Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point

Resea ch A icle Jose Diblík* and Mi osla a Růžičko á Vanishing and blow-up solu ions o a class o nonlinea complex diffe en ial equa ions nea he singula poin h ps://doi.o g/10.1515/anona-2023-0120 ecei ed July 16, 2023; accep ed Decembe 1, 2023 Abs ac : A singula nonlinea diffe en ial equa ion ()=+zw zaw zw z w d d, , σ whe e >σ 1 , is conside ed in a neighbou hood o he poin =z0loca ed ei he in he complex plane  i σ is a na u al numbe , in a Riemann su ace o a a ional unc ion i σ is a a ional numbe , o in he Riemann su ace o loga i hmic unc ion i σ is an i a ional numbe . I is assumed ha ( ) =wwz ,{}∈⧹ a 0, and ha he unc ion is analy ic in a neighbou hood o he o igin in  × . Conside ing σ o be an in ege , a a ional, o an i a ional numbe , o each o he abo e-men ioned cases, he exis ence is p o ed o analy ic solu ions ( ) =wwz in a domain ha is pa o a neighbou hood o he poin =z0in  o in he Riemann su ace o ei he a a ional o a loga i hmic unc ion. Wi hin his domain, he p ope y ()= →wzlim 0 z0is p o ed and an asymp o ic beha iou o ()wz is es ablished. Se e al examples and figu es illus a e he esul s de i ed. The blow-up phenomenon is discussed as well. Keywo ds: analy ic solu ion, asymp o ic beha iou , blow-up phenomenon, complex plane, diffe en ial equa- ion, singula poin MSC 2020: 34M35, 34M30, 34M10, 34A25 1 In oduc ion A singula nonlinea diffe en ial equa ion ()=+zw zaw zw z w d d, , σ(1) whe e >σ 1 , is conside ed in a neighbou hood o he poin =z0ei he in he complex plane  i { } ∈≔σ1,2, … is a na u al numbe , in a Riemann su ace o a a ional unc ion i ∈⧹σis a a ional numbe , o in he Riemann su ace o loga i hmic unc ion i ∈≔⧹σis an i a ional numbe . In (1), z is an independen a iable, ( ) =wwz , and {}∈⧹ a 0. The unc ion → :is assumed o be analy ic in a neighbou hood  o he poin ()∈×0,0 ha ing he o m:  * Co esponding au ho : Jose Diblík, Facul y o Ci il Enginee ing, B no Uni e si y o Technology, Ve eří 331/95, 602 00 B no, Czech Republic; Facul y o Elec ical Enginee ing and Communica ion, Technická 3058/10, 616 00 B no, Czech Republic, e-mail: [email p o ec ed], [email p o ec ed] Mi osla a Růžičko á: Facul y o Ma hema ics, Uni e si y o Białys ok, K. Ciołkowskiego 1M, 15-245 Białys ok, Poland, e-mail: [email p o ec ed], [email p o ec ed] Ad ances in Nonlinea Analysis 2024; 13: 20230120 Open Access. © 2024 he au ho (s), published by De G uy e . This wo k is licensed unde he C ea i e Commons A ibu ion 4.0 In e na ional License.   {( ) ∣∣ ∣ ∣ }=∈×<<zw z ρ w e,:, , k (2) whe e ()∈∞ρ0, and ()∈−∞∞ k ,a e he fixed cons an s such ha he e exis s a fini e numbe > M 0 sa is ying ∣( )∣ () ≥∈ M zwsup , . zw, A p oo is gi en o he exis ence o analy ic solu ions ( ) =wwz defined in a mul iple connec ed domain in a neighbou hood o he singula poin =z0and anishing o →z0 ( he poin =z0i sel does no belong o his domain being i s bounda y poin ). Such a domain lies in he complex plane  i ∈σ, in a Riemann su ace o a a ional unc ion i ∈⧹σ, o in he Riemann su ace o he loga i hmic unc ion i ∈σ. Whe he he complex plain o he Riemann su ace is chosen is de e mined by he domain o he e m z σ in (1). Asymp o ic analysis o equa ions in a neighbou hood o a singula poin in he complex plane has a long his o y. In a pionee ing a icle [3], a sys em o nonlinea diffe en ial equa ions and an ini ial p oblem () ()′= =zw h z w w,and00 (3) a e conside ed wi h a unc ion h being holomo phic in a neighbou hood o ( ) 0,0 and sa is ying ()= h 0,0 0. The au ho s p o e ha a solu ion ( ) =wwz such ha ()=w00 can be cons uc ed as a powe se ies con e gen in a neighbou hood o =z0. In [31], assuming ()(∣∣ ) = h zoz,0 m, he au ho s use unc ional-analy ical me hods o show ha (3) admi s an analy ic solu ion exp essed by a powe se ies wi h an ini ial powe z m .Asa pa icula case o sys em (3), a ising when he Jacobian ma ix ( ) ′ h 0,0 w is singula , sys ems ()′=zy hzyw,, , σ1 (4) ()′=zw h z y w,, , 2(5) and hei modifica ions a e in es iga ed whe e >σ 1 is an in ege , z is an independen complex a iable, ()= y yz ,( ) =wwz , h i, =i1,2 , a e holomo phic ec o - unc ions in a neighbou hood o ()0,0,0 anishing he e. We e e o [13–17,19–23,25–27,30] and o e e ences he ein. As he p incipal me hod o in es iga ion is o en used a cons uc ion o o mal solu ions in he o m o special se ies ( ha a e no necessa ily exac powe se ies) con e gen in a subse o a neighbou hood o =z0. A subs an ial es ic ion used in he abo e-men ioned esul s is he assump ion ha σ in (4) is an in ege as he me hods used by hei au ho s a e no applicable in cases o σ being a ional o i a ional. In his s udy, we sugges a “geome ical”me hod connec ed wi h he p ope ies o he unc ion z σ and allowing us o omi his es ic ion in he case o scala equa ion (1). I ∈σ, equa ion (1) is conside ed in he complex plane  , while, in he case o =∕∈⧹σmm 12 , whe e >>mm 1 12and ∈mm, 12 a e ela i ely p ime, we use he Riemann su ace o he unc ion = ∕ wz m12and, i ∈σ, he Riemann su ace o he loga i hmic unc ion. Mo eo e , he p ope ies o solu ions o sys ems (4) and (5) ha e only been s udied in a subse o he o igin (in a sec o wi h i s e ex a he poin =z0). Ou in es iga ion o he asymp o ic p ope ies o solu ions o equa ion (1) co e , in a sense, he whole neighbou hood o he singula poin =z0(in  o in he a o emen- ioned Riemann su aces). In he igh -hand side o (1), a “pe u ba ion”()zw z w,o a linea equa ion ′=zw a w σis conside ed. Such a o m o nonlinea i y is qui e na u al because, i we use, o example, he e m ( ) zw *,ins ead, he assump ions o ou esul s educe such a gene al o m o he o m used in (1), i.e. o () ( ) ≔ zw zw zw *,, . In some o mulas h oughou his a icle, i no ambigui y can a ise, a simplified no a ion is used o he dependen a iables no indica ing hei dependence on independen a iables. The geome ical me hod o in es iga ion sugges ed in his s udy is qui e diffe en om he me hods used p e iously and can be used o analysing o he classes o equa ions in he complex domain. Fo he eade ’s con enience, in Sec ion 2, we ecall some auxilia y no ions and concep s well known om he heo y o unc ions o complex a iable. T ans o ma ions applied o equa ion (1) wi h ocus on sys ems equi alen on gi en cu es and ays o (1) a e discussed in Sec ion 3, while in Sec ion 4, he beha iou o solu ions is s udied o he sys ems de i ed. In Sec ion 5, he esul s o his a icle (Theo ems 1–4) a e o - mula ed. Thei p oo s a e gi en in Sec ion 6. Since a majo pa o he p oo s o Theo ems 1–3 is iden ical o an 2Jose Diblík and Mi osla a Růžičko á a bi a y alue o σ , we conside only one a ian o he p oo whe e he diffe ences depending on na u al, a ional, o i a ional alues o σ a e emphasized. The p oo o Theo em 4 is a consequence o he common pa o he p oo . Each o Examples 1–5 accompanies he cons uc ions pe o med in Sec ion 6. Ne e heless, in Sec ion 7, a mo e complex example is conside ed. Concluding ema ks and open p oblems o mula ed a e gi en in Sec ion 8. A close connec ion o he findings o his a icle wi h he unlimi ed g ow h o moduli o solu ions nea he singula poin =z0( he so-called blow-up phenomenon) is men ioned and discussed as well. 2 P elimina ies Conside an ini ial p oblem ()()′= =wFzw wz w,, , 00 (6) whe e z and w a e he complex a iables and F is a complex- alued unc ion. By a special case o he well known Cauchy-Ko ale skaya heo em i he unc ion F is analy ic in a neighbou hood o he poin ( ) zw, 00 , p oblem (6) has a unique analy ic solu ion ( ) =wwz in a neighbou hood o he poin z0(we e e , e.g., o [8]). Recall also ha an analy ic unc ion is a unc ion ha can be exp essed by a con e gen powe se ies. A holomo phic unc ion is a unc ion ha is diffe en iable in each neighbou hood o he poin o i s domain. Fo complex unc ions, he no ions o an analy ic and a holomo phic unc ion a e equi alen . Le   ⊂ be a pa h-connec ed domain. A cu e lying in  is said o be simple i i does no c oss i sel . A domain  is simply connec ed i any simple closed cu e in  can be con inuously sh unk in o a poin while emaining in  . A domain  ha is no simply connec ed is called a mul iply-connec ed domain. The symbol  ∂ deno es he bounda y o  , and  s ands o he closu e o  (i.e. o he se  ∪∂ ). The concep o analy ic con inua ion (ex ension) is used in his a icle as well. I means he ollowing. Le 1 and 2be wo analy ic unc ions in open domains  1 and  2 o he complex plane  , espec i ely. Le  ∩≠∅ 12 and  ≢ 12 .I ≡ 1 2 in  ∩ 12 , 2is called an analy ic con inua ion o 1 o  2 , and ice e sa. The analy ic con inua ion is unique. 3 T ans o ma ions o equa ion (1) Se e al auxilia y ans o ma ions o equa ion (1) a e necessa y o i s analysis. In Sec ion 3.1, we show why he coefficien a in (1) can be assumed o be eal and posi i e. Then, wo ypes o eal wo-dimensional sys ems equi alen o (1) a e conside ed. In Sec ion 3.2, we de i e an equi alen wo-dimensional eal sys em along gi en cu es s a ing and ending a he poin =z0, while in Sec ion 3.3, an equi alen eal wo-dimensional eal sys em along gi en ays leading off he poin =z0is ob ained. 3.1 On he coefficien a in (1) Le he complex coefficien a in (1) be gi en in i s exponen ial o m: ∣∣ [ )=∈ a ae θ π,0,2 . iθ Then, a subs i u ion, geome ically exp essing a o a ion, {} () =∈⧹ ∕− z e ,0 , iθ σ 1(7) whe e is a new independen a iable, changes equa ion (1) in o one o a simila o m: Vanishing and blow-up solu ions 3 ∣∣ ( ) () ()() =+ ∕− − −∕− w aw w e we d d, , σiθσiθσσ121 wi h a posi i e coefficien ∣ ∣ ains ead o he p e ious complex coefficien a . In he ollowing in es iga ion, we will assume ha ∣() ∣ zw z w, in (1)issufficien ly small. This is ue i ei he ∣∣zo ∣∣wis small enough. Then, due o (7), his p ope y emains in o ce also o he exp ession: ∣( ) ∣∣( )∣ () ()() () = ∕− − −∕− ∕− w e w e w e w,, . iθ σ iθ σ σ iθ σ121 1 The e o e, he coefficien a in (1) can be assumed, wi hou loss o gene ali y, eal and posi i e, i.e. a can be eplaced by i s modulus ∣ ∣ a. We implici ly use his p ope y in he in es iga ions in he ollowing and, when- e e compu a ions in his a icle depend on a , we assume ha i is a posi i e numbe . 3.2 Real sys em equi alen o (1) on he loops o a gi en cu e The beha iou o solu ions o (1) in a small neighbou hood o he poin =z0will be s udied along he loops o a cu e defined as: () (()) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅ ∕− zφ νσφ σc e cos 1 1 , σiφ 011 (8) whe e ν 0is a fixed eal numbe , > c 0 is a eal pa ame e , and φ is a eal independen a iable. I c and ν 0a e fixed, we will assume ha φ a ies in such a way ha (())−− >νσφ c os 1 0 , 0 (9) and, mo eo e , as we need ∣()∣<zφ ρ , ha (()) () () ⎜⎟ ⎛ ⎝ −− −⎞ ⎠< ∕− νσφ σc ρ cos 1 1 . σ 011 Inequali y (9) will hold i and only i Figu e 1: Loops o (8) specified by =∕ ν π32 0, =σ4 , = s 0,1,2, and = c 3,2, 1 . 4Jose Diblík and Mi osla a Růžičko á −⎛ ⎝−+ ⎞ ⎠<<−⎛ ⎝++ ⎞ ⎠=± σνπsπ φ σνπsπ s 112 2112 2, 0,1,… 00 (10) and, ob iously, ()==±zφ k0, 0, 1,… k (11) o ≔−⎛ ⎝++⎞ ⎠=± φ σνππk k 112,0,1,…. k0 (12) The closu e o each loop o he cu e (8) sepa a ed by an inequali y (10) is a closed cu e passing h ough he o igin. I ∈σ, hen he e a e −σ 1 diffe en cu e a cs (8) lying in he complex plane  ,defined, e.g., by: =− s σ0,…, 2 in (10). I =∕∈⧹σmm 12 , whe e >>mm 1 12and ∈mm, 12 a e ela i ely p ime, hen he e a e −m 1 2 diffe en cu e a cs (8) on he Riemann su ace o he unc ion = ∕ wz m12defined, e.g., by =− s m0,…, 2 2in (10). I ∈σ, hen he e is a coun able se o disjunc loops o cu e (8) defined by =± s 0, 1,… in (10). We conside hem on he Riemann su ace o he loga i hmic unc ion. Figu e 1 shows diffe en loops o (8) in he z plane as defined by angles φ sa is ying (10), whe e = s 0,1,2, =∕ ν π3 2 0, and ela ed o he alues = c1 (g een loops), = c 2 (b own loops), and = c 3 (blue loops). We will ans o m equa ion (1) in o a sys em o wo o dina y diffe en ial equa ions on pa s o he closu es o he loops o cu e (8) wi h independen a iable φ sa is ying inequali ies (10). Then, () (() ) =wz wzφ .To educe he compu a ions, we do no always w i e he a gumen () z φ o w o he a gumen φ o z (simila ly, we p oceed when new dependen a iables α and β in he ollowing a e used). By he chain ule, we de i e (()) (()) () () ()[ () (() )]() == + −− − wzφ φwzφ zφ zφ φe z φwae zφe zφ w zφ φ dddddd,dd, iν σ iν iν 000 (13) whe e () ( )(( )) (( ( ))) ( ( ( )))( ) (()) () () (()) (()) () (()) () (( )) () (()) ⎜⎟ ⎜⎟ =−− −− ×−−−+ ⎛ ⎝ −− −⎞ ⎠ =⎛ ⎝ −− −− +⎞ ⎠ =−− ∕− ∕−− ∕− −−− zφ φσσc νσφ νσφσe νσφ σc ie zφ νσφ νσφi iz φ νσφ e dd1 11 cos 1 sin 1 1 cos 1 1 sin 1 cos 1 cos 1 . σσ iφ σiφ iν σ φ 11 0111 0011 0 0 0 1 0 (14) Finally, using (8), (13), and (14), (()) ()[ () (() )] () (())() [()(())] (()) (() ) [()(())] (())(()) () ()[ ()(())] (()) ⎜⎟ =+ −− =+ −− =+ −− ⎛ ⎝ −− −⎞ ⎠ =−+ −− −− − −− −− −− −− − −− wzφ φe z φwae zφe zφ w iz φ νσφ ee wae zφe zφ w i νσφ zφe wae zφe zφ w i νσφ νσφ σc iσ cwae zφe zφ w νσφ dd,cos 1 ,cos 1 ,cos 1 cos 1 1 1, cos 1 . iν σ iν iν iν iφ σ iν iν iφ σ iν iν iν iν 0 1 0 1 0 01 20 000 0 00 00 00 (15) Le (())=wwzφ in (15) be ep esen ed by i s algeb aic o m, i.e. (()) () ( ) =+wzφ y φ iy φ 12 wi h () y φ 1 and () y φ 2 being he eal and imagina y pa s o (() ) wzφ , espec i ely. Then, Vanishing and blow-up solu ions 5 (()) () () ()[ ()(())] (()) ()(()())( ) (()) ( )( () ())[ (() (() )) (() (() ))] (()) =+ =−+ −− =−+ − −− +−+ + −− −− −− wzφ φyφ φiyφ φ iσ cwae zφe zφ w νσφ iσ cy φ iy φ a ν i ν νσφ iσ cy φ iy φ zφe zφ w i zφe zφ w νσφ dddddd 1, cos 1 1cossin cos 1 1Re,Im, cos 1 . iν iν iν iν 12 20 12 00 20 12 20 00 00 Equalling he eal and imagina y pa s, we see ha y1 and y 2sa is y he ollowing sys em o o dina y diffe en ial equa ions equi alen o equa ion (1) on he gi en loop segmen o he cu e (8): () (()) (( ) ( ) ) () (()) ( ( ()) ( ())) ′=− −− − +− −− −− −− yσc νσφ aνy aνy σc νσφ y ze zw y ze zw 1 cos 1 sin cos 1 cos 1 Re , Im , , iν iν 1200102 2021 00 (16) () (()) (( ) ( ) ) () (()) (( ()) ( ())) ′=− −− + +− −− − −− yσc νσφ aνy aνy σc νσφ y ze zw y ze zw 1 cos 1 cos sin 1 cos 1 Re , Im , . iν iν 2200102 2012 00 (17) 3.3 Real sys em equi alen o (1) on a sys em o ays Conside ays unning om he o igin ()=<<=z e ρ ν,0 , cons , iν (18) and ans o m equa ion (1) along hese in o a sys em o wo eal equa ions. Since == − w zw zw e d dd dd dd d, iν (1) can be w i en as: (()) () =+ −− w ewa e ew d d, . σ i σ ν iν iν1 Assuming (() ) wz by i s algeb aic o m (()) () ( ) =+wz x ix 12 , we de i e () () (())( ( (()))) ( () ())( ( (()))) (( ) ( ))(() ()) ( ( ( (())) ( ( (()))))) () () =⎛ ⎝+⎞ ⎠ =+ =++ =−−− + ⋅+ + −− −− w x ix ewz a e ewz ex ix a e ewz σνiσνx ix a e e w z i e e w z d ddddd , , cos 1 sin 1 Re , Im , . σσ iσ ν iν iν iσ ν iν iν iν iν iν iν 12 1 112 12 Func ions x1 and x 2sa is y he ollowing sys em o o dina y diffe en ial equa ions: ( ( )) ( ( )) ( ( ))( ( ) ( )) (( ))(() ()) ′=−+−+− − +− + x a σ νx a σ νx σ ν x e x e σ νx e x e cos 1 sin 1 cos 1 Re Im sin 1 Im Re , σiν iν iν iν 11212 12 (19) ( ( )) ( ( )) ( ( ))( ( ) ( )) (( ))( () ()) ′=− − + − + − + +−− + x a σ νx a σ νx σ ν x e x e σ ν x e x e sin 1 cos 1 cos 1 Im Re sin 1 Re Im . σiν iν iν iν 21212 12 (20) 6Jose Diblík and Mi osla a Růžičko á 4 Auxilia y esul s on he beha iou o solu ions In his sec ion, we p o e wo lemmas used in p o ing he esul s o his a icle. In Sec ion 4.1, he beha iou o solu ions o (1) is conside ed along loop segmen s o he cu e (8), while in Sec ion 4.2, he beha iou o solu ions o (1) is conside ed along ays (18) defined in Sec ion 3.2. 4.1 Beha iou o solu ions along segmen s o cu e (8) Assume, in he defini ion o cu es (8), ≠ ν m π 0,∈m, i.e. ≠ν s in 0 . 0 (21) In he ollowing, we conside sys em (16), (17) in he domain {( ) ∣()∣ }≔∈<<+<φy y zφ ρy y e Ω ,, :0 , , k c 12 312222 whe e k and ρ a e he same as in he defini ion o he domain  gi en by (2), ()zφ is defined by (8), and assume ha (9) holds as well. The alues φ k,defined by (12), ha e p ope y (11), i.e. hese alues define singula poin s = φ φ k o sys em (16), (17) because (())−− =νσφ c os 1 0 k 0. I is clea ha he condi ions a e me a e e y poin o Ω c o he well-known heo ems on he exis ence and uniqueness o solu ions o ini ial Cauchy p oblem as well as on he con inuous dependence o solu ions on he ini ial da a. Le ( )ε ν0be a posi i e numbe such ha () ∣ ∣≤ ε νενsin , 000 (22) whe e ε 0 is a fixed numbe sa is ying << ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ ερ a M 0min, 2 . 0(23) Define cylinde s  ( ) λas se s  () {( ) } ( )≔∈+==∈−∞λφyy yyeλ λ k,, Ω: , cons , ,. λ 12 c 12222 (24) Lemma 1. Assume ha φ a ies in a fixed domain defined by (10). Le c and ν 0,sa is ying (21), be fixed such ha ∣()∣ ()<≤zφ εν0 . 0(25) Then, any in eg al cu e (()())φy φ y φ,, 12 o sys em (16), (17) in e sec ing a a alue = φ φ * a cylinde  ( ) λ,i.e. i (()() ) φyφ yφ *,*,* 12 sa is y () ()+= y φyφe ** , λ 12222 beha es as ollows. The in eg al cu e (()())φy φ y φ,, 12 ,as φ inc eases, is passing (i) om domain +> y ye λ 12222 (26) in o domain +< y ye λ 12222 (27) i <ν s in 0 0and inequali y ∣(())∣ () ()=+<<wzφ yφyφe e λk2122222 (28) holds o e e y admissible > φ φ * . (ii) om domain (27)in o domain (26) i >ν s in 0 0and inequali y (28) holds o e e y admissible < φ φ * . Vanishing and blow-up solu ions 7 P oo . Conside he beha iou o in eg al cu es o sys em (16), (17) in e sec ing cylinde s  ( ) λ. To do his, compu e he scala p oduc ( ) →→ NT,a an a bi a y poin o cylinde  ( ) λwi h fixed λ , whe e → N is i s no mal ec o di ec ed ou wa ds and → T is a ec o o he ec o field defined by sys em (16), (17). As () ⎜⎟ →=→=⎛ ⎝⎞ ⎠ N yy T y φy φ 0, , and 1, d d,d d , 12 12 we ha e () ( ) ( ( ( ()))) (()) →→=+= −− −− − NT y y φyy φσceaν ze wzφ νσφ ,d dd d1sinIm , cos 1 . λiν 112220 20 0(29) We will show ha (29) implies () →→=NT ν s gn , sgnsin 0 (30) whene e (25) holds. Indeed, o =+zzizRe Im and ()(())(())=+ −− − e zw e zw i e zw,Re , Im , , iν iν iν 00 0 we ha e ( ( )) ( ( )) ( ( ))=⋅ +⋅ −−− ze zw z e zw z e zwIm , Re Im , Im Re , . iν iν iν 000 The e o e, om (22), (23), and (25), i ollows ∣ ( ( ))∣ ∣∣ ∣∣ ∣∣ () ∣ ∣ ∣ ∣≤+= ≤ ≤ < − ze z w z M z M z M ε ν M ε ν M a νIm , 2 2 2 sin sin . iν 000 0 0 Then, aking in o accoun ha () (()) − −− > σce νσφ 1 cos 1 0 , λ2 20 om (29), we de i e ( ) ( ( ( ))) →→=− = − NT a ν ze zw ν s gn , sgn sin Im , sgnsin iν 0 0 0 and (30) holds. Finally, we ema k ha (22), (23), and (25) imply ∣()∣<zφ ρ , i.e. he alues o z used a e wi hin he domain  defined by (2) and ha o mula (30) is independen o he alue o λ . The geome ical meaning o equa ion (30) is as gi en in pa s (i) and (ii) o he lemma. □ Rema k 1. No e ha he φ -axis i sel (i.e. he se o poin s ( ) φ,0,0 , whe e φ sa isfies (10)) is also an in eg al cu e o (16) and (17); he e o e, no o he in eg al cu e in e sec s he φ -axis. Fo a sufficien ly la ge c , he conside ed loops o (8), as i ollows om (25), a e comple ely con ained in he ( )ε ν0-neighbou hood o he poin =z0.I c is sufficien ly small, hen loops o (8) a e no connec ed in he ( )ε ν0-neighbou hood o he poin =z0wi h his ( )ε ν0-neighbou hood con aining hei pa s (Figu e 2). In mos o he ollowing figu es, he alue ( )ε ν0can be seen on he plo s scaled down o fi in o a single figu e. 4.2 Beha iou o solu ions along sys em o ays (18) Assume, in he defini ion o ays (18), ≠−⎛ ⎝+⎞ ⎠=± ν σπmπ m 112 ,0,1,…, i.e. ()−≠σν c os 1 0 . (31) In his pa , we will examine he beha iou o solu ions o Sys ems (19) and (20) in he domain: 8Jose Diblík and Mi osla a Růžičko á {( ) }≔∈<<+< xx ρxxe Ω ,, :0 , , s k 12 3122 22 whe e k and ρ a e he same as in he defini ion o he domain  gi en by (2) and ( ) z is defined by (18), i.e. ∣()∣=z . Fo sys em (19), (20), he hypo heses o he well-known heo ems on he exis ence and uniqueness o solu ions o ini ial Cauchy p oblem as well as on he con inuous dependence o solu ions on he ini ial da a a e ue a e e y poin o Ω s .Define cones  ()δas se s  () ( ) () ≔⎧ ⎨ ⎩∈+= ⎛ ⎝−−⎞ ⎠⎫ ⎬ ⎭ − δ xx xxδ aσν ,, Ω: exp2cos 1 , s p 12 122 221(32) whe e δ and p a e he fixed pa ame e s sa is ying >δ 0 and ()∈ p σ1, . Pu ∣( )∣≔− ∕ ε σν ** cos 1 , νp p μ ,1(33) whe e μ sa isfies { } << −μσp0min1, and ε* p is a posi i e numbe sa is ying () ⎜⎟ << ⎧ ⎨ ⎩ ⎛ ⎝+− ⎞ ⎠ ⎫ ⎬ ⎭ −−− ∕ ερ a Mρ p aρ 0*min , 41 . pμσpμ μ 1 1 (34) I an in eg al cu e (()()) x x ,, 12 o he sys em (19), (20) in e sec s a fixed cone  ()δa a poin (]=∈ *0, * νp,, hen ∣(())∣ () () () () ⎜⎟ =+=⎛ ⎝−−⎞ ⎠ − wz x x δ aσν *** exp 2cos 1 * , p 2122 221 whe e δ sa isfies () () ⎜⎟ <⎛ ⎝+−⎞ ⎠ − δk aσν exp cos 1 *p1(35) by he defini ions o  ()δand Ω s . Figu e 2: Loops o (8) specified by =∕ ν π2 0and =σ3 i ()= ε ν 1 0,= c 3, 2, 1, 0.5, and 0.3. Vanishing and blow-up solu ions 9 () () ( ) () ⎜⎟ ⎛ ⎝−⎞ ⎠≤≤ − ∕− σc εν ε σ 11sin 1 . σ11 00 (63) Assume ha (63) holds. Then, bo h he abo e-men ioned segmen s in e sec he ay (18), whe e ν is gi en by (50) and sa isfies (55), a he poin () ()()() ⎜⎟ =⎛ ⎝− −⎞ ⎠ ∕− zσ σc e *cos 1 1, n σiψ n 11 p o ided ha (deduc ing om (55)) ∣∣[ ]∈zωε *,* , np (64) whe e ∣∣ () () () ⎜⎟ =⎛ ⎝− −⎞ ⎠ ∕− zσ σc *cos 1 1 . n σ11 (65) I he pa ame e c is fixed (being sufficien ly la ge) and i ω is sufficien ly small, hen, simul aneously, he se o alues ∣∣z* n sa is ying (64) will be non-emp y and (63) will hold as well. This is shown in he ollowing. Le ω be fixed and sufficien ly small. Define he numbe (() ) ∈∕− πσ0, 2 1 2as he solu ion o he equa ion: ∣∣ = ≤≤ max . ωz ε 2** np (66) This maximum will be achie ed o ∣∣=z ω * n because () () −= −→∕ σ lim cos 1 0 σ π12 . Then, om (65), we de i e () () ( ) ⎜⎟ =⎛ ⎝− −⎞ ⎠ ∕− ωσ σc cos 1 1 σ 211 (67) o (( ) )=−− − σσcω 11a ccos 1 . σ 21 (68) F om o mulas (67) and (68) we deduce he ollowing. I , o an inc easing c and dec easing ω , he p oduc − c ωσ 1 emains he same, hen he alues 2 , 1 will be fixed as well. Inequali y (63) will be sa isfied i ()(())()(()) ≥−− ≥−− −− c σεσ σεσ 1 1sin 1 1 1sin 1 σσ 01 101 wi h (63) s ill holding o all []∈ , 12 . Ob iously, i is possible o apply he abo e-men ioned easoning e en a e eplacing ω wi h an <ω ω 1 and c wi h an > cc 1such ha = −− c ωcω σσ 111 1 . Then, he se o alues ∣∣z* n such ha (64) holds is nonemp y. The e o e, i ω is sufficien ly small, hen he e exis s a numbe (]∈ ε ωε ** ,* pp such ha each poin ( ) z o he segmen o he ay (18)defined by: ≤≤ω ε * * p (69) wi h ν gi en by (50) is in e sec ed o e e y fixed []∈ , 12by he loops o he cu e (8), co esponding o ( ) =± ν νn , 00, and (63) holds. These loops a e defined by he o mula: () (()( )) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠ ±∕− zφ νn σ φ σc e cos , 1 1 , n σiφ 011 (70) whe e φ a ies wi hin domains (57). Mo eo e , i is easy o show ha he a cs (70) a e symme ic wi h espec o he gi en ay. Le be fixed. By he abo e-men ioned cons uc ion, h ee cu es a e passing h ough he poin () =zωe iψ n , he ay i sel and wo loops o he cu e (8), specified by (70). By Lemma 2, (ii), o mula (38), he inequali y 16 Jose Diblík and Mi osla a Růžičko á ∣( )∣ () <w e e niψ n k 0 holds o e e y []∈ ωε,* p . In he ollowing, we use segmen s o cu e loops (70)defined by angles φ (wi hin domains (57)) such ha () ( )≤< + ψn φ ν n , (71) and () ()<≤ − ν n φ ψn, . (72) Along a cs (70), whe e φ sa isfies (71) i he alue ( ) + ν n , 0 is conside ed o (72) i he alue ( ) − ν n , 0 is used, he e exis s he analy ic con inua ion o solu ion ( ) wz n 0 o (1), sa is ying ini ial condi ion (53). I domain (71) is conside ed, hen, by (62), ()==−−< + νν σ s in sin sin 1 0 00 and, by Lemma 1, whe e ( ) = φ ψn *, pa (i), o mula (28), ∣ (())∣<wzφ e . nk 0 I domain (72) is conside ed, hen, by (62), ()==−> − νν σ s in sin sin 1 0 00 and by pa (ii) o Lemma 1, whe e ( ) = φ ψn *, he same inequali y holds. We p o e ha his solu ion con e ges o ze o i ( ) → ± φ νn , . Fo ()zφ defined by (70), we ha e () () = →±zφlim 0 φ νn ,(73) because (()) () −− = ∕= → ± ± νσφ πlim cos 1 cos 2 0 . φ νn ,0 Mo eo e , by (50), (47), and (48), we ha e (( ) ( )) ( )−=−> ± σνn σ c os 1 , sin 1 0 . (74) Then, pu ing ( ) =± ν νn ,in equa ion o ays (18), we see ha (31) holds. As i ollows om o mulas (36) and (37) in Lemma 2, (i), whe e, by (33) and (74), (( )) (( )) () ≔−≥− ∕∕ ± εσ εσ **sin 1 *sin 1 , νn p pμpμ ,, 111 he solu ion ( ) wz n 0sa isfies ()= →wzlim 0 zn 00. Due o he a iabili y o []∈ , 12 and he analy ic con inua ion o ( ) wz n 0, his p ope y also holds on he cu e loops i ( ) → ± φ νn , (p ope y (73)). Mo eo e , ∣()∣ ()() ∣∣ ⎜⎟ ≤⎛ ⎝−−⎞ ⎠ ± − wz δ aσνn z exp cos 1 , , np 01 o sufficien ly small ∣∣z. Rema k 4. The ollowing p ope y should be added o he p e ious conside a ion. Le = νφ in (18)befixed, whe e ei he () () ( ) +−<≤ + ψn π σφνn 21 , (75) o () ()() ≤< − − − ν n φ ψn π σ ,21 . (76) Since, in bo h cases, ()−>σφ c os 1 0, om Lemma 2, (i), o mulas (36) and (37), we ha e Vanishing and blow-up solu ions 17 ∣()∣ () ∣∣ ⎜⎟ ≤⎛ ⎝−−⎞ ⎠ − wz δ aσφ z exp cos 1 , np 01 o a sufficien ly small ∣∣z,∣∣ ( ( ( )) ) ∈− ∕ zεσφ0, *cos 1 pμ1and ()= →wzlim 0 z0. This is ue o all ays (18) wi h a fixed φ wi hin in e als (75) and (76). Example 1. Assume ha , in equa ion (1), we ha e =σ2 ,= a1 , and = M2 . Le =ρ 1 and = k0 . The ollowing cons uc ions a e isualized by Figu e 3. In acco dance wi h (23), pu {}<= < ⎧ ⎨ ⎩ ⎫ ⎬ ⎭==ερ a M 00.02min, 2min 1, 0.25 0.25 . 0 Mo eo e , o ν 0defined by o mulas (45) and (46) wi h = n 0 , we ha e, by (50), ei he ()()==+=+ + ν ν ψ π 0, 0 00 o ()()==−=− − ν ν ψ π 0, 0 00 and, by (22), we can se (in he o mula, he dependence on is emphasized and no a ion () ε ν 0is used ins ead o ( )ε ν0)( ) ∣ ( )∣ ∣ ( )∣==±= ± ε νεν π sin 0, 0.02 sin 0.02sin . 000 In acco dance wi h (56), define a ange o by numbe s 1 and 2 sa is ying =∕ π3 2and =∕ π 6 1(i =∕ ζ 12 is aken). Al hough we ha e defined he numbe 2 “ad hoc”(no using i s defini ion (66)) o be e illus a e all compu a ions, his choice is in acco dance wi h (66) as shown in he ollowing. Le us ake a numbe ( )ε ν0 sui able o an a bi a y []∈ , 12 . Since () [] [] ==∕=⋅∕= ∈∈ εν πmin min 0.02sin 0.02sin 6 0.02 1 2 0.01 , ,0, 12 12 we can pu , independen ly o ,()≔ ε ν0.01 . 0 Now, conside a alue ε* p . By o mula (34), wi h ()=∕∈ p σ32 1, and { } =∕< −μσp14 min1, , {[( )]}( )<∕+∕=∕≐ ε *min 1, 1 8 1 2 2 17 0.00019 , p44 and we pu = ε *0.00018 . p Mo eo e , le ( o he p ope ies o ω , we e e o (51) and (52)) =< =ωεc0.0001 *and 5,000 . p Then, by (65), ∣∣ () ()() ⎜⎟ =⎛ ⎝− −⎞ ⎠== = ∕− zσ σc c *cos 1 1cos cos 5,000 0.0002cos , n σ11 and he solu ion o equa ion (66) ∣∣ == ≤≤ ≤ ≤ max max ωz ε 212cos 1.8 ** np gi es he alue =∕ π3 2. Following he abo e-men ioned ecommenda ion, i we eplace ω wi h ω 1 and c wi h c1 p ese ing he p ope y == −− c ωcω0.5 σσ 111 1 , say =ω0.00005 1and = c 10,000 1, we can de i e he same solu- ion. Since, o = 1 , ∣∣ () () () ⎜⎟ =⎛ ⎝− −⎞ ⎠== ∕=≐<= ∕− zσ σc cπε *cos 1 1cos cos 6 5,000 3 10,000 0.00017 *0.00018 , n σ p 111 1 we can pu ( e e ing o (69)) 18 Jose Diblík and Mi osla a Růžičko á =∕ ε ** 3 10,000 . p(77) Then, ≐<= ε ε ** 0.00017 *0.0001 8 pp . Inequali y (63) holds as well since () () () ⎜⎟ ⎛ ⎝−⎞ ⎠== = < = ∕− σc c εν 1111 5,000 0.0002 0.01 . σ11 0 The cu e a cs (70) a e defined by he angles φ wi hin domains (71) and (72), i.e. () () () ()=≤< = + = −<≤ = +− ψπφν π ν π φψ π 00, 32and 0, 20 (78) ha a e pa s o domains (60) and (61), i.e. +<< + −<< − π φ π π φ π 232and 232 , espec i ely. Figu e 3 shows he segmen s o he cu e loops defined by (70), i.e. () (() ) () =−=±− ± zφ ν φ ceπ φ e cos 0, cos 5,000 iφ iφ 00 passing h ough poin s =zεe ** pi π ,=z e i π , whe e =∕ ≐ 2 10,000 0.00014 , and =zωe i π and he domains o φ a e gi en by inequali ies (78) wi h ==∕ π 6 1,==∕ π * 4 , and ==∕ π3 2. Fo he ed segmen , he compu a ions wi h () − ν 0, and ( ) − ν 0, 0 a e ele an , while, o he g een one, he alues ( ) + ν 0, and ( ) + ν 0, 0 a e used. Along hese segmen s, he solu ion o he ini ial p oblem (see (51) and (53)): ()=≔=wz w z z ωe,whe e , ωiπ 0 000 000 is analy ically con inued. No e ha , o diffe en alues o ≠ n 0 , we ob ain iden ical cons uc ions. 6.1.2 Auxilia y lemma The ollowing Lemma 3 and Rema k 5 on he mu ual posi ions o diffe en segmen s o he loops o cu e (8) will be used in u he cons uc ions. Ins ead o 1 and 2 in hei o mula ions, a bi a y alues * 1 and * 2 sa is ying ( ) <<<∕− πσ0** 21 12 can be used. Fo an illus a ion, we e e o Example 2 and Figu e 4. Lemma 3. Le numbe s 1 and 2 be fixed such ha ()<<<∕ − πσ021 . 12 (79) Conside wo cu e a cs ( ) zφ n1and ( ) zφ n2 o (8) defined by he ollowing o mulas: () (( )( )) () () ( ) () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅≤< +∕− + zφ νn σ φ σc eψnφνn cos , 1 1,,, n σiφ 101 1 11 1(80) () (( )( )) () () ( ) () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅≤< +∕− + zφ νn σ φ σc eψnφνn cos , 1 1,, , n σiφ 202 2 11 2(81) whe e c1 and c 2 a e he posi i e cons an s, ( ) + ν n , 01and () + ν n , 02 a e defined by (45), i.e. ()()(())()()(())=− + =− + ++ ν n σ ψn ν n σ ψn ,1 ,,1 , 011 022 (82) and () + ν n , 1, ( ) + ν n , 2a e defined by (47), i.e. ()() () ()() () =++ −=++ − ++ ν n ψn π σand ν n ψ n π σ ,21 ,21 . 11 22 I Vanishing and blow-up solu ions 19 (()) (())=zψn zψn, 12 (83) hen ∣ ()∣ ∣ ()∣ () ( )<<< + zφ zφ o e e yψn φ νn ,, . nn12 1 (84) P oo . F om (80)–(82), we see ha (83) implies () ()−=−σ cσ c cos 1 cos 1 . 1 1 2 2 (85) Now, on he in e al () ( ) ≤< + ψn φ ν n ,1, we will in es iga e he p ope ies o he unc ion:  () (( )( )) (( )( )) =−− −−− ++ φνn σ φ cνn σ φ c cos , 1 cos , 1 . 02 2 01 1(86) By (85),  (()) () () =−−−=ψn σ cσ c cos 1 cos 1 0 . 2 2 1 1 (87) Due o (79), we ha e ()()<− <− <∕σ σ π01 1 2 12 . Then, () ()<−<−σ σ 0 cos 1 cos 1 , 21 and, analysing (87), we ha e < c c 2 1 . Mo eo e ,  () ()(()())()(()()) ′=−−− −−−− ++ φσνn σφ cσνn σφ c 1sin , 1 1sin , 1 02 2 01 1 and  (()) ()()()() ′=≔ −− −−− ψn A σσ cσσ c 1sin 1 1sin 1 . 2 2 1 1 Due o (79), we ha e () ()−> −>σ σ s in 1 sin 1 0, 21 Figu e 4: Visualiza ion – o Lemma 3, Rema k 5, and Example 2. 20 Jose Diblík and Mi osla a Růžičko á and, consequen ly,  (())′=>ψn A 0. Compu a ion o he second-o de de i a i e leads o  () ()(()())()(()()) ″=− −−− +−−− ++ φσνn σφ cσνn σφ c 1cos , 1 1cos , 1 . 202 2 201 1 The e o e,  ( ) φsa isfies he ollowing Cauchy ini ial p oblem o a diffe en ial second-o de equa ion:   ()( )() (()) (()) ″+− = = ′ =φσ φ ψn ψnA10, 0,and . 2(88) The gene al solu ion o (88)is  () () ()=−+−φC σ φC σ φsin 1 cos 1 , 12 whe e, o he specifica ion o a bi a y cons an s C 1 and C 2 by he ini ial condi ions, we use he equa ions  (()) ()() ()()=−+−=−=ψn C σ ψn C σ ψn Csin 1 cos 1 0, 12 2  (())()()()()′=− − =−−=ψn C σ σ ψn C σ A1cos 1 1 . 11 The solu ion o he p oblem (88)is  () ( )=− −−φA σσφ 1sin 1 . (89) The angle φ a ies wi hin he in e al indica ed in (84), which implies ()()()() ()+<−<++−+∕<++nπσφnπσ π nππ21 1 21 1 221 . 1 Fo such alues o φ , we ha e ()−<σφ s in 1 0 ; he e o e, o  ( ) φexp essed by (86) and (89),  () (( )( )) (( )( )) ()=−− −−− =− −−> ++ φνn σ φ cνn σ φ cA σσφ cos , 1 cos , 1 1sin 1 0 . 02 2 01 1 F om his inequali y, we deduce ha  ()>φ0 also holds in he same in e al, whe e  () (( )( )) () (( )( )) () () () ⎜⎟⎜⎟ ≔⎛ ⎝ −− −⎞ ⎠−⎛ ⎝ −− −⎞ ⎠ +∕− +∕− φνn σ φ σc νn σ φ σc cos , 1 1cos , 1 1 . σσ 02 2 11 01 1 11 This inequali y is equi alen wi h (84). □ Rema k 5. A p ope y simila o ha o mula ed in Lemma 3, i.e. he inequali y ∣ ()∣ ∣ ()∣ ( ) ( ) <<≤ − zφ zφ νn φ ψn, o e e y , nn12 1 can be p o ed in much he same way, i he ollowing segmen s o loops a e conside ed ins ead o (80) and (81): () (( )( )) () () () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅<≤ −∕− − zφ νn σ φ σc eνn φψn cos , 1 1,, , n σiφ 101 1 11 1 () (( )( )) () () () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅<≤ −∕− − zφ νn σ φ σc eνn φψn cos , 1 1,, , n σiφ 202 2 11 2 ( ) − ν n , 01and () − ν n , 02 a e defined by (46) and () − ν n ,1and ( ) − ν n ,2a e defined by (48). Example 2. We use some cons uc ions om Example 1 o illus a e Lemma 3 and Rema k 5. We e e o Figu e 4 whe e he wo ed and wo g een loop segmen s can be seen passing h ough he poin =z e i π , whe e =∕ ≐ 2 10,000 0.00014 . Pu = c 2,500 6 1 and = c 2,500 2 2. The inne g een segmen is defined by he o mula: () (( ) ) () =−=∕− + zφ ν φ ceπφ e cos 0, cos 7 6 2,500 6 , ig iφ iφ 001 1 Vanishing and blow-up solu ions 21 whe e () ( )=≤<∕= + ψπφπν 0530, , 1 while he ou e g een one is defined by he o mula: () (( ) ) () =−=∕− + zφ ν φ ceπφ e cos 0, cos 4 3 2,500 2 , og iφ iφ 002 2 whe e () ( )=≤< ∕= + ψπφπν 01160, . 2 The inne ed segmen is defined by he o mula: () (( ) ) () =−=∕− − zφ ν φ ceπφ e cos 0, cos 5 6 2,500 6 , i iφ iφ 001 1 whe e ()=∕< ≤ − ν π φπ0, 3 , 1 while he ou e ed one is defined by he o mula: () (( ) ) () =−=∕− − zφ ν φ ceπφ e cos 0, cos 2 3 2,500 2 , o iφ iφ 002 2 whe e ()=∕< ≤ − ν π φπ0, 6 . 2 Assump ion (83) holds since () () () ()====∕zπ zπ zπ zπ 1 5,000 2 ig og i o 0000 and ∣ ()∣ ∣ ()∣<<<∕zφ zφ πφ π,i 53 ig og00 and ∣ ()∣ ∣ ()∣<∕<<zφ zφ π φ π,i 3 . i o This is in acco dance wi h he asse ions o Lemma 3 and Rema k 5. Rema k 6. Le us poin ou ano he p ope y o he mu ual posi ion o diffe en segmen s o loops o he cu e (8), which is ob ious. I wo segmen s () (()( )) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅ +∕− zφ νn σ φ σc e cos , 1 1 σiφ 10 0 1 11 and () (()( )) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅ +∕− zφ νn σ φ σc e cos , 1 1, σiφ 20 0 2 11 whe e []∈ , 12 is fixed, a e defined o () ( )≤< + ψn φ ν n , and < c c 2 1 , hen ∣ ()∣ ∣ ()∣<zφ zφ . 10 20 The same p ope y holds i ( ) + ν n , 0 is eplaced by ( ) − ν n , 0 and domain o φ is () ( ) <≤ − ν n φ ψn, . This p ope y is isualized in Figu e 5, whe e = 2 wi h de ails desc ibed in Example 3. Example 3. In he ollowing, some cons uc ions om Example 1 a e used o illus a e he asse ions o Rema k 6. We e e o Figu e 5 whe e he wo ed and wo g een loop segmen s a e isualized passing h ough he poin s =zεe ** pi π and =zωe i π . The inne g een loop segmen is defined by he o mula: 22 Jose Diblík and Mi osla a Růžičko á () (( ) ) () =−=∕− + zφ ν φ ceπφ e cos 0, cos 4 3 5,000 , ig iφ iφ 002 1 whe e ()≤< ∕= + πφ π ν 11 6 0, , 2 while he ou e g een one is defined by he o mula: () (( ) ) () =−=∕− ∕ + zφ ν φ ceπφ e cos 0, cos 4 3 5,000 3 , og iφ iφ 002 2 whe e ()≤< ∕= + πφ π ν 11 6 0, . 2 The inne ed loop segmen is defined by he o mula: () (( ) ) () =−=∕− − zφ ν φ ceπφ e cos 0, cos 2 3 5,000 , i iφ iφ 002 1 whe e ()=∕< ≤ − ν π φπ0, 6 , 2 while he ou e ed one is defined by he o mula: () (( ) ) () =−=∕− ∕ − zφ ν φ ceπφ e cos 0, cos 2 3 5,000 3 , o iφ iφ 002 2 whe e ()=∕< ≤ − ν π φπ0, 6 . 2 We see ha ∣ ()∣ ∣ ()∣<≤<∕zφ zφ πφ πi 11 6 ig og00 and Figu e 5: Visualiza ion – o Rema k 6 and Example 3. Vanishing and blow-up solu ions 23 ∣ ()∣ ∣ ()∣<∕<≤zφ zφ π φπi 6 . i o 00 This is in acco dance wi h he asse ions o Rema k 6. 6.1.3 Con inua ion o (()) w z n 0 o a domain P n ω Le n be fixed. Va ying and in admissible bounda ies, by he heo em on he exis ence o a unique solu ion o he ini ial p oblem, we ob ain an analy ic con inua ion o he solu ion ( ) wz n 0,defined by ini ial p oblem (53) wi h he ini ial poin ( ) wz 0, whe e =zz ω n 0is de e mined by (51), on a domain P n ω ,defined as a domain co e ed by all he abo e-men ioned loop segmen s o he cu e (8). The bounda ies o and a e he ollowing. The alue a ies wi hin he in e al (69), whe e ω is sufficien ly small and ε * * p is defined in such a way ha he segmen s o loops (70) o cu e (8) a e defined by angles φ sa is ying (71) and (72). The alue a ies as indica ed in (56) wi h 2 defined by (66) and wi h 1 defined by (58). Then, among o he s, he inequali y ∣()∣<wz e nk 0holds. F om he me hod o cons uc ion and (56), (71), (72), and (73), i ollows ha he domain P n ω is simply connec ed and lies in he sec o  ( ) φ n, cen ed a he poin =z0,defined as:  () () () () () ⎜⎟⎜⎟ ≔⎧ ⎨ ⎩−⎛ ⎝+−⎞ ⎠≤≤ + ⎛ ⎝+−⎞ ⎠ ⎫ ⎬ ⎭ φφψn π σφψn π σ :21 21 . n22 The sec o  ( ) φ nis loca ed ei he in he complex plane  o in a Riemann su ace and does no con ain he o igin. Applying (56), he leng h o he in e al o φ defining  ( ) φ nsa isfies ()() +−<− π σπ σ 2121 . 2 Assume now ha n is no fixed. T acing ca e ully compu a ions in Pa s 6.1.1 and 6.1.2, we see ha hese do no depend on any o he alues n . So, o e e y n , he abo e-men ioned conside a ions a e co ec . This means ha wo domains P n ω 1and P n ω 2 , whe e ≠ n n 12 , a e geome ically iden ical (ha ing iden ical o ms). These domains can diffe only by hei loca ion in he complex plane o in a Riemann su ace and, in such a case, hey do no in e sec . The ollowing ob ious s a emen is ue abou he numbe o domains P n ω wi h diffe en loca ions ( wo domains P n ω 1and P n ω 2 a e diffe en i ∩=∅ P P nω nω 12 o ≠ n n 12 ). (a) I ∈σ, hen he e exis −σ 1 diffe en domains ()⊂ P φ nω n ,=− n σ0,…, 2, whe e he sec o s  ( ) φ n, =− n σ0,…, 2 a e loca ed in he complex plane  . (b) I ∈⧹σ, =∕σmm 12 , and ∈mm, 12 a e ela i ely p ime, hen he e exis −mm 12 diffe en domains ()⊂ P φ nω n , =−− n mm0,…, 1 12 , whe e he sec o s  ( ) φ na e loca ed on he Riemann su ace o he unc ion ∕ zm12. (c) I ∈σ, hen he e exis s an infini e coun able se o diffe en domains ()⊂ P φ nω n ,=± n 0, 1,… , whe e he sec o s  ( ) φ na e loca ed on he Riemann su ace o he loga i hmic unc ion. Now, we desc ibe in de ail he cons uc ion o he domain P n ω based on he p ope ies o he cu e segmen s gi en in Lemma 3, Rema k 5, and Rema k 6. Fo a isualiza ion, we e e o Figu e 6, which illus a es he ele an cons uc ions o Example 4. The domain P n ω con ains all poin s be ween he ollowing wo cu es including all bounda y poin s excep o he poin =z0. The bounda y o P n ω is o med by wo closed, simple, and con inuous cu es “embedded”in each o he (below called he inne and ou e bounda ies) wi h a unique common poin =z0. The inne bounda y is o med by wo segmen s o loops (70) specified by ( o he defini ions o ± ν 0 and ± ν , we e e o o mulas (45)–(48)): ()()(())() ()==−+ ≤< ++ ν νn σ ψn ψn φ νn ,1 , , , 0011 1 (90) ()()(()) () ()==−− <≤ −− ν νn σ ψn νn φ ψn,1 ,, , 00111 (91) 24 Jose Diblík and Mi osla a Růžičko á and passing h ough he poin (() ) ==zz ω iψnexp ωn0, de e mined by (51), while he ou e bounda y is defined by wo segmen s o loops (70) specified by: ()()(())() ()==−+ ≤< ++ ν νn σ ψn ψn φ νn ,1 , , , 0022 2 (92) ()()(()) () ()==−− <≤ −− ν νn σ ψn νn φ ψn,1 ,, , 00222 (93) and passing h ough he poin (() ) ≔zε iψn **exp ** εp p . Fo he defini ion o ε * * p , we e e o he explana ion accompanying inequali ies (69). This cons uc ion is co ec since, by Lemma 3 and Rema k 5, a cs defined by (90) and (92) ha e no in e sec ion o () ( ) << + ψn φ ν n ,1and a cs defined by (91) and (93) ha e no in e sec ion o () ()<< − ν n φ ψn, 1 . Example 4. Using se e al cons uc ions o Example 1 again, we will cons uc he domain P ω 0 . The inne bounda y o P ω 0 consis s o wo segmen s o loops (70) passing h ough he poin =zωe i π wi h = 1 and = c 5,000 3. The ed segmen in Figu e 6 is defined as: () (( ) ) () =−=∕− − zφ ν φ ceπφ e cos 0, cos 5 6 5,000 3 , iφ iφ 001 whe e () ()=∕< ≤ = − ν π φπψ0, 3 0 , 1 while he g een one in Figu e 6,isdefined as: () (( ) ) () =−=∕− + zφ ν φ ceπφ e cos 0, cos 7 6 5,000 3 , iφ iφ 001 whe e () ( )=≤<∕= + ψπφπν 0530, . 1 The ou e bounda y o P ω 0 consis s o wo segmen s o loops (70) passing h ough he poin =zεe ** pi π wi h = 2 and =∕ c 5,000 3 . The ed segmen in Figu e 6 is defined as: Figu e 6: To Example 4 –domain P ω0. Vanishing and blow-up solu ions 25 whe e () ()−=−≤< = −− ν ππ φν π 0, 23 0, 6 . 22 Finally, by equa ion (117), he segmen o he ay ( ) − z 10 is defined by: () ∣ ( ( ))∣ () == <≤ = −∕−− − z e e z ν ε,0 0, ** . iν πi p10 0, 3 11 1 The domain − Ω ω 0 and i s bounda y can be seen in Figu e 8. The union ∪ +− Ω Ω ω ω 00 is hen isualized in Figu e 9. 6.3 Domain (())Pn and p ope y ∅ ∅ (())∩∩((++))≠≠Pn Pn 1 F om he p e ious conside a ions, we conclude ha he solu ion ( ) wz n 0is analy ically con inued on he domain: ()≔∪∪ −+ P nPΩΩ . nω nω nω P o ided ha () ( )≢+ P nPn1, we ha e () ( )∩+≠∅ P nPn1since ∩≠∅ ++ − Ω Ω . nω n ω1, (121) The las p ope y ollows om (47), (48), (56), (96), and (112) because, in he opposi e case, he inequali y ()() () ()() () () (( ) ) ()() ⎜⎟ ⎜⎟ +−=+ −+⎛ ⎝+−⎞ ⎠+− <+− − =++ −−⎛ ⎝+−⎞ ⎠−− + − νn π σnπ σ π σπ σ νn π σ nπ σ π σπ σ ,2121 12121 1, 21 211 12121 (122) mus hold. Simpli ying (122), we ob ain < 0, which con adic s (56), i.e. () ( )∩+ P nPn1con ains an open se and, by cons uc ion, is a simply connec ed domain. Figu e 9: To Example 5 –domain ∪+ Ω Ω ωω0 ‒0. 32 Jose Diblík and Mi osla a Růžičko á Rema k 7. In he ollowing, we e e o Example 5. The domain () P 0is shown in Figu e 10. We do no isualize he p ope y (121) in an addi ional figu e since i can be explained by he same one. I was shown abo e ha wo disjunc P n ω 1and P n ω 2 wi h ≠ n n 12 a e iden ical up o hei loca ions in he complex plane o Riemann su ace. In much he same way, one can p o e ha he domains − Ω n ω 1and − Ω n ω 2a e iden ical wi h he domains + Ω n ω 1and + Ω n ω 2. They can diffe only by hei loca ions. Since, in Figu e 9, we ha e ∩≠∅ +− Ω Ω ωω00 ( his in e - sec ion is highligh ed in blue), he ollowing will be ue as well: ∩≠∅ − +− Ω Ω ωω10, ∩≠∅ +− Ω Ω ωω01 . No e ha , o =σ2 o o a a ional =∕σmm 12 , whe e −=mm 1 12 , he e exis s only a single domain ( )P n. All o he s ha e iden ical o ms and loca ions. Since =σ2 , we ha e ⋃= = − n σn 0 2110 and   ≔⧹ 110 . A ci cle neighbou hood o he o igin is d awn in Figu e 11. I s adius equals ∣() ∣ − z0 20 , whe e he cu e () − zφ 20 is defined by o mula (120) and by: ∣()∣ () == = ∕∕≐ −− z ν cπ 0cos 0, cos 6 20,000 3 0.00013 . 20 2 II Inequali y (49) is now  ∣()∣ () ∣∣ ∣∣ ⎟ ⎜ ⎜⎟ ≤⎧ ⎨ ⎩ ⎛ ⎝−−⎞ ⎠ ⎫ ⎬ ⎭=⎧ ⎨ ⎩ ⎛ ⎝−⎞ ⎠ ⎫ ⎬ ⎭ ∈− wz e δ aσ zδz min , exp sin 1 min 1, exp 1 2 , zkp1 1 1(123) whe e ( ) <∕δ exp 1 2 * and ()=−= ε σ **sin 1 0.00009 p1. We will show ha he in e sec ion () ( )∩+ P nPn1con ains a pa o he ay (18), whe e () () () =≔+ −=+ − ν νn ψn π σnπ σ *121 1 . (124) This is a consequence o he ac ha he angle ( ) = ν νn *belongs o wo in e als, namely, (97) and (113), whe e n is eplaced by + n1 since he chain o inequali ies () ()() ( ) () +− −<< + − −+ ν n π σνn νn π σ 1, 21*,21 holds being equi alen wi h − << 0 . This ay is defined o ( ) ∈′ 0, , whe e ′ is defined by he in e sec ion o loop segmen s (() ) + zνn * n2and (()) + − zνn * n2, 1 , i.e. Figu e 10: To Example 5 and Rema k 7 –domain ()≔∪∪ + P P0Ω Ω ωω ω 0 ‒00. Vanishing and blow-up solu ions 33 ∣ ( ())∣ ∣ ( ())∣ (( ) ) () () ⎜⎟ ′= = =⎛ ⎝−−∕ −⎞ ⎠ ++ −∕− zνn z νn σ π σc ** cos 1 2 1 . nn σ 22,1 2 II 11 Mo eo e , le γ0be a fixed numbe such ha <≤γ 0 , 01(125) whe e 1 sa isfies (56). Then, he domain () ( )∩+ P nPn1con ains a sec o a ound he angle ( )ν n *wi h cen e a he poin =z0defined by he inequali y: () () () ()−= + −−≤≤ + −+= + ν nγ nπ σγφ nπ σγνnγ *21 121 1* 00 0 0 (126) since he inequali y () ()()() +− −≤+ − −+ ν n π σνn π σ 1, 21 ,21 is equi alen o () ()−< + ν n νn ** and, he e o e, () () () ()−< −< +< + ν n νnγνnγνn **** . 00 (127) Inequali ies (56) and (127) imply (126). 6.4 Solu ions analy ically con inuable o (())∪∪((++))Pn Pn 1 The ollowing explana ion can be ollowed in Figu es 12 and 13, Example 5, and Rema k 8. Le ℓ () ( ) ⊂nPn 1and ℓ () ( ) ⊂+nPn1 2be he segmen s o wo loops o he cu e (8) symme ic wi h espec o he ay (18), defined by (124), s a ing a he poin =z0and such ha , o e e y φ sa is ying () ()−<< + ν nγφνnγ ** , 00 whe e γ0sa isfies (125), we ha e Figu e 11: To Example 5 and Rema k 7 –a ci cle neighbou hood o =z0 . 34 Jose Diblík and Mi osla a Růžičko á ℓ () () ( ) { }∩ ∩ + ∩ = < <∞ ≠∅ =nPnPn z e j1,0 ,1,2 . jiφ Assume as well ha , on he cu e ℓ ( ) n 1, we ha e () () = →+ zφlim 0 , φ νnγ *0 while on he cu e ℓ ( ) n 2, () () = →− zφlim 0 . φ νnγ * 0 To define he cu e ℓ ( ) n 1, we pu () ℓ =≔−∕+− ν νπσγ21 , 00 0 1(128) and o define he cu e ℓ ( ) n 2, we pu () ℓ =≔∕−− ν νπ σγ21 00 0 2(129) in (8). Then, he poin o in e sec ion ℓ() ℓ( ) =∩ M nn n12 lies on he ay (18) defined by angle (124). This is a consequence o he ollowing compu a ion i ( ) = φ νn * . Fo ℓ ( ) n 1, we ha e (()())( ()) ℓ −− = −∕+−νσνn πσγ c os 1 *cos 2 1 . 00 1 In he case o ℓ ( ) n 2, we ob ain he same esul since (()())(()) ℓ −− = ∕−−νσνn πσγ c os 1 *cos 2 1 . 00 2 The symme y p ope y can be p o ed in much he same way. The equa ions defining he cu es ℓ ( ) n 1and ℓ ( ) n 2a e Figu e 12: To Example 5 and Rema k 8 –a ci cle neighbou hood o =z0 . Vanishing and blow-up solu ions 35 () ( ( )( )) () ℓ() () ⎜⎟ =⎛ ⎝ −∕+ − − −⎞ ⎠⋅ ∕− zφ πσγφ σc e cos 2 1 1 n σi φ 011 1 and () ( ( )( )) () ℓ() () ⎜⎟ =⎛ ⎝ ∕− − + −⎞ ⎠⋅ ∕− zφ πσγφ σc e cos 2 1 1 , n σiφ 011 2 whe e () ()−<≤ + ν nγφνnγ ** 0 0 . In bo h cases, we assume ha he pa ame e c is iden ical and sufficien ly la ge. A sui able alue o c can be de e mined, e.g., as he solu ion o equa ion: ∣ (() ( ))∣ ∣ ( ( ))∣ ℓ() +∕ − = ++ zψn πσ z νn 1, . nn 22 1(130) Rema k 8. We use Example 5 again o isualize cu es ℓ ( ) 0 1and ℓ ()0 2. Knowing ha domains o he ype ( )P n ha e he same o m, wi hou loss o gene ali y, we can use he p e ious cons uc ions. The e o e, o cla i y, we do no eplace he alue = n 0 wi h = n1 (excep o () P 1). Bo h cu es a e isualized in Figu es 12 and 13 (highligh ed in blue and in ed), whe e, by o mula (124), ()= νπ *02 and ==∕γ π 6 01 . Cu es ( ) ± zφ 20 a e defined by o mulas (119) and (120). Sol ing equa ion (130), whe e = n 0 , leads o = c cII. The o mulas o ℓ ( ) 0 1and ℓ ()0 2a e () () () ℓ =−∕+ − ⋅= −∕− ∕⋅zφ πγφ ceπφ e cos 2 cos 3 20,000 3 iφ iφ 0 II 1 and () ()() ℓ=∕− − ⋅= ∕− ∕⋅zφ πγφ ceπφ e cos 2 cos 3 20,000 3 , iφ iφ 0 II 2 whe e − ∕< ≤∕πφπ6 6 . Assuming ha φ a ies wi hin he domain desc ibed by he inequali ies: () ()−≤≤ + ν nγφνnγ ** , whe e ( ) ∈γγ0, 0is fixed, conside he beha iou o solu ions o in eg al cu es gi en by (1)on ℓ ( ) n 1and ℓ ( ) n 2 wi h espec o a fixed cylinde  ( ) λdefined by (24), i.e. wi h espec o he cylinde : Figu e 13: To Example 5 and Rema k 8 –a ci cle neighbou hood o =z0 –zoom. 36 Jose Diblík and Mi osla a Růžičko á += α βe . λ222 (131) Ini ially, we will examine he case o he cu e ℓ ( ) n 1showing ha any in eg al cu e in e sec ing he la e al su ace o cylinde (131) o i s lowe base defined by he plane ()=− φ νn γ * (132) emains inside he cylinde i φ inc eases. We assign o each poin on he lowe base o he cylinde a poin lying in he plane ( ) = φ νn * (133) being defined as i s mapping by he co esponding in eg al cu e when φ s a s wi h he alue (132) and ends wi h he alue (133). Such a beha iou o in eg al cu es is a consequence o Lemma 1, (i) whe e he geome- ical meaning is gi en by inequali ies (26) and (27), since (()) ℓ ==−∕+−<νν πσγ s in sin sin 2 1 0 . 00 1 This inequali y holds, we e e o (125), (128), and also o Rema k 1. The a o emen ioned mapping is con inuous implemen ing a one- o-one co espondence be ween he lowe base, i.e. he se { () () } =−+≤φαβ φ ν n γα β e,, : *,λ222 and a simply connec ed nonemp y domain  ( ) n *such ha  () {( ) () }⊂=+≤nφαβφνnαβe *,, : *, . λ222 In addi ion, o each poin o he domain  ( ) n *, he e co esponds a solu ion o (1), which is analy ical on ℓ ( ) n 1. On ℓ ( ) n 2, one can p oceed in much he same way. Any in eg al cu e in e sec ing he la e al su ace o cylinde (131) o i s uppe base defined by he plane ()=+ φ νn γ * , (134) emains inside he cylinde i φ dec eases. We assign o each poin on he uppe base o he cylinde a poin on he plane ( ) = φ νn * (135) as a esul o mapping along he co esponding in eg al cu e when φ s a s wi h he alue (134) and ends wi h he alue (135). Such a beha iou o in eg al cu es is a consequence o Lemma 1, (ii) whe e he geome ical meaning is cha ac e ized by inequali ies (26) and (27) since, due o (125) and (129), (()) ℓ ==∕−−>νν πσγ s in sin sin 2 1 0 . 00 2 In his case, he uppe base o cylinde (131), i.e. he se { () () }=+ +≤φαβ φ ν n γ α β e,, : *, λ222 is mapped in o a simply connec ed nonemp y domain  ( ) n ** such ha  () {( ) () }⊂=+≤nφαβφνnαβe ** ,, : *, . λ222 By cons uc ion, o each poin o he domain  ( ) n ** , he e co esponds a solu ion o (1) analy ical on ℓ ( ) n 2. Nei he o he domains  ( ) n *and  ( ) n ** will degene a e o a poin being a closed neighbou hood o he poin (() )νn *,0,0 in he plane  { () () }=∈φαβ φ ν n αβ,, : *,, since he sys em (16), (17) has a i ial solu ion ()( ) =φαβ φ, , ,0,0 o () ()−≤≤ + ν nγφνnγ ** . Thus, he se  () () ( ) ≔∩nn n *** is compac wo-dimensional. By he cons uc ion, o each poin o he domain  ()n, he e co esponds a solu ion o (1) analy ical on ℓ ( ) n 1and ℓ ( ) n 2. This means ha he solu ion is Vanishing and blow-up solu ions 37 analy ical on () ( )∪+ P nPn1. Wi hin he domain ( )P n, his solu ion has p ope ies desc ibed p e iously o a solu ion ( ) wz n 0and, wi hin domain ( ) + P n1, such a solu ion ( ) + wz n1 0 has, as ou cons uc ions do no depend on n , he same p ope ies as indica ed o he solu ion ( ) wz n 0defined on ( )P n. Deno ing such a solu ion by () + wz nn,1 0 , we eplace by i he p e iously conside ed solu ion ( ) wz n 0. Since  ()nis a wo-dimensional se , he e exis infini ely many solu ions o his ype. 6.5 Solu ions analy ically con inuable o (())∪∪((++))∪∪((++))Pn Pn Pn12 Le us epea he conside a ions o he p e ious sec ion o he domain: ()()+∪ + P nPn12 . Now, adap ing he p e ious no a ion, he se  ()() ( ) +≔ +∩ +nn n1*1** 1is compac wo-dimensional. By he cons uc ion, o each poin o he domain  ()+n1, he e co esponds a solu ion o (1) analy ical on ℓ ( ) +n1 1and ℓ ( ) +n1 2. Thus, such a solu ion is analy ical on ()( ) +∪ + P nPn12 . Wi hin he domain ( ) + P n1, his solu ion has he abo e-desc ibed p ope ies and, wi hin he domain ( ) + P n2, such a solu ion has, as ou cons uc ions do no depend on n , he same p ope ies as indica ed o ( ) + P n1. Then, he e exis s a compac wo-dimensional se  () ()⊂nn 1(we a gue in much he same way as ea lie ) such ha , h ough e e y poin o he domain  ()n 1, a solu ion o (1) passes analy ical on () ( ) ( ) ∪+∪+ P nPn Pn12 . Deno ing such a solu ion by () + wz nn,2 0, eplace by i he p e iously conside ed solu ion () + wz nn,1 0 . Since  ()n 1is a wo-dimensional se , he e exis infini ely many solu ions o his ype. 6.6 Solu ions analy ically con inuable o (())∪∪⋯⋯∪∪((++))Pn Pn N Le he abo e-men ioned cons uc ion be ca ied ou o each fixed in ege n and le N be an a bi a y na u al numbe . As a esul , one can always find an infini e se o solu ions o (1) ha a e analy ical on he domain: () ( ) ( ) ( )∪+∪+∪⋯∪+ P nPn Pn PnN12 . (136) I σ is an in ege , hen, o = n 0 and =− N σ 1 , he cycle (136) closes since ()( ) −= P σP10 . Then, we ob ain an infini e se o solu ions ha a e analy ical in he domain: () () () ( ) ( )∪∪∪⋯∪−∪− P P P Pσ Pσ012 2 1 , whe e () ( ) =− P Pσ01 . I σ is a a ional numbe , =∕σmm 12 , hen, o = n 0 and =− N mm 12 , he cycle closes on he Riemann su ace o he unc ion ∕ zm12since ()( ) −= P mm P0 12 . Then, we ob ain an infini e se o solu ions ha a e analy ical in he domain: () () ( ) ( )∪∪⋯∪ −−∪ − P PPmmPmm01 1 , 12 12 whe e () ( ) =− P Pm m012 . I σ is an i a ional numbe , hen he cycle does no close. The e o e, we can only say ha , o an a bi a y na u al numbe N and an in ege = n n 0 , he e exis s an infini e se o solu ions o (1) ha a e analy ical in he union o a fini e numbe o domains: () ( ) ( ) ( ) ∪+∪+∪⋯∪+ P nPn Pn PnN12 00 0 0 loca ed on he Riemann su ace o he loga i hmic unc ion. 38 Jose Diblík and Mi osla a Růžičko á 7 Example Le , in an equa ion (1), =σ3 ,= a1 , and = M2 . Le =ρ 1 and = k0 . By (50), we ha e () ()() == + −=+ ν ψn nπ σnπ21 121 2, and he e a e wo diffe en alues o = n 0, 1 in he complex plane  , ()=∕ψπ0 2 and ()=∕ψπ13 2 . The espec i e cons uc ions can be seen in Figu e 14. In acco dance wi h (23), pu {}<= < ⎧ ⎨ ⎩ ⎫ ⎬ ⎭==ερ a M 00.02min, 2min 1, 0.25 0.25 . 0 Nex , we apply o mula (34) wi h ()=∕∈ p σ32 1, and {)=∕< −μσp14 min1, pu ing ()=< ⎧ ⎨ ⎩⎛ ⎝+∕⎞ ⎠⎫ ⎬ ⎭=∕ ≐ ε *0.00018 min 1, 1 812 2 17 0.00019 . p 44 Le =<ωε0.0001 * p and =∕ c 10 4 8. Then, by (65), ∣∣ () () () ⎜⎟ =⎛ ⎝− −⎞ ⎠=⎛ ⎝∕⎞ ⎠= ∕− ∕− zσ σc *cos 1 1cos2 10 2 10 2cos2 , n σ11 8 12 4 and he solu ion o equa ion (66) ∣∣ == ≤≤ ≤ ≤ max max ωz ε 212cos21.9 ** np gi es he alue =∕ π 6 2 . In (58), we se =∕ ζ 12 . Then, =∕=∕ π212 12 . Fo ± ν 0 defined by o mulas (45) and (46) wi h = n 0, 1 , we ha e, by (50), () () () () () () () () =∕ = ∕ =∕ = ∕ =∕ = ∕ =∕ =∕ ++ −− ++ −− ν π ν π ν π ν π ν π ν π ν π ν π 0,76,1,196,0,56,1,176, 0,43,1,103,0,23,1,83. 01010101 02020202 Figu e 14: Case =σ3 –domains P ω0and P ω1. Vanishing and blow-up solu ions 39 By (22), we can pu ()∣()∣∣()∣== ±± ε νεν ν sin 0, 0.02 sin 0, , 0000 whe e a dependence on is emphasized. Because () ∣ ( )∣ [] [] ==∕= ∈∈ ± εν ν πmin min 0.02 sin 0, 0.02sin 6 0.01 , ,0,0 12 12 we can, independen ly o , pu ()≔ ε ν0.0 1 0. Since, o = 1 , ∣∣ () () () ⎜⎟ ⎜⎟ =⎛ ⎝− −⎞ ⎠=⎛ ⎝∕ ∕⎞ ⎠=⎛ ⎝ ∕ ∕⎞ ⎠≐<= ∕− ∕∕ zσ σc πε *cos 1 1cos 6 10 2 32 10 2 0.0001316 *0.00019 , n σ p 111 8 12 8 12 we can pu =∕ ≐ <= ε ε ** 3 10,000 0.00013 *0.0001 8 pp 4. Inequali y (63) holds as well since () () () ⎜⎟ ⎛ ⎝−⎞ ⎠=⎛ ⎝⎞ ⎠=∕⋅ ≐ < = ∕− ∕− σc c εν 111 22 2 10 0.0000707 0.01 . σ11 12 40 Fo comple eness, we compu e he angles, defined by (47) and (48): () () () () () () () () =∕ = ∕ =∕ =∕ =∕ =∕ =∕ =∕ ++−− ++−− ν π ν π ν π ν π ν πν πν πν π 0,56, 1,116,0, 6,1,76, 0, 11 12, 1, 23 12, 0, 12, 1, 13 12. 11 11 22 22 Now, le us cons uc he domains P ω 0 and P ω 1 . Pu =∕⋅ c 34 10 i 8 and =∕ ⋅ c 112 10 . o8The inne bounda y o P ω 0 consis s o wo segmen s o loops (70) passing h ough he poin =∕ zωe iπ 2 wi h = 1 . The ed segmen is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ ν σφ σc eπφ ce cos 0, 1 1cos 5 6 2 2, i σiφ i iφ 00111 12 whe e () ()=∕< ≤∕= − ν π φπ ψ0, 6 2 0 , 1 while he g een one is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ ν σφ σc eπφ ce cos 0, 1 1cos 7 6 2 2 , ig σiφ i iφ 00111 12 whe e () ( )=∕≤ < ∕=+ ψπφπν 02 560, . 1 The ou e bounda y o P ω 0 consis s o wo segmen s o loops (70) passing h ough he poin =∕ zεe ** piπ 2wi h = 2 . The ed segmen is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ ν σφ σc eπφ ce cos 0, 1 1cos 2 3 2 2, o σiφ o iφ 00211 12 whe e () ()=∕ < ≤∕= − ν π φπ ψ0, 12 2 0 , 2 while he g een one is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ ν σφ σc eπφ ce cos 0, 1 1cos 4 3 2 2, og σiφ o iφ 00211 12 whe e () ( )=∕≤ < ∕ = + ψπφπν 0 2 11 12 0, . 2 The inne bounda y o P ω 1 consis s o wo segmen s o loops (70) passing h ough he poin = ∕ zωe iπ32 wi h = 1 . The ed segmen is defined as: 40 Jose Diblík and Mi osla a Růžičko á () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ ν σ φ σc eπφ ce cos 1, 1 1cos 17 6 2 2 , i σiφ i iφ 10111 12 whe e () ()=∕<≤∕= − ν πφπψ1, 7 6 3 2 1 , 1 while he g een one is defined as: () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ ν σ φ σc eπφ ce cos 1, 1 1cos 19 6 2 2 , ig σiφ i iφ 10111 12 whe e () ( )=∕≤< ∕= + ψπφπν 132 116 1, . 1 The ou e bounda y o P ω 1 consis s o wo segmen s o loops (70) passing h ough he poin =∕ zεe ** piπ32 wi h = 2 . The ed segmen is defined as: () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ ν σ φ σc eπφ ce cos 1, 1 1cos 8 3 2 2 , o σiφ o iφ 10211 12 whe e () ()=∕<≤∕= − ν πφπψ1, 13 12 3 2 1 , 2 while he g een one is defined as: () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ ν σ φ σc eπφ ce cos 1, 1 1cos 10 3 2 2 , og σiφ o iφ 10211 12 whe e () ( )=∕≤< ∕= + ψπφπν 132 2312 1, . 2 Figu e 15 shows a ci cle neighbou hood o he o igin in  whe e he e exis solu ions wi h he p ope ies indica ed in Theo em 1. The adius o he ci cle ∣()∣==⋅ +− z01.510 21 4is compu ed using he cu es con- s uc ed in Pa 6.2.1. We ha e ∣ ( ( ))∣ (()( )()) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠ =⎛ ⎝∕− ∕⎞ ⎠=⋅ +++ ∕− ∕− zν ν σ ν σc ππ c 1, cos 1, 1 1, 1 cos 10 3 11 3 2310. og σ o 101020111 12 4 The cu e ( ) + zφ 11 is defined as (we e e o o mulas (101) and (102)) () (( ) ( ) ( )) () (() ) () () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −−− −⎞ ⎠ =⎛ ⎝−⎞ ⎠≤≤ ++∕− +∕++ zφ σν σφ σc e ν φ ce ν n φ ν n cos 1 1, 1 1 cos 2 1, 2 2,, ,, σiφ iφ 11 1 I 11 1 I 12 12 whe e he alue o he pa ame e cI is gi en by he equa ions: (()( )()) () () () () ( ) ⎜⎟⎜⎟ ⎜⎟ ⎛ ⎝ −− −⎞ ⎠=⎛ ⎝∕− ∕⎞ ⎠=⎛ ⎝−⎞ ⎠ ++∕− ∕∕− ν σ ν σc ππ cσc cos 1, 1 1, 1cos 10 3 11 3 211 o σσ 021 11 0 12 I 11 and Vanishing and blow-up solu ions 41