Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem
We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media.
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| Date: | 2006 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3512 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509614970241024 |
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| author | Degtyarev, S. P. Дегтярев, С. П. Дегтярев, С. П. |
| author_facet | Degtyarev, S. P. Дегтярев, С. П. Дегтярев, С. П. |
| author_sort | Degtyarev, S. P. |
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| datestamp_date | 2020-03-18T19:56:35Z |
| description | We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media. |
| first_indexed | 2026-03-24T02:43:55Z |
| format | Article |
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UDK 517.946
S.�P.�Dehtqrev (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
VLYQNYE NEODNORODNOSTY PORYSTOJ SREDÁ
NA MHNOVENNUG KOMPAKTYFYKACYG
NOSYTELQ REÍENYQ ZADAÇY FYL|TRACYY
We study the instantaneous support shrinking phenomenon and the initial behavior of support of a
solution of filtration equation in an inhomogeneous porous medium.
Vyvça[t\sq mytt[va kompaktyfikaciq ta poçatkova povedinka nosiq rozv’qzku rivnqnnq fil\tra-
ci] u neodnoridnomu porystomu seredovywi.
1. Vvedenye. Cel\g dannoj rabot¥ qvlqetsq yzuçenye qvlenyq mhnovennoj
kompaktyfykacyy nosytelq reßenyq sledugwej zadaçy dlq uravnenyq fyl\-
tracyy v porystoj srede:
ρ ( x ) ut = ∆ um – up
, x ∈ RN
, t > 0,
(1)
u ( x , 0 ) = u0 ( x ) > 0, x ∈ RN
,
hde m > 0, 0 < p < min { 1 , m } , ρ ( x ) — neprer¥vno dyfferencyruemaq polo-
Ωytel\naq funkcyq s zadann¥m rostom na beskoneçnosty:
B–1 ≤ ρ ( x ) / ρ0 ( | x | ) ≤ B dlq | x | ≥ 1, B ≥ 1, (2)
svojstva funkcyy ρ0 ( r ) budut utoçnen¥ nyΩe.
Qvlenye mhnovennoj kompaktyfykacyy nosytelq k nastoqwemu vremeny
yzuçeno dovol\no polno, y v πtom napravlenyy poluçeno mnoho hlubokyx
rezul\tatov, vklgçaq yzuçenye takoho typa qvlenyj dlq uravnenyj v¥sokoho
porqdka (sm., naprymer, [1]). V;dannoj rabote m¥ budem sledovat\ metodu y tex-
nyke rabot [2 – 4] y rassmotrym vlyqnye neodnorodnosty sred¥ kak dlq stepen-
noho, tak y dlq nestepennoho povedenyq funkcyj ρ0 ( r ) y u0 ( x ) yz (1), (2).
Rassmotrym snaçala sluçaj stepennoho povedenyq ukazann¥x funkcyj. Pust\
ρ0 ( r ) ≡ rq, q > 0 y, sleduq [3], predpoloΩym, çto naçal\naq funkcyq u0 ( x )
udovletvorqet sledugwym predpoloΩenyqm:
H1 ) u0 ( x ) neprer¥vna, ohranyçena, neotrycatel\na y u0 ( x ) → 0 ravnomer-
no pry | x | → ∞ ;
H2 ) | x | a u0 ( x ) → A ravnomerno pry | x | → ∞ dlq nekotor¥x a , A > 0.
Dlq zadaçy (1) s ρ ( x ) ≡ 1 y m > 0 v [2, 3] opysano naçal\noe povedenye
nosytelq reßenyq, a ymenno
( , ) : ( )x t x c t a p| | <
−
−
1
1
1 ⊂ S ( t ) ⊂ ( , ) : ( )x t x c t a p| | <
−
−
2
1
1 ,
hde S ( t ) — nosytel\ u ( x , t ) . Okaz¥vaetsq, çto v sluçae neodnorodnoj sred¥,
kohda ρ ( x ) ≠ const y vedet sebq, kak opysano v (2), spravedlyva sledugwaq
teorema.
Teorema 1. Pust\ dlq zadaçy (1) v¥polnen¥ uslovyq (2), H1 , H2 , a takΩe
sformulyrovann¥e nyΩe uslovyq H3 , H4 . Tohda esly – a ( 1 – p ) + q < 0, to
( , ) : ( )x t x c t a p q| | <
− − +
3
1
1 ⊂ S ( t ) ⊂ ( , ) : ( )x t x c t a p q| | <
− − +
4
1
1
dlq slaboho reßenyq u ( x , t ) zadaçy (1) s nekotor¥my postoqnn¥my 0 < c3 <
© S.;P.;DEHTQREV, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1035
1036 S.;P.;DEHTQREV
< c4 . Esly Ωe – a ( 1 – p ) + q ≥ 0 y m ≥ 1 – ( q + 2 ) / a , to qvlenye mhnovennoj
kompaktyfykacyy nosytelq moΩet otsutstvovat\.
2. Predvarytel\n¥e svedenyq. Sformulyruem nekotor¥e analohy teorem
suwestvovanyq y sravnenyq yz [2, 3, 5 – 7] dlq sluçaq neodnorodnoj sred¥, t.;e.
kohda ρ ( x ) ≠ const .
Pust\ u0 : RN → R — neprer¥vnaq neotrycatel\naq ohranyçennaq funkcyq.
Opredelenye 1. Neotrycatel\naq funkcyq u ( x , t ) , opredelennaq na RN ×
× [ 0 , T ] , naz¥vaetsq neprer¥vn¥m slab¥m reßenyem zadaçy (1) na [ 0 , T ] , esly:
1) u ( x , t ) neprer¥vna na RN × [ 0 , T ] , neotrycatel\na y ohranyçena;
2) dlq lgboj oblasty Ω ⊂ RN s hladkoj hranycej y dlq lgboj probnoj
funkcyy ς ∈ C1
( Ω × [ 0 , T ] ) ∩ C2
( Ω × ( 0 , T ) ) takoj, çto ζ ≥ 0 n a Ω ×
× [ 0 , T ] y ζ = 0 na ∂ Ω × [ 0 , T ] , u ( x , t ) udovletvorqet yntehral\nomu toΩ-
destvu
ρ ζ( ) ( , ) ( , )x u x t x t dx
Ω
∫ = ρ ζ τ ζ τν( ) ( ) ( , ) ( , )x u x x dx u x ds dm
t
0
0
0
Ω Ω
∫ ∫∫− ∂
∂
+
+ u u u dx dt
m p
t
ς ∆
Ω
+ −[ ]∫∫∫ ζ ζ τ
0
(3)
dlq lgboho 0 ≤ t ≤ T. Zdes\ ∂ν oboznaçaet proyzvodnug po normaly, vneßnej
k Ω .
Narqdu s zadaçej Koßy (1) rassmotrym takΩe sootvetstvugwug zadaçu
Dyryxle
ρ ( x ) ut = ∆ um – up
, x ∈ Ω × ( 0 , T ) ,
u ( x , t ) = U ( x , t ) , x ∈ ∂Ω × ( 0 , T ) , (4)
u ( x , 0 ) = u0 ( x ) ≥ 0, x ∈ Ω ,
hde T > 0 y Ω ⊂ RN — ohranyçennaq svqznaq oblast\ s kompaktnoj hranycej
∂Ω , kusoçno prynadleΩawej C1 y udovletvorqgwej uslovyg vneßnej sfer¥
[8], U ( x , t ) ∈ C (∂Ω × [ 0, T ] ) , u0 ∈ C (Ω ) , u0 ( x ) = U ( x , 0) pry x ∈ ∂Ω , u0 , U ≥ 0.
Opredelenye 2. Neotrycatel\naq funkcyq u ( x , t ) , opredelennaq na Ω ×
× [ 0 , T ] , naz¥vaetsq slab¥m reßenyem zadaçy (4) s dann¥my u0 y U, esly:
1) u ∈ C ([ 0, T ] : L1
( Ω ) ) ∩ L∞
(Ω × [ 0, T ] ) ;
2) dlq lgboj probnoj funkcyy ς ∈ C1,0
( Ω × [ 0 , T ] ) ∩ C2,1
( Ω × ( 0 , T ) ) ,
ζ ≥ 0 v Ω × [ 0 , T ] y ζ = 0 n a ∂ Ω × [ 0 , T ] , u ( x , t ) udovletvorqet ynte-
hral\nomu toΩdestvu
ρ ζ( ) ( , ) ( , )x u x t x t dx
Ω
∫ = ρ ζ τ ζ τν( ) ( ) ( , ) ( , )x u x x dx U x ds dm
t
0
0
0
Ω Ω
∫ ∫∫− ∂
∂
+
+ u u u dx dt
m p
t
ς ∆
Ω
+ −[ ]∫∫∫ ζ ζ τ
0
(5)
dlq lgboho 0 ≤ t ≤ T.
Zamenqq ravenstvo v (5) na neravenstvo ≤ , poluçaem opredelenye slaboho
subreßenyq u ( x , t ) zadaçy (4) na [ 0 , T ] s dann¥my u0 y U.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
VLYQNYE NEODNORODNOSTY PORYSTOJ SREDÁ … 1037
Opredelenye 3. Obobwenn¥m superreßenyem zadaçy (4) s dann¥my u0 y U
naz¥vaetsq funkcyq u ( x , t ) takaq, çto suwestvugt funkcyy u0
∗ , U ∗, udov-
letvorqgwye tem Ωe predpoloΩenyqm, y funkcyq h ∈ L∞
(Ω × ( 0, T ) ) , t a k
çto u0 ≤ u0
∗ v Ω , U ≤ U∗ na ∂Ω × ( 0, T ) , h ≥ 0 poçty vezde v Ω × ( 0, T ) y
u ( x , t ) — slaboe reßenye zadaçy (4) s dann¥my u0
∗ , U ∗ y so slahaem¥m
h x x dxd
t
( , ) ( , )τ ζ τ τ
Ω
∫∫
0
, dobavlenn¥m v pravug çast\ (5).
Analohyçno opredelqgtsq obobwenn¥e sub- y superreßenyq zadaçy Ko-
ßy;(1).
Teorema 2. Zadaçy (1) y (4) dopuskagt edynstvennoe slaboe reßenye na
[ 0 , T ] .
Teorema 3. Pust\ u ( x , t ) — slaboe reßenye zadaçy (4) na [ 0 , T ] s dann¥-
my u0 , U . Esly v ( x , t ) — slaboe subreßenye zadaçy (4) na [ 0 , T ] s temy Ωe
dann¥my, to v ≤ u poçty vezde v Ω dlq 0 ≤ t ≤ T. Esly w ( x , t ) — obobwen-
noe superreßenye zadaçy (4) na [ 0 , T ] s temy Ωe dann¥my, to w ≥ u poçty
vezde v Ω dlq 0 ≤ t ≤ T.
Analohyçnaq teorema sravnenyq ymeet mesto y dlq zadaçy Koßy (1). Doka-
zatel\stva teorem 2,;3 dlq m < 1 analohyçn¥ dokazatel\stvam sootvetstvug-
wyx teorem yz [2], a dlq sluçaq m ≥ 1 — dokazatel\stvam yz [5 – 7], pryçem yz
rezul\tatov [9] sleduet neprer¥vnost\ slab¥x reßenyj pry neprer¥vn¥x
dann¥x. Otmetym, çto trebovanye neprer¥vnoj dyfferencyruemosty funkcyy
ρ ( x ) svqzano s metodom dokazatel\stva teorem 2,;3 y neobxodymost\g prymene-
nyq pry πtom klassyçeskyx rezul\tatov o razreßymosty kraev¥x zadaç dlq
kvazylynejn¥x parabolyçeskyx uravnenyj (sm. [8]).
Çtob¥ sformulyrovat\ analoh lemm¥;3.1 yz [3], vvedem opysannug nyΩe
zamenu peremenn¥x. Pust\
f ( k ) = 1
0sup ( )
| |>x k
u x
, k > 0,
hde v sylu predpoloΩenyq H2 f ( k ) � ka
/ A pry k → ∞ . Opredelym funkcyy
uk ( x , t ) sledugwym obrazom:
uk ( x , t ) = f k u kx f k k tp q( ) , ( )( )( )− −1 , (6)
hde u ( x , t ) — reßenye zadaçy (1). Kak sleduet yz opredelenyq uk ( x , t ) ,
Lk ( uk ) ≡ ρk ( x ) ( uk )t – D k u uk
m
k
p( )∆ + = 0, x ∈ RN
, t > 0,
uk ( x , 0 ) = u0k ( x ) = f ( k ) u0 ( k x ) , x ∈ RN
.
Zdes\ ρk ( x ) = ρ ( kx ) k–q
, D ( k ) = f ( k )
p
–
m
k–2. Otmetym, çto f ( k ) → ∞ , D ( k ) → 0
pry k → ∞ .
Lemma 1. Esly dlq zadannoj poloΩytel\noj postoqnnoj C u0k ( x ) < C
dlq | x | > R > 0, k > k ( R ) , to uk ( x , t ) < C dlq | x | > R + d ( k ) , k > k ( R ) , hde
d ( k ) < C ( m , p , N , M ) k–1, M = || u0 ||∞ .
Dokazatel\stvo analohyçno dokazatel\stvu lemm¥;4.1 yz [2] y v¥tekaet yz
sledugweho utverΩdenyq (lemm¥ 3.1 yz [3]).
Lemma 2. Suwestvuet d = d ( m , p , N , M ) > 0 takoe, çto esly v zadaçe (1)
0 ≤ u0 ( x ) ≤ ε pry | x | ≥ R dlq nekotor¥x ε , R > 0, to 0 ≤ u ( x , t ) ≤ ε pry
| x | ≥ R + d ( m , p , N , M ) , t > 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1038 S.;P.;DEHTQREV
Sformulyruem teper\ sledugwee uslovye:
H3 ) dlq zadannoho R > 0 suwestvuet funkcyq U 0( | x | ) , ub¥vagwaq k
nulg pry | x | → ∞ , takaq, çto u0k ( x ) < U0( | x | ) pry | x | > R , k > k ( R ) .
Teorema 4. Pry v¥polnenyy uslovyj H1 – H3 – a ( 1 – p ) + q < 0 y dlq fyk-
syrovannoho t0 > 0 suwestvuet konstanta C, ne1zavysqwaq ot k , takaq,
çto uk ( x , t0 ) ≡ 0 pry | x | > C, k > k .
Sledstvye 1. u ( x , t ) ≡ 0 dlq 0 < t < T0 , | x | > ct a p q
1
1− − +( ) .
Dejstvytel\no, kak sleduet yz teorem¥;4, f k u kx f k k tp q( ) , ( )( )( )− −1
0 ≡ 0 dlq
| x | > C. Oboznaçaq f k k tp q( ) ( )− −1
0 = t ∈ ( 0 , T0 ] , k x = ξ , y uçyt¥vaq, çto f ( k ) �
� ka
/ A , poluçaem u ( ξ , t ) ≡ 0 dlq 0 < t < T0 , | ξ | > ct a p q
1
1− − +( ) .
Sleduq [3], vvedem sledugwee predpoloΩenye:
H4 ) dlq zadann¥x l1 < l2 suwestvuet konstanta B1 takaq, çto u0k ( x ) > B1
dlq l1 ≤ | x | ≤ l2 , k > k ( l1 , l2 ) .
Teorema 5. Pry v¥polnenyy predpoloΩenyq H4 dlq lgboho l1 < | x0 | < l2
v¥polneno
uk ( x0 , t ) ≥ B tr r
1
1/ −[ ]+λ , 0 ≤ t ≤ T0 ,
hde r = ( min ( 1 , m ) – p )–1 y λ , T0 — nekotor¥e poloΩytel\n¥e postoqnn¥e,
zavysqwye ot m , p , N , B , B1 .
Yz πtoj teorem¥ sleduet, çto dlq t = T0 / 2 uk ( x0 , T0 / 2 ) > B Tr r
1
1
0 2/ /−[ ]λ =
= c > 0, k > k , hde C ne;zavysyt ot k .
Sledovatel\no,
f k u kx T f k kp q( ) , ( ) ( )( / )( )
0 0
12 − − ≥ c .
Polahaq k x0 = ξ , ( ) ( )/ ( )T f k kp q
0
12 − − = t , poluçaem, çto u ( ξ , t ) > 0 pry
c t a p q
1
1
1− − +( ) < | ξ | < c t a p q
2
1
1− − +( ) , hde uçten¥ svojstva f ( k ) .
3. Dokazatel\stva teorem 4, 5 y 1. Yz;lemm¥;1 y predpoloΩenyq H3
sleduet, çto dlq dostatoçno bol\ßyx k uk ( x , t ) < U0 ( R ) dlq | x | > R + 1, k >
> k ( R , p , m , N , q , M ) . Rassmotrym sluçay m ≥ 1 y m < 1 otdel\no. Zafyksy-
ruem t0 > 0 y dlq lgboho k > k opredelym Sk ( t0 ) — nosytel\ uk ( x , t0 ) .
Rassmotrym sluçaj m ≥ 1. Zafyksyruem x0 y rassmotrym funkcyg srav-
nenyq
w ( x , t ) = h t t x y x xm m
0 0
1
−( ) + | − |( )[ ],
/
, 0 < t < t0 ,
hde h ( t , x ) — reßenye zadaçy
B x h z xq
z| | ′( , ) = 1
2
h z xp( , ), h ( 0 , x ) = 0
y funkcyq y — reßenye uravnenyq
∆y = 1
2
y
p
m ,
t.;e.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
VLYQNYE NEODNORODNOSTY PORYSTOJ SREDÁ … 1039
h t t x p B x t tp p
q
p p0
1
1
1
1 1 0
1
1
1
2
1−( ) = −
−( )−
−
− −
− −, ( ) ,
y x x C N
p
m
x x p m| − |( ) =
− −0 0
2
1, / .
Dlq funkcyy w ( x , t ) ymeem
Lk ( w ) = ρk ( x ) wt – D ( k ) ∆wm + wp =
= – ρk
m m m m m
p
mx
m
h y mh h D k h y h y( ) ( )1 1 1 1+[ ] ′ − +[ ] + +[ ]− − ∆ =
= – 1
2
1 1 1ρk
q
m m m p m m
p
m
x
B x
h y h D k y D k h h y
( )
( ) ( )+[ ] − − + +[ ]− − + ∆ ∆ ≥
≥ – 1
2
2
1 1 1
1
h y h D k y D k h h ym m m p m
p
m p
p
m+[ ] − − + +
− − + −
( ) ( )∆ ∆ ,
hde yspol\zovano to, çto h ym
p
m+[ ] ≥ 2
1
p
m p
p
mh y
−
+
.
Sledovatel\no,
Lk ( w ) ≥ 1
2
1 1
2
1
2 1
2
1
1 1h h
h y
y D k h y D k hp
m
m m
p
m
p
m
p
p
m m−
+[ ]
+ −( ) + − +
−
−
−
( ) ( )∆ ≥
≥ c h D k c h xp m
1 2
2− | |−( )
dlq dostatoçno bol\ßyx k , tak kak ∆ hm = ch xm | |−2 .
Poπtomu
Lk ( w ) ≥ h c D k c h x h c D k c t xp m p p
m p
p
q m p
p
1 2
2
1 3 0
1 1
2
−[ ] = −
− −
−
−
− −
−
−
| | | |( ) ( )
( )
≥ 0
pry | x | ≥ R1 = R1 ( t0 , m , p , q ) .
V¥berem R bol\ße, çem R1 , y zametym, çto po opredelenyg y ( r ) suwest-
vuet R0 , zavysqwee ot U 0 ( R ) , N , p / m , takoe, çto y ( R0 ) = U0 ( R ) , y ( r ) >
> U0 ( R ) dlq r > R0 .
RassuΩdaq ot protyvnoho, predpoloΩym, çto dlq nekotoroho dostatoçno
bol\ßoho k suwestvuet x0 takoe, çto | x0 | ≥ R + 1 + R0 y uk ( x0 , t0 ) > 0 ( esly
πto nevozmoΩno, to dokazatel\stvo zaverßeno s C = R + 1 + R0 ) . V;ßare
B ( x0 , R0 ) = { | x – x0 | < R0 } ymeem
1) na ∂B ( x0 , R0 ) = { | x – x0 | = R0 } : uk ( x , t ) < U0 ( R ) ≤ y ≤ w dlq 0 ≤ t ≤ t0 ;
2) uk ( x0 , t0 ) – w ( x0 , t0 ) = uk ( x0 , t0 ) > 0;
3) Lk ( w ) ≥ 0, Lk ( uk ) = 0 v B ( x0 , R0 ) × ( 0 , t0 ) .
Sledovatel\no, uk – w dolΩna dostyhat\ svoeho (poloΩytel\noho) mak-
symuma na naçal\nom mnoΩestve B ( x0 , R0 ) × { t = 0 } y poπtomu suwestvuet
toçka x ∈ B ( x0 , R0 ) takaq, çto u x w xk0 0( ) ( , )− > 0, otkuda sleduet, çto
c t x R
q
p( )0 0 0
1| |+( )
−
− ≤ c x R a| | −( )−0 0 , çto, v svog oçered\, pry dostatoçno bol\-
ßyx | x0 |, | x0 | > C ( R0 ) , daet | |
− −
−x
a p q
p
0
1
1
( )
≤ C. Y;tak kak a (1 – p ) – q > 0, | x0 | ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1040 S.;P.;DEHTQREV
≤ C0 = C ( t0 , R0 , a , q , p ) , çto dokaz¥vaet dannoe utverΩdenye dlq sluçaq m ≥ 1
s C = max {R + 1 + R0 , C0 } .
V sluçae m < 1 rassmotrym bar\ernug funkcyg w = h ( t0 – t , x) + y x xm1
0
/ | − |( ),
hde y ( r ) y h ( z , x ) takye Ωe, kak y v¥ße. Ymeem
Lk ( w ) = − − + +[ ]ρk
q
p m m px
B x
h D k w h y
( )
( ) /1
2
1∆ ≥
≥ − − + +
1
2
2
2
h D k w h yp m
p
p
p
m( )∆ ≥ − + +
D k w c h ym p
p
m( )∆ .
Rassmotrym
∆ wm = 1 1
1
1 1 1 2
2
1 2
2 1 1
2
m
h y y y
m
m
h y y ym
m
m m
m
m−
+
∇( ) + −
+
∇( )
−
−
−
−
+
+ h y y y m h y y h ym
m
m m
m
m+
+ − +
∇ ∇( )
−
−
−
−1 1 1 1 1 2 1 1
2 1∆ ( ) +
+ m m h y h m h y hm
m
m
m
( )− +
∇( ) + +
− −
1
1 2
2
1 1
∆ = A1 + A2 + A3 + A4 + A5 + A6 .
Ocenym kaΩdoe slahaemoe otdel\no, uçyt¥vaq pry πtom, çto
| ∇y | = c x x p m| |− −
−
0
2
1
1
/ = c y
p m1
2
+ /
, | ∇h | = c x h| |−1 .
Poskol\ku m < 1, to
| A1 | ≤ c y y y
m
m m
p m− − +
1 1 2 1
2
2/
= c y
p
m .
A2 y A3 ocenyvagtsq analohyçno. Dalee,
| A4 | ≤ c h y h y x ym
m
m
p m
+
−
− −
+
| |
1 2 1 1
1
1
2
/
≤ c
x
h y y x ym
m
m
p m
| |
| |+
−
− −
+1 1 1 1
1
1
2
/
≤
≤ c
x x
x
y y
p m p m| |
| |
− − − +
0
1
2
1
2
/ /
≤ c y
p
m
pry | x0 | , dostatoçno bol\ßyx po sravnenyg s R0 . Zdes\ m¥ vospol\zovalys\
tem, çto | x – x0 | –1 = c y
p m− −1
2
/
.
Zametym, çto A5 ≤ 0,
| A6 | ≤ c h y h xm
m
+
−
−| |
1 1
2 ≤ ch xm | |−2 .
Krome toho, pry dostatoçno bol\ßyx x y fyksyrovannom t0 moΩno sçytat\
h ≤ 1, tak çto | A6 | ≤ ch xp | |−2 ≤ c hp.
Yz yzloΩennoho v¥ße sleduet, çto
Lk ( w ) ≥ − − + +( )c D k y c D k h c y hp m p p m p
1 2 3( ) ( )/ / ≥ 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
VLYQNYE NEODNORODNOSTY PORYSTOJ SREDÁ … 1041
pry k > k dlq dostatoçno bol\ßyx k y dostatoçno bol\ßom | x0 | .
Kak y v sluçae m ≥ 1, sravnyvaq funkcyg uk ( x , t ) s w ( x , t ) na mnoΩestve
B ( x0 , t0 ) × [ 0 , t0 ] , vydym, çto pry nekotorom x ∈ B ( x0 , t0 ) w ( x , 0 ) < u0k ( x ) , y
dokazatel\stvo zaverßaetsq, kak y v sluçae m ≥ 1. Tem sam¥m teorema;4
dokazana.
Dokazatel\stvo teorem¥;5 analohyçno dokazatel\stvu teorem¥;2.4 yz [3] s
toj Ωe bar\ernoj funkcyej
G
x x
L
t
r
1 0
2
2− −
−
| |
+
l , 1
r
= min ( 1 , m ) – p , [ x ]+ ≡ max { x , 0 },
pry podxodqwem v¥bore konstant G , L , l , l1 < | x0 | < l2 .
Obratymsq teper\ k sluçag – a ( 1 – p ) + q ≥ 0. Rassmotrym funkcyg f ( x , t ) =
=
( )
( ) /
t t
x a
0
2 21
−
+
γ
, hde a — konstanta yz uslovyq H2 .
Prost¥e v¥çyslenyq pokaz¥vagt, çto pry uslovyy – a ( 1 – p ) + q ≥ 0 moΩno
v¥brat\ t0 dostatoçno mal¥m, çtob¥ poluçyt\ f ( x , 0 ) ≤ u0 ( x ) . Zatem moΩno
v¥brat\ γ dostatoçno bol\ßym, çtob¥ poluçyt\ L ( f ) ≤ 0, tak kak v sylu (2)
L ( f ) ≤
ρ γ( )
( ) /
( ) ( )( )/ ( )/x
x
t c x t c xa
m a m q p a p q
1
1 12 2 0 1
2 1 2 2
0 2
2 1 2
+
− + +( ) + +( )
− − − − − −
.
Takym obrazom, f qvlqetsq subreßenyem y u ( x , t ) ≥ f ( x , t ) pry t dostatoçno
mal¥x, çto oznaçaet otsutstvye mhnovennoj kompaktyfykacyy nosytelq. Tem
sam¥m teorema;1 dokazana.
4. Nestepennoj sluçaj. Rassmotrym teper\ sytuacyg, kohda funkcyy ρ ( x )
y u0 ( x ) mohut vesty sebq nestepenn¥m obrazom. Pry πtom m¥ ohranyçymsq
sluçaem m > 1 y predpoloΩym, çto suwestvuet takaq monotonno ub¥vagwaq
do 0 funkcyq v ( r ) , r ∈ [ 0, ∞ ) , çto
B
–1 ≤ u0 ( x ) / v ( | x | ) ≤ B , B ≥ 1.
PredpoloΩym dalee, çto funkcyq v ( r ) y funkcyq ρ 0 ( r ) yz (2) udovletvorq-
gt sledugwym uslovyqm: v v( ) ( ) / ( ) / ( )r r r rp
r
p
rr r rr
1 1
0 01 1− −( )′ + ( )′′ + ( )′ + ( )′′ρ ρ ≤
≤ C ; funkcyq F ( r ) ≡ v ( r )1–
p ⋅ ρ0 ( r ) monotonno ub¥vaet do nulq pry r → ∞
( analoh uslovyq – a ( 1 – p ) + q < 0 yz teorem¥;1 ) . PredpoloΩym takΩe, çto
m + p > 2.
Teorema 6. Pry sdelann¥x v¥ße predpoloΩenyqx suwestvugt konstant¥
0 < C1 < C2 y t0 > 0 takye, çto
{ ( x , t ) : | x | < F
–1
( C2 t ) } ⊂ S ( t ) ⊂ { ( x , t ) : | x | < F
–1
( C1 t ) }
pry t ∈ [ 0 , t0 ] , hde F
–1 — funkcyq, obratnaq k F.
Dokazatel\stvo. Yspol\zovav podxod yz [4], rassmotrym sledugwug ra-
dyal\no symmetryçnug funkcyg: h ( x , t ) = y x t p( , ) /( )[ ] −1 1 , hde
y ( x , t ) ≡ f x x( ) ( )| | − | |[ ]+1
2
tg , f ( r ) ≡ ( ) ( )B r p+[ ] −1 1v , g ( r ) ≡ 1 / B ρ0 ( r ) .
Proverym, çto h ( x , t ) qvlqetsq superreßenyem uravnenyq yz (1):
L h ≡ ρ ( x ) ht – ∆ hm + hp =
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1042 S.;P.;DEHTQREV
= − −
−
− +
−
′ | | − ′ | |
−
− +
−1
2 1
1
1
1
20
1
2 2
1
2ρ
ρ
( )
( )
( ) ( )x
B x
y m
p
m p
p
y f x x
p
p
m p
p tg –
– m
p
y f x x
N
x
m
p
y f x x
m p
p
m p
p
1
1
2
1
1
1
2
1
1
1
1
−
′′ | | − ′′ | |
−
−
| | −
′ | | − ′ | |
− +
−
− +
−( ) ( ) ( ) ( )tg tg + y
p
p1− ≥
≥ y
x
B x
c t y c t y
c t
x
y
p
p
m p
p
m
p
m
p1
0
1
2
1
2
1
1 3
1
11 1
2
−
− +
−
−
−
−
−− − − −
| |
ρ
ρ
( )
( )
( ) ( )
( )
,
hde m¥ vospol\zovalys\ radyal\noj symmetryej y ohranyçennost\g proyzvod-
n¥x ot funkcyj f y g . V;sylu sdelann¥x predpoloΩenyj netrudno vydet\,
çto suwestvugt takye dostatoçno bol\ßoe r0 > 0 y dostatoçno maloe t0 > 0,
çto dlq | x | > r0 , t ∈ [ 0 , t0 ] y ∈ [ 0 , y0 ] pry y0 dostatoçno malom, otkuda polu-
çaem L h ≥ 0 pry | x | ≥ r0 , t ∈ [ 0 , t0 ] .
Pry πtom h ( x , 0 ) = ( B + 1 ) v ( | x | ) > u0 ( x ) , | x | ≥ r0 y, v çastnosty,
h x x r( , )0
0| |= = ( B + 1 ) v ( r0 ) > u x
x r0
0
( ) | |= , tak çto pry dostatoçno mal¥x t
h x t x r( , ) | |= 0
> u x t x r( , ) | |= 0
, hde u ( x , t ) — reßenye zadaçy Koßy (1). Umen\ßaq,
esly neobxodymo, t0 , budem sçytat\, çto poslednee neravenstvo v¥polnqetsq
pry t ∈ [ 0 , t0 ] . Takym obrazom, h ( x , t ) qvlqetsq superreßenyem dlq u ( x , t ) v
oblasty | x | ≥ r0 , t ∈ [ 0 , t0 ] , y, sledovatel\no, 0 ≤ u ( x , t ) ≤ h ( x , t ) v πtoj ob-
lasty. Yz;opredelenyq funkcyy h sleduet, çto ee nosytel\ soderΩytsq v
mnoΩestve f x x( ) ( )| | − | |1
2
tg ≥ 0, t.;e. v ( | x | )1–
p
ρ0 ( | x | ) ≤ t
B B p2 1 1( )+ − , yly | x | ≤
≤ F t
B B p
−
−+
1
12 1( )
, çto dokaz¥vaet pravoe vklgçenye v teoreme;6. Levoe vklg-
çenye dokaz¥vaetsq analohyçno, esly v kaçestve subreßenyq rassmotret\ ana-
lohyçnug funkcyg h(x , t ), tol\ko v kaçestve y(x , t) vzqt\ y = f x x( ) ( )| | − | |[ ]+2tg ,
pryçem f ( r ) =
1
2
1
B
r
p
v( )
−
, g ( r ) = B / ρ0 ( r ) .
Teorema dokazana.
5. Ysçeznovenye reßenyq za koneçnoe vremq.
Teorema 7. Pust\ v¥polnen¥ uslovyq teorem¥;1 yly;6. Tohda suwestvuet
takoe T 0 > 0, çto reßenye zadaçy Koßy;(1) xarakteryzuetsq svojstvom:
u ( x , t ) ≡ 0 pry t ≥ T0 , x ∈ RN
.
Dokazatel\stvo πtoj teorem¥ pry nalyçyy rastuwej ρ ( x ) sleduet yz
teorem;1 y 6, utverΩdagwyx nalyçye πffekta mhnovennoj kompaktyfykacyy
nosytelq reßenyq zadaçy Koßy;(1), y lemm¥;2 (o „lokalyzacyy”). Prymenqq
lemmu;2, netrudno ustanovyt\, çto reßenye, ymegwee v kakoj-to moment vreme-
ny t0 > 0 kompaktn¥j nosytel\, soderΩawyjsq v nekotorom ßare radyusa R0 >
> 0, soxranqet svojstvo kompaktnosty nosytelq y pry vsex t > t0 , pryçem eho
nosytel\ soderΩytsq v ßare radyusa R1 = R0 + d ( m , p , | u ( x , t0 ) |∞ ) . Rassmatry-
vaq teper\ reßenye u ( x , t ) zadaçy;(1) kak reßenye zadaçy Dyryxle v ßare
radyusa R1 s nulev¥my hranyçn¥my uslovyqmy, vydym, çto prostranstvenno-
odnorodnaq funkcyq vyda h = h ( T0 – t ) budet superreßenyem ukazannoj zadaçy
Dyryxle, esly v kaçestve h ( y ) yspol\zovat\ reßenye zadaçy M h′ ( y ) = hp
( y ) ,
h ( 0 ) = 0, hde M ≡ max ρ ( x ) po;ßaru radyusa R1 , a T0 v¥brano dostatoçno
bol\ßym.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
VLYQNYE NEODNORODNOSTY PORYSTOJ SREDÁ … 1043
6. O neobxodym¥x uslovyqx kompaktyfykacyy. V;zaklgçenye rassmotrym
vopros o neobxodym¥x uslovyqx mhnovennoj kompaktyfykacyy, naklad¥vaem¥x
na naçal\nug funkcyg. V;çastnosty, A.;Íyßkov¥m y R.;Kersnerom v [1] b¥l
postavlen vopros o tom, moΩet ly nablgdat\sq qvlenye mhnovennoj kompak-
tyfykacyy nosytelq, esly naçal\naq funkcyq ymeet, naprymer, predstavlenye
u0 ( x ) = c x nn
n Z
δ( )−
∈
∑ + f ( x ) , Z — mnoΩestvo cel¥x çysel, | |
∈
∑ cn
n Z
< ∞ , x ∈ R1,
δ ( x ) — del\ta-funkcyq, f > 0, f ∈ L1 ( R1
) . ∏tot vopros b¥l sformuly-
rovan;;dlq sluçaq, kohda dyffuzyq v uravnenyy opys¥vaetsq p -laplasyanom
∇ ( | ∇u | l
–
1
∇u ) vmesto ∆ ( um
) , no on Ωe ymeet mesto y dlq uravnenyq porystoj
sred¥ v;(1). V;rassmatryvaemom sluçae zadaçy;(1) s ρ ( x ) ≡ 1 y v prostejßem
sluçae m = 1 otvet poloΩytelen, t.;e. qvlenye mhnovennoj kompaktyfykacyy
nosytelq ymeet mesto. Bolee toho, qvlenye mhnovennoj kompaktyfykacyy
nablgdaetsq daΩe pry naçal\n¥x dann¥x s neohranyçenn¥m maksymumom y
ne;prynadleΩawyx L 1 ( RN
) . Rassmotrym, naprymer, dlq sluçaq razmernosty
N = 1 prostejßug zadaçu:
ut = uxx – up
, x ∈ RN
, t > 0,
(7)
u ( x , 0 ) = u0 ( x ) , x ∈ RN
,
hde u0 = 1
| |
−
∈
∑
n
x na
n Z
δ( ), a ∈ ( 0, 1 ) .
Netrudno pokazat\, çto πta zadaça ymeet neotrycatel\noe pry t > 0 reßenye
u ( x , t ) , kotoroe v sylu pryncypa sravnenyq maΩoryruetsq reßenyem uravnenyq
teploprovodnosty s temy Ωe naçal\n¥my dann¥my:
u ( x , t ) ≤ v ( x , t ) ≡ 1
4
2
4
πt
e
x
t
−
∗ u0 ( x ) = C t
n
ea
x n
t
n Z
( )
( )
1
2
4
| |
−
−
∈
∑ .
NesloΩn¥e v¥çyslenyq pokaz¥vagt, çto pry t > 0 v ( x , t ) ≤ C ( t ) ( 1 + | x | )–
a y,
sledovatel\no, u ( x , t ) ≤ C ( t ) ( 1 + | x | )–
a. No tohda yz [3] yly [4] sleduet, çto
pry lgbom τ > 0 funkcyq u ( x , t + τ ) ymeet ohranyçenn¥j nosytel\. Sle-
dovatel\no, v sylu proyzvol\nosty t y τ nosytel\ u ( x , t ) ohranyçen pry
lgbom t > 0.
Esly Ωe koπffycyent¥ pry del\ta-funkcyqx v poslednem v¥raΩenyy dlq
u0( x ) v (7) ne;stremqtsq k nulg, naprymer u0 ( x ) = δ( )x n
n Z
−
∈
∑ , to qvlenye mhno-
vennoj kompaktyfykacyy otsutstvuet. Dejstvytel\no, zafyksyruem proyz-
vol\noe x0 ∈ R1 y rassmotrym ßar K s centrom v toçke x0 radyusa 1/2. Pust\
ζ = ζ ( x ) — poloΩytel\naq sobstvennaq funkcyq zadaçy Dyryxle dlq operato-
ra Laplasa v πtom ßare, normyrovannaq kakym-lybo obrazom ( naprymer, moΩno
poloΩyt\ ζ ( x ) = sin ( π ( x – x0 + 1 / 2 ) ) dlq N = 1 ) . UmnoΩym obe çasty uravne-
nyq v (7) na ζ ( x )
s, s ≥ 2, y proyntehryruem po nosytelg ζ ( x ) (predel¥ ynte-
hryrovanyq opuskaem):
d
dt
u x dx s u x dx s s u x us
xx
s
x
s p sζ ζ ζ ζ ζ ζ( ) ( ) ( ) ( )∫ ∫ ∫ ∫= + − −− −1 2 21 .
Prynymaq vo vnymanye, çto ζ xx = – π2
ζ , u dxp sζ∫ ≤ u dxp psζ∫ ≤ u dxs p
ζ∫( ) , y
oboznaçaq f ( t ) ≡ u dxsζ∫ , ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1044 S.;P.;DEHTQREV
d f t
dt
( )
+ c1 f ( t ) + f ( t ) p ≥ 0, c1 = const > 0.
Reßaq πto dyfferencyal\noe neravenstvo, dlq f ( t ) poluçaem ocenku
f ( t ) ≥ f c ep c t p( )0 11
2
1
13− − −− −( )[ ] ,
çto oznaçaet, çto f ( t ) > f ( 0 ) / 2 pry t ∈ [ 0 , t0 ] , hde t0 ne;zavysyt ot v¥bora x0
v opredelenyy ζ y f , esly f ( 0 ) otdeleno ot nulq. ∏to y dokaz¥vaet otsut-
stvye kompaktyfykacyy.
1. Kersner R., Shishkov A. Instantaneous shrinking of the support of energy solutions // J. Math. Anal.
and Appl. – 1996. – 198. – P. 729 – 750.
2. Borelli M., Ughi M. The fast diffusion equation with strong absorption: the instantaneous shrinking
phenomenon // Rend. Ist. mat. Univ. Trieste. – 1994. – 26, Fasc. I E, II. – P. 109 – 140.
3. Ughi M. Initial behavior of the free boundary for a porous media equation with strong absorption //
Adv. Math. Sci. and Appl. Gakkotosho, Tokyo. – 2001. – 11, #;1. – P. 333 – 345.
4. Abdullaev1U.1H. O mhnovennom sΩatyy nosytelq reßenyq nelynejnoho v¥roΩdagwehosq
parabolyçeskoho uravnenyq // Mat. zametky. – 1998. – 63, v¥p.;3. – S.;323 – 331.
5. Berstch M. A class of degenerate diffusion equations with a singular nonlinear term // Nonlinear
Analysis, Methods and Appl. – 1983. – 7, #;1. – P. 117 – 127.
6. Berstch M., Kersner R., Peletier L. A. Positivity versus localization in degenerate diffusion equa-
tions // Ibid. – 1985. – 9, # 9. – P. 987 – 1008.
7. Aronson D., Crandall M. G., Peletier L. A. Stabilization of solutions of a degenerate nonlinear
diffusion problem // Ibid. – 1982. – 6, # 10. – P. 1001 – 1022.
8. Lad¥Ωenskaq1O.1A., Solonnykov1V.1A., Ural\ceva1N.1N. Lynejn¥e y kvazylynejn¥e uravne-
nyq parabolyçeskoho typa. – M.: Nauka, 1967. – 736;s.
9. DiBenedetto E. Continuity of weak solutions to a general porous medium equation // Indiana Univ.
Math. J. – 1983. – 32, #;1. – P. 83 – 118.
Poluçeno 28.12.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
|
| id | umjimathkievua-article-3512 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:43:55Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/84ac587f47697f0171818761c433642a.pdf |
| spelling | umjimathkievua-article-35122020-03-18T19:56:35Z Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem Влияние неоднородности пористой среды на мгновенную компактификацию носителя решения задачи фильтрации Degtyarev, S. P. Дегтярев, С. П. Дегтярев, С. П. We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media. Вивчається миттєва компактифікація та початкова поведінка носія розв'язку рівняння фільтрації у неоднорідному пористому середовищі. Institute of Mathematics, NAS of Ukraine 2006-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3512 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 8 (2006); 1035–1044 Український математичний журнал; Том 58 № 8 (2006); 1035–1044 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3512/3761 https://umj.imath.kiev.ua/index.php/umj/article/view/3512/3762 Copyright (c) 2006 Degtyarev S. P. |
| spellingShingle | Degtyarev, S. P. Дегтярев, С. П. Дегтярев, С. П. Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| title | Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| title_alt | Влияние неоднородности пористой среды на мгновенную компактификацию носителя решения задачи фильтрации |
| title_full | Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| title_fullStr | Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| title_full_unstemmed | Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| title_short | Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| title_sort | influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3512 |
| work_keys_str_mv | AT degtyarevsp influenceoftheinhomogeneityofporousmediaontheinstantaneouscompactificationofthesupportofsolutionofthefiltrationproblem AT degtârevsp influenceoftheinhomogeneityofporousmediaontheinstantaneouscompactificationofthesupportofsolutionofthefiltrationproblem AT degtârevsp influenceoftheinhomogeneityofporousmediaontheinstantaneouscompactificationofthesupportofsolutionofthefiltrationproblem AT degtyarevsp vliânieneodnorodnostiporistojsredynamgnovennuûkompaktifikaciûnositelârešeniâzadačifilʹtracii AT degtârevsp vliânieneodnorodnostiporistojsredynamgnovennuûkompaktifikaciûnositelârešeniâzadačifilʹtracii AT degtârevsp vliânieneodnorodnostiporistojsredynamgnovennuûkompaktifikaciûnositelârešeniâzadačifilʹtracii |