Boundedness of weak solutions of a nondiagonal parabolic system of two equations
We study the problem of boundedness of weak solutions of a general nondiagonal parabolic system of nonlinear differential equations whose matrix of coefficients satisfies special structural conditions. To do this, we use a procedure based on the estimation of a certain function of unknowns.
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| Дата: | 2009 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3028 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509047360323584 |
|---|---|
| author | Portnyahin, D. V. Портнягін, Д. В. |
| author_facet | Portnyahin, D. V. Портнягін, Д. В. |
| author_sort | Portnyahin, D. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:35Z |
| description | We study the problem of boundedness of weak solutions of a general nondiagonal parabolic system of nonlinear differential equations whose matrix of coefficients satisfies special structural conditions. To do this, we use a procedure based on the estimation of a certain function of unknowns. |
| first_indexed | 2026-03-24T02:34:53Z |
| format | Article |
| fulltext |
UDK 517.956.45
D. V. Portnqhin (In-t fizyky kondensovanyx system NAN Ukra]ny, L\viv)
OBMEÛENIST| SLABKYX ROZV�QZKIV NEDIAHONAL|NO}
PARABOLIÇNO} SYSTEMY DVOX RIVNQN|
We study boundedness of weak solutions of a general nondiagonal parabolic system of nonlinear
differential equations with a matrix of coefficients, which satisfies special structural hypotheses. To do
this, we use a technique basing on the estimation of a certain function of unknowns.
Yzuçaetsq ohranyçennost\ slab¥x reßenyj obwej nedyahonal\noj parabolyçeskoj system¥ ne-
lynejn¥x dyfferencyal\n¥x uravnenyj s matrycej koπffycyentov, udovletvorqgwej specy-
al\n¥m strukturn¥m uslovyqm. Pry πtom prymenqetsq texnyka, osnovannaq na ocenke oprede-
lennoj funkcyy neyzvestn¥x.
1. Vstup. U danij roboti budemo vyvçaty obmeΩenist\ slabkyx rozv�qzkiv neli-
nijno] nediahonal\no] paraboliçno] systemy dvox rivnqn\ dyverhentnoho vyhlqdu
za special\nyx prypuwen\ na ]] strukturu.
Isnu[ kil\ka vidomyx kontrprykladiv, qki pokazugt\, wo ocinky De DΩordΩi
� Neßa � Mozera, vzahali kaΩuçy, ne spravdΩugt\sq dlq eliptyçno] systemy,
kotru moΩna rozhlqdaty qk çastkovyj vypadok paraboliçno]. Pryklad neobme-
Ωenoho rozv�qzku linijno] eliptyçno] systemy z obmeΩenymy koefici[ntamy bu-
lo pobudovano De DΩordΩi v [1]. Isnu[ we odyn pryklad J. Neçasa i J. Suçeka
nelinijno] eliptyçno] systemy z dostatn\o hladkymy koefici[ntamy, ale iz roz-
v�qzkom, wo ne naleΩyt\ navit\ do W
2,
2.
Ci, a takoΩ bahato inßyx prykladiv svidçat\ pro te, wo problema rehulqr-
nosti rozv�qzku dlq systemy [ nabahato skladnißog, niΩ dlq odnoho rivnqnnq
druhoho porqdku.
Stosovno system dyferencial\nyx rivnqn\ ocinky De DΩordΩi uzahal\neno
til\ky na ]x special\nyj klas, tak zvani slabkozv�qzani systemy. Systema nazy-
va[t\sq slabkozv�qzanog, qkwo vona [ perepletenog lyße u dodankax, wo ne
mistqt\ poxidnyx.
Tomu predstavlq[ interes znaxodΩennq syl\nozv�qzanyx system, tobto ta-
kyx, wo [ perepletenymy u dodankax z poxidnymy i rozv�qzky qkyx magt\ pevnu
rehulqrnist\.
Texnika, qku my budemo vykorystovuvaty, zastosovuvalasq v [2] dlq slabko-
nelinijnyx system (dyv. takoΩ [3 – 5]) i polqha[ v perexodi do novo] funkci],
dlq qko] ocinky vstanovlggt\sq zvyçajnym çynom, zvidky vyplyva[ ocinka dlq
koΩno] komponenty vektor-funkci] rozv�qzku. Cq texnika dozvolq[ doslidΩu-
vaty nelinijni nediahonal\ni systemy.
Osnovna ideq polqha[ v nastupnomu: zamist\ toho, wob namahatysq vstanovy-
ty ocinky dlq koΩno] z komponent rozv�qzku ( u, v ), vvesty qkus\ novu funkcig
komponent rozv�qzku H ( u, v ), z ocinok qko] moΩna bude vyvesty ocinky dlq
koΩno] komponenty rozv�qzku ( u, v ).
U danij roboti, obmeΩugçys\ systemamy rivnqn\ druhoho porqdku iz speci-
al\nog strukturog, my pokazu[mo obmeΩenist\ rozv�qzku nelinijno] paraboliç-
no] systemy dvox rivnqn\, qka zv�qzana u najstarßyx poxidnyx i v qko] starßi
koefici[nty zaleΩat\ vid x ta nevidomyx u ta v.
2. Osnovni poznaçennq ta prypuwennq. Budemo rozhlqdaty systemu dvox
rivnqn\ vyhlqdu
u
x
A x u u B x t u u ut
i
x x x x− ∂
∂
( ) = ( )( )( ) ( ), , , , , , , , , ,1 1
v v v v ,
(1)
v v v v vt
i
x x x xx
A x u u B x t u u u− ∂
∂
( ) = ( )( )( ) ( ), , , , , , , , , ,2 2 , ( x, t ) ∈ Q
© D. V. PORTNQHIN, 2009
400 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
OBMEÛENIST| SLABKYX ROZV�QZKIV NEDIAHONAL|NO} … 401
(tut i dali Q � oblast\, v qkij rozhlqda[t\sq zadaça), dlq qko] modellg [
systema
u
x
a x u u b x u f x t
ut
i
− ∂
∂
( )∇ + ( )∇ = ( )
+ +
( )1 1 1
1
1
, , , , ,v v v
v
,
(2)
v v v v
v
t
ix
a x u u b x u f x t
u
− ∂
∂
( )∇ + ( )∇ = ( )
+ +
( )2 2 2
1
1
, , , , , , ( x, t ) ∈ Q,
f x t L Qj ( ) ∈ ( ), τ , τ >
n + 2
2
. (3)
Prypustymo, wo isnu[ funkciq dvox zminnyx
˜ ,H u( )v taka, wo dlq bud\-qkyx u,
v, x ∈ R
C u H u C u1
2 2
2
2 2( + ) ≤ ( ) ≤ ( + )v v v˜ , , (4)
0 ≤
˜ ,H uu( )v ,
˜ ,H uv v( ) ≤
C u2 +( )v , (5)
C1 ≤ ˜ ,H uuu( )v , ˜ ,H uuv v( ) , ˜ ,H uvv v( ) ≤ C2 , (6)
de C1 > 0, C2 < ∞ � konstanty, i
a x u H u a x u H u x u H uu u1 2( ) ( ) + ( ) ( ) = ( ) ( ), , ˜ , , , ˜ , , , ˜ ,v v v v v vv Λ ,
(7)
b x u H u b x u H u x u H uu1 2( ) ( ) + ( ) ( ) = ( ) ( ), , ˜ , , , ˜ , , , ˜ ,v v v v v vv vΛ
ta
a H a Huu u1 2
˜ ˜+ v ≥ 0, (8)
2
2
1 2 1 2 1 2
1 2 1 2 1 2
( + ) ( + ) + +
( + ) + + ( + )
a H a H a b H b H a H
a b H b H a H b H b H
uu u u uu
u uu u
˜ ˜ ˜ ˜ ˜
˜ ˜ ˜ ˜ ˜
v v vv
v vv v vv
≥ 0. (9)
Tut Λ � ( Ω × R × R → R )-vymirna funkciq taka, wo
0 < Λ1 ≤ Λ( )x u, , v ≤ Λ2 ∀u, v, x ∈ R, (10)
Λ1 , Λ2 � çysla. Krim toho, prypuska[mo, wo koefici[nty a1 , a2 , b1 , b2 zado-
vol\nqgt\ umovu eliptyçnosti (12).
Pryklad. Navedemo paraboliçnu systemu, wo zadovol\nq[ vvedeni umovy:
a u u
a u
1
2( ) = ( ) − ( )
, ,
,
v v
vΛ
α
, b u u b u2 1( ) = ( ) − ( ), , ,v v vΛ α ,
α =
H
H
u
v
, H = u u2 2+ +v vε ,
C1 ≤ Λ( )u, v ≤ C2 , a
C
2
3
1
≤
+
α
α
,
b
C
1
3
1
≤
+ α
, ε ≤ 1
10
, C1 ≥ 5C3 > 0.
Zadamo hranyçni umovy typu Dirixle:
( − − )( ) ∈ ( )u g g x t W p
1 2 0
1, , ,
v Ω dlq majΩe usix t ∈ ( 0, T ),
(11)
( )( ) = ( )( )u x u x, , ,v v0 0 0 .
Rozv�qzok systemy (1) z danymy Dirixle (11) rozumi[mo u slabkomu sensi, qk
u [6].
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
402 D. V. PORTNQHIN
Oznaçennq 1. Vymirna vektor-funkciq ( , )u u1 2 = ( u, v ) nazyva[t\sq slab-
kym rozv�qzkom zadaçi (2.1) � (2.11), qkwo
u C T L L T Wj ∈ ( ) ( )( ) ( )0 02 2 1 2, ; , ; ,Ω ΩI
i dlq vsix t ∈ ( 0, T ] i probnyx funkcij
ϕ ∈ ( ) ( )( ) ( )W T L L T W1 2 2 2
0
1 20 0, ,, ; , ;Ω ΩI
vykonu[t\sq
u x t dx u A dx dj
j
j
jt i
j
jx
t
i
ϕ ϕ ϕ τ( ) + {− + }∫ ∫∫
×( ]
,
,Ω Ω 0
=
= u x dx B dx dj
j
j
j
t
0
0
0ϕ ϕ τ( ) +∫ ∫∫
×( ]
,
,Ω Ω
.
Hranyçna umova (11) rozumi[t\sq u slabkomu sensi.
Oznaçymo hranyçni normy funkcij, wo znadoblqt\sq u podal\ßomu rozhlqdi.
Oznaçennq 2. Nexaj Ω � mnoΩyna v R n ( n � dovil\ne natural\ne çys-
lo) i ∂Ω � çastyna ]] meΩi, W( )Ω � dovil\nyj sobol[vs\kyj prostir. Dlq
funkci] u ( x ), zadano] na ∂Ω, oznaçymo
u W W(∂ ) ( )=Ω Ωinf
ψ
ψ ,
de infimum beret\sq po vsix funkciqx ψ ∈ W ( Ω ) takyx, wo ψ ( x ) = u ( x ) maj-
Ωe skriz\ na ∂Ω. Poznaçymo çerez W ( ∂Ω ) funkcional\nyj prostir, dlq qkoho
vywevkazana norma [ skinçennog.
Opyßemo poznaçennq, velyçyny ta funkci], wo vxodqt\ do system (1), (2) ta
budut\ zustriçatysq u cij roboti.
Tut i dali Q = ( 0, T ] × Ω, S = ∂Ω × ( 0, T ], ∂Q ≡ { Ω × { 0 } } U { ∂Ω × ( 0, T ] },
Ω � obmeΩena oblast\ v Rn z kuskovo-hladkog meΩeg, x ∈ Ω, T > 0, t ∈ ( 0,
T ], n ≥ 2, i = 1, … , n, j = 1, 2 i za indeksamy, wo povtorggt\sq, rozumi[mo pidsu-
movuvannq, u, v ∈ C T L L T W( ) ( )( ) ( )0 02 2 1 2, ; , ; ,Ω ΩI , W0
1 2, ( )Ω � prostir funk-
cij iz W1 2, ( )Ω , wo znykagt\ na ∂Ω u sensi slidiv dlq majΩe vsix t ∈ ( 0, T ].
Pid paraboliçnistg systemy (1) budemo rozumity, wo ]] çastyna bez çasovyx
poxidnyx [ eliptyçnog. Ponqttq eliptyçnosti systemy dyferencial\nyx riv-
nqn\ druhoho porqdku polqha[ u nastupnomu (qk joho bulo vvedeno v [7]):
∃ λ > 0, 0 < G = G ( x ) ∈ L2
( Q ) | ∀s ∈ R2n ∀r ∈ R2 ∀x ∈ Rn :
A x r s s s Gi
j
j
i( ) ≥ −, , λ 2 . (12)
Prypuska[mo, wo koefici[nty zahal\no] systemy A x r si
j( ), , [ ( Ω × R 2 ×
× R2n → R )-vymirnymy funkciqmy Karateodori, wo zadovol\nqgt\ umovu elip-
tyçnosti i umovy zrostannq:
∃ Λ2 > 0 | ∀s ∈ R2n ∀r ∈ R2 ∀x ∈ Rn : A x r s si
j( ) ≤, , Λ2 , (13)
a takoΩ strukturni umovy
∃ a x r b x rj j( ) ( ), , , | ∀s ∈ R2n ∀r ∈ R2 ∀x ∈ Rn :
A x r s a x r s b x r s G x r s( ) , , , , , ,1
1
1
1
2
1( ) − ( ) − ( ) ≤ ( ) , (14a)
A x r s a x r s b x r s G x r s( ) , , , , , ,2
2
1
2
2
2( ) − ( ) − ( ) ≤ ( ) , (14b)
de a x rj( ), , b x rj( ), zadovol\nqgt\ (4) � (10) i Gj = G x r sj( ), , > 0 [ deqkymy
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
OBMEÛENIST| SLABKYX ROZV�QZKIV NEDIAHONAL|NO} … 403
vidpovidno ( Ω × R2 → R )- ta ( Ω × R2 × R2n → R )-vymirnymy funkciqmy Kara-
teodori vid x, u, v, na qki nakladagt\sq umovy zrostannq
∃ G ( x ) | ∀r ∈ R2 ∀s ∈ R2n ∀x ∈ Rn :
0 < G x r s
G x
r s
j( ) ≤ ( )
+( ) +( )
, ,
1 12 σ
z σ >
n
n
−
+
2
2
ta G, wo zadovol\nq[ umovy
G ( x ) ∈ Lθ( )Ω , θ >
2 2
4 1 2
( + )
− ( − )( + )
n
nσ
.
ZauvaΩennq 1. NevaΩko pereviryty, beruçy do uvahy toj fakt, wo G ∈
∈ L
n
n
2 2
4 1 2
( + )
− ( − )( + )σ , wo iz strukturnyx umov (14a), (14b) vyplyva[ umova eliptyç-
nosti (12) z çyslom λ i funkci[g G, zaleΩnymy til\ky vid danyx zadaçi.
Prypuska[mo, wo pravi çastyny B x t rj( ), , [ ( Ω × R × R2 → R )-vymirnymy
funkciqmy, wo zadovol\nqgt\ umovu
∀r ∈ R2 ∀x ∈ Rn ∀t ∈ R : B x t r
f
r
j j( ) ≤
+
, ,
1
, (15)
de fj zadovol\nq[ (3).
Dlq prostoty vykladu vykorystovuvatymemo poznaçennq
˜
, , ,
, , , , ,
u
u x x t
g x t x t T0
0
1
0
0
=
( ) ∈ =
( ) ∈ ∂ ∈ ( )
Ω
Ω
˜
, , ,
, , , , .
v
v
0
0
2
0
0
=
( ) ∈ =
( ) ∈ ∂ ∈ ( )
x x t
g x t x t T
Ω
Ω
Krim toho, vvedemo funkcional\nyj prostir
˜ , ; , ;, ,W Q L W T L T W( ) = ( ) ( )( ) ( )2 1 2 2 1 20 0Ω ΩI ,
tobto funkciq u naleΩyt\ do W̃ Q( ), qkwo intehral
u u ut
T
2 2 2
0
+ ∇ +( )∫∫
Ω
[ skinçennym.
Prypuska[mo, wo dlq funkcij g x tj( ), , ( )( )u x0 0, v u hranyçnyx umovax (11)
vykonugt\sq umovy
˜ ˜u W Q0 ∈ (∂ ), ̃
˜v0 ∈ (∂ )W Q ,
do toho Ω
g x t L Sj( ) ∈ ( )∞, , ( )( ) ∈ ( × { })∞u x L0 0 0, v Ω .
3. Ocinka sumy kvadrativ. Dlq podal\ßoho rozhlqdu potribni ocinky
intehrala vid sumy kvadrativ prostorovyx poxidnyx vid komponent rozv�qzku za-
daçi (1) � (11).
Teorema 1. Nexaj ( u, v ) � rozv�qzok zadaçi (1) � (11) i prypuwennq (14a),
(14b) ta (18) [ vykonanymy, todi ma[ misce ocinka
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
404 D. V. PORTNQHIN
∇ + ∇( )∫∫ u
T
2 2
0
v
Ω
≤ C (16)
iz stalog C, wo zaleΩyt\ vid f , G , ˜ ˜u W Q0 (∂ ) ,
˜ ˜v0 W Q(∂ ), n , Λ 1 , Λ 2 , ε ,
mes Q i ne zaleΩyt\ vid u ta v.
ZauvaΩennq 2. Pid ũ0 ta ̃v0 u formulgvanni teoremy i u podal\ßomu do-
vedenni rozumi[mo dovil\ni funkci] z W̃ Q( ), wo nabuvagt\ znaçennq ũ0 çy ̃v0
na paraboliçnij meΩi oblasti. Tomu kinceve tverdΩennq zalyßa[t\sq pravyl\-
nym dlq hranyçnyx norm.
Dovedennq. PomnoΩymo perße z rivnqn\ (1) na ( − )u ũ0 , druhe na ( − )v ṽ0 .
Zintehruvavßy po oblasti Ω × ( 0, t ) z uraxuvannqm umovy eliptyçnosti (12),
umovy zrostannq (13), umovy na B
j (15), hranyçnyx umov, nerivnosti Gnha ta ne-
rivnosti Sobol[va, otryma[mo ocinku (16).
4. Ocinky L∞∞∞∞-norm. Rozhlqnemo pytannq obmeΩenosti slabkyx rozv�qzkiv
systemy, koefici[nty qko] zadovol\nqgt\ prypuwennq (14a), (14b). Osnovnym
rezul\tatom statti [ nastupna teorema.
Teorema 2. Nexaj ( u, v ) � rozv�qzok systemy (1). Dlq funkci] H, oznaçe-
no] v (4) � (8), ma[ misce ocinka
H L Q∞( ) ≤ C.
Taki sami ocinky vykonugt\sq i dlq komponent rozv�qzku
u L Q∞( ) ≤ C1 ,
v L Q∞( ) ≤ C2 ,
de stali C1, C2 zaleΩat\ til\ky vid n , f j , G , Λ1, mes Q , g
L S1 2, ∞( )
,
u L0 0, v ∞( )Ω , konstant v teoremax vkladennq i ne zaleΩat\ vid u ta v.
Dlq dovedennq ci[] teoremy potribna vidoma lema Stampak�q.
Lema 1. Nexaj ψ ( y ) � nevid�[mna nezrostagça funkciq, oznaçena na
[ k0, ∞ ), qka zadovol\nq[ umovu
ψ ( m ) ≤
C
m k
k
( − )
( ){ }ϑ
δψ dlq m > k ≥ k0
z ϑ > 0 ta δ > 1. Todi
ψ ( k0 + d ) = 0,
de d = C k1
0
1 12/ / /ϑ δ ϑ δ δψ{ }( ) ( − ) ( − ) .
Dovedennq dyv. u roboti [7, s. 8] (lema 4.1).
TakoΩ my vykorystovu[mo nastupnu lemu (dyv. [6, s. 7], tverdΩennq 3.1).
Lema 2. Qkwo u ∈ L T L L T W∞( ) ( )( ) ( )0 02 2
0
1 2, ; , ; ,Ω ΩI , to vykonu[t\sq ne-
rivnist\
u C u uq
TT
t T
n
Ω Ω Ω
∫ ∫∫∫ ∫≤ ∇
< <
2
00 0
2
2
ess sup
/
z q =
2 2( + )n
n
ta stalog C, wo zaleΩyt\ lyße vid n.
Dovedennq teoremy 2. DomnoΩymo perße rivnqnnq (1) na Hu i dodamo
druhe, domnoΩene na Hv ( H bude oznaçene pizniße). Vyberemo ( H – k ) + v
qkosti probno] funkci] z k ≥ k0 =
max , , ,H g g H uL S L( ) ( )[ ]∞ ∞( ) ( )1 2 0 0v Ω . Z po-
dal\ßoho bude vydno, wo taka funkciq [ dopustymog. Skorystavßys\ struk-
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
OBMEÛENIST| SLABKYX ROZV�QZKIV NEDIAHONAL|NO} … 405
turnymy umovamy (14a), (14b), umovamy na pravi çastyny (15) ta zintehruvavßy po
τ vid 0 do t, t ≤ T, ta po x po oblasti Ω, otryma[mo
1
2
2
1 1
0
( − ) + ∇ + ∇ ( − )∇ + ( − )∇( )
( )
∫ ∫∫ {〈H k a u b H H k u H H kA k
t
uu u
t
χ
Ω Ω
v vv, +
+
H u H H a u b H H k u H H ku u uu u
2
2 2∇ + ∇ + ∇ + ∇ ( − )∇ + ( − )∇〉 〈v vv v v, +
+
H u H H f H f H
u
H ku u A k u A k
t
2
1 2
0
1
1
∇ + ∇ ≤ ( + )
+ +
( − )〉} ( ) ( )∫∫v vv
v
χ χ
Ω
+
+
{ ( ) ( )}∇ ( − ) + ∇ ( − )∫∫ ( )G H H k G H H ku
t
A k1 2
0
v
Ω
χ ,
χA k( ) � xarakterystyçna funkciq mnoΩyny A ( k, t ) = { x ∈ Ω | H – k ≥ 0 }. Dali
〈 〉∇ + ∇ ( − )∇ + ( − )∇ ∇ + ∇a u b H H k u H H k H u H Huu u u u1 1
2
v v + vv v, +
+ 〈 〉∇ + ∇ ( − )∇ + ( − )∇ ∇ + ∇a u b H H k u H H k H u H Huu u u u2 2
2
v v + vv v, =
=
{[ + ] ∇ + ( + ) + + (∇ ∇ )[ ]a H a H H u a b H H b H a H uu u u u1
2
2
2
1 2 1
2
2
2
v v v v +
+
[ + ] ∇ } + {[ + ] ∇b H H b H a H a H uu uu u1 2
2 2
1 2
2
v v vv +
+
[ ]( + ) + + (∇ ∇ ) + [ + ] ∇ }( − )a b H b H a H u b H b H H ku uu u1 2 1 2 1 2
2
v vv v vvv v .
Vykonavßy pidstanovku
F x x( ) = ,
H F H= ( )˜ ,
H F Hu u= ′ ˜ , H F Hv v= ′ ˜ ,
H F H F Huu u uu= ′′ + ′˜ ˜2 ,
H F H H F Hu u uv v v= ′′ + ′˜ ˜ ˜ ,
H F H F Hv v v= ′′ + ′˜ ˜2 ,
zhidno z (7) dlq perßo] hrupy dodankiv u fihurnyx duΩkax znaxodymo
{…} = ( ) ∇ + ( ) (∇ ∇ ) + ( ) ∇Λ Λ Λx u H u x u H H u x u Hu u, , , , , ,v v v v vv v
2 2 2 2 =
=
Λ Λ( ) ∇ ( ) = ( ) ′ ∇ ( )x u H u x u F H u, , , , , ˜ ,v v v v
2 2 2
.
Na pidstavi (8) dlq druho] hrupy dodankiv u fihurnyx duΩkax ma[mo
{…}( − ) = ( ) ′′ ∇ ( ) ( − ) + {[ + ] ∇H k x u F H u H k a H a H uuu uΛ , , ˜ , ˜ ˜v v v
2
1 2
2 +
+ [ ]( + ) + + (∇ ∇ ) + [ + ] ∇ } ′( − )a b H b H a H u b H b H F H ku uu u1 2 1 2 1 2
2˜ ˜ ˜ ˜ ˜
v vv v vvv v ≥
≥ Λ( ) ′′ ∇ ( ) ( ) −( )x u F H u F H k, , ˜ , ˜v v
2
.
Zvidsy, vykorystovugçy prypuwennq (5), (6), otrymu[mo
1
2 4
2
0
3 2
2( )( ) − + ( ) ∇ ( )( )
( )
( )∫ ∫∫F H k x u
k
H
H uA k
t
t
A k
˜ , , ˜
˜ ,/χ χ
Ω Ω
Λ v v ≤
≤
C f F F H k C G uA k
t
A k
t
′ ( ) − + ∇ + ∇( )( ) ( ) ( )∫∫ ∫∫˜ χ χ
Ω Ω0 0
v ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
406 D. V. PORTNQHIN
de poznaçeno f = f1 + f2 . Pryhadugçy oznaçennq H̃ i vykonugçy deqki
peretvorennq, perepysu[mo ce takym çynom:
k H k k H kA k
t
t
A k( ) ( )− + ∇ −( )
( )
( )∫ ∫∫˜ ˜4 2
0
1
4 2
χ χ
Ω Ω
Λ ≤
≤
C f H k C G uA k
t
A k
t
( )− + ∇ + ∇( )( ) ( )∫∫ ∫∫˜4
0 0
χ χ
Ω Ω
v ,
de χA k( ) � xarakterystyçna funkciq mnoΩyny A k t( ), = { ∈x HΩ ˜4 –
– k ≥ }0 . Oskil\ky t ∈ ( 0, T ] [ dovil\nym, to beruçy supremum, znaxodymo
k H k k H k
t T
A k
t
T
A ksup ˜ ˜
0
4 2
0
1
4 2
< <
( )
( )
( )( ) ( )− + ∇ −∫ ∫∫χ χ
Ω Ω
Λ ≤
≤
C f H k C G uA k
T
A k
t
( )− + ∇ + ∇( )( )
−
( )∫∫ ∫∫˜4
0
1
0
χ χσ
Ω Ω
v . (17)
Zastosovugçy uzahal\nenu nerivnist\ Hel\dera do pravo] çastyny (17), ma[mo
k w k w
t T t
T
sup
0
2
0
1
2
< < ( )
∫ ∫∫+ ∇
Ω Ω
Λ ≤ C w fq Q r Q
T
A k
q p
, ,
/ /
Ω
∫∫ ( )
− −
0
1 1 1
χ +
+
C u GQ Q
T
A k∇ + ∇
∫∫ ( )
−( − ) −
v 2
0
1 1 2 1
, ,
/ /
ε
σ ε
χ
Ω
,
de w = ( )− +H̃ k4 , p ta ε bulo vybrano takym çynom, wo τ > p > ( n + 2 ) / 2 i
θ > ε > 2 2 4 1 2( + ) − ( − )( + )[ ]n n/ σ , oskil\ky nevaΩko pereviryty, wo ostannq
nerivnist\ vykonu[t\sq. Z lemy 2 vyplyva[, wo
w w wq Q
t T
T
,
/
sup≤ + ∇
< <
∫ ∫∫
0
2
0
2
1 2
Ω Ω
.
Oskil\ky, ne zmenßugçy zahal\nosti, moΩna prypustyty, wo k ≥ 1, to na pid-
stavi ci[] nerivnosti i enerhetyçno] ocinky (16) otrymu[mo
w C w f k C G kq Q q Q r Q
q p
Q, , ,
/ /
,
/ /2 1 1 1 1 1 2 1≤ ( ) + ( ){ } { }− − −( − ) −ψ ψε
σ ε .
(18)
Tut
ψ( ) = ( )∫k A k t dt
T
mes ,
0
.
Zastosovugçy nerivnist\ Gnha do pravo] çastyny (18), oderΩu[mo
w C k C kq Q
q p
,
/ / / / /≤ ( ) + ( ){ } { }− − [ −( − ) − ]ψ ψ σ ε1 1 1 1 1 2 1 2 . (19)
Ocinymo
( − ) ( ) = ( − )
{ } ∫∫ ( )m k m m kq
T
A m
q
ψ χ1
0
1
/
/
Ω
<
<
Ω
∫∫ ( )
<
0
1T
q
A m
q
q Qw wχ
/
, ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
OBMEÛENIST| SLABKYX ROZV�QZKIV NEDIAHONAL|NO} … 407
de m > k ≥ k0
. Pidstavlqgçy ce v (19), znaxodymo
( − ) ( ) ≤ ( ) + ( ){ } { }( − − ) ( − )m k m C k C kq q q p qψ ψ ψ ε1 1 1 1 2 4/ / / / =
= C k C k{ } { }( ) + ( )ψ ψδ δ1 2 . (20)
Z prypuwen\ wodo fj ta vnaslidok vyboru p ma[mo
τ > p >
n + 2
2
,
zvidky
2 2
2
1
2 2
1( + ) −
( + )
−
n n
n p
> 1, i, takym çynom, δ1 > 1.
Z prypuwen\ wodo G ta vnaslidok vyboru ε
θ > ε >
2 2
4 1 2
( + )
− ( − )( + )
n
nσ
,
zvidky
2 2
2
1
1
2
1( + ) − − −
n
n
σ
ε
> 1, i, takym çynom, δ2 > 1.
Na pidstavi lemy 1 iz spivvidnoßennq (20) moΩemo zrobyty vysnovok, wo
ψ( + )k d0 = 0
dlq deqkoho d, dostatn\o velykoho, ale skinçennoho, wo zaleΩyt\ lyße vid n,
f j, G , Λ1 , g
L S1 2, ∞ ( )
, u
L0 0, v ∞ ( )Ω , stalyx u teoremax vkladennq ta ne zale-
Ωyt\ vid u ta v. I, takym çynom,
H̃
L Q∞ ( )
≤ C.
NevaΩko baçyty, wo zavdqky (4) taki sami ocinky magt\ misce i dlq komponent
rozv�qzku ( u, v ). Vlasne,
u L Q∞ ( ) ≤ C1 , v L Q∞ ( ) ≤ C2 .
Teoremu dovedeno.
1. De Giorgi E. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico //
Boll. Unione mat. ital. – 1968. – P. 135 – 137.
2. Pozio M. A., Tesei A. Global existence of solutions for a strongly coupled quasilinear parabolic
system // Nonlinear Anal. – 1990. – 12, # 8. – P. 657 – 689.
3. Dung L. Hölder regularity for certain strongly coupled parabolic systems // J. Different. Equat. –
1999. – 151. – P. 313 – 344.
4. Wiegner M. Global solutions to a class of strongly coupled parabolic systems // Math. Ann. – 1992.
– 292. – P. 711 – 727.
5. Portnyagin D. A generalization of the maximum principle to nonlinear parabolic systems // Ann.
pol. math. – 2003. – 81, # 3. – P. 217 – 236.
6. DiBenedetto E. Degenerate parabolic equations. – New York: Springer, 1993.
7. Chen Y. Z., Wu L. C. Second order elliptic equations and elliptic systems. – Providence, RI: Amer.
Math. Soc., 1998.
OderΩano 24.04.07,
pislq doopracgvannq � 25.12.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
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| id | umjimathkievua-article-3028 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:53Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/37/1d4c25b3ca23063e9f9b1a211138a137.pdf |
| spelling | umjimathkievua-article-30282020-03-18T19:43:35Z Boundedness of weak solutions of a nondiagonal parabolic system of two equations Обмеженість слабких розв'язків недіагональної параболічної системи двох рівнянь Portnyahin, D. V. Портнягін, Д. В. We study the problem of boundedness of weak solutions of a general nondiagonal parabolic system of nonlinear differential equations whose matrix of coefficients satisfies special structural conditions. To do this, we use a procedure based on the estimation of a certain function of unknowns. Изучается ограниченность слабых решений общей недиагональной параболической системы нелинейных дифференциальных уравнений с матрицей коэффициентов, удовлетворяющей специальным структурным условиям. При этом применяется техника, основанная на оценке определенной функции неизвестных. Institute of Mathematics, NAS of Ukraine 2009-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3028 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 3 (2009); 400-407 Український математичний журнал; Том 61 № 3 (2009); 400-407 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3028/2805 https://umj.imath.kiev.ua/index.php/umj/article/view/3028/2806 Copyright (c) 2009 Portnyahin D. V. |
| spellingShingle | Portnyahin, D. V. Портнягін, Д. В. Boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| title | Boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| title_alt | Обмеженість слабких розв'язків недіагональної параболічної системи двох рівнянь |
| title_full | Boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| title_fullStr | Boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| title_full_unstemmed | Boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| title_short | Boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| title_sort | boundedness of weak solutions of a nondiagonal parabolic system of two equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3028 |
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