Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem
We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined.
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| Date: | 2006 |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509545588064256 |
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| author | Tedeev, A. F. Тедеев, А. Ф. Тедеев, А. Ф. |
| author_facet | Tedeev, A. F. Тедеев, А. Ф. Тедеев, А. Ф. |
| author_sort | Tedeev, A. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:54:47Z |
| description | We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined. |
| first_indexed | 2026-03-24T02:42:48Z |
| format | Article |
| fulltext |
UDK 517.946
A. F. Tedeev (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
NAÇAL|NO-KRAEVÁE ZADAÇY DLQ KVAZYLYNEJNÁX
VÁROÛDAGWYXSQ PARABOLYÇESKYX URAVNENYJ
S DEMPFYROVANYEM. ZADAÇA NEJMANA*
We investigate the behaviour of the total mass of a solution of the Neumann problem for a wide class of
degenerate parabolic equations with damping in a space with a noncompact boundary. We find new
critical indexes in the problem considered.
DoslidΩu[t\sq povedinka total\no] masy rozv’qzku zadaçi Nejmana dlq ßyrokoho klasu vyrod-
Ωenyx paraboliçnyx rivnqn\ z dempfiruvannqm u prostorax iz nekompaktnog meΩeg. Znajdeno
novi krytyçni pokaznyky v doslidΩuvanij zadaçi.
1. Vvedenye. V dannoj rabote rassmatryvaetsq zadaça Nejmana
ut =
i
N
i
m
x
q
x
u Du u a x Du
i
=
− −∑ ∂
∂ ( ) −
1
1 1λ ν( ) , ( x, t ) ∈ Ω × ( 0, ∞ ), (1.1)
i
N
m
x iu Du u n
i
=
− −∑
1
1 1λ = 0, ( x, t ) ∈ ∂ Ω × ( 0, ∞ ), (1.2)
u ( x, 0 ) = u0 ( x ), x ∈ Ω, (1.3)
hde Ω ⊂ R
N, N ≥ 2, — neohranyçennaq oblast\ s dostatoçno hladkoj nekom-
paktnoj hranycej, n = ( n i ) —7vneßnqq normal\ k ∂ Ω, Du = u ux xN1
, ,…( ). V
dal\nejßem predpolahaem, çto a ( x ) y u0 ( x ) — neotrycatel\n¥e yzmerym¥e
funkcyy, pryçem u0 ∈ L1 ( Ω ), t. e. u0 ymeet koneçnug massu. Krome toho,
m + λ – 2 ≥ 0, λ > 0, 1 < q < λ + 1, ν q > m + λ – 1. (1.4)
Budem predpolahat\ takΩe dopolnytel\n¥e uslovyq na dann¥e zadaçy. Uravne-
nye ut = ∆u Du uq p− + δ , δ > 0, b¥lo vperv¥e yssledovano v rabote ·1‚ s
cel\g yzuçenyq vlyqnyq çlena − Du q
(yly, ynaçe, dempfyrovanyq) na
problemu suwestvovanyq yly nesuwestvovanyq hlobal\n¥x po vremeny reßenyj
zadaçy Dyryxle. Podrobn¥j analyz rezul\tatov v πtom napravlenyy moΩno
najty v obzornoj rabote ·2‚. Otmetym takΩe nedavnyj cykl rabot ·3 – 6‚, hde
ymegtsq dal\nejßye ss¥lky.
Cel\g dannoj rabot¥ qvlqetsq naxoΩdenye uslovyj na q, pry kotor¥x
massa reßenyq (1.1) – (1.3) u t( , ) ,⋅ 1 Ω ≡ u t L( , ) ( )⋅
1 Ω stremytsq k nulg pry t →
→ ∞. Otmetym, çto esly a ( x ) ≡ 0 , to dlq poçty vsex t > 0 u t( , ) ,⋅ 1 Ω ≡
≡ u0 1, Ω y, sledovatel\no, massa ne stremytsq k nulg pry t → ∞. Odnako, kak
v¥qsnylos\, daΩe nalyçye syl\noho stoka, t. e. dempfyrovanyq, v (1.1) ne vseh-
da harantyruet stremlenye k nulg mass¥ reßenyq (1.1) – (1.3). V sluçae a ( x ) ≡
≡ const, Ω = R
N
(zadaça Koßy) zadaça (1.1), (1.3) yssledovalas\ v rabote ·7‚, hde
dan otvet na vopros: pry kakyx uslovyqx na parametr¥ zadaçy massa reßenyq
stremytsq k nulg? A ymenno, najden krytyçeskyj pokazatel\ q* =
= N m N( ) ( )/+ − + +( ) +λ λ ν1 1 1 . ∏to oznaçaet, çto v sluçae q ≤ q*
u t RN( , ) ,⋅ 1 → 0 pry t → ∞, a v sluçae q > q* u t RN( , ) ,⋅ 1 > c > 0 pry dosta-
toçno bol\ßyx t > t0
. Krome toho, naprymer, esly supp u0 < ∞, to pry dosta-
toçno bol\ßyx znaçenyqx t dokazan¥ sledugwye ocenky:
*7
V¥polnena pry podderΩke INTAS (hrant 03-51-5007).
© A. F. TEDEEV, 2006
272 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
NAÇAL|NO-KRAEVÁE ZADAÇY DLQ KVAZYLYNEJNÁX VÁROÛDAGWYXSQ … 273
u t RN( , ) ,⋅ 1 ≤ c t
–
A pry q < q *, (1.5)
hde
A =
q q
H
N
* − +( )ν 1 , H = ( )( ) ( )λ ν λ+ − − + −1 1 2q q m ·
y
u t RN( , ) ,⋅ 1 ≤ C t q(ln )
−
−
1
1ν
pry q = q *. (1.6)
Otmetym, çto zadaça (1.1) – (1.3) s a ( x ) ≡ 0 yzuçalas\ v rabotax ·8 – 11‚, hde
b¥ly dan¥ toçn¥e ocenky u t( , ) ,⋅ ∞ Ω y heometryy nosytelq. Kombynyruq
podxod¥, yzloΩenn¥e v πtyx rabotax y rabote ·7‚, m¥7ustanavlyvaem ocenky ty-
pa (1.5) y (1.6), kotor¥e, kak πto budet vydno yz dal\nejßeho, suwestvenno zavy-
sqt ot heometryy oblasty y povedenyq a ( x ) na beskoneçnosty. Dlq formuly-
rovky osnovn¥x rezul\tatov y yx dokazatel\stv nam potrebuetsq rqd opredele-
nyj y vspomohatel\n¥x predloΩenyj. Vsgdu v dal\nejßem çerez c budem obo-
znaçat\ postoqnn¥e, zavysqwye lyß\ ot parametrov zadaçy y ne zavysqwye ot
razmera oblasty Qt .
2. Vspomohatel\n¥e utverΩdenyq y formulyrovky osnovn¥x rezul\ta-
tov. V¥delym klass¥ oblastej s nekompaktnoj hranycej, udovletvorqgwyx
uslovyqm yzoperymetryçeskoho typa. Budem sçytat\, çto naçalo koordynat
prynadleΩyt Ω. Pust\ l ( v ) = inf mes N Q− ∂( ){ }1 ∩ Ω , hde ynfymum beretsq po
vsem otkr¥t¥m mnoΩestvam Q ⊂ Ω s lypßycevoj hranycej; mesN Q = v. Bu-
dem hovoryt\, çto Ω prynadleΩyt klassu B1 ( g ), esly suwestvuet neub¥vag-
waq neprer¥vnaq dlq vsex v > 0 funkcyq g ( v ) takaq, çto
v
v
( )
( )
N N
g
− /1
ne ub¥va-
et dlq vsex v > 0. Dalee, pust\ rk ( x ) = x xk1
2 2 1 2
+…+( ) /
, 1 ≤ k ≤ N, y dlq zadan-
noho ρ > 0 Ω ( ρ ) = Ω ∩ r xk ( ) <{ }ρ , V ( ρ ) = mesN Ω ( ρ ). Oboznaçym çerez R ob-
ratnug k V ( ρ ) funkcyg. Budem hovoryt\, çto Ω prynadleΩyt klassu B2 ( g ),
esly Ω ∈ B1 ( g ) y suwestvuet postoqnnaq c0 > 0 takaq, çto
R ( v ) ≥ c
g0
v
v( )
(2.1)
dlq vsex v > 0. Lehko vydet\, çto esly Ω ∈ B1 ( g ), to spravedlyvo obratnoe k
(2.1) neravenstvo
R ( v ) ≤ N
g
v
v( )
(2.2)
dlq vsex v > 0. Krome toho, yz (2.1) y (2.2) sleduet, çto
1
N
g Vρ ρ( )( ) ≤ V ( ρ ) ≤
1
0c
g Vρ ρ( )( ) (2.3)
dlq vsex ρ > 0. V svog oçered\, yz (2.3) v¥tekaet, çto mesN Ω = ∞. Klass¥ ob-
lastej B1 ( g ), B2 ( g ) b¥ly vveden¥ v [12], hde poluçen¥ toçn¥e ocenky skoros-
ty stabylyzacyy reßenyq zadaçy Nejmana dlq lynejn¥x parabolyçeskyx urav-
nenyj vtoroho porqdka. Typyçn¥m prymerom oblastej klassa B2 ( g ) qvlqetsq
oblast\ typa paraboloyda Ω
h = x R x xN h∈ ′ <{ }: 1 , hde ′x = x xN2
2 2 1 2
+…+( ) /
,
x1 > 1, 0 ≤ h ≤ 1. V πtom sluçae (sm. [9] dlq N ≥ 2 y [12] pry N = 2 )
g ( v ) =
c N Nmin ,( )v v− /( )1 γ , γ =
h N
h N
( )
( )
−
− +
1
1 1. (2.4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
274 A. F. TEDEEV
Dalee, çtob¥ yzbeΩat\ hromozdkyx formulyrovok rezul\tatov, budem pred-
polahat\, çto a ( x ) ≡ a ( rk ( x ) ). Bolee toho, predpoloΩym, çto a ( ρ ) — rastu-
waq funkcyq dlq vsex ρ > 0. Esly a1 ( s ) — ub¥vagwaq perestanovka funkcyy
1 / a ( rk ( x ) ), to sohlasno opredelenyg
a1 ( s ) =
1
a r xk ( )
*
( )
=
1
a R s( )( ) . (2.5)
Napomnym, çto pod ub¥vagwej perestanovkoj yzmerymoj funkcyy f ( x ) pony-
maetsq sledugwee: f ( s )
* = inf ( )τ µ τ <{ }s , µ ( τ ) = mesn x f x∈ >{ }Ω: ( ) τ (sm.,
naprymer, [13]). PredpoloΩym, çto
s
a s
q
( ) vozrastaet dlq vsex s > 0. (2.6)
Vvedem teper\ ponqtye reßenyq zadaçy (1.1) – (1.3) v Q∞ = Ω × ( 0, ∞ ). Budem
hovoryt\, çto u ( x, t ) — reßenye zadaçy (1.1) – (1.3) v Q∞ , esly u ≥ 0,
u ∈ L QT∞, ( )loc ∩ C T L0 2, ; ( ),( )( )loc Ω , | D u
σ
| λ
+
1
, a ( x ) | D u
ν
|
q ∈ L QT1, ( )loc , σ =
= ( m + λ – 1 ) / λ, dlq lgboho T > 0 y dlq lgboj funkcyy η ∈ C QT0
1( )
Q
t
m q
T
u u Du Du D a Du∫∫ − + +{ }− −η η ηλ ν1 1
d x d t = 0 . (2.7)
Krome toho, u ( x, t ) → u0 pry t → 0 v L1 ( Ω ).
Vopros o suwestvovanyy reßenyq zadaçy (1.1) – (1.3) predstavlqet samostoq-
tel\n¥j ynteres y budet rassmotren v otdel\noj rabote.
PreΩde çem perejty k toçn¥m formulyrovkam osnovn¥x rezul\tatov, vve-
dem nekotor¥e oboznaçenyq. Pust\ P y ϕ — obratn¥e sootvetstvenno k
V s sm( ) + − +λ λ2 1
y s a sH m q m+ − − + −( )[ ] /λ ν λ2 1 1
( )
( )
funkcyy. Zdes\ H = ( λ +
+ 1 ) ( ν q – 1 ) – q ( m + λ – 2 ). Osnovn¥my rezul\tatamy rabot¥ qvlqgtsq sledu-
gwye teorem¥.
Teorema 2.1. Pust\ u ( x, t ) — reßenye zadaçy (1.1) – (1.3) v Q∞ , suppu0 ⊂
⊂ Ω( )ρ0 , ρ0 < ∞, Ω ∈ B2 ( g ) y v¥polnen¥ uslovyq (1.4) s m + λ – 2 > 0 y (2.6).
Tohda spravedlyv¥ ocenky
E ( t ) ≡
Ω
∫ ⋅u t dx( , ) ≤ cV t
t
a t t
q q
ϕ ϕ
ϕ
ν
( )
( )
( )
( )
( )
( )
/ −1 1
, (2.8)
E ( t ) ≤ c
a P
V P P
d
t
q q
q
1
1
1 1
∫
( )
( )
−
− −/
( )
( ) ( )
( )
τ
τ τ
τν
ν
, (2.9)
u t( , ) ,⋅ ∞ Ω ≤ c
t
t
m
ϕ λ λ
( )
( )+ + −
/1 1 2
(2.10)
dlq vsex t > t0 = t u m
0 0 0 1
2ρ λ, ,Ω
+ −( ) .
Teorema 2.2. Pust\ u ( x, t ) — reßenye zadaçy (1.1) – (1.3) v Ω
h × ( 0, ∞ ),
a ( x ) ≡ x1
α
, 0 ≤ α < q, suppu0 ⊂ Ωh( )ρ0 , y v¥polnen¥ uslovyq (1.4) s m + λ –
– 2 > 0,
q > q
* =
N m
N
h
h
( )+ − + + +
+
λ λ α
ν
1 1
1
, (2.11)
hde Nh = ( N – 1 ) h + 1. Tohda dlq dostatoçno bol\ßyx t > t1 = t u1 0 0 1ρ , ,Ω( )
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
NAÇAL|NO-KRAEVÁE ZADAÇY DLQ KVAZYLYNEJNÁX VÁROÛDAGWYXSQ … 275
E ( t ) ≥ c > 0. (2.12)
Teorema 2.3. Pust\ u ( x, t ) — reßenye zadaçy (1.1) – (1.3) v Q∞ , Ω ∈ B2 ( g )
y v¥polnen¥ uslovyq (1.4) y (2.6) s ν = σ = ( m + λ – 1 ) / λ. Tohda spravedlyva
ocenka
E ( t ) ≤ c u dx
V t
t a tt
q q q
Ω Ω\ ( )
( ) ( )
( )
( ) ( )ρ
λ λ λ
ρ
ρ ρ( )
− −∫ + ( )
( )
/ /0 , (2.13)
hde ρ ( t ) dlq vsex t > 0 opredelqetsq sledugwym obrazom:
ρ ρλ λ λ λ λq m q ma( ) ( )( ) ( )+ − + + −( ) + −/2 1 2 = t q m( )( )− + − /λ λ λ1
. (2.14)
Pryvedem prymer¥. Pust\ Ω = Ω
h
, α ≡ x1
α
dlq x ≥ 1 y a ≡ 1 dlq 0 < x <
< 1, 0 ≤ α < q. Tohda yz (2.8) poluçaem
E ( t ) ≤ c t
–
Λ
, (2.15)
hde Λ =
( )( )*q q N
H
h− +ν 1
1
, H1 = H + α ( m + λ – 2 ); q
*
opredeleno v (2.11). Yz
(2.15) vydno, çto esly q < q
*
, to E ( t ) → 0, t → ∞ . V tex Ωe predpoloΩenyqx
pry q = q
*
yz (2.9) sleduet ocenka
E ( t ) ≤ c t qln ( )[ ] − −/1 1ν
. (2.16)
Dalee, ocenka (2.10) prynymaet vyd
u t( , ) ,⋅ ∞ Ω ≤ Ct q H− + + − /( )λ α1 1
. (2.17)
Yz pryvedenn¥x rezul\tatov sleduet, çto q = q
*
yhraet rol\ krytyçeskoho
pokazatelq v zadaçe (1.1) – (1.3). Kak vydno, πtot pokazatel\ zavysyt ne tol\ko
ot m, λ, ν, no y ot heometryy oblasty y funkcyy a ( x ). Zametym pry πtom, çto
esly h = 1, a ( x ) ≡ 1, to nov¥j krytyçeskyj pokazatel\ sovpadaet s poluçen-
n¥m v [7] dlq sluçaq zadaçy Koßy. Ponqtno, çto sluçaj zadaçy Koßy ne ys-
klgçaetsq yz dannoho yssledovanyq. Otmetym takΩe sledugwyj fakt: ocenky
(2.17) y (2.10) ne zavysqt ot heometryy oblasty. Toçnost\ pryvedenn¥x rezul\-
tatov podtverΩdaetsq tem, ç t o f u n k c y q u ( x, t ) =
= ( ) ( )( ) ( )t t f x t tq H q m H+ +( )− + + − − − + −( )/ /
0
1
0
11 1λ α ν λ
qvlqetsq reßenyem uravne-
nyq (1.1) s a = x α
. Pry πtom f ( r ) udovletvorqet uravnenyg
− + + − + − + −
λ α ν λ1 1
1 1
q
H
f r
q m
H
r fr( )
( )
= r
d
dr
r f f fN N m
r r
− − − − −( )( )1 1 1 1λ –
– r f r
qα ν( ) .
Dalee, teorema 2.3 spravedlyva y v nev¥roΩdennom sluçae. Naprymer, esly
m + λ – 2 = 0, to ρ ( t ) = t1 1/ +( )λ
, y esly suppu0 < ∞, to yz (2.3) sleduet
E ( t ) ≤ cV t t a tq q q1 1 1 1 1/ / / /+ − − +[ ] + − −( ) ( )[ ]( ) ( )( ) ( ) ( )λ λ λ λ λ λ
. (2.18)
Zametym, çto yz uslovyj (1.4) pry ν = ( m + λ – 1 ) / λ poluçaem λ < q < λ + 1.
Krytyçnost\ Ωe pokazatelq q v (2.18) opredelqetsq uslovyem stremlenyq k
nulg pry t → ∞ pravoj çasty v (2.18). Po-vydymomu, ocenka (2.18) qvlqetsq
novoj daΩe v sluçae λ = 1.
V zaklgçenye πtoho punkta pryvedem vspomohatel\noe utverΩdenye, qvlqg-
weesq çastn¥m sluçaem lemm¥ 3.1 [14].
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
276 A. F. TEDEEV
Lemma 2.1. Pust\ Ω ∈ B1 ( g ) y v¥polneno uslovye:
s
a s
p
( )
vozrastaet dlq
s > 0, p > 1. Tohda ymeet mesto neravenstvo
Ω
∫ ( )a r x Du dxk
p( ) ≥ c
E
G E E
p
p p
p
p
β
β β β/ − / −( )( ) ( ) (2.19)
dlq
0 < β < p, G ( s ) =
s
g s a R s
p
( ) ( )
( )
1
, Eγ =
Ω
∫ u dxγ
.
Esly sup ( )pu x ⊂ Ω( )ρ , to oçevydn¥m sledstvyem (2.19) qvlqetsq neraven-
stvo Puankare
Ω( )ρ
∫ u d xp ≤ c
a
a Du d x
p
pρ
ρ
ρ
( )
( )Ω
∫ . (2.20)
3. Dokazatel\stvo osnovn¥x rezul\tatov. Dokazatel\stvo teore-
m¥�2.1. DokaΩem snaçala, çto
Z ( t ) = inf { r > 0 : u ( x, t ) = 0, p. v. x ∈ Ω \ Ω ( r ) } ≤ c φ ( t ). (3.1)
Rassmotrym posledovatel\nost\ rn = 2 ρ ( 1 – 2
–
n
–
1
), n = 0, 1, … , ρ > 2ρ0 .
Pust\ ζn kr x( )( ) — posledovatel\nost\ hladkyx funkcyj, udovletvorqgwyx
uslovyqm: ζn = 0 pry x ∈ Ω ( rn ), ζn ≡ 1 pry x ∈ Ω \ Ω rn( ) , hde rn =
r rn n+ +1
2
, y
| D ζn | ≤ c 2
n
ρ
–
1
. Tohda, umnoΩaq obe çasty uravnenyq (1.1) na ζλ θ
n u+1
, θ > 0, y
yntehryruq po Qt , poluçaem
yn + 1 ≡ sup
0
1
< <
+∫
τ
θ
t Un
u d x +
0
2 1
t
U
m
n
u Du d x d∫ ∫ + − +θ λ τ +
+
0
t
U
q
n
u Du d x d∫ ∫ θ ν τ ≤ c u d x d
n t
U U
m
n n
2 1
1
0
1
( )
\
λ
λ
λ θ
ρ
τ
+
+
+ + −∫ ∫ , (3.2)
hde Un = Ω \ Ω ( rn ), Un = Ω \ Ω rn( ) . Toçno tak Ωe, kak v rabote [9], dokaz¥vaet-
sq neravenstvo
yn + 1 ≤ c t y f t y
n
K
n
m K m2 1
1
1 1 1 2 1
0 0
2 11 1
( )
( )( ) ( )( ) ( ) ( )
λ
λ
θ λ λ λ λ θ
ρ
θ θ
+
+
+ + + + − + + − +/ / /+ + ( ), (3.3)
hde
f s0( ) = F
s
sN K
m
1
1 1
2 11
1( ) ( )
( ) ( )
− +
+ − +
/ /
+θ
λ θ
θ
,
K1 + θ = N ( m + λ – 2 ) + ( 1 + θ )( λ + 1 ),
F1
1( )−
— obratnaq k F1 ( s ) = s g sλ λ β λ+ + − +/( ) ( )1 1 1
funkcyq. Perepyßem (3.3) v
vyde
y
A
n
a
+1 ≤ c n2 1( )λ+
, (3.4)
hde
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
NAÇAL|NO-KRAEVÁE ZADAÇY DLQ KVAZYLYNEJNÁX VÁROÛDAGWYXSQ … 277
a = 1
2 1
1
1
+ + − +
+
−
( )( )m
K
λ λ
θ
,
A = t f t yK m a( )( ) ( ) ( )1 1 1
0 0
2 11+ + − − + − + −/ /+ ( )[ ]θ λ λ λ θθ ρ .
Nam potrebuetsq ewe odno rekurrentnoe neravenstvo. Poskol\ku yzvestno [9],
çto
Z ( t ) ≤ c P t u m
0 1
2
0,Ω
+ −( ) +[ ]λ ρ , (3.5)
to, yspol\zuq neravenstvo Puankare (2.20) s p = q, poluçaem
0
t
U
q
n
w d x d∫ ∫ τ ≤ c a a Dw d x dq
t
U
q
n
ρ ρ τ( )( )− ∫ ∫1
0
≤
≤ c
Z t
a Z t
y
q
n
( )
( )( )
, w = u q q( )ν θ+ /
. (3.6)
Takym obrazom, prymenqq k pravoj çasty (3.2) neravenstvo Hel\dera, v sylu (3.6)
naxodym
yn + 1 ≤ c u d x d
n t
U
m
n
2 1
1
0
1
( )λ
λ
λ θ
ρ
τ
+
+
+ + −∫ ∫ ≤
≤ c tV
Z t
a Z t
y
n
q m q
q m q
n
m q2 1
1
1
1
1
( )
( ) ( )
( ) ( )
( ) ( )( )
( )
( )
λ
λ
ν λ ν θ
λ θ ν θ
λ θ ν θ
ρ
ρ
+
+
− + −( ) +
+ + − +
+ + − +[ ]
( )
/
/
/
.
(3.7)
Perepyßem neravenstvo (3.7) v vyde
y
B
n
b
+1 ≤ c n2 1( )λ+ yn , (3.8)
hde b =
ν θ
λ θ
q
m
+
+ + − 1
> 1, poskol\ku
ν q > m + λ – 1, B = tV
Z t
a Z t
q m q
q b
( )
( )
( )
( ) ( )ρ ρν λ ν θ λ[ ]
( )
− + −( ) + − −/1 1
.
Obæedynqq teper\ neravenstva (3.4) y (3.8), s uçetom neravenstva Gnha
poluçaem
y
A B
n+
−
1
1
1
1 1
ε
ε ε ≤ c
y
A
y
B
n
a
n
b
+ ++
1 1 ≤ c n2 1( )λ+ yn ,
hde ε1 =
b
b a+ −1
< 1. Sledovatel\no, v sylu yteratyvnoj lemm¥ 5.6 [15, c. 113]
zaklgçaem, çto yn → 0, n → ∞,
A y B a b( )( )
0
1− / ≤ c0 . (3.9)
Dlq ocenky y0 rassmotrym posledovatel\nost\
y
(
n
) = sup ( , )
( )0
1
< < >
+∫ ⋅
τ ρ
θτ
t r xk n
u d x +
0
2 1
t
r x
m
k n
u Du d x d∫ ∫
>
+ − +
( ) ρ
θ λ τ +
+
0
t
r x
q
k n
u Du d x d∫ ∫
>( ) ρ
θ ν τ , ρn =
ρ( )1 2
2
+ n
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
278 A. F. TEDEEV
RassuΩdaq toçno tak Ωe, kak pry dokazatel\stve (3.8), ymeem y n b( )( ) ≤
≤ c n2 1( )λ+ y
(
n
+
1
)B. Yteryruq poslednee neravenstvo, poluçaem ocenku
y0 = y
(
0
) ≤
≤ ctV
Z t
a Z t
q m q m
q q m( )
( )
( )
( ) ( )
( )( ) ( )ρ ρ
λ θ ν λ
λ ν θ ν λ
( )
+ + − − + −( )
− + + − + −( )
/
/
1 1
1 1
. (3.10)
Podstavlqq (3.10) v (3.9) y uçyt¥vaq oçevydnoe neravenstvo Z ( t ) ≤ 2 ρ, pryxo-
dym k v¥vodu, çto u ≡ 0, esly ρ
H
a ( ρ )
m
+
λ
–
2 ≥ c t q m
1
1ν λ− + −( )
. ∏to y dokaz¥vaet
ocenku (3.1).
Dalee, umnoΩym obe çasty (1.1) na u
θ
y proyntehryruem rezul\tat po Ω ( ρ )
s ρ = Z ( t ). V rezul\tate poluçym
1
1
1
θ τ
ρ
θ
+ ∫ +d
d
u d x
Ω( )
= – θ
ρ
θ λ
Ω( )
∫ + − +u Du d xm 2 1 –
– c a Du d xq q q
Ω( )
( )
ρ
ν θ∫ + / ≤ – c a Du d xq q q
Ω( )
( )
ρ
ν θ∫ + /
. (3.11)
Prymenqq teper\ neravenstva Hel\dera y Puankare, poluçaem
Ω( )ρ
θ∫ +u d x1 ≤ V u d xq q q
q
( )( ) ( )
( )
( ) ( )
ρ ν ν θ
ρ
ν θ
θ ν θ
− + +
+ +
/
/
∫
1
1
Ω
≤
≤ cV a a Du d xq q q q q q q
q
( ) ( )( ) ( ) ( ) ( )
( )
( )
( ) ( )
ρ ρ ρν ν θ θ ν θ
ρ
ν θ
θ ν θ
− + − + + +
+ +
/ / /
/
[ ]
∫1 1 1
1
Ω
.
Sledovatel\no, yz (3.11) ymeem
d
d
u d x
τ
ρ
θ
Ω( )
∫ +1 ≤ −
− − + − +
+ +
/
/
∫cV a u d xq q
q
( ) ( )( ) ( )
( )
( ) ( )
ρ ρ ρν θ
ρ
θ
ν θ θ
1 1 1
1
Ω
.
Yntehryruq πto neravenstvo, lehko naxodym
Ω( )ρ
θ∫ +u d x1 ≤ cV a tq q
( ) ( )
( ) ( )
ρ ρ ρ
θ ν− − + −[ ] /1 1 1 1
.
Nakonec, prymenqq neravenstvo Hel\dera, s uçetom pred¥duweho neravenstva
poluçaem
Ω Z t
u t d x
( )
( , )
( )
∫ ⋅ ≤
Ω Z t
u d x V Z t
( )
( )
( )( )
( )
+
+
+∫
( )
/
/1
1 1
1θ
θ
θ θ ≤
≤ V Z t Z t a Z t t V Z tq q
( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( )( )[ ] ( )− − + − +
+/ / /1 1 1 1 1 1
1θ ν θ
θ θ ≤
≤ cV t t a t tq q
ϕ ϕ ϕ
ν
( ) ( ) ( )
( )( ) ( )[ ]− − −/1 1 1 1
.
DokaΩem ocenku (2.9). Yntehryruq uravnenye (1.1) po Ω ( ρ ), ρ = Z ( t ),
poluçaem
dE t
dt
( )
= – D ( t ), (3.12)
hde
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E ( t ) =
Ω( )
( , )
ρ
∫ ⋅u t d x , D ( t ) =
Ω( )ρ
ν∫ a Du d x
q
.
Prymenqq neravenstva Hel\dera y Puankare, ymeem
E ( t ) ≤
Ω( )
( )
( ) ( )( )
ρ
ν
ν
ν νρ∫
/
/−u d x Vq
q
q q
1
1 ≤ c
a
D t V
q q
q qρ
ρ
ρ
ν
ν ν
( )
( ) ( )
( )
( ) ( )
/
/−
1
1
. (3.13)
Poskol\ku yz (3.5) pry dostatoçno bol\ßyx t > t0 sleduet, çto Z ( t ) ≤ c P ( t ), yz
neravenstv (3.12) y (3.13) v¥tekaet
dE t
dt
( )
≤ – c
a P t
V P t P t
E tq q
q( )
( ) ( )
( )
( )
( )
−ν
ν
1 . (3.14)
Sledovatel\no, yntehryruq (3.14) ot t0 do t, poluçaem
E ( t ) ≤ c
a P
V P P
d
t
t
q q
q
0
1
1 1
∫ ( )
( )
−
− −/
( )
( ) ( )
( )
τ
τ τ
τν
ν
.
Teorema 2.1 dokazana.
Dokazatel\stvo teorem¥ 2.2. Dlq lgb¥x 0 < t1 < t ymeem
E ( t1 ) = E ( t ) +
t
t
q
x Du d x d
1
1∫ ∫
Ω
α ν τ . (3.15)
Prymenqq neravenstvo Hel\dera, poluçaem
t
t
q
x Du d x d
1
1∫ ∫
Ω
α ν τ ≤
t
t
m
q
u Du d x d
1
2 1
1
∫ ∫ + − +
+
/
Ω
θ λ
λ
τ
( )
×
×
t
t
q q m q
q
x u d x d
1
1
1 1 1 1 1
1 1
∫ ∫ + + − + − + − −( ) + −
+ − +
/ /
/
Ω
α λ λ λ ν λ θ λ
λ λ
τ( ) ( ) ( ) ( ) ( )
( ) ( )
≡
≡ J Jq q
1
1
2
1 1/ /+ + − +( ) ( ) ( )λ λ λ
, (3.16)
hde θ > 0 — dostatoçno maloe çyslo. UmnoΩaq obe çasty (1.1) na u
θ
y ynteh-
ryruq po Ω × ( t1 , t ), ymeem
J1 ≤ c
Ω
∫ + ⋅u t d x1
1
θ( , ) . (3.17)
Dalee nam potrebuetsq sledugwaq ocenka dlq Ω ∈ B1 ( g ) [8, 9]:
u t( , ) ,⋅ ∞ Ω ≤ c
t E d
t t E d
t
t
t
t m
−
−
+ −
/
/
∫
∫
1
2
1
2
2
( )
( )
τ τ
ψ τ τ
λ , (3.18)
hde ψ — obratnaq k Ψ ( z ) = z z g zm+ − +/( )λ λ2 1( ) funkcyq.
V çastnosty, esly Ω = Ω
h
, to v sylu (2.4) dlq z > 1 ψ ( z ) =
= cz m N Nh h( )+ − + +( ) /λ λ2 1
. Sledovatel\no, s uçetom toho, çto E ( t ) ≤ u0 1,Ω ∀ t >
> 0, yz (3.18) dlq t > 1 poluçaem
u t( , ) ,⋅ ∞ Ω ≤ c u tK N Kh h h
0 1
1
,Ω
λ+( ) −/ /
, (3.19)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
280 A. F. TEDEEV
hde Kh = ( m + λ – 2 ) Nh + λ + 1. Tohda
Ω
∫ + ⋅u t d x1 θ( , ) ≤ E t u t( ) ( , ) ,1 1⋅ ∞ Ω
θ ≤ cE t u tK N Kh h h( ) ,1 0 1
1
Ω
λ θ θ+( ) −/ /
. (3.20)
Otmetym ewe, çto dlq Ω = Ω
h
Z t( ) ≤ c u tm K Kh hρ λ
0 0 1
2 1+( )+ − / /
,
( )
Ω ≤
≤ 2 0 1
2 1c u tm K Kh h
,
( )
Ω
+ − / /λ ≡ ˜ ( )Z t dlq t > t0 = ρ λ
0 0 1
2K mh u ,Ω
+ −
. Sledovatel\no, dlq
t0 < t1 < t ymeem
J2 ≤ c E Z u d
t
t
q H q q
1
1 1 1∫ + + −
∞
− + −/ /⋅( ) ˜( ) ( , )( ) ( )
,
( ) ( )τ τ τ τα λ λ θ λ
Ω ≤
≤ c u E t dm H q K q
t
t
HN qN q Kh h h h
0 1
2 1 1
1
1 1
1
,
( ) ( )( ) ( ) ( ) ( )( )Ω
+ − + − +( ) + −( ) − − − +( ) + −[ ]/ /∫λ α θ λ λ θ α λ λτ τ .
(3.21)
Poskol\ku q > q
*
, yntehral v pravoj çasty (3.21) sxodytsq pry θ <
<
( )( )*q q N
qN
h
h
− +ν 1
. Obæedynqq ocenky (3.15) – (3.17), (3.21), naxodym
E ( t1 ) ≤ E ( t ) + c u t E tH K q q N Kh h h*
,
( )*
( )0 1 1 1Ω
/ /− −
(3.22)
dlq vsex t1 > t0 . V¥berem teper\ t1 :
t1 = max , *
,
( )( )*
t c u H K K q q N
h h h
0 0 1
1
2 1
Ω
/ /( )
− +[ ]ν
.
Tohda yz (3.22) lehko v¥vodym ocenku 2 E ( t ) ≥ E ( t1 ) dlq vsex t > t1 . Otmetym
takΩe [7] (lemma 4.1), çto sluçaj u ( x, t1 ) ≡ 0 dlq vsex x ∈ R
N
y nekotoroho
t1 > 0 ne ymeet mesta.
Teorema 2.2 dokazana.
Dokazatel\stvo teorem¥ 2.3. Ymeem
E ( t ) =
Ω
∫ ⋅u t d x( , ) =
Ω( )
( , )
ρ
∫ ⋅u t d x +
Ω Ω\ ( )
( , )
ρ
∫ ⋅u t d x ≡ I1 + I2 . (3.23)
Sohlasno neravenstvu Hel\dera
I1 ≤
Ω( )
( )
( ) ( )( )
ρ
ν
ν
ν νρ∫
/
/−u d x Vq
q
q q
1
1
. (3.24)
Pust\ u
ν = v. Tohda, prymenqq lemmu 2.1 s β = ν
–
1 < q, p = q, poluçaem
D ( t ) ≡
Ω
∫ a D d xqv ≥ c
F t
G E t F t
q
q q
q
q
( )
( ) ( )( ) ( )ν ν ν/ /− − −( )1 1 1 , (3.25)
hde
Fq ( t ) =
Ω
∫ ⋅v( , )t d xq
, G ( s ) =
s
g s
a R s
q
( )
( )
( )( )−1
.
Napomnym, çto
s
g s( )
∼ R ( s ) dlq Ω ∈ B2 ( g ). Yz (3.25) sleduet, çto
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NAÇAL|NO-KRAEVÁE ZADAÇY DLQ KVAZYLYNEJNÁX VÁROÛDAGWYXSQ … 281
Fq ( t ) ≤ c E t G
D t
E t
q
q
q
( )
( )
( )
( )ν
ν
ν
1
1
1
−
−
, (3.26)
hde G1 ( s ) =
s
G s
qν −
−
1
1( )
, G1
1( )−
— obratnaq k G1 funkcyq. Dalee, yntehryruq
(1.1) po Ω, ymeem
d
dt
E t( ) = – D ( t ). Takym obrazom, yz (3.23), (3.24), (3.26) polu-
çaem neravenstvo
E ( τ ) ≤ c E V G
d
d
E Eq q q
q q
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( )
τ ρ
τ
τ τν ν ν
ν ν
− − −
−
/
/
−
{ }1
1
1
1
+ I2 ( τ ).
(3.27)
Dlq ocenky I2 ( τ ) postupym sledugwym obrazom. Pust\ ζ ( x ) = ζ ( rk ( x ) ) —
hladkaq funkcyq, ravnaq 1 pry rk ( x ) > ρ, nulg pry rk ( x ) < ρ / 2, 0 ≤ ζ ≤ 1, 0 <
< rk < ∞. Krome toho, | D ζ | ≤ c / ρ. UmnoΩym obe çasty (1.1) na ζ
s
, s ≥ λ + 1, y
rezul\tat proyntehryruem po Ω. V rezul\tate poluçym
d
d
u d xs
τ
ζ τ
Ω
∫ ⋅( , ) +
Ω
∫ a Du d x
q sν ζ ≤
c
Du d xs q
ρ
ζ
ρ ρ
ν
Ω Ω( ) \ \ 2
1
( )
−∫ ,
ν =
m + −λ
λ
1
.
Prymenqq k pravoj çasty neravenstvo Gnha, poluçaem
d
d
u x d xs
τ
ζ τ
Ω
∫ ( , ) +
Ω
∫ a Du d x
q sν ζ ≤
≤ c
q
a Du d xq qλ ε λ ν/ ∫
Ω
+ c
q
q
a r x d x
q
q q k
q− ( )
− −
−
( )
− −
/
/
/∫λ ε
ρ
λ λ
λ
ρ ρ
λ λ
( )
( )
( )
( )
\ \
( )
Ω Ω 2
.
Polahaq c
q
qλ ε λ/ = 1 / 2, yz posledneho neravenstva ymeem
d
d
u d xs
τ
ζ τ
Ω
∫ ⋅( , ) ≤ c
V
aq q q
( )
( )( ) ( )
ρ
ρ ρλ λ λ/ /− − .
Nakonec, yntehryruq πto neravenstvo, naxodym
Ω
∫ ζ τsu x d x( , ) ≤
Ω
∫ ζsu d x0 + c
V
aq q q
τ ρ
ρ ρλ λ λ
( )
( )( ) ( )/ /− − ≡
≡ ˜ ( , )F ρ τ ≤ ˜ ( , )F tρ , t1 < τ < t. (3.28)
Dalee, otmetym, çto G1 ( s ) = s s g sN N qε0 1 1( ) ( )− −/[ ] , ε0 =
N q q
N
( )ν − +1
. Znaçyt,
G s
s
1
0
( )
ε vozrastaet. ∏to oznaçaet, çto funkcyq s G s q q q
1
1 1( ) ( ) ( )
( )− − −( ) /ν ν ν
=
= s G s sqN q q q q/ / /− − − −( )[ ]ε ν ν ν ν ε0 0
1
1 1 1( ) ( ) ( ) ( )( ) takΩe vozrastaet. Takym obrazom, yz
(3.27) y (3.28) sleduet
dy
dτ
≤ −
/ −
−c E t G
y
E t
Vq
q q
( )
( )
( )
( )
( )
1 1
1
1
1ν
ν ντ ρ ≤
≤ − ( )− − − −/ /c E t y G Vq q q q q( ) ( ) ( )( ) ( )
1
1 1
1
10 0ν ν ε ν ν ε ντ ρ , (3.29)
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282 A. F. TEDEEV
hde y ( τ ) = E ( τ ) – ˜ ( , )F tρ .
Ytak, pry t1 = σ t, σ ∈ (0, 1), yntehryruq (3.29) ot σ t do t, poluçaem
E ( t ) ≤ ≤ c E t tq q q q q q( ) ( ) ( ) ( ) ( )σ σδ ν ν ε ν ν ν ε ν− − − + − − − +/ /1 1 1 10 0 ×
× G V
q q q
1
1 1 10( )
( ) ( )
ρ
ν ν ε ν− − − +( ) /
+ ˜ ( , )F tρ , (3.30)
hde δ =
ν
ν ν
q
q N q
2
2 21+ −( )
< 1. Nakonec, yteryruq (3.30) po σ, ymeem
E ( t ) ≤ ct G Vq q− − − − −/ /( )1 1
1
1 1 1( ) ( )
( )ν ν
ρ + c ˜ ( , )F tρ .
Zameçaq teper\, çto G V1
1( )ρ −( ) ∼ V aq q q( ) ( )( ) ( )ρ ρ ρν ν/ /− − −1 1 1
y naxodq ρ yz
uslovyq
ρ ρλ λ λ λ λ( ) ( )( ) ( )( )t a tq m q m+ − + + −( ) + −/ ( )2 1 2 = t q m( )( )− + − /λ λ λ1
,
pryxodym k trebuemomu utverΩdenyg.
Teorema 2.3 dokazana.
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term // SIAM J. Math. Anal. – 1989. – 20. – P. 886 – 907.
2. Souplet Ph. Resent results and open problems on parabolic equations with gradient nonlinearities //
J. Different. Equat. – 2001. – # 20. – P. 1 – 19.
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2002. – Preprint.
4. Ben-Arti M., Souplet Ph., Wessler F. B. The local theory for viscous Hamilton – Jacobi equations
in Lebesgus spaces // J. Math. Pures and Appl. – 2002. – 81. – P. 343 – 378.
5. Souplet Ph. Gradient blow-up for multidimensional nonlinear parabolic equations with general
boundary conditions. – 2002. – Preprint.
6. Benachour S., Laurencot Ph., Schmidt D., Souplet Ph. Extinction and non-extincion for viscous
Hamilton – Jacobi equations in R
N. – 2002. – Preprint.
7. Andreucci D., Tedeev A. F., Ughi M. The Cauchy problem for degenerate parabolic eqautions with
source and damping // Ukr. Math. Bull. – 2004. – 1, # 1. – P. 1 – 23.
8. Tedeev A. F. Ocenky skorosty stabylyzacyy pry t → ∞ reßenyq vtoroj smeßannoj zadaçy
dlq kvazylynejnoho parabolyçeskoho uravnenyq vtoroho porqdka // Dyfferenc. uravnenyq.
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narrowing domains // Proc. Poy. Soc. Edinburgh A. – 1998. – 128, # 6. – P. 1163 – 1180.
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12. Huwyn A. K. Ob ocenkax reßenyj kraev¥x zadaç dlq parabolyçeskoho uravnenyq vtoroho
porqdka // Tr. Mat. yn-ta AN SSSR. – 1973. – 126. – S. 5 – 45.
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Neumann problem for a quasilinear second order degenerate parabolic equations in domains with
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nyq parabolyçeskoho typa. – M.: Nauka, 1967. – 736 s.
Poluçeno 10.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
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| id | umjimathkievua-article-3451 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:42:48Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/db/44664b316dfeabd105dc69cb3a5c66db.pdf |
| spelling | umjimathkievua-article-34512020-03-18T19:54:47Z Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem Начально-краевые задачи для квазилинейных вырождающихся параболических уравнений с демпфированием. Задача Неймана Tedeev, A. F. Тедеев, А. Ф. Тедеев, А. Ф. We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined. Досліджується поведінка тотальної маси розв'язку задачі Неймана для широкого класу вироджених параболічних рівнянь з демпфіруванням у просторах із некомпактною межею. Знайдено нові критичні показники в досліджуваній задачі. Institute of Mathematics, NAS of Ukraine 2006-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3451 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 2 (2006); 272–282 Український математичний журнал; Том 58 № 2 (2006); 272–282 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3451/3640 https://umj.imath.kiev.ua/index.php/umj/article/view/3451/3641 Copyright (c) 2006 Tedeev A. F. |
| spellingShingle | Tedeev, A. F. Тедеев, А. Ф. Тедеев, А. Ф. Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem |
| title | Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem |
| title_alt | Начально-краевые задачи для квазилинейных вырождающихся параболических уравнений с демпфированием. Задача Неймана |
| title_full | Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem |
| title_fullStr | Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem |
| title_full_unstemmed | Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem |
| title_short | Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem |
| title_sort | initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. neumann problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3451 |
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