On blow-up solutions and dead zones in semilinear equations
We study semilinear elliptic equations of the form div(A(z)∇u) = f(u) in Ω⊂ C, where A(z) stands for a symmetric 2×2 matrix function with measurable entries, det A =1, and such that 1/ K |ξ|² ≤ 〈A(z)ξ,ξ〉 ≤ K |ξ|², ξ ∈ R², 1≤ K < ∞. Making use of our Factorization theorem, we give some explicit so...
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Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. 2018-08-04T18:02:13Z 2018-08-04T18:02:13Z 2018 On blow-up solutions and dead zones in semilinear equations / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 4. — С. 9-15. — Бібліогр.: 12 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2018.04.009 https://nasplib.isofts.kiev.ua/handle/123456789/141139 517.5 We study semilinear elliptic equations of the form div(A(z)∇u) = f(u) in Ω⊂ C, where A(z) stands for a symmetric 2×2 matrix function with measurable entries, det A =1, and such that 1/ K |ξ|² ≤ 〈A(z)ξ,ξ〉 ≤ K |ξ|², ξ ∈ R², 1≤ K < ∞. Making use of our Factorization theorem, we give some explicit solutions for the above equation if f = e^u or f = e^q, when matrices A(z) are chosen in an appropriate form. Досліджено напівлінійне диференціальне рівняння виду div(A(z)∇u)=f(u) в Ω⊂C, де A(z) — симетрична 2×2 матрична функція з вимірними коефіцієнтами, detA=1, і така, що 1/K|ξ|2⩽⟨A(z)ξ,ξ⟩⩽K|ξ|2,ξ∈R2,1⩽K<∞. Із застосуванням теореми про факторизацію, доведену нами раніше, наведено явні розв’язки для зазначеного рівняння, якщо матриці A(z) обрані належним чином і f=e^u або f=u^q. Исследовано полулинейное дифференциальное уравнение вида div(A(z)∇u)=f(u) в Ω⊂C, где A(z) симметричная 2 Ч 2 матричная функция с измеримыми коэффициентами, detA =1 и такая, что 1/K|ξ|2⩽⟨A(z)ξ,ξ⟩⩽K|ξ|2,ξ∈R2,1⩽K<∞. С применением теоремы о факторизации, доказанной нами ранее, приведены явные решения для указанного уравнения, если матрицы A(z) выбраны надлежащим образом и f=e^u или f=u^q. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика On blow-up solutions and dead zones in semilinear equations Вибухові розв’язки та мертві зони для напівлінійних рівнянь О взрывающихся решениях и мертвых зонах для полулинейных уравнений Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On blow-up solutions and dead zones in semilinear equations |
| spellingShingle |
On blow-up solutions and dead zones in semilinear equations Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Математика |
| title_short |
On blow-up solutions and dead zones in semilinear equations |
| title_full |
On blow-up solutions and dead zones in semilinear equations |
| title_fullStr |
On blow-up solutions and dead zones in semilinear equations |
| title_full_unstemmed |
On blow-up solutions and dead zones in semilinear equations |
| title_sort |
on blow-up solutions and dead zones in semilinear equations |
| author |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| topic |
Математика |
| topic_facet |
Математика |
| publishDate |
2018 |
| language |
English |
| container_title |
Доповіді НАН України |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| format |
Article |
| title_alt |
Вибухові розв’язки та мертві зони для напівлінійних рівнянь О взрывающихся решениях и мертвых зонах для полулинейных уравнений |
| description |
We study semilinear elliptic equations of the form div(A(z)∇u) = f(u) in Ω⊂ C, where A(z) stands for a symmetric 2×2 matrix function with measurable entries, det A =1, and such that 1/ K |ξ|² ≤ 〈A(z)ξ,ξ〉 ≤ K |ξ|², ξ ∈ R², 1≤ K < ∞. Making use of our Factorization theorem, we give some explicit solutions for the above equation if f = e^u or f = e^q, when matrices A(z) are chosen in an appropriate form.
Досліджено напівлінійне диференціальне рівняння виду div(A(z)∇u)=f(u) в Ω⊂C, де A(z) — симетрична 2×2 матрична функція з вимірними коефіцієнтами, detA=1, і така, що 1/K|ξ|2⩽⟨A(z)ξ,ξ⟩⩽K|ξ|2,ξ∈R2,1⩽K<∞. Із застосуванням теореми про факторизацію, доведену нами раніше, наведено явні розв’язки для зазначеного рівняння, якщо матриці A(z) обрані належним чином і f=e^u або f=u^q.
Исследовано полулинейное дифференциальное уравнение вида div(A(z)∇u)=f(u) в Ω⊂C, где A(z)
симметричная 2 Ч 2 матричная функция с измеримыми коэффициентами, detA =1 и такая, что 1/K|ξ|2⩽⟨A(z)ξ,ξ⟩⩽K|ξ|2,ξ∈R2,1⩽K<∞. С применением теоремы о факторизации, доказанной нами ранее,
приведены явные решения для указанного уравнения, если матрицы A(z) выбраны надлежащим образом
и f=e^u или f=u^q.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/141139 |
| citation_txt |
On blow-up solutions and dead zones in semilinear equations / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 4. — С. 9-15. — Бібліогр.: 12 назв. — англ. |
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2025-11-26T20:25:55Z |
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2025-11-26T20:25:55Z |
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| fulltext |
9ISSN 10256415. Допов. Нац. акад. наук Укр. 2018. № 4
1. Introduction. In this paper, we give new applications of the quasiconformal mappings theory,
see e.g. [1—6], to the study of semilinear partial differential equations in the plane.
Let Ω be a domain in the complex plane .C. Denote by 2 2( )M × Ω the class of two by two sym
metric matrices ( ) { }jkA z a= with measurable entries and det ( ) 1A z = almost everywhere in Ω
satisfying the uniform ellipticity condition
2 21
| | ( ) , | | a.e. in ,A z K
K
ξ 〈 ξ ξ〉 ξ Ω� � (1)
for every .ξ∈C The factor K can be either a constant 1 K < ∞� or a measurable function ( ) ( ),K z L∈ Ω
( ) ( ),K z L∞∈ Ω with 1 ( )K z < ∞� a.e. in Ω. Every such matrix function A generates a quasi con for mal
mapping ω as a homeomorphic solution of the Sobolev class 1, 2
loc ( )W Ω to the Beltrami equation
( ) ( ) ( ) a.e. in ,z zz z zω = µ ω Ω (2)
where the complex dilatation ( )zµ is given by
22 11 12
1
( ) ( 2 ).
det( )
z a a ia
I A
µ = − −
+
(3)
The condition of ellipticity (1) is written now as
1
| ( ) | a.e. in .
1
K
z
K
−
µ Ω
+
� (4)
© V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, 2018
doi: https://doi.org/10.15407/dopovidi2018.04.009
UDC 517.5
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk
Email: vgutlyanskii@gmail.com, staro@ukr.net, vl.ryazanov1@gmail.com
On blowup solutions and dead zones in semilinear equations
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ
We study semilinear elliptic equations of the form div( ( ) ) ( )A z u f u∇ = in Ω ⊂C, where ( )A z stands for a sym met
ric 2 2× matrix function with measurable entries, det 1A = , and such that 2 2 21/ | | ( ) , | | , ,K A z Kξ 〈 ξ ξ〉 ξ ξ ∈R� �
1 .K < ∞� Making use of our Factorization theorem, we give some explicit solutions for the above equation if uf e=
or qf u= , when matrices ( )A z are chosen in an appropriate form.
Keywords: quasiconformal mappings, semilinear PDE, blowup solutions.
10 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2018. № 4
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Vice versa, given a measurable complexvalued function µ satisfying (4), we can invert the al
gebraic system (3) to obtain
2
2 2
2
2 2
| 1 | 2Im
1 | | 1 | |
( ) .
2Im |1 |
1 | | 1 | |
A z
−µ − µ
− µ − µ =
− µ + µ
− µ − µ
(5)
In this case, we say that the matrix function A and the corresponding quasiconformal mapping
ω are agreed.
Let :f →R R be a continuous function. In [7] we have proven the following Factorization
theorem, cf. the corresponding result for the smooth case in [8].
Theorem 1. Let 2 2( ) ( ).A z M ×∈ Ω Then every weak solution 1, 2
loc ( ) ( )u W C∈ Ω ∩ Ω of the se
milinear equation
div [ ( ) ( )] ( ( )), ,A z u z f u z z∇ = ∈Ω (6)
can be expressed as
( ) ( ( )),u z T z= ω (7)
where : Gω Ω → is a Kquasiconformal mapping agreed with the matrix function 2 2( ) ( )A z M ×∈ Ω
and 1, 2
loc ( ) ( )T W G C G∈ ∩ is a weak solution to the equation
( ) ( ( )), . . .T J w f T w a e in G∆ = (8)
Here, ( )J w stands for the Jacobian determinant of the inverse mapping 1( ).z w−= ω
Among the quasiconformal mappings : Gω Ω → , there are a variety of the socalled volumepre
serving maps, for which ( ) 1,J zω ≡ .z ∈Ω In this partial case, we arrive at the following statement:
Corollary 1. Let 2 2( ) ( )A z M ×∈ Ω be a matrix function that generates a volumepreserving qua
siconformal mapping ω(z). Then every weak solution 1, 2
loc ( ) ( )u W C∈ Ω ∩ Ω of the semilinear equation
div [ ( ) ( )] ( ( )), ,A z u z f u z z∇ = ∈Ω (9)
can be expressed as
( ) ( ( )),u z T z= ω (10)
where 1, 2
loc ( ) ( )T W G C G∈ ∩ is a weak solution to the equation
( ( )), . . .T f T w a e in G∆ = (11)
Some applications of the Factorization theorems that we are going to give below are based
just on Corollary 1.
2. Explicit blowup solutions. Let Ω be a bounded domain in C and let ∂ Ω denote its
boundary. In this section, we study the problem
div [ ( ) ( )] ( ( ))A z u z f u z∇ = , (12)
( ) , as ( ) : dist( , ) 0,u z d z z→ ∞ = ∂Ω → (13)
11ISSN 10256415. Допов. Нац. акад. наук Укр. 2018. № 4
On blowup solutions and dead zones in semilinear equations
see, e.g., [9] and [10], as well as its Laplace’s counterpart:
( ) ( ( ))u z f u z∇ = , (14)
( ) , as ( ) : dist( , ) 0.u z d z z→ ∞ = ∂Ω → (15)
Solutions to these problems are called boundary blowup solutions or large solutions. If f(u) = eu,
then (14) is a classical Liouville—Bieberbach semilinear equation that was first investigated by
Bieberbach in his pioneering work [11] related to the study of automorphic functions in the plane.
The corresponding equation (12) with f(u) = eu, can be viewed as a divergent counterpart to the
Liouville—Bieberbach semilinear equation. Recall that if f is a conformal mapping of Ω onto the
unit disk, then the boundary blowup solutions for the Liouville—Bieberbach semilinear equation
are expressed explicitly by the formula
2
2 2
8 | ( ) |
( ) log .
(1 | ( ) | )
f z
u z
f z
′
=
−
(16)
Theorem 2. Let Ω be the annulus | | 1r z< < in the complex plane C and let the matrix function
2 2( ) ( )A z M ×∈ Ω is generated by the formula (5) with the complex dilatation
2 2( ) (| |) (| |) 1 (| |) ,
z
z z z z
z
µ = ν ± ν − ν
(17)
where ( ),tν 0 1t <� , stands for an arbitrary measurable function. If | ( ) | 1,t qν <� then there exists
one and only one boundary blowup solution to the semilinear equation
div [ ( ) ] uA z u e∇ = in the annulus | | 1,r z< < (18)
which is given explicitly by the formula
2
22 2
2
( ) log .
| | log | |log sin
log
u z
z r z
r
π
=
π
⋅
(19)
Indeed, if the complex dilatation ( )zµ has the form
( ) (| |) ,
z
z k z
z
µ = (20)
where ( ) :k τ →R C is a measurable function such that | ( ) | <1,k kτ � then the formula
| |
1
1 ( )
( ) exp
| | 1 ( )
z
z k d
z
z k
+ τ τ ω = − τ τ
∫ (21)
represents a unique quasiconformal mapping of the unit disk, as well as the whole complex plane,
onto itself with complex dilatation µ and the normalization: ω(0) = 0, ω(1) = 1, see, e.g.,
[4, p. 82], and [12].
Analyzing formula (21) with specified as above ( ),k t we see that the Jacobian ( ) 1,J zω ≡ i.e.,
the mapping ω is volumepreserving, and | ( ) | | |z zω = for .z ∈C Mapping conformally the given
12 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2018. № 4
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
annulus onto the unit disk and applying the Bieberbach explicit formula (16), we see that the
function
2
22 2
2
( ) log
| | log | |log sin
log
T w
w r w
r
π
=
π
⋅
(22)
represents the blowup solution to the semilinear Liouville—Bieberbach equation in the annulus
| | 1.r w< < It remains to apply Corollary 1. The uniqueness follows from a fundamental result by
Marcus and Véron, see [10], Theorem 5.3.7.
Our next example deals with the study of the blowup solutions to the Liouville—Biberbach
type equation defined in an unbounded domain of the complex plane.
Theorem 3. Let H + be the right halfplane { : Re 0}z z > in the complex plane C and let the mat
rix function 2 2( ) ( )A z M ×∈ Ω have the entries 11 1,a = 2
12 2 ( ) / 1 ( ),a x x= ± ν − ν 2 2
22 (1 3 ( )) / (1 ( ))a x x= + ν − ν
2 2(1 3 ( )) / (1 ( ))a x x= + ν − ν , where ( ),xν ,x ∈C stands for an arbitrary measurable realvalued function such that
| ( ) | 1.x qν <� Then there exist boundary blowup solutions to the semilinear equation
div [ ( ) ] , ,uA z u e z H +∇ = ∈ (23)
which are written explicitly:
2
2
( ) log , ,u z z x iy
x
= = + (24)
2 2( ) log 8 2 2log(1 ), 0.xu z x e− λ= λ − λ − − λ > (25)
Indeed, the matrix function ( )A z with the above entries generates, by formula (3) the com
plex dilatation
2 2( ) ( ( ) ( ) 1 ( ))z x i x xµ = ν ± ν − ν (26)
which, as we see, depends on x only. By Proposition 5.23 in [4], see also [12], a unique quasicon
formal mapping of the right halfplane onto itself with the complex dilatation µ and the norma
lization ω(0) = 0, ω (i) = i and ω(∞) = ∞, is represented explicitly by the formula
0
1 ( )
( ) .
1 ( )
x
t
z dt iy
t
+ µ
ω = +
− µ∫ (27)
Analyzing formula (27), we see that the Jacobian ( ) 1,J zω ≡ i. e., the mapping ω is volume
preserving, and Re ( )z xω = for .z ∈C By Corollary 1, a solution 1, 2
loc ( ) ( )u W C∈ Ω ∩ Ω of the sem
ilinear equation
div [ ( ) ( )] , ,uA z u z e z H +∇ = ∈ (28)
is expressed as
( ) ( ( )),u z T z= ω (29)
where 1, 2
loc ( )T W G∈ is a solution to the equation
( )( ) , .T wT w e in H +∆ = . (30)
13ISSN 10256415. Допов. Нац. акад. наук Укр. 2018. № 4
On blowup solutions and dead zones in semilinear equations
Since the function
1
( )
1
w
F w
w
−
=
+
maps conformally the right halfplane H + onto the unit disk ,D we see that, by the Bieberbach
formula (16), the function
2
2 2
8 | ( ) |
( ) log 2logRe log 2
(1 | ( ) | )
F w
T w w
F w
′
= = − +
−
gives us a blowup solution to Eq. (30) in H + . Now, by formula (29), we have that the first re
quired solution has the form
( ) ( ) 2logRe ( ) log 2 2logRe log 2.u z T z x= ω = − ω + = − +
The second solution can be obtained in the same way.
3. Free boundary. The effect of the “dead zone” very important for applications to solutions
of some partial differential equations, see, e.g., [9], the Introduction and § 1, is that the solution of
the corresponding differential equation vanishes on some nonempty open set of the domain of
definition. For example, it is well known that the solution of the semilinear equation
qu u∆ =
may have the “dead zone” only when 0 < <1,q see, e.g., [9, p. 15].
We confine ourselves to only one result in this direction, which is again a simple consequence
of Corollary 1.
Theorem 4. Let C be the complex plane and let the matrix function
2
2
22
2 ( )
1
1 ( )
( )
2 ( ) 1 3 ( )
1 ( )1 ( )
x
x
A z
x x
xx
ν
− ν = ν + ν
− ν− ν
∓
∓
, (31)
where ( )xν , ,x ∈R stands for an arbitrary measurable realvalued function such that | ( ) | 1.x qν <�
Then the semilinear equation
div[ ( ) ] , 0 1, ,qA z u u q z∇ = < < ∈C (32)
has the following solution with the “dead zone” in the complex plane:
2
1
2
0
2 ( )
, ( ), ,( , )
1 ( )
0 ( ).
x q
t
y dt if y x xu x y
t
if x x
−
ν γ ± > ϕ ∈ = − ν
ϕ
∫ R
�
(33)
14 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2018. № 4
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Here,
1
2 1(1 )
,
2(1 )
qq
q
− −
γ = +
and
2
0
2 ( )
( ) , ,
1 ( )
x
t dt
y x x
t
ν
= ϕ = ± ∞ < < +∞
− ν
∫
stands for the corresponding free boundary parametrization.
Indeed, the matrix function ( )A z generates a quasiconformal automorphism of the complex
plane ω , which is expressed explicitly by formula (27). Since the mapping ω is volumepres
erving, we can apply Corollary 1 in order to represent solutions to Eq. (32) in the form
( ) ( ( )),u z T z= ω
where ( )T w satisfies the equation
( ) ( ), .qT w T w w i∆ = = ξ + η
It remains to note that the function
2
1( ) , 0qT w if−= γη η >
and ( ) 0T w = , if 0,η� satisfy the above equation.
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15ISSN 10256415. Допов. Нац. акад. наук Укр. 2018. № 4
On blowup solutions and dead zones in semilinear equations
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Received 07.12.2017
В.Я. Гутлянський, О.В. Нєсмєлова, В.І. Рязанов
Інститут прикладної математики і механіки НАН України, Слов’янськ
Email: vgutlyanskii@gmail.com, staro@ukr.net, vl.ryazanov1@gmail.com
ВИБУХОВІ РОЗВ’ЯЗКИ ТА МЕРТВІ ЗОНИ
ДЛЯ НАПІВЛІНІЙНИХ РІВНЯНЬ
Досліджено напівлінійне диференціальне рівняння виду div( ( ) ) ( )A z u f u∇ = в Ω ⊂C , де ( )A z — симетрична
2 × 2 матрична функція з вимірними коефіцієнтами, det 1A = , і така, що 2 2 21/ | | ( ) , | | , ,K A z Kξ 〈 ξ ξ〉 ξ ξ∈R� �
1 .K < ∞� Із застосуванням теореми про факторизацію, доведену нами раніше, наведено явні розв’язки для
зазначеного рівняння, якщо матриці ( )A z обрані належним чином і uf e= або qf u= .
Ключові слова: квазіконформні відображення, напівлінійні рівняння в частинних похідних, вибухові роз в’яз ки.
В.Я. Гутлянский, О.В. Несмелова, В.И. Рязанов
Институт прикладной математики и механики НАН Украины, Славянск
Email: vgutlyanskii@gmail.com, staro@ukr.net, vl.ryazanov1@gmail.com
О ВЗРЫВАЮЩИХСЯ РЕШЕНИЯХ И МЕРТВЫХ ЗОНАХ
ДЛЯ ПОЛУЛИНЕЙНЫХ УРАВНЕНИЙ
Исследовано полулинейное дифференциальное уравнение вида div( ( ) ) ( )A z u f u∇ = в Ω ⊂C , где ( )A z —
симметричная 2 × 2 матричная функция с измеримыми коэффициентами, det 1A = и такая, что 2 2 21/ | | ( ) , | | , ,K A z Kξ 〈 ξ ξ〉 ξ ξ∈� �
2 2 21/ | | ( ) , | | , ,K A z Kξ 〈 ξ ξ〉 ξ ξ∈R� �
1 .K < ∞� С применением теоремы о факторизации, доказанной нами ранее,
приведены явные решения для указанного уравнения, если матрицы ( )A z выбраны надлежащим образом
и uf e= или qf u= .
Ключевые слова: квазиконформные отображения, полулинейные уравнения в частных производных, взры
вающиеся решения.
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