9 ( ) 7:30 8:30 ( ) 3 : ( ) 9:00 9:5 ( ) Schrödinger 9:30 9:55 ( ) Penrose-Fife 0:00 0:5 ( ) ( ) : ( ) 0:40 :0 ( ) :0 3:00 ( ) 4 : ( ) 3:00 3:5 ( ) 3:

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1 ( ) ( ) TEL ( ) URL ( ) ( ) URL ( ) 3:30 4:00 ( ) : ( ) 4:0 4:35 ( ) 4:40 5:05 ( ) Instability of bound states of nonlinear Schrödinger equations 5:0 5:35 ( ) Navier-Stokes : ( ) 5:50 6:5 ( ) Porous Media 6:0 6:45 ( ) 6:50 7:5 ( ) Convergence rate to the nonlinear waves for viscous conservation laws on the half space 8:00 9:00 ( ) 9:00 :00 ( 0 ) : ( ) ( )

2 9 ( ) 7:30 8:30 ( ) 3 : ( ) 9:00 9:5 ( ) Schrödinger 9:30 9:55 ( ) Penrose-Fife 0:00 0:5 ( ) ( ) : ( ) 0:40 :0 ( ) :0 3:00 ( ) 4 : ( ) 3:00 3:5 ( ) 3:30 3:55 ( ) Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation 4:00 4:5 ( ) 5 : ( ) 4:40 5:05 ( ) 5:0 5:35 ( ) Wellposedness of the fifth order KdV equation 5:40 6:05 ( ) 6 ; ( ) 6:0 6:45 ( ) 6:50 7:5 ( ) 8:00 9:00 ( )

3 9 ( ) 7:30 8:30 ( ) 7 : ( ) 9:00 9:5 ( ) 9:30 9:55 ( ) Schrödinger-Poisson 0:00 0:5 ( ) ABP type estimates for nonlinear elliptic systems ( ) : ( ) 0:40 :0 ( ) :0 3:00 ( ) 3:00 3:30 ( ) 8 : ( ) 3:30 3:55 ( ) 4:00 4:5 ( ) On smoothing effect for higher order curvature flow equations 9 : ( ) 4:40 5:05 ( ) 5:0 5:35 ( ) 5:40 6:05 ( ) Decay property for second order hyperbolic systems of viscoelastic materials 0 : ( ) 6:0 6:45 ( ) Modulation Navier-Stokes 6:50 7:5 ( ) The two constants and tensors of the original Navier-Stokes equations 8:00 0:00 3

4 9 3 ( ) 7:30 8:30 ( ) : ( ) 9:00 9:5 ( ) 9:30 9:55 ( ) cone 0:00 0:5 ( ) Strichartz estimates for wave equations with a potential in an exterior domain 0:30 0:55 ( ) :00 :30 4

5 ' ", ' Schrödinger!#"$&% '( )#*+ Schrödinger,-. { i t u = (NLS) xu + λn (u), u(0, x) = u 0 (x) (t, x) R R? λn (u) 687 9;:<>= #4 I GKJ H4= i 4LD&M4N 6POJQ=SRTU)*+V <;[ + 6P\] I G^J_H4= /043 5 u(t, x) EFD4 4W&XAY& p ' 6&GH &Z )#*+V" ]a`b b m H&G λ R, N (u) = u p u ( < p) u "#cdefg 68hi8T I e iθ u ( Tjki θ R) 6l N (e iθ u) = e iθ N (u) (.) Gnò<qpras? H=4pQr (.) Qtu# #v G8Tw u(t, x) s (NLS) xa? H G8sy#z{HG Tjki e iθ u(t, x), θ }D#~ I R#T (NLS) x" [##b`{h G syz{h4= b pr (.) 6 ƒ & ˆ kg8š &=,-. (NLS) ŒŽ& [4 " ]&`b4 J_H4= )*+ Schrödinger,-. TR " p = 3 p = 5 ~ s ŒŽ & i š #œ4 žÿk d 68 J_H ª ~& H8±{`&68²³ "4 #µ i8b`h= «8 # 4& 4 s &¹ de 6Pº» b x u(t, x) 6P¼½ &¾ H Z < " ipt` n Q d#sà#3 GÂÁ m [skã À 68Ä m b#`kåâ=k # # dsæ#çqè Q { t # d"é QTËÊ NÌÎÍ 68Oi8b t 9 x + 6 µ J_HÏÊÑÐÒ ÍÔÓ_Õ ¹ d Ö 6PÖ i8b`&h= D &"t H t, x 4ØÙ G Ú jsaûü& [` Z < "t4ý ` i T`= 44&#ßÞà [479K:&` 6PâãiPb,-.á, t 64Ð4Ò Ó Õ ¹ d

6 + Ö x 6 NÌÓ_Õ ¹ d Ö GPŠ4 G " J_H4= R#T x u(t, x) " ipb u(t, x) iå ž J H & Á * 6PO ipb` H4= ipb8)*+v Õ # s 8")*Q+ #" J H )*#+ Kerr ~ 6UO i8b t 9 λ 6nÖ i8b`h [] = b (NLS) D { [ #D Å "Z Qb Ð %"!x{#$ pg )#$ p Ð %%"&x{' G )"' λn (u), Ð %%"&x{( ²) ž *,+-/.08x [/,34 x5} p G6 5{} p [/ 3 s78 "9: Ã,;b`H4= 4 Ð %<= G # &>? B bc4 J : b D"E&" ] `b F"G " J H G H ip`?ji <q= &4 3, " K Ð %!xl#$ pg )#$ pnm *OK Ð %%&#x' G )' M *PK x( ²) ž M "#C J_H À3a"QR 6AS_H= # %" yt_9vuwkgxnw T YZ " [ b`h4s UJW 4& Ð %%"&x&"c J H\[ B& ] &R 4x"^ i XJW,, #x( ²) ž"c J_HA[ B{] " ]{`#b x^ J_H. Ð %"_` s? ;a λ s EF#D& "x{( ²) ž s/<;[h z ^b J_HAc }? H=[ t)* de [fgginb [3, 8] 6A)hb t ÅÂ= Schrödinger,#-. i j k l m n o p i j q r m (NLS) Sx 6 YZ J Hs ", RLt x suj H CDQ${% 6wv } inb t zxaq[kã8[ `= yz /03 6 x Ås " `] I K Ð %%&x s ' J Hz{zM G8`< 3 } 6n\&b`H # Ð %!x 6P][` ~"~ G x 6 &¾ H Z < " <^RkÅ CD$a% 6A ƒinb t Å as? H= x 6A #JkH "Q, u(t, x) 8ÈNˆ s t L Š GG I " {"Œ [#` G8sŽ{Rki8`Q= { iut` I s š N #? H= t λ sœ D (NLS) x u(t, x) s t, x " ]`b Ãnz x G ¾4"ž"Ÿ ipb#`j " J " " «ª, ± ²³ A "µ/ ¹ Jº»¼ u(t) L (R) (½¾ ), (.) x u(t) L (R) + λ p + u(t) p+ L p+ (R) ( ÀÁÂà ) (.) Ä ¹ Ä ³ AÇÉÈN ÅÊ ³ÎÅÏÐÆÑÒÅÓ ½Å¾ ÅÀÁÆÂà Schrödinger ËÅÌÅÍ " J \ ÔÖÕ Ñ" J,ØJÙ ¼ Ú Û \ ± ²J³ \ÜJÝ" "Þ Öß à"á"ß" âãänå æº µjç/è H (R),, H (R) = {f(x) L (R); x f L < }

7 e Ñ ÈN»¼ / á u(t, ) H (R) Sobolev ß ç / Ê / ¼ ÀÁÂà u(t) L p+ A (NLS) æwº H (R) ß âlã w»¹ t Ñ! #" " J J»J J "Ù#%$ÖàÖ¼'& æ\º" )(#*#,+!-Û "æ\º ÜJÝJ \ µjà"á"ß"ê" (*/. ß'/43ßÉµÆ Æ Æ w '56 Å Ñ Ã0 u(t, x) C(I; H (R)) 7/ Ñ / #" "»¼ Ñ I Ê98 :#;<!= 9> à 0? Ñ ¼BA Aß (NLS) ß7Nç t u x u C D 溹 çbevü ÝN µæàáåßê w» Í F/GH Ñ IɵBJL N 'K ¼ Ê u(t, x) C (I; H (R)) H (R) H (R) LNM änå Ñ ¼ Ñ u(t, x) C(I; H (R))!O P9Q9R Ñ ßÞTSÜ æº9u P¼ λn (u) C(I; H (R)) C(I; H (R)) [ T, T ] Theorem. (VNW X9Y/Z [ Ginibre-Velo [7]). u 0 H (R) /¼J L / T > 0 ²#\ æ\º" (NLS)! u C([ T, T ]; H (R)) C ([ T, T ]; H (R)) 9] ^µ ²9\ _¼ [ Ginibre-Velo [7]). u 0 H (R) _¼ç"à λ > 0 /! < p < Theorem. (V#W `9aNZ λ < 0 /! < p < 5 V b_¼ L L ^µ ²N\ _¼ (NLS) /8 å c/d u C(R; H (R)) C (R; H (R)) /] Theorem. f gh (i j )k (NLS) l7 m/n ß ob p æwº q/r ËÌÍ s tj unv wj» _ ¼ Ñ %l J m#n \Ê Ë"ÌÍ (NLS) ß Duhamel u#v x w æaº!y _Þ Ñ u(t) = Φ(u) U(t)u 0 iλ t 0 U(t τ)n (u(τ)) dτ (.3) N Ñ _¼"à/z æa Ê U(t) = exp(it x ) PNQ N{/wN Schrödinger ËÌÍ è (.3) }9~ß7 º» _ 8 å m9n t dτ Ê ß / B_ âã ß H (R) M _ m#n Ñ _¼ 0 Bochner æú > å u 0 H (R) ρ 0 I = [0, T ] ß M æ º (.3) %s t Ö Φ L (I; H (R)) 9ƒ N B ρ0 = {u; u L (I;H (R)) ρ 0 } 3

8 e Ñ Í Î Ñ q#rs tjß _ J " ¼"à9z/æA p ß" _ O P9QNR N (u) u = 0 Ñ Þ ß _ Ñ 9ƒ 9 ßÊ» äå B ρ0 L (R; L (R)) _¼ U(t) L (R) Ñ Ñ _ J " 0\Ã, O P9Q#R Ñ Sobolev á, H (R) L (R) U Φ(u) L (I;H (R)) ρ 0 + CT (ρ 0 ) p ρ 0 T 9nNr A9Eæà Φ(u ) Φ(u ) L (I;L (R)) CT ρ p 0 u u L (I;L (R)) u u L (I;L (R)) T 9n9r A9Eæ\à (.4) " è"!äµ Ù'/_ ¼$#'$ĺ Φ Ê)q r/s)t ßL )$ ºÉ»_ļ / C(I; L (R)) L (I; H (R)) ß&% 4_ (.3) ) u ²'\ Ù'/_ ¼ Ê (.3) )}'~ Ê C(I; H (R)) ß% _ Ñ ')( w _¼ ^Ü *NßµN»ºÊ u C(I; H (R)) (.4) + Ew l-,. / N / 0ß _¼ NU 9> å ß9U J9_ N ²9\ I = [ T, 0] ^ Ü *ßµJ»ºÞ l3nß " -4 Ñ 9_¼65!,#8 : Ñ9 6:"ß; <Ûæ\ "»=4J Theorem. f!g,h 7 i j68 k u(t) H (R) J N»¼#& J 98 å ( Ñ (9*N. & µ Ã,0? è.a º»9 _!8 :ß9U»ºÞ J ²N\ _ 4 _¼ I = ( T, T ) U E ¼ u C(I; H (R)) C (I; H (R)) ß M æúº (.) (.) ± ² _64N B6CJß" à"á"ßê" äå ËD cut-off χ ν (x) = χ(νx) FEHG η ε (t), η ε (x) w» _ ¼ u ν,ε η ε (t) η ε (x) (χ ν u) Ê t, x ß µ» ºJI Ñ x / K6L _ 4 Aß S"ܼ9&/æAº ʪ âm Nà ż u ν,ε à9z/æa ß _¼ _ error ν,ε Ê i t u ν,ε = x u ν,ε + λn (u ν,ε ) + error ν,ε (.5) ν 0, ε 0 O)P ÑQ N _ L loc (I; L (R)) R)S Ñ 0 (.5) )T ~Nß UbJLº ߵƻº7mNn¹æw µn» ÑV)W N 7_¼ u ν,ε x d dt u ν,ε(t) L (R) = Im(error ν,ε, u ν,ε (t)) t = u ν,ε (t) L (R) = u ν,ε(0) L (R) + Im(error ν,ε, u ν,ε (τ)) dτ 0 _ ¼=4H4 " N ν 0, ε 0 L X ± ² Z4 Ñ 9_¼ Á Y À"ÁJÂ"à (.) ± ² ß µjº"ê" (.5) 6T!~Jß U J º" t u ν,ε x ß µj»"º!m#nûæ\ W _ 4 Ñ l3ß -4 A Ñ 9_ ¼ 4

9 à O U Í Awº λ > 0 L ± ² A ½¾ ÀÁNÂà u(t) H (R) C w _)4 \ "Ù#,_ ¼ # $"ºÖ 8 å (6># \µj º c#d #" _64J \ Ñ #_¼ ªJß" λ < 0 O P Q W n ß À Á  à Gagliardo-Nirenberg %C w» _ Ú x u L (R) C u α L (R) xu β L (R) C (.6) _ ¼=4 4 Ñ ÑÅ _L¼ Æ α = + (p + )/, β = (p + )/ < p < 5 " Ñ Ñ" β < _64J \ß!S"ÜJ¼ Jß" (.6) ß Young #C TwJ» N & -4 A Ñ Í º #8 å c9d N ²9\ &+ E4 Ñ u(t) H (R) < C _¼ 5 J Ñ µ Ñ" $"º Þ ß Ü. Theorem. λ < 0 5 p u 0 H (R) Ý Ñ r ANA _N 8 å c9d 3! ²9\ _¼ æ /æ c. ß M¹æ à 0 ºÊ N8 å ¹ ß ; < _ 4N A V Nº» _ u(t) H (R) [8,, 7] ¼ #-ßNá"º4 ' _ N»/ 49KÊ c EA _ P9Q HJ Ñ N!#"$ &% E èv 9& 4Lß " E(' E.» Kerr m) _ 4+* ß.-/0 B_ 4*ß _¼4 3 ;, 'NÊ" * 3 Np4Jß ã Ð 3 5 J 67à8*Þ9 N_ Ñ K/ ¼ ± ²N³ 6 wj»º98 å c#d 869" : _ 4*A " #$"º»_. C7& ># "ß ± ²J³ Ê (.), (.) > ÞT5#6 Ñ" K GF > 9 H *J IJ E D E ºJK± ²³ zj6w»ºn8 å c/d 3 L69"#: _ 4+*AÊ Ñ» Ñ K (.) MF µjç è(c 4ONJÊ >!#, #" : _!+!-Jß",_ ¼ 4 >#+!-JÊ C L (R) H>P vjß ABNº» _ Q/ ßR+P93ß S ATNº» _¼ Theorem.3 (Y. Tsutsumi [9]). < p < 5 C u 0 L (R) * : _¼4&>.* &C m9n ËÌ"Í t u(t) = U(t)u 0 iλ U(t τ)n (u(τ)) dτ (.7) 0 6 N à : u(t, x) C(R; L (R)) L r loc (R; /] ^µ ²N\#: _¼à/z ævc Ñ Lp+ (R)) /r = / /(p + ) è(c >!}9~Nß (.7) _ 8 å m9nnê H (R) ß 86* _ âã ß M : _ m9n Ñ _¼ Bochner Theorem.3 >WX > Yß _Þ >Ê C Strichartz > #C Í [5, 6, 3] U t U(t)u 0 L r (R;L q (R)) C u 0 L (R), 0 U(t τ)f (τ) dτ L r (R;L q (R)) C F L r (R;L q (R)) 5

10 ' Ç C ( ( ½ é ô u ÑÅ _ɼ4 4 Ñ C ÑÅ 0 /r j = / /q j < (j =, ) è(c Ñ" _ ¼ /q + /q = /r + /r = Strichartz > #C Ê C78 :J : _* P#Q Í Schrödinger >7+> 9#H * BEw _H4G* 6 : Þ&> Ñ C < < ÞLà : ËÌÍ x n < * >» -? E Eß,$º æç * 6ß : _¼ > 9 H *86!" âãäjå Ñ 86 #": _H4*\Ê# $ Ñ" K&% z&)c u C(I; H s (R)) (( >+Q*%,+-.6/ &%8.F9: s < 0), >; H AB :C DE -FG HI)KJL!MN < FO P&4QRS?C-T UV < W -XYZT [\ M]^_ <` Fa bc&4qrkfdfegpjh4 [4, 6]. 3 i j k l m n o*prqks6t?u Tv = o wx Fy zc&4e {PJ }?:*~KT E ƒf)kjh* /T ˆ Šr? : Œ?Ž +r = f J h 4 T C Er?š 4 -?T t œ ž&=kÿ E *e P*4K? =K h ªT E «I K+ ±?² E ³ c 4 œ ž E µ =! 7 ¹ 3 + u ± E³ c 4 œ ž E {?h J : C ºr» +¹¼ ½ TCQ Q: Æ +h (pseudo-conformal identity) Tr¾r8rE¹]À Á+¹ÂrÃ?F Ä Å E + 4 [6, 0, ] ÈÉ Schrödinger Ï = R Ñ!Ò R 3 EC ÊËÌ ÍÎ T - : ~!T + Ó 4KÔ?h = c t (NLS) u(t, x) 45?7& 8 -T L Õ!Ö F ØÙc 4 Q?R =KÚ Û c*4?r/c¹ü Ý} Þ T ß N (u) à F*~!á&~Ká f K+ Jh* KT C T-?: œ E Ý} (NLS) Schrödinger ½T-E œä QR/Få æi P*4 ªQ Cç èý} â ã (LS) { i t u = xu, u(0, x) = u 0 (x) Schrödinger â ã T - F TêR3 Eê~/Tê g+ëó&4/ôh = c&4 T68 ì6j Q& Ý } U(t)u 0 t ½:í îï ð?ñå+ òó =!ô Ò*Ñ!Jh4 F C?QT Š : o&p q Shrödinger s tu Tõ6 E ö J âã =K J6h4T6CQ Q :* Ñ,è Ü Ý } Kerr TÁøh oêpq/s tùu = y Æ 4 o TÁ] î = ì6j6h4ùrgú J3ëÒ6ûù3ëÒ6h6 & J6C 6ü = Yh4R T - :ëý Fourier Fϕ(ξ) (π) / e iξx ϕ(x) dx, (LS) u(t, x) T* ge þÿ34 u(t, x) = F exp(itξ )Fu 0 = (4πit) / exp(i x y /4t)u 0 (x) dx 6

11 ) Ç ½ QTþ ÿ?8*ekcýtqr!f Æ 84 ÈÉ Schrödinger ÊËÌÍ ÎÍ L q -L q q, /q + /q = Rc*4 QT*R3C ýt ¼½F 0 {. U(t)u 0 L q (R) Ct (/ /q) u 0 L q (R) (3.) Dollard Î Mf(x) = exp(ix /4t)f(x), Df(x) = (it) / f(x/t) R,c 4 Q T*R 3CýT¼ ½F0 { t T*R 3 U(t)u 0 = MDFMu 0 (3.) M ENc*4R/C?QT¼½8&e Òû*Ñ!C U(t)u 0 = MDFu 0 + o(t / ) in L (R) as t. u 0 L (R) Rgc*4 È?É Schrödinger ÊË?Ì ÍÎÍ small data Í ýe E6 h J, t (NLS) T Fê~/T& geëó Ô gt 8 = & u(t, x) ÜÝ} ÞT F *3h¾*~/C :ØÙ ÑScI 4TC?T œ ž*= p N (u)!" S cf?4 Q?R!: #?E%$&fḡP 4 Áç Ò C¹Ü Ý } ½?T%'(?C Schrödinger ½ â ã º * T + Ÿ V%, - F 0 Ò htc œ ž T.:%/ 0 âã âã S Ö 3 u 4 5 ½E4%6 798:;8< = Y?h4QR!E 4 %= >: C œ ž T.?F#fÑKh%'( 8 e & =<? J C ªT {f~ÿ (super critical case) A = B c*4* CDP? E, EñÅ %' ( =K I g (p=3 : critical case) Ü Ý } ÞT%I A. FGIH JKLMKNOP QR 4 < p : super critical case F S 0 3h*R3C T T œ ž : = > T*RK (NLS) Theorem < p Rgcê4 ç6òc u 0 H (R) L (R) : C u 0 H (R) L (R) < ρ 0 =T Òc*LT&Rc*4 Q T*R3C (NLS) T Ï B U u E³IÑ/JýTQR F V { () t = ( + t ) / R ÑKJC u(t) L (R) C t /. () 54 ϕ H (R) FGHIÑKJ C u(t) = U(t)ϕ + o() in L (R) as t. p 7

12 J E X ½ ½ Theorem 3. Í ( ) E ³ Ñ JëC () C 0 > 0 T R h J C E46QR = sup{t ; sup 0 t<t 3C 0 ρ 0 } T = = c E ³ t [0, T ) Ñ J6C /0 TE â ã (.3) L Õ!Ö = R C T 4 5 = Y crkc U(t) L L t / u(t) L (R) < u(t) L (R) C 0 t / ρ 0 + C C 0 t / ρ 0 + C t 0 t R< f!h u ± T%' (?C Ö 3 u G e ep T C = S 0 I IR!C ρ 0 u(t) L (R) C 0 ρ 0 t / + Cρ p 0 0 t τ / N (u(τ)) L (R) H (R) dτ t τ / u(τ) p L (R) u(τ) H (R) dτ (3.3) t 0 (3.3) T?E u(τ) H (R) t τ / τ (p )/ dτ C 0 ρ 0 t / (3.4) R! Ò û*ñkc (3.4) T&R" C t 4 < p # Jh?QQC?LÑ RcR!C¼ ½ 0 TE =$ š T / < (3.4) t t T T%& = R'QR6C RKhI )(+*F,.-KJ*Ñ çi / 3C 0 ρ 0 C 0 ρ 0 T =. t τ / τ (p )/ dτ C t / = F T*R"E = () U( t)u(t) t H (R) 5!6 h / 0 âã *e (.3) t U( t)u(t) = u 0 iλ U(t τ)n (u(τ)) dτ 0 7 Q?R C?8 e E% N () N (u(τ)) H (R) C τ (p )/ ((p )/ > )! Q?RKE9 = {&š?rkc TGH lim t U( t)u(t) = c*q?rkf!"+. : E, ð T, p = 3 ' (F <; 'c4t CL R T>= = >? JT 6œ p ž*=! "KÄ C TA!B ÜÝ}ÞT 4 5 FC?hT Theorem 3.! ýt &! C?QTÒ =DE c B. Î ÍGFIHKJGL NGM PONQP N O PGROS 3 < p : super critical caseütwv (NLS) u(t) Z"[Z\ Ý } ½ X ] œ ä ^\_[f a`bcd XOe t Y Schrödinger ârã \_ &Zf_ghi[jlkmZn V u(t) U(t)ϕ e7op U( t)u(t) ϕ \_qr\ 8

13 @ Z, V P & V k k o S i, P f o i[!. r> u(t) = U(t)(U( t)u(t)) U(t)v(t) (3.5) \ " i f o i j!\ V Theorem 3. X!n e X f e "! o #%$ r!\ j Y &' \()jm+* Theorem < p V u 0 H (R), xu 0 L (R) \ o f.- */ V ex \ 0j7r X \ u 0 H (R) + xu 0 L (R) < ρ 0 ' V (NLS) 4 5 X% /3 k8 9 o i u C(R; H (R)) C (R; H (R)) Y 67 V xu C(R; L (R)) f -* o V \.:; X< = n?k!j Theorem 3. Y> Theorem 3. qo ACBEDGFIHEJLKNM WVYX r[z>f \][^W_j*` & P v(t) L (R) QRS &UT P L acb[de Fourier f Zhg%i _%O F Q/j & O, v(t) L (R) C( v(t) L (R) + xv(t) hk L (R)) ml%n O L a.bod ts VvX0Qr ZGf.\% &/prq, xv(t) L (R) Q%RuS &ut rr &, w%x y f z%{ J = U(t)xU( t) = x + it x = M(it x )M. u(t) L (R) = U(t)v(t) L (R) Ct / v(t) L (R). Q > _j xv L (R) = Ju L (R), JN (u) = p + u p Ju p u p 3 u Ju n?>krzck } j ~Z e ƒ k W Oˆ f Š (.3) O% x X[^ JΦ(u) L (R) Zqr[Z k %}!j ρ 0 + C ρ 0 + C P Ju(t) L (R) t 0 t 0 u(τ) p L (R) Ju(τ) L (R) dτ τ (p )/ ( u(τ) H (R) + Ju(τ) L (R)) p dτ (3.6) OZ 3 < p O ' %3 4 q τ (p )/ Q Q/ Œ/ q Ž k o i _+r VYXj C. W šwa œ žwÿ[ U ŸU W F p = 3 : critical ku + E casek q P p(ë O + 5 OLªE«+ Ē±² jrlo + Ou³ 0Z.:0µh p = 3 &+P = Theorem 3. e/ q%_j qu q0v P O/¹ (3.6) % º k%»0vhx O/ _*+h k τ (p )/ = τ Q¼ P k τ O kqvq _ Q½¾ 9

14 k Q Z Q 0 & q, QV, q Q o k ¼ & o R Q ½ S P &, V P & P S j, q P V rz ~ >j < = k ± Z P u(t) e % Schrödinger ƒ O 5 k«v X!i _>j!r.x Q Q P p = 3 Z ~j S q%_ ZY >jm* Theorem 3.3 (Hayashi-Naumkin Z [0]). p = 3 P u 0 H (R).-, *u/o0zh0kj r O xu 0 L (R) P u 0 H (R) + xu 0 L (R) < ρ 0 'P O %3 45 k 8%9 o i (NLS) -* u C(R; H (R)) C (R; H (R)) O!rZ Q 6/7 n?k!j xu C(R; L (R)) P Q > () u(t) L (R) C t /. () ϕ L (R), Fϕ L (R) -* º ϕ Q 8 9 o i u(t) = U(t)F exp ( i(λ/) Fϕ log t ) Fϕ + o() in L (R) as t. O Theorem 3.3 X Theorem 3.3 AuBoD LU V U +O -* º5 ƒ z /» Z 5 O/ª «% k () {+i _ j!r X P e F exp ( i(λ/) Fϕ log t) F & O! ~[j 5 OuªL«#"7ho%$+?Qi OL'&)( o j+* F HmJLKNM e P _-, /. ²Z r3 kl~e-4 '0 (NLS), 'P Dollard 5 _ Z U(t) = MDFM x P v(t) = U( t)u(t) Zq 4rr i t v = λu( t)n (U(t)v) = λmf D MN (MDFMv) = λ t MF N (FMv) & P OZ t ' k }Z M P i t Fv = λ t N (Fv) + R(t) (3.7) Z 674!rr & P R(t) = λ t (FMF N (FMv) N (Fv)) e8 p r % X0:9;% ~ 74 O7<=k%~ O > (3.7) λ t N O r X0 :C =k D Q o:ef Z & P t Zq 4Orr ~ 4 &%P Φ = τ Fv(τ) dτ & +r[z O &' &P e iλφ Fv e OZ t r X ' P u(t) Oª «" rz 74 &' t e iλφ Fv = ie iλφ R(t) (3.8) (3.8) O< = e7g 8 p % kh o V O º k IKJ 4 0

15 , ' x & q q Q q P S & & 3 Q ', * V 0 Q Q Q P & l & rr>m O = & e 9 ; P R(t) Q 8 p 0rZ E _ * R(t) L (R) Ct 5/4 ( u(t) L (R) + Ju(t) L (R)) 3 Z _ ƒ z 6E O < O EU3 4[ q r OU³ &P Ju(t) L (R) -WZ' ^^q V q _ ZGr)ukQqm'4Eµ q e P Ju(t) e l L (R) O k µ * l"! l 3 r Z ]74 &# Q $%'& (*) 3. +*, & e O 5 O. E λ Q-%S Q P λ = λ + iλ (λ, λ e -S O0Yk k+q) E 5 Oª%«% %± V X E _ 40 ) λ Q/ y S λ (< 0) e 6 l"7 k8%9 o:e _[ "9 k ;:"< P3'45 b*= OA"BCO Ohm Ž _ ED E _4 k Z Shimomura [5] P p = 3, λ < 0 OZ k/ q l k3f D E * ` D u(t) L (R) C t / (log( + t)) / ( P t 0) (3.9) Z _ \%r[z ³ X E _4r X e P λ kg[o0, {Z % *!l kwoh I X K[ZU } LU4 M* Q QJ P kw Kita-Shimomura [7] [Z, p = 3, λ < 0, λ ON O P][^ l krf D E 3 λ 5 O P 'Q L a b d Q Z :²µ; l!l E KWZ E _W34*KTK (3.9) Q S, &P λ Z._rVU"WNO 3 λ e O3 ".\% %k d dt Ju(t) L (R) 0 (Liskevich-Pelermuter ƒ _0VhX [9]) x P / y Ginzburg-Lndau % %ƒ O %/3 4 5 O 5"X *Y[Z X E _ \ ] ^ _ [3] 4 [] G. P. Agrawal, Nonlinear fiber optics, Academic Press, Inc. (995). [] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 0, American Mathematical Society (003). [3] T. Cazenave and A. Haraux, An introduction to Semilinear Evolution Equations, Oxford Science Publications (998). [4] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 5 (003), [5] J. Ginibre, An introduction to nonlinear Schrödinger equations, in Nonlinear Waves, (R. Agemi, Y. Giga and T. Ozawa, Eds.), GAKUTO International Series, Mathematical Sciences and Applications 0 (997),

16 [6] J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations. II. Scattering theory, general case, J. Funct. Anal. 3 (979), [7] J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 3 (979), 3. [8] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 8 (977), [9] N. Hayashi, E. Kaikina and P. Naumkin, Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation, Discrete and Continuous Dynamical Systems 5 (999), [0] N. Hayashi and P.I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math. 0 (998), [] N. Hayashi and P.I. Naumkin, Modified wave operator for Schrödinger type equations with subcritical dissipative nonlinearities, Inverse Problems and Imaging (007), [] N. Hayashi and M. Tsutsumi, L (R n )-decay of classical solutions for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A 04 (986), [3] T. Kato, Nonlinear Schrödinger equations, in Schrödinger Operators, (H. Holden and A. Jensen Eds.), Lecture Notes in Phys. 345, Springer-Verlag (989), [4] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46(986), 3 9. [5] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 0 (998), [6] C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 06(00), [7] N. Kita and A. Shimomura, Large time behavior of solutions to Schrodinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan 6 (009),

17 [8] N. Kita and A. Shimomura, Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differential Equations 4 (007), 9 0. [9] V.A. Liskevich and M.A. Perelmuter, Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc. 3 (995), [0] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrodinger equations in spatial dimensions and, J. Funct. Anal. 69 (999), 0 5. [] K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrodinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (00), [] H. Nawa and M. Tsutsumi, On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation, Funkcial. Ekvac. 3 (989), [3] N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with p-laplacian, J. Differential Equations 8 (00), [4] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 39 (99), [5] A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. P. D. E. 3 (006), [6] R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of wave equation, Duke Math. J., 44(977), [7] M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrodinger equations, SIAM J. Math. Anal. 5 (984), [8], ƒ =, (004). [9] Y. Tsutumi, L solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30(987), 5 5. [30] Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal. TMA, (987), [3] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 0(987),

18 ( ) Hale-Raugel [], Raugel []. ε 0 ( ), g C ([0, ]), 0 < ε ε 0, Q ε R, Q ε := {(x, y) R ; 0 < y < εg(x), 0 < x < }. : u + f(u, λ) = 0 in Q ε, ( P) ε u = 0 ν ε on Q ε. λ, ν ɛ Q ε., f : R R C 3, 3, f(0, λ) 0. Q := (0, ) (0, ) (x, y) (x, εg(x)y) Q ε ( P) ε,., (P) ε L ε := g divb ε, B ε := L ε u + f(u, λ) = 0 in Q, u := B ε u ν = 0 ν Bε on Q g x g xy y g x y x + { } g ε + (g xy) y, ν Q. ε = 0 (P) ε : (P) 0 g(x) (g(x)v x) x + f(v, λ) = 0 in (0, ), v x (0) = v x () = 0., [],. (P) ε (P) 0., (P) 0 ( 0 ) (P) ε, (P) 0, ( 0 ), (P) ε.

19 λ, (P) ε, (P) 0 u v ( λ. u := u H (Q) + ) u ε y H (Q). ( )... I R, {(v(λ), λ) H (0, ) R; λ I} (P) 0., θ, C ε (0 < ε ε 0 ), 0 < ε ε, (P) ε {(u ε (λ), λ) H (Q) R; λ I}, max λ I u ε(λ) v(λ) C ε. {(u, λ) H (Q) R; u v(λ) θ, λ I} (P) ε.... (v 0, λ 0 ) H (0, ) R (P) 0 ( (P) 0 {(v(s), λ(s)) H (0, ) R; s [ s 0, s 0 ]} (v(0), λ(0)) = (v 0, λ 0 ) dv dλ (0) 0, ds ds (0) = 0, d λ ds (0) 0 )., θ, C ε (0 < ε ε 0 ), 0 < ε ε, (P) ε {(u ε (s), µ ε (s)) H (Q) R; s [ s 0, s 0 ]}, { } d j max s 0 s s 0 ds (u ε(s) v(s)) j + d j ds (µ ε(s) λ(s)) j C ε j=0,,. s 0 < τ ε < s 0, (u ε (τ ε ), µ ε (τ ε )) (P) ε. {(u, λ) H (Q) R; u v 0 + λ λ 0 θ } (P) ε...3. (0, λ 0 ) (P) 0 ( (P) 0 {(tv(t), λ(t)) H (0, ) R; t [ t 0, t 0 ]} λ(0) = λ 0 t [ t 0, t 0 ] v(t) 0 )., θ 3, C 3 ε 3 (0 < ε 3 ε 0 ), 0 < ε ε 3, (P) ε {(tu ε (t), µ ε (t)) H (Q) R; t [ t 0, t 0 ]}, max t 0 t t 0 ( uε (t) v(t) + µ ε (t) λ(t) ) C 3 ε. (0, µ ε (0)) (P) ε. {(u, λ) H (Q)\{0} R; u + λ λ 0 θ 3 } (P) ε. L (Q) [] J. K. Hale, G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl., 7 (99), [] G.Raugel, Dynamics of partial differential equations on thin domains, Lecture Notes in Math., 609 (994), 08-35, Springer-Verlag.

20 Instability of bound states of nonlinear Schrödinger equations ( ) Schrödinger iu t = u + f(x, u )u, (t, x) R +N (). f(x, s) C (R N R + ; R) u H F (x, u ) dx < R N., F (x, s) = τ f(x, τ) dτ. 0 Schrödinger () e iωt φ ω (x). e iωt φ ω (x) () φ ω. φ ω + ωφ ω + f(x, φ ω )φ ω = e iωt φ ω, ε > 0, δ > 0 inf s R u 0 e is φ ω H < δ u 0 H sup inf u(t) t>0 s R eis φ ω H < ε.., u u 0 (). e iωt φ ω. Schrödinger (). E(u) := u dx + F (x, u ) dx, R N R N Q(u) := u dx. R N.. Schrödinger (), E, Q., ω, ω ω (ω, ω ) e iωt φ ω, ω φ ω C. S ω S ω (u) = E(u) + ωq(u)..3. S ω(φ ω ) 0. KerS ω(φ ω ) = span(iφ ω ), S ω(φ ω ). n(s ω(φ ω )) S ω(φ ω ). d(ω) = S ω (φ ω ). S ω(φ ω )..4 ([]). n(s ω(φ ω )) =., d (ω) > 0 e iωt φ ω, d (ω) < 0 e iωt φ ω..

21 .5 ([3]). n(s ω(φ ω )) d (ω) > 0, n(s ω(φ ω )) d (ω) < 0. e iωt φ ω., g φω (x, ρ) H C ρ +α H, α > 0. () g φω (x, ρ) = f(x, φ ω + ρ )(φ ω + ρ) f(x, φ ω )φ ω f(x, φ ω )ρ i s f(x, φ ω )Re(φ ω ρ)φ ω. e iωt φ ω..6. f(x, s) = s α, α > 0, (). 3 H Colin-Colin-Ohta []...7. n(s ω(φ ω )) =. S ω(φ ω ) φ ω., d (ω) > 0 e iωt φ ω..8. n(s ω(φ ω )) = :, d (ω) > 0.5.,.7 () i t u = u u α u γu u 3, i t u = u u α u γu u 3, i t u 3 = u 3 u 3 α u 3 γu u., N 5, α (0, /N)., Φ ω = (0, 0, e iωt φ)., φ 0 = φ + ωφ φ α < γ (ω) < γ (ω) γ(ω) < γ < γ (ω) n(s ω(φ ω )) =... d (ω) > 0.7,., Φ ω. 3 [] γ (ω) < γ. [] Mathieu Colin, Thierry Colin, and Masahito Ohta, Instability of solitary waves for a system of nonlinear Schrödinger equations with three-wave interaction. preprint. [] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal., 74 (987), no., [3] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal., 94 (990), no.,

22 Navier-Stokes ( ) Ω R 3, Ω Ω =. Γ 0,..., Γ L C -surfaces Γ,... Γ L Γ 0,. Navier-Stokes. u u + u u + p = f, in Ω (0, T ), t div u = 0, in Ω (0, T ), (N-S) u Ω = β, on Ω (0, T ), u(0) = a, in Ω, u = u(x, t), p = p(x, t), f = f(x, t), β = β(x) a = a(x),. β, β general flux condition :. V har (Ω) L j=0 L j=0 Γ j Γ j β νds = 0, V har (Ω) := {h C (Ω) ; div h = 0, rot h = 0 in Ω, h ν = 0 on Ω} (G.F.). [3] dim V har (Ω) = L V har (Ω) = span{ψ,..., ψ L } {ψ j } L j= Ω., V har (Ω) = span{ q,..., q L }., { q j } L j= Ω q j Γ0 = 0 q j Γi = δ ij., [3],.., (N-S)... a L σ, f L loc ([0, ); L ). β H / ( Ω) (G.F.), L ( ) β νds ψ k <, () k= Γ k 4C s 3

23 . C s = 3 / /3 π /3 H0(Ω) L 6 (Ω) Sobolev. u L loc ([0, ); L (Ω)) L loc ([0, ); H (Ω)) (0, T ) (N-S). (N-S) reproduct property... f β.. 0 < T <, a L σ (0, T ) T < T (N-S) u u(t ) = u(0) = a L.., β b, Ω div b = 0 b Ω = β,, Galerkin.,, f β, (N-S) u, L p -L q..3. < l <, < p < 3/, δ = δ(l, p) > 0. β W /p, p ( Ω). f BC(R; L l ) T,, t R f(t) = f(t + T ). f L 3 R Hölder. β f β W /p,p ( Ω) + sup f(s) l < δ, p = 3p s R 3 p,, (N-S) u BC(R; L 3 σ)., t R u(t) = u(t + T )..4. (i), f, Navier-Stokes reproduct property.,, β. (ii) 3 f β. [] H.Kozono, M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (), 48 (996), 33 50, [] H. Kozono, M. Yamazaki, On a larger class of stable solutions to the Navier-Stokes equations in exterior domains, Math. Z., 8, (998), [3] H. Kozono, T. Yanagisawa, Leray s problem for the stationary Navier-Stokes equations and the harmonic vector fields I, Math. Z., 6 (009), 7 39 [4] T. Miyakawa, Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., (98), [5] M. Yamazaki, The Navier-Stokes equations in the weak-l n space with time-dependent external force, Math. Ann., 37 (000),

24 POROUS MEDIUM ( D3) () : { t u u α = divf(t, x, u) in (0, ) R n, u(0, x) = u 0 (x) 0 in R n., α >, f = f(t, x, z) : (0, ) R R R n. (), αu α u = 0 0,.,,., f 0, Barenblatt U () ( ) : U (t, x) = t γ (C kt γ n x ) α +, γ = n γ(α ), k =, n(α ) + αn, C > 0, a + := max{a, 0}. U {x R n ; C kt γ n x = 0} Barenblatt U. α =, ()., Hölder, Hölder. f 0 Hölder, Caffarelli-Friedman []., ε > 0, () t u ε u α ε = 0, u ε (0, x) = u 0 (x) + ε, Aronson-Benilan u α C(n, α) C(n, α) ε, t u ε u ε t t, u ε ε Hölder, u Hölder., Aronson-Benilan, (, f u )., DiBenedetto-Friedman [], p >, p-laplace () t v div( v p v) = 0 Hölder.,,., De Giorgi,., () v, v f 0 ()., f 0 () DiBenedetto-Friedman, Hölder,., DiBenedetto- Friedman, [], f 0 () Hölder

25 , Hölder, Hölder.,, f, () Hölder, u Hölder. Theorem. Let u be a bounded non-negative weak solution of (). Let us assume for some p > n and let p f(t, x, u) L (0, ; L p (R n )) =. Then u is uniformly Hölder continuous and n p u α (t, x) u α (s, y) C u α p n σ L ((0, ) R n ) f p n σ L (0, ; L p (R n )) ( u (α ) L ((0, ) R n ) t s σ + x y σ ) for (t, x), (s, y) (0, ) R n, where the constant 0 < σ < and C > 0 are depending only on n, α and p., u u = 0., u () αu α,., u, u., (), ũ(s, x) = M u(t, x) t = M αβ s, β = α, ) Q ρ,m = Q ρ,m (t 0, x 0 ) = (t 0, x 0 ) + ( ρ M, 0 B β ρ, ρ, M > 0. M > 0 u,,. Lemma. Let ρ 0 > 0 enough small depending only on n, α, p, u L ((0, ) R n ) and f. Then there exist 0 < c < and 0 < η < depending only on n, α, p such that sup u α M j, Q ρj,m j osc u α = sup u α Q ρj,m j Q ρj,m j inf Q ρj,m j u α η j u α L ((0, ) R n ) and ρ j = c j ρ 0 for all j N, where {M j } j N is some monotone decreasing sequence depending on n, α, p and u., De Giorgi, u M j., ρ 0, f L ((0, ) ; L p (R n )) u L ((0, ) R n )., σ log c = log η, Hölder Hölder (cf. Ladyženskaja-Solonnikov-Ural ceva [3, pp. 96, Lemma 5.8]). REFERENCES [] Caffarelli, L. A. and Friedman, A., Indiana Univ. Math. J., 9(980), [] DiBenedetto, E. and Friedman, A., J. Reine Angew. Math., 357(985),. [3] Ladyženskaja, O. A. and Solonnikov, V. A. and Ural ceva, N. N., Linear and quasilinear equations of parabolic type, American Mathematical Society, 967. [4] Porzio, M. M. and Vespri, V., J. Differential Equations, 03(993),

26 ( ) κ θ + h(θ, u) = f in Ω Find (θ, u) H0(Ω) ( H (Ω) ) θ = 0 on Ω u s.t. K(θ) a.e. in Ω ν u (u z)dx + (g(u) l) (u z) 0 Ω Ω for z (H (Ω)), z K(θ) a.e. in Ω [3], [4] [5] X Banach, X X, X, X A X X ( ), X, <, > X X duality pairing.. Ã : X X X (SM) (SM) (SM) Ã(v, ) ;maximal monotone, D(Ã(v, )) = X for v X (SM) {v n } X, v n v weakly in X = u Ã(v, u), u n Ã(v n, u) s.t. u n u in X, Ã semi-monotone semi-monotone.. (Y. M. [4]) Ã : D(Ã) = X X X ;,semi-monotone, Au := Ã(u, u) for u X, g X, K 0 X;, φ:x X R { }, φ(v, ); for v X D(φ(v, )) K 0 for v K 0,, (K) (K) {v n } K 0, v n v weakly in X = φ(v n, ) φ(v, ) on X in the sense of Mosco., u { φ(u, u) <, u Au; < u g, u v > +φ(u, u) φ(u, v) for v X

27 3 κ θ + h(θ, u) = f in Ω Find (θ, u) H0(Ω) ( H (Ω) ) θ = 0 on Ω s.t. u + g(u) + I K(θ)λ (u) = l u n = l on Ω I K(θ)λ ( ) I K(θ) ( ) semi-monotone A(, ) : X X X := H (Ω) (H (Ω) ) [A((ζ, v), (θ, u)), (ξ, z)] := κ θ ξ + h(θ, v)ξ + ν u z + g(v) z [, ] X X duality pairing Ω [5] () f n f () (3) Ω Ω Ω [] K. H. Hoffmann, M. Kubo, N. Yamazaki, Optimal control problems for elliptic-parabolic variational inequalities with time-dependent constraints; Numer. Funct. Anal. Optim. 7 (006), no. 3-4, [] N. Kenmochi, Monotoniocity and Compactness methods for Nonlinear Variational Inequalities: HANDBOOK OF DIFFERENTIAL EQUATIONS, Stationary Partial Differential Equations;, volume 4, Elsevier B.V., (008), pp [3] R. Kano, N. Kenmochi, Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces; Recent Advance in Nonlinear Analysis: Proceedings of the International Conference on Nonlinear Analysis, (008), pp [4] Y. Murase, Abstract Quasi-Variational Inequality of Elliptic type and Applications; Banach Center Publications Vol.86 Nonlocal and abstract parabolic equations and their applications (009) pp [5] R. Kano, N. Kenmochi, Y. Murase, Elliptic quasi-variational inequalities and applications; Proceedings of the 7th AIMS conference (to be published)

28 Convergence rate to the nonlinear waves for viscous conservation laws on the half space Yoshihiro Ueda In this talk, we study the convergence rate of solutions to the initial-boundary value problem for scalar viscous conservation laws on the half line: u t + f(u) x = u xx, x > 0, t > 0, u(0, t) = u, lim u(x, t) = u +, t > 0, x u(x, 0) = u 0 (x), x > 0. Here the flux f = f(u) is a given smooth function of u satisfying f(0) = f (0) = 0 and u ± are given constants. In this problem, we assume that the initial function u 0 (x) satisfies u 0 (0) = u and lim x u 0 (x) = u + as the compatibility conditions. Throughout this talk, we impose the following condition on the flux f(u): Either or () f (u) > 0 for u R, () f (0) > 0, f(u) > 0 for u [u, 0). (3) It is known that the asymptotic behavior of solutions to () is closely related to the solution of the Riemann problem for the corresponding hyperbolic equation: u t + f(u) x = 0, x R, t >, { u, x < 0, u(x, ) = u +, x > 0. In the case where the flux f(u) in () satisfies the convexity condition () and the Riemann problem (4) has the rarefaction wave solution, Liu-Matsumura-Nishihara [] showed that the large-time behavior of the solutions depends on the signs of the characteristic speeds f (u ± ). More precisely, it is shown that the asymptotic behavior of the solutions in this case is classified into the following three cases: (a) f (u ) < f (u + ) 0, (b) 0 f (u ) < f (u + ) and (c) f (u ) < 0 < f (u + ). In the case (a) where u < u + 0, the solutions of () converge to the stationary solution. Here the stationary solution φ(x) is defined by the solution of the stationary problem corresponding to (): f(φ) = φ x, φ(0) = u, lim x φ(x) = u +. (5) (4)

29 In the case (b) where 0 u < u +, the asymptotic state of the solutions is described by the rarefaction wave. Here the rarefaction wave ψ R (x, t) is given as the solution of the Riemann problem (4) and is given explicitly for t > by ψ R (x, t) = u, x f (u )(t + ), (f ) ( x t+), f (u )(t + ) x f (u + )(t + ), u +, f (u + )(t + ) x. In the final case (c) where u < 0 < u +, the asymptotic state of the solutions is given by the superposition of the stationary solution φ(x) satisfying (5) with u + = 0 and the rarefation wave ψ R (x, t) given by (6) with u = 0. Our first theorem gives the convergence rate in this last case (c) under the convexity assumption (), and is stated as follows. Theorem. Suppose that () and u < 0 < u + hold. Assume that u 0 u + H L. Then the initial-boundary value problem () has a unique global solution u(x, t) with u u + C([0, ); H ). Moreover, the solution satisfies (u φ ψ R )(t) L p C( + t) γ log ( + t), (u φ ψ R )(t) L C ɛ ( + t) +ɛ (7) for each p with p < and any ɛ > 0, where γ = (/)( /p), and C and C ɛ are positive constants; C ɛ is depending on ɛ. Here φ(x) is the stationary solution satisfying (5) with u + = 0 and ψ R (x, t) is the rarefaction wave given by (6) with u = 0. On the other hand, the case (c) has also been studied very recently by Hashimoto- Matsumura [] when the flux f(u) satisfies the weaker convexity condition (3). They proved the asymptotic stability of the superposition of the stationary solution and the rarefaction wave under the smallness condition both on u + and the initial perturbation. The second purpose of this talk is to get the convergence rate under the weaker convexity condition (3). Namely, we show the following theorem. Theorem. Assume (3), u < 0 < u + and u 0 u + H L. Then there is a positive constant ε 0 such that, if u + ε 0 and u 0 φ ψ R (, 0) H ε 0, then the initial-boundary value problem () has a unique global solution u(x, t) with u u + C([0, ); H ). Moreover, the solution satisfies the quantitative estimates in (7). References [] I. Hashimoto and A. Matsumura, Large time behavior of solutions to an initial boundary value problem on the half line for scalar viscous conservation law, Methods Appl. Anal., 4 (007), [] T.-P. Liu, A. Matsumura and K. Nishihara, Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal., 9 (998), (6)

30 Schrödinger ( ) Schrödinger : u + V (x)u = µ u 3 + βu u in R N, u + V (x)u = βu u + µ u 3 in R N, u (x), u (x) 0 as x., µ, µ, β, N =,, 3., u = (u, u ) (E), u (E) u (x), u (x) > 0. (E) [, 4, 6, 7]., V (x), V (x), : 0 < β < β, β [0, β ) (β, ) (E)., V j (x) x, V j (x) (E)., (E), β > 0, (E), V j (x) (V) V j (x) = V j ( x ) = V j (r), V j C ([0, )). (V) 0 < inf V j (r) sup V j (r) <. (V3) V j (r) 0 for all r 0. (V4) N = 3 : {V (r)r + 33 V (r)r + 83 } V (r) > 0. inf r>0 (E).. (i) V j (x) const. > 0, (V) (V4) (ii) Busca Sirakov[], (V) (V3) β > 0 (E) : u(x) = u( x ) = (u ( x ), u ( x )). (V) (V4) :.. V j (x) (V) (V4)., β 0 > 0, β [0, β 0 ) (E)..... V (x), V (x)., β [0, β 0 ) (E).

31 .3. (i) Wei Yao[8].. (ii) (E) β > 0. [6] V (x) = V (x) =, µ = µ = β =, ω ω + ω = ω 3 in R N, θ (0, π/) u = (ω cos θ, ω sin θ) (E) : Step (E) Step x Step 3 β = 0 (E) Step 4 Step, β [0, β 0 ], (E) L Brow-up argument Gidas[3]. Step Step β [0, β 0 ],., Step. Step 3, β = 0 { uj + V j (x)u j = µ j u 3 j in R N, u j > 0, u j H (R N ). Kabeya Tanaka[5]. Step 4 Φ(β, u) = I β(u), I β (u) = u + u + V (x)u + V (x)u dx µ u 4 + βu R 4 u + µ u 4 dx N R N.. [] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (), 75 (007), [] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space. J. Differential Equations, 63 (000), [3] B. Gidas, Symmetry and isolated singularities of positive solutions of nonlinear elliptic equations. Nonlinear Partial Differential Equations in Engineering and Applied Science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 979), Lecture Notes in Pure and Appl. Math., 54 (980), Dekker, New York, [4] N. Ikoma, Existence of standing waves for coupled nonlinear Schrödinger equations. to appear in Tokyo J. Math.. [5] Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in R N and Séré s non-degeneracy condition. Comm. Partial Differential Equations, 4 (999), [6] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in R n. Comm. Math. Phys., 7 (007), 99. [7] G.-M. Wei, Existence and concentration of ground states of coupled nonlinear Schrödinger equations. J. Math. Anal. Appl., 33 (007), [8] J. Wei and Yao, Note on uniqueness of positive solutions for some coupled nonlinear Schrödinger equations. to appear in Methods Anal. Appl..

32 Penrose-Fife (P) e t α = f, e = u + λ(w), α α(u) in (0, T ) Ω, (.) w t + ψ t (w) + g(w) αλ (w) 0 in (0, T ) Ω, (.) α = h on (0, T ) Γ, (.3) e(0) = e 0, w(0) = w 0 in Ω. (.4) Ω R N ( N 3) Γ = Ω f, h Ω α : R R ψ t L (Ω) ψ t ψ t g C (R) λ C (R) Hilbert H A : H H A : R(I + A) = H (I : H H ) ϕ : H [, ] ϕ : H H ϕ(z) := {z H : (z, v z) H ϕ(v) ϕ(z), v H} ϕ ϕ ( ) 990 O.Penrose P.C.Fife e, w, u w = 0 w = 0 w α(u) = (u > 0), u ψt (w) = κ w κ > 0 (.) ( u ) (.) - g g(w) ( u )λ (w) κ (.) 4 (.) ψ t (w) = κ w + I [0,] w

33 I [0,] (z) = { 0 z [0, ] + otherwise w < 0, w > w [0, ] I [0,] ψ t (w) = κ w + I [σ (t), σ (t)]w ( ) (.4) (.5) (P)... There exists a unique solution of (P) (e, w) : [0, T ] H (Ω) L (Ω) such that the following items are satisfied : (a) e W, (0, T ; H (Ω)) L (0, T ; L (Ω)), sup 0 t T ˆα(u(t))dx < + and Ω w W, (0, T ; L (Ω)) L (0, T ; H (Ω)). (b) There exists α L (0, T ; H (Ω)) such that α α(u) a.e. in (0, T ) Ω and the following evolution equation holds: e (t) + F ( α(t) h(t)) = f(t) + h(t) in H (Ω) for a.e. t (0, T ). (c) The following evolution equation holds: w (t) + ψ t (w(t)) + g(w(t)) α(t)λ (w(t)) 0 in L (Ω) for a.e. t (0, T ). (d) e(0) = e 0 and w(0) = w 0. F H 0(Ω) H (Ω)(= {H 0(Ω)} ) [] A. Ito and M. Kubo, Well-posedness for an extended Penrose-Fife phase field model with energy balance supplied by Dirichlet boundary conditions, Nonlinear Anal. 9 (008), [] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal. (994), [3] O. Penrose and P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D 43 (990), 44 6.

34 ( ) ; t u = u, x Ω, t > 0, ν u = u p, x Ω, t > 0, u(x, 0) = φ(x), x Ω. () Ω = {x = (x, x N ) R N : x N > 0}, N, t = / t, ν = / x N, ( ) } φ X {f L (Ω) L Ω, e x /4 dx : f 0 in Ω, () + /N < p, (N )p < N (3).. t u = u + u p in R N (0, ), u(x, 0) = λφ 0 in R N (4) Kawanago [3] (c.f. Kavian []).. λ > 0, φ L (R N ) L (R N, e x /4 dx), p > + /N, (N )p < N +. λ > 0 : (i) λ > λ, (4) u ; (ii) λ = λ, (4) u, u(t) L (R N ) t p as t ; (iii) 0 < λ < λ, (4) u, u(t) L (R N ) t N as t. ()... τ > 0, Ω [0, τ) u (), u C(Ω (0, τ)) L (0, σ : L (Ω)), 0 < σ < τ, (x, t) Ω (0, τ) u(x, t) = Ω G(x, t, y)φ(y)dy + t. [ ) G(x, y, t) = (4πt) N x y exp ( + exp 4t 0 Ω G(x, y, t s)u p (y, s)dσ y ds ( x y. y = (y, y N ) Ω y = (y, y N ). 4t )], x, y Ω, t > 0

35 () φ L (Ω), (). () u T M (φ), lim sup t TM (φ) 0 u(t) L (Ω) =, T M (φ) blow-up time., p = L p (Ω). X X + L (Ω,e x /4 dx), K = {φ X ; T M (φ) = }, B = X \ K = {φ X ; T M (φ) < }, Int (K) X K, K X K. u F [u](t) F [u](t) = u dx u p+ dσ Ω p + Ω... () (3) (). K 0 Int (K) X. φ = λϕ, ϕ X \ {0}, λ ϕ, λϕ Int (K) if λ (0, λ ϕ ), K if λ = λ ϕ, B if λ > λ ϕ. : (i) φ Int(K) \ {0}, q [, ],., c = lim u(x, t)dx = t Ω u(t) q t N ( q ) as t ( φ(x)dx + Ω 0 Ω ) u p (x, t)dσdt t N ( q ) u(t) c g(t) q Ct + Ct N (p N ), t. C, g(x, t) = (4πt) N/ exp( x /4t); (ii) φ K, u(t) t /(p ) as t ; (iii) φ B, lim t TM (φ) 0 F [u](t) =. References [] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (987), [] T. Kawakami, Global existence of solutions for the heat equation with a nonlinear boundary condition, preprint. [3] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (996), 5.

36 Asymptotic behavior of solutions for the shadow reaction-diffusion system with the nonlinearity of the FitzHugh-Nagumo type Yasuhito Miyamoto Department of Mathematics, Tokyo Institute of Technology Meguro-ku, Tokyo, 5-855, Japan September, 009 Let Ω R N (N {,, 3}) be a bounded domain with smooth boundary, and let R + := {t R; t > 0} and N := {,,...}. In this article we study the asymptotic behavior of solutions to the shadow reaction-diffusion system u t = u + f(u) αξ in Ω R +, τξ t = γ(u) βξ in R +, (SS) ν u = 0 on Ω R + and (u(x, 0), ξ(0)) = (u 0 (x), ξ 0 ) and its reduced system obtained by letting τ = 0 in (SS) u t = u + f(u) αξ in Ω R +, 0 = γ(u) βξ in R +, (RE) ν u = 0 on Ω R + and u(x, 0) = u 0 (x). Here α, β, τ are positive constants, γ(u) := Ω u(x, t)dx, and ν denotes the outer normal derivative on Ω. We assume that f satisfies the following (A) or (A): f(u) := a 0 u(u a )(a u), where a 0 > 0 and a, a R, and Ω α Ω π β N, (A) f C 5, and there { are p >, C 0 > 0, and C > 0 such that C 0 C u p if u 0; f(u) C 0 + C u p if u 0. (A) Note that if (A) holds, then (A) holds.

37 Let, denote the L -inner product, and let X := HN R. We see that (SS) (resp. (RE)) has a unique solution in C 0 ([0, + ), X) (resp. C 0 ([0, + ), HN )). Therefore, (SS) (resp. (RE)) generates a global semiflow in X (resp. HN ). By Φ τ (t)(u 0, ξ 0 ) (resp. Ψ(t)u 0 ) we denote the semiflow generated by (SS) (resp. (RE)). We denote the set consisting of all the equilibrium points of Φ τ (t) (resp. Ψ(t)) by E (SS) (resp. E (RE) ). We say that ū( HN ) is an exponentially asymptotically stable steady state of Ψ(t) if ū E (RE) and if sup λ σ(l Re(λ) < 0, where L 0ū) 0 ū denotes the linearized operator of (RE) at ū and σ(l 0 ū) denotes the spectral set. Theorem A. Suppose that N = and that (A) holds. ( i ) Let τ := β /(α Ω ). If τ (0, τ ), then the ω-limit set of (u 0, ξ 0 ) H N R for Φ τ (t), ω Φ τ ((u 0, ξ 0 )), is a singleton. (ii) The ω-limit set of u 0 H N for Ψ(t), ω Ψ(u 0 ), is a singleton. The assumptions of N = and (A) come from the difficulty in proving the non-degeneracy of the solutions to a scalar elliptic equation. We see that (SS) (resp. (RE)) is a gradient-like system, namely, the ω-limit set of any precompact orbit belongs to E (SS) (resp. E (RE) ). Therefore, this theorem means that every orbit Φ τ (t)(u 0, ξ 0 ) (resp. Ψ(t)u 0 ) converges to one of the equilibrium points of (SS) (resp. (RE)) provided that all the assumptions in Theorem A are satisfied. We define an injective mapping Γ from HN to X by Γu := (u, β ) γ(u). The functional space to which Φ τ (t)(u 0, ξ 0 ) belongs is different from one to which Ψ(t)u 0 belongs. However, using the mapping Γ, we can measure the distance in X between Φ τ (t)(u 0, ξ 0 ) and ΓΨ(t)u 0 and show the closeness of the two orbits. The second result is Theorem B. Suppose that N {,, 3} and that (A) holds. Let u 0 H N. Suppose that ω Ψ (u 0 ) is an exponentially asymptotically stable steady state. For any ξ 0 R, ε > 0, and t 0 > 0, there is τ 0 > 0 such that if τ (0, τ 0 ), then ω Φ τ ((u 0, ξ 0 )) = Γω Ψ (u 0 ), sup Φ τ (t)(u 0, ξ 0 ) ΓΨ(t)u 0 X < ε, t>t 0 and ω Φ τ ((u 0, ξ 0 )) is an exponentially asymptotically stable steady state. In particular, for u 0 H N, there is τ > 0 such that if τ (0, τ ), then ω Φ τ (Γu 0 ) = Γω Ψ (u 0 ).

38 Global Existence and Asymptotic Behavior of Solutions for Quasi-linear Dissipative Plate Equation Y. Liu ( ), S. Kawashima ( ) Faculty of Mathematics, Kyushu University In this paper we consider the initial value problem of the following quasilinear dissipative plate equation in multi-dimensional space R n with n : n u tt u tt + b ij ( xu) xi x j + u t = 0. () The initial data are given as i,j= u(x, 0) = u 0 (x), u t (x, 0) = u (x). () Here u = u(x, t) is the unknown function of x = (x,, x n ) R n and t > 0, and represents the transversal displacement of the plate at the point x and the time t. Also, b ij = b ij (V ) are smooth functions of V = (V ij ) S n satisfying the following structural conditions, where S n denotes the totality of n n real symmetric matrices: [A] There exists φ = φ(v ) such that b ij (V ) = ( φ/ V ij )(V ). [A] ij αβ bij αβ (O)ω iω j ω α ω β > 0 for ω = (ω,, ω n ) S n. Here we put b ij bij αβ (V ) = (V ) = V αβ φ V ij V αβ (V ), i, j, α, β =,, n. For simplicity we consider the case where the functions b ij αβ (V ) verify bij αβ (O) = δ ij δ αβ. The equation () has the decay property of regularity-loss type, which is indicated by the exponent σ(k, n) = max{σ 0 (k), σ (k, n)}, where σ 0 (k) = k + [ k+ ], σ (k, n) = k + [ n+k 4 ]. For n, we define s(n) by s(n) = 3[ n 4 8, n =, 6, n = 3, ] + 5, n 4, which indicates the regularity of the initial data.

39 Theorem. Suppose that the conditions [A] and b ij αβ (O) = δ ijδ αβ are satisfied; the latter implies [A]. Let n and s s(n). Assume that u 0 H s+ (R n ) L (R n ) and u H s (R n ) L (R n ), and put E := u 0 H s+ + u H s + (u 0, u ) L. Then if E is suitably small, then the problem (), () has a unique global solution u(x, t) satisfying the following optimal decay estimates: for k with σ(k, n) s, and for k with σ(k, n) s 4. k xu(t) H s σ(k,n) CE ( + t) n 8 k 4 k xu t (t) H s 4 σ(k,n) CE ( + t) n 8 k 4 Theorem. Suppose that [A] and b ij αβ (O) = δ ijδ αβ are satisfied. Let n and s s(n). Assume that u 0 H s+ (R n ) L (R n ) and u H s (R n ) L (R n ), and put E := u 0 H s+ + u H s + (u 0, u ) L. Let u(x, t) be the global solution to the problem (), () which is constructed in Theorem. Then we have the following asymptotic relations: x{u(t) k MG 0 (t + )} H s σ(k,n) CE ( + t) n 8 k+ 4 for k with σ(k, n) s, and x k t {u(t) MG 0 (t + )} H s 6 σ(k,n) CE ( + t) n 8 k+5 4 for k with σ(k, n) s 6. Here M is a constant given by M = R n (u 0 + u )(x)dx, and G 0 (x, t) = F [e ξ 4t ](x) is the fundamental solution to the fourth-order parabolic equation u t + u = 0.

40 ( ) ( ) U(t) = e it Gf(t) = t 0 U(t s)f(s) ds, t R. φ, f u(t) = U(t)φ, v(t) = igf(t) { i t u + u = 0, (t, x) R R n, u(0, x) = φ(x), x R n, () { i t v + v = f, (t, x) R R n, v(0, x) = 0, x R n,. n. u, v.. (Stricharz, [0,,, 4, 6], etc.). (i) C = C(q, n) > 0 (ii) Uφ L r (R;L q (R n )) C φ. (q, r) (admissible pair). q + nr =, q n n. C = C(q, q, n) > 0 Gf L r (I;L q (R n ) C f L r (I;L q (R n )). (q i, r i ), i =,, q i + q i. () =. I = [0, T ]. (Kato-smoothing, [9,,, 5], etc.). (i) C = C(n) > 0 ϕuφ L (I;H / (R n )) C φ. ϕ(x) = x / ε, ε > 0, x = + x. I = [0, T ]. (ii) C = C(n) > 0. I = [0, T ]. ϕgf L (I;H / (R n )) C f L (I;L (R n )).3... ( ).., [3] KdV ([8] ).

41 . { i t u + u = λ u p u, (t, x) R R n, u(0, x) = u 0 (x), x R n, (NLS) λ > 0, < p <... ([, 7], etc.). u 0 H (R). n 6 < p < p (n), n 7 < p + n 4. p (n) = (n =, ), = + 4 n (n 3). u C(R; H (R n )) (NLS), T > 0 ϕu L (0, T ; H 3/ (R n )). [] Constantin, P. and Saut, J.-C., Local smoothing properties of dispersive equations. J. Amer. Math. Soc., (988), [] Ginibre, J. and Velo, G., The global Cauchy problem for the non linear Schrödinger equation revisited. Ann. Inst. Henri Poincaré, analyse non linéaire, (985), [3] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in Applied Math. Adv. Math. Suppl. Studies, 8 (983), [4] Kato, T., Nonlinear Schrödinger equations. Lecture Notes in Physics, 345 Schrödinger operators, (H. Holden and A. Jensen eds.) Springer-Verlag, Berlin-New York (989), [5] Kato, T. and Yajima, K., Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys., (989), [6] Keel, M., and Tao, T., Endpoint Strichartz estimate. Amer. J. Math., 0 (998), [7] Nakamura, Y., Regularity of solutions to nonlinear Schrödinger equations with H initial data. Yokohama Math. J., 47 (999), [8] Segata, J., Well-posedness for the fourth-order nonlinear Schrodinger-type equation related to the vortex filament. Differential Integral Equations, 6 (003), [9] Sjölin, P., Regularity of solutions to the Schrödinger equations. Duke Math. J., 55 (987), [0] Strichartz, R. S., Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44 (977), [] Vega, L., The Schrödinger equation : pointwise convergence to the initial data. Proc. Amer. Math. Soc., 0 (988), [] Yajima, K., Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys., 0 (987),

42 ( ) u j t u j + u j t = m u k p j,k, t > 0, x R n ( j m), () k= u j (0, x) = a j (x), t u j (0, x) = b j (x), x R n ( j m)., m m, p j,k p j,k = 0 (j, k =,,, m). () {p j,k } m j,k=,., ( ), Fujita. u u u + t t = u p, t > 0, x R n, u(0, x) = a(x), t u(0, x) = b(x), x R n, p = + n, ; p + n, p > + n, ([5], [7])., () m =, p, = p, = 0, p, >, p, >, ([3])... p j,k m P, m E m : p, p,... p,m p, p,... p,m P =......, E m =..... p m, p m,... p m,m α = t (α, α, α m ) R m, (P E m ) α = t (,, ).,, m = () ([3]),.. ([]). n =,, 3. (a j, b j ) W, (R n ) W, (R n ) L (R n ) L (R n ) ( j m), m det(p E) 0, p j,k > ( j m), 0 < α j < n k= ( j m)

43 , (). u j (t) C([0, ); L (R n ) L (R n )) ( j m). α, ;. u t u + u t = u m p, t > 0, x R n, u t u + u t = u p, t > 0, x R n,. () u m u t m + u m = u m p m, t > 0, x R n, t u j u j (0, x) = a j (x), t (0, x) = b j(x), x R n ( j m).. ([4]). n. max j m α j (a j, b j ) W, (R n ) W, (R n ) L (R n ) L (R n ) ( j m), a j (x)dx 0, b j (x)dx 0 ( j m), R n R n a j0 (x)dx > 0 b j0 (x)dx > 0 R n R n, p j > ( j m), ) α j0 (= max α j n j m = α j0,, {u j (t)} m j=: (),.... [], [6]. [] Ogawa, T., Takeda, H., Global existence of solutions for a system of nonlinear damped wave equations, preprint. [] Renclawowicz, J., Applicationes Mathematicae, 7, (000), no., [3] Sun, F., Wang, M., Nonlinear Analysis :, 66 (007), no., [4] Takeda, H., Global existence and nonexistence of solutions for a system of nonlinear damped wave equations, to appear. [5] Todorova, G., Yordanov, B., J. Differential Equations, 74 (00), [6] Umeda, N., Tsukuba J. Math. 7 (003), no., [7] Zhang, Q., C. R. Acad. Sci. Paris, 333 (00), 09-4.

44 Well-posedness for the fifth order KdV equation KdV 5 KdV. { t u 5 xu 0 x (u 3 ) + 5 x ( x u) + 0 x (u xu) = 0 in (0, T ) R u(0, x) = u 0 (x) (), S(R) ().,.,, () (LWP)., S. Kwon [] Sobolev H s, s > 5 () LWP. C. E. Kenig, G. Ponce and L. Vega [],, KdV LWP., x (u xu) (Picard ). iteration step.., H s,a := {u S (R); u H s,a := ξ s a ξ a û(ξ) L < }, (a < 0)., () LWP., s, a, iteration step 3. s 4, s a 3 < a 4 (), s, a () H s,a () LWP., x (u xu). τ ξ 5 ξf (ξ g) C f ˆXs,a,b g ˆXs,a,b. (3) ˆXs,a,b ˆX s,a,b H s,a Bourgain. ˆX s,a,b := {f S (R ) : f := ξ s a ξ a τ ξ 5 b f ˆXs,a,b L ξ,τ < }. () LWP ˆX s,a,b,., (3). ˆXs,a,b, High-High-Low interaction High-High-High interaction,

45 . High-High-Low interaction (3), () (3). { ξ } f Ŷ a := ξ a τ ξ ε f L ξ,τ, (0 < ε ). High-High-High interaction, b = s = 4 log (3)., { ξ } Besov. f ˆXs := { ξ s τ ξ 5 } f L ξ,τ (A j B k ) A j, B k. j, k 0 l j lk A j := {(ξ, τ) R ; j ξ < j+ }, B k := {(ξ, τ) R ; k τ ξ 5 < k+ }. f h (ξ) := f(ξ) ξ, f l (ξ) := f(ξ) ξ,. Ẑ s,a := {f S (R ); f Ẑs,a := f h ˆXs + f l Ŷ a + ξ a f l L ξ L τ < }, (). τ ξ 5 ξf (ξ g) Ẑs,a C f Ẑs,a g Ẑs,a 3 x (u) 3 [k, Z]-multiplier norm method (). Fourier,. Theorem. s, a s 4, s a 3 < a 4, () LWP.. Thorem. s, a s, a 4 Hs,a (). Lemma 3. a 4, T = T ( u 0 H,a) C > 0. ( ) ) sup u(t) H (R) + u(t) C ( u Ḣ t [0,T ] a (R) 0 Ḣ a(r) + u H (R) + T u H (R) [] C.E.Kenig, G.Ponce, and L.Vega, Higher-order nonlinear dispersive equations, Comm. PureAppl. Math. Soc,, 994, no., [] S. Kwon, On the fifth order KdV equation: Local well-posedness and lack of uniform continuity of the solution map, J. Differential Equation, 45, 008, no.9,

46 (S) i =,, M (u i ) t + A i (x, t, P (t), u i ) + B i (x, t, P (t), u) = β i (u i ), (x, t) Ω i (0, T ), (S) β i (u i ) n (x, t) = 0, (x, t) Ω i (0, T ), u i (x, 0) = u i0 (x), u i0 L (Ω i ) BV (Ω i ), P (t) = (P (t),, P M (t)), P i (t) = w i (x)u i (x, t)dx, Ω i w i Lip(Ω i ). Ω i R N Lipschitz = ( / x,..., / x N ), = N i= / x i R N spatial nabla, Laplacian [0,T] A i (x, t, ξ) = (A i,..., A N i )(x, t, ξ) Ω [0, T ] R R N B i (x, t, ξ) Ω [0, T ] R β i R Lipschitz n Ω β i β i(ξ) = 0 (S) Stefan ([]) (S) BV - BV - Definition i =,..., M u i0, u i L (Ω i ) BV (Ω i ) u = (u i ) i u i u (S) BV - () u i C([0, T ]; L (Ω i )), L -lim t 0 u i (, t) = u i0 ; () β(u i ) L (0, T ; L (Ω i ) N ), ϕ D + (R N (0, T )) k R T 0 Ω i sgn(u i k)[(u i k)ϕ t β i (u i ) ϕ + [A i (x, t, P (t), u i ) A i (x, t, P (t), k)] ϕ [B i (x, t, P (t), u i ) + A i (x, t, P (t), k)]ϕ]dxdt T + sgn(t r u i k)[a i (x, t, P (t), u i ) A i (x, t, P (t), k)] n(x)ϕdh N dt 0. 0 Ω i T r BV (Ω) S.N.Kruzkov [3] C. Bardos, A. Y. Leroux and J. C. Nedelec [] Dirichlet BV (S) BV

47 u t + A i (x, t, u) + B i (x, t, u) = β i (u), (x, t) Ω (0, T ), β i (u) (IBV P ) i (x, t) = 0, (x, t) Ω (0, T ), n u(x, 0) = u 0 (x), u 0 L (Ω) BV (Ω). filtration Stefan ([4], [5]) (IBVP) BV - [4] BV - L A i, B i, β i, i =, Theorem u, v u 0, v 0 L (Ω) BV (Ω) (IBV P ),(IBV P ) BV - u(, t) v(, t) L (Ω) e α t u 0 v 0 L (Ω) + ( tc + α e α t C ) β β L (I) [ + eα t sup A α (, t, ξ) A (, t, ξ) BV (Ω) + sup B (, t, ξ) B (, t, ξ) L (Ω) t (0,T ),ξ I t (0,T ),ξ I ] + sup u(, t) BV (Ω) t (0,T ) sup t (0,T ),ξ I ξ A (, t, ξ) ξ A (, t, ξ) L (Ω) I R u, v C C sup t u(, t) BV, sup t v(, t) BV, sup t,ξ A (, t, ξ) BV (Ω), sup t,ξ ξ A (, t, ξ) Lip(Ω), sup t,ξ B (, t, ξ) BV (Ω), β L (I), β L (I), β β L (I) α B(x, t, ξ) ξ α BV [5] Theorem Schauder Theorem i =,, M u i0 L (Ω i ) BV (Ω i ) (S) BV - u = (u i ) i References [] C. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equations with boundary conditions, Comm. In Partial differential equations, 4(9), , (979) [],, 009. [3] S. N. Kru zkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 0 (970), [4] H. Watanabe, S. Oharu, BV -entropy solutions to nonlinear strongly degenerate parabolic equations -A uniqueness theorem-, submitted. [5] H. Watanabe, S. Oharu, Unique existence of BV -entropy solutions for strongly degenerate convective diffusion equations,, 640 (009), ,

48 ( ) < s < T < +, Ω R N (N =, 3) Γ := Ω. (P):= {() (8)} : ψ θ ψ in Q := (s, T ) Ω, () θ t + v θ κ θ = 0 in Q(θ) := {(t, x) Q; ψ < θ < ψ }, () θ t + v θ κ θ 0 in Q (θ) := {(t, x) Q; θ = ψ }, (3) θ t + v θ κ θ 0 in Q (θ) := {(t, x) Q; θ = ψ }, (4) v + (v )v ν v = g(θ) p t in Q, (5) divv = 0 in Q, (6) θ = 0, v = 0 on Σ := (s, T ) Γ, (7) θ(s) = θ 0, v(s) = v 0 in Ω, (8) θ = θ(t, x), v := (v (t, x),, v N (t, x)), p := p(t, x), κ, ν. (P) (), ψ := ψ (t, x) ψ := ψ (t, x) (), Q (θ) (3), Q (θ) (4). θ,. g : R R N, θ 0 : Ω R, v 0 : Ω R N,,,. Navier-Stokes Boussinesq. Boussinesq,, (99), Foias, Manley, Temam (987). N =, (008), (009), (P), : u (t) κ 0 u(t) + G(u(t), u(t)) + I K(t) (u(t)) 0, u (t) + νau(t) + B(u(t), u(t)) = g(u(t)). Crauel, Debussche, Flandoli(997),,, (00).

49 3 H := L (Ω), V := H (Ω), V V V H V., Navier-Stokes D σ (Ω) := {u C 0 (Ω) := (C 0 (Ω)) ; divu = 0 in Ω}, H := L σ(ω), V := H σ(ω). L σ(ω), H σ(ω) D σ (Ω), L (Ω), H (Ω). V H V. H K(t) := {z H; ψ (t) z ψ (t) a.e. on Ω},.. (A) ψ i L loc (R; H (Ω)) L (R; V ) L (R Ω), ψ i L (R; H) L (R; H) R Ω ψ ψ R Γ ψ 0 ψ ; (i =, ) (A) g i Lipschitz (i =,, N); (A3) θ 0 K(s), v 0 H. 3.. N =. (A), (A), (A3), (P). H := H H, V := V V K(s) := K(s) H. 3. (P) (θ 0, v 0 ) K(s) (u(t), u(t)) E(t, s) := {E (t, s), E (t, s)} : K(s) K(t), : (E) E(t, s) = E(t, t ) E(t, s) for all s, t, t with < s t t < + ; (E) E(s, s) is identity; (E3) If < s n t n < +, Z n K(s n ) with s n s, t n t in R, Z n Z in H as n +. Then E(t n, s n )Z n E(t, s)z in H as n A(t) H (T), (T) t : (T) A(t) t : E(τ, t 0 )A(t 0 ) = A(τ) for all τ, t 0 with τ t 0 t; (T) A(t) t B H : dist(e(t, s)(b K(s)), A(t)) 0 as s,, ε > 0 s ε,t t inf E(t, s)u Z H < ε for all s s ε,t and U B K(s). Z A(t) N =. (A), (A), (A3), {A(t)} t R H, A(t) (P) t.

50 ( ) u : [0, T ] [0, L] R N u t ( b(x) div a(x) u ) x = 0, a.a. (t, x) (0, T ) (0, L), u x () u(t, 0) = g 0, u(t, L) = g L, a.a. t (0, T ), u(0, x) = u 0 (x), a.a. x (0, L), u 0 g 0, g L R N g 0 g L a(x), b(x) [0, L] [4] () H = L (0, L; R N ) b f, g b = L 0 b(x)(f(x), g(x)) dx L (0, L; R N ) φ : H (, + ] L a(x) v φ(v) = x + a(0) v(0) g 0 + a(l) v(l) g L if v H BV (0, L; R N ), 0 + otherwise, BV (0, L; R N ) (0, L) R N φ g 0, g L R N [0, L] H φ +.. H φ : H (, + ] H φ : H H φ φ(u) = {f H φ(u + h) φ(u) f, h H for any h H} () du + φ(u) 0 in H a.a. t > 0, dt u(0) = u 0, () () u(t) u 0 u(t)

51 a(x) x, x,..., x m (0, L) : 0 = x 0 < x < x < < x m < x m = L (m ) { m } H = h i χ (xi,x i+ ) h i R N, h 0 = g 0, h m = g i=0 ε > 0 φ φ ε : H (, + ] m φ ε a(x (v) = i ) m h i h i + ε if v = h i χ (xi,x i+ ) H, i= i=0 + otherwise, φ ε () u ε (t) ε 0 () u(t) C([0, T ]; H)... a(x) 0 φ ε (u ε ) in H (3) φ ε u ε h i u ε (x) = h i R N in (x i, x i+ ), i = 0,,,... m (4) h i = ( s i )g 0 + s i g L, 0 = s 0 < s < s < < s m =. (3) (4) (3) h i h i+ h a(x i+ ) i hi+ h i + ε a(x h i h i) i = 0, i =,,..., m hi h i + ε h 0 = g 0, h m = g L [] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam, 973. [] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations. Proc. Taniguchi Conf. on Math., Advanced Studies in Pure Mathematics, 3(00), [3] H. Kuroda and N. Yamazaki, Approximating problems of vectorial singular diffusion equations with inhomogeneous terms and numerical simulations. AIMS Journals, to appear. [4] G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, Cambridge, 00.

52 ( D), f f., Ω R,. u t u = f, (t, x) (0, T ) Ω, (H) u(0, x) = 0, x Ω, u = 0, (t, x) (0, T ) Ω. f 0 L q (0, T ; L p (Ω)) ( < p, q < ), () u t L q (0,T ;L p (Ω)) + D u L q (0,T ;L p (Ω)) C f L q (0,T ;L p (Ω)) () (H) p = L () f L q (0, T ; L (Ω)) Nagai-Ogawa [4] (Nagai-Ogawa[4]). f L (0, T ; L (Ω)) u (H) 0 α < 4π C > 0 () sup 0<t<T Ω ( ) α u(t, x) exp dx C Ω f L (L ) (H) Poisson { u = f in Ω, (P) u = 0 on Ω, Brezis-Merle [] f L (Ω) (P) 0 α < 4π C > 0 ( ) α u(x) (3) exp dx C Ω f L (Ω) Ω Dolzmann-Hungerbühler-Müller [3] f L (Ω) BMO (Bounded Mean Oscillation) (4) [u] BMO C f L

53 [u] BMO := sup u(x) u B dx, u B := B Ω B B B u(x)dx John-Nirenberg u BMO u Dolzmann-Hungerbühler- Müller [3] (4) Brezis-Merle (3) Brezis-Merle 4π Dolzmann-Hungerbühler-Müller [3] BMO 4π BMO. f µ f (λ) := {x ; f(x) > λ}, f f f (r) := inf {λ ; µ f (λ) B r } f f f (r) := B B r r f (y)dy BMO W { } (5) W (B R ) := f L (B R ); [f] W (BR ) := sup (f (r) f (r)) < 0<r<R 3. f (r) f (r) f W BMO f W f BMO W W Bennett-Sharpley [] W BMO 4. f L (0, T ; L (B R )) u (H) sup [u(t)] W (BR ) 0<t<T 4π f L (0,T ;L (B R )) 5. Nagai-Ogawa (H) u u References [] Brezis, H., Merle, F., Uniform estimates and blow-up behavior for solutions of u = V (x)e u in two dimentions, Comm. Partial Differential Equations, 6 (99), [] Bennett, C., Sharpley, R., Interpolation of operators, Pure and applied mathematics, 988. [3] Dolzmann, G., Hungerbühler, N., Müller, S., Uniqueness and maximal regularity for nonlinear elliptic systems of n-laplace type with measure valued right hand side, J. reine angew. Math., 50 (000), 35. [4] Nagai, T., Ogawa, T., Brezis-Merle inequalities and application to the global existence of the Cauchy problem of the Keller-Segel and self-interacting systems, preprint. address:

54 Schrödinger-Poisson ( PD) Schrödinger-Poisson i t u + u = λp u, P = u, u(0, x) = u 0 (x). (SP) (t, x) R + λ R u 0 H s (R ) (s > ) Poisson : P L (R ), P 0 as x, P (0) = 0. background( doping profile) u ( [4] neutrality ) P ( ( )) x y P (t, x) = log u(t, y) dy y R y log x y log( x y / y ) O( y ) P u(t) L p (R ) ( p (, 4)) welldefined. u ( u C0 ) P x (SP) (SP) u(t) = u 0 iλ t 0 e i t s (P u)(s)ds Lebesgue 3 [3].. s > u 0 H s (R ) T > 0 (SP) u C([ T, T ]; H s ) u 0 u H s C([ T, T ]; H s ) u(t, x) = a(t, x)e iϕ(t,x), i t a + ( a = i ϕ a + ) a ϕ, a(0) = u 0, t ϕ + ϕ + λp = 0, ϕ(0) = 0, P = a,

55 t a Schrödinger Hamilton-Jacobi a C([0, T ]; H s ) ϕ C([0, T ]; C ) s > H s L (s > ) a(t) H s L L P ϕ u = ae iϕ ϕ ( ) u C([0, T ]; H s ) u t t ϕ u t H s (SP).. u 0 H s (R ) (s > ).. L - W (t) = (log x ) u(t, x) dx v(t, x) := u(t, x) exp( iλ t W (s)ds) v 0 i t v + ( ) v = λ (log x y ) v(y) v, v(0, x) = u 0 (x). E(t) = v(t) L + λ = u(t) L + λ (log x y ) v(t, y) v(t, x) dy dx (log x y ) u(t, y) u(t, x) dy dx ( ).3. (SP) h ih t u h + h uh = λp h u h, P h = u h, u h (0, x) = u h 0(x). u h WKB u h = e iϕ0/h (a 0 + ha + + h N a N + o(h N )) (h 0) 3 [, ] a j (j ) P u h = a h e iϕh /h. a h, ϕ h h a j ϕ h h j ϕ j j a j Taylor e ihj ϕ e j+ = + h j ϕj+ + o(h j ) ( ) ϕ j+ [] T. Alazard and R. Carles, Semi-classical limit of Schrödinger Poisson equations in space dimension n 3, J. Differential Equations 33 (007), no., [] R. Carles and S. Masaki, Semiclassical analysis for Hartree equations, Asymptotic Analysis 58 (008), no. 4, 7. [3] Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 0, New York University Courant Institute of Mathematical Sciences, New York, 003. [4] P. Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson equations, SIAM J. Math. Anal. 34 (00), no. 3, (electronic).

56 ABP type estimates for nonlinear elliptic systems ( ) L p regularity ABP regularity Caffarelli [C] L p Caffarelli-Crandall-Kocan-Świȩch[CCKS] L p ABP extremal Koike-Świȩch[KS] extremal N[N] regularity Harnack Koike-Świȩch[KS3] Lp Koike-Świȩch[KS3] [N] Ishii( 9) Ishii- Koike( 9) Busca-Sirakov[BS] L p ABP Harnack ( ) ABP Ω R n (n ) ; { P (D u i ) µ i (x) Du i c i (x, u,..., u l ) = f i (x) in Ω i =,, l. () S n n n 0 < λ Λ S n λ,λ := {A Sn ; λi A ΛI} q p > n/, q > n f i L p +(Ω), µ i L q +(Ω) L p +(Ω) L p (Ω) Ω B (0) x Ω i {,..., l} c i (x, 0,..., 0) = 0

57 (H0) c(x, u) u R l Lipschitz (Lipschitz ν) Lebesgue null set N Ω x Ω\N (H) u v u, v R l u j = v j j {,..., l} c j (x, u) c j (x, v) for a.e.x Ω... (H0), (H) n p q, n < q (H): i {,..., l} l j= c i u j (x, u) 0 a.e.in Ω R l. (H3) m ij := sup ess. c i (x, u) ( m ij ν < ) (x,u) Ω R u M := ( m ij ) l i,j= l j negative semi-definite u = (u,..., u l ) C( Ω, R l ) () L p ABP i.e. C > 0 ( sup(u... u l ) C Ω sup(u u l ) + f f l L n (Ω) Ω.. (H), (H3) { c (x, u) = a(x)(u u ), { c (x, u) = u + 3 arctan u, c (x, u) = a(x) ( u + u ), c (x, u) = arctan u u. a(x) Ω [/, ]..3. (H) ABP B (0) R n { u + v = 0 v = 0. u = x, v = n ABP ). [BS] Busca, J., and B. Sirakov, Ann. Inst. H. Poincaré Anal. Non Lineaire (004). [C] Caffarelli, L. A., Ann. Math. 30 (989). [CCKS] Caffarelli, L. A., M. G. Crandall, M. Kocan, and A. Świȩch, Comm. Pure Appl. Math. 49 (996). [KS] Koike, S., and A. Świȩch, function method, Math. Ann., 339 (007). [KS3] Koike, S., and A. Świȩch, to appear in J. Math. Soc. Japan [N] Nakagawa, K., to appear in Adv. Math. Sci. Appl.

58 ( ), Cauchy. t u u + (u ψ) = 0, t > 0, x R 3, () ψ = u, t > 0, x R 3, u(0, x) = u 0 (x) 0, x R 3., t := / t, j := / x j, := 3 j= j, := (,, 3 )., u = u(t, x), ψ = ψ(t, x),,.,,., convectiondiffusion, Keller-Segel, Navier-Stokes., convection-diffusion, ([]). Navier-Stokes, ([, 3]). Keller-Segel, Navier-Stokes ([4, 7]).,,, ([9]).,,., ([5, 6])., ([8]).,,,,. (, ). u 0 L (R 3 ) L (R 3 ), p, γ := 3, : p () u(t) p C( + t) γ, t > 0. ([5, 8])., L., u (), Hardy-Littlewood-Sobolev ( ), ψ., 3 < p <, γ := 3, : p ψ(t) p C u(t) 3p/(3+p) C( + t) γ+/, t > 0., t., Cauchy () ([6]). u(t) = e t u 0 + t 0 e (t s) (u ( ) u)(s)ds, t > 0, x R 3., G(t, x) := (4πt) 3/ e x /(4t), e t ϕ(x) := R 3 G(t, x y)ϕ(y)dy.,.

59 ( ) (3) V 0 (t, x) := G(t, x) u 0 (y)dy, V (t, x) := G(t, x) R 3 J(t, x) := t 0 e (t s) (V 0 ( ) V 0 ) (s)ds, R 3 yu 0 (y)dy, K(t, x) := 3 log( + t) G(t, x) R 3 y (V 0 ( ) J + J ( ) V 0 ) (, y)dy,, λ > 0, J. λ 4 J(λ t, λx) = J(t, x), t > 0, x R 3., Cauchy (),.. u 0 L (R 3 ) L (R 3 ). ( u ) (), V 0, V, J, K (3)., p, γ := 3 : p u(t) V0 ( + t) V ( + t) J( + t) K( + t) p =o ( t γ log( + t) ) as t., R 3 u 0 (y)dy 0, J, K 0., L p m(r 3 ) := { ϕ L p (R 3 ) x m ϕ L p (R 3 ) }.,., L p (R 3 ),. ( ). u 0 L (R 3 ) L (R 3 ). p, γ := 3 p, : xu(t) p C( + t) γ+.,,.. : u(t) V 0 ( + t) V ( + t) J( + t) p C( + t) γ log( + t).,. References [] Carpio, A., SIAM J., Math. Anal. 7 (996), [] Escobedo, M., Zuazua, E., J. Funct. Anal. 00 (99), 9-6. [3] Fujigaki, Y., Miyakawa, T., SIAM J. Math. Anal. 33 (00), [4] Kato, M., Differential Integral Equations (009), [5] Kawashima, S., Kobayashi, R., Funkcial. Ekvac. 5 (008), [6] Kurokiba, M., Ogawa, T., J. Math. Anal. Appl. 34 (008), [7] Nagai, T., Yamada, T., J. Math. Anal. Appl. 336 (007), [8] Ogawa, T., Yamamoto, M., Math. Models Methods Appl. Sci., 9 (009), [9] Yamada, T., Higher-order asymptotic expansions for a parabolic system modeling chemotaxis in the whole space, to appear in the Hiroshima Math. J. address:

60 On smoothing effect for higher order curvature flow equations ( ) D { V = Γ(t) H Γ(t), t > 0 Γ(0) = Γ 0 () Willmore { V = Γ(t) H Γ(t) H3 Γ(t) + H Γ(t)K Γ(t), Γ(0) = Γ 0 t > 0 () Γ = {Γ(t); t > 0} V Γ(t) H Γ(t), K Γ(t) Γ(t) Γ(t) Laplace-Beltrami Laplacian Γ Γ(t) = {(x, y); y = w(x, t)} w Γ 0 Γ 0 = {(x, y); y = w 0 (x)} Willmore w t = ( + wx) w xxxx + 0w xw xx w xxx + 3w3 xx ( + wx) 3 ( + wx) 8w xwxx 3 3 ( + wx), (3) 4 w t = ( + wx) w xxxx + 0w xw xx w xxx + 3w3 xx ( + wx) 3 ( + wx) 8w xwxx 3 3 ( + wx) wxx 3 4 ( + wx), (4) 4.. w 0 h +4θ (R)(0 < θ < /4) T > 0 w C([0, T ]; BC (R)) C ((0, T ]; BC (R)) C((0, T ]; BC 5 (R)) (3) w w w x C θ((0, T ]; BC 4 (R)) B ((0, T ]; C 4+4θ (R)); w x Cθ ([0, T ]; BC(R)) B([0, T ]; C 4θ (R))... w 0 h +4θ (R)(0 < θ < /4) T > 0 w C([0, T ]; BC (R)) C ((0, T ]; BC (R)) C((0, T ]; BC 5 (R)) Willmore (4) w w w x C θ((0, T ]; BC 4 (R)) B ((0, T ]; C 4+4θ (R)); w x Cθ ([0, T ]; BC(R)) B([0, T ]; C 4θ (R)).

61 Willmore [], [3] [], [3] h +β (0 < β < ) Hölder h +4θ (0 < θ < /4). { du = A(U)U + G(U), dt (5) U(0) = U 0 [] { du (t) = Λu(t) + f(t), 0 < t T, dt u(0) = u 0 X Banach Λ : D(Λ) X X X D Λ (θ, ) = {x X; sup t θ Λe tλ x < }, t>0 D Λ (θ) = {x D Λ (θ, ); lim t θ Λe tλ x = 0} t 0 (3),(4) [] A. Buttu, On the Evolution Operator for a Class of Non-autonomous Abstract Parabolic Equations. J. Math. Anal. Appl., 70 (99), [] J. Escher, U. F. Mayer, G. Simonett, The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal., 9 (998), [3] G. Simonett, The Willmore flow near spheres. Differential Integral Equations, 4 (00),

62 T = R/πZ t u + ( ) k+ x k+ u + x(u ) = 0, (t, x) R T u(0, x) = ϕ(x), ϕ H s (T). k N, H s (T) := {f H s (T) ˆf(0) = 0}.., ϕ H s (T) T > 0,.. [0, T ] u.. [0, T ] u. 3. t [0, T ] u(t) H s (T). 4. ϕ u., k = KdV,. [] Bourgain, [], [3] k = s., k.. k, s k. k =, [], [3]. k =, Kawahara,., u(t) = U(t)ϕ t 0 U(t t ) x (u(t ) )dt, U(t)f := Fn [e ink+ t ˆf]. u,.,,.,,.

63 . Y λ, F (x, ) S(R) (x T λ ) Y λ := F : R T λ R x F (x, ) C. F (τ, 0) = 0 (τ R), X s,b,λ, F Xs,b,λ := τ n k+ b n s F l n (λ)l τ Y λ., T λ := R/(πλ)Z, ( f l n (λ) = f(n) dn Z λ ) ( := f n Z ) ( n λ)., F F.,.. λ. k, s [ k, 0], 0 < ϵ < k + s, x (uv) Xs,,λ λϵ u Xs,,λ v X s,,λ. 3. k, s < k, b R. x (uv) Xs,b, u Xs,b, v Xs,b,,., s. [] J.Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, GAFA.3(993),07-56, [] C.Kenig, G.Ponce and L.VEGA, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc.9 (996), [3] J.Colliander, M.Keel, G.Staffilani, H.Takaoka and T.Tao, Sharp global wellposedness for KdV and modified KdV on R and T, preprint(00).

64 ( ) D3 : u t = u a u p u in Ω (0, ), ν u = u q u on Ω (0, ), u(x, 0) = u 0 (x) in Ω. () Ω R n, p, q >, a > 0., (p = q a = q). (), []([3]), [], [4]. p > q p = q, a > q, (), p < q p = q, a < q (a, p, q )., (p = q a = q) n =,, []. φ = qφ q in (, ), φ(±) = +.,. (n ). p = q, a = q.. v = u (q )., v. v t = v + n v q ( + (q ) v ), 0 < r < R, t > 0, r (q )v () v (0, t) = 0, v (R, t) = (q ), t > 0., () u, () v 0. u, v (3) v. v (r) (q )(R r) C(R r). (3) () ψ ( ψ = qψ q in B R, ψ = + on B R ), (3) ψ : v (r) ψ(r) (q ) C(R r),, ψ(r) (q ) ().. u T. v C[0, R] C[0, R),, (3) v(r, t) v (r) uniformly on [0, R] as t T 0., u 0(r) u 0 (r) q, W, (0, R).

65 , (n ). Theorem,. 3 (n, Ω = B R ). 0. () 0, Theorem 3,. ( ), v 0., () r = R v : v t (R, t) = v (R, t) (n )(q ). R, v ( ), v (R, t) 0. v 0.,. ( )., () /v. ( 3),..,.,.., {w η } η>0. w η + n w η = qwη q in (0, R), w(r) = η, w (R) = η q. r [] F. Andreu, J. M. Mazón, J. Toledo, J. D. Rossi, Porous medium equation with absorption and a nonlinear boundary condition, Nonlinear Anal. TMA 49 (00) [] M. Chipot, M. Fila, P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenianae 60 (99) [3] J. López Gómez, V. Márquez, N. Wolanski, Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary, Rev. Unión Mat. Argent. 38 (993) [4] A. Rodríguez-Bernal, A. Tajdine, Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up, J. Differential Equations 69 (00)

66 DECAY PROPERTY FOR HYPERBOLIC SYSTEMS OF VISCOELASTIC MATERIALS P.M.N. DHARMAWARDANE GRADUATE SCHOOL OF MATHEMATICS, KYUSHU UNIVERSITY FUKUOKA 8-858, JAPAN We consider the following second order hyperbolic systems with dissipation: n n () u tt B jk u xj x k + K jk u xj x k + Lu t = 0 with the initial data j,k= j,k= () u(x, 0) = u 0 (x), u t (x, 0) = u (x). Here the unknown u is an m-vector function of x = (x,, x n ) R n and t 0, B jk are m m real constant matrices satisfying (B jk ) T = B kj, K jk (t) are smooth m m real matrix functions of t 0 satisfying K jk (t) T = K kj (t), and L is an m m real constant matrix; the symbol denotes the convolution with respect to t. The system () is a model system of viscoelasticity. We introduce the symbols of the differential operators: n n B ω = B jk ω j ω k, K ω (t) = K jk (t)ω j ω k, j,k= where ω = (ω,, ω n ) S n. Notice that B ω and K ω (t) are real symmetric matrices. We impose the following structural conditions. [A] B ω is real symmetric and positive definte, K ω (t) is real symmetric and nonneagtive definite, and L is real symmetric and nonnegative definite. [A] B ω K ω (t) is real symmetric and positive definte uniformlly in t 0, where K ω (t) = t K 0 ω(s)ds. [A3] K ω (0) + L is real symmetric and positive definite. [A4] There are positive constants C 0 and c 0 such that C 0 K ω (t) K ω(t) c 0 K ω (t) and C 0 K ω (t) K ω(t) C 0 K ω (t), where K ω(t) = t K ω (t) and K ω(t) = t K ω (t). We are interested in the decay property of the system (). Theorem (Decay estimate). Under the conditions [A] [A4], the solution u to the problem (), () satisfies the decay estimate k xu t (t) L + k+ x u(t) L j,k= C( + t) n/4 k/ u L + C( + t) n/4 k/ / u 0 L + Ce ct ( k xu L + k+ x u 0 L ).

67 Theorem (Energy estimate). Under the conditions [A] [A4], the solution to the problem (), () satisfies the energy estimate k xu t (t) H + k+ x u(t) H + t C( k xu H + k+ x u 0 H ). 0 k+ x u t (τ) L + k+ x u(τ) L dτ Note that there is no regularity-loss in the decay estimate and the energy estimate stated in Theorems and, respectively. This shows that the dissipative property of the system () is of the standard type. Next we consider the following modifications of the system (): n n (3) u tt B jk u xj x k + ( ) θ/ K jk u xj x k + Lu t = 0, (4) u tt j,k= n B jk u xj x k + j,k= j,k= n K jk u xj x k + ( ) θ/ Lu t = 0, j,k= where θ > 0 is a parameter. The introduction of the operator ( ) θ/ weakens the dissipation and this gives the following weaker decay estimate. Theorem 3 (Decay estimate). Under the conditions [A] [A4], the solution u to the problem (3) (or (4)), () satisfies the decay estimate k xu t (t) L + k+ x u(t) L C( + t) n/4 k/ u L + C( + t) n/4 k/ / u 0 L + C( + t) l/θ ( x k+l u L + x k+l+ u 0 L ). Theorem 4 (Energy estimate). Under the conditions [A] [A4], the solution u to the problem (3) (or (4)), () satisfies the energy estimate k xu t (t) H +θ/ + k+ x u(t) H +θ/ + C( k xu H +θ/ + k+ x u 0 H +θ/ ). t 0 k+ x u t (τ) L + k+ x u(τ) L dτ Note that we have the regularity-loss in the decay estimate in Theorem 3. A similar regularity-loss occurs also in the energy estimate in Theorem 4. This means that the dissipative property of the systems (3) and (4) is of the regularity-loss type. Such a regularity-loss property was also observed for other intersting systems. See the references below. References [] T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 6 (006), [] K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 8 (008),

68 Modulation Navier-Stokes Navier-Stokes Modulation. n. t u u + (u )u + π = 0 for t (0, ), x R n, (NS) div u = 0 for t (0, ), x R n, u(0, x) = u 0 (x) for x R n. (Modulation ). F Fourier, F Fourier. {ϕ k } k Z n C0 (R n ). ϕ C0 (R n ) supp ϕ { ξ R n ξ n }, ϕ(ξ k) = for any ξ R n k Z n, ϕ k (ξ) := ϕ(ξ k). < s <, q, σ, Modulation Mq,σ(R s n ). { } Mq,σ(R s n ) := f S (R n ) f M s q,σ (R n ) <, ( ) ( + k ) sσ F ϕ k Ff σ σ L q (R n ) for σ <, f M s q,σ (R n ) := k Z n sup ( + k ) sσ F ϕ k Ff L q (R n ) for σ =. k Z n, Lebesgue Sobolev, Modulation. Modulation, Feichtinger [],. Modulation Wang, Zhao, Guo [8], Schrödinger Navier-Stokes u 0 M 0,(R n )., Wang, Hudzik [7] Schrödinger Klein-Gordon., [3], Navier-Stokes u 0 M 0 q,σ(r n ) (, q, σ n/(n )), u 0 M 0 q,σ(r n ) (, q n, σ n/(n ))., [3] Navier-Stokes.. n, < s <, q, σ <,. s + n(σ ). σ div u 0 = 0 u 0 [M s q,σ(r n )] n T > 0, (NS) u [ C([0, T ], M s q,σ(r n )) ] n., q n, u 0,.. (), s = 0 σ n/(n ), [3].

69 (), s, s <. Bejenaru, Tao [], M,σ(R n ) ( σ < ) (NS). (3), s = M,(R n ), [5]. Miura [5] u 0 vmo gmo,,. M,(R n ) vmo gmo., s = + n(σ )/σ >,. (4), s 0 [4]. Koch, Tataru [4] u 0 BMO,,. M s q,σ(r n ) BMO if q n, + n(σ ) σ s 0., s = + n(σ )/σ > 0,., (NS), Banach. u(t) = e t u 0 t 0 P e (t τ) (u u)dτ., P := + ( ) div., < s 0.. s, s R, q, q, q, r, σ, σ, σ, ν. (i) q r e t u M s q,σ (R n ) C( + t) n ( r q ) u M s r,σ (R n ) ( ) (ii) s s e t u M s q,σ (R n ) C + t s es u M s q,σ (R n ) ) (iii) σ ν e t u M s q,σ (R n ) C ( + t n ( σ ν ) u M s r,ν (R n ) (iv) [Toft [6]] /q = /q + /q, /σ = /σ + /σ uv M 0 q,σ (R n ) C u M 0 q,σ (R n ) v M 0 q,σ (R n ). [] I. Bejenaru, T. Tao, J. Funct. Anal., 33, (006) [] H. G. Feichtinger, Technical Report, University of Vienna, 983. [3] T. Iwabuchi, preprint. [4] H. Koch, D. Tataru, Adv. Math., 57 (00), 35. [5] H. Miura, J. Funct. Anal., 8 (005), 0 9. [6] J. Toft, J. Funct. Anal., 07 (004) [7] B. Wang, H. Hudzik, J. Differential Equations, 3 (007), [8] B. Wang, L. Zhao, B. Guo, J. Funct. Anal., 33 (006), -39.

70 The two constants and tensors of the original Navier-Stokes equations ( ) Abstract The two-constants theory is the one now accepted for isotropic, linear elasticity. The original Navier-Stokes equations [ NS equations ] or Navier equations were introduced in deducing the two-constants theory. From the view of NS equations, we would like to report the deduction of tensor or equations by Navier, Poisson, Cauchy, Saint-Venant, and Stokes and the concurrence between each other. Especially, we would like to take up a subject for discussion on Saint-Venant, however his idea on tensor is, we think, an epock-making for taking the concurrence among three pioneers of NS equations and contributing to Stokes tensor and equations, which strengthens the frame of NS equations. A genealogy of tensor Navier[3, 4] : t e ij = ε(δ ij u k,k + u i,j + u j,i ), t f ij = (p εu k,k)δ ij ε(u i,j + u j,i ) (Euler) = Poisson[5, 6] = Saint-Venant[7] = Stokes[8] Cauchy[, ] : t e,f ij = λv k,k δ ij + µ(v i,j + v j,i ) molecular deduction non-molecular deduction Navier-Poisson-Cauchy-pattern Poisson-Saint-Venant-Stokes Poisson : t e ij = a (δ 3 iju k,k + u i,j + u j,i ), t f ij = pδ ij + λv k,k δ ij + µ(v i,j + v j,i ) Saint-Venant : t f ij = ( (P 3 xx+p yy +P zz ) εv 3 k,k)δ ij +ε(v i,j +v j,i ), (P 3 xx+p yy +P zz ) = p Stokes : t f ij = ( p µv 3 k,k)δ ij + µ(v i,j + v j,i ), Two constants Now, we would like to propose the uniformal methods to describe the kinetic equations such as : The partial differential equations of the elastic solid or elastic fluid are expressed by using one or the pair of C and C such that : in the elastic solid : u (C t T + C T ) = f, In the elastic fluid : u (C t T + C T ) + = f, where T, T, are the tensors or terms consisting our equations. For example, in modern notation of the incompressible Navier-Stokes equations, the kinetic equation with the equation of continuity are conventionally described as follows : u t µ u + u u + p = f, div u = 0. Moreover, C and C are described as follows : { { C Lr g S, S = g 3 C 3, C Lr g S, S = g 4 C 4, { C = C 3 Lr g = π 5 Lr g, C = C 4 Lr g = π 3 Lr g.

71 Table : The two contsnts in the kinetic equations no name problem C C C 3 C 4 L r r g g remark Navier π elastic solid ε [3] 5 dρ ρ 4 fρ ρ : radius 0 Navier π fluid motion of fluid ε 5 dρ ρ 4 f(ρ) ρ : radius 0 [4] 3 Poisson [5] 4 Poisson [6] 5 Cauchy [] Saint-Venant [7] Stokes [8] 3 Stokes [8] E elastic solid k π 5 K motion of fluid k 30 system of particles R π 3 π 3 K 6 G fluid ε ε 3 fluid µ µ 3 π 5 π 3 0 dρ ρ F (ρ) α 5 r 5 d. r fr dr α 5 r 3 fr ε r 3 d. 3 r fr dr C 3 = π 4π 5 = 30 ε r fr C 3 4 = π 4π 3 = 6 dr r 3 f(r) f(r) ±[rf (r) f(r)] 0 0 dr r 3 f(r) f(r) f(r) elastic solid A B A = 5B C and C are two coefficients, for example, k and K by Poisson, or ε and E by Navier, or R and G by Cauchy, and which are expressed by the infinite operator L ( 0 or ) 0 by personal principles or methods, where r and r are the functions related to the radius of the active sphere of the molecules, rised to the power of n. [] A.L.Cauchy, Sur les équations qui expriment les conditions de l équilibre ou les lois du mouvment intérieur d un corps solide, élastique ou non élastique, Exercises de Mathématique, 3(88); Œuvres complètes D Augustin Cauchy, (Ser. ) 8(890), [] A.L.Cauchy, Sur l équilibre et le mouvement d un système de points matériels sollicités par des forces d attraction ou de répulsion mutuelle, Exercises de Mathématique, 3(88); Œuvres complètes D Augustin Cauchy (Ser. ) 8(890), 7-5. [3] C.L.M.H.Navier, Mémoire sur les lois de l équilibre et du mouvement des corps solides élastiques, Mémoires de l Academie des Sience de l Institute de France, 7(87), (Lu : 4/mai/8. ) [4] C.L.M.H.Navier, Mémoire sur les lois du mouvement des fluides, Mémoires de l Academie des Sience de l Institute de France, 6(87), ( Lu : 8/mar/8. ) [5] S.D.Poisson, Mémoire sur l Équilibre et le Mouvement des Corps élastiques, Mémoires de l Academie royale des Siences, 8(89), , ( Lu : 4/apr/88. ) [6] S.D.Poisson, Mémoire sur les équations générales de l équiblibre et du mouvement des corps solides élastiques et des fluides, J. École Polytech., 3(83), -74. ( Lu : /oct/89. ) [7] J.C.Saint-Venant, Note à joindre au Mémoire sur la dynamique des fluides. (Extrait.), Académie des Sciences, Comptes-rendus hebdomadaires des séances, 7(843), ( Lu : 4/apr/834. ) [8] G.G.Stokes, On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, 849, ( read 845 ).

72 ( ) 0 < T < +, Ω R N (N =,, 3) Γ := Ω. []. n t = {K ( ) n λ( )n f} + µn( n f) in Q(T ) := Ω (0, T ), f t = δmf in Q(T ), m t = K ( ) m + C n C m in Q(T ), (P) 0 n + f, m 0, f 0, n 0 in Q(T ), n = 0, in Σ(T ) := Γ (0, T ), m = 0 in Σ(T ), n n(0) = n 0, f(0) = f 0, m(0) = m 0 in Ω., K i ( ) (i =, ), λ( ) (0, T ), µ, δ, C, C. n, n 0, m 0, f 0. [], n, m, f., (P) (QVI). n L (0, T ; L (Ω)), S : L (0, T ; L (Ω)) L (0, T ; L (Ω)). m t = K (t) m + C n C m a.e. in Q(T ), Sn = m m = 0 a.e. in Σ(T ), n m(0) = m 0 a.e. in Ω., T : L (0, T ; L (Ω)) W, (0, T ; L (Ω)) ( [Tm](x, t) := f 0 (x) exp δ t 0 ) m(x, s)ds for (x, t) Q(T ),., Λn := (T S)(n), (P)

73 (QVI). 0 n Λn a.e. in Q(T ), T (QVI) + 0 Ω T 0 ( n t Ω n(0) = n 0 in Ω. ) µn( n [Λn]) (n v)dxdt ( λ(t) { n [Λn] } + K (t) n) (n v)dxdt 0, for v L (0, T ; H 0(Ω)) with 0 v Λn a.e. in Q(T ), Definition. (S)-(S4) {n, m, f} (P). (S) n W, (0, T ; L (Ω)) L (0, T ; H 0(Ω)) (S) 0 n f a.e. in Q(T ), T ( ) n µn( n f) (n v)dxdt 0 Ω t T ( + λ(t) { n f } ) + K (t) n (n v)dxdt 0, 0 Ω for v L (0, T ; H 0(Ω)) with 0 v f a.e. in Q(T ), (S3) n(0) = n 0 a.e. in Ω, (S4) m = Sn, f = Λn, in L (Ω). Theorem. (A), (A). (A) 0 < K 0 K i (t), (i =, ) K 0, (A) n 0 H (Ω), 0 n <, m 0 H (Ω), 0 m <, f 0 H (Ω), 0 f <., (P) {n, m, f} [0, T ] (0 < T T ). [] Mark A.J. Chaplain and Alexander R.A. Anderson, Mathematical Modelling of Tissue Invasion, Cancer Modelling and Simulation, Chapter 0, A CRC Press UK 3 Blades Court Deodar Road London SW5 NU UK. [] Risei Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems, Banach Center Publication, Volume 86, (009), pp [3] Risei Kano, Nobuyuki Kenmochi, Yusuke Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publication, Volume 86, (009), pp

74 cone ( ) D = { x R N ; x 0 and x/ x Ω } Ω S N Ω D R N cone u t = u + K (x, t)v p, x D, t > 0, v t = v + K (x, t)u p, x D, t > 0, u(x, t) = v(x, t) = 0, x D, t > 0, u(x, 0) = u 0 (x) 0, v(x, 0) = v 0 (x) 0, x D, p, p p p > u 0 (x) v 0 (x) D D u 0 (x) = v 0 (x) = 0 K i (x, t) (i =, ) D (0, ) Ω Ω {ω n } n= Ω {ψ n (θ)} n= ( θ = x/ x ) {ω n} n= γ + γ(γ + N ) = ω () γ + = (N ) + (N ) + 4ω K i (x, t) (i =, ) C U, C L > 0 (A) K i (x, t) C U x σ i (t + ) q i (A) K i (x, t) C L x σ i t q i σ i, q i 0 (i =, ) α, α α = ( + σ + q ) + ( + σ + q )p, α = ( + σ + q ) + ( + σ + q )p p p p p ( [4]). K i (x, t) (i =, ) (A) ; (i) max{α, α } N + γ + (ii) H a = { ξ C( D); ξ(x) M x a ψ (x/ x ) for x D with some M > 0 } u 0 H a with a < α v 0 H a with a < α

75 () ( ). max{α, α } < N + γ + K i (x, t) (i =, ) (A) H a = { ξ C( D); 0 ξ(x) m x a ψ (x/ x ) for x D with small m > 0 } (u 0, v 0 ) H a H a with a > α, a > α (). () u(x, t) = S(t)u 0 (x) + v(x, t) = S(t)v 0 (x) + t 0 t 0 S(t)ξ(x) = D S(t s)k (x, s)v(x, s) p ds, S(t s)k (x, s)u(x, s) p ds, G(x, y, t)ξ(y)dy G(x, y, t) = G(r, θ, ρ, ϕ, t) (r = x, ρ = y, θ = x/ x, ϕ = y/ y Ω) cone D Levine-Meier [7] G(r, θ, ρ, ϕ, t) = ) t (rρ) (N )/ exp ( ρ + r ( rρ ) I νn ψ n (θ)ψ n (ϕ) () 4t t I ν ( z ν (z/) I ν (z) = ) k k!γ(ν + k + ) ν n = [(N ) /4 + ω n ] /, Γ(z) = k=0 0 n= s z e s ds [] T. Hamada, Nonexistence of global solutions of parabolic equations in conical domains, Tsukuba J. Math. 9 (995), 5 5. [] T. Hamada, On the existence and nonexistence of global solutions of semilinear parabolic equations with slowly decaying initial data, Tsukuba J. Math. (997), [3] T. Igarashi and N. Umeda, Existence and nonexistence of global solutions in time for a reaction-diffusion system with inhomogeneous terms, Funkcialaj Ekvacioj 5 (008), [4] T. Igarashi and N. Umeda, Nonexistence of global solutions in time for reaction-diffusion systems with inhomogeneous terms in cones, Tsukuba J. Math., Vol. 33, No. (009), to appear. [5] H. A. Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, J. Appli. Math. Phys. (ZAMP) 4 (99), [6] H. A. Levine and P. Meier, The value of critical exponent for reaction-diffusion equation in cones, Arch. Ratl. Mech. Anal. 09 (990), [7] H. A. Levine and P. Meier, A blowup result for the critical exponent in cones, Israel J. Math. 67 (989), [8] G. N. Watson, A Treatise on the Theory of Bessel Functions, nd Ed., Cambridge University Press, London/New York 944.

76 STRICHARTZ ESTIMATES FOR WAVE EQUATIONS WITH A POTENTIAL IN AN EXTERIOR DOMAIN : t u u + c(x)u = F (t, x), (t, x) R Ω, (P) u(0, x) = f 0 (x), t u(0, x) = f (x), x Ω, u(t, x) = 0, (t, x) R Ω, Ω R n \ Ω R n.. Assumption A. c(x) C 0 (Ω) Ω. (P) Strichartz. Strichartz, Smith and Sogge [9] (see also [, 6]) c(x) = 0, n = 3. Strichartz,, Strichartz. (P) Strichartz + c(x). Mochizuki [7, 8], Wilcox [0], Isozaki [], Kerler [5]. Iwashita [3], Iwashita and Shibata [4],. : p, q, r, σ admissible n 3, q, r > (n ) p, σ < n 3, q = n ( ), p r = n ( σ ), q + n p = r + n σ,. /r + / r =, /σ + / σ =. L (Ω) H = ( + c(x)) / χ(h) χ(ξ) C 0 (R 3 ) ξ R, ξ R +, R > 0 0..

77 . n = 3, R 3 \ Ω. Assumption A. p, q, r, σ admissible. (P) u : ( ) χ(h)u L q (R;L p (Ω)) C f k Ḣ(mp/) k (Ω) + F L r (R;L σ (Ω)), k=0, [ χ(h)]h δp/ u L q (R;L p (Ω)) ( ) C f k Ḣ(mp/) k (Ω) + F L r (R;Ẇ δp/,σ (Ω)), δ p = M p m p, m p = 4 k=0, ( ( ) ), M p p = σ 0, σ p 0 > 6. References [] N. Burq, Global Strichartz estimates for nontrapping geometry: A remark about an article by H. Smith and C. Sogge, Comm. Partial Differential Equations 8 (003), [] H. Isozaki, Differentiability of generalized Fourier transforms associated with Schrödinger operators, J. Math. Kyoto Univ. 5 (985), [3] H. Iwashita, L q L r estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in L q spaces, Math. Ann. 85 (989), [4] H. Iwashita and Y. Shibata, On the analyticity of spectral functions for some exterior boundary value problems, Gras. Mat. III 85 (989), [5] C. Kerler, Perturbations of the Laplacian with variable coefficients in exterior domains and differentiablility properties of the resolvent, Asymptotic Anal. 9 (999), [6] J.L. Metcalfe, Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle, Trans. Amer. Math. Soc. 356 (004), [7] K. Mochizuki, Spectral and scattering theory for second order elliptic differential operators in an exterior domain, Lecture Notes Univ. Utah, Winter and Spring 97. [8] K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 984. [9] H. Smith and C.D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 5 (000), [0] C.H. Wilcox, Scattering Theory for the D Alembert Equation in Exterior Domains, Lecture Notes Math. 44 (975).

78 ( ) BZ ([]) U t = D U + F (U), t > 0, x = (x, y) R. () U = U(t, x) R N N D N N F : R N R N F S ± R N U S ± () S ± () ϕ(x ct) (ϕ, c) Dϕ ξξ + cϕ ξ + F (ϕ) = 0, ξ R. () S + S () U t = ε D U + F (U), t > 0, x = (x, y) R. (3) 0 < ε D F ε (3) core (i) spiral tip S S + S + S (ii) ϕ (3) (i) core 0 spiral tip (ii) (3) (a) U S + c (b) S + U c (c) y c y S x S U = ϕ(x ct) x S + S x (a) (b)x (c)

79 S S S + S back front core S + S + S + S (S S + ) (S + S ) 0 core U t = DU xx + F (U; k) = L (U; k), t > 0, x R. (4) (4) k S(x) k c Pitchfork k > k c S(x) k < k c S(x) ϕ ± (x c ± t) k = k c η (0 < η ) U(t, x) = Ξ(r(t); η)(x l(t)) (4) l(t), r(t) ([]) { l = r + O( r 3 + η 3/ ), ṙ = (M r M η)r + O( r 4 + η (5) ). M, M (5) r l r Ξ(r(t); η)(x l(t)) ϕ ± (x c ± t) U t = ε D U + F (U; k c η), t > 0, x = (x, y) R. (6) x = Γ(t, σ) + λν(t, σ) Γ U ν σ = Σ(t, x), λ = Λ(t, x) (6) U (4) Ξ U(t, x) = Ξ(r(t, σ); η)(µ)+εw (t, σ, λ) µ = λ/ε (4) { V = εr + ε γ κ + O(ε 3 + r 3 + η 3/ ), D t r = (M r M η)r + εγ κ + ε γ 3 r σσ + O(ε + r 4 + η (7) ). V = V (t, σ) D t r = r t + Σ t r σ (7) (5) κ(t, σ) (7) core [] Fife, P. C., Understanding the Patterns in the BZ Reagent. J. Stat. Phys., 39 (985), [] Ei, S.-I., Mimura, M. and Nagayama, M., Pulse-Pulse Interaction in Reaction-Diffusion Systems. Physica D., 65 (00), [3] Hagberg, A. and Meron, E., Order parameter equations for front transitions: Nonuniformly curved fronts. Phisica D., 3 (998),