Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables
We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the existence and uniqueness of t...
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| Date: | 2007 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2007
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3408 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509493344862208 |
|---|---|
| author | Lavrenyuk, S. P. Pukach, P. Ya. Лавренюк, С. П. Пукач, П. Я. |
| author_facet | Lavrenyuk, S. P. Pukach, P. Ya. Лавренюк, С. П. Пукач, П. Я. |
| author_sort | Lavrenyuk, S. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:28Z |
| description | We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain
unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the
existence and uniqueness of the solution of this problem independent of the behavior of solution as $|x| \rightarrow +\infty$.
The indicated classes of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the domain in no way limits the order of nonlinearity. |
| first_indexed | 2026-03-24T02:41:59Z |
| format | Article |
| fulltext |
UDK 517.95
S. P. Lavrengk (L\viv. nac. un-t, Ûeßiv. un-t, Pol\wa),
P. Q. Pukaç (Nac. un-t „L\viv. politexnika”)
MIÍANA ZADAÇA DLQ NELINIJNOHO
HIPERBOLIÇNOHO RIVNQNNQ V NEOBMEÛENIJ
ZA PROSTOROVYMY ZMINNYMY OBLASTI
We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with
power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an
arbitrary number of space variables. We obtain conditions for the existence and uniqueness of the
solution of this problem independent of the behavior of solution as x → +∞. The indicated classes
of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the
domain in no way limits the order of nonlinearity.
Yssledovana pervaq smeßannaq zadaça dlq kvazylynejnoho hyperbolyçeskoho uravnenyq vto-
roho porqdka so stepennoj nelynejnost\g v neohranyçennoj po prostranstvenn¥m peremenn¥m
oblasty. Rassmotren sluçaj proyzvol\noho kolyçestva prostranstvenn¥x peremenn¥x. Polu-
çen¥ uslovyq suwestvovanyq y edynstvennosty reßenyq πtoj zadaçy nezavysymo ot povedenyq
reßenyq pry x → +∞. Ukazann¥e klass¥ suwestvovanyq y edynstvennosty qvlqgtsq
prostranstvamy lokal\no yntehryruem¥x funkcyj, pryçem razmernost\ oblasty nykak ne ohra-
nyçyvaet stepen\ nelynejnosty.
Vidomo [1, 2], wo dlq linijnyx rivnqn\ hiperboliçnoho typu rozv’qzok zaleΩyt\
vid poçatkovyx umov i pravo] çastyny lyße v obmeΩenij oblasti – vseredyni xa-
rakterystyçno] poverxni. Krim toho, na samij xarakterystyçnij poverxni vyko-
nugt\sq spivvidnoßennq dlq intehrala enerhi] [2]. Cej fakt budemo vykorysto-
vuvaty pry doslidΩenni rozv’qzku mißano] zadaçi v neobmeΩenij oblasti dlq
pevnoho klasu nelinijnyx hiperboliçnyx rivnqn\ druhoho porqdku.
Zadaçi v neobmeΩenyx oblastqx dlq nelinijnyx hiperboliçnyx rivnqn\
vyhlqdu utt – ∆u + A u γ = f, γ > 0, vyvçaly u bahat\ox robotax [3 – 8]. Praci
[9, 10] prysvqçeno doslidΩenng perßo] mißano] zadaçi dlq slabko nelinijnoho
hiperboliçnoho rivnqnnq ta systemy takyx rivnqn\ druhoho porqdku, wo uzaha-
l\nggt\ rivnqnnq vyhlqdu utt – ∆u + ut
ρ = f v neobmeΩenij za prostorovymy
zminnymy oblasti. Pry c\omu peredbaçalosq, wo stepin\ nelinijnosti ρ > 1
sutt[vo zaleΩyt\ vid rozmirnosti oblasti, u qkij doslidΩugt\ zadaçu. U cij
statti umovy na stepin\ nelinijnosti ρ > 0 Ωodnym çynom ne zaleΩat\ vid roz-
mirnosti oblasti. Zaznaçymo takoΩ, wo v [11] vyvçeno mißanu zadaçu dlq slab-
ko nelinijno] systemy hiperboliçnyx rivnqn\ perßoho porqdku z dvoma nezaleΩ-
nymy zminnymy.
V oblasti Q = Ω × (0; T), Ω ⊂ Rn , rozhlqnemo zadaçu
u a x u u u f x ttt
i j
n
ij x x t
p
ti j
− ( ) + =
=
−∑
,
( ) ( , )
1
2
, p > 1, (1)
u u xt = =0 0( ), (2)
u u xt = =0 1( ) , (3)
u S = 0 , (4)
de S = ∂ Ω × (0; T) — biçna poverxnq oblasti Q, 0 < T < + ∞. Prypustymo, wo
Ω — neobmeΩena oblast\ z meΩeg ∂ Ω klasu C1
, ΩR = Ω ∩ x x Rn∈ <{ }R :
— zv’qzna mnoΩyna dlq dovil\noho R > 1 z rehulqrnog za Kal\deronom [12,
c. 45] meΩeg ∂ ΩR. ZauvaΩymo, zokrema, wo opukla oblast\ Ω zadovol\nq[ usi
zaznaçeni umovy [12, c. 46] (zauvaΩennq 1.11). Bez obmeΩennq zahal\nosti pry-
© S. P. LAVRENGK, P. Q. PUKAÇ, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11 1523
1524 S. P. LAVRENGK, P. Q. PUKAÇ
puska[mo, wo O ∈ Ω, de O — poçatok koordynat. Poznaçymo dali QR,τ = ΩR ×
× (0; τ), Qτ = Ω × (0; τ), ∂ ΩR = H H
1 2
R R∪ ,
H
1
R = ∂Ω Ω∩ R ,
H
2
R = ∂ ΩR \
H
1
R
, SR,τ =
= ∂ ΩR × (0; τ), SR = ∂ ΩR × (0; T) dlq dovil\nyx R > 1, τ ∈ (0; T ] .
Budemo vykorystovuvaty taki prostory funkcij:
H R R0
1
,
( )
H
1
Ω =
u u H uR R: ( ),∈ ={ }1 0Ω
H
1
,
H0
1
, ( )loc Ω =
u u H RR R: ( )
,
∈ ∀ >{ }0
1 1
H
1
Ω ,
H0
2
, ( )loc Ω =
u u H H RR RR: ( ) ( )
,
∈ ∀ >{ }2
0
1 1Ω Ω∩
H
1
,
Lr
loc( )Ω = u u L Rr
R: ( )∈ ∀ >{ }Ω 1 ,
L Qr
loc( ) = u u L Q Rr
R T: ( ),∈ ∀ >{ }1 , r ∈ (1, + ∞) .
Oznaçennq. Rozv’qzkom zadaçi (1) – (4) nazyva[mo funkcig u , wo zado-
vol\nq[ vklgçennq
u C T H∈ [ ]( )0 0
1; ; ( ), loc Ω ,
u C T L L Q L T Ht
p∈ [ ]( ) ( )0 02 2
0
1; ; ( ) ( ) ( ; ); ( ),loc loc locΩ Ω∩ ∩ , (5)
u L T Ltt ∈ ( )∞ ( ; ); ( )0 2
loc Ω ,
umovy (2) – (4) i rivnqnnq (1) v sensi rozpodiliv.
Poznaçymo r = min ,2 ′{ }p , de ′p = p / (p – 1), s = max ,2 2 2p −{ } .
Teorema 1. Nexaj funkci] aij naleΩat\ do prostoru C( )Ω , a xij ( ) =
= a xji( ) , pryçomu a xij i ji j
n
( )
,
ξ ξ=∑ 1
≥ a0
2ξ , a0 > 0, dlq dovil\noho vektora
ξ = ( ξ1, … , ξ n ) ∈ R
n
, dlq vsix x ∈ Ω ta dlq dovil\nyx i, j = 1, … , n; funkci]
aij , ai j x j, naleΩat\ do prostoru L∞( )Ω dlq dovil\nyx i, j = 1, … , n; f ∈
∈ L Qr
loc( ), f L Qt ∈ loc
2 ( ) ; u H0 0
2∈ , ( )loc Ω , u1 ∈ Ls
loc( )Ω ∩ H0
1
, ( )loc Ω . Todi isnu[
[dynyj rozv’qzok u zadaçi (1) – (4).
Dovedennq. Vyberemo dovil\ne fiksovane çyslo R > 1. Rozhlqnemo dopo-
miΩnu zadaçu v oblasti QR T, dlq rivnqnnq (1) z pravog çastynog f R
i umova-
my
u SR
= 0 , (6)
u u xt
R
= =0 0 ( ), u u xt t
R
= =0 1 ( ), x ∈ ΩR, (7)
de u xR
0 ( ) = u x xR0( ) ( )ζ , u xR
1 ( ) = u x xR1( ) ( )ζ , ζ R ∈ NC n2( )R , 0 ≤ ζR
( x ) ≤ 1,
ζR
( x ) = 1 pry x ≤ R – 1, ζR
( x ) = 0 pry x ≥ R, f x tR( , ) = f ( x, t ) pry ( x, t ) ∈
∈ QR T, , f x tR( , ) = 0 pry ( x, t ) ∈ Q QR T\ , .
Dlq dovedennq isnuvannq rozv’qzku dopomiΩno] zadaçi vykorysta[mo sxemu
dovedennq teoremy 6.1 [13, c. 234]. Rozhlqnemo v oblasti QR T, poslidovnist\
hal\orkins\kyx nablyΩen\ u x tN ( , ) =
k
N
k
N
kc t x=∑ 1
( ) ( )ϕ , de N = 1, 2, … , { ϕk
} —
ortonormovana v L2 ( Ω ) systema linijno nezaleΩnyx elementiv prostoru
H R
2( )Ω ∩ H R0
1( )Ω ∩ LS
R( )Ω takyx, wo linijni kombinaci] { ϕk
} [ wil\nymy v
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
MIÍANA ZADAÇA DLQ NELINIJNOHO HIPERBOLIÇNOHO … 1525
c\omu prostori. Pry c\omu funkci] ck
N
vyznaçagt\ qk rozv’qzky zadaçi Koßi
dlq systemy zvyçajnyx dyferencial\nyx rivnqn\
u a x u u u f x t dxtt
N
k ij x
N
k x t
N p
t
N
k
R
k
i j
n
i j
R
ϕ ϕ ϕ ϕ+ + −
=
−
=
∑∫ ( ) ( , ),
,
2
1
0
Ω
, (8)
c u xk
N
k
R N( ) ( ),
,0 0= , c u xkt
N
k
R N( ) ( ),
,0 1= , k = 1, … , N, (9)
u x u xR N
k
R N
k
k
N
0 0
1
,
,
,( ) ( )=
=
∑ ϕ , u x u xR N
k
R N
k
k
N
1 1
1
,
,
,( ) ( )=
=
∑ ϕ ,
u uR N R
H R
0 0 2 0,
( )
− →
Ω
, u uR N R
LS
R H R
1 1
0
1
0,
( ) ( )
− →
Ω Ω∩
, N → +∞ .
Za teoremog Karateodori [14] isnu[ neperervnyj rozv’qzok zadaçi (8), (9), wo
ma[ absolgtno neperervnu poxidnu, vyznaçenyj na deqkomu promiΩku 0 0, t[ ],
t0N≤ T. Z apriornyx ocinok, otrymanyx nyΩçe, vyplyvatyme, wo t0 = T. Krim to-
ho, za umov teoremy na funkcig f systemu (8) moΩna poçlenno dyferencigvaty
za zminnog t.
PomnoΩymo koΩne rivnqnnq (8) vidpovidno na funkci] ckt
N
, pidsumu[mo po k
vid 1 do N ta zintehru[mo po promiΩku 0, τ[ ], τN≤ T. Pislq vykonannq cyx ope-
racij oderΩymo rivnist\
u u a x u u u f x t u dx dttt
N
t
N
ij x
N
x t
N
t
N p R
t
N
i j
n
Q
i j
R
+ + −
=
=
∑∫ ( ) ( , )
,, 1
0
τ
. (10)
Oskil\ky
u u dx dttt
N
t
N
QR, τ
∫ = 1
2
2
u x dxt
N
R
( , )τ
Ω
∫ – 1
2 1
2
u x dxR N
R
, ( )
Ω
∫ ,
a x u u dx dtij x
N
x t
N
i j
n
Q
i j
R
( )
,, =
∑∫
1τ
=
= 1
2 1i j
n
ij x
N
x
Na x u x u x dx
i j
R ,
( ) ( , ) ( , )
=
∑∫ τ τ
Ω
– 1
2
0
1i j
n
ij x
R N
x
R Na u x u x dx
i j
R ,
, ,( ) ( ) ( )
=
∑∫
Ω
≥
≥
a
u x dx
R
N0 2
2 Ω
∫ ∇ ( , )τ –
a n
u x dx
R
R N1
0
2
2 Ω
∫ ∇ , ( ) ,
a a x
i j x
i j1 =
∈
max sup ( )
, Ω
,
f x t u dx dtR
Q
t
N
R
( , )
, τ
∫ ≤ 1
q
u dx dt
Q
t
N q
R, τ
∫ + 1
r
f x t dx dtR r
QR
( , )
, τ
∫ ,
de q = p pry p ≥ 2 i q = 2 pry p ∈ ( 1, 2 ) , to z (10) (vykorystovugçy lemu
Hronuolla pry p ∈ ( 1, 2 ) ) oderΩymo ocinky
u x u x dx Ct
N N
R
( , ) ( , )τ τ
2 2
1+ ∇( ) ≤∫
Ω
, τ ∈ [ 0; T ] , (11)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1526 S. P. LAVRENGK, P. Q. PUKAÇ
u x dx dt Ct
N p
QR T
( , )
,
τ ≤∫ 1 . (12)
Zaznaçymo, wo stala C1 ne zaleΩyt\ vid N.
Zdyferencig[mo systemu (8) za zminnog t, pomnoΩymo na cktt
N
, pidsumu[mo
po k vid 1 do N ta zintehru[mo rezul\tat po promiΩku [ 0; τ ] , τN≤ T. OderΩymo
rivnist\
u u a x u u p u f x t u dx dtttt
N
tt
N
ij x t
N
x tt
N
t
N p
t
R
tt
N
i j
n
Q
i j
R
+ + − −
=
=
∑∫ ( ) ( ) ( , )
,,
1 0
1τ
. (13)
Peretvorymo j ocinymo intehraly rivnosti (13):
u u dx dtttt
N
tt
N
QR, τ
∫ = 1
2
0
2 2
u x u x dxtt
N
tt
N
R
( , ) ( , )τ −( )∫
Ω
,
i j
n
ij x t
N
x tt
N
Q
a x u u dx dt
i j
R ,
( )
, =
∑∫
1τ
=
1
2 1
a x u u dx dtij x t
N
x t
N
i j
n
tQ
i j
R
( )
,
,
=
∑∫
τ
=
= 1
2
0 0
11
a x u x u x a x u x u x dxij x t
N
x t
N
ij x t
N
x t
N
i j
n
i j
n
i j i j
R
( ) ( , ) ( , ) ( ) ( , ) ( , )
,,
τ τ −
==
∑∑∫
Ω
≥
≥
a
u x dxt
N
R
0 2
2
∇∫ ( , )τ
Ω
–
a n
u x dxR N
R
1
1
2
2
∇∫ , ( )
Ω
,
f x t u dx dtt
R
Q
tt
N
R
( , )
, τ
∫ ≤ 1
2
2
u dx dttt
N
QR, τ
∫ + 1
2
2
f x t dx dtt
R
QR
( , )
, τ
∫ .
Vykorystovugçy navedeni ocinky, oderΩu[mo
u x a u x dxtt
N
t
N
R
( , ) ( , )τ τ
2
0
2
+ ∇[ ]∫
Ω
≤
≤ u dx dttt
N
QR
2
, τ
∫ + f x t dx dtt
R
QR
( , )
,
2
τ
∫ + u x dxtt
N
R
( , )0
2
Ω
∫ +
+ na u x dxR N
R
1 1
2
∇∫ , ( )
Ω
. (14)
Rozhlqnemo rivnist\ (8) pry t = 0 i pomnoΩymo ]] na cktt
N ( )0 . Pislq pidsumo-
vuvannq po k vid 1 do N oderΩymo
u x a x u x u x dxtt
N
ij x
N
x tt
N
i j
n
i j
R
( , ) ( ) ( , ) ( , )
,
0 0 0
2
1
− ( )
=
∑∫
Ω
+
+ u x u x u x f x u x dxt
N p
t
N
tt
N R
tt
N
R
( , ) ( , ) ( , ) ( , ) ( , )0 0 0 0 0 0
2−
−[ ] =∫
Ω
. (15)
Ocinymo intehraly rivnosti (15):
i j
n
ij x
N
x tt
N
R
i j
a x u x u x dx
,
( ) ( , ) ( , )
=
∑∫ ( )
1
0 0
Ω
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
MIÍANA ZADAÇA DLQ NELINIJNOHO HIPERBOLIÇNOHO … 1527
= a x u x u x dxij x x
N
i j
n
tt
N
j i
R
,
,
( ) ( , ) ( , )0 0
1=
∑∫
Ω
+ a x u x u x dxij x x
N
i j
n
tt
N
i j
R
( ) ( , ) ( , )
,
0 0
1=
∑∫
Ω
≤
≤ 1
8
0
2
u x dxtt
N
R
( , )
Ω
∫ + 2 02
2 3 2
a n u x dxN
R
∇∫ ( , )
Ω
+
+ 1
8
0
2
u x dxtt
N
R
( , )
Ω
∫ + 2 1
2 2
0
2
1
a n u x dxx x
R N
i j
n
i j
R
,
,
,
( )( )
=
∑∫
Ω
,
de a2 = max essup ( )
,
,
i j x
ij xa x
j
∈Ω
;
u x u x u x dxt
N p
t
N
tt
N
R
( , ) ( , ) ( , )0 0 0
2−
∫
Ω
≤
≤ 1
8
0 2 0
2 2 2
u x u x dxtt
N
t
N p
R
( , ) ( , )+
−
∫
Ω
,
f x u x dxR
tt
N
R
( , ) ( , )0 0
Ω
∫ ≤ 1
8
0
2
u x dxtt
N
R
( , )
Ω
∫ + 2 0
2
f x dxR
R
( , )
Ω
∫ .
Na pidstavi navedenyx ocinok z (15) oderΩu[mo nerivnist\
u x dxtt
N
R
( , )0
2
Ω
∫ ≤ 4 01
2 2
0
2 2
1
a n u x f x dxx x
R N R
i j
n
i i
R
,
,
,
( ) ( , )( ) +
=
∑∫
Ω
+
+ 4 2
2 3
1
2
1
2 2
ΩR
a n u x u x dxR N R N p
∫ ∇ +[ ]−, ,( ) ( ) . (16)
Vykorystovugçy lemu Hronuolla, z (14) ta (16) ma[mo ocinky
u x dx Ctt
N
R
( , )τ
2
2≤∫
Ω
, ∇ ≤∫ u x dx Ct
N
R
( , )τ
Ω
2
2 , τ ∈ [0; T ], (17)
de stala C2 ne zaleΩyt\ vid N. Krim toho, vykorystovugçy (12), otrymu[mo
u u dx dt Ct
N p
t
N
p
QR T
− ′
≤∫
2
3
,
, (18)
de C3 ne zaleΩyt\ vid N.
OtΩe, na pidstavi ocinok (11), (12), (17), (18) isnu[ taka pidposlidovnist\
uNk{ } poslidovnosti uN{ } , wo
u uN Rk → ∗-slabko v L T H R
∞( )( ; ); ( )0 0
1 Ω i slabko v L T H R
2
0
10( ; ); ( )Ω( ),
ut
Nk → v slabko v L T H R
2
0
10( ; ); ( )Ω( ) i slabko v L Qp
R T( ), .
ZauvaΩymo, wo v = ut
R
[15, c. 123]. Krim toho,
u ut
N
t
Rk → ∗-slabko v L T H R
∞( )( ; ); ( )0 0
1 Ω ,
u utt
N
tt
Rk → ∗-slabko v L T L R
∞( )( ; ); ( )0 2 Ω i slabko v L T L R
2 20( ; ); ( )Ω( ),
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1528 S. P. LAVRENGK, P. Q. PUKAÇ
u ut
N p
t
N Rk k
−
→
2
χ slabko v L Q
p
R T
′
( ),
pry Nk → ∞. Vykorystovugçy teoremu 5.1 [13, c. 70], bez obmeΩennq zahal\nos-
ti moΩemo vvaΩaty, wo
u ut
N
t
Rk → syl\no v L QR T
2( ),
i majΩe skriz\ u ut
N
t
Rk → v QR T, pry Nk → ∞. OtΩe, χR
t
R p
t
Ru u=
−2
.
Analohiçno do toho, qk ce zrobleno pry dovedenni teoremy 1.1 [13, c. 20],
oderΩu[mo, wo uR ∈ C T H R0 0
1; ; ( )[ ]( )Ω , u C T Lt
R
R∈ [ ]( )0 2; ; ( )Ω i funkciq uR
za-
dovol\nq[ umovyN(7). Znovu vykorystovugçy sxemu dovedennq teoremyN1.1 [13,
c. 20], z otrymanyx vywe zbiΩnostej lehko otrymu[mo, wo dlq funkci] uR
vy-
konu[t\sq totoΩnist\
u a x u u u x t dx dttt
R
ij x
R
x t
R p
t
R R
i j
n
Q
i j
R
v v v - f v+ +
=
−
=
∑∫ ( ) ( , )
,,
2
1
0
τ
(19)
dlq vsix v ∈ L T H L QR
p
R T
2
0
10( ; ); ( ) ( ),Ω( ) ∩ i vsix τ ∈ ( 0; T ] . Takym çynom, z (19)
vyplyva[, wo uR
[ rozv’qzkom rivnqnnq
u a x u u u f x ttt ij x x
i j
n
t
p
t
R
i j
− ( ) + =
=
−∑ ( ) ( , )
, 1
2
(20)
v oblasti QR T, v sensi rozpodiliv. Krim toho, funkciq uR
zadovol\nq[ umo-
vuN(6) i vklgçennq
u L T H L Qt
R
R
p
R T∈ ( )2
0
10( ; ); ( ) ( ),Ω ∩ , u L T Ltt
R
R∈ ( )∞ ( ; ); ( )0 2 Ω .
ProdovΩymo funkcig uR
z nulem na oblast\ Q QR T\ , i zbereΩemo dlq ne]
te Ω poznaçennq. Nexaj R nabuva[ znaçennq m z mnoΩyny N \ 1{ } . Todi oder-
Ωymo poslidovnist\ funkcij um{ }. Nexaj R0 > 1 — dovil\ne fiksovane çyslo.
Rozhlqnemo mnoΩynu
BR
nx t x R t T
0
1
0= ∈ < ={ }+( , ) : ,R .
Poznaçymo
BR Q0, = BR Q
0
∩ . Pobudu[mo xarakterystyçnyj konus rivnqnnq (1)
z verßynog v dovil\nij toçci ( , )x T0 mnoΩyny BR Q0, . Nexaj poverxnq
c\ohoNkonusa zada[t\sq rivnqnnqm ω( , )x t = 0 u prostori R
n +1
. Vidomo
[1, c. 125], wo funkciq ω [ rozv’qzkom dyferencial\noho rivnqnnq ωt
2
–
–N a xij x xi j
n
i j
( )
,
ω ω=∑ 1
= 0. Poznaçymo çerez K x TR0 0( , ) = ( , )x t n∈{ +
R
1
, ω( , )x t ≤
≤N 0} konus z verßynog v dovil\nij toçci ( , )x T0 , DR0
τ = ∪
( , ) ,
( , )
x T
R
R Q
K x T
0 0
0 0∈( )
B
∩
∩ Qτ , τ ∈ (0; T ] . Zaznaçymo, wo hranycq oblasti DR0
τ
sklada[t\sq z çastyny
ΩR0
0
hiperplowyny t ={ }0 , çastyny ΩR0
τ
hiperplowyny t ={ }τ , çastyny
SR0
τ
poverxni Sτ i xarakterystyçno] poverxni
R0
τ∑ rivnqnnq (1). Isnu[ take
mR0
∈ N, wo DR
T
0
⊂ Qm TR0
, . PokaΩemo, wo dlq vsix m > mR0
+ 1 majΩe skriz\
v DR
T
0
pravyl\nog [ rivnist\ u x tm( , ) = u x t
mR0
1+
( , ). Dlq c\oho zapyßemo riv-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
MIÍANA ZADAÇA DLQ NELINIJNOHO HIPERBOLIÇNOHO … 1529
nqnnq (20) dlq funkcij u
mR 0
1+
i um
, vidnimemo vid druhoho rivnqnnq perße,
po mnoΩymo riznycg na funkcig ut
m – ut
mR 0
1+
i zintehru[mo rezul\tat po
oblasti DR0
τ , τ ∈ (0; T ] . Pislq vykonannq cyx operacij otryma[mo rivnist\
u u u u a x u utt
m
tt
m
t
m
t
m
ij x
m
x
m
xi j
n
D
R R
i i
R
j
R
−
−( ) − −
+ + +
=
∑∫ 0 0 0
0
1 1 1
1
( )
,τ
u ut
m
t
mR−( )+
0
1 +
+ u u u u u u dx dtt
m p
t
m
t
m p
t
m
t
m
t
mR R R− + − + +−
−( )
=
2 1 2 1 1
0 0 0 0, (21)
oskil\ky f x tm( , ) = f x t
mR 0
1+
( , ) v oblasti DR
T
0
. Krim toho,
u x u x u xm mR
0 0
1
0
0( ) ( ) ( )= =
+
, u x u x u xm mR
1 1
1
1
0( ) ( ) ( )= =
+
, x R∈Ω
0
0
.
Peretvorymo j ocinymo dodanky rivnosti (21):
D
tt
m
tt
m
t
m
t
m
R
R Ru u u u dx dt
0
0 0
1 1
τ
∫ −( ) −( )[ + +
=
=
1
2
0
0
1 2
u u dxt
m
t
mR
R
−
+
∫
Ωτ
+ 1
2
0
0
1 2
R
Ru u t dSt
m
t
m
x t
τ
ν
∑
−∫ +
cos( , ) , –
–
D i j
n
ij x
m
x
m
x
t
m
t
m
R
i i
R
j
Ra x u u u u dx dt
0
0 0
1
1 1
τ
∫ ∑
=
+ +−
−( )
,
( )( ) =
=
1
2
0
0
01
1
1
a x u u u u dxij x
m
x
m
i j
n
x
m
x
m
i i
R
R
j j
R
( )
,
−( ) −
+
=
+
∑∫
Ωτ
+
+
1
2
0
0 01
1
1
R
i i
R
j j
R
a x u u u u t dSij x
m
x
m
i j
n
x
m
x
m
x t
τ
ν
∑
−( ) −
∫ ∑ +
=
+
( ) cos( , )
,
, –
– a x u u u u x dSij x
m
x
m
i j
n
t
m
t
m
j x ti i
R
R
R( ) cos( , )
,
,−( )
∑
−
+
=
+∑∫ 0
0
01
1
1
τ
ν ≥
≥
a
u u dxm mR
R
0 1 2
2
0
0
∇ −( )+∫
Ωτ
+
+
1
2
0
0 01
1
1
R
i i
R
j j
R
a x u u u u t dSij x
m
x
m
i j
n
x
m
x
m
x t
τ
ν
∑
−( ) −
∫ ∑ +
=
+
( ) cos( , )
,
, –
–
R
i i
R Ra x u u u u x dSij x
m
x
m
i j
n
t
m
t
m
j x t
0
0 0
1
1
1
τ
ν
∑
−
−( )∫ ∑
+
=
+
( ) cos( , )
,
, ,
ν — zovnißnq normal\ do R
T
0
∑ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1530 S. P. LAVRENGK, P. Q. PUKAÇ
u u u u u u dx dtt
m p
t
m
t
m p
t
m
D
t
m
t
mR R
R
R− + − + +−
−( ) ≥∫
2 1 2 1 1
0 0
0
0 0
τ
.
Na pidstavi otrymanyx ocinok z (21) oderΩymo nerivnist\
u u a u u dxt
m
t
m m mR R
R
− + ∇ −( )
≤
+ +∫ 0 0
0
1 2
0
1 2
0
Ωτ
, τ ∈ [0; T ], (22)
oskil\ky poverxnq
R
T
0
∑ [ xarakterystyçnog dlq rivnqnnq (1), i, otΩe, beruçy
do uvahy [2], ma[mo
R
Ru u t dSt
m
t
m
x t
0
0
1 2
τ
ν
∑
−∫ +
cos( , ) , +
+ a x u u u u t dSi j x
m
x
m
i j
n
x
m
x
m
x ti i
R
R
j j
R( ) cos( , )
,
,−
∑
−( )+
=
+∑∫ 0
0
0
1
1
1
τ
ν –
– 2 00
0
0
1
1
1
a x u u u u x dSij x
m
x
m
i j
n
t
m
t
m
j x ti i
R
R
R( ) cos( , )
,
,−( )
∑
−( ) ≥
+
=
+∑∫
τ
ν .
Todi z (22) otrymu[mo u x tm( , ) = u x t
mR0
1+
( , ), ( x, t ) ∈ DR
T
0
, dlq vsix m >
> mR 0 1+ .
Nexaj teper R0 nabuva[ znaçennq z mnoΩyny N \ 1{ } . Vyznaçymo funkcig
u takym çynom: u x t( , ) = u x tmk +1( , ) , ( , )x t Dk
T∈ , k = 2, 3, 4,N…N.
Dlq dovedennq [dynosti rozv’qzku prypuska[mo isnuvannq dvox rozv’qzkiv
u( )1
i u( )2
zadaçi (1) – (4). Todi, vybyragçy dovil\nu oblast\ DR
T
0
, oderΩu[mo
rivnist\ vyhlqdu (22), z qko] j vyplyva[, wo u x t( )( , )1 = u x t( )( , )2
v DR
T
0
. Na pid-
stavi dovil\nosti R0 ma[mo u x t( )( , )1 = u x t( )( , )2
v Q.
Teoremu dovedeno
OderΩymo teper dostatni umovy isnuvannq ta [dynosti periodyçnoho za pros-
torovymy zminnymy rozv’qzku zadaçi (1) – (4).
Poznaçymo çerez ei = ( 0, … , 0, 1, 0, … , 0) n-vymirnyj vektor, i-ta koordy-
nata qkoho dorivng[ 1, a vsi inßi [ nulqmy ( i ∈ { 1, … , n}).
Teorema 2. Nexaj vykonugt\sq umovy teoremy 1 ta isnugt\ taki çysla
T0 > 0 i k n∈ …{ }1, , , wo:
a) x T ek± ∈0 Ω dlq dovil\nyx x ∈ Ω ;
b) a x T eij
k( )+ 0 = a xij ( ) dlq vsix x ∈ Ω , i, j = 1, … , n;
v) f x T e tk( , )+ 0 = f x t( , ) dlq majΩe vsix ( , )x t Q∈ ;
h) u x T ek
0 0( )+ = u x0( ), u x T ek
1 0( )+ = u x1( ) dlq majΩe vsix x ∈ Ω.
Todi isnu[ [dynyj rozv’qzok u zadaçi (1) – (4), wo [ periodyçnog funkci[g
za zminnog xk z periodom T0.
Dovedennq. Oskil\ky vykonugt\sq umovy teoremy 1, to isnu[ [dynyj roz-
v’qzok u zadaçi (1) – (4). Funkciq u x T e tk( , )+ 0 , ( , )x t Q∈ , takoΩ [ rozv’qzkom
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
MIÍANA ZADAÇA DLQ NELINIJNOHO HIPERBOLIÇNOHO … 1531
zadaçi (1) – (4). Todi z [dynosti rozv’qzku vyplyva[, wo u x T e tk( , )+ 0 = u ( x, t )
dlq majΩe vsix ( x, t ) ∈ Q.
Teoremu dovedeno.
1. Petrovskyj Y. H. Lekcyy ob uravnenyqx s çastn¥my proyzvodn¥my. – M.: Hostexyzdat,
1950. – 303 s.
2. Hodunov S. K. Uravnenyq matematyçeskoj fyzyky. – M.: Nauka, 1971. – 416Ns.
3. Carpio A. Existence of global solutions to some nonlinear dissipative wave equations // J. math.
pures et appl. – 1994. – 73, # 5. – P. 471 – 488.
4. Rubino B. Weak solutions to quasilinear wave equations of Klein – Gordon or Sine – Gordon type
and relaxation to reaction-diffusion equations // Nonlinear Different. Equat. and Appl. – 1997. –
# 4. – P. 439 – 457.
5. Vittilaro E. Global nonexistence theorems for a class of evolution equation with dissipation //
Arch. Ration. Mech. and Anal. – 1999. – 149, # 2. – P. 155 – 182.
6. Pecher H. Sharp existence results for self-similar solutions of semilinear wave equations //
Nonlinear Different. Equat. and Appl. – 2000. – # 7. – P. 323 – 341.
7. Agre K., Rammaha M. A. Global solutions to boundary value problems for a nonlinear wave
equation in high space dimensions // Different. and Integr. Equat. – 2001. – 14. – P. 1315 – 1331.
8. Todorova G., Yordanov B. Critical exponent for a nonlinear wave equations with damping //
J. Different. Equat. – 2001. – # 174. – P. 464 – 489.
9. Pukaç P. Q. Mißana zadaça v neobmeΩenij oblasti dlq slabko nelinijnoho hiperboliçnoho
rivnqnnq zi zrostagçymy koefici[ntamy // Mat. metody ta fiz.-mex. polq. – 2004. – 47, # 4.
– S.149 – 154.
10. Pukaç P. Q. Mißana zadaça v neobmeΩenij za prostorovymy zminnymy oblasti dlq nelinij-
no] hiperboliçno] systemy druhoho porqdku // Visn. L\viv. un-tu. Ser. mex.-mat. – 2005. –
Vyp.N64. – S. 214 – 231.
11. Lavrengk S. P., Oliskevyç M. O. Metod Hal\orkina dlq hiperboliçnyx system perßoho po-
rqdku z dvoma nezaleΩnymy zminnymy // Ukr. mat. Ωurn.N– 2002. – 54, # 10. – S. 1356 – 1370.
12. Haevskyj X., Hreher K., Zaxaryas K. Nelynejn¥e operatorn¥e uravnenyq y operatorn¥e
dyfferencyal\n¥e uravnenyq. – M.: Myr, 1978. – 336 s.
13 Lyons Û.-L. Nekotor¥e metod¥ reßenyq nelynejn¥x kraev¥x zadaç. – M.: ∏dytoryal
URSS, 2002. – 587 s.
14. Koddynhton ∏.A., Levynson N. Teoryq ob¥knovenn¥x dyfferencyal\n¥x uravnenyj. – M.:
Yzd-vo ynostr. lyt., 1958. – 475 s.
15. Myxajlov V. P. Dyfferencyal\n¥e uravnenyq v çastn¥x proyzvodn¥x. – M.: Nauka, 1976.
– 392 s.
OderΩano 13.03.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
|
| id | umjimathkievua-article-3408 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:59Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/09/bb3f61d6c0b6b816a6dfa7af47759e09.pdf |
| spelling | umjimathkievua-article-34082020-03-18T19:53:28Z Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables Мішана задача для нелінійного гіперболічного рівняння в необмеженій за просторовими змінними області Lavrenyuk, S. P. Pukach, P. Ya. Лавренюк, С. П. Пукач, П. Я. We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the existence and uniqueness of the solution of this problem independent of the behavior of solution as $|x| \rightarrow +\infty$. The indicated classes of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the domain in no way limits the order of nonlinearity. We investigate the first mixed problem for a quasilinear Исследована первая смешанная задача для квазилинейного гиперболического уравнения второго порядка со степенной нелинейностью в неограниченной по пространственным переменным области. Рассмотрен случай произвольного количества пространственных переменных. Получены условия существования и единственности решения этой задачи независимо от поведения решения при $|x| \rightarrow +\infty$ Указанные классы существования и единственности являются пространствами локально интегрируемых функций, причем размерность области никак не ограничивает степень нелинейности. Institute of Mathematics, NAS of Ukraine 2007-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3408 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 11 (2007); 1523–1531 Український математичний журнал; Том 59 № 11 (2007); 1523–1531 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3408/3559 https://umj.imath.kiev.ua/index.php/umj/article/view/3408/3560 Copyright (c) 2007 Lavrenyuk S. P.; Pukach P. Ya. |
| spellingShingle | Lavrenyuk, S. P. Pukach, P. Ya. Лавренюк, С. П. Пукач, П. Я. Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| title | Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| title_alt | Мішана задача для нелінійного гіперболічного рівняння в необмеженій за просторовими змінними області |
| title_full | Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| title_fullStr | Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| title_full_unstemmed | Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| title_short | Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| title_sort | mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3408 |
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