NONLINEAR SYSTEM OF SINGULAR PARTIAL DIFFERENTIAL EQUATIONS

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1 Jourl of Mhemcl Sceces: dvces d pplcos Volume 43, 27, Pges 3-53 vlble hp://scefcdvces.co. DOI: hp://d.do.org/.8642/ms_72748 OLIER SYSTEM OF SIGULR PRTIL DIFFERETIL EQUTIOS PTRICE POGÉRRD Mhemcs Lborory E 458 Uversy of L Réuo 2 rue Joseph Wezell, 9749 Frce e-ml: prce.pogerrd@uv-reuo.fr bsrc Ths pper eplores some oler sysems of sgulr prl dfferel equos wre he form DU Λ(, U + f (,, ζu, DU. Uder ssumpo o Λ, uque solvbly heorems re provded he spce of fucos h re holomorphc o ope se, dffereble wh respec o o rel ervl ], r ] d eedg o couous fuco. The suded sysems co Fuchs sysems. Iroduco k Cosder sysem of dfferel equos D U f (, U, where k s eger 2 d f s holomorphc eghbourhood of { } C. We kow h such problem geerlly does o hve lyc soluo, see, for emple, [2, 3, 9, 3]. y forml soluo belogs o Gevrey clss of order >, we could refer o [6, 7, 8, 2] mog ohers. The purpose of hs pper s o vesge oler sysems of he ype 2 Mhemcs Subec Clssfco: 35F5, 35F2, 35, 35B6, 352. Keywords d phrses: oler sysem, sgulr PDE, prlly holomorphc fucos. Receved December 4, Scefc dvces Publshers

2 32 PTRICE POGÉRRD DU Λ(, U + f (,, ζu, D U, (. where s rel dgol mr, Λ (, M ( C d f s fuco whch s couous wh respec o rel ervl [, r ] d holomorphc he remg vrbles. Ths regulry ssumpo lso ppers [, 4, 4, 5]. The ler prs of our equos re rregulr he sese of [5]. However, we re eresed soluos eedg couously. Uder resoble ssumpo o Λ, we show h (. hs uque soluo (, U(, holomorphc o ope se, dffereble wh respec o o rel ervl ], r ] d couous o [, r ]. To cheve our seme, we frs ver he D operor Λ(,, whch he leds us o fed po problem. We prepre some esmos h llow o pply he corco mppg prcple. Our m resuls re Theorem. d Theorem.4. Prlly holomorphc sysem We wll mke use of he followg oos:, (, R, C, D, D, {,, 2,...}, (,,,, D D D.. Seme of Resuls Gve ervl I R, ope se Ω C, Bch spce E k, d eger k, we deoe by C ( I Ω; E he lgebr of fucos u D l u : I Ω E such h for l k, he prl dervve : I Ω E ess, s couous d for y I, he mppg l Ω D u(, E s holomorphc. I s esly checked, usg Cuchy s egrl formul, h hs spce s sble by dffereo wh respec o d h we hve

3 OLIER SYSTEM OF SIGULR PDE 33 l l D D u D D u for y, l k. k, I Whe E C, he prevous spce wll be smply deoed by C ( Ω. Le us cosder sysem of prl dfferel equos of he form DU (, Λ(, U(, + f (,, ( ζu (,, ( DU (,, (. whch U (,, s he ukow, dg(,, u u s dgol mr wh rel coeffces ( ƒ, h re ll, Λ s upper rgulr mr of order whose coeffces re fucos of (, R C, ζu ( ζ u,, ζ u where ech ζ s fuco of (, R C ssfyg ζ (,, D U deoes he ƒ ƒ, -uple (( Du,, f s fuco of he vrbles R, C, y ( y C z (( z ƒ, ƒ C. ƒ,,, We ssume here re r > d ope eghbourhood Ω (resp., O of he org C ( resp., C y C z such h he coeffces of Λ, lke he fucos, ζ belog o ([, r ] Ω C d f, C ([, r ] ( Ω O ; C. Le Z be he zero se of he polyom P ( λ de( λi Λ(,. Theorem.. Suppose Z s cluded he hlf-ple R eλ <. The, here es ], r ] org r d ope eghbourhood Ω Ω of he, C such h sysem (. hs uque soluo U C ( ], r] Ω; C ;, C ([, r] Ω C d ecessrly D U C ([, r] Ω; C. Suppose ll, he Theorem. c be eeded s follows.

4 34 PTRICE POGÉRRD Le I deoe he se of ƒ, such h d ssume ƒ, \ I s o empy. Suppose ζ (, for I d ζ for I. (.2 For ech ƒ, \ I, le w be holomorphc fuco o Ω. We cosder sysem (. uder (.2 wh he l codos Impose u (, w ( for ƒ, \ I. (.3 The se I s uquely wre s D w ( oly f. (.4 I {, } wh < < <., p 2 p The, we ssoce wh he mr Λ ( λ, he squre submr of order p Λ M I, ( Λ ( λ., I k l < k, l< p Le us me Z he zero se of he polyom P ( λ de( λi Λ(,. We he hve he followg resul: Theorem.2. Suppose Z s cluded he hlf-ple R e λ <. The, here es ], r ] r d ope eghbourhood Ω Ω of he org C such h he problem (.-(.2-(.3 hs, uque soluo U ( ], r ] Ω; C C ([, r] Ω; C C d so D U ([, r] Ω; C., C Here s emple bou hs heorem.

5 OLIER SYSTEM OF SIGULR PDE Emple.3. For ll (, b C d (, β R, here s r > d ope eghbourhood Ω of he org C such h he problem 2 Du u + u2 + [ + u + ( u2 ], 2 β Du2 bu2 + [ + u2 + ( u ], u (,, 2, 2 2 hs uque soluo ( u, u C (], r] Ω; C C ([, r] Ω C 2 2 ; 2 C s resul ( D u, D u ([, r] Ω; C. ow we ur our eo o sysem of he form: D U(, Λ(, U(, + f (,, ( ζu (,, ( D U (,, (.5 where s posve rel umber d Λ s squre mr of order whose coeffces belog o he spce ([, r ] Ω. ble o se he followg resul. 2 C We re he Theorem.4. ( If <, ke ζ (,, d le W : Ω C be holomorphc fuco (wh D W ( oly whe. The, here es r ], r ] d ope eghbourhood Ω Ω of he org C such h sysem (.5 wh he l d U (, W (,, hs uque soluo U C (], r] Ω; C C ([, r] Ω; C, d D U C ([, r] Ω; C. (2 If d f he zero se of he polyom λ de( λi Λ(, s cluded he hlf-ple eλ <, R he, here es ], r ] ope eghbourhood Ω Ω of he org r d C such h sysem (.5, hs uque soluo U C ( ], r] Ω; C C ([, r] Ω; C, d D U C ([, r] Ω; C.

6 36 PTRICE POGÉRRD Remrk.5. Whe, we ob more geerl sysem of equos h Fuchs sysem; deed, we do o eed o ke ζ (, bu we smply hve ζ (,. Recll ([, ] h he Fuchs cse we hve lredy suded oler equos spces of holomorphc fucos. 2. Reformulo I order o prove Theorem., we frs rsform he problem. By wrg Λ Λ( + E, where E Λ Λ( ssfes ll he sme ssumpos s ζ, we my suppose h Λ s upper rgulr cos mr, mely, Λ ( λ M ( C, where λ f. (2., < Furhermore, R e λ < for,,. (2.2 e, we wll eed he followg resul: P Le D, λ C d le P be he elemery operor λ. Lemm 2.. Suppose R e λ <. Le r > d le Ω be ope eghbourhood of he org C. The, for every v C ([, r] Ω, he, equo P u v hs uque soluo u C (], r] Ω C ([, r] Ω. I ddo, D u C ([, r] Ω wh D u(,. s defed by Ths soluo λ e u(, ( P v(, e ϕ( v( τ, λϕ( τ dτ, where ϕ τ ( l f f >,. (2.3 Moreover, D u P D v for y.

7 OLIER SYSTEM OF SIGULR PDE 37, Proof. Le us sr wh he uqueess. Suppose u C (], r] Ω, C ([, r] Ω ssfes P u. Le Ω. Sce Dϕ (, he λϕ dervve of he mppg ( ( u, e s equl o zero o ], r ], so here ess c C such h λϕ ( ( u, c e for ], r]. Ths fuco hs fe lm oly f c becuse lm ϕ. I follows h u o [, ] Ω. r Cocerg he esece, we shll prove h he forml soluo (2.3 s of clss, (], r] Ω C ([, r] Ω. C Le be ope polydsk such h Ω; deoe M m v. [, r] The formul (2.3 c be wre equvlely λϕ ( ( e u, e v( σ, ( σ λϕ( σ dσ. (2.4 We frs cosder he cse >. The, λϕ( e ( Re λ ϕ( e, d we my se c λϕ( τ e m τ [, r] τ. λϕ( σ e The mppg ( σ,, ], ] ], r] v( σ, ( σ s couous, holomorphc wh respec o d bouded by Mc. Thus, from (2.4, we hve (], r] Ω. u C ow we show h u hs uque couous

8 38 PTRICE POGÉRRD eeso o fuco of he spce ([, r] Ω. Cosderg C Le b Ω. λϕ e ( λϕ( τ e dτ τ λ, we oce h λϕ ( ( ( e u, + v, b λ e [ v( τ, v(, b ] Le ε >. There ess > d λϕ( τ dτ. τ δ such h: le ( [, r] Ω b δ for, he we hve v(, v(, b ( R e λ ε, herefore,, wh δ ( ( ( Re λ ϕ,, ( + λ e u v b e ( Re λ ε d he seme follows wh ( Re λ ϕ( τ dτ τ ε, u (, v(, λ, Ω. (2.5 We e cosder he cse. The (2.4 s reduced o he well-kow formul v( σ, u (, dσ, (2.6 λ+ σ v( σ, where he mppg ( σ,, ], ] [, r] C s λ+ σ couous, holomorphc wh respec o d bouded by he egrble λ+ I drecly resuls h u ([, r] Ω fuco σ M σ. (2.5 ew sce C wh u (, v(, dσ λ+ σ v(, λ, Ω.

9 OLIER SYSTEM OF SIGULR PDE 39 I ll cses, we urlly hve D u P D v for y. If ], r], he he prl dervve of (2.3 ess d s gve by D u(, [ λu(, v(, ], +, whch leds o u (], r] Ω, D u C ([, r] Ω D u(, see (2.5; furher, he proof of our lemm. C wh, P u v o [, r ] Ω. Ths complees ow we c cosder he sysem of dfferel equos: D U(, ΛU(, + V (,, (2.7 where Λ s mr ssfyg (2.-(2.2 d V ( v,, s, ssumed o be of clss ([, r] Ω; C. v C For every ƒ,, deoe P D λ. The (2.7 s wre s P u λu + v,. (2.8 + Gve ƒ,, for every p ƒ, +, we se p G p { γ ( γ, γ ƒ, ; γ < γ < < γ }., p 2 p The crdl of p G s equl o he umber of ( p -combos from p ƒ +,, h s o sy (. From Lemm 2., oe hs u P v d, by fe duco, u + p p γ G cγqγv, where cγ p λγ d Q V v. lγl γ P P + γ γ γ p p l (2.9

10 4 PTRICE POGÉRRD These cosderos d Lemm 2. prove he followg resul. Lemm 2.2. Problem (2.7 hs uque soluo U ( u,, u, C (], r] Ω; C C ([, r] Ω; C d, D u C ([, r] Ω wh Du (, for y ƒ,. Furhermore, ech u s fe ler combo of up o 2 erms of he form P P P vγ, where γ2,, γ γ2 γ p p p ƒ, d p ƒ, +. Deog by (2. R V he soluo of problem (2.7, we defe, edomorphsm R of he vecor spce ([, r] Ω; C. C If we se U (, D U(, ΛU(,,.e., U RU, problem (. s covered o U F U, where F deoes he operor ( F U (, f (,, ( ζru (,, ( D RU (,. (2. The e seco ms o pply he corco mppg prcple o F Bch spce h we re gog o roduce. 3. Frmework d Esmes for he Operor F Gve mor fuco R { ξ} wh rdus of covergece φ + R >, le ρ be prmeer d le r > be such h ρ r < R. Defo 3.. We defe Ω ΩR, ρ, r { C ; < R ρr}, d B by he se of fucos u ([, r] Ω; E B φ, R, ρ, r C for whch here ess c such h

11 OLIER SYSTEM OF SIGULR PDE 4 [, r], u(, cφ( ρ + ξ, where ξ. (3. Ths precsely mes D u(, E cd φ( ρ, for ll d ll [, r]. Obvously, B s vecor subspce of ([, r ] Ω; E, C d he smlles c for whch (3. s ssfed s orm o B deoed by or smply f o cofuso s possble. φ, R, ρ, r Lemm 3.2. The spce B s Bch spce. Proof. Le ( U be Cuchy sequece B d le ε >. There ess such h, for ll, d ll [, r], ( U U (, εφ( ρ + ξ. (3.2 If K s compc subse of Ω, he we hve m U [, r] K U E εck, where C m φ( ρr + s < + sce he mppg φ( ρr + K K s couous o Ω. Ths shows h ( U s Cuchy sequece ([, r ] Ω; E C so coverges compcly o fuco U C, ([, r] Ω; E ; foror, for ll [, r] d, he sequece ( D U (, coverges o D U(,. By leg ed o fy o (3.2, we ge U U B d U U ε, herefore U B d ( U coverges o U B.

12 42 PTRICE POGÉRRD The followg lemm wll be useful o sudy he forhcomg operors. Le d λ <. If >, we se λϕ ( ( Sk e k τ λϕ( τ e dτ, τ k. (3.3 Recll h S λ. Whe k, gve Lemm 2., S k > eeds couously o wh S (. k Lemm 3.3. There ess c c ( λ such h, for ll > k S ( c, k. (3.4 k There ess c c ( λ, r such h, for ll [, r ], > k + Sk ( c, k. (3.5 k + k k Proof. Sce τ, we hve (3.4 wh c. To show (3.5, we oce h S ( τ ( ( τ λϕ k + λϕ e e λϕ e k λ k + τ dτ + τ dτ, k + 2 τ k + k + τ (3.6 s log s k +. We he cosder egers k >,.e., k whch s he smlles eger lrger h or equl o ; sce λ s <, esues h Sk ( k + c, k + where c + + >. Le k be eger. From (3.4, oe hs, for ll [, r ] k + + k k + (, where Sk c c r c c cr >. k + The resul follows wh c m( c, c.

13 OLIER SYSTEM OF SIGULR PDE 43 Remrk 3.4. Whe, he c m( 2, λ does o deped o r d (3.5 s vld for ll. Oherwse, oe c see h lm S ( +. + k Here re he esmes volvg operor P of Lemm 2.. Lemm 3.5. There ess c c(, λ, r > such h, for R >, ρ r < R d ([, r] Ω, ssfes u C he fuco P u C ([, r] Ω [, r], u(, φ( ρ + ξ [, r], P u(, cφ( ρ + ξ, d (3.7 [, r], u(, Dφ( ρ + ξ [, r], P u(, cρ φ( ρ + ξ. Proof of (3.7. For ll [, r] d, oe hs (3.8 D u(, D φ( ρ ( ρ k k D k + φ(, k! d from Lemm 2., D P u P D u, hece D P k D u(, ρ S ( k k k + φ(, k! where S k s defed by (3.3 whch we subsue ssero s cofrmed by (3.4. Proof of (3.8. s bove, we ge hs cse R e λ o λ. The k + + k D φ( k + D P u(, ρ Sk ( cρ ( ρ k! k k from (3.5, d he cocluso follows. k + + D φ( ( k +!

14 44 PTRICE POGÉRRD by We he cosder he epressos ζ RU d DR U. Le us deoe R he -h compoe of R. From ow o, we wll ke ( u,, u m u. E E C d Lemm 3.6. There ess c c(, Λ(,, r > such h, for R >, ρr < R d U B, we hve, for every ƒ,, ƒ, d [, r], R U (, c U φ( ρ + ξ, (3.9 [, r], D R U(, cρ U φ( ρ + ξ. (3. Proof. From Lemm 2.2, hs resul mus be esblshed wh erms lke P P P γ u γ 2 γ p p for ( 3.9, d erms lke P P P D u γ γ 2 γ p p for ( 3., where p ƒ, d γ 2,, γ p ƒ,. Le U B, he uγ p (, U φ( ρ + ξ for ll [, r]. Usg (3.7 p-mes, we ob (3.9. Oherwse, oe hs D u (, U Dφ( ρ + ξ for ll [, r]. γ p pplyg (3.7 ( p -mes d (3.8 oce, we ge (3.. Le us specfy herefer he mor fuco we shll employ. Gve R >, we cosder he ere sere (2. of [5] φ( ξ K p ( ξ R ( p + p 2, (3.

15 OLIER SYSTEM OF SIGULR PDE 45 2 where he cos K > s such h φ φ. Recll h φ lso ssfes he followg properes. Le η >, here ess c c( η >, such h ηr ( ηr cφ d ecessrly ηr cφ( ρ + ξ for ll [, r]. (3.2 ηr ( ρ + ξ Lemm 3.7. Le u ([, r] Ω C d c < be such h u(, cφ( ρ + ξ for ll [, r], he u s bouded by c o [, r ] Ω, he fuco ( u belogs o he spce C ([, r] Ω d c K + φ( ρ + ξ for ll [, r]. (3.3 u(, c Cocerg he operor F, we re gog o se up Proposo 3.8. There s > such h, for ll, he followg holds: here es R ], R ], ρ, r ], r ] wh ρ r < R such h he mppg F s src corco he closed bll B (, of he Bch spce B. Le us observe ow h we c wre f (,, y, z f (,, y, z g (,, y, z, y, z ( y y where For R >, we se ƒ, + h,,, y, z, y, z z, z, (, ƒ, ƒ, ([, r ] Ω O O ; C. g, h, C R { C ; m < R}, ƒ, ( (, (3.4

16 46 PTRICE POGÉRRD d {( y z C OR, C ; m y < R, m z, < ƒ, (, ƒ, ƒ, R}. We f, oce d for ll, η >, R > d R > such h ηr Ω d O O. Cosequely, he fucos f C ([, r ] ( ηr O R ; C d g, h C ([, r ] ( O O ; C re bouded, sy by cos M >. We pu, ηr ε( r R ( ( ] ] ] ]., m ζ (, [, r] ηr, for r, R, r, R ƒ, Ths fuco hs lm s ( r, R eds o (,. From Cuchy s equles d Lemm 3.7, oe hs ηr ζ (, ε( r, R c( η ε( r, R φ( ξ, ηr ξ d, gve φ( ξ φ( ρ + ξ (sce φ d ρ r < R, comes ζ, (, ƒ,. (3.5 ( c( η ε( r, R φ( ρ + ξ for ll [, r] Smlrly, we hve d R R f (,, y, z c( η Mφ( ρ + ξ, y z (3.6, g, h, y (,, y, z, y, z c( η Mφ( ρ + ξ y. (3.7 z, z

17 OLIER SYSTEM OF SIGULR PDE 47 Proof of Proposo 3.8. Le > d U B be such h U. I wh follows, y cos h does o deped o he prmeers we hve, R, ρ, r wll be deoed by c. From (3.9-(3.5 d (3., ζru (, cε( r, R φ( ρ + ξ, DRU (, cρ φ( ρ + ξ. (3.8 The, uder codo lke c ( r, R ε 2 d cρ 2, (3.9 follows from (3.6 d Lemm 3.7 h [, r ] Ω, belogs o ([, r ] Ω; C C d F U s well-defed o [, r], F U(, cφ( ρ + ξ. Ths proves he esece of > suffcely lrge ( > c, such h F ( B (, B (, for ll. (3.2 Le U B (,. s epled for f, f (3.9 s ssfed, we lso hve,, (,,,, g h ζr U D RU ζru, D RU cφ( ρ + ξ. Thece, from (3.4 d Lemm 3.6, we ob U U F F c( ε( r, R + ρ U U. Le. We frs ke ρ such h c ρ 2 d ρ c < 2. e, we choose ( r, R ], r ] ] R ], wh ρ r < R (for sce r R 2ρ, such h c ε ( r, R 2 d c ε( r, R < 2. Thus we hve (3.9, (3.2 d c ( ε( r, R + ρ <. We ge he desred resul herefrom.

18 48 PTRICE POGÉRRD 4. Proof of Theorem. U B By Proposo 3.8, he mppg F hs uque fed po C, (, C ([, r] ; C d U R U C (], r] ; C Ω ([, r] Ω; C Ω s soluo of (.. Le us show he uqueess, of hs soluo. Le U (], r] Ω; C C ([, r] ; C C be 2 Ω soluo of (., he U D U ΛU ([, r] ; C C s 2 2 Ω fed po of F. There s R such h ηr Ω, so, from Cuchy > equles, U B φ, R,,. r We ke m (, U φ, R,,. r Usg S, R, ρ, s, r wh Proposo 3.8 g, here es ] ] ] ] ρ s < S such h U U o [, s] Ω,.e., o [, s] Ω S, ρ, s coeced ope se. We shll prove h he rel umber { ], r] ; U U o [ ] Ω} m 2, s equl o r. For hs purpose, we ssume sce Ω s < < r d we se W ( U (, U (,. Ths fuco W s holomorphc o Ω d he fucos 2 U belog o ([ r] Ω;., C, C Thus, wrg U (, W ( + ( U (,,, we defe uquely C (], r] Ω; C C ([, r] Ω; C we fd h hese U d U re soluos of (( D + I U Λ( U + f (,, ζw + ζ( U, DW mely, + ( D, (4. U (( D + I U g(, ( U, ( D, (4.2, U where C ([, r] Ω O;, O O of he org g C s ope eghbourhood C y C z defed, from (3.9, les for

19 OLIER SYSTEM OF SIGULR PDE 49 m y < 2ε( r, R d m z, ƒ, (, ƒ, ƒ, < 2. (4.3 By rslo, (4.2 s reduced o wh g C ([, r ] Ω O; C h s o sy o sysem lke (.. s bove, here es s ], r ] d ope eghbourhood d 2 U cocde o [, ], [ + s ],.e., o [ + s ]., Ω Ω Ω of he org C such h U s Ω hece U d U 2 cocde o Ths llows us o coclude h, Ω r. 5. Proof of Theorem.2 Suppose U ( u,, s fuco ssfyg Theorem.2. Le u ƒ, \ I, we defe uquely u, C (], r] Ω by he relo Le us show h [, [, we observe h u u (, w ( + u (,. (5. u belogs ecessrly o ([, r] Ω. C s (, u (, τ v ( τ, dτ, where v D u C ([, r] Ω, herefore, u (, w ( + σ v ( σ, dσ. (5.2 If s ope polydsk such h Ω, he mppg ( σ,, ], ] [, r] σ v( σ, C s couous, holomorphc wh respec o d bouded by Mσ, where M m v. I follows h he ls [, r], egrl belogs o C ([, r] Ω ; ulmely we hve u C (], r] Ω C ([, r] Ω.

20 5 PTRICE POGÉRRD Deog by f he -h compoe of f, sysem (. s lso wre he form Du (, λ (, u (, + f (,, ( ζ U (,, DU(,, Iecg (5., we ob ƒ,. (5.3 Du λu I + λ I ( w + u + f (,, ζu, D U, for I, (5.4 d D u I I ow we se ( ( u + λ u + λ w + u + f (,, ζu, D U, whe I. (5.5 dg(,,, where f I, f o, d U ( u,, u wh u u for I. Po ou h U C,, (], r] Ω; C C ([, r] Ω; C. We deoe by ( M λ, he squre mr of order, where he λ re fucos of (, R C defed by λ (, λ (, f f f I, I d I d,. (5.6

21 OLIER SYSTEM OF SIGULR PDE 5 Usg mr represeo d epdg log colums, we oe h de( λi M (, ± P( λ [ λ ( ]. (5.7 I Le g ( g,, where g (, y ƒ λ ( w ( + y, I d δ ( δ, where δ ( ƒ, f f I, o. Equos (5.4 d (5.5 c he be wre s he followg sysem: D U M U + g(, δu + f (,, ζu, D U. Regrdg f, by pug W ( w,, wh w for I, oe hs f (,, ζ U, D U f (,, ζw + U, D W + D U h(,, U, D U, where w (, (, ƒ, f f I, o, d, sce ζ W d DW vsh he org of R C hks o (.2 d (.4, here es r > d ope eghbourhood Ω Ω (resp., O O of he org C (resp., C y C z such h, h C ([, r ] ( Ω O ; C. fer ll, leg f (,, y, y, z g(, y + h(,, y, z, U ssfes D U M U + f (,, δu, U, D U.

22 52 PTRICE POGÉRRD Cosderg he proof of Theorem., c lso be wre for such f,, U C, r Ω; C hece we hve esece d uqueess for (] ] ([, r] Ω; C C whch complees he proof of Theorem Proof of Theorem.4 s epled Seco 2, mr Λ c be cosdered cos. Sce he dgol mr I commues wh y mr of order, foror, wh verble oe, follows, fer chgg he oos, h s eough o sudy sysem (. for upper rgulr cos + mr Λ T ( C. By pplyg Theorem.2 for such mr d for ll equl o, we cheve our epeced resul. Refereces [] M. S. Boued d C. Goulouc, Sgulr oler Cuchy problems, J. Dff. Equ. 22 (976, [2] W.. Hrrs Ju, Y. Sbuy d L. Weberg, Holomorphc soluos of ler dfferel sysems sgulr pos, rch. Ro. Mech. l. 35 (969, [3] Y. Hsegw, O he l-vlue problems wh d o double chrcersc, J. Mh. Kyoo-Uv. (2 (97, [4].. Ledev, ew mehod for solvg prl dfferel equos, M. Sbork 22(64 (948, [5] T. Md, Esece d o-esece of ull-soluos for some o-fuchs prl dfferel operors wh -depede coeffces, goy Mh. J. 22 (99, [6] M. Myke d Y. Hshmoo, ewo polygos d Gevrey dces for ler prl dfferel operors, goy Mh. J. 28 (992, [7] M. Myke, ewo polygo d Gevrey herrchy he de formuls for sgulr sysem of ordry dfferel equos, Fukcl Ekvco 55 (22, [8] S. Ouch, Geue soluos d forml soluos wh Gevrey ype esmes of oler prl dfferel equos, J. Mh. Sc., Tokyo 2(2 (995,

23 OLIER SYSTEM OF SIGULR PDE 53 [9] J. Persso, Sgulr holomorphc soluos of ler prl dfferel equos wh holomorphc coeffces d olyc soluos of equos wh lyc coeffces, sérsque, Soc. Mh. Frce, 89(9 (98, [] P. Pogérrd, Sur ue clsse d équos de Fuchs o léres, J. Mh. Sc. Uv. Tokyo 7 (2, [] P. Pogérrd, Soluos rmfées à crossce lee de ceres équos de Fuchs qus-léres, Osk Jourl of Mhemcs 47 (2, [2] J. P. Rms, Théorèmes d dces Gevrey pour les équos dfféreelles ordres, Mem. mer. Mh. Soc. 48 (984, 296. [3] Y. Sbuy, Covergece of forml power seres soluos of sysem of oler dfferel equos rregulr sgulr po, Geomercl pproches o dfferel equos, Proc. 4h Schevege Cof., 979, Lec. oes Mh. 8 (98, [4] H. Thr, Uqueess of he soluo of o-ler sgulr prl dfferel equos, J. Mh. Soc. Jp 48 (996, [5] C. Wgschl, Le problème de Gours o lére, J. Mh. Pures ppl. 58 (979, g