nrnaional Jornal of Morn Nonlinar Thory an Alicaion, 3,, -9 h://oiorg/36/imna35 Pblish Onlin Jn 3 (h://irorg/ornal/imna) abiliy olion of h Nonlinar chröingr Eqaion Mahi Ab Elm M-Ali Darmn of Mahmaics, Facly of Ecaion, Kassala Univrsiy, Kassala, an Email: mahi@mailsccn Rciv Fbrary 9, 3; rvis March 8, 3; acc Aril 3, 3 Coyrigh 3 Mahi Ab Elm M-Ali This is an on accss aricl isrib nr h Craiv Commons Aribion icns, hich rmis nrsric s, isribion, an rrocion in any mim, rovi h original ork is rorly ci ABTRACT n his ar iss sabiliy hory of h mass criical, mass-srcriical an nrgy-sbcriical of solion o h nonlinar chröingr qaion n gnral, ak car in vloing a sabiliy hory for nonlinar chröingr qaion By sabiliy, iss h rory: h aroima solion o nonlinar chröingr qaion obying i F ih small in a siabl sac an small in H an hn hr iss a vriabl solion o nonlinar chröingr qaion hich rmains vry clos o in criical norms Kyors: N; Wllos nrocion n his ar, sy h sabiliy hory of solions o h nonlinar chröingr qaion (N) W consir h Cachy roblm for h nonlinar chröingr qaion i F (), hr F, h solion, is a coml-val fncion in Th Eqaion () is call mass-criical or - criical if, an i is call mass-srcriical an nrgy-sbcriical hn Th solions o () hav h invarian aling,, () Dfiniion (olion) sch ha A fncion : is a srong solion o () if an only if i blongs o C, H, an for all saisfis h ingral qaion i r i i F r r (3) A fncion : is a ak solion o () if an only if, H, an for all saisfis h ingral Eqaion (3) Th solions o () hav h mass hr M M M, Enrgy E E hr, E (, ), Dfiniion Th roblm () is locally llos H if for any H hr is a im an an on ball B in H sch ha in T B, an a sbs X of C T, T, H, sch ha for ach B hr iss a niq solion X o h Eqaion (3), an frhrmor, h ma is coninos from B o X f T can b akn arbirarily larg T, h roblm is globally llos Dfiniion 3 A global solion o () is aring in H as if hr iss s s H sch ha lim i s H H for s imilarly, can fin aring in Coyrigh 3 cirs JMNTA
M A E M-A 3 For mor finiion of criical cas s [-3] n his ar iss sabiliy hory of h mass criical, mass-srcriical an nrgy-sbcriical of solion o h nonlinar chröingr qaion n scion hr iss h sabiliy of h mass criical solions an in scion for mass-srcriical an nrgysb-criical solions ar iss Thorm an Thn hr iss a niq maimal-lifsan solion : C o () ih an iniial aa Morovr: ) Th inrval is an on sbs of ) For all, hav M M so, fin M M 3) f h solion os no blo forar in im, hn s, an morovr ars forar in i im o for som Convrsly, if hn hr iss a niq maimallifsan solion hich ars forar in im o i ) f h solion os no blo backar in im, hn inf an morovr ars backar in i im o for som Convrsly, if hn hr iss a niq maimallifsan solion hich ars backar in im o i 5) f M c hr a consan c ning only on hn c M n ariclar, no blo occrs an hav global isnc an aring boh ays 6) For vry A an hr iss Wih rory: if : C is a solion (no ncssarily maimal-lifsan) sch ha M, A an, v ar sch ha M v, hn hr iss a solion v: C v v v an M v for all For roof: [-6] No in h folloing ill iss anar local ll-osnss horm Thorm, an l s c H Assm ha if is no an vn ingr Thn hr iss sch ha if bn an hr is a comac inrval conaining zro sch ha ih sch ha i () hn hr iss a niq solion o () on Frhrmor, hav h bons (5) (6) (7) hr for h closr of all s fncions nr his norm richarz Esima n his scion is som noaion an richarz sima om Noaion W ri X Y anyhr in his ork hnvr hr iss a consan c innn of h aramrs, so ha X cy Th shorc O X nos a fini linar gahring of rms ha look lik X, b ossibly ih som facors chang by hir coml congas W sar by h finiion of sac-im norms q r r q r 3 q r, RR s Th inhomognos obolv norm is an ingr) is fin by: as: f f f H s H s Whn s is any ral nmbr as H s q H (hn s s f f f ˆ s s H H fins s Th homognos obolv norm s f f f ˆ s s H H For any sac im slab K R, q r K R o no h Banach sac of W s fncion KR C hos norm is q q q r r KR K Wih h sal asmns hn q or r is qal o infiniy Whn q r abbrvia q r q as, A Gagliaro-Nirnbrg y inqaliy for chröingr qaion h gnraor of h so conformal Coyrigh 3 cirs JMNTA
M A E M-A ransformaion iffrniaion f richarz Esima J i lays h rol of arial Dfiniion Th onn air qr, is says h chröingr-amissibl if, an qr,, qr,,,, q r Dfiniion Th onn air qr, is says h chröingr-accabl if q, r, q r or,, qr i b h fr chröingr volion From h lici formla i iy, f f y y πi obain h sanar isrsiv inqaliy i f f, () for all n ariclar, as h fr roagaor consrvs h norm, i f f For all an, solvs h inhomognos Eqa- f : C ion () for som, an - P () hr in h ingral Dhaml (3) Thn hav, C C F, F,, (3) for som consan C ning only on h imnsion For som consan C ning only on hav h Holr inqaliy, F F v C v,,, v W no rrn o rov Thorm Proof Thorm Th horm follos from a conracion maing argmn Mor accra, fin i i s i F s s, sing h richarz simas, ill sho ha h ma Φ is a conracion on h s B B hr : B H H C, : s H c, B W C an, nr h mric givn by, v v Hr C, nos a consan ha changs from lin o lin No ha h norm aaring in h mric als lik No also ha boh B an B ar clos (an hnc coml) in his mric Using h richarz inqaliy an obolv mbing, fin ha for B B Φ s c H C, F H C, s H c C, C, s c H s c s c C, C H, An similarly, Coyrigh 3 cirs JMNTA
M A E M-A 5 Φ,,, C C F, C C Arging as abov an invoking (), obain Φ C, F C, Ths, choosing scinly small, s ha for, h fncional Φ mas h s B B back o islf To s ha Φ is a conracion, ra h abov calclaions o obain Φ Φv C, FFv C, v Thrfor, choosing vn smallr (if ncssary), can nsr ha Φ is a conracion on h s B B By h conracion maing horm, i follos ha Φ has a fi oin in B Frhrmor, B noing ha Φ mas ino CH (no s H ) W no rn or anion o h niqnss inc niqnss is a local rory, i nogh o sy a nighborhoo of By Dfiniion of solion (an h richarz inqaliy), any solion o () blongs o B B on som sch nighborhoo Uniqnss hs follos from niqnss in h conracion maing horm Th claims (6) an (7) follo from anohr alicaion of h richarz inqaliy Rmark By h richarz inqaliy, kno ha i Ths, () hols ih for iniial aa ih scinly small norm insa ha, by h monoon convrgnc horm, () hols rovi is chosn scinly small No ha by aling, h lngh of h inrval ns on h fin roris of, no only on is norm 3 abiliy of h Mass Criical n his scion iss h sabiliy hory a mass criical cas Consir h iniial-val roblm () ih An imoran ar of h local ll-os nss hory is h sy of ho h srong solions bil in h as sbscion n on h iniial aa Mor accra, an o kno if h small rrbaion of h iniial aa givs small changs in solion n gnral, ak car in vloing a sabiliy hory for nonlinar chröingr Eqaion () Evn hogh sabiliy is a local qsion, i lays an imoran rol in all ising ramns of h global ll-osnss roblm for nonlinar chröingr qaion a criical cas, for mor s [7] has also rov sfl in h ramn of local an global qsions for mor oic nonlinariis [8,9] n his scion, ill only is h sabiliy hory for h mass-criical N mma 3 b a comac inrval an l b an aroima solion o () maning ha i F, for som fncion os ha M for som osiiv consan M b sch ha (3) an l M (3) for som M os also h smallnss coniions,, i N (33) (3) (35) for som M, M is a small consan Thn, hr iss a solion o () on ih iniial aa a im saisfying hr, (36) M (37) M M (38) N F F (39) Proof: By symmry, may assm inf : Thn saisfis h iniial val roblm i F F For fin Coyrigh 3 cirs JMNTA
6 M A E M-A By (33), : N A F F, A F F,,,,,,,,,,,, Frhrmor, by richarz, (3), an (35), g,, A i, N, A, Combining (3) an (3), obain A A A (3) (3) A sanar coniniy argmn hn shos ha if is akn sfficinly small, for any, A hich imlis (39) Using (39) an (3), obain (36) Frhrmor, by richarz, (3), (35), an (39), F F N N M, hich sablishs (37) for M sfficinly small To rov (38), s richarz, (3), (3), (39), an (33): F FF F N N N M M M M Choosing M M,, sfficinly small, his finishs h roof Bas on h rvios rsl, ar no abl o rov sabiliy for h mass-criical N Thorm 3 b a comac inrval an l b an aroima solion o () in h sns ha i F for som fncion Assm ha coniion (3) in mma 3 hols an,, (3) for som osiiv consan M an N an l oby (3) for som M Frhrmor, sos h smallnss coniions (3), (35) in mma 3 For som hr M, M, is a small consan Thn, hr iss a solion o () on ih iniial aa a im saisfying,, C M M, (33) C M, M, M (3) C M M,, (35) Proof: bivi ino J ~ sbinrvals,, J, sch ha hr M M,, is as in mma 3 W rlac M by M as h mass of h iffrnc migh gro slighly in im By choosing sfficinly small ning on J, M an M, can aly mma 3 o obain for ach an all C, C M C M M F F C N Provi, can rov ha hir conrars of (3) an (3) hol ih rlac by To vrify his, s an inciv argmn By richarz, (3), (35), an h inciv hyohsis, N, N, F F k M C k Coyrigh 3 cirs JMNTA
M A E M-A 7 imilarly, by richarz, (3), (3), an h inciv hyohsis, i i,, F F,, N N k C k Choosing sfficinly small ning on J, M, an M, can nsr ha hyohss of mma 3 conin o hol as varis mma 33 (abiliy) Fi an For vry A an hr iss ih h rory: if : C is sch ha A an ha aroimaly solvs () in h sns ha, i F (36) An, v ar sch ha i v, Thn hr iss a solion v: C o () ih v v sch ha v No ha, h masss of an v o no aar immialy in his lmma, alhogh i is ncssary ha hs masss ar fini imilar sabiliy rsls for h nrgy-criical N (in H ) insa of, of cors) hav aar in [-] Th mass-criical cas i is acally slighly simlr as on os no n o al ih h isnc of a rivaiv in h rglariy class For mor s [5] Proof: (kch) Firs l rov h claim hn A is scinly small ning on v: C b h maimal-lifsan solion ih iniial aa v v Wriing v on h inrval, s ha i F F i F an Ths, if s i X, by h riangl inqaliy, (3), an (36), hav X C F F,, hnc, by () an h hyohsis A, X C A X X hr C ns only on f A is scinly small ning on, an is scinly small ning on an, hn sanar coniniy argmns giv X as sir To al ih h cas hn A is larg, simly ira h cas hn A is small (shrinking, ra ly) afr a sbivision of h im inrval abiliy of h Mass-rcriical an Enrgy-bcriical n his scion iss h abiliy hory of h mass-srcriical an nrgy-sbcriical o h nonlinar chröingr qaion Consir h iniial-val roblm () ih an 3 chos n his cas h iniial-val roblm, is locally ll-os in H No rri () as i () 3, H R * W iss h sabiliy by h folloing roosiion Bfor bginning n fin h Kao inhomognos richarz sima [6] i f, c f H H () Proosiion For ach N hr iss N, an c c N, sch ha h folloing hols, H for all an solv f An Thn i, f H for all an fin i N, H H i () H c c N H Proof: b fin by solvs h qaion i hn (3) Coyrigh 3 cirs JMNTA
8 M A E M-A sinc H N Can b ivi, ino A A N in inrvals, ch ha for all, h qaniy H, is Aroria small, (δ o b slc blo) ngraion (3) ih iniial im is hr i is i, s s () Alying h Kao richarz sima () on obain i an H, c H, 5 5 3 3 3 5 3 c c No ha 5 imilarly, 3 H, 5 5 3 H H,, H, 5 3 H, 3 3 5 3 H,, o (5) bsiing h abov simas in (5), o g, i H, c H, H, (6) 3 c c c,, H H As long as min, 6c an i c min, H, 6c (7) W obain i, H c (8) H, i Takn no in () an aly o boh sis o obain i i is i, s s (9) inc h Dhaml ingral is rsric o,, by again alying h Kao sima, similarly o (6) obain, i i H, c,, H H 3 c c c,, H H By (8) an (9), bon h Formr of rssion o obain i i c H, H, ar iras ih, obain i H, i c H, c To absorb h scon ar of (7) for all inrvals, A, rqir A c min, 6 c () W rvi ha h nnc of aramrs δ is an absol consan chosn o m h firs ar of (7) Th inqaliy () rmins ho h small ns o b akn in rms of A (an hs, in rms of N ) W r givn N hich hn rmin A REFERENCE [] Kraani, On h Blo-U Phnomnon of h Criical Nonlinar chröingr Eqaion, Jornal of Fncional Analysis, Vol 35, No, 6, 7-9 oi:6/fa55 [] R Killi, M Visan an X Zhang, Enrgy-Criical N Coyrigh 3 cirs JMNTA
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