and regular solutions of a boundary value problem are established in a weighted Sobolev space.

Transcription

1 Ieraoal Joral of heorecal ad Appled Mahemacs 5; (: -9 Pblshed ole Je 5 5 (hp:// do:.648/j.jam.5. he olvably of a New Bodary Vale Problem wh ervaves o he Bodary Codos for Forward- Backward ear ysems Mxed of Keldysh ype Mlvarae meso Mahammad A. Nrmammadov eparme of Naral ceces ad s eachg Mehods of Azerbaja eachers Ise (Brch Gba Azerbaja Bak eparme of Mahemacs ad eparme of Psychology of Khazar Uversy Azerbaja Bak Emal address: rmamedov@mal.r o ce hs arcle: Mahammad A. Nrmammadov. he olvably of a New Bodary Vale Problem wh ervaves o he Bodary Codos for Forward-Backward ear ysems Mxed of Keldysh ype Mlvarae meso. Ieraoal Joral of heorecal ad Appled Mahemacs. Vol. No. 5 pp. -9. do:.648/j.jam.5. Absrac: he solvably of he bodary vale problem for lear sysems of he mxed hyperbolc-ellpc eqaos of Keldysh ype he mlvarae doma wh he chagg me dreco are sded. Applyg mehods of fcoal aalyss -reglarzg coao by he parameer ad by meas of pror esmaes he exsece ad qeess of geeralzed ad reglar solos of a bodary vale problem are esablshed a weghed obolev space. Keywords: Chagg me reco Weghed obolev pace ysem Eqaos of Mxed ype Weak rog ad Reglar olo Forward-Backward ear ysems Mxed of Keldysh ype. Irodco Ieres of vesgaos of o-classcal eqaos arses applcaos he feld of hydro-gas dyamcs modelg of physcal processes (see e.g. [6] [7] [] [] [3] [8] [][] ad he refereces gve here. No-classcal model s defed as he model of mahemacal physcs whch s represeed he form of he eqao or sysems of paral dffereal eqaos ha does o f o oe of he classcal ypes-ellpc parabolc or hyperbolc. I parclar o-classcal models are descrbed by eqaos of mxed ype (for example he rcom eqao degeerae eqaos (for example he Keldysh eqao or he eqaos of obolev ype (e.g. he Barebla-Zsol-Kacha eqao he eqao of he mxed ype wh he chagg me dreco ad forwardbackward eqaos. I rece years he aeo of may scholars has red o he sdy of well-posed bodary vale problems for oclasscal eqaos of mahemacal physcs parclar for forward-backward eqaos of he parabolc ype ( e.g. [6] [9] ad he refereces gve here. I he heory of bodary vale problems for degeerae eqaos ad eqaos of mxed-ype s a well-kow fac ha he well-posedess ad he class of s correcess esseally deped o he coeffce of he frs order dervave (yoger member of eqaos (e.g. [3] [4] [8][9] [4][8] ad he refereces gve here. I he paper [8] was rodced he ew called Fchera's fco order o defy sbses of he bodary of he doma where he bodary vale problem for sch kd of eqaos s posed where s ecessary or o o specfy he bodary codo. A amely bodary codos deped from sg of he Fchera's fco Ф(x. I he work [3] (see Chaper p ad Chaper 3 p ad papers [4][] ew bodary codos (so called ype of problem E whch some par of he bodary shall be exemp from he bodary codos were sded. I he paper [7] [8] varos rchle problems whch ca be formlaed for eqaos of Keldysh ype oe of he wo ma classes of lear ellpc hyperbolc eqaos were vesgaed. Ope bodary codos ( whch daa are prescrbed o oly par of he bodary ad closed bodary codos ( whch daa are prescrbed o he ere bodary were boh cosdered. Emphass s o he formlao of bodary codos for whch solos ca be show o exs a approprae fco space.

2 Mahammad A. Nrmammadov: he olvably of a New Bodary Vale Problem wh ervaves o he Bodary Codos for Forward-Backward ear ysems Mxed of Keldysh ype Mlvarae meso Bodary vale problems for eqaos of mxed hyperbolc-ellpc ype wh chagg me dreco had bee sded deals []-[].Grea dffcles come o beg he vesgao of lear sysems of degeerae ellpc ad hyperbolc eqaos. I mahemacal modelg paral dffereal eqaos of he mxed ype are sed ogeher wh bodary codos specfyg he solo o he bodary of he doma. I some cases classcal bodary codos cao descrbe process or pheomeo precsely. herefore mahemacal models of varos physcal chemcal bologcal or evromeal processes ofe volve o-classcal codos. ch codos sally are defed as olocal bodary codos ad reflec saos whe he daa o he doma bodary cao be measred drecly or whe he daa o he bodary deped o he daa sde he doma. I hs case bodary codo parclarly maybe gve for some par of he bodary wh dervaves. Coseqely hs paper cosdered bodary codos correspods o he so- called well-posed bodary codo of Fchera s ad Keldysh a applcao ew approaches form preseao. I mercal mehods for solvg hese eqaos he problem of sably has receved a grea deal of mporace ad aeo. Fally he problem for he sysem of eqaos of mxed hyperbolc-ellpc of Keldysh ype cldg propery of chagg me dreco has o bee exesvely vesgaed. herefore prese paper we wll sdy hs problem.. Problem aeme Noao ad Prelmares e G be a boded doma he Ecldea space R of x = x... x cldg a par of hyper-plae he po x = ad wh smooh bodary { } = { < } G = G x > G C G G x. he bodary of G x = for x > ad cosss of a par of hyper-plae smooh srface G. Aalogcally he bodary G cosss of a par of hyper-plae x = for x < ad smooh srface G.Assme ha = G > ; = G where Γ = s a bodary of doma. I he doma cosder he sysem of eqaos: ( ( ( υ = k k x x a x x ( a x υx b ( x b ( x υ c ( x c ( x υ = f ( x ( ( ( υ = k υ xυ a x x a ( x υx b ( x b ( x υ c ( x c ( x υ = f ( x (. where he x s aplace operaor x =... x x Everywhere we wll assme ha he coeffces of he sysem of eqao (. are sffcely smooh. Moreover he codos ( k ( > for ( = ; x x = ( x... x G R x k x < for are sasfed. As far as s kow ha qadrac form of he eqaos of sysem (. chages he hs sysem coas paros degeerag ellpc degeerag hyperbolc mxed ad compose ype dffereal eqaos a he same me cldg chagg dreco me of varable he doma. Assme he oaos: Γ = {( x Γ : x > = } Γ = {( x Γ : x < = } Γ = {( x Γ : x > = } Γ = {( x Γ : x < = } = G [ ] = G [ ] = { x > } = { x < }. he bodary vale problem Fd he solo of sysem eqaos (. he doma sasfyg he codos: Γ = Γ = = _ (. Γ υ Γ = υ Γ = υ = _. (.3 Γ Remark.. I hs sao he Γ Γ Γ Γ se are carrers as bodary codos whch depedg o he sgs ( b ( x b ( x k ( = whe he b k ( δ < ( ( b k ( δ < codos ms be sasfed everywhere. hs dcaed bodary vale problems for he sysem of eqaos (. are pg he form (. (.3 ad seg he bodary codos (. (.3 correspods o ad cosse wh he approach ced above(e.g. [3] [4] [8][7] []ec.. By he symbol C we deoe a class of wce coosly dffereable fcos he closed doma sasfyg he bodary codos (. ad (.3 by H ( H ( obolev s space wh weghed spaces obaed by he class C whch s closed by he orms:

3 Ieraoal Joral of heorecal ad Appled Mahemacs 5; (: -9 3 = H x ( k ( x d = H xx x x ( k ( x k ( x k ( x d k respecvely. Irodce he space orm (e.g. [][5]: α k W ( K α k = = dxd W obolev s wh α α α α = α... α =... = =. x ce k ( x for x by he obolev s embeddg heorems [][5] he fcos from he spaces H ( wll sasfy he bodary codos (. (.3. emma.. Assme ha he followg codos ( (a b ( x k ( δ < for = x G ; ( (b b ( x k ( δ < for = (c ( a x M k ( x ( a kx 3 ( x G ; a x M k ( x M k ( x where M M M 3 M- are sffcely large cosas ( (d c( x a b( x ( x (e a M k x (f xc ( x for x ( c( x ' = c ( x ( x or αc αc αc cα ( x are holds. If α ( x = x M 4 where cosa 4 he for all fcos ( eqaly x υ ( x M s sffcely large C he followg ( υ αd αυ ( υ d α ( υ d υ ( υ d m υ H (.4 holds re. Where he cosa m s o depede from fcos ( x ad υ ( x. Proof. e ( x υ( x C ad cosder he followg egrals: ; J = ( υ αd α ( υ d. J = ( υ αυd ( υ αυd Afer egrao by pars ad allowg for bodary codos of (. (.3 ad akg o acco oegave bodary egrals we ge: ( ( ( J = ( υ αd α ( υ d ( b k ( α αk ( ( a α ( kα x x kαx a ( x υ α x b αυ } ( α α αυ c c c d k ( α ( x dx G x x ( k ( x α ( x dx k ( x α ( x dx J = αυ ( υ d αυ ( υ d ( ( ( ( b k α αk υ αx υυx υx α αc αc υ ( ( ( a x αυ a υx αυ c υ bυ d ( k ( α ( x dx G G G ( k ( α ( x dx. ( k ( α ( x dx Now sg eqales of Cachy-Byakovsky Pocare ad codos of emma. for coeffces of sysem eqaos (. ad akg o acco he fac ha ( he coeffces k = are homogeeos o he bodares ad he smmarzg esmaes for J ad J obas he valdy of eqaly (.4. efo.. We say ha ( x ad υ( x are reglar solo of problem ((.-(.3 f he fcos ( x υ( x H ( sasfy eqao of (. almos everywhere doma. We eed o seek ew srcre sep of proof or oclasscal mehod for solvably of problem ((.-(.3. For

4 4 Mahammad A. Nrmammadov: he olvably of a New Bodary Vale Problem wh ervaves o he Bodary Codos for Forward-Backward ear ysems Mxed of Keldysh ype Mlvarae meso hs reaso frs of all beg o formlae he heory of exsece frs ake he decayg sysem eqaos he followg form: = x = k ( k ( x a b c f ( x. (.5 ( υ = υ υ υx υ υ = k ( a b c f ( x. (.6 For provg solvably of he problem ((.5 (. we se he mehod of reglarzao ad s he fac ha he hyper-plae x = s a characersc for eqao (.5. herefore we ca cosder he bodary vale problem ((.5 (. he followg form: Bodary vale problem. Fd he solo of eqao (.5 he doma sasfyg he bodary codos = Γ Γ = Γ = s =. (.7 Bodary vale problem. Fd he solo of eqao (5 he doma sasfyg he bodary codos = Γ Γ = = Γ s =. (.8 By С ( С ( we deoe a class of fely dffereable fcos he closed domas sasfyg he bodary codos (.7 ad (.8 respecvely. 3. Uqeess olo of Problem ((.- (.3 pace H ( heorem3.. Assme ha he codos of emma. hold he he reglar solo of he problem ((.-(.3 s qe. Proof. Ideed le υ ad υ be wo solos of problem ((-(3 whch s sasfyg he sysems eqaos (. e = υ = υ υ. he he fcos υ wll be sasfyg eqaos: ( υ = ad ( υ = he doma. ppose ha υ be sasfed. e s ake seqece fcos { }{ } υ C =... ec sch ha H for υ υ. By he eqaly of (.4 we have H for ( υ ( υ ( υ H H m where he cosa m depede from he fcos ( x ad υ ( x ( υ ( υ. herefore we ca asser ha ( υ ( υ By he vre of eqaly of (.4 we have for. ( ( υ α ( ( υ α ( ( υ αυ ( ( υ αυ m ( υ. H ( H ( ( ( ( Hece passg o lm as las eqaly we ge H. O he oher sdes we have υ space H H ( υ υ υ υ H ( H ( for. Hece υ. ha s proof of heorem 3.. Now we eed he proof of solvably problem ((.- ( he Exsece Weak (Reglar olo of Problems ((.5 (.7 ((.5 (.8 emma4.. Assme ha he codo (a-(c (e (f of emma. are holds he for ay fcos ( ( ( x C x C followg eqales ( ( α m ( H ( ( ( α m ( H (4. are vald. Proof. e s cosder he egrals: ( αd = fαd ( αd = f αd. Afer egrao by pars allowg for bodary codos ad akg o acco oegave bodary

5 Ieraoal Joral of heorecal ad Appled Mahemacs 5; (: -9 5 egrals we ge b x k k a x k x ( ( ( ( α ( ( α α ( α ( ( α ( x x [( ] } ( kα x c α c α d k ( α ( x dx ( x C ( G b x k k a x k x ( ( ( ( α ( ( α α ( α ( ( α ( x x [( ] } k α x cα c α d Hece sg Cachy-Byakovsky ad Pocare eqales akg o acco codos (a (c (e(f of emma. for chose cosas wh he fac ha coeffces k s homogeeos o he bodares he we ge he rh of eqales (4.. Moreover sg eqaly Holder s we have f ( H ( m f m ( x c ( G ( H ( where he cosas m m are depede from he fco ( x.ha s proof of emma 4. efo4..he ad we sae for he bodary vale problem ( k ( α ( x dx fco. ( x c (. ( ( ( x x H x H s sad o be reglar solo of problem ((.5 (.7 ((.5 (.8 f s geeralzed solo sasfes almos everywhere eqao (.5 doma (. emma 4.. e he codos of emma 4. be flflled. he reglar solo of problem ((.5 (.7 ((.5 (.8 s qe. Proof. he emma4. s proved smlarly way o he emma. ad emma 4..ce he eqao of (.5 s also degeerag he de o reglarzg effec o apply for eqao (.5 I he doma reglarzed eqao of mxed ype ( = k ( ( ( k b a ( c = f ( x (4. = = = = =. Γ Γ = (4.3 Γ x Aalogcally we wll cosder he followg bodary vale problem x x ( = k ( ( ( k b a ( c = f ( x (4.4 = = = = Γ =. Γ = (4.5 Γ Proceedg from he kow resls of he papers [4] we ca affrm he followg proposo. Remark4.. If he codos of emma 4. emma 4. ad b ( x k δ < ( x are sasfed he for ay rgh-had sde f ( x f ( x ( ( f ( x f ( x ( here exss a qe solo of bodary vale problem (4. (4.3 ((4.4 (4.5 from he space W ( ( W ( followg esmaes ad hs solo allows f f ( m3 W ( where he cosas m 3 ad m4 are depede of he fco ( x. Proof of hs proposo proves smlarly o emma. emma4. ad heorem3.. heorem 4.. (o he solvably of problem ( (.5 (.7 Assme ha he codos of emma4. hold. If k k M k ( x f ( x f ( x ( x xj f f ( m4 W ( (4.6 ( δ < for b x k x j =... are sasfed he here exss a qe reglar solo of problem ((.5 (.7 from he space H (. heorem 4.. (o he solvably of problem (.5 (.8 Assme ha he codos of emma. hold. If k k M k x f x f x ( x xj

6 6 Mahammad A. Nrmammadov: he olvably of a New Bodary Vale Problem wh ervaves o he Bodary Codos for Forward-Backward ear ysems Mxed of Keldysh ype Mlvarae meso ( δ < b x k x j =... are sasfed he here exss a qe reglar solo of problem ((.5 (.8 from he space H (. Proof of heorem4. ad 4. he followg a pror esmaes f m ( 5 k x d f m ( 6 k x d (4.7 hold for he fcos ( x W ( ( x W ( ad beg he solo of bodary vale problems ((4. (4.3 ((4.4 (4.5 respecvely. Where he cosas m 5 ad m6 are depede of ad ( x. he proof of hese saemes easly obaed by egrao by pars ad sg he Cachy eqaly. Frher for obag he secod a pror esmao we ake he fco ξ ( sch ha ξ( for ( η > η > ; η ξ for η η for ξ. he we cosder he fco W ( x ξ ( ( x Obvosly he fco W ( x =. wll sasfy he eqao ξf k ξ k ( ξ F W = =. (4.8 Hece by vre of eqales (4.6 ad (4.7 he se of F x. fcos { } are formly boded space η I oher sde doma η = x < < he eqao W = F belogs o ellpcal ype of eqao. herefore mlply eqao of (4.8 by W egrae by pars he doma allowg bodary codos se he Cachy-Byakovsky eqaly we ge F m ( 7 W W W k Wx k Wxx d where cosa m 7 s depede of ( x. Now le s cosder he fco ξ ( C ( sch ha ( < < η ξ ( η < <. ce ξ ( ad ξ ( ξ ( he akg φ ( x ξ ( ( x see ha he fcos φ ( x sasfy he eqao φ ξ ξ ξ ξ for =. I s easy o = f x k ( x k ( = Φ x. (4.9 Hece clde ha he fcos Ф ( x Ф ( x herefore we ca ake fe dfferece. are formly boded wh respec o he space φ φ = I s easy o see ha he fco φ ( x sasfy he eqaos h ( x h φ ( x φh = ξ ξ ξ = Φh h f k k ( x Usg he resls o smoohess of he solo of problem ((4. (4.3 ad a pror esmaes (4.6 (4.7 ad passg o lm as h he obaed eqales ϕh m ( 8 φhh φ φh k φxh k φxx d ad esablshg relao bewee he fcos f ( x ad Ф ( x we ge f f m9 k x (. k x d x C From he represeaos of fco ( x φ ad from eqao (4. by sadard esmao mehod we ge

7 Ieraoal Joral of heorecal ad Appled Mahemacs 5; (: -9 7 xx k. Hece by sadard compacess mehod we ca coclde ha ( x s geeralzed solo of problem ((.5 (.7 ad belogs o he space H ( ad a he same me sasfy he eqao (.5 ad codo (.7 almos everywhere. I a smlar way repeag all he seps carred o for he doma for also we ca esablsh ha problem ((.5 (.8 has a geeralzed solo ad belogs o he space H (. 5. Ma Resl of Exsece ad Uqeess rog (Reglar olo of Problems ((.5 (.7 ((.5 (.8 efo5.. (followg [][4][] he fco ( ( ( x H ( x H s sad o be a srog solo of bodary vale problem ( ( (( (3 f here exss a seqeces of fcos { } C ( ({ } C ( sch ha eqaly ( ( H ( lm f x = lm = s flflled he doma as well f sead of he doma ake. he followg heorem o he exsece of srog solo holds. heorem 5.. Assme ha he codos of emma. hold. If kxkx M k x j =... j ( δ b k < x are sasfed he for ay fco ( f ( f here exss a qe srog solo of bodary vale problem (.5 (.7 from he H. space H (for he problem (.5 (.8 from Proof. From hese heorem 3. heorem 4. heorem4. x solo of problem ((.5 (.7 here exss ( x solo of problem ((.5 (.8 he domas ad respecvely ad belogg respecvely o he H. he by he cosrco spaces H ( ad of sch spaces here exss seqeces { } C ( ({ } C ( sch ha lm = lm =. H ( H ( From he obvos eqaly H ( m m H ( follows ha { ( } { ( } f ( f ( f ( f ( for. for.. hs sppose ha he reglar solos ad are srog solo. We are cosrcg he seqeces of fcos f W ( f W ( { f} f ( { f} f sch ha ( for.he for he fcos f ad f here exss srog solo problem of ((.5 (.7 ad ((.5 (.8from he space H ad H ( o by eqaly of emma. we have f m H ( Hece we ca clde ha H ( f respecvely. m. H ( H for ad hese fcos are srog of problem ((5 (7 ad((5 (8 respecvely. 6. he olvably of Problem ((.5 (. heorem 6.. (Glg solos he spaces Assme ha H H holdhe he cosrced fco ( x ( ( x x = x x x H =. wll also be from he class (6. Proof. he heorem6. proved exacly ad smlarly way o he Remark6. (e.g. []. hs we have he proof of he followg heorem accordace esseally a combao of he proof of heorems ad emmas ad heorem 6.. Now we ca proof he ma heorem of solvably of problem ((.5 (.. heorem6.. (O he solvably of problem ((5 ( Assme ha he codos of emma. emma4. ad heorems are sasfed he for ay f f here exss a qe geeralzed fcos solo of problem ((.5 (. from he space H (. Proof. ce o he base of heorem 4. heorem 4. ad x heorem 5. here exss a qe solo ( x space of problems ((.5 (.7 ad ((.5 (.8from he H ad H ( respecvely. he fco ( x whch s cosrced by formla (6. wll also be

8 8 Mahammad A. Nrmammadov: he olvably of a New Bodary Vale Problem wh ervaves o he Bodary Codos for Forward-Backward ear ysems Mxed of Keldysh ype Mlvarae meso from he class ( x H ( ad a he same me s geeralzed solo of eqao (.5 moreover he x x s srog geeralzed solo fcos ad of problem ((.5 (..Coseqely meas ha he srog ad weak solos of correspodg problems are dey (e.g. [][]. I follows ha he problem ((.5 (. s solvably. he qeess of problem ((.5 (. follows by meas of eqaly of emma.. ha s proof of heorem 3.. Aalogcally he exsece srog solo of problem ((.5 (. from he space H ca be proved. 7. O he olvably of Problem ((.- (.3 For provg he solvably of problem ((.-(.3 we se he mehod of coao by parameer. I holds. heorem 7.. (o he solvably problem of (.6 (.3 Assme ha he codos ( δ (7. c ( x a ( x b ( x < x ( a x M k ( x (7. holds he for ay fcos of f ( x f ( x ( here exss qe solo of problem ((.6 (.3 he H. ( case sead of codo of (7. space replaced smalles of coeffce ( a x he here exss W. qe solo of problem ((.6 (.3 space Proof. By vre of codo (7. ad ( b k δ < he operaor ( ( = x υ k υ υ a υ b υ c υ ( s coercve. ce he coeffce of k ( s sg fxed (accordg o [4] he here exss qe solo of υ( x W problem ((.6 (.3 space W (. If he (accordace o [5] ay solo of problem ((.6 W. Aalogcally (.3 wll be eleme of space repeag all he seps carred o for he solo υ( x H ( ad also we ca esablsh ha problem ((.6 (.3 has geeralzed solo f he codo (7. s sasfed. herefore he heorem 7. s proved. Now we ms prove solvably of problem ((.-(.3. e M = K A B C N = x ( k where K = ( k ( a A = ( B a P x Q R b = b ( k c a b C = P = ( Q = c a b c f R = = f =. c υ f he he sysem eqaos ( ca be wre he form: = M N = f. (7.3 heorem7.. Assme ha he codos of emmas ad heorems moreover f f f f ( f ( x = a M k x are flflled he here exss a qe solo of problem (.-(.3 space H (. I case of a s smalles he here exss a qe solo of problem ((.-(.3 from he space H ( W (. Proof. Mlply he eqao (7.3 by he vecor η = α υ doma afer egrao by pars ad sg he Cachy eqaly allowg for bodary codo (by aalogcally aco o he emma. we ge he followg esmaes m or m ( H ( ( H ( W ( (7.4 Now le H - s he space of vecor fco φ = ( φ φ sch ha ( φ φ φ ad φ ( x =.he orm of space H s defed by φ = φ φ From he resls of he heorems follows he followg a pror esmaes m M or 6 H ( m M 7 H ( W ( (7.5 where m m 6 m7 cosa are o depede from ( x.i remas o show ha aalogcal esmaes(7.4 (7.5 are also have o for operaor. Ideed we may rewre M = N he 8 ( or m9 ( N H ( W ( m N H ( are vald. Now we cosder he se of eqaos: = M τn where τ. Obvosly he τ followg a pror esmae s formly boded respec o parameer of τ : m where m τ H ( depede from parameer τ ad ( x. Oher sde for τ = we have = M. I hs case were cosdered problem s solvable. Noce ha f τ = he =.he as well as kow mehod of coao by parameer (for example see [5]ec. wh he sadard approaches he

9 Ieraoal Joral of heorecal ad Appled Mahemacs 5; (: -9 9 solvably of problem (. (.3 (.4 ca be proved. Ahor sggess he ope problem (7.6 (7.7: K ( K ( x a( x. b( x c( x = f ( x (7.6 xx x > K ( xk x < x x ( α β α < β > ρ > ρ he coeffces of eqao (7.6 are sffcely smooh. he bodary vale problem. Fd he solo of eqao = α x β sasfyg he (7.6 doma { } codos: = = = =. ( Coclso x= α x= β = x> = x< he solvably of he bodary vale problem for lear sysems of he mxed hyperbolc-ellpc ype he mlvarae doma wh he chagg me dreco are sded. he exsece ad qeess of geeralzed ad reglar solos of a bodary vale problem are esablshed a weghed obolev space. I hs case applyg dea of resl works (e.g. [] [][] ad heorem prove ha weak ad srog solos of he bodary vale problem for lear sysems eqaos of he mxed hyperbolc-ellpc ype he mlvarae doma wh he chagg me dreco are dey.. Refereces []. araso O weak ad srog solos of bodary vale problems Comm. PreAppl. Mah. 5 ( MR7 #46. [] Adams R. obolev paces ecod Ed. Academc Press Elsever cece 3. [3] A.V. Bsadze ome Classes of Paral ffereal Eqaos Gordo ad Breach: New York 988. [4] V.N. Vragov Bodary Vale Problems for he Noclasscal Eqaos of Mahemacal Physcs Novosbrsk: NU 983. ( Rssa. [5].. obolev Applcaos of Fcoal Aalyss Mahemacal Physcs; Eglsh rasl. Amer. Mah. oc Provdece R.I [6]. Bers Mahemacal Aspecs of bsoc ad rasoc Gas yamcs rveys Appled Mahemacs vol. 3 Joh Wley & os Ic. New York; Chapma & Hall d. odo 958. [7] F. I. Frakl eleced Works Gas yamcs Naka Moscow Rssa 973. [8] G. Fchera O a fed heory of bodary vale problems for ellpc-parabolc eqaos of secod order Bodary Problems ffereal Eqaos Uv. of Wscos Press Madso pp [9] Fredrchs K. O. ymmerc posve lear dffereal eqaos. Comm. Pre Appl. Mah. ( [] Fredrchs K. O. he dey of weak ad srog dffereal operaors. ras. Amer. Mah. oc. 55 ( []. Cac B.. Keyfz E. H. Km Mxed hyperbolc-ellpc sysem self-smlar flows Bol. oc. Brasl. Ma.(N.. vol. 3 o. 3 pp [] C. omglaa. sseme smmerc d eqazo a dervae parzal A. Mah. Pre e Appl. II v. pp [3] B. P U Problem Valor ol Cooro Por eqazoal a ervave Pzzal ef erro Arde Co Paro Prcpale po Compose Red. em. Fas. c. Uv. Gaglaro [4] M.V. Keldysh O cera classes of ellpc eqaos wh sglary o he bodary of he doma okl. Akad. Nak R 77 pp [5] O.A.adyzheskaya (Eglsh raslao: he Bodary Vale Problems of Mahemacal Physc Appled Mahemacal ceces 49. prger- Verlag New York 985. [6] erseov.a. Abo a forward-backward eqao of parabolc ype. Novosbrsk Naka 985 ( Rssa. [7].H. Oway he rechle Problem For Ellpc-Hyperbolc Eqaos of Keldysh ype ecre Noes Mahemacs IN pr edo: prger Hedelberg ordech odo New York. [8]. po C.. Morawez K.K. Peye O closed bodary vale problems for eqaos of ellpc-hyperbolc ype Comm. Pre. Appl. Mah. vol. 6 pp [9] a k N.A Novkov V.A.ad Yoeko N.N. Nolear eqaos of varable ype. Novosbrsk 983 Naka. [] C.. Morawez A weak solo for a sysem of eqaos of ellpc-hyperbolc ype Comm. Pre Appl. Mah. Vol. pp [] Nrmamedov M.A. O he solvably of he frs local bodary vale problems for lear sysems eqaos of oclasscal ype wh secod order. Joral oklad (Adgey Ieraoal Academy Nalchk 8 v. p.5-58 ( Eglsh [] Nrmammadov M.A. he Exsece ad Uqeess of a New Bodary Vale Problem (ype of Problem E for ear ysem Eqaos of he Mxed Hyperbolc-Ellpc ype he Mlvarae meso wh he Chagg me reco. Hdav Pblshg Cooperao Absrac ad Appled Aalyss Volme 5 Research Arcle I pp. - ( Eglsh