HQ; h|) 1 2∥ϕ′( t h|HQ; h| −j)χu∥L2L2, bQ r2 ;2i ∥˜vj∥Lp_Lq ! 1 (h|HQ; h|)1/2∥ϕ′( t h|HQ; h| −j)χu∥L2L2.

G2i mb MQr 2biBKi2 ∥ ˜wj∥Lp_LqX ;BM- #2+mb2 Q7 S`QTQbBiBQM dX9- r2 ?p2

∥ ˜wj∥Lp_Lq !∥ " (j+1)h|HQ; h| (j−1)h| HQ; h| eis∆D_G j(s)ds∥L2. N3

dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v "mi Gj Bb HQ+HBx2/ rv 7`QK i?2 T2`BQ/B+ `vX q2 iF2 ˜χ ∈ C₀∞ 2[mH iQ QM2 QM i?2

bmTTQ`i Q7 ∇χ M/ pMBb?BM; M2` i?2 T2`BQ/B+ `vX h?2M Gj = ˜χGjX >2M+2 r2 +M

mb2 i?2 /mH 2biBKi2 Q7 i?2 bKQQi?BM; 2z2+i rBi?Qmi HQbb BM i?2 MQM i`TTBM; `2;BQM Q7 S`QTQbBiBQM dX8 iQ ;2i ∥ ˜wj∥Lp_Lq ! ∥ " (j+1)h|HQ; h| (j−1)h| HQ; h| eis∆D_G j(s)ds∥L2 =∥ " (j+1)h|HQ; h| (j−1)h| HQ; h| eis∆D_χG_˜ j(s)ds∥L2 ! ∥Gj∥L2_H−1/2 ! ∥ϕ(_h|_{HQ; h| −}t j)_∇χu∥_L2_H1/2.

q2 mb2 i?2 *?`Bbi@EBb2H2p H2KK irB+2- iF2 i?2 b[m`2 M/ bmK iQ Q#iBM ) j∈Z ∥uj∥2Lp_Lq ! ) j∈Z + 1 h|HQ; h|∥ϕ ′₍ t h|HQ; h| −j)χu∥ 2 L2_L2 +∥ϕ( t h|HQ; h| −j)∇χu∥ 2 L2_H1/2 , ! 1 h|HQ; h|∥χu∥ 2 L2_L2+∥∇χu∥2_L2_H1/2

i?2 }`bi i2`K Bb +QMi`QHH2/ mbBM; i?2 bKQQi?BM; 2biBKi2 rBi? HQ;`Bi?KB+ HQbb UdXkX9V- M/ i?2 b2+QM/ i?2 bKQQi?BM; 2biBKi2 QM i?2 MQM@i`TTBM; `2;BQM UdXkXjVX >2M+2 r2 ;2i

)

j∈Z

∥uj∥2Lp_Lq !∥ψ(−h2∆)u₀∥2_L2.

"mi- #2+mb2 Q7 i?2 +QMiBMmQmb 2K#2//BM; l2_{(Z) ֒}_{→ l}p_(Z)7Q` p ≥ 2 r2 FMQr i?i

∥χu∥Lp_Lq ∼ ⎛ ⎝) j_∈Z ∥uj∥p_Lp_Lq ⎞ ⎠ 1/p ! ⎛ ⎝) j_∈Z ∥uj∥2Lp_Lq ⎞ ⎠ 1/2 M/ r2 i?mb +M +QM+Hm/2, ∥ ˜ψ(_−h2∆_)χ_`v_χ_Q#bi_ψ(_−h2∆_)eit∆D_u 0∥Lp_Lq !∥ψ(−h2∆)u₀∥2 L2. dXkXjX *QM+HmbBQM

q2 +QM+Hm/2 7`QK i?2 irQ T`2pBQmb bm#b2+iBQMb i?i S`QTQbBiBQM dX9 BKTHB2b i?2 2biBKi2 ∥e−it∆D_ψ(_−h2_∆_)u

0∥Lp_(R,Lq_(Ω)) ≤ ∥ψ(−h2∆)u₀∥_L2.

GBF2 BM (ApRy)- r2 +M `2KQp2 i?2 7`2[m2M+v +mi@Qz iQ ;2i ∥e−it∆D_u

0∥Lp_(R,Lq_(Ω)) ≤ ∥u₀∥_L2.

>2M+2 S`QTQbBiBQM dX9 BKTHB2b i?2Q`2K dXRX h?mb- i?2 `2bi Q7 i?2 TT2` rBHH #2 /2pQi2/ iQ T`Qp2 S`QTQbBiBQM dX9X

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

6B;m`2 dXjXRX, h?2 i`TT2/ b2i

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb

G2i D #2 M2B;?#Q`?QQ/ Q7 i?2 i`TT2/ `vX 6Q` i2+?MB+H `2bQMb- r2 bmTTQb2 i?i D Bb M QT2M +vHBM/2` rBi? i?2 i`TT2/ `v 7Q` tBbX 6Q` T > 0- r2 /2}M2 i?2 i`TT2/ b2i Q7 D BM iBK2 T ,

.2}MBiBQM dXdX q2 bv i?i (x, ξ) ∈ T⋆_Ω_{∩ (Ω × {|ξ| ∈ [α}

0, β0]}) #2HQM;b iQ i?2 i`TT2/

b2i Q7 D BM iBK2 T - /2MQi2/ TT(D) B7 M/ QMHv B7 i?2`2 2tBbib #`QF2M #B+?`+i2`BbiB+

γ bi`iBM; 7`QK D M/ t < −T bm+? i?i γ(t) = (x, ξ)X JQ`2Qp2`- r2 /2}M2 ˆTT(D) :=

TT(D)∩ {D × Rn}X

AM Qi?2` rQ`/b- TT(D) Bb +QKTQb2/ Q7 i?2 TQBMib Q7 Ω × {|ξ| ∈ [α0, β0]} i?i HB2 BM D

7i2` bQK2 iBK2 #B;;2` i?M T - M/ ˆTT(D)Bb +QKTQb2/ Q7 i?2 TQBMib Q7 D ×{|ξ| ∈ [α0, β0]}

i?i biBHH HB2 BM D 7i2` iBK2 T U};m`2 dXjXRVX

q2 rBHH bv i?i u ∈ L2 _{Bb KB+`Q@HQ+HHv bmTTQ`i2/ BM U ⊂ T}⋆_Ω B7 PT(a)u = u 7Q` HH

a_{∈ C}₀∞(T⋆_Ω₎bm+? i?i a = 1 BM UX h?2 BK Q7 i?Bb b2+iBQM Bb iQ b?Qr i?i Bi Bb bm{+2Mi

iQ T`Qp2 UdXkXRV 7Q` /i KB+`Q@HQ+HHv bmTTQ`i2/ BM ˆT2%|HQ; h|(D)X

dXjXRX aQK2 T`QT2`iB2b Q7 i?2 #BHHB`/ ~Qr

q2 }`bi M22/ bQK2 T`QT2`iB2b Q7 i?2 #BHHB`/ ~Qr bbQ+Bi2/ rBi? Rn_{\ (Θ}

1∪ Θ2) . JQ`2

T`2+Bb2Hv- r2 `2 BMi2`2bi2/ BM i?2 `2;mH`Biv BM (x, ξ) Q7 i?2 ~Qr Φt(x, ξ)X q2 }`bi b?Qr,

G2KK dX3X G2i Θ #2 bKQQi?- bi`B+iHv +QMp2t +QKT+i bm#b2i Q7 Rn- rBi? MQ BM}MBi2

Q`/2` +QMi+i TQBMi- M/

W(Θ) := {(x, ξ), x∈ Rn_{\Θ, ξ}_{∈ R}n_, _{bXiX ∃t ≥ 0, x + tξ ∈ Θ} .}

q2 /2MQi2 #v t i?2 TTHB+iBQM

t: (x, ξ)_{∈ W (Θ) −→ i?2 bKHH2bi t}′ bm+? i?i x + t′ξ _{∈ ∂Ω,}

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb M/ #v WiM(Θ) = {(x, ξ)_{∈ W bXiX ξ · n(x + t(x, ξ)ξ) = 0},} i?2 iM;2Mi `vbX h?2M, RX QM W (Θ)\WiM(Θ)- t Bb C∞, kX i Bb HQ+HHv >ƺH/2` QM W (Θ)X

S`QQ7X G2i g ∈ C∞_(Rn_{, R)} #2 bm+? i?i ∂Θ Bb ;Bp2M #v i?2 BKTHB+Bi `2HiBQM g(x) = 0X q2

/2MQi2- 7Q` x, ξ ∈ Rn- t ∈ R

h(x, ξ, t) = g(x + tξ).

rv 7`QK i?2 iM;2Mi `vb, r2 TB+F (x1, ξ1) ∈ W \WiMX G2i t1 = t(x1, ξ1)X h?2M-

∂th(x1, ξ1, t1) = ξ1·∇g(x1+ t1ξ1)̸= 0 #v /2}MBiBQM Q7 W \WiMX h?2`27Q`2- #v i?2 BKTHB+Bi

7mM+iBQMb i?2Q`2K- t Bb C∞ _{7mM+iBQM Q7 (x, ξ) 7Q` (x, ξ) BM M2B;?#Q`?QQ/ Q7 (x}₁_{, ξ}₁_)X

:2M2`H +b2, G2i (x1, ξ1) ∈ W M/ t1 = (x1, ξ1)X h?2M h(x1, ξ1, t1) = 0X G2i k ≥ 1 #2

i?2 bKHH2bi BMi2;2` bm+? i?i

h((x1, ξ1), t1) = 0, ∂th((x1, ξ1), t1) = 0, . . . , ∂tkh((x1, ξ1), t1)̸= 0.

LQi2 i?i k 2tBbib #2+mb2 Θ ?b MQ BM}MBi2 Q`/2` +QMi+i TQBMiX "v i?2 C∞ T`2T`@

iBQM i?2Q`2K /m2 iQ JH;`M;2 (JHe9)- i?2`2 2tBbib a1, . . . , ak−1 ∈ C∞(Rn× Rn)- M/

c _{∈ C}∞(R_×Rn _{× R}n₎ MQi pMBb?BM; QM ((x 1, ξ1), t1)- bm+? i?i- QM M2B;?#Q`?QQ/ Q7 ((x1, ξ1), t1)- h r`Bi2b h(x, ξ, t) = c(x, ξ, t)(tk+ ak−1(x, ξ)tk−1+ · · · + a0(x, ξ)). h?2M- 7Q` (x, ξ, t) BM M2B;?#Q`?QQ/ Q7 (x1, ξ1, t1)- t= t(x, ξ) _{⇐⇒ t}k+ ak−1(x, ξ)tk−1+ · · · + a0(x, ξ) = 0.

"mi- #v ("`BRy)- i?2 `QQib Q7 KQMB+ TQHvMQKBH `2 >ƺH/2` rBi? `2bT2+i iQ i?2 +Q2{+B2Mib @ Q7 TQr2` QM2 Qp2` i?2 KmHBTHB+Biv Q7 i?2 `QQiX "2+mb2 i?2 ai, 0≤ i ≤ k − 1 `2 C∞rBi?

`2bT2+i iQ (x, ξ)- r2 +QM+Hm/2 i?i t Bb >ƺH/2` BM M2B;?#Q`?QQ/ Q7 (x1, ξ1)X

_2K`F dXNX h?2 rQ`bi TQr2` BM i?2 >ƺH/2` BM2[mHBiv BM M2B;?#Q`?QQ/ Q7 iM;2Mi `v

Bb 1

k- r?2`2 k Bb i?2 Q`/2` Q7 +QMi+i Q7 i?2 iM;2Mi `vX LQi2 i?i BM Qm` 7`K2rQ`F Q7

bi`B+iHv ;2Q/2bB+HHv +QMp2t Q#bi+H2b- k = 2X

G2i η > 0X q2 /QTi i?2 7QHHQrBM; MQiiBQMb 7Q` i?2 iM;2Mib b2ib iQ Θ1∪ Θ2 M/ i?2B`

η@M2B;?#Q`?QQ/,

W_iM = (W_iM(Θ1)∪ WiM(Θ2))∩ {|ξ| ∈ [α0, β0]}

W_iM,i,η = {(x, ξ)_{∈ W (Θ}i), |ξ|∈ [α0, β0]bXiX |ξ · n(x + t(x, ξ)ξ)| ≤ η},

WiM,η = WiM,1,η∪ WiM,2,η.

q2 b?Qr i?i `v +MMQi Tbb bKHH 2MQm;?@M2B;?#Q`?QQ/ Q7 i?2 iM;2Mi b2i KQ`2 i?M irB+2- i?i Bb

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

G2KK dXRyX h?2`2 2tBbib η > 0 bm+? i?i Mv `v +MMQi +`Qbb WiM,η KQ`2 i?M irB+2X

S`QQ7X A7 Bi Bb MQi i?2 +b2- 7Q` HH n ≥ 0- i?2`2 2tBbib (xn, ξn) ∈ K × S2- r?2`2 K Bb

+QKT+i b2i bi`B+iHv +QMiBMBM; i?2 Q#bi+H2b- bm+? i?i Φt(xn, ξn) +`Qbb W_iM,1

n i H2bi

i?`22 iBK2bX 1ti`+iBM; 7`QK (xn, ξn) +QMp2`;BM; bm#b2[m2M+2- #v +QMiBMmBiv Q7 i?2 ~Qr-

H2iiBM; n ;QBM; iQ BM}MBiv r2 Q#iBM `v i?i Bb iM;2Mi iQ Θ1 ∪ Θ2 BM i H2bi i?`22

TQBMibX h?2`27Q`2- Bi bm{b2b iQ b?Qr i?i bm+? `v +MMQi 2tBbibX

_2K`F i?i- #2+mb2 i?2`2 Bb QMHv irQ Q#bi+H2b- B7 (x, ξ) ∈ WiM- B7 r2 +QMbB/2` i?2

`v biìBM; 7`QK (x, ξ) M/ i?2 `v biìBM; 7`QK (x, −ξ)- QM2 Q7 i?2 irQ /Q MQi +`Qbb Mv Q#bi+H2 BM TQbBiBp2 iBK2bX "mi- B7 i?2`2 Bb `v iM;2Mi iQ i?2 Q#bi+H2b BM i H2bi i?`22 TQBMib- B7 r2 +QMbB/2` i?2 b2+QM/ iM;2Mi TQBMi (x0, ξ0)- #Qi? `vb biìBM; 7`QK (x0, ξ0)

M/ (x0,−ξ0) ?p2 iQ +`Qbb M Q#bi+H2- i?2`27Q`2- i?Bb Bb MQi TQbbB#H2X

LQr- r2 +M +QMi`QH ?Qr Km+? irQ `vb bi`iBM; 7`QK /Bz2`2Mi TQBMib M/ /B`2+iBQMb +M /Bp2`;2,

G2KK dXRRX G2i V #2 #QmM/2/ QT2M b2i +QMiBMBM; i?2 +QMp2t ?mHH Q7 Θ1∪ Θ2X h?2M-

i?2`2 2tBbib α > 0- C > 0 M/ τ > 0 bm+? i?i- 7Q` HH x, ˜x ∈ V - HH ξ, ˜ξ bm+? i?i

|ξ|, |ξ′|_{∈ [α}0, β0]- 7Q` HH t > 0 i?2`2 2tBbib t′ p2`B7vBM; |t′− t| ≤ τ bm+? i?i

d(Φt′(˜x, ˜ξ), Φ_t′(x, ξ)))≤ Ct ′

d((˜x, ˜ξ), (x, ξ))α. UdXjXRV

S`QQ7X S`2HBKBM`v MQiiBQMb M/ `2K`FbX 6Q` i = 1, 2- H2i ti #2 i?2 TTHB+iBQM

bbQ+Bi2/ iQ W (Θi) #v H2KK dX3X ++Q`/BM; iQ H2KK dXRy- r2 +?QQb2 η > 0 bKHH

2MQm;? bQ i?i Mv `v +MMQi +`Qbb WiM,η KQ`2 i?M irB+2X LQi2 i?i- #v H2KK dX3-

#2+mb2 Wi Bb +QKT+i- ti Bb ;HQ#HHv >ƺH/2` QM WiX q2 /2MQi2 #v µ > 0 i?2 bKHH2bi

Q7 i?2 irQ >ƺH/2` TQr2`bX JQ`2Qp2`- ti Bb C∞ QM W \WiM-B-kη i?mb BM T`iB+mH` ;HQ#HHv

GBTb+?Bix QM W \WiM,i,ηX

*b2 , (˜x, ˜ξ), (x, ξ) `2 7`X q2 TB+F .0>0iQ #2 +?QQb2M Hi2`X LQi2 i?i r2 Hrvb

?p2 d(Φt(˜x, ˜ξ), Φt(x, ξ))≤ d(x, ˜x) + 2β0t+ 2β0, i?2`27Q`2- B7 d((˜x, ˜ξ), (x, ξ)) ≥ .0- d(Φt(˜x, ˜ξ), Φt(x, ξ))≤ d(x, ˜x) + d((˜x, ˜ξ), (x, ξ)) 2β0 .₀ (t + 1), UdXjXkV M/ i?2 2biBKi2 ?QH/bX *b2 ", (˜x, ˜ξ), (x, ξ) `2 +HQb2X LQr- r2 bmTTQb2 i?i d((˜x, ˜ξ), (x, ξ)) < .0X G2i t0-

`2bTX t1- #2 i?2 }`bi iBK2 r?2`2 {Φt(x, ξ), t≥ 0}- `2bTX

Φ_t_(˜_{x, ˜}_{ξ), t}_{≥ 0}%- +`Qbb M Q#bi+H2- rBi? i?2 +QMp2MiBQM i?i t0 = +∞- `2bTX t1 = +∞ B7 i?Bb /Q2b MQi ?TT2MX q2 /2MQi2

#v X0 M/ X1 i?2 2p2MimH TQBMib Q7 BMi2`b2+iBQM rBi? i?2 Q#bi+H2bX q2 bmTTQb2 i?i

t0 ≤ t1X LQi2 i?i- 7Q` 0 ≤ t ≤ t0- Φt(˜x, ˜ξ) = (˜x+ t ˜ξ, ˜ξ)- Φt(x, ξ) = (x + tξ, ξ)M/ i?mb

d(Φt(˜x, ˜ξ), Φt(x, ξ))≤ (1 + t)d((˜x, ˜ξ), (x, ξ)), 7Q` 0 ≤ t ≤ t0. UdXjXjV

LQr- r2 rQmH/ HBF2 iQ mM/2`biM/ r?i ?TT2Mb 7Q` t ≥ t0.

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb

6B;m`2 dXjXkX, *b2 "XR

*b2 "XR, t0, t1 <∞ M/ X0 M/ X1 #2HQM;b iQ i?2 bK2 Q#bi+H2 ΘiX

q2 ?p2- #2+mb2 t Bb µ@>ƺH/2` QM Wi

|t0− t1|≤ Cd((˜x, ˜ξ), (x, ξ))µ≤ C.µ0 UdXjX9V

q?2`2 C /2T2M/b QMHv Q7 i?2 ;2QK2i`v Q7 i?2 Q#bi+H2bX q2 +?QQb2 .0 bKHH 2MQm;? bQ

i?i

C.µ₀ _≤ 1 4

β₀ UdXjX8V

r?2`2 d Bb i?2 /BbiM+2 #2ir22M i?2 Q#bi+H2bX h?2M- i i?2 iBK2 t1- {Φt(x, ξ), t≥ 0} ?b

MQi +`Qbb2M MQi?2` Q#bi+H2 v2iX h?mb- Φt1(x, ξ) = (x′, ξ′) M/ Φt1+(˜x, ˜ξ) = (˜x′, ˜ξ′) `2

;Bp2M #v- 7i2` `2~2+iBQM ! (x′, ξ′) = (x + t0ξ+ (t1− t0)ξ′, ξ− 2(n · ξ)ξ) (˜x′, ˜ξ′) =/x˜+ t1ξ, ˜˜ ξ− 2(˜n · ˜ξ) ˜ξ 0 rBi? _! n= n(x + t0ξ) ˜ n= n(˜x+ t1ξ)˜ . LQi2 i?i- #2+mb2 X ∈ ∂Θi→ n(X) Bb C∞- |n_{− ˜n| ≤ |n(x + t}0ξ)− n(˜x + t0ξ)| + |n(˜˜ x+ t0ξ)˜ − n(˜x + t1ξ)|˜ ≤ C|x + t0ξ− ˜x − t0ξ| + C|t˜ 1− t0|| ˜ξ| ≤ C|x − ˜x| + |t0||ξ− ˜ξ| + C|t1− t0|| ˜ξ|, KQ`2Qp2`- MQi2 i?i |ti|≤ /BK(V ) α₀ , UdXjXeV Ryj

dX ai`B+?ìx 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b i?mb- #2+mb2 Q7 H2KK dX3- r2 ;2i |n_{− ñ| ≤ |x − ˜x| + |t}0||ξ− ˜ξ| + C| ˜ξ|d((˜x, ˜ξ), (x, ξ))µ0 ≤ Cd((˜x, ˜ξ), (x, ξ))µ0_. UdXjXdV r?2`2 _! µ0= µ B7 (˜x, ˜ξ) ∈ WiM,η Q` (x, ξ) ∈ WiM,η, µ0= 1 2Hb2. LQr- iQ +QMi`QH d((x′_{, ξ}′_{), (˜}_x′_{, ˜}_ξ′₎₎ _{r2 QMHv ?p2 iQ r`Bi2} ! ˜ξ′_{− ξ}′ _{= ˜}_ξ_{− ξ − 2(ñ · ˜}_{ξ)( ˜}_ξ_{− ξ) + 2(ñ − n) · ˜}_ξξ_{+ n · ( ˜}_ξ_{− ξ)ξ} ˜ x′_{− x}′ = ˜x_{− x + (t}1− t0) ˜ξ− t0(ξ− ˜ξ)− (t1− t0)ξ′

M/ #2+mb2 Q7 UdXjXdV- UdXjXeV- M/ H2KK dX3 r2 Q#iBM

d(Φt1(x, ξ), Φt1+(˜x, ˜ξ))≤ Cd((x, ξ), (˜x, ˜ξ))

µ0 UdXjX3V

rBi?- #2+mb2 Q7 UdXjX9V M/ UdXjX8V

|t0− t1|≤ 1 4 d β0 . *b2 "Xk, t0 < ∞ M/ t1 = +∞c Q` t0, t1 < ∞ M/ X0 M/ X1 #2HQM;b iQ /Bz2`2Mi

Q#bi+H2bX amTTQb2 7Q` 2tKTH2 i?i X0∈ Θ1X 6Q` i?2 bF2 Q7 bBKTHB+Biv- r2 /Q i?2 T`QQ7

BM /BK2MbBQM k M/ i?2M 2tTHBM ?Qr iQ /Ti Bi BM /BK2MbBQM jX

amTTQb2 i?i i?2 `v bi`iBM; 7`QK (x, ˜ξ) /Q MQi +`Qbb Θ1X q2 +M Hrvb iF2 ξ1 iQ #2

bm+? i?i (x, ξ1)∈ WiM,1 rBi? |ξ1| = |ξ|- M/

|ξ_{− ξ}1|≤ |ξ − ˜ξ|, | ˜ξ− ξ1|≤ |ξ − ˜ξ|, UdXjXNV

i?i Bb- r2 +?QQb2 ξ1 bm+? i?i i?2 `v bi`iBM; 7`QK (x, ξ1) Bb iM;2Mi iQ Θ1 M/ #2ir22M

i?2 `v bi`iBM; 7`QK (x, ξ) M/ i?2 `v bi`iBM; 7`QK (x, ˜ξ) 7Q` t ≤ t0 U};m`2 dXjXjVX

_2K`F i?i- b +QMb2[m2M+2 Q7 UdXjXNV #2+mb2 (x, ξ1)∈ WiM M/ |ξ − ˜ξ| ≤ .0- iFBM;

.0 ≤ 1₂η bbm`2 i?i- M2+2bb`BHv- (x, ξ) ∈ WiM,ηX

G2i t′

0 #2 i?2 iBK2 7Q` r?B+? i?2 `v bi`iBM; 7`QK (x, ξ1) Bb iM;2Mi iQ Θ1X LQi2 i?i-

#2+mb2 i?2 TTHB+iBQM t Bb >ƺH/2`- BM i?2 bK2 rv i?i UdXjX9V- M/ #2+mb2 Q7 UdXjXNV- iFBM; ;BM .0 bKHH 2MQm;? bQ i?i UdXjX8V Bb p2`B}2/- r2 ?p2

|t0− t′0|≤ 1 4 d β₀. P#pBQmbHv d(Φt_0′(x, ξ), Φt_0′(x, ˜ξ))≤ d(Φt′ 0(x, ξ), Φt′0(x, ξ1)) + d(Φt0′(x, ξ1), Φt′0(x, ˜ξ)). UdXjXRyV "mi- mbBM; i?2 +b2 "XR d(Φt′ 0(x, ξ), Φt′0(x, ξ1))≤ Cd((x, ξ), (x, ξ1)) µ _UdXjXRRV Ry9

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb

6B;m`2 dXjXjX, *b2 "XkX @ i?2 `v 7`QK (x, ˜ξ) /Q MQi +`Qbb Θ1

M/ KQ`2Qp2`- BM i?2 bK2 rv i?M UdXjXjV- #2+mb2 Q7 UdXjXeV- d(Φt′

0(x, ˜ξ), Φt′0(x, ξ1))≤ Cd((x, ˜ξ), (x, ξ1)). UdXjXRkV

*QK#BMBM; UdXjXNV- UdXjXRyV- UdXjXRRV M/ UdXjXRkV r2 Q#iBM d(Φt′

0(x, ˜ξ), Φt′0(x, ξ))≤ C|ξ − ˜ξ|

µ_.

"mi- BM i?2 bK2 rv i?M UdXjXjV ;BM- #2+mb2 MQM2 Q7 i?2 `vb bi`iBM; 7`QK (˜x, ˜ξ) Q` (x, ˜ξ) +`Qbb Q#bi+H2b- d(Φt′ 0(˜x, ˜ξ), Φt′0(x, ˜ξ))≤ Cd((x, ˜ξ), (x, ξ1)) M/ r2 +QM+Hm/2 i?i d(Φt′ 0(˜x, ˜ξ), Φt′0(x, ξ))≤ Cd((˜x, ˜ξ), (x, ξ)) µ_.

LQr- bmTTQb2 i?i i?2 `v bi`iBM; 7`QK (x, ˜ξ) /Q +`Qbb Θ1X h?2M- i?2`2 2tBbib x1∈ [x, ˜x]

bm+? i?i i?2 `v bi`iBM; 7`QK (x1, ˜ξ)Bb iM;2Mi iQ Θ1U};m`2 dXjX9VX q2 /Q i?2 2t+i bK2

bim/v b BM i?2 T`2pBQmb +b2- MQr iFBM; i?2 `v bi`iBM; 7`QK (x1, ˜ξ) b BMi2`K2/B`v iQ

Q#iBM ;BM d(Φt′ 0(˜x, ˜ξ), Φt′0(x, ξ))≤ Cd((˜x, ˜ξ), (x, ξ)) µ_, _|t 0− t′0|≤ 1 4 d β0 . UdXjXRjV

_2K`F i?i- iFBM; ;BM .0 ≤ 1₂η- r2 FMQr BM i?2 bK2 rv i?i M2+2bb`BHv-(x, ξ) ∈

W_iM,ηX

AM /BK2MbBQM j- r2 /Q i?2 bK2 T`QQ7 rBi? MQi?2` BMi2`K2/Bi2 TQBMi- BM Q`/2` iQ i`2i QMHv +QTHM` `vb i?`22 #v i?`22X

*QM+HmbBQM, hQ +QM+Hm/2- r2 Bi2`i2 i?Bb `;mK2Mi mT iQ iBK2 t, b bQQM b d(ΦT0(x, ξ), ΦT0(˜x, ˜ξ))≥

.0- r2 mb2 UdXjXkV 7Q` iBK2b H`;2` i?M T0X "2ir22M `2~2+iBQMb- r2 mb2 UdXjXjV iQ;2i?2`

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

6B;m`2 dXjX9X, *b2 "XkX# @ i?2 `v 7`QK (x, ˜ξ) +`Qbb Θ1

rBi? UdXjXeVX i `2~2+iBQMb- r2 mb2 UdXjX3V U+b2 "XRV Q` UdXjXRjV U+b2 "XkVX LQi2 i?i #2+mb2 i?2 `vb +MMQi +`Qbb WiM,η KQ`2 i?M irB+2- r2 `2 BM i?2 +b2 r?2`2 µ0= µi

KQbi 7Qm` iBK2bX aQ r2 ;2i- rBi? N(t) i?2 MmK#2` Q7 `2~2+iBQMb r2 ?p2 2M+QmMi2`2/ BM iBK2 t,

d(Φt(˜x, ˜ξ), Φt(x, ξ))≤ CN(t)(1 + t)d((x, ξ), (˜x, ˜ξ))µ

, UdXjXR9V

7Q` HH t ≥ 0 2t+2Ti i?Qb2b BM i?2 BMi2`pHb [t0, t1] r?2M i?2 +b2 "XRX Bb 2M+QmMi2`2/ M/

i?Qb2b BM i?2 BMi2`pHb [t0, t′0]r?2M i?2 +b2 "Xk Bb 2M+QmMi2`2/X LQi2 i?i r2 Hrvb ?p2

|t0− t1|, |t0− t′0|≤ 1 4 d β0 ,

M/ i?2 iBK2 b2T`BM; i?2 t1Ƕb U`2bT t0ǶbV 7`QK MQi?2` bm+? 7Q`#B/2M BMi2`pH @ i?i Bb-

7`QK MQi?2` `2~2+iBQM Q7 QM2 Q7 i?2 irQ `vb @ Bb i H2bi d

β0 − |t0− t1|≥ d β0 − 1 4 d β0 >0 U`2bT d β0 − |t0− t ′

0|Vc i?2`27Q`2 UdXjXR9V ?QH/b 7Q` HH t 2t+2Ti BM /BbDQBMib BMi2`pHb Q7 H2M;?i

i KQbi τ 2 = d 4β0X hQ +QM+Hm/2- Bi bm{b2 iQ MQi2 i?i N(t) ≤ 2 β0 dtX

_2K`F dXRkX "2+mb2 i?2 /B`2+iBQMH +QKTQM2Mi Q7 i?2 ~Qr Bb MQi +QMiBMmQmb rBi? `2bT2+i

iQ iBK2- r2 +M MQi ?p2 UdXjXRV 7Q` HH iBK2X >Qr2p2`- r2 +M iF2 τ > 0 b bKHH b r2 rMi i i?2 +Qbi iQ iF2 #B;;2` +QMbiMi C BM UdXjXRV @ i?i +Q``2bTQM/b iQ iF2 .0

bKHH2`X

_2K`F dXRjX AM Qm` 7`K2rQ`F Q7 bi`B+iHv ;2Q/2bB+HHv +QMp2t Q#bi+H2b- ++Q`/BM; iQ

_2K`F dXN M/ Q7 UdXjXR9V- r2 ?p2 α = 24_{= 16X}

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb dXjXkX S`QT2`iB2b Q7 i?2 i`TT2/ b2i

q2 MQr BMp2biB;i2 bQK2 T`QT2`iB2b Q7 i?2 i`TT2/ b2i r2 ?p2 /2}M2/ BM i?2 #2;BMBM; Q7 i?2 b2+iBQMX

G2KK dXR9X 6Q` HH #B+?`+i2`BbiB+ γ bi`iBM; 7`QK D rBi? bT22/ BM [α0, β0]- r2 ?p2

d(γ(t), TT(D)c) > 0 ∀t ∈ [−T − 1, −T ]

S`QQ7X LQiB+2 i?i- #2+mb2 Q7 i?2 +QMiBMmBiv Q7 i?2 ~Qr- TT(D) Bb QT2MX LQr- H2i γ #2

#B+?`+i2`BbiB+ bi`iBM; 7`QK D rBi? bT22/ BM [α0, β0]X h?2 b2i {γ(t), t ∈ [−T −1, −T]} Bb

+QKT+i M/ TT(D)c Bb +HQb2/- bQ i?2 /BbiM+2 #2ir22M i?2K Bb iiBM2/X "mi #v /2}MBiBQM

Q7 TT(D)- i?2b2 irQ b2ib M2p2` +`QbbX h?2`27Q`2- i?2 T`QTQbBiBQM ?QH/bX

h?2 7QHHQrBM; +`m+BH H2KK Bb +QMb2[m2M+2 Q7 H2KK dXRR

G2KK dXR8X 6Q` HH D, ˜D- i?2`2 2tBbib T⋆ _>₀- c > 0 bm+? i?i 7Q` HH T ≥ 0,

d(TT−T⋆(D)c,T_T(D))≥ e−cT, UdXjXR8V M/- B7 D ⊂ ˜D d(TT( ˜D)c,TT(D))≥ 1 4e−cTd( ˜D c_{, D).} _UdXjXReV

S`QQ7X G2i mb b?Qr UdXjXR8VX G2i T⋆ _> 0X q2 `;m2 #v +QMi`/B+iBQMX amTTQb2 i?i

i?2 T`QT2`iv Bb 7Hb2X h?2M- 7Q` HH n ≥ 1 i?2`2 2tBbib Tn ≥ 0- (xn, ξn) ∈ TTn(D) M/

(˜xn, ˜ξn)∈ TTn−T⋆(D) c _{bm+? i?i} d((˜xn, ˜ξn), (xn, ξn))≤ e−nTn. UdXjXRdV "v H2KK dXRR- i?2`2 2tBbib T′ n∈ [Tn− τ, Tn+ τ ] bm+? i?i d(ΦT′ n(˜xn, ˜ξn), ΦTn′(xn, ξn)))≤ d((˜xn, ˜ξn), (xn, ξn)) α_CT′ n_, i?2M- #2+mb2 Q7 UdXjXRdV- d(ΦT′ n(˜xn, ˜ξn), ΦTn′(xn, ξn)))≤ e −αnTn_CTn+τ _{−→ 0} UdXjXR3V b n ;Q2b iQ BM}MBivX q2 rQmH/ HBF2 iQ FMQr ?Qr 7` ΦT′ n(˜xn, ˜ξn) +M #2 7`QK DX G2i (yn, ηn) = ΦT′ n−T⋆+τ(˜xn, ˜ξn).

"v /2}MBiBQM Q7 i?2 i`TT2/ b2i- Φt(˜xn, ˜ξn) /∈ D 7Q` HH t ≥ Tn− T⋆X AM T`iB+mH`- i?Bb Bb

i`m2 7Q` HH t ≥ T′

n− T⋆+ τX h?mb- i?2 `v bi`iBM; 7`QK (yn, ηn)M2p2` +`Qbb DX h?2`27Q`2-

#2+mb2 r2 ?p2 +?Qb2M D iQ #2 +vHBM/2` rBi? i?2 T2`BQ/B+ i`D2+iQ`v 7Q` tBb- (yn, ηn)Bb

BM i?2 +b2 r?2`2 ΦT′

n(˜xn, ˜ξn) = ΦT⋆−τ(yn, ηn)Bb i?2 +HQb2bi iQ D- bm+? i?i U};m`2 dXjX8V,

Ç ηnBb T`HH2H iQ i?2 T2`BQ/B+ i`D2+iQ`v-

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

6B;m`2 dXjX8X, (yn, ηn) BM i?2 rQ`bi +b2 b+2M`BQ

Ç |ηn| = |ξn| = α0-

Ç yn∈ ∂Θi

BM i?Bb rQ`bi +b2 b+2M`BQ- 7i2` iBK2 T⋆_{− τ- Φ}

Tn(˜xn, ˜ξn) = ΦT⋆−τ(yn, ηn)Bb i /BbiM+2

i H2bi c(T⋆_{) > 0} Q7 D- rBi? c(T⋆₎ /2T2M/BM; QMHv Q7 D- i?2 KBMBKH +m`pim`2 Q7

i?2 Q#bi+H2b- α0- β0- M/ bm+? i?i c(T⋆) → ∞ b T⋆ → ∞X PM i?2 Qi?2` ?M/-

ΦTn(xn, ξn) ∈ D- bQ- #2+mb2 |Tn′ − Tn|≤ τ- ΦTn′(xn, ξn) Bb i /BbiM+2 i KQbi β0τ Q7 DX

q2 +M i?mb +?QQb2 T⋆ _>₀ H`;2 2MQm;? bQ i?i 7Q` HH n ≥ 0

d(ΦT′

n(˜xn, ˜ξn), ΦTn′(xn, ξn))≥ 1. UdXjXRNV

r?B+? +QMi`/B+ib UdXjXR3V M/ UdXjXR8V ?QH/bX

G2i mb MQr b?Qr UdXjXReVX A7 (x, ξ) ∈ TT( ˜D)c M/ (˜x, ˜ξ) ∈ TT(D)r2 ?p2 ΦT(x, ξ) /∈ ˜D

M/ ΦT(˜x, ˜ξ)∈ D- i?2`27Q`2

d(ΦT(x, ξ), ΦT(˜x, ˜ξ)) > d( ˜Dc, D). UdXjXkyV

PM i?2 Qi?2` ?M/- #v H2KK dXR9- i?2`2 2tBbi T′ _{bm+? i?i |T − T}′_|_{≤ τ p2`B7vBM;}

d(ΦT′(x, ξ), Φ_T′(˜x, ˜ξ))≤ CT ′

d((x, ξ), (˜x, ˜ξ)). UdXjXkRV ++Q`/BM; iQ _2K`F dXRk- r2 +M bmTTQb2- mT iQ 2MH`;2 i?2 +QMbiMi C- i?i τ Bb bKHH 2MQm;? bQ i?i β0τ ≤ 1₄d( ˜Dc, D)X h?2M UdXjXkyV BKTHB2b d(ΦT′(x, ξ), Φ_T′(˜x, ˜ξ))≥ 1 2d( ˜D c_{, D).} _UdXjXkkV h?2`27Q`2- #v UdXjXkRV M/ UdXjXkkV- B7 d((x, ξ), (˜x, ˜ξ)) ≤ 1 4C−Td( ˜D c_{, D)- r2 +MMQi ?p2} (x, ξ)_{∈ T}T( ˜D)c M/ (˜x, ˜ξ) ∈ TT(D)X h?mb UdXjXReV Bb p2`B}2/X Ry3

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb dXjXjX _2/m+iBQM Q7 i?2 T`Q#H2K

q2 `2 MQr BM TQbBiBQM iQ b?Qr i?i r2 +M `2/m+2 Qm`b2Hp2b iQ T`Qp2 i?2 7QHHQrBM; T`QTQbBiBQM,

S`QTQbBiBQM dXReX h?2`2 2tBbib . > 0- bKHH QT2M M2B;?#Q`?QQ/ D Q7 i?2 i`TT2/ `v-

bm+? i?i- ∀u0 ∈ L2 KB+`QHQ+HHv bmTTQ`i2/ BM ˆT2%|HQ; h|(D)- M/ rv 7`QK bKHH 2MQm;?

M2B;?#Q`?QQ/ Q7 ∂ (Θ1∪ Θ2)- r2 ?p2 7Q` HH χ ∈ C0∞ bmTTQ`i2/ BM D

∥χe−it∆ψ(_−h2∆_)u₀_∥_Lp_(0,%h|HQ; h|)Lq ≤ C∥ψ(−h2∆)u₀∥_L2. UdXjXkjV

S`QQ7 Q7 S`QTQbBiBQM dXRe BKTHB2b i?2Q`2K dXRX q2 bmTTQb2 i?i S`QTQbBiBQM dXRe ?QH/b

M/ b?Qr Qm` KBM `2bmHiX "v i?2 rQ`F Q7 i?2 T`2pBQmb b2+iBQM M/ i?2 b2KB@+HbbB+H +?M;2 Q7 p`B#H2- Bi bm{b2b iQ b?Qr i?i i?2`2 2tBbib .′ _>₀_{bm+? i?i}

∥χe−ith∆D_ψ(_−h2_∆_)u

0∥Lp_(0,%′|HQ; h|)Lq_(Ω) ≤ Ch−1/p∥u₀∥_L2, UdXjXk9V

7Q` χ bmTTQ`i2/ BM M2B;?#Q`?QQ/ Q7 i?2 i`TT2/ `vX

q2 `2 ?2`2 BMbTB`2/ #v (ApRy)- a2+iBQM k- TXke8@keeX b (ApRy) `2+HHb- #v i?2 b2KB+Hb@ bB+H }MBi2 bT22/ Q7 T`QT;iBQM /m2 iQ (G2#Nk)- r2 +M `2bi`B+i Qm`b2Hp2b iQ #B+?`+i2`BbiB+ i`p2HH2/ i bT22/ BM [α0, β0]X q2 `272` iQ (G2#Nk)- (\) TT2M/Bt "- M/ (ApRy) TT2M/Bt

7Q` i?2 T`QT;iBQM T`QT2`iB2b Q7 i?2 a+?`ƺ/BM;2` ~Qr BM i?2 b2KB@+HbbB+H `2;BK2 7Q` i?2 T`Q#H2K rBi? #QmM/`B2bX

G2i D1- D2- D3 #2 QT2M M2B;?#Q`?QQ/b Q7 i?2 i`TT2/ `v bm+? i?i

D3! D2 ! D1! D.

G2i T = 2.| HQ; h|X q2 rBHH b?Qr i?i- 7Q` χ bmTTQ`i2/ BM D3 M/ 7Q` bQK2 .3< .2 < .1 < .,

RX B7 UdXjXk9V ?QH/b 7Q` HH /i KB+`QHQ+HHv bmTTQ`i2/ BM D1∩ TT+1(D1) rBi? .′ = .1

i?2M Bi ?QH/b 7Q` HH /i bmTTQ`i2/ BM D2 rBi? .′ = .2-

kX i?2`2 2tBbib bKHH M2B;?#Q`?QQ/ Q7 i?2 #QmM/`v V bm+? i?i- B7 UdXjXk9V ?QH/b 7Q` HH /i KB+`QHQ+HHv bmTTQ`i2/ BM D ∩ TT(D)∩ Vc rBi? .′ = .i?2M Bi ?QH/b 7Q` HH

/i bmTTQ`i2/ BM D1∩ TT+1(D1)rBi? .′ = .1-

jX B7 UdXjXk9V ?QH/b 7Q` HH /i bmTTQ`i2/ BM D2 rBi? .′ = .2 i?2M Bi ?QH/b 7Q` HH /i

rBi? .′ _{= .} 3-

M/ i?2 T`QTQbBiBQM rBHH 7QHHQrX

URV _2/m+iBQM iQ i?2 i`TT2/ b2ibX G2i ψ(−h2_∆_)u

0 #2 bmTTQ`i2/ BM D2X "v i?2

b2KB+HbbB+H }MBi2 bT22/ Q7 T`QT;iBQM- i?2`2 2tBbib τ > 0 bKHH 2MQm;? bm+? i?i- KQ/mHQ O(h∞)MQM +QMi`B#miBM; i2`Kb- e−ith∆ψ(_−h2_∆_)u

0 Bb bmTTQ`i2/ BM D1 7Q` HH t ∈ [−τ, 0]X

G2i T⋆#2 ;Bp2M #v H2KK dXR8X G2i a ∈ C∞_(Rn_×Rn₎#2 bm+? i?i a = 1 BM T

T+1+T⋆(D₁)

M/ a = 0 QmibB/2 TT+1(D1)X 6Q` i?2 +QMp2MB2M+2 Q7 i?2 `2/2`- H2i mb /2MQi2

T := T + 1 + T⋆.

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b

1_{− a Bb bmTTQ`i2/ BM T}Tˆ(D1)c - bQ- 2p2`v #B+?`+i2`BbiB+ γ bi`iBM; 7`QK D1 i iBK2 t = 0

rBi? bT22/ BM [α0, β0]p2`B}2b #v H2KK dXR9

d(γ(t),amTT(1 − a)) > 0 ∀t ∈ [− ˆT − τ, − ˆT]. UdXjXk8V G2i Ψ ∈ C∞_(R)_{bm+? i?i Ψ(t) = 0 7Q` t ≤ −τ- Ψ(t) = 1 7Q` t ≥ 0- M/ Ψ}′ _{≥ 0- M/ b2i}

Ψ_T_{(t) = Ψ(t + ˆ}_T)X q2 /2}M2

w(t, x) = ΨT(t)e−i(t+ ˆT)h∆ψ(−h2∆)u0.

h?2M w biBb}2b

i∂tw− h∆w = iΨ′T(t)e−i(t+ ˆT)h∆ψ(−h2∆)u0

w_|∂Ω = 0, w|_t_{≤− ˆ}_T_−τ = 0

#v i?2 .m?K2H 7Q`KmH- b w(t) = e−i ˆT h∆_{u(t) := e}−i(t+ ˆT)h∆_ψ(_−h2_∆_)u

0 7Q` t ≥ − ˆT M/ b Ψ′ T Bb bmTTQ`i2/ BM [− ˆT − τ, − ˆT]- r2 ?p2- 7Q` t ≥ − ˆT e−i ˆT h∆u(t, x) = " _{− ˆ}T − ˆT−τ

e−i(t−s)h∆iΨ′_T(s)e−i(s+ ˆT)h∆ψ(_−h2∆_)u₀_ds. .2MQi2- 7Q` Q ∈ {PT(a), PT(1 − a)}

uQ(t, x) =

" _{− ˆ}T − ˆT−τ

e−i(t−s)h∆iΨ′_T(s)Q/e−i(s+ ˆT)h∆ψ(_−h2∆_)u₀0_ds. r?B+? Bb bQHmiBQM Q7

i∂tuQ− h∆uQ= iΨ′T(t)Q

e−i(t+ ˆT)h∆ψ(_−h2∆_)u₀0_.

q2 rBHH b22 i?i i?2 i2`K χu1−A /Q2b MQi +QMi`B#mi2X hQ i?Bb Tm`TQb2- r2 mb2 i?2 b@rp2

7`QMi b2i Q7 u1−A- W Fb(u1−A)X q2 `272` iQ (ApRy) 7Q` i?2 /2}MBiBQM Q7 i?Bb MQiBQMX "v

T`QTQbBiBQM X3 Q7 (ApRy)- B7 ρ0∈ W Fb(u1−A)- i?2M i?2 #`QF2M +?`+i2`BbiB+ bi`iBM; 7`QK

ρ0 ∈ W Fb(u1−A)Kmbi BMi2`b2+i i?2 rp2 7`QMi b2i

W Fb(PT(1 − a)e−i ˆT h∆u)∩ {t ∈ [− ˆT − τ, − ˆT]}. UdXjXkeV

q2 `2 BMi2`2bi2/ BM 2biBKiBM; u QMHv QM D3bQ Bi Bb 2MQm;? iQ +QMbB/2` ρ0 ∈ W Fb(χu1−A)X

AM T`iB+mH`- B7 γ Bb #`QF2M +?`+i2`BbiB+ bi`iBM; 7`QK ρ0 ∈ W Fb(χu1−A)- i?2`2 2tBbib

t_{∈ [− ˆ}T_{− τ, − ˆ}T]bm+? i?i γ(t) BMi2`b2+i UdXjXkeVX "2+mb2 Q7 UdXjXk8V- i?Bb Bb MQi TQbbB#H2- M/

∥χu1−A∥Hσ_(Ω_×R)= O(h∞)∥ψ(−h2∆)u₀∥_L2 UdXjXkdV

7Q` HH σ ≥ 0- M/ i?Bb i2`K /Q2b MQi +QMi`B#mi2X RRy

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb PM i?2 Qi?2` ?M/- `2K`F i?i

∥χuA∥_Lp₍_{− ˆ}_{T ,}_{− ˆ}_T_−τ+T₁_)Lq ≤ " _{− ˆ}T − ˆT−τ∥χe −i(t−s)h∆_Ψ′ T(s)A / e−i(s+ ˆT)h∆ψ(_−h2∆_)u₀0_∥ Lp₍_{− ˆ}_{T ,}_{− ˆ}_T_−τ+T 1)Lqds = " _{− ˆ}T − ˆT_−τ∥χe −ith∆Ψ′ T(s)A / e−i(s+ ˆT)h∆ψ(_−h2∆_)u₀0_∥ Lp₍_{− ˆ}_T_{−s,− ˆ}_T_−τ+T 1−s)Lqds ≤ " _{− ˆ}T − ˆT_−τ∥χe −ith∆Ψ′ T(s)A / e−i(s+ ˆT)h∆ψ(−h2∆_)u₀0_∥_Lp_(0,T₁_)Lqds.

h?mb- B7 ai`B+?`ix 2biBKi2b UdXjXkjV BM iBK2 T1 = .1|HQ; h| ?QH/ i`m2 7Q` Mv /i KB@

+`QHQ+HHv bmTTQ`i2/ r?2`2 a ̸= 0 M/ bTiBHHv bmTTQ`i2/ BM D1- i?i Bb BM TT+1(D1)∩ D1-

r2 ?p2- #2+mb2 Ψ′ T(s)A

e−i(s+ ˆT)h∆ψ(_−h2∆_)u₀0 Bb- 7Q` s ∈ [− ˆT − τ, − ˆT] M/ KQ/mHQ MQM +QMi`B#miBM; i2`Kb- bm+? /i

∥χuA∥_Lp_{(− ˆ}_{T ,}_{− ˆ}_T_−τ+T₁_)Lq ! h−1/p " _{− ˆ}T − ˆT−τ∥Ψ ′ T(s)A / e−i(s+ ˆT)h∆ψ(−h2∆_)u₀0_∥_L2ds ≤ h−1/p " _{− ˆ}T − ˆT−τ

Ψ′_T_(s)_∥PT(a)∥_L2_→L2∥e−i(s+ ˆT)h∆ψ(−h2∆)u₀∥_L2ds

= h−1/p " _{− ˆ}T − ˆT−τ Ψ′_T_(s)_∥PT(a)∥_L2_→L2∥ψ(−h2∆)u₀∥_L2ds = h−1/p_∥PT(a)∥L2_→L2∥ψ(−h2∆)u₀∥_L2. q?2`2 r2 mb2/ i?i8_{− ˆ}T − ˆT−τΨ′T(s)ds = 1. M/ i?2`27Q`2- #2+mb2 Q7 UdXjXkdV ∥χe−i ˆT h∆_u_∥ Lp_{(− ˆ}_{T ,}_{− ˆ}_T_−τ+T 1)Lq ! h −1/p_∥PT(a)∥ L2_→L2∥ψ(−h2∆)u₀∥_L2. Ai `2KBMb iQ 2biBKi2 ∥PT(a)∥L2_→L2X q2 ?p2- ++Q`/BM; iQ (wrQRk), ∥PT(a)∥L2_→L2 ! bmT |a| + h1/2 ) |α|≤2n+1 bmT |∂|α|_a| ≤ 1 + h1/2 ) |α|≤2n+1 bmT |∂|α|_a|. "v H2KK dXR8- a +M #2 +?Qb2M BM bm+? rv i?i |∂|α|a|_{≤ 2e}c|α|T bQ- 7Q` T = 2.| HQ; h|, |∂|α|a| ! h−2c|α|% r2 iF2 . > 0 bKHH 2MQm;? bQ i?i 2c(2n + 1). ≤ 1 2 M/ r2 ;2i ∥χe−i ˆT h∆_u_∥ Lp_{(− ˆ}_{T ,}_{− ˆ}_T_−τ+T₁_)Lq ! h−1/p∥ψ(−h2∆)u0∥L2, RRR

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b i?i Bb ∥χe−ith∆ψ(−h2∆_)u₀_∥_Lp_(0,T 1−τ)Lq ! h−1/p∥ψ(−h 2_∆_)u 0∥L2, M/ i?mb ∥χe−ith∆ψ(_−h2∆_)u₀_∥_Lp_(0,% 2|HQ; h|)Lq ! h−1/p∥ψ(−h 2_∆_)u 0∥L2 7Q` .2 < .1 bKHH 2MQm;?X

UkV _2bi`B+iBQM iQ /i bmTTQ`i2/ rv 7`QK i?2 #QmM/`vX G2i η > 0X q2

`2K`F i?i i?2`2 2tBbib t1(η) > 0- t2(η) > 0 bm+? i?i t1,2(η) → 0 b η → 0 M/- B7 r2

/2MQi2

E1(η) = {d(x, ∂(Θ1∪ Θ2) < η} , E2(η) = {d(x, ∂(Θ1∪ Θ2)≥ η} ,

i?2M HH #B+?`+i2`BbiB+ bi`iBM; 7`QK E1∩ D rBi? bT22/ BM [α0, β0]p2`B}2b

d(γ(t), V_{∩ D) > 0 ∀t ∈ [−2t}1,−t1],

M/ HH #B+?`+i2`BbiB+ bi`iBM; 7`QK E2∩ D rBi? bT22/ BM [α0, β0]p2`B}2b

d(γ(t), V_{∩ D) > 0 ∀t ∈ [−t}2,0],

rBi? V(η) bKHH M2B;?#Q`?QQ/ Q7 i?2 #QmM/`vX G2i ψ(−h2_∆_)u

0KB+`QHQ+HHv bmTTQ`i2/ BM TT+1(D1)∩{D1× Rn}X G2i τ0(η) =Kt(2t1, t2)X

q2 iF2 η > 0 bm{+2MiHv bKHH /2T2M/BM; QMHv Q7 D M/ D1 bQ i?i τ0 ≤ 1/2 M/ KQ/mHQ

MQM +QMi`B#miBM; i2`Kb- e−ith∆_ψ(_−h2_∆_)u

0 Bb KB+`QHQ+HHv bmTTQ`i2/ BM TT+1−2τ0(D1)∩

{D_{× R}n_} 7Q` HH t ∈ [−τ

0,0]- M/ i?mb BM TT(D)∩ {D × Rn}7Q` HH t ∈ [−τ0,0]X

q2 /2}M2 χb ∈ C0∞ bm+? i?i χb(x) = 1 B7 x ∈ E1(η/2) M/ χb(x) = 0 B7 x /∈ E1(η)X

q2 2biBKi2 χχbu QM i?2 QM2 ?M/ rBi? i?2 T`2pBQmb bi`i2;v M/ i?2 i`MbHiBQM QM

[_−2t1,−t1]- M/ χ(1 − χb)u QM i?2 Qi?2` ?M/ rBi? i?2 i`MbHiBQM [−t2,0]X

UjV _2/m+iBQM iQ /i bmTTQ`i2/ BM DX q2 `2K`F i?i 7Q` HH #B+?`+i2`BbiB+

bi`iBM; 7`QK D3! D2 rBi? bT22/ BM [α0, β0]- r2 ?p2 7Q` t0 ≥ 0 bKHH 2MQm;?

d(γ(t), D₂c) > 0, _{∀t ∈ [0, t}0],

M/ r2 7QHHQr i?2 T`2pBQmb bi`i2;B2bX

dX9X TT`QtBKiBQM Q7 i?2 bQHmiBQM M2` i?2 i`TT2/ b2i

dX9XRX S?b2 7mM+iBQMb

Dans le document Dispersive effects and long-time asymptotics for wave equations in exterior domains (Page 93-135)

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dX9X TT`QtBKiBQM Q7 i?2 bQHmiBQM M2` i?2 i`TT2/ b2i

dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v

dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb

dX9X TT`QtBKiBQM Q7 i?2 bQHmiBQM M2` i?2 i`TT2/ b2i