Barrier functions for one class of semilinear parabolic equations
For a parabolic quasilinear equation with monotone convex potential, we construct superparabolic and subparabolic barrier functions by the method of decomposition.
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2008
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| author | Bondarenko, V. G. Prokopenko, Yu. Yu. Бондаренко, В. Г. Прокопенко, Ю. Ю. Бондаренко, В. Г. Прокопенко, Ю. Ю. |
| author_facet | Bondarenko, V. G. Prokopenko, Yu. Yu. Бондаренко, В. Г. Прокопенко, Ю. Ю. Бондаренко, В. Г. Прокопенко, Ю. Ю. |
| author_sort | Bondarenko, V. G. |
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| collection | OJS |
| datestamp_date | 2020-03-18T19:49:31Z |
| description | For a parabolic quasilinear equation with monotone convex potential, we construct superparabolic and subparabolic barrier functions by the method of decomposition. |
| first_indexed | 2026-03-24T02:39:09Z |
| format | Article |
| fulltext |
UDK 517.956
V. H. Bondarenko, G. G. Prokopenko (Nac. texn. un-t Ukrayn¥ „KPY”, Kyev)
BAR|ERNÁE FUNKCYY DLQ ODNOHO KLASSA
POLULYNEJNÁX PARABOLYÇESKYX URAVNENYJ
For a parabolic quasilinear equation with a monotone convex potential, we construct superparabolic and
subparabolic barrier functions by using the decomposition method.
Dlq paraboliçnoho kvazilinijnoho rivnqnnq z monotonnym opuklym potencialom metodom de-
kompozyci] pobudovano super- ta subparaboliçni bar’[rni funkci].
1. Postanovka zadaçy y predvarytel\n¥e svedenyq. Lynejnoe paraboly-
çeskoe uravnenye kak matematyçeskaq model\ processa dyffuzyy obladaet rq-
dom nedostatkov. Tak, reßenyg u ( t, x ) zadaçy Koßy
∂
∂
=u
t
Lu , u ( 0, x ) = f ( x ), x ∈ Rn
,
s πllyptyçeskym operatorom
Lu a x
u
x x
A x u t xjk
j k
= ( ) ∂
∂ ∂
≡ ( )∇ ( )
2
2tr ,
sootvetstvuet beskoneçnaq skorost\ rasprostranenyq vozmuwenyj.
Bolee estestvennoj matematyçeskoj model\g qvlqetsq nelynejnoe parabo-
lyçeskoe uravnenye (sm. obzor [1]), naprymer polulynejnoe uravnenye
∂
∂
= + ( ∇ )u
t
Lu x u uΦ , , v fazovom prostranstve E = Rn.
V çastnosty, otmetym rabotu [2], posvqwennug yssledovanyg svojstv reßenyq
zadaçy Koßy dlq uravnenyq „reakcyq–dyffuzyq”
∂
∂
= − ( )u
t
u g t x u up∆ , sgn , u ( 0, x ) = f ( x ), 0 < p < 1, g ( t, x ) ≥ 0,
v kotoroj pryveden¥ uslovyq mhnovennoj kompaktyfykacyy nosytelq (MKN)
reßenyq: pry nekotor¥x uslovyqx na funkcyy f y g supp u ( t, x ) ⊂ [ a; b ] dlq
lgboho t > 0.
Analohyçn¥e rezul\tat¥ poluçen¥ y dlq bolee obwyx kvazylynejn¥x urav-
nenyj [3].
V nastoqwej rabote rassmotrena zadaça Koßy
∂
∂
= − ( )u
t
Lu uΦ , u ( 0, x ) = f ( x ) ≥ 0, x ∈ Rn, (1)
y pry nekotor¥x uslovyqx na potencyal Φ dlq reßenyq u ( t, x ) postroen¥ ba-
r\ern¥e funkcyy — superparabolyçeskaq y subparabolyçeskaq. Sxema postro-
enyq takyx funkcyj — metod rasweplenyq (dekompozycyy) uravnenyq (1): upo-
mqnut¥e funkcyy qvlqgtsq superpozycyej reßenyj uravnenyj
∂
∂
=v
v
t
L ,
dw
dt
w= − ( )Φ .
Ydeq metoda rasweplenyq b¥la ewe v 70-x hodax predloΩena G. L. Dalec-
kym. ∏tot metod prymenqlsq v rabotax [4, 5] dlq lynejnoho uravnenyq
∂
∂
= +u
t
Lu L u1 ,
hde vozmuwenye L1 — πllyptyçeskyj operator: v upomqnut¥x rabotax poluçe-
© V. H. BONDARENKO, G. G. PROKOPENKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1449
1450 V. H. BONDARENKO, G. G. PROKOPENKO
no predstavlenye
e e e B tt L L tL tL( + ) = + ( )1 1
y yzuçen¥ svojstva semejstva operatorov B ( t ).
Vsgdu nyΩe predpolahaetsq, çto operatornoe pole A ( x ) dyffuzyy ohrany-
çeno y dyfferencyruemo.
Vvedem rqd oboznaçenyj y opredelenyj.
Fundamental\noe reßenye nevozmuwennoho parabolyçeskoho uravnenyq obo-
znaçym çerez p ( t, x, y ) , t > 0, x, y ∈ E, a sootvetstvugwug perexodnug veroqt-
nost\
P t x p t x y dy( ) = ( )∫, , , ,Γ
Γ
nazovem dyffuzyonnoj meroj. Tohda
v( ) = ( ) ( )∫t x f y p t x y dy
E
, , ,
est\ reßenye zadaçy Koßy dlq uravnenyq
∂
∂
=v
v
t
L , v ( 0, x ) = f ( x ) ≥ 0.
Çerez w ( t, x ) oboznaçym neotrycatel\noe reßenye zadaçy Koßy dlq ob¥k-
novennoho dyfferencyal\noho uravnenyq
dw
dt
w= − ( )Φ , w ( 0, x ) = f ( x ) ≥ 0.
Opredelenye 1. Pust\ µ — veroqtnostnaq mera na borelevskoj σ -al-
hebre metryçeskoho prostranstva X, f — neprer¥vnaq vewestvennaq funkcyq
na X. Nazovem meru µ:
neprer¥vnoj otnosytel\no f, esly
lim ,
ε
εµ
→
( )
0
Dc = 0;
lypßycevoj otnosytel\no f, esly otnoßenye
µ
ε
ε( )Dc,
ohranyçeno, hde proobraz
D x c f x cc, :ε ε= < ( ) < +{ }.
Neprer¥vnost\ otnosytel\no f oznaçaet absolgtnug neprer¥vnost\ µ ot-
nosytel\no ynducyrovannoj mer¥ µf
.
Prymerom lypßycevoj mer¥ qvlqetsq haussova mera v R n
, esly f ( x ) =
= ϕ x( ), hde ϕ ( s ) — monotonna s ohranyçenyqmy snyzu na skorost\ vozrasta-
nyq yly ϕ ymeet ohranyçennug varyacyg na [ 0; l ] dlq vsex l. Otsgda sleduet
lypßycevost\ dyffuzyonnoj mer¥ P ( t, x, Γ ) dlq takyx Ωe funkcyj, tak kak
fundamental\noe reßenye p ( t, x, y ) dopuskaet dvustoronngg ocenku hausso-
v¥my plotnostqmy. Odno yz dostatoçn¥x uslovyj lypßycevosty dlq nekoto-
r¥x klassov mer y funkcyj pryvedeno v [6, s. 193 – 194].
Obobwenye. Pust\ ϕ ( s ) — vozrastagwaq funkcyq,
∆c x c f x c, :ε ϕ ϕ ε= ( ) < ( ) < ( + ){ }.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
BAR|ERNÁE FUNKCYY DLQ ODNOHO KLASSA … 1451
Sootnoßenyq
lim ,
ε
εµ
→
( )
0
∆c = 0,
1
ε
µ ε( )∆c, < const
oznaçagt sootvetstvenno neprer¥vnost\ y lypßycevost\ µ otnosytel\no kom-
pozycyy ϕ–
1 � f. Esly ϕ — lypßyceva, to yz neprer¥vnosty y lypßycevosty µ
otnosytel\no f sledugt te Ωe svojstva otnosytel\no ϕ–
1 � f.
UtverΩdenye 1. Pust\
g t f x t dx
Dt
( ) = ( ) − ( )( )∫ µ , D x f x tt = ( ) >{ }: .
Esly f neprer¥vna, µ lypßyceva otnosytel\no f, to g′ ( t ) = 0.
2. Osnovn¥e rezul\tat¥. 2.1. Stepennoj potencyal. Rassmotrym vna-
çale zadaçu Koßy
∂
∂
= −u
t
Lu buα , x ∈ Rn
, u ( 0, x ) = f ( x ) ≥ 0, b > 0, α > 0,
y yssleduem otdel\no sluçay α > 1 y 0 < α < 1.
Pust\ α > 1. PoloΩym
w t x f x bt f x( ) = ( ) + ( − ) ( )( )− − ( − ), /1 1 1 1 1α α α
y oboznaçym çerez u1 y u2 kompozycyy, sostavlenn¥e yz funkcyj v ( t, x ) y
w ( t, x ):
u t x w t y p t x y dy
E
1( ) = ( ) ( )∫, , , , ,
u t x bt t x2
1 1 11 1= ( ) + ( − ) ( )( )− − ( − )v v, , /α α α
.
Lehko vydet\, çto
dw
dt
= – b wα
, w ( 0, x ) = f ( x ).
Pust\ 0 < α < 1.
Osobennost\ takoj zadaçy — obnulenye u ( t, x ) dlq t > t0 pry nekotor¥x
uslovyqx na naçal\nug funkcyg. Çerez ˜ ,w t x( ) oboznaçym neotrycatel\noe
reßenye zadaçy Koßy:
dw
dt
bw
˜
˜= − α
, ˜ ,w x( )0 = f ( x ) ≥ 0,
t. e.
˜ ,
, ,
, ,
/
w t x
f x b t t
f x
b
t
f x
b
( ) =
( ) − ( − ) < ( )
( − )
≥ ( )
( − )
( )− ( − )
−
−
1 1 1
1
1
1
1
0
1
α α
α
α
α
α
α
y poloΩym
u t x w t y p t x y dy
E
3( ) = ( ) ( )∫, ˜ , , , = ( )− ( − )( ) − ( − ) ( )∫ f y bt p t x y dy
Dt
1 1 11α αα / , , ,
u t x
t x bt t
t x
b
t
t x
b
4
1 1 1
1
1
1
1
0
1
( ) =
( ) − ( − ) < ( )
( − )
≥ ( )
( − )
( )− ( − )
−
−
,
, ,
,
,
,
,
,
/v
v
v
α α
α
α
α
α
α
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1452 V. H. BONDARENKO, G. G. PROKOPENKO
hde
D y f y btt = ( ) > ( − ){ ( ) }( − ): /1 1 1α α
.
Teorema 1. Funkcyq u1 — subparabolyçeskaq, funkcyq u 2 — superpara-
bolyçeskaq, t. e. ymegt mesto neravenstva
u1 ( t, x ) ≤ u ( t, x ) ≤ u2 ( t, x ), t > 0, x ∈ Rn, α > 1.
Esly dyffuzyonnaq mera P ( t, x, Γ ) lypßyceva, to funkcyq u 3 — super-
parabolyçeskaq, funkcyq u4 — subparabolyçeskaq, t. e.
u4 ( t, x ) ≤ u ( t, x ) ≤ u3 ( t, x ), t > 0, x ∈ Rn, 0 < α < 1.
Dokazatel\stvo sostoyt v yspol\zovanyy pryncypa maksymuma, t. e. v
ustanovlenyy znaka nevqzok
h
u
t
Lu buk
k
k k= ∂
∂
− + α
.
Tak,
h t x b w t y p t x y dy w t y p t x y dy
E E
1( ) = ( ) ( )
− ( ) ( )
∫ ∫, , , , , , ,
α
α ≤ 0
v sylu neravenstva Hel\dera dlq veroqtnostnoj mer¥ P ( t, x, dy ).
Vtoraq nevqzka
h t x2( ), =
= α α αα α α( − ) ( ) + ( − ) ( ) ( )∇ ( ) ∇ ( )− − − ( − )−( ) ( )1 1 12 1 1 1 2bt t x bt t x A x t x t xv v v v, , , , ,/ ≥ 0
v sylu poloΩytel\nosty matryc¥ dyffuzyy.
Pry v¥çyslenyy sledugwej nevqzky yspol\zuetsq utverΩdenye 1:
h t x b w t y p t x y dy w t y p t x y dy
E E
3( ) = ( ) ( )
− ( ) ( )
∫ ∫, ˜ , , , ˜ , , ,
α
α ≥ 0
v sylu neravenstva Hel\dera.
Nakonec,
h t x4( ), =
= –
bt
t x
t x bt A x t x t x
α α αα
α α( − )
( )
( ) − ( − ) ( )∇ ( ) ∇ ( )+
− ( − )−( ) ( )1
11
1 1 1 2
v
v v v
,
, , , ,/ ≤ 0,
esly t <
v1
1
− ( )
( − )
α
α
t x
b
,
.
Yz poluçenn¥x neravenstv sleduet utverΩdenye teorem¥.
V çastnom sluçae α =
1
2
teorema 1 dokazana v rabote [7].
2.2. V¥pukl¥j potencyal. V pryvedenn¥x v¥ße v¥çyslenyqx znak nevqz-
ky faktyçesky opredelqetsq napravlenyem v¥puklosty funkcyy Φ. Obobwym
rezul\tat¥ p. 2.1, rassmotrev uravnenye (1), hde potencyal Φ — stroho vozras-
tagwaq funkcyq, Φ ( 0 ) = 0, y vvedem dva sluçaq:
2.2.1. Φ v¥pukla vnyz na ( 0; ∞ ),
dz
zΦ( )
∞
∫σ
< ∞ dlq lgboho σ > 0. Po-
loΩym
ψ( ) =
( )
∞
∫s
dz
z
s
Φ
, w t x bt f x( ) = + ( ( ))− ( ), ψ ψ1
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
BAR|ERNÁE FUNKCYY DLQ ODNOHO KLASSA … 1453
y opredelym funkcyy
u t x w t y p t x y dy
E
1( ) = ( ) ( )∫, , , , ,
u t x bt t x2
1( ) = + ( ( ))− ( ), ,ψ ψ v .
Zametym, çto v sylu ub¥vanyq ψ spravedlyvo neravenstvo
u t x t x2( ) ≤ ( ), ,v .
2.2.2. Pust\ Φ v¥pukla vverx na ( 0; ∞ ), dz
zΦ( )∫0
σ
< ∞ dlq lgboho σ > 0.
PoloΩym
ψ̃( ) =
( )∫s
dz
z
s
Φ
0
,
˜ ,
˜ ˜ ,
˜
,
,
˜
,
w t x
f x bt t
f x
b
t
f x
b
( ) =
( ( )) − < ( )
≥ ( )
− ( ) ( )
( )
ψ ψ ψ
ψ
1
0
u t x w t y p t x y dy
E
3( ) = ( ) ( )∫, ˜ , , , = ˜ ˜ , ,ψ ψ− ( )( ( )) − ( )∫ 1 f y bt p t x y dy
Dt
,
u t x
t x bt t
b
dz
z
t
b
dz
z
t x
t x4
1
0
0
1
0
1
( ) =
( ( )) − <
( )
≥
( )
−
( )
( )
( ) ∫
∫
,
˜ ˜ , , ,
, ,
,
,
ψ ψ v
v
v
Φ
Φ
hde
D y f y btt = ( ) > ( ){ }−: ψ̃ 1
.
Otmetym oçevydnoe neravenstvo u4 ≤ v.
Teorema 2. V sluçae 2.2.1 spravedlyva dvustoronnqq ocenka
u1 ( t, x ) ≤ u ( t, x ) ≤ u2 ( t, x ), t > 0, x ∈ Rn
.
Esly dyffuzyonnaq mera P ( t, x, Γ ) lypßyceva otnosytel\no funkcyy ψ ( f ),
to v sluçae 2.2.2 ymegt mesto neravenstva
u4 ( t, x ) ≤ u ( t, x ) ≤ u3 ( t, x ), t > 0, x ∈ Rn
.
Dokazatel\stvo opqt\ Ωe sostoyt v ustanovlenyy znaka nevqzok. Tak,
h t x
b
w t y p t x y dy w t y p t x y dy
E E
1( ) = ( ) ( )
− ( ) ( )∫ ∫ ( ),
, , , , , ,Φ Φ ≤ 0
v sylu v¥puklosty Φ vnyz;
h t x A x t x t x
u
u2
2
2 2( ) = ( )∇ ( ) ∇ ( ) ( )
( )
′( ) − ′( )( ) ( ), , , ,v v
v
v
Φ
Φ
Φ Φ ≥ 0
v sylu vozrastanyq Φ′;
V¥çyslenye nevqzky h3 opyraetsq na utverΩdenye 1:
h t x
b
w t y p t x y dy w t y p t x y dy
E E
3( ) = ( ) ( )
− ( ) ( )∫ ∫ ( ), ˜ , , , ˜ , , ,Φ Φ ≥ 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1454 V. H. BONDARENKO, G. G. PROKOPENKO
v sylu v¥puklosty Φ vverx;
h t x A x t x t x
u
u4
4
2 4( ) = ( )∇ ( ) ∇ ( ) ( )
( )
′( ) − ′( )( ) ( ), , , ,v v
v
v
Φ
Φ
Φ Φ ≤ 0
v sylu vozrastanyq Φ′.
Rys. 1
Yz ustanovlenn¥x dlq nevqzok neravenstv v¥tekaet utverΩdenye teorem¥ 2.
3. Rezul\tat¥ v¥çyslytel\noho πksperymenta. V kaçestve prymera b¥la
rassmotrena zadaça Koßy
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
BAR|ERNÁE FUNKCYY DLQ ODNOHO KLASSA … 1455
∂
∂
= ∂
∂
+ ∂
∂
−u
t
u
x
u
x
bu
2
1
2
2
2
2
α
, u x x
x x
( ) =
+ +
0
1
1
2
1
2
2
2
, ,
pry α = 2 y pry α = 0.5.
Rys. 2
Dlq dannoho naçal\noho uslovyq b¥ly çyslenno poluçen¥ znaçenyq bar\er-
n¥x funkcyj u t x x1 1 2( ), , y u t x x2 1 2( ), , pry α = 2 y u t x x3 1 2( ), , y u t x x4 1 2( ), ,
pry α = 0.5, a takΩe metodom koneçn¥x raznostej poluçeno reßenye πtoho
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1456 V. H. BONDARENKO, G. G. PROKOPENKO
uravnenyq u t x x( ), ,1 2 .
Na rys. 1 pokazan¥ hrafyky çyslennoho reßenyq dlq α = 2, a takΩe razny-
ca meΩdu çyslenn¥m reßenyem y subparabolyçeskoj funkcyej y raznyca meΩ-
du çyslenn¥m reßenyem y superparabolyçeskoj funkcyej. Hrafyky sootvet-
stvugt t = 0.5. Parametr b vzqt ravn¥m 2.
Na rys. 2 pokazan¥ hrafyky çyslennoho reßenyq dlq α = 0.5, a takΩe raz-
nyca meΩdu çyslenn¥m reßenyem y subparabolyçeskoj funkcyej y raznyca
meΩdu çyslenn¥m reßenyem y superparabolyçeskoj funkcyej. Hrafyky soot-
vetstvugt t = 0.5. Parametr b vzqt ravn¥m 2. Na rys. 2 vydno nalyçye MKN.
Vsledstvye πtoho πffekta bar\ern¥e funkcyy y reßenye za koneçnoe vremq
okaz¥vagtsq toΩdestvenno ravn¥my nulg, pryçem tem b¥stree, çem bol\ße
znaçenye parametra b. Pry b = 2 u t3 0 0( ), , = 0 naçynaq s t = 1, u t( ), ,0 0 = 0
naçynaq s t = 0.774, u t4 0 0( ), , = 0 naçynaq s t = 0.767.
1. Kalaßnykov A. S. Nekotor¥e vopros¥ kaçestvennoj teoryy nelynejn¥x v¥roΩdagwyxsq
parabolyçeskyx uravnenyj vtoroho porqdka // Uspexy mat. nauk. – 1987. – 42, #V3. – S. 135 –
176.
2. Kalaßnykov A. S. Ob uslovyqx mhnovennoj kompaktyfykacyy nosytelej reßenyj poluly-
nejn¥x parabolyçeskyx uravnenyj y system // Mat. zametky. – 1990. – 47, # 1. – S. 74 – 80.
3. Galaktionov V. A., Shishkov A. E. Saint-Venant’s principle in blow-up for higher-order quasi-linear
parabolic equations // Proc. Royal Soc. Edinburgh. – 2003. – 113A. – P. 1075 – 1119.
4. Bondarenko V. H. Vozmuwennoe parabolyçeskoe uravnenye na rymanovom mnohoobrazyy //
Ukr. mat. Ωurn. – 2003. – 55, # 7. – S. 977 – 982.
5. Bondarenko V. H. Postroenye fundamental\noho reßenyq vozmuwennoho parabolyçeskoho
uravnenyq // Ukr. mat. Ωurn. – 2003. – 55, # 8. – S. 1011 – 1021
6. Federer H. Heometryçeskaq teoryq mer¥. – M: Nauka, 1987. – 760 s.
7. Bondarenko V. H., Selyn A. N. Ocenky reßenyq uravnenyq typa „reakcyq – dyffuzyq” //
Systemni doslidΩennq ta informacijni texnolohi]. – 2007. – # 1. – S. 124 – 128.
Poluçeno 22.05.07,
posle dorabotky — 28.08.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
|
| id | umjimathkievua-article-3260 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:39:09Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/09/ed2cf9ca90a75b0f8d141526b7b9e409.pdf |
| spelling | umjimathkievua-article-32602020-03-18T19:49:31Z Barrier functions for one class of semilinear parabolic equations Варьерные функции для одного класса полулинейных параболических уравнений Bondarenko, V. G. Prokopenko, Yu. Yu. Бондаренко, В. Г. Прокопенко, Ю. Ю. Бондаренко, В. Г. Прокопенко, Ю. Ю. For a parabolic quasilinear equation with monotone convex potential, we construct superparabolic and subparabolic barrier functions by the method of decomposition. Для параболічного квазілінійного рівняння з монотонним опуклим потенціалом методом декомпозиції побудовано супер- та субпараболічні бар'єрні функції. Institute of Mathematics, NAS of Ukraine 2008-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3260 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 11 (2008); 1449–1456 Український математичний журнал; Том 60 № 11 (2008); 1449–1456 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3260/3266 https://umj.imath.kiev.ua/index.php/umj/article/view/3260/3267 Copyright (c) 2008 Bondarenko V. G.; Prokopenko Yu. Yu. |
| spellingShingle | Bondarenko, V. G. Prokopenko, Yu. Yu. Бондаренко, В. Г. Прокопенко, Ю. Ю. Бондаренко, В. Г. Прокопенко, Ю. Ю. Barrier functions for one class of semilinear parabolic equations |
| title | Barrier functions for one class of semilinear parabolic equations |
| title_alt | Варьерные функции для одного класса полулинейных параболических уравнений |
| title_full | Barrier functions for one class of semilinear parabolic equations |
| title_fullStr | Barrier functions for one class of semilinear parabolic equations |
| title_full_unstemmed | Barrier functions for one class of semilinear parabolic equations |
| title_short | Barrier functions for one class of semilinear parabolic equations |
| title_sort | barrier functions for one class of semilinear parabolic equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3260 |
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