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

Research Article Josef Diblík* and Miroslava Růžičková Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point https://doi.org/10.1515/anona-2023-0120 received July 16, 2023; accepted December 1, 2023 Abstract: A singular nonlinear differential equation ()=+zw zaw zwf z w d d, , σ where >σ 1 , is considered in a neighbourhood of the point =z0located either in the complex plane  if σ is a natural number, in a Riemann surface of a rational function if σ is a rational number, or in the Riemann surface of logarithmic function if σ is an irrational number. It is assumed that ( ) =wwz ,{}∈⧹ a 0, and that the function f is analytic in a neighbourhood of the origin in  × . Considering σ to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions ( ) =wwz in a domain that is part of a neighbourhood of the point =z0in  or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property ()= →wzlim 0 z0is proved and an asymptotic behaviour of ()wz is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well. Keywords: analytic solution, asymptotic behaviour, blow-up phenomenon, complex plane, differential equation, singular point MSC 2020: 34M35, 34M30, 34M10, 34A25 1 Introduction A singular nonlinear differential equation ()=+zw zaw zwf z w d d, , σ(1) where >σ 1 , is considered in a neighbourhood of the point =z0either in the complex plane  if { } ∈≔σ1,2, … is a natural number, in a Riemann surface of a rational function if ∈⧹σis a rational number, or in the Riemann surface of logarithmic function if ∈≔⧹σis an irrational number. In (1), z is an independent variable, ( ) =wwz , and {}∈⧹ a 0. The function →f:is assumed to be analytic in a neighbourhood  of the point ()∈×0,0 having the form:  * Corresponding author: Josef Diblík, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic; Faculty of Electrical Engineering and Communication, Technická 3058/10, 616 00 Brno, Czech Republic, e-mail: [email protected], [email protected] Miroslava Růžičková: Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland, e-mail: [email protected], [email protected] Advances in Nonlinear Analysis 2024; 13: 20230120 Open Access. © 2024 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License.   {( ) ∣∣ ∣ ∣ }=∈×<<zw z ρ w e,:, , k (2) where ()∈∞ρ0, and ()∈−∞∞ k ,are the fixed constants such that there exists a finite number > M 0 satisfying ∣( )∣ () ≥∈ M fzwsup , . zw, A proof is given of the existence of analytic solutions ( ) =wwz defined in a multiple connected domain in a neighbourhood of the singular point =z0and vanishing for →z0 (the point =z0itself does not belong to this domain being its boundary point). Such a domain lies in the complex plane  if ∈σ, in a Riemann surface of a rational function if ∈⧹σ, or in the Riemann surface of the logarithmic function if ∈σ. Whether the complex plain or the Riemann surface is chosen is determined by the domain of the term z σ in (1). Asymptotic analysis of equations in a neighbourhood of a singular point in the complex plane has a long history. In a pioneering article [3], a system of nonlinear differential equations and an initial problem () ()′= =zw h z w w,and00 (3) are considered with a function h being holomorphic in a neighbourhood of ( ) 0,0 and satisfying ()= h 0,0 0. The authors prove that a solution ( ) =wwz such that ()=w00 can be constructed as a power series convergent in a neighbourhood of =z0. In [31], assuming ()(∣∣ ) = h zoz,0 m, the authors use functional-analytical methods to show that (3) admits an analytic solution expressed by a power series with an initial power z m .Asa particular case of system (3), arising when the Jacobian matrix ( ) ′ h 0,0 w is singular, systems ()′=zy hzyw,, , σ1 (4) ()′=zw h z y w,, , 2(5) and their modifications are investigated where >σ 1 is an integer, z is an independent complex variable, ()= y yz ,( ) =wwz , h i, =i1,2 , are holomorphic vector-functions in a neighbourhood of ()0,0,0 vanishing there. We refer to [13–17,19–23,25–27,30] and to references therein. As the principal method of investigation is often used a construction of formal solutions in the form of special series (that are not necessarily exact power series) convergent in a subset of a neighbourhood of =z0. A substantial restriction used in the above-mentioned results is the assumption that σ in (4) is an integer as the methods used by their authors are not applicable in cases of σ being rational or irrational. In this study, we suggest a “geometrical”method connected with the properties of the function z σ and allowing us to omit this restriction in the case of scalar equation (1). If ∈σ, equation (1) is considered in the complex plane  , while, in the case of =∕∈⧹σmm 12 , where >>mm 1 12and ∈mm, 12 are relatively prime, we use the Riemann surface of the function = ∕ wz m12and, if ∈σ, the Riemann surface of the logarithmic function. Moreover, the properties of solutions to systems (4) and (5) have only been studied in a subset of the origin (in a sector with its vertex at the point =z0). Our investigation of the asymptotic properties of solutions to equation (1) cover, in a sense, the whole neighbourhood of the singular point =z0(in  or in the aforementioned Riemann surfaces). In the right-hand side of (1), a “perturbation”()zwf z w,of a linear equation ′=zw a w σis considered. Such a form of nonlinearity is quite natural because, if we use, for example, the term ( ) fzw *,instead, the assumptions of our results reduce such a general form to the form used in (1), i.e. to () ( ) ≔fzw zwfzw *,, . In some formulas throughout this article, if no ambiguity can arise, a simplified notation is used of the dependent variables not indicating their dependence on independent variables. The geometrical method of investigation suggested in this study is quite different from the methods used previously and can be used for analysing other classes of equations in the complex domain. For the reader’s convenience, in Section 2, we recall some auxiliary notions and concepts well known from the theory of functions of complex variable. Transformations applied to equation (1) with focus on systems equivalent on given curves and rays to (1) are discussed in Section 3, while in Section 4, the behaviour of solutions is studied of the systems derived. In Section 5, the results of this article (Theorems 1–4) are formulated. Their proofs are given in Section 6. Since a major part of the proofs of Theorems 1–3 is identical for an 2Josef Diblík and Miroslava Růžičková arbitrary value of σ , we consider only one variant of the proof where the differences depending on natural, rational, or irrational values of σ are emphasized. The proof of Theorem 4 is a consequence of the common part of the proof. Each of Examples 1–5 accompanies the constructions performed in Section 6. Nevertheless, in Section 7, a more complex example is considered. Concluding remarks and open problems formulated are given in Section 8. A close connection of the findings of this article with the unlimited growth of moduli of solutions near the singular point =z0(the so-called blow-up phenomenon) is mentioned and discussed as well. 2 Preliminaries Consider an initial problem ()()′= =wFzw wz w,, , 00 (6) where z and w are the complex variables and F is a complex-valued function. By a special case of the well known Cauchy-Kovalevskaya theorem if the function F is analytic in a neighbourhood of the point ( ) zw, 00 , problem (6) has a unique analytic solution ( ) =wwz in a neighbourhood of the point z0(we refer, e.g., to [8]). Recall also that an analytic function is a function that can be expressed by a convergent power series. A holomorphic function is a function that is differentiable in each neighbourhood of the point of its domain. For complex functions, the notions of an analytic and a holomorphic function are equivalent. Let   ⊂ be a path-connected domain. A curve lying in  is said to be simple if it does not cross itself. A domain  is simply connected if any simple closed curve in  can be continuously shrunk into a point while remaining in  . A domain  that is not simply connected is called a multiply-connected domain. The symbol  ∂ denotes the boundary of  , and  stands for the closure of  (i.e. for the set  ∪∂ ). The concept of analytic continuation (extension) is used in this article as well. It means the following. Let f 1 and f2be two analytic functions in open domains  1 and  2 of the complex plane  , respectively. Let  ∩≠∅ 12 and  ≢ 12 .If ≡ff 1 2 in  ∩ 12 ,f2is called an analytic continuation of f 1 to  2 , and vice versa. The analytic continuation is unique. 3 Transformations of equation (1) Several auxiliary transformations of equation (1) are necessary for its analysis. In Section 3.1, we show why the coefficient a in (1) can be assumed to be real and positive. Then, two types of real two-dimensional systems equivalent to (1) are considered. In Section 3.2, we derive an equivalent two-dimensional real system along given curves starting and ending at the point =z0, while in Section 3.3, an equivalent real two-dimensional real system along given rays leading offthe point =z0is obtained. 3.1 On the coefficient a in (1) Let the complex coefficient a in (1) be given in its exponential form: ∣∣ [ )=∈ a ae θ π,0,2 . iθ Then, a substitution, geometrically expressing a rotation, {} () =∈⧹ ∕− zve v,0 , iθ σ 1(7) where v is a new independent variable, changes equation (1) into one of a similar form: Vanishing and blow-up solutions 3 ∣∣ ( ) () ()() =+ ∕− − −∕− vw vaw vwf ve we d d, , σiθσiθσσ121 with a positive coefficient ∣ ∣ ainstead of the previous complex coefficient a . In the following investigation, we will assume that ∣() ∣ zwf z w, in (1)issufficiently small. This is true if either∣∣zor ∣∣wis small enough. Then, due to (7), this property remains in force also for the expression: ∣( ) ∣∣( )∣ () ()() () = ∕− − −∕− ∕− vwf ve w e vwf ve w,, . iθ σ iθ σ σ iθ σ121 1 Therefore, the coefficient a in (1) can be assumed, without loss of generality, real and positive, i.e. a can be replaced by its modulus ∣ ∣ a. We implicitly use this property in the investigations in the following and, whenever computations in this article depend on a , we assume that it is a positive number. 3.2 Real system equivalent to (1) on the loops of a given curve The behaviour of solutions to (1) in a small neighbourhood of the point =z0will be studied along the loops of a curve defined as: () (()) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅ ∕− zφ νσφ σc e cos 1 1 , σiφ 011 (8) where ν 0is a fixed real number, > c 0 is a real parameter, and φ is a real independent variable. If c and ν 0are fixed, we will assume that φ varies in such a way that (())−− >νσφ c os 1 0 , 0 (9) and, moreover, as we need ∣()∣<zφ ρ , that (()) () () ⎜⎟ ⎛ ⎝ −− −⎞ ⎠< ∕− νσφ σc ρ cos 1 1 . σ 011 Inequality (9) will hold if and only if Figure 1: Loops of (8) specified by =∕ ν π32 0, =σ4 , = s 0,1,2, and = c 3,2, 1 . 4Josef Diblík and Miroslava Růžičková −⎛ ⎝−+ ⎞ ⎠<<−⎛ ⎝++ ⎞ ⎠=± σνπsπ φ σνπsπ s 112 2112 2, 0,1,… 00 (10) and, obviously, ()==±zφ k0, 0, 1,… k (11) for ≔−⎛ ⎝++⎞ ⎠=± φ σνππk k 112,0,1,…. k0 (12) The closure of each loop of the curve (8) separated by an inequality (10) is a closed curve passing through the origin. If ∈σ, then there are −σ 1 different curve arcs (8) lying in the complex plane  ,defined, e.g., by: =− s σ0,…, 2 in (10). If =∕∈⧹σmm 12 , where >>mm 1 12and ∈mm, 12 are relatively prime, then there are −m 1 2 different curve arcs (8) on the Riemann surface of the function = ∕ wz m12defined, e.g., by =− s m0,…, 2 2in (10). If ∈σ, then there is a countable set of disjunct loops of curve (8) defined by =± s 0, 1,… in (10). We consider them on the Riemann surface of the logarithmic function. Figure 1 shows different loops of (8) in the z plane as defined by angles φ satisfying (10), where = s 0,1,2, =∕ ν π3 2 0, and related to the values = c1 (green loops), = c 2 (brown loops), and = c 3 (blue loops). We will transform equation (1) into a system of two ordinary differential equations on parts of the closures of the loops of curve (8) with independent variable φ satisfying inequalities (10). Then, () (() ) =wz wzφ .To reduce the computations, we do not always write the argument () z φ of w or the argument φ of z (similarly, we proceed when new dependent variables α and β in the following are used). By the chain rule, we derive (()) (()) () () ()[ () (() )]() == + −− − wzφ φwzφ zφ zφ φe z φwae zφe fzφ w zφ φ dddddd,dd, iν σ iν iν 000 (13) where () ( )(( )) (( ( ))) ( ( ( )))( ) (()) () () (()) (()) () (()) () (( )) () (()) ⎜⎟ ⎜⎟ =−− −− ×−−−+ ⎛ ⎝ −− −⎞ ⎠ =⎛ ⎝ −− −− +⎞ ⎠ =−− ∕− ∕−− ∕− −−− 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 fzφ w iz φ νσφ ee wae zφe fzφ w i νσφ zφe wae zφe fzφ w i νσφ νσφ σc iσ cwae zφe fzφ 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) Let (())=wwzφ in (15) be represented by its algebraic form, i.e. (()) () ( ) =+wzφ y φ iy φ 12 with () y φ 1 and () y φ 2 being the real and imaginary parts of (() ) wzφ , respectively. Then, Vanishing and blow-up solutions 5 (()) () () ()[ ()(())] (()) ()(()())( ) (()) ( )( () ())[ (() (() )) (() (() ))] (()) =+ =−+ −− =−+ − −− +−+ + −− −− −− wzφ φyφ φiyφ φ iσ cwae zφe fzφ w νσφ iσ cy φ iy φ a ν i ν νσφ iσ cy φ iy φ zφe fzφ w i zφe fzφ 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 the real and imaginary parts, we see that y1 and y 2satisfy the following system of ordinary differential equations equivalent to equation (1) on the given loop segment of the curve (8): () (()) (( ) ( ) ) () (()) ( ( ()) ( ())) ′=− −− − +− −− −− −− yσc νσφ aνy aνy σc νσφ y zefzw y zefzw 1 cos 1 sin cos 1 cos 1 Re , Im , , iν iν 1200102 2021 00 (16) () (()) (( ) ( ) ) () (()) (( ()) ( ())) ′=− −− + +− −− − −− yσc νσφ aνy aνy σc νσφ y ze fzw y ze fzw 1 cos 1 cos sin 1 cos 1 Re , Im , . iν iν 2200102 2012 00 (17) 3.3 Real system equivalent to (1) on a system of rays Consider rays running from the origin ()=<<=zt te t ρ ν,0 , const , iν (18) and transform equation (1) along these into a system of two real equations. Since == − w zw tt zw te d dd dd dd d, iν (1) can be written as: (()) () =+ −− tw tewateftew d d, . σ i σ ν iν iν1 Assuming (() ) wzt by its algebraic form (()) () ( ) =+wzt xt ix t 12 , we derive () () (())( ( (()))) ( () ())( ( (()))) (( ) ( ))(() ()) ( ( ( (())) ( ( (()))))) () () =⎛ ⎝+⎞ ⎠ =+ =++ =−−− + ⋅+ + −− −− tw ttxt tixt t ewztateftewzt extixtateftewzt σνiσνxtixt a te f te w z t i te f te w z t 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 Functions x1 and x 2satisfy the following system of ordinary differential equations: ( ( )) ( ( )) ( ( ))( ( ) ( )) (( ))(() ()) ′=−+−+− − +− + tx a σ νx a σ νx σ ν x tef x tef σ νx tef x tef cos 1 sin 1 cos 1 Re Im sin 1 Im Re , σiν iν iν iν 11212 12 (19) ( ( )) ( ( )) ( ( ))( ( ) ( )) (( ))( () ()) ′=− − + − + − + +−− + tx a σ νx a σ νx σ ν x tef x tef σ ν x tef x tef sin 1 cos 1 cos 1 Im Re sin 1 Re Im . σiν iν iν iν 21212 12 (20) 6Josef Diblík and Miroslava Růžičková 4 Auxiliary results on the behaviour of solutions In this section, we prove two lemmas used in proving the results of this article. In Section 4.1, the behaviour of solutions to (1) is considered along loop segments of the curve (8), while in Section 4.2, the behaviour of solutions to (1) is considered along rays (18) defined in Section 3.2. 4.1 Behaviour of solutions along segments of curve (8) Assume, in the definition of curves (8), ≠ ν m π 0,∈m, i.e. ≠ν s in 0 . 0 (21) In the following, we consider system (16), (17) in the domain {( ) ∣()∣ }≔∈<<+<φy y zφ ρy y e Ω ,, :0 , , k cr 12 312222 where k and ρ are the same as in the definition of the domain  given by (2), ()zφ is defined by (8), and assume that (9) holds as well. The values φ k,defined by (12), have property (11), i.e. these values define singular points = φ φ k of system (16), (17) because (())−− =νσφ c os 1 0 k 0. It is clear that the conditions are met at every point of Ω cr of the well-known theorems on the existence and uniqueness of solutions to initial Cauchy problem as well as on the continuous dependence of solutions on the initial data. Let ( )ε ν0be a positive number such that () ∣ ∣≤ ε νενsin , 000 (22) where ε 0 is a fixed number satisfying << ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ ερ a M 0min, 2 . 0(23) Define cylinders  ( ) λas sets  () {( ) } ( )≔∈+==∈−∞λφyy yyeλ λ k,, Ω: , const, ,. λ 12 cr 12222 (24) Lemma 1. Assume that φ varies in a fixed domain defined by (10). Let c and ν 0,satisfying (21), be fixed such that ∣()∣ ()<≤zφ εν0 . 0(25) Then, any integral curve (()())φy φ y φ,, 12 of system (16), (17) intersecting at a value = φ φ * a cylinder  ( ) λ,i.e. if (()() ) φyφ yφ *,*,* 12 satisfy () ()+= y φyφe ** , λ 12222 behaves as follows. The integral curve (()())φy φ y φ,, 12 ,as φ increases, is passing (i)from domain +> y ye λ 12222 (26) into domain +< y ye λ 12222 (27) if <ν s in 0 0and inequality ∣(())∣ () ()=+<<wzφ yφyφe e λk2122222 (28) holds for every admissible > φ φ * . (ii)from domain (27)into domain (26) if >ν s in 0 0and inequality (28) holds for every admissible < φ φ * . Vanishing and blow-up solutions 7 Proof. Consider the behaviour of integral curves of system (16), (17) intersecting cylinders  ( ) λ. To do this, compute the scalar product ( ) →→ NT,at an arbitrary point of cylinder  ( ) λwith fixed λ , where → N is its normal vector directed outwards and → T is a vector of the vector field defined by system (16), (17). As () ⎜⎟ →=→=⎛ ⎝⎞ ⎠ N yy T y φy φ 0, , and 1, d d,d d , 12 12 we have () ( ) ( ( ( ()))) (()) →→=+= −− −− − NT y y φyy φσceaν zefwzφ νσφ ,d dd d1sinIm , cos 1 . λiν 112220 20 0(29) We will show that (29) implies () →→=NT ν s gn , sgnsin 0 (30) whenever (25) holds. Indeed, for =+zzizRe Im and ()(())(())=+ −− − e fzw e fzw i e fzw,Re , Im , , iν iν iν 00 0 we have ( ( )) ( ( )) ( ( ))=⋅ +⋅ −−− ze fzw z e fzw z e fzwIm , Re Im , Im Re , . iν iν iν 000 Therefore, from (22), (23), and (25), it follows ∣ ( ( ))∣ ∣∣ ∣∣ ∣∣ () ∣ ∣ ∣ ∣≤+= ≤ ≤ < − ze f 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, taking into account that () (()) − −− > σce νσφ 1 cos 1 0 , λ2 20 from (29), we derive ( ) ( ( ( ))) →→=− = − NT a ν ze fzw ν s gn , sgn sin Im , sgnsin iν 0 0 0 and (30) holds. Finally, we remark that (22), (23), and (25) imply ∣()∣<zφ ρ , i.e. the values of z used are within the domain  defined by (2) and that formula (30) is independent of the value of λ . The geometrical meaning of equation (30) is as given in parts (i) and (ii) of the lemma. □ Remark 1. Note that the φ -axis itself (i.e. the set of points ( ) φ,0,0 , where φ satisfies (10)) is also an integral curve of (16) and (17); therefore, no other integral curve intersects the φ -axis. For a sufficiently large c , the considered loops of (8), as it follows from (25), are completely contained in the ( )ε ν0-neighbourhood of the point =z0.If c is sufficiently small, then loops of (8) are not connected in the ( )ε ν0-neighbourhood of the point =z0with this ( )ε ν0-neighbourhood containing their parts (Figure 2). In most of the following figures, the value ( )ε ν0can be seen on the plots scaled down to fit into a single figure. 4.2 Behaviour of solutions along system of rays (18) Assume, in the definition of rays (18), ≠−⎛ ⎝+⎞ ⎠=± ν σπmπ m 112 ,0,1,…, i.e. ()−≠σν c os 1 0 . (31) In this part, we will examine the behaviour of solutions to Systems (19) and (20) in the domain: 8Josef Diblík and Miroslava Růžičková {( ) }≔∈<<+<txx tρxxe Ω ,, :0 , , rs k 12 3122 22 where k and ρ are the same as in the definition of the domain  given by (2) and ( ) zt is defined by (18), i.e. ∣()∣=zt t . For system (19), (20), the hypotheses of the well-known theorems on the existence and uniqueness of solutions to initial Cauchy problem as well as on the continuous dependence of solutions on the initial data are true at every point of Ω rs .Define cones  ()δas sets  () ( ) () ≔⎧ ⎨ ⎩∈+= ⎛ ⎝−−⎞ ⎠⎫ ⎬ ⎭ − δ txx xxδ aσν t ,, Ω: exp2cos 1 , rs p 12 122 221(32) where δ and p are the fixed parameters satisfying >δ 0 and ()∈ p σ1, . Put ∣( )∣≔− ∕ tε σν ** cos 1 , νp p μ ,1(33) where μ satisfies { } << −μσp0min1, and ε* p is a positive number satisfying () ⎜⎟ << ⎧ ⎨ ⎩ ⎛ ⎝+− ⎞ ⎠ ⎫ ⎬ ⎭ −−− ∕ ερ a Mρ p aρ 0*min , 41 . pμσpμ μ 1 1 (34) If an integral curve (()())txt xt,, 12 of the system (19), (20) intersects a fixed cone  ()δat a point (]=∈tt t *0, * νp,, then ∣(())∣ () () () () ⎜⎟ =+=⎛ ⎝−−⎞ ⎠ − wzt xtxtδ aσν t *** exp 2cos 1 * , p 2122 221 where δ satisfies () () ⎜⎟ <⎛ ⎝+−⎞ ⎠ − δk aσν t exp cos 1 *p1(35) by the definitions of  ()δand Ω rs . Figure 2: Loops of (8) specified by =∕ ν π2 0and =σ3 if ()= ε ν 1 0,= c 3, 2, 1, 0.5, and 0.3. Vanishing and blow-up solutions 9 () () ( ) () ⎜⎟ ⎛ ⎝−⎞ ⎠≤≤ − ∕− σc εν ε σ r 11sin 1 . σ11 00 (63) Assume that (63) holds. Then, both the above-mentioned segments intersect the ray (18), where ν is given by (50) and t satisfies (55), at the point () ()()() ⎜⎟ =⎛ ⎝− −⎞ ⎠ ∕− zσr σc e *cos 1 1, n σiψ n 11 provided that (deducting from (55)) ∣∣[ ]∈zωε *,* , np (64) where ∣∣ () () () ⎜⎟ =⎛ ⎝− −⎞ ⎠ ∕− zσr σc *cos 1 1 . n σ11 (65) If the parameter c is fixed (being sufficiently large) and if ω is sufficiently small, then, simultaneously, the set of values ∣∣z* n satisfying (64) will be non-empty and (63) will hold as well. This is shown in the following. Let ω be fixed and sufficiently small. Define the number (() ) ∈∕−rπσ0, 2 1 2as the solution of the equation: ∣∣ = ≤≤ rrmax . ωz ε 2** np (66) This maximum will be achieved for ∣∣=z ω * n because () () −= −→∕ σrlim cos 1 0 σrπ12 . Then, from (65), we derive () () ( ) ⎜⎟ =⎛ ⎝− −⎞ ⎠ ∕− ωσr σc cos 1 1 σ 211 (67) or (( ) )=−− − rσσcω 11arccos 1 . σ 21 (68) From formulas (67) and (68) we deduce the following. If, for an increasing c and decreasing ω , the product − c ωσ 1 remains the same, then the values r 2 ,r 1 will be fixed as well. Inequality (63) will be satisfied if ()(())()(()) ≥−− ≥−− −− c σεσr σεσr 1 1sin 1 1 1sin 1 σσ 01 101 with (63) still holding for all []∈rrr, 12 . Obviously, it is possible to apply the above-mentioned reasoning even after replacing ω with an <ω ω 1 and c with an > cc 1such that = −− c ωcω σσ 111 1 . Then, the set of values ∣∣z* n such that (64) holds is nonempty. Therefore, if ω is sufficiently small, then there exists a number (]∈ ε ωε ** ,* pp such that each point ( ) zt of the segment of the ray (18)defined by: ≤≤ωtε * * p (69) with ν given by (50) is intersected for every fixed []∈rrr, 12by the loops of the curve (8), corresponding to ( ) =± ν νnr, 00, and (63) holds. These loops are defined by the formula: () (()( )) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠ ±∕− zφ νnr σ φ σc e cos , 1 1 , n σiφ 011 (70) where φ varies within domains (57). Moreover, it is easy to show that the arcs (70) are symmetric with respect to the given ray. Let r be fixed. By the above-mentioned construction, three curves are passing through the point () =zωe iψ n , the ray itself and two loops of the curve (8), specified by (70). By Lemma 2, (ii), formula (38), the inequality 16 Josef Diblík and Miroslava Růžičková ∣( )∣ () <wte e niψ n k 0 holds for every []∈tωε,* p . In the following, we use segments of curve loops (70)defined by angles φ (within domains (57)) such that () ( )≤< + ψn φ ν nr, (71) and () ()<≤ − ν nr φ ψn, . (72) Along arcs (70), where φ satisfies (71) if the value ( ) + ν nr, 0 is considered or (72) if the value ( ) − ν nr, 0 is used, there exists the analytic continuation of solution ( ) wz n 0to (1), satisfying initial condition (53). If domain (71) is considered, then, by (62), ()==−−< + νν σr s in sin sin 1 0 00 and, by Lemma 1, where ( ) = φ ψn *, part (i), formula (28), ∣ (())∣<wzφ e . nk 0 If domain (72) is considered, then, by (62), ()==−> − νν σr s in sin sin 1 0 00 and by part (ii) of Lemma 1, where ( ) = φ ψn *, the same inequality holds. We prove that this solution converges to zero if ( ) → ± φ νnr, . For ()zφ defined by (70), we have () () = →±zφlim 0 φ νnr,(73) because (()) () −− = ∕= → ± ± νσφ πlim cos 1 cos 2 0 . φ νnr,0 Moreover, by (50), (47), and (48), we have (( ) ( )) ( )−=−> ± σνnr σr c os 1 , sin 1 0 . (74) Then, putting ( ) =± ν νnr,in equation of rays (18), we see that (31) holds. As it follows from formulas (36) and (37) in Lemma 2, (i), where, by (33) and (74), (( )) (( )) () ≔−≥− ∕∕ ± tεσrεσr **sin 1 *sin 1 , νnrp pμpμ ,, 111 the solution ( ) wz n 0satisfies ()= →wzlim 0 zn 00. Due to the variability of []∈rrr, 12 and the analytic continuation of ( ) wz n 0, this property also holds on the curve loops if ( ) → ± φ νnr, (property (73)). Moreover, ∣()∣ ()() ∣∣ ⎜⎟ ≤⎛ ⎝−−⎞ ⎠ ± − wz δ aσνnr z exp cos 1 , , np 01 for sufficiently small ∣∣z. Remark 4. The following property should be added to the previous consideration. Let = νφ in (18)befixed, where either () () ( ) +−<≤ + ψn π σφνnr 21 , (75) or () ()() ≤< − − − ν nr φ ψn π σ ,21 . (76) Since, in both cases, ()−>σφ c os 1 0, from Lemma 2, (i), formulas (36) and (37), we have Vanishing and blow-up solutions 17 ∣()∣ () ∣∣ ⎜⎟ ≤⎛ ⎝−−⎞ ⎠ − wz δ aσφ z exp cos 1 , np 01 for a sufficiently small ∣∣z,∣∣ ( ( ( )) ) ∈− ∕ zεσφ0, *cos 1 pμ1and ()= →wzlim 0 z0. This is true for all rays (18) with a fixed φ within intervals (75) and (76). Example 1. Assume that, in equation (1), we have =σ2 ,= a1 , and = M2 . Let =ρ 1 and = k0 . The following constructions are visualized by Figure 3. In accordance with (23), put {}<= < ⎧ ⎨ ⎩ ⎫ ⎬ ⎭==ερ a M 00.02min, 2min 1, 0.25 0.25 . 0 Moreover, for ν 0defined by formulas (45) and (46) with = n 0 , we have, by (50), either ()()==+=+ + ν νrψ rπr0, 0 00 or ()()==−=− − ν νrψ rπr0, 0 00 and, by (22), we can set (in the formula, the dependence on r is emphasized and notation () ε ν r0is used instead of ( )ε ν0)( ) ∣ ( )∣ ∣ ( )∣==±= ± ε νενr πr rsin 0, 0.02 sin 0.02sin . r000 In accordance with (56), define a range for r by numbers r 1 and r 2 satisfying =∕rπ3 2and =∕rπ 6 1(if =∕ ζ 12 is taken). Although we have defined the number r 2 “ad hoc”(not using its definition (66)) to better illustrate all computations, this choice is in accordance with (66) as shown in the following. Let us take a number ( )ε ν0 suitable for an arbitrary []∈rrr, 12 . Since () [] [] ==∕=⋅∕= ∈∈ εν r πmin min 0.02sin 0.02sin 6 0.02 1 2 0.01 , r rr rrrr,0, 12 12 we can put, independently of r ,()≔ ε ν0.01 . 0 Now, consider a value ε* p . By formula (34), with ()=∕∈ p σ32 1, and { } =∕< −μσp14 min1, , {[( )]}( )<∕+∕=∕≐ ε *min 1, 1 8 1 2 2 17 0.00019 , p44 and we put = ε *0.00018 . p Moreover, let (for the properties of ω , we refer to (51) and (52)) =< =ωεc0.0001 *and 5,000 . p Then, by (65), ∣∣ () ()() ⎜⎟ =⎛ ⎝− −⎞ ⎠== = ∕− zσr σc r crr *cos 1 1cos cos 5,000 0.0002cos , n σ11 and the solution of equation (66) ∣∣ == ≤≤ ≤ ≤ rr r max max ωz ε r 212cos 1.8 ** np gives the value =∕rπ3 2. Following the above-mentioned recommendation, if we replace ω with ω 1 and c with c1 preserving the property == −− c ωcω0.5 σσ 111 1 , say =ω0.00005 1and = c 10,000 1, we can derive the same solution. Since, for =rr 1 , ∣∣ () () () ⎜⎟ =⎛ ⎝− −⎞ ⎠== ∕=≐<= ∕− zσr σc r cπε *cos 1 1cos cos 6 5,000 3 10,000 0.00017 *0.00018 , n σ p 111 1 we can put (referring to (69)) 18 Josef Diblík and Miroslava Růžičková =∕ ε ** 3 10,000 . p(77) Then, ≐<= ε ε ** 0.00017 *0.0001 8 pp . Inequality (63) holds as well since () () () ⎜⎟ ⎛ ⎝−⎞ ⎠== = < = ∕− σc c εν 1111 5,000 0.0002 0.01 . σ11 0 The curve arcs (70) are defined by the angles φ within domains (71) and (72), i.e. () () () ()=≤< = + = −<≤ = +− ψπφνr πrνr πrφψ π 00, 32and 0, 20 (78) that are parts of domains (60) and (61), i.e. +<< + −<< − πrφ πrπrφ πr 232and 232 , respectively. Figure 3 shows the segments of the curve loops defined by (70), i.e. () (() ) () =−=±− ± zφ νrφ ceπrφ e cos 0, cos 5,000 iφ iφ 00 passing through points =zεe ** pi π ,=zte i π , where =∕ ≐t2 10,000 0.00014 , and =zωe i π and the domains for φ are given by inequalities (78) with ==∕rrπ 6 1,==∕rr π * 4 , and ==∕rrπ3 2. For the red segment, the computations with () − ν r0, and ( ) − ν r0, 0 are relevant, while, for the green one, the values ( ) + ν r0, and ( ) + ν r0, 0 are used. Along these segments, the solution of the initial problem (see (51) and (53)): ()=≔=wz w z z ωe,where , ωiπ 0 000 000 is analytically continued. Note that, for different values of ≠ n 0 , we obtain identical constructions. 6.1.2 Auxiliary lemma The following Lemma 3 and Remark 5 on the mutual positions of different segments of the loops of curve (8) will be used in further constructions. Instead of r 1 and r 2 in their formulations, arbitrary values r* 1 and r* 2 satisfying ( ) <<<∕−rrπσ0** 21 12 can be used. For an illustration, we refer to Example 2 and Figure 4. Lemma 3. Let numbers r 1 and r 2 be fixed such that ()<<<∕ −rrπσ021 . 12 (79) Consider two curve arcs ( ) zφ n1and ( ) zφ n2 of (8) defined by the following formulas: () (( )( )) () () ( ) () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅≤< +∕− + zφ νnr σ φ σc eψnφνnr cos , 1 1,,, n σiφ 101 1 11 1(80) () (( )( )) () () ( ) () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅≤< +∕− + zφ νnr σ φ σc eψnφνnr cos , 1 1,, , n σiφ 202 2 11 2(81) where c1 and c 2 are the positive constants, ( ) + ν nr, 01and () + ν nr, 02 are defined by (45), i.e. ()()(())()()(())=− + =− + ++ ν nr σ ψn r ν nr σ ψn r,1 ,,1 , 011 022 (82) and () + ν nr, 1, ( ) + ν nr, 2are defined by (47), i.e. ()() () ()() () =++ −=++ − ++ ν nr ψn r π σand ν n r ψ n r π σ ,21 ,21 . 11 22 If Vanishing and blow-up solutions 19 (()) (())=zψn zψn, 12 (83) then ∣ ()∣ ∣ ()∣ () ( )<<< + zφ zφ foreveryψn φ νnr,, . nn12 1 (84) Proof. From (80)–(82), we see that (83) implies () ()−=−σr cσr c cos 1 cos 1 . 1 1 2 2 (85) Now, on the interval () ( ) ≤< + ψn φ ν nr,1, we will investigate the properties of the function:  () (( )( )) (( )( )) =−− −−− ++ φνnr σ φ cνnr σ φ c cos , 1 cos , 1 . 02 2 01 1(86) By (85),  (()) () () =−−−=ψn σr cσr c cos 1 cos 1 0 . 2 2 1 1 (87) Due to (79), we have ()()<− <− <∕σrσrπ01 1 2 12 . Then, () ()<−<−σr σr0 cos 1 cos 1 , 21 and, analysing (87), we have < c c 2 1 . Moreover,  () ()(()())()(()()) ′=−−− −−−− ++ φσνnrσφ cσνnrσφ c 1sin , 1 1sin , 1 02 2 01 1 and  (()) ()()()() ′=≔ −− −−− ψn A σσr cσσr c 1sin 1 1sin 1 . 2 2 1 1 Due to (79), we have () ()−> −>σr σr s in 1 sin 1 0, 21 Figure 4: Visualization –to Lemma 3, Remark 5, and Example 2. 20 Josef Diblík and Miroslava Růžičková and, consequently,  (())′=>ψn A 0. Computation of the second-order derivative leads to  () ()(()())()(()()) ″=− −−− +−−− ++ φσνnrσφ cσνnrσφ c 1cos , 1 1cos , 1 . 202 2 201 1 Therefore,  ( ) φsatisfies the following Cauchy initial problem for a differential second-order equation:   ()( )() (()) (()) ″+− = = ′ =φσ φ ψn ψnA10, 0,and . 2(88) The general solution of (88)is  () () ()=−+−φC σ φC σ φsin 1 cos 1 , 12 where, for the specification of arbitrary constants C 1 and C 2 by the initial conditions, we use the equations  (()) ()() ()()=−+−=−=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 solution of the problem (88)is  () ( )=− −−φA σσφ 1sin 1 . (89) The angle φ varies within the interval indicated in (84), which implies ()()()() ()+<−<++−+∕<++nπσφnπσrπ nππ21 1 21 1 221 . 1 For such values of φ , we have ()−<σφ s in 1 0 ; therefore, for  ( ) φexpressed by (86) and (89),  () (( )( )) (( )( )) ()=−− −−− =− −−> ++ φνnr σ φ cνnr σ φ cA σσφ cos , 1 cos , 1 1sin 1 0 . 02 2 01 1 From this inequality, we deduce that  ()>φ0 also holds in the same interval, where  () (( )( )) () (( )( )) () () () ⎜⎟⎜⎟ ≔⎛ ⎝ −− −⎞ ⎠−⎛ ⎝ −− −⎞ ⎠ +∕− +∕− φνnr σ φ σc νnr σ φ σc cos , 1 1cos , 1 1 . σσ 02 2 11 01 1 11 This inequality is equivalent with (84). □ Remark 5. A property similar to that formulated in Lemma 3, i.e. the inequality ∣ ()∣ ∣ ()∣ ( ) ( ) <<≤ − zφ zφ νnr φ ψn, for every , nn12 1 can be proved in much the same way, if the following segments of loops are considered instead of (80) and (81): () (( )( )) () () () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅<≤ −∕− − zφ νnr σ φ σc eνnrφψn cos , 1 1,, , n σiφ 101 1 11 1 () (( )( )) () () () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅<≤ −∕− − zφ νnr σ φ σc eνnrφψn cos , 1 1,, , n σiφ 202 2 11 2 ( ) − ν nr, 01and () − ν nr, 02 are defined by (46) and () − ν nr,1and ( ) − ν nr,2are defined by (48). Example 2. We use some constructions from Example 1 to illustrate Lemma 3 and Remark 5. We refer to Figure 4 where the two red and two green loop segments can be seen passing through the point =zte i π , where =∕ ≐t2 10,000 0.00014 . Put = c 2,500 6 1 and = c 2,500 2 2. The inner green segment is defined by the formula: () (( ) ) () =−=∕− + zφ νrφ ceπφ e cos 0, cos 7 6 2,500 6 , ig iφ iφ 001 1 Vanishing and blow-up solutions 21 where () ( )=≤<∕= + ψπφπνr0530, , 1 while the outer green one is defined by the formula: () (( ) ) () =−=∕− + zφ νrφ ceπφ e cos 0, cos 4 3 2,500 2 , og iφ iφ 002 2 where () ( )=≤< ∕= + ψπφπνr01160, . 2 The inner red segment is defined by the formula: () (( ) ) () =−=∕− − zφ νrφ ceπφ e cos 0, cos 5 6 2,500 6 , ir iφ iφ 001 1 where ()=∕< ≤ − ν rπ φπ0, 3 , 1 while the outer red one is defined by the formula: () (( ) ) () =−=∕− − zφ νrφ ceπφ e cos 0, cos 2 3 2,500 2 , or iφ iφ 002 2 where ()=∕< ≤ − ν rπ φπ0, 6 . 2 Assumption (83) holds since () () () ()====∕zπ zπ zπ zπ 1 5,000 2 ig og ir or0000 and ∣ ()∣ ∣ ()∣<<<∕zφ zφ πφ π,if 53 ig og00 and ∣ ()∣ ∣ ()∣<∕<<zφ zφ π φ π,if3 . ir or This is in accordance with the assertions of Lemma 3 and Remark 5. Remark 6. Let us point out another property of the mutual position of different segments of loops of the curve (8), which is obvious. If two segments () (()( )) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅ +∕− zφ νnr σ φ σc e cos , 1 1 σiφ 10 0 1 11 and () (()( )) () () ⎜⎟ =⎛ ⎝ −− −⎞ ⎠⋅ +∕− zφ νnr σ φ σc e cos , 1 1, σiφ 20 0 2 11 where []∈rrr, 12 is fixed, are defined for () ( )≤< + ψn φ ν nr, and < c c 2 1 , then ∣ ()∣ ∣ ()∣<zφ zφ . 10 20 The same property holds if ( ) + ν nr, 0 is replaced by ( ) − ν nr, 0 and domain of φ is () ( ) <≤ − ν nr φ ψn, . This property is visualized in Figure 5, where =rr 2 with details described in Example 3. Example 3. In the following, some constructions from Example 1 are used to illustrate the assertions of Remark 6. We refer to Figure 5 where the two red and two green loop segments are visualized passing through the points =zεe ** pi π and =zωe i π . The inner green loop segment is defined by the formula: 22 Josef Diblík and Miroslava Růžičková () (( ) ) () =−=∕− + zφ νrφ ceπφ e cos 0, cos 4 3 5,000 , ig iφ iφ 002 1 where ()≤< ∕= + πφ π ν r11 6 0, , 2 while the outer green one is defined by the formula: () (( ) ) () =−=∕− ∕ + zφ νrφ ceπφ e cos 0, cos 4 3 5,000 3 , og iφ iφ 002 2 where ()≤< ∕= + πφ π ν r11 6 0, . 2 The inner red loop segment is defined by the formula: () (( ) ) () =−=∕− − zφ νrφ ceπφ e cos 0, cos 2 3 5,000 , ir iφ iφ 002 1 where ()=∕< ≤ − ν rπ φπ0, 6 , 2 while the outer red one is defined by the formula: () (( ) ) () =−=∕− ∕ − zφ νrφ ceπφ e cos 0, cos 2 3 5,000 3 , or iφ iφ 002 2 where ()=∕< ≤ − ν rπ φπ0, 6 . 2 We see that ∣ ()∣ ∣ ()∣<≤<∕zφ zφ πφ πif 11 6 ig og00 and Figure 5: Visualization –to Remark 6 and Example 3. Vanishing and blow-up solutions 23 ∣ ()∣ ∣ ()∣<∕<≤zφ zφ π φπif 6 . ir or00 This is in accordance with the assertions of Remark 6. 6.1.3 Continuation of (()) w z n 0to a domain P n ω Let n be fixed. Varying t and r in admissible boundaries, by the theorem on the existence of a unique solution to the initial problem, we obtain an analytic continuation of the solution ( ) wz n 0,defined by initial problem (53) with the initial point ( ) wz 0, where =zz ω n 0is determined by (51), on a domain P n ω ,defined as a domain covered by all the above-mentioned loop segments of the curve (8). The boundaries for t and r are the following. The value t varies within the interval (69), where ω is sufficiently small and ε * * p is defined in such a way that the segments of loops (70) of curve (8) are defined by angles φ satisfying (71) and (72). The value r varies as indicated in (56) with r 2 defined by (66) and with r 1 defined by (58). Then, among others, the inequality ∣()∣<wz e nk 0holds. From the method of construction and (56), (71), (72), and (73), it follows that the domain P n ω is simply connected and lies in the sector  ( ) φ n, centred at the point =z0,defined as:  () () () () () ⎜⎟⎜⎟ ≔⎧ ⎨ ⎩−⎛ ⎝+−⎞ ⎠≤≤ + ⎛ ⎝+−⎞ ⎠ ⎫ ⎬ ⎭ φφψnr π σφψn r π σ :21 21 . n22 The sector  ( ) φ nis located either in the complex plane  or in a Riemann surface and does not contain the origin. Applying (56), the length of the interval for φ defining  ( ) φ nsatisfies ()() +−<− rπ σπ σ 2121 . 2 Assume now that n is not fixed. Tracing carefully computations in Parts 6.1.1 and 6.1.2, we see that these do not depend on any of the values n . So, for every n , the above-mentioned considerations are correct. This means that two domains P n ω 1and P n ω 2 , where ≠ n n 12 , are geometrically identical (having identical forms). These domains can differ only by their location in the complex plane or in a Riemann surface and, in such a case, they do not intersect. The following obvious statement is true about the number of domains P n ω with different locations (two domains P n ω 1and P n ω 2 are different if ∩=∅ P P nω nω 12 for ≠ n n 12 ). (a) If ∈σ, then there exist −σ 1 different domains ()⊂ P φ nω n ,=− n σ0,…, 2, where the sectors  ( ) φ n, =− n σ0,…, 2 are located in the complex plane  . (b) If ∈⧹σ, =∕σmm 12 , and ∈mm, 12 are relatively prime, then there exist −mm 12 different domains ()⊂ P φ nω n , =−− n mm0,…, 1 12 , where the sectors  ( ) φ nare located on the Riemann surface of the function ∕ zm12. (c) If ∈σ, then there exists an infinite countable set of different domains ()⊂ P φ nω n ,=± n 0, 1,… , where the sectors  ( ) φ nare located on the Riemann surface of the logarithmic function. Now, we describe in detail the construction of the domain P n ω based on the properties of the curve segments given in Lemma 3, Remark 5, and Remark 6. For a visualization, we refer to Figure 6, which illustrates the relevant constructions of Example 4. The domain P n ω contains all points between the following two curves including all boundary points except for the point =z0. The boundary of P n ω is formed by two closed, simple, and continuous curves “embedded”in each other (below called the inner and outer boundaries) with a unique common point =z0. The inner boundary is formed by two segments of loops (70) specified by (for the definitions of ± ν 0 and ± ν , we refer to formulas (45)–(48)): ()()(())() ()==−+ ≤< ++ ν νnr σ ψn r ψn φ νnr,1 , , , 0011 1 (90) ()()(()) () ()==−− <≤ −− ν νnr σ ψn r νnr φ ψn,1 ,, , 00111 (91) 24 Josef Diblík and Miroslava Růžičková and passing through the point (() ) ==zz ω iψnexp ωn0, determined by (51), while the outer boundary is defined by two segments of loops (70) specified by: ()()(())() ()==−+ ≤< ++ ν νnr σ ψn r ψn φ νnr,1 , , , 0022 2 (92) ()()(()) () ()==−− <≤ −− ν νnr σ ψn r νnr φ ψn,1 ,, , 00222 (93) and passing through the point (() ) ≔zε iψn **exp ** εp p . For the definition of ε * * p , we refer to the explanation accompanying inequalities (69). This construction is correct since, by Lemma 3 and Remark 5, arcs defined by (90) and (92) have no intersection for () ( ) << + ψn φ ν nr,1and arcs defined by (91) and (93) have no intersection for () ()<< − ν nr φ ψn, 1 . Example 4. Using several constructions of Example 1 again, we will construct the domain P ω 0 . The inner boundary of P ω 0 consists of two segments of loops (70) passing through the point =zωe i π with =rr 1 and = c 5,000 3. The red segment in Figure 6 is defined as: () (( ) ) () =−=∕− − zφ νrφ ceπφ e cos 0, cos 5 6 5,000 3 , iφ iφ 001 where () ()=∕< ≤ = − ν rπ φπψ0, 3 0 , 1 while the green one in Figure 6,isdefined as: () (( ) ) () =−=∕− + zφ νrφ ceπφ e cos 0, cos 7 6 5,000 3 , iφ iφ 001 where () ( )=≤<∕= + ψπφπνr0530, . 1 The outer boundary of P ω 0 consists of two segments of loops (70) passing through the point =zεe ** pi π with =rr 2 and =∕ c 5,000 3 . The red segment in Figure 6 is defined as: Figure 6: To Example 4 –domain P ω0. Vanishing and blow-up solutions 25 where () ()−=−≤< = −− ν rππ φν r π 0, 23 0, 6 . 22 Finally, by equation (117), the segment of the ray ( ) − zt r10 is defined by: () ∣ ( ( ))∣ () == <≤ = −∕−− − z t te te t z ν r ε,0 0, ** . riν r πi p10 0, 3 11 1 The domain − Ω ω 0 and its boundary can be seen in Figure 8. The union ∪ +− Ω Ω ω ω 00 is then visualized in Figure 9. 6.3 Domain (())Pn and property ∅ ∅ (())∩∩((++))≠≠Pn Pn 1 From the previous considerations, we conclude that the solution ( ) wz n 0is analytically continued on the domain: ()≔∪∪ −+ P nPΩΩ . nω nω nω Provided that () ( )≢+ P nPn1, we have () ( )∩+≠∅ P nPn1since ∩≠∅ ++ − Ω Ω . nω n ω1, (121) The last property follows from (47), (48), (56), (96), and (112) because, in the opposite case, the inequality ()() () ()() () () (( ) ) ()() ⎜⎟ ⎜⎟ +−=+ −+⎛ ⎝+−⎞ ⎠+− <+− − =++ −−⎛ ⎝+−⎞ ⎠−− + − νnr π σnπ σrπ σπ σ νn r π σ nπ σrπ σπ σ ,2121 12121 1, 21 211 12121 (122) must hold. Simplifying (122), we obtain <r0, which contradicts (56), i.e. () ( )∩+ P nPn1contains an open set and, by construction, is a simply connected domain. Figure 9: To Example 5 –domain ∪+ Ω Ω ωω0 ‒0. 32 Josef Diblík and Miroslava Růžičková Remark 7. In the following, we refer to Example 5. The domain () P 0is shown in Figure 10. We do not visualize the property (121) in an additional figure since it can be explained by the same one. It was shown above that two disjunct P n ω 1and P n ω 2 with ≠ n n 12 are identical up to their locations in the complex plane or Riemann surface. In much the same way, one can prove that the domains − Ω n ω 1and − Ω n ω 2are identical with the domains + Ω n ω 1and + Ω n ω 2. They can differ only by their locations. Since, in Figure 9, we have ∩≠∅ +− Ω Ω ωω00 (this intersection is highlighted in blue), the following will be true as well: ∩≠∅ − +− Ω Ω ωω10, ∩≠∅ +− Ω Ω ωω01 . Note that, for =σ2 or for a rational =∕σmm 12 , where −=mm 1 12 , there exists only a single domain ( )P n. All others have identical forms and locations. Since =σ2 , we have ⋃= = − n σn 0 2110 and   ≔⧹ 110 . A circle neighbourhood of the origin is drawn in Figure 11. Its radius r equals ∣() ∣ − z0 20 , where the curve () − zφ 20 is defined by formula (120) and by: ∣()∣ () == = ∕∕≐ −− rz νr cπ 0cos 0, cos 6 20,000 3 0.00013 . 20 2 II Inequality (49) is now  ∣()∣ () ∣∣ ∣∣ ⎟ ⎜ ⎜⎟ ≤⎧ ⎨ ⎩ ⎛ ⎝−−⎞ ⎠ ⎫ ⎬ ⎭=⎧ ⎨ ⎩ ⎛ ⎝−⎞ ⎠ ⎫ ⎬ ⎭ ∈− wz e δ aσr zδz min , exp sin 1 min 1, exp 1 2 , zkp1 1 1(123) where ( ) <∕δtexp 1 2 * and ()=−=tε σ r **sin 1 0.00009 p1. We will show that the intersection () ( )∩+ P nPn1contains a part of the ray (18), where () () () =≔+ −=+ − ν νn ψn π σnπ σ *121 1 . (124) This is a consequence of the fact that the angle ( ) = ν νn *belongs to two intervals, namely, (97) and (113), where n is replaced by + n1 since the chain of inequalities () ()() ( ) () +− −<< + − −+ ν nr π σνn νnr π σ 1, 21*,21 holds being equivalent with − <<r r 0 . This ray is defined for ( ) ∈′tt0, , where ′tis defined by the intersection of loop segments (() ) + zνn * n2and (()) + − zνn * n2, 1 , i.e. Figure 10: To Example 5 and Remark 7 –domain ()≔∪∪ + P P0Ω Ω ωω ω 0 ‒00. Vanishing and blow-up solutions 33 ∣ ( ())∣ ∣ ( ())∣ (( ) ) () () ⎜⎟ ′= = =⎛ ⎝−−∕ −⎞ ⎠ ++ −∕− tzνn z νn σrπ σc ** cos 1 2 1 . nn σ 22,1 2 II 11 Moreover, let γ0be a fixed number such that <≤γr0 , 01(125) where r 1 satisfies (56). Then, the domain () ( )∩+ P nPn1contains a sector around the angle ( )ν n *with centre at the point =z0defined by the inequality: () () () ()−= + −−≤≤ + −+= + ν nγ nπ σγφ nπ σγνnγ *21 121 1* 00 0 0 (126) since the inequality () ()()() +− −≤+ − −+ ν nr π σνnr π σ 1, 21 ,21 is equivalent to () ()−< + ν nrνn r ** and, therefore, () () () ()−< −< +< + ν nrνnγνnγνnr **** . 00 (127) Inequalities (56) and (127) imply (126). 6.4 Solutions analytically continuable to (())∪∪((++))Pn Pn 1 The following explanation can be followed in Figures 12 and 13, Example 5, and Remark 8. Let ℓ () ( ) ⊂nPn 1and ℓ () ( ) ⊂+nPn1 2be the segments of two loops of the curve (8) symmetric with respect to the ray (18), defined by (124), starting at the point =z0and such that, for every φ satisfying () ()−<< + ν nγφνnγ ** , 00 where γ0satisfies (125), we have Figure 11: To Example 5 and Remark 7 –a circle neighbourhood of =z0 . 34 Josef Diblík and Miroslava Růžičková ℓ () () ( ) { }∩ ∩ + ∩ = < <∞ ≠∅ =nPnPn zte t j1,0 ,1,2 . jiφ Assume as well that, on the curve ℓ ( ) n 1, we have () () = →+ zφlim 0 , φ νnγ *0 while on the curve ℓ ( ) n 2, () () = →− zφlim 0 . φ νnγ * 0 To define the curve ℓ ( ) n 1, we put () ℓ =≔−∕+− ν νπσγ21 , 00 0 1(128) and to define the curve ℓ ( ) n 2, we put () ℓ =≔∕−− ν νπ σγ21 00 0 2(129) in (8). Then, the point of intersection ℓ() ℓ( ) =∩ M nn n12 lies on the ray (18) defined by angle (124). This is a consequence of the following computation if ( ) = φ νn * . For ℓ ( ) n 1, we have (()())( ()) ℓ −− = −∕+−νσνn πσγ c os 1 *cos 2 1 . 00 1 In the case of ℓ ( ) n 2, we obtain the same result since (()())(()) ℓ −− = ∕−−νσνn πσγ c os 1 *cos 2 1 . 00 2 The symmetry property can be proved in much the same way. The equations defining the curves ℓ ( ) n 1and ℓ ( ) n 2are Figure 12: To Example 5 and Remark 8 –a circle neighbourhood of =z0 . Vanishing and blow-up solutions 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 where () ()−<≤ + ν nγφνnγ ** 0 0 . In both cases, we assume that the parameter c is identical and sufficiently large. A suitable value of c can be determined, e.g., as the solution of equation: ∣ (() ( ))∣ ∣ ( ( ))∣ ℓ() +∕ − = ++ zψn πσ z νnr1, . nn 22 1(130) Remark 8. We use Example 5 again to visualize curves ℓ ( ) 0 1and ℓ ()0 2. Knowing that domains of the type ( )P n have the same form, without loss of generality, we can use the previous constructions. Therefore, for clarity, we do not replace the value = n 0 with = n1 (except for () P 1). Both curves are visualized in Figures 12 and 13 (highlighted in blue and in red), where, by formula (124), ()= νπ *02 and ==∕γrπ 6 01 . Curves ( ) ± zφ 20 are defined by formulas (119) and (120). Solving equation (130), where = n 0 , leads to = c cII. The formulas for ℓ ( ) 0 1and ℓ ()0 2are () () () 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 where − ∕< ≤∕πφπ6 6 . Assuming that φ varies within the domain described by the inequalities: () ()−≤≤ + ν nγφνnγ ** , where ( ) ∈γγ0, 0is fixed, consider the behaviour of solutions of integral curves given by (1)on ℓ ( ) n 1and ℓ ( ) n 2 with respect to a fixed cylinder  ( ) λdefined by (24), i.e. with respect to the cylinder: Figure 13: To Example 5 and Remark 8 –a circle neighbourhood of =z0 –zoom. 36 Josef Diblík and Miroslava Růžičková += α βe . λ222 (131) Initially, we will examine the case of the curve ℓ ( ) n 1showing that any integral curve intersecting the lateral surface of cylinder (131) or its lower base defined by the plane ()=− φ νn γ * (132) remains inside the cylinder if φ increases. We assign to each point on the lower base of the cylinder a point lying in the plane ( ) = φ νn * (133) being defined as its mapping by the corresponding integral curve when φ starts with the value (132) and ends with the value (133). Such a behaviour of integral curves is a consequence of Lemma 1, (i) where the geometrical meaning is given by inequalities (26) and (27), since (()) ℓ ==−∕+−<νν πσγ s in sin sin 2 1 0 . 00 1 This inequality holds, we refer to (125), (128), and also to Remark 1. The aforementioned mapping is continuous implementing a one-to-one correspondence between the lower base, i.e. the set { () () } =−+≤φαβ φ ν n γα β e,, : *,λ222 and a simply connected nonempty domain  ( ) n *such that  () {( ) () }⊂=+≤nφαβφνnαβe *,, : *, . λ222 In addition, to each point of the domain  ( ) n *, there corresponds a solution to (1), which is analytical on ℓ ( ) n 1. On ℓ ( ) n 2, one can proceed in much the same way. Any integral curve intersecting the lateral surface of cylinder (131) or its upper base defined by the plane ()=+ φ νn γ * , (134) remains inside the cylinder if φ decreases. We assign to each point on the upper base of the cylinder a point on the plane ( ) = φ νn * (135) as a result of mapping along the corresponding integral curve when φ starts with the value (134) and ends with the value (135). Such a behaviour of integral curves is a consequence of Lemma 1, (ii) where the geometrical meaning is characterized by inequalities (26) and (27) since, due to (125) and (129), (()) ℓ ==∕−−>νν πσγ s in sin sin 2 1 0 . 00 2 In this case, the upper base of cylinder (131), i.e. the set { () () }=+ +≤φαβ φ ν n γ α β e,, : *, λ222 is mapped into a simply connected nonempty domain  ( ) n ** such that  () {( ) () }⊂=+≤nφαβφνnαβe ** ,, : *, . λ222 By construction, to each point of the domain  ( ) n ** , there corresponds a solution to (1) analytical on ℓ ( ) n 2. Neither of the domains  ( ) n *and  ( ) n ** will degenerate to a point being a closed neighbourhood of the point (() )νn *,0,0 in the plane  { () () }=∈φαβ φ ν n αβ,, : *,, since the system (16), (17) has a trivial solution ()( ) =φαβ φ, , ,0,0 for () ()−≤≤ + ν nγφνnγ ** . Thus, the set  () () ( ) ≔∩nn n *** is compact two-dimensional. By the construction, to each point of the domain  ()n, there corresponds a solution to (1) analytical on ℓ ( ) n 1and ℓ ( ) n 2. This means that the solution is Vanishing and blow-up solutions 37 analytical on () ( )∪+ P nPn1. Within the domain ( )P n, this solution has properties described previously for a solution ( ) wz n 0and, within domain ( ) + P n1, such a solution ( ) + wz n1 0 has, as our constructions do not depend on n , the same properties as indicated for the solution ( ) wz n 0defined on ( )P n. Denoting such a solution by () + wz nn,1 0 , we replace by it the previously considered solution ( ) wz n 0. Since  ()nis a two-dimensional set, there exist infinitely many solutions of this type. 6.5 Solutions analytically continuable to (())∪∪((++))∪∪((++))Pn Pn Pn12 Let us repeat the considerations of the previous section for the domain: ()()+∪ + P nPn12 . Now, adapting the previous notation, the set  ()() ( ) +≔ +∩ +nn n1*1** 1is compact two-dimensional. By the construction, to each point of the domain  ()+n1, there corresponds a solution to (1) analytical on ℓ ( ) +n1 1and ℓ ( ) +n1 2. Thus, such a solution is analytical on ()( ) +∪ + P nPn12 . Within the domain ( ) + P n1, this solution has the above-described properties and, within the domain ( ) + P n2, such a solution has, as our constructions do not depend on n , the same properties as indicated for ( ) + P n1. Then, there exists a compact two-dimensional set  () ()⊂nn 1(we argue in much the same way as earlier) such that, through every point of the domain  ()n 1, a solution to (1) passes analytical on () ( ) ( ) ∪+∪+ P nPn Pn12 . Denoting such a solution by () + wz nn,2 0, replace by it the previously considered solution () + wz nn,1 0 . Since  ()n 1is a two-dimensional set, there exist infinitely many solutions of this type. 6.6 Solutions analytically continuable to (())∪∪⋯⋯∪∪((++))Pn Pn N Let the above-mentioned construction be carried out for each fixed integer n and let N be an arbitrary natural number. As a result, one can always find an infinite set of solutions to (1) that are analytical on the domain: () ( ) ( ) ( )∪+∪+∪⋯∪+ P nPn Pn PnN12 . (136) If σ is an integer, then, for = n 0 and =− N σ 1 , the cycle (136) closes since ()( ) −= P σP10 . Then, we obtain an infinite set of solutions that are analytical in the domain: () () () ( ) ( )∪∪∪⋯∪−∪− P P P Pσ Pσ012 2 1 , where () ( ) =− P Pσ01 . If σ is a rational number, =∕σmm 12 , then, for = n 0 and =− N mm 12 , the cycle closes on the Riemann surface of the function ∕ zm12since ()( ) −= P mm P0 12 . Then, we obtain an infinite set of solutions that are analytical in the domain: () () ( ) ( )∪∪⋯∪ −−∪ − P PPmmPmm01 1 , 12 12 where () ( ) =− P Pm m012 . If σ is an irrational number, then the cycle does not close. Therefore, we can only say that, for an arbitrary natural number N and an integer = n n 0 , there exists an infinite set of solutions to (1) that are analytical in the union of a finite number of domains: () ( ) ( ) ( ) ∪+∪+∪⋯∪+ P nPn Pn PnN12 00 0 0 located on the Riemann surface of the logarithmic function. 38 Josef Diblík and Miroslava Růžičková 7 Example Let, in an equation (1), =σ3 ,= a1 , and = M2 . Let =ρ 1 and = k0 . By (50), we have () ()() == + −=+ ν ψn nπ σnπ21 121 2, and there are two different values for = n 0, 1 in the complex plane  , ()=∕ψπ0 2 and ()=∕ψπ13 2 . The respective constructions can be seen in Figure 14. In accordance with (23), put {}<= < ⎧ ⎨ ⎩ ⎫ ⎬ ⎭==ερ a M 00.02min, 2min 1, 0.25 0.25 . 0 Next, we apply formula (34) with ()=∕∈ p σ32 1, and {)=∕< −μσp14 min1, putting ()=< ⎧ ⎨ ⎩⎛ ⎝+∕⎞ ⎠⎫ ⎬ ⎭=∕ ≐ ε *0.00018 min 1, 1 812 2 17 0.00019 . p 44 Let =<ωε0.0001 * p and =∕ c 10 4 8. Then, by (65), ∣∣ () () () ⎜⎟ =⎛ ⎝− −⎞ ⎠=⎛ ⎝∕⎞ ⎠= ∕− ∕− zσr σc rr *cos 1 1cos2 10 2 10 2cos2 , n σ11 8 12 4 and the solution of equation (66) ∣∣ == ≤≤ ≤ ≤ rr r max max ωz ε r 212cos21.9 ** np gives the value =∕rπ 6 2 . In (58), we set =∕ ζ 12 . Then, =∕=∕rr π212 12 . For ± ν 0 defined by formulas (45) and (46) with = n 0, 1 , we have, by (50), () () () () () () () () =∕ = ∕ =∕ = ∕ =∕ = ∕ =∕ =∕ ++ −− ++ −− νr π νr π νr π νr π νr π νr π νr π νr π 0,76,1,196,0,56,1,176, 0,43,1,103,0,23,1,83. 01010101 02020202 Figure 14: Case =σ3 –domains P ω0and P ω1. Vanishing and blow-up solutions 39 By (22), we can put ()∣()∣∣()∣== ±± ε νενr νrsin 0, 0.02 sin 0, , r0000 where a dependence on r is emphasized. Because () ∣ ( )∣ [] [] ==∕= ∈∈ ± εν ν r πmin min 0.02 sin 0, 0.02sin 6 0.01 , r rr rrrr,0,0 12 12 we can, independently of r , put ()≔ ε ν0.0 1 0. Since, for =rr 1 , ∣∣ () () () ⎜⎟ ⎜⎟ =⎛ ⎝− −⎞ ⎠=⎛ ⎝∕ ∕⎞ ⎠=⎛ ⎝ ∕ ∕⎞ ⎠≐<= ∕− ∕∕ zσr 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 put =∕ ≐ <= ε ε ** 3 10,000 0.00013 *0.0001 8 pp 4. Inequality (63) holds as well since () () () ⎜⎟ ⎛ ⎝−⎞ ⎠=⎛ ⎝⎞ ⎠=∕⋅ ≐ < = ∕− ∕− σc c εν 111 22 2 10 0.0000707 0.01 . σ11 12 40 For completeness, we compute the angles, defined by (47) and (48): () () () () () () () () =∕ = ∕ =∕ =∕ =∕ =∕ =∕ =∕ ++−− ++−− νr π νr π νrπ νr π νr πνr πνrπνr 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, let us construct the domains P ω 0 and P ω 1 . Put =∕⋅ c 34 10 i 8 and =∕ ⋅ c 112 10 . o8The inner boundary of P ω 0 consists of two segments of loops (70) passing through the point =∕ zωe iπ 2 with =rr 1 . The red segment is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ νr σφ σc eπφ ce cos 0, 1 1cos 5 6 2 2, ir σiφ i iφ 00111 12 where () ()=∕< ≤∕= − ν rπ φπ ψ0, 6 2 0 , 1 while the green one is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ νr σφ σc eπφ ce cos 0, 1 1cos 7 6 2 2 , ig σiφ i iφ 00111 12 where () ( )=∕≤ < ∕=+ ψπφπνr02 560, . 1 The outer boundary of P ω 0 consists of two segments of loops (70) passing through the point =∕ zεe ** piπ 2with =rr 2 . The red segment is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ νr σφ σc eπφ ce cos 0, 1 1cos 2 3 2 2, or σiφ o iφ 00211 12 where () ()=∕ < ≤∕= − ν rπ φπ ψ0, 12 2 0 , 2 while the green one is defined as: () (( )( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ νr σφ σc eπφ ce cos 0, 1 1cos 4 3 2 2, og σiφ o iφ 00211 12 where () ( )=∕≤ < ∕ = + ψπφπνr0 2 11 12 0, . 2 The inner boundary of P ω 1 consists of two segments of loops (70) passing through the point = ∕ zωe iπ32 with =rr 1 . The red segment is defined as: 40 Josef Diblík and Miroslava Růžičková () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ νr σ φ σc eπφ ce cos 1, 1 1cos 17 6 2 2 , ir σiφ i iφ 10111 12 where () ()=∕<≤∕= − ν rπφπψ1, 7 6 3 2 1 , 1 while the green one is defined as: () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ νr σ φ σc eπφ ce cos 1, 1 1cos 19 6 2 2 , ig σiφ i iφ 10111 12 where () ( )=∕≤< ∕= + ψπφπνr132 116 1, . 1 The outer boundary of P ω 1 consists of two segments of loops (70) passing through the point =∕ zεe ** piπ32 with =rr 2 . The red segment is defined as: () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ −∕− ∕ zφ νr σ φ σc eπφ ce cos 1, 1 1cos 8 3 2 2 , or σiφ o iφ 10211 12 where () ()=∕<≤∕= − ν rπφπψ1, 13 12 3 2 1 , 2 while the green one is defined as: () (()( )) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠=⎛ ⎝ ∕− ⎞ ⎠ +∕− ∕ zφ νr σ φ σc eπφ ce cos 1, 1 1cos 10 3 2 2 , og σiφ o iφ 10211 12 where () ( )=∕≤< ∕= + ψπφπνr132 2312 1, . 2 Figure 15 shows a circle neighbourhood of the origin in  where there exist solutions with the properties indicated in Theorem 1. The radius of the circle ∣()∣==⋅ +− rz01.510 21 4is computed using the curves constructed in Part 6.2.1. We have ∣ ( ( ))∣ (()( )()) () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −− −⎞ ⎠ =⎛ ⎝∕− ∕⎞ ⎠=⋅ +++ ∕− ∕− zν r νr σ νr σc ππ c 1, cos 1, 1 1, 1 cos 10 3 11 3 2310. og σ o 101020111 12 4 The curve ( ) + zφ 11 is defined as (we refer to formulas (101) and (102)) () (( ) ( ) ( )) () (() ) () () () () ⎜⎟ ⎜⎟ =⎛ ⎝ −−− −⎞ ⎠ =⎛ ⎝−⎞ ⎠≤≤ ++∕− +∕++ zφ σνrσφ σc e νr φ ce ν nr φ ν nr cos 1 1, 1 1 cos 2 1, 2 2,, ,, σiφ iφ 11 1 I 11 1 I 12 12 where the value of the parameter cI is given by the equations: (()( )()) () () () () ( ) ⎜⎟⎜⎟ ⎜⎟ ⎛ ⎝ −− −⎞ ⎠=⎛ ⎝∕− ∕⎞ ⎠=⎛ ⎝−⎞ ⎠ ++∕− ∕∕− νr σ νr 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 solutions 41