Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry
By using the method of a priori estimates, we establish differential inequalities for energetic norms in $W^l_{2,r}$ of solutions of problems with a free bound in media with the fractal geometry for one-dimensional evolutionary equation. On the basis of these inequalities, we obtain estimates for...
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| author | Berezovsky, A. A. Mitropolskiy, Yu. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. |
| author_facet | Berezovsky, A. A. Mitropolskiy, Yu. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. |
| author_sort | Berezovsky, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:01:15Z |
| description | By using the method of a priori estimates, we establish differential inequalities for energetic norms in $W^l_{2,r}$
of solutions of problems with a free bound in media with the fractal geometry for one-dimensional evolutionary equation.
On the basis of these inequalities, we obtain estimates for the stabilization time $T$. |
| first_indexed | 2026-03-24T02:46:36Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 517.946.9
G. A. Mytropol\skyj, (Yn-t matematyky NAN Ukrayn¥, Kyev),
M. X. Íxanukov-Lafyßev (Kabardyno-Balkar. un-t, Yn-t ynformatyky y problem
rehyon. upravlenyq KBNC RAN, Nal\çyk, Rossyq)
STABYLYZACYQ ZA KONEÇNOE VREMQ
V ZADAÇAX SO SVOBODNOJ HRANYCEJ
DLQ NELYNEJNÁX URAVNENYJ
V SREDAX S FRAKTAL|NOJ HEOMETRYEJ
By using the method of a priori estimates, we establish differential inequalities for energetic norms
in W r2
1
, of solutions of problems with a free bound in media with the fractal geometry for one-dimen-
sional evolutionary equation. On the basis of these inequalities, we obtain estimates for the stabilization
time T.
Metodom apriornyx ocinok vstanovleno dyferencial\ni nerivnosti dlq enerhetyçnyx norm v
W r2
1
, rozv’qzkiv zadaç iz vil\nymy meΩamy v seredovywi z fraktal\nog heometri[g dlq
odnovymirnoho evolgcijnoho rivnqnnq i na ]x osnovi otrymano ocinky dlq çasu stabilizaci] T.
V poslednye hod¥ dlq opysanyq struktur¥ neuporqdoçenn¥x sred y protekag-
wyx v nyx processov ßyroko yspol\zuetsq teoryq fraktalov [1 – 4]. Voznyka-
gwye pry πtom dyfferencyal\n¥e uravnenyq naz¥vagtsq dyfferencyal\n¥-
my uravnenyqmy v sredax s fraktal\noj heometryej [2]. V çastnosty, pry reße-
nyy rqda problem πkolohyy voznykaet neobxodymost\ yssledovanyq πvolgcyon-
n¥x zadaç so svobodn¥my hranycamy dlq nelynejn¥x uravnenyj v sredax s
fraktal\noj strukturoj. V dal\nejßem budem rassmatryvat\ tol\ko odnomer-
n¥e zadaçy, kohda yskomaq funkcyq zavysyt ot odnoj prostranstvennoj pere-
mennoj. V πtom sluçae dlq opredelenyq funkcyj u = u ( x, t ) y s = s ( t ) , t > 0,
poluçaem sledugwug zadaçu so svobodnoj hranycej dlq odnomernoho πvolgcy-
onnoho uravnenyq:
∂
∂
u
t
=
1
x x
x u
u
x
f uα
α ψ∂
∂
∂
∂
−( ) ( ) , x0 < x < s ( t ) , t > 0,
u ( x, 0 ) = u0 ( x ) , x0 ≤ x ≤ s ( 0 ) ,
ψ( )u
u
x
∂
∂
= γ u, x = x0 , t ≥ 0, (1)
u = 0, ψ( )u
u
x
∂
∂
= 0, x = s ( t ) , t ≥ 0,
hde α = 0, 1, 2 sootvetstvenno pry ploskoj, cylyndryçeskoj y sferyçeskoj
symmetryy; x0 = 0 pry 0 ≤ α < 1 y otlyçno ot nulq pry 1 ≤ α < 2, s ( t ) — mo-
notonno nevozrastagwaq neprer¥vno dyfferencyruemaq funkcyq; s ′ ( t ) ≤ 0,
s ( 0 ) = b > 0. V rabote predpolahaetsq suwestvovanye neotrycatel\noho reße-
nyq zadaçy (1). Tol\ko pry πtyx uslovyqx budet poluçena ocenka dlq vremeny
stabylyzacyy.
Predstavlqet ynteres πvolgcyq prostranstvenno lokalyzovannoho naçal\-
noho raspredelenyq u ( x, 0 ) = u0 ( x ) . V kaçestve u0 ( x ) moΩet b¥t\ prynqto toç-
noe reßenye sootvetstvugwej (1) stacyonarnoj zadaçy dlq ψ ( u ) = u
σ
, f ( u ) =
= u
β
, σ ≥ 0, 0 ≤ β < 1 [5].
© G. A. MYTROPOL|SKYJ, A. A. BEREZOVSKYJ, M. X. ÍXANUKOV-LAFYÍEV, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7 997
998 G. A. MYTROPOL|SKYJ, , M. X. ÍXANUKOV-LAFYÍEV
1. Sluçaj slaboho v¥roΩdenyq. PoloΩym v uravnenyy (1)
ψ ( u ) = u
σ
, 0 ≤ α < 1, γ = 0.
Tohda poluçym zadaçu
∂
∂
u
t
=
1
x x
x u
u
x
f uα
α σ∂
∂
∂
∂
− ( ), 0 < x < s ( t ) , t > 0 , (2)
u ( x, 0 ) = u0 ( x ) , 0 ≤ x ≤ s ( 0 ) , (3)
u
u
x x
σ ∂
∂ =0
= 0, t ≥ 0, (4)
u = 0, u
u
x
σ ∂
∂
= 0, x = s ( t ) , t ≥ 0. (5)
UmnoΩym dyfferencyal\noe uravnenye (2) skalqrno na x u t
α σ( )1+
:
( ) ( ) ( ), ( ) ( ) , ( ) ( ), ( )u x u x u u u f u x ut t x x t t
α σ α σ σ α σ1 1 1+ + +− + = 0 (6)
y preobrazuem yntehral¥, vxodqwye v toΩdestvo (6). Dlq pervoho yntehrala
poluçaem
( ), ( )u x ut t
α σ1+ = ( ) / /1 2 2
0
2
+ σ α σx u ut , (7)
hde u 0
2 = ( u, u ) , ( u, ϑ ) t = u dx
s t
ϑ
0
( )
∫ . V dal\nejßem v oboznaçenyy ska-
lqrnoho proyzvedenyq t budem opuskat\. Yntehryruq po çastqm s uçetom hra-
nyçn¥x uslovyj (4), (5) y predpolahaq, çto s ′ ( t ) = O ( 1 ) , t ≥ 0, vtoroj ynteh-
ral preobrazuem k vydu
( )( ) , ( )x u u ux x t
α σ σ1+ = –
1
2 1
2 1
0
2
( )
( )/
+
+
σ
α σd
dt
x u x . (8)
Podstavlqq (7), (8) v toΩdestvo (6), poluçaem
( ) ( ), ( )
( )
( )/ / /( )1
1
2 1
2 2
0
2 1 2 1
0
2
+ + +
+
+ +σ
σ
α σ α σ α σx u u f u x u
d
dt
x ut t x = 0. (9)
Yspol\zuq dyfferencyal\noe uravnenye (2), pryvodym summu perv¥x dvux
slahaem¥x yz toΩdestva (9) k vydu
( ) ( ), ( )/ / ( )1 2 2
0
2 1+ + +σ α σ α σx u u f u x ut t =
= ( ) ( ) ( ) ( )
( ) ( )
1
1
0
2
0
+ −
∫ ∫σ α
α σ σ α σ σ
s t
x x
s t
x x
x
x u u u dx x u u f u u dx . (10)
Podstavlqq (10) v toΩdestvo (9), posle yntehryrovanyq po çastqm v poslednem
yntehrale naxodym
1
2 1
1
1
2 1
0
2 1 2
0( )
( ) ( ) ( )/
( )
+
+
+
+ +∫σ σ
α σ α σd
dt
x u x u g u dxx x
s t
+
+
σ
σ
α σ
1
1 2
0+
+∫ x u
f u
u
dxx
s t
( )
( )
( )
≤ 0, (11)
hde g ( u ) = ′f uu( ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
STABYLYZACYQ ZA KONEÇNOE VREMQ V ZADAÇAX … 999
Ysxodq yz fyzyçeskoho sm¥sla, predpolahaem, çto f ( u ) — neotrycatel\naq
monotonno vozrastagwaq funkcyq y f ( 0 ) = 0, a g ( u ) ≥ 0 — monotonno ub¥-
vagwaq funkcyq. Yntehral¥, soderΩawyesq v (11), voobwe hovorq, nesobst-
venn¥e, tak kak v sylu hranyçn¥x uslovyj x = s ( t ) , u = 0, f ( u ) = 0, g ( u ) = ∞ .
Poπtomu pryvedenn¥e rassuΩdenyq spravedlyv¥ v klasse funkcyj, hde ras-
smatryvaem¥e yntehral¥ koneçn¥.
Yz (11) sleduet neravenstvo
d
dt
x u x u g u dxx x
s t
α σ α σ/
( )
( ) ( ) ( )2 1
0
2 1 2
0
2+ ++ ∫ ≤ 0. (12)
Poskol\ku dlq lgboj neprer¥vno dyfferencyruemoj po x na [ 0, s ( t ) ] , t > 0,
funkcyy u ( x, t ) ≥ 0, obrawagwejsq v nul\ na hranyce x = s ( t ) ( u ( s ( t ), t ) =
= 0 ) , v¥polnqetsq neravenstvo
u ( x, t ) ≤
b
x ux
1
01
−
−
α
α
α
, b = s ( 0 ) = max ( )
t
s t
≥0
,
to
g ( u ) ≥ g
b
x ux
1
2
01
−
−
α
α
α
/ . (13)
1.1. Pust\ σ = 0. Tohda
d
dt
x u x u g u dxx x
s t
α α/
( )
( )2
0
2 2
0
2+ ∫ ≤ 0. (14)
S pomow\g (13) yz neravenstva (14) naxodym
d
dt
x u x u g
b
x u dxx x x
s t
α α
α
α
α
/ /
( )
2
0
2 2
1
2
0
0
2
1
+
−
−
∫ ≤ 0,
yly
dy
dt
g
b
y y+
−
−
2
1
1 α
α
≤ 0, y = x ux
α /2
0
2
,
otkuda dlq vremeny stabylyzacyy, t. e. toho znaçenyq t = T , pry kotorom
y ( T ) = 0, poluçaem ocenku
T ≤
1
2
1
0
0
1
y
dy
g
b
y y
( )
∫ −
−
α
α
< ∞ .
1.2. Pust\ σ > 0 , f ( u ) = cu
β, c = const > 0, β > 0 . Tohda neravenstvo
(12) prynymaet vyd
d
dt
x u c x u u dxx x
s t
α σ α σ βσ β/
( )
( ) ( ) ( )2 1
0
2 1 2 1
0
2+ + −+ + ∫ ≤ 0. (15)
Poskol\ku
x u
u
dxx
s t
α σ
β( )
( )
1 2
1
0
1+
−∫ ≥ x u dx
b x u
x
s t
x
α σ
α α σ
β σ
α
( )
( )
( )
/
( )/ ( )
1 2
0
1 2 1
0
2
1 2 1
1 1+
− +
− +
∫ ⋅ −
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
1000 G. A. MYTROPOL|SKYJ, , M. X. ÍXANUKOV-LAFYÍEV
yz (15) naxodym
dy
dt
c yk+ +2 ( )σ β ≤ 0, k =
1 2
2 1
+ +
+
β σ
σ( )
. (16)
Yz (16) dlq vremeny stabylyzacyy T poluçaem ocenku
T ≤
1
2 0
0
c
dy
yk
y
( )
( )
σ β+ ∫ < ∞ , k < 1, β < 1, y ( t ) = x ux
α /2
0
2
.
2. Sluçaj syl\noho v¥roΩdenyq. Rassmotrym zadaçu
∂
∂
u
t
=
1
x x
x u
u
x
f uα
α σ∂
∂
∂
∂
− ( ), x0 < x < s ( t ) , 0 < t < T , (17)
u ( x, 0 ) = u0 ( x ) , x0 ≤ x ≤ s ( 0 ) , (18)
u
u
x x x
σ ∂
∂ = 0
= 0, 1 ≤ α ≤ 2 , t > 0, (19)
u = 0, u
u
x
σ ∂
∂
= 0, x = s ( t ) , t ≥ 0. (20)
UmnoΩaq dyfferencyal\noe uravnenye (17) skalqrno na x u t
α σ( )1+
y pro-
vodq preobrazovanyq, analohyçn¥e yzloΩenn¥m v¥ße, s uçetom uslovyj (18) –
(20) poluçaem neravenstvo
d
dt
x u x u g u dxx x
x
s t
α σ α σ/
( )
( ) ( ) ( )2 1
0
2 1 22
0
+ ++ ∫ ≤ 0. (21)
2.1. Pust\ σ = 0. Poskol\ku
u2
( x ) ≤
dx
x
x u dx
x
s t
x
x
s t
α
α
0 0
2
( ) ( )
∫ ∫ ≤ M x ux
α /2
0
2
,
hde
M =
ln ,
,
b
x
b
0
1
1
1
1
pry
pry
α
α
α
α
=
−
>
−
yz (21), uçyt¥vaq svojstva funkcyy g ( u ) , dlq vremeny stabylyzacyy poluçaem
ocenku
T ≤
dy
g My y
y
20
0
( )∫
( )
< ∞ , y ( t ) = x ux
α /2
0
2
.
2.2. Pust\ σ > 0 , f ( u ) = cu
β, c = const > 0. Tohda neravenstvo (21) pry-
nymaet vyd
d
dt
x u c x u
u
dxx x
x
s t
α σ α σ
ββ/
( )
( ) ( )2 1
0
2 1 2
12
1
0
+ +
−+ ∫ ≤ 0. (22)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
STABYLYZACYQ ZA KONEÇNOE VREMQ V ZADAÇAX … 1001
Yz neravenstva (22) sohlasno yzloΩennomu v¥ße alhorytmu naxodym ocenku
dlq vremeny stabylyzacyy
T ≤ v
dy
yk
y
0
0( )
∫ < ∞ , k < 1 ( β < 1 ) ,
hde k =
1 2
2 1
+ +
+
β σ
σ( )
, v =
1
2 0
1 2 1
c
b
xβ α
β σ
− +( )/ ( )
.
1. Dynaryev O. G. Fyl\tracyq v trewynovatoj srede s fraktal\noj heometryej trewyn //
Mexanyka Ωydkosty y haza. – 1990. – S. 66 – 70.
2. Mal\ßakov A. V. Uravnenye hydrodynamyky dlq poryst¥x sred so strukturoj porovoho
prostranstva, obladagweho fraktal\noj heometryej // YnΩ.-fyz. Ωurn. – 1985. – 62, # 3. –
S. 405 – 410.
3. Zel\dovyç Q. B., Sokolov D. D. Fraktal¥, podobye, promeΩutoçnaq asymptotyka // Uspexy
fyz. nauk. – 1985. – 146, # 3. – S. 493 – 506.
4. Nyhmatullyn R. R. Reßenye obobwennoho uravnenyq perenosa v sredax s fraktal\noj
heometryej // Phys. status. solidi. – 1986. – 133.
5. Mytropol\skyj G. A., Berezovskyj A. A., Íxanukov-Lafyßev M. X. Stabylyzacyq za ko-
neçnoe vremq v zadaçax so svobodn¥my hranycamy dlq nekotor¥x klassov nelynejn¥x
uravnenyj vtoroho porqdka // Ukr. mat. Ωurn. – 1999. – 51, # 2. – S. 214 – 223.
Poluçeno 05.06.2001,
posle dorabotky — 04.11.2002
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
|
| id | umjimathkievua-article-3658 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:36Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/98/ccffba331ebfc2d1d14826e16543d498.pdf |
| spelling | umjimathkievua-article-36582020-03-18T20:01:15Z Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry Стабилизация за конечное время в задачах со свободной границей для нелинейных уравнений в средах с фрактальной геометрией Berezovsky, A. A. Mitropolskiy, Yu. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. By using the method of a priori estimates, we establish differential inequalities for energetic norms in $W^l_{2,r}$ of solutions of problems with a free bound in media with the fractal geometry for one-dimensional evolutionary equation. On the basis of these inequalities, we obtain estimates for the stabilization time $T$. Методом апріорних оцінок встановлено диференціальні нерівності для енергетичних норм в $W^l_{2,r}$ розв'язків задач із вільними межами в середовищі з фрактальною геометрією для одновимірного еволюційного рівняння і на їх основі отримано оцінки для часу стабілізації $T$. Institute of Mathematics, NAS of Ukraine 2005-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3658 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 7 (2005); 997–1001 Український математичний журнал; Том 57 № 7 (2005); 997–1001 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3658/4046 https://umj.imath.kiev.ua/index.php/umj/article/view/3658/4047 Copyright (c) 2005 Berezovsky A. A.; Mitropolskiy Yu. A.; Shkhanukov-Lafishev M. Kh. |
| spellingShingle | Berezovsky, A. A. Mitropolskiy, Yu. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Митропольский, Ю. А. Шхануков-Лафишев, М. Х. Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry |
| title | Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry |
| title_alt | Стабилизация за конечное время в задачах со свободной границей для нелинейных уравнений в средах с фрактальной геометрией |
| title_full | Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry |
| title_fullStr | Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry |
| title_full_unstemmed | Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry |
| title_short | Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry |
| title_sort | finite-time stabilization in problems with free boundary for nonlinear equations in media with fractal geometry |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3658 |
| work_keys_str_mv | AT berezovskyaa finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT mitropolskiyyua finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT shkhanukovlafishevmkh finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT berezovskijaa finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT mitropolʹskijûa finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT šhanukovlafiševmh finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT berezovskijaa finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT mitropolʹskijûa finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT šhanukovlafiševmh finitetimestabilizationinproblemswithfreeboundaryfornonlinearequationsinmediawithfractalgeometry AT berezovskyaa stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT mitropolskiyyua stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT shkhanukovlafishevmkh stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT berezovskijaa stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT mitropolʹskijûa stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT šhanukovlafiševmh stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT berezovskijaa stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT mitropolʹskijûa stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej AT šhanukovlafiševmh stabilizaciâzakonečnoevremâvzadačahsosvobodnojgranicejdlânelinejnyhuravnenijvsredahsfraktalʹnojgeometriej |