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