Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations
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1 Ieraioa Jora of Moder Noiear Theory ad Aicaio -6 h://ddoiorg/46/ijmaa Pbihed Oie March (h://wwwcirorg/jora/ijma) Effec of Weigh Fcio i Noiear Par o Goba Sovabiiy of Cachy Probem for Semi-Liear Hyerboic Eaio Akbar B Aiev Aar A Kazimov Iie of Mahemaic ad Mechaic of NAS of Azerbaija Bak Azerbaija Nakhchiva Sae Uiveriy Nakhchiva Azerbaija Emai: aievakbar@mahabaz aarkazimov979@gmaicom eceived December ; revied Jaary 9 ; acceed Jaary ABSTACT I hi aer we iveigae he effec of weigh fcio i he oiear ar o goba ovabiiy of he Cachy robem for a ca of emi-iear hyerboic eaio wih damig Keyword: Cachy Probem; Wave Eaio; Goba Sovabiiy; Weigh Fcio; Semi-Liear Hyerboic Eaio Irodcio Coider he Cachy robem for he emi-iear wave eaio wih damig a () () a L I he cae whe a i ideede of he eiece ad oeiece of he goba oio wa iveigaed i he aer [-8] The ahor iere are foced o o caed criica eoe c which i he mber defied by he foowig roery: if c he a ma daa oio of correodig Cachy robem have a goba oio whie c a oio wih daa oiive o bow i fiie ime regarde of he mae of he daa I he ree aer we iveigae he effec of he weigh fcio a o goba ovabiiy of Cachy robem () ad () Saeme of Mai e We coider he Cachy robem for a ca of emiiear hyerboic eaio f () (4) Throgho hi aer we ame ha he oiear erm ) f ad f o fcio i he domai ) f ad f f f aifie he foowig codiio: a are coi- (5) a L (6) for (7) 4 for (8) I he ee by we deoe he a L - orm For imiciy of oaio i aricar we wrie iead of The coa C c ed hrogho hi aer are oiive geeric coa which may be differe i vario occrrece Theorem Soe ha he codiio (5)-(8) are aified The here ei a rea mber ch ha if : W L L L W L L L Coyrigh Scie
2 A B ALIEV A A KAZIMOV The robem () ad (4) admi a ie oio ; ; C W С L aified he decay roery r D c d r r 4 (9) c d () 4 4 c C mi Proof of Theorem I i we kow ha if ma () W L c T he Tma ie robem () ad (4) have a goba oio (ee for eame [9]) Uig he Forier raformaio Pachere heorem ad he Hadorff-Yog ieaiy for he oio we have he foowig ieaiie (ee []): L 4 D c E L 4 с d; 4 c E L 4 с d 4 c E 4 с d o L L E k W L f L L f O he oher had by vire of codiio L o o () () (4) (5) f c a d (6) ad f c a d (7) Uig he Hoder ieaiy from (6) we have d d L f c a By vire of codiio (7) (8) ad he miicaive ieaiy of Gagiardo-Nireberg ye we have L f ca D L (8) (ee[]) (9) Aaogoy from (7) we have f ca D L () () From () (6) ad () we have he foowig eimae 4 D c E с 4 D D d 4 c E 4 с D D d I foow from () ad () ha () () Coyrigh Scie
3 4 A B ALIEV A A KAZIMOV 4 4 G cd c G G G G d ; G 4 4 G cd c G G G d G ad G are defied by ad 4 (4) (5) G (6) 4 G D (7) (8) 4 The we have from (9) () ad (8) ha (9) 4 () I i cear from codiio (7) (8) ad (9) () ha Aowig for (4) (5) we obai ha G G () c T () ma Th he a riori eimae (9) i aified o T From (4) ad () we yied he ieaiy () 4 Noeiece of Goba Soio Ne e dic he coerar of he codiio (7) ad (8) To hi ed we coidered he Cachy robem for he emi-iear hyerboic ieaiie f f () () The weak oio of ieaiy () wih iiia daa () W L i caed a fcio L which ad aifie he foowig ieaiy: d d dd f d d for ay fcio C From Theorem i foow ha if ad (4) he here ei ch ha for ay U robem () ad () have a i- e oio ad Theorem Le ; С ; L C W (5) d (6) The robem () ad () have o orivia oio 5 Proof of Theorem We ame ha () Le ; ad chooe i a goba oio of () ad C be ch ha r r r r Takig ch a (ee [8]) a he e fcio i Defiiio we ge ha Coyrigh Scie
4 A B ALIEV A A KAZIMOV 5 d The chooe of d dd d d imie ha (7) d (8) Defie by he choice of i i eay o how ha dd C dd C dd C Agai Take caed variabe i yi i he we have d dd (9) c y (4) dd y c c y dd y c c y dd y c (4) 4 (4) (44) Leig i (9) owig o (5) (4) (4) we ge d dd C (45) Takig io acco codiio (6) from (45) i foow ha dd C (46) Frher by ayig he Hoder ieaiy from (7) we obai 4 h d 4 dd dd 4 dd Leig i (47) owig o (45) we ge (47) d dd Fiay akig io codiio (6) we have ha 6 Ackowedgme Thi work wa ored by he Sciece Deveome Fodaio der he Preide of he ebic of Azerbaija Gra No EIF--()-8/8- (4) EFEENCES [] A B Aiev ad A A Kazymov Goba Weak Soio Coyrigh Scie
5 6 A B ALIEV A A KAZIMOV of he Cachy Probem for Semi-Liear Pedo-Hyerboic Eaio Differeia Eaio Vo 45 No 9 - [] A B Aiev ad B H Lichaei Eiece ad No-Eiece of Goba Soio of he Cachy Probem for Higher Semi-Liear Pedo-Hyerboic Eaio Noiear Aayi Theory Mehod ad Aicaio Vo 7 No [] B Ikehaa Y Miaaka ad Y Nakaake Decay Eimae of Soio of Diiaive Wave Eaio i wih Lower Power Noieariie Jora of he Mahemaica Sociey of Jaa Vo 56 No [4] T Li ad Y Zho Breakdow of Soio Dicree ad Coio Dyamica Syem Vo No doi:94/dcd9955 [5] I E Sega Dierio for No-Liear eaiic Eaio II Jora of he America Mahemaica Sociey Vo 4 No [6] Q S Zhag A Bow-U e for a Noiear Wave Eaio wih Damig The Criica Cae da Comered de Académie de Sciece de Pari Serie I Vo 9-4 [7] G Todorova ad B Yordaov Criica Eoe for a Noiear Wave Eaio wih Damig da Come red de Académie de Sciece de Pari Serie I Vo [8] E Miidieri ad S I Pokhozhaev A Priori Eimae ad he Abece of Soio of Noiear Paria Differeia Eaio ad Ieaiie Proceedig of he Sekov Iie of Mahemaic Vo 4-6 [9] A B Aiev Sovabiiy i he Large of he Cachy Probem for Qaiiear Eaio of Hyerboic Tye Dokady Akademii Nak SSS Vo 4 No [] O V Beov V P Ii ad S M Nikoki Iegra ereeaio of Fcio ad Embeddig Theorem VH Wio ad So Wahigo DC 978 Coyrigh Scie
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