Inverse problem for a parabolic equation with strong power degeneration
We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solu...
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509660753166336 |
|---|---|
| author | Ivanchov, N. I. Saldina, N. V. Іванчов, М. І. Салдіна, Н. В. |
| author_facet | Ivanchov, N. I. Saldina, N. V. Іванчов, М. І. Салдіна, Н. В. |
| author_sort | Ivanchov, N. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:57:29Z |
| description | We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. |
| first_indexed | 2026-03-24T02:44:38Z |
| format | Article |
| fulltext |
UDK 517.95
M. I. Ivançov, N. V. Saldina (L\viv. nac. un-t)
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ
Z SYL|NYM STEPENEVYM VYRODÛENNQM
An inverse problem of determining a time-dependent coefficient of the higher derivative in the complete
parabolic equation is considered. At initial time, this coefficient is equal to zero. Conditions for the
existence and uniqueness of the classical solution of the considered problem are established.
Rozhlqnuto obernenu zadaçu vyznaçennq zaleΩnoho vid çasu koefici[nta pry starßij poxidnij u
povnomu paraboliçnomu rivnqnni, qkyj dorivng[ nulg u poçatkovyj moment çasu. Vstanovleno
umovy isnuvannq ta [dynosti klasyçnoho rozv’qzku vkazano] zadaçi.
Cq stattq [ prodovΩennqm roboty avtoriv po vyvçenng obernenyx paraboliçnyx
rivnqn\ z vyrodΩennqm [1]. Taki zadaçi magt\ velyke praktyçne zastosuvannq v
naftodobuvnij promyslovosti, biolohi], medycyni, finansax ta inßyx, tobto v
tyx haluzqx, de z deqkyx oçevydnyx pryçyn nemoΩlyvo toçno vymirqty paramet-
ry doslidΩuvanoho procesu i de matematyçnyj aparat obernenyx zadaç di[ osob-
lyvo efektyvno. Odnym iz perßyx, xto doslidyv obernenu zadaçu vyznaçennq
koefici[nta pry starßij poxidnij v paraboliçnomu rivnqnni, buv F. DΩons [2].
Oberneni zadaçi z vyrodΩennqm po prostorovyx zminnyx rozhlqdalys\ u robotax
[3 – 5] dlq rivnqn\ hiperboliçnoho ta eliptyçnoho typiv z nevidomym vil\nym
çlenom ta molodßym koefici[ntom. Prote pytannq znaxodΩennq nevidomoho
koefici[nta, qkyj vyrodΩu[t\sq po çasovij zminnij, zalyßa[t\sq vidkrytym.
Zadaçi zi slabkym ta syl\nym vyrodΩennqmy [ sutt[vo riznymy, oskil\ky vidtvo-
rennq nevidomoho koefici[nta ta joho povedinka zaleΩat\ vid riznyx vyxidnyx
danyx. I qkwo v slabkomu vyrodΩenni, qke malo vidriznq[t\sq vid nevyrodΩeno-
ho vypadku, vplyv molodßyx çleniv ne zming[ rezul\tativ, otrymanyx dlq riv-
nqnnq teploprovidnosti [6], to u vypadku povnoho paraboliçnoho rivnqnnq z
syl\nym vyrodΩennqm sytuaciq istotno zming[t\sq.
1. Formulgvannq zadaçi ta osnovni rezul\taty. V oblasti QT = { ( x, t ) :
0 < x < h, 0 < t < T } rozhlqnemo paraboliçne rivnqnnq
ut = a ( t ) uxx + b ( x, t ) ux + c ( x, t ) u + f ( x, t ), ( x, t ) ∈ QT , (1)
z nevidomym koefici[ntom a ( t ) > 0, t ∈ ( 0, T ], poçatkovog umovog
u ( x, 0 ) = ϕ ( x ), x ∈ [ 0, h ], (2)
krajovymy umovamy
u ( 0, t ) = µ1 ( t ), u ( h, t ) = µ2 ( t ), t ∈ [ 0, T ], (3)
ta umovog perevyznaçennq
a ( t ) ux ( 0, t ) = µ3 ( t ), t ∈ [ 0, T ]. (4)
Poznaçymo çerez Gk ( x, t, ξ , τ ) funkci] Hrina perßo] ( k = 1 ) ta druho] ( k =
= 2 ) krajovyx zadaç dlq rivnqnnq teploprovidnosti
ut = a ( t ) uxx + f ( x, t ) . (5)
Vony magt\ vyhlqd
Gk ( x, t, ξ , τ ) = 1
2
2
4
2
π ( ) − ( )
− ( − + )
( ) − ( )
( ) ( )= − ∞
∞
∑θ θ τ
ξ
θ θ τt
x nh
tn
exp +
+ (− ) − ( + + )
( ) − ( )
( )
1
2
4
2
k x nh
t
exp
ξ
θ θ τ
, k = 1, 2, (6)
de θ ( t ) = a d
t
( )∫ τ τ
0
.
© M. I. IVANÇOV, N. V. SALDINA, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11 1487
1488 M. I. IVANÇOV, N. V. SALDINA
Prypuskagçy, wo funkciq a ( t ) [ vidomog, ta vvodqçy poznaçennq v ( x, t ) ≡
≡ ux ( x, t ), prqmu zadaçu (1) – (3) zaminymo ekvivalentnog systemog rivnqn\
u ( x, t ) = u0 ( x, t ) +
+
G x t b c u d d
ht
1
00
( ) ( ) ( ) + ( ) ( )( )∫∫ , , , , , , ,ξ τ ξ τ ξ τ ξ τ ξ τ ξ τv , ( x, t ) ∈ QT , (7)
v ( x, t ) = v0 ( x, t ) +
+ G x t b c u d dx
ht
1
00
( ) ( ) ( ) + ( ) ( )( )∫∫ , , , , , , ,ξ τ ξ τ ξ τ ξ τ ξ τ ξ τv , ( x, t ) ∈ QT , (8)
de u0 ( x, t ) — rozv’qzok rivnqnnq (5) z umovamy (2), (3), qkyj ma[ vyhlqd
u0 ( x, t ) = G x t d G x t a d
h t
1
0
1 1
0
0 0( ) ( ) + ( ) ( ) ( )∫ ∫, , , , , ,ξ ϕ ξ ξ τ τ µ τ τξ –
– G x t h a d G x t f d d
t ht
1 2
0
1
00
ξ τ τ µ τ τ ξ τ ξ τ ξ τ( ) ( ) ( ) + ( ) ( )∫ ∫∫, , , , , , , . (9)
Vyraz v0 ( x, t ) otrymu[t\sq z (9) dyferencigvannqm ta intehruvannqm çastyna-
my z zastosuvannqm vlastyvostej funkci] Hrina:
v0 ( x, t ) = G x t d G x t f d
h t
2
0
2 1
0
0 0 0( ) ′( ) + ( ) ( ) − ′( )∫ ∫ ( ), , , , , , ,ξ ϕ ξ ξ τ τ µ τ τ +
+ G x t h f h d G x t f d d
t ht
2 2
0
2
00
( ) ′ ( ) − ( ) + ( ) ( )( )∫ ∫∫, , , , , , , ,τ µ τ τ τ ξ τ ξ τ ξ τξ . (10)
Pidstavyvßy (8) v (4), otryma[mo
a ( t ) =
µ3
0
( )
( )
t
tv ,
, t ∈ [ 0, T ]. (11)
Obernenu zadaçu (1) – (4) zvedeno do ekvivalentno] systemy rivnqn\ (7), (8), (11).
Osnovnym rezul\tatom roboty [ nastupna teorema.
Teorema isnuvannq ta [dynosti. Prypustymo, wo vykonugt\sq umovy:
1) ϕ ∈ C1
[ 0, h ]; µi ∈ C1
[ 0, T ], i = 1, 2; µ3 ∈ C [ 0, T ]; b, c, f ∈ C1, 0
(QT );
2) f ( 0, t ) – ′( )µ1 t > 0, t ∈ [ 0, T ]; µ 3 ( t ) > 0, t ∈ ( 0, T ], isnu[ hranycq
lim /t
t
t→ + ( + )
( )
0
3
1 2
µ
β ≡ M, | b ( x, t ) | ≤ Bt( − ) +β γ1 2 0/
, | c ( x, t ) | ≤ Ctγ1
, ( x, t ) ∈ QT , γ i > 0,
i = 0, 1, β > 1, M, C, B > 0 — deqki stali;
3) ϕ ( 0 ) = µ1 ( 0 ), ϕ ( h ) = µ2 ( 0 ).
Todi moΩna vkazaty take çyslo t0 , 0 < t0 ≤ T, qke vyznaça[t\sq vyxidnymy
danymy zadaçi, wo isnu[ [dynyj rozv’qzok zadaçi (1) – (4) ( a ( t ), u ( x, t ) ) z klasu
C [ 0, t0 ] × C2, 1
( Qt0
) ∩ C Qt( )
0
, ux ( 0, t ) ∈ C ( 0, t0 ], a ( t ) > 0, t ∈ ( 0, t0 ], pryçomu
isnu[ skinçenna dodatna hranycq lim
t
a t
t→ +
( )
0 β .
2. Vstanovlennq apriornyx ocinok. Isnuvannq rozv’qzku dovedemo za dopo-
mohog teoremy Íaudera pro neruxomu toçku cilkom neperervnoho operatora.
Dlq c\oho vyznaçymo apriorni ocinky rozv’qzku systemy.
Za pryncypom maksymumu [7, s. 25]
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ … 1489
| u ( x, t ) | ≤ U < ∞ v QT . (12)
Vstanovymo povedinku | v ( x, t ) |. Na pidstavi rivnosti G x t d
h
20
( )∫ , , ,ξ τ ξ = 1 ro-
bymo vysnovok pro obmeΩenist\ perßoho ta çetvertoho dodankiv rivnosti (10)
stalymy, wo zaleΩat\ vid vyxidnyx danyx:
G x t d
h
2
0
0( ) ′( )∫ , , ,ξ ϕ ξ ξ ≤ C1
, G x t f d d
ht
2
00
( ) ( )∫∫ , , , ,ξ τ ξ τ ξ τξ ≤ C2
. (13)
Dlq ocinky dvox nastupnyx vyraziv z (10) vykorysta[mo qvnyj vyhlqd funkci]
Hrina ta vidomu ocinku [8, s. 12]
G2 ( x, t, ξ , τ ) ≤
C
t
C3
4θ θ τ( ) − ( )
+ .
Todi
G x t f d C C d
t
t t
2 1
0
5 6
0
0 0( ) ( ) − ′( ) ≤ +
( ) − ( )
( )∫ ∫, , , ,τ τ µ τ τ τ
θ θ τ
,
(14)
G x t h f h d C C d
t
t t
2 2
0
7 8
0
( ) ′ ( ) − ( ) ≤ +
( ) − ( )
( )∫ ∫, , , ,τ µ τ τ τ τ
θ θ τ
.
Ostatoçno otrymu[mo
| v0 ( x, t ) | ≤ C C d
t
t
9 10
0
+
( ) − ( )∫ τ
θ θ τ
. (15)
Vraxovugçy nerivnist\
G x t d
C
tx
h
1
0
11( ) ≤
( ) − ( )∫ , , ,ξ τ ξ
θ θ τ
ta vvodqçy poznaçennq V ( t ) ≡
max ,
,x h
x t
∈[ ]
( )
0
v , z rivnqnnq (8) znaxodymo
V ( t ) ≤ C C d
t
C
V
t
d
t t
12 13
0
14
1 2
0
0
+
( ) − ( )
+ ( )
( ) − ( )∫ ∫
( − ) +τ
θ θ τ
τ τ
θ θ τ
τ
β γ/
+ C
d
t
t
15
0
1τ τ
θ θ τ
γ
( ) − ( )∫ .
(16)
Poznaçymo
a0 ( t ) ≡ a t
t
( )
β , amax ( t ) ≡ max
0
0
≤ ≤
( )
τ
τ
t
a , amin ( t ) ≡ min
0
0
≤ ≤
( )
τ
τ
t
a . (17)
Vraxovugçy oznaçennq funkci] θ ( t ) ta (17), zvedemo (16) do vyhlqdu
V ( t ) ≤ C
C
a t
d
t
C
a t
V
t
d
t t
12
16
1 1
0
17
1 2
1 1
0
0 1
+
( ) −
+
( )
( ) +
−+ +
( − ) +
+ +∫ ∫
min min
/τ
τ
τ τ τ
τ
τ
β β
β γ γ
β β
.
(18)
Vykorystavßy zaminu z = τ
t
, ocinymo intehral
d
t t
dz
z t
dz
z
C
t
t
τ
τβ β β β β β+ + ( − ) + ( − ) ( − )−
=
−
≤
−
=∫ ∫ ∫1 1
0
1 2 1
0
1
1 2
0
1
18
1 2
1
1
1
1/ / / . (19)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1490 M. I. IVANÇOV, N. V. SALDINA
Z ohlqdu na (19) lehko baçyty, wo druhyj intehral u (18) ma[ porqdok osobly-
vosti menßyj, niΩ
1
1 2t( − )β / . OtΩe, funkciq v ( x, t ) povodyt\ sebe qk
C
t
19
1 2( − )β /
pry t → + 0.
Ocinymo v ( 0, t ) znyzu. V rivnqnni (8) poklademo x = 0. Z vyhlqdu funkci]
G2 ( 0, t, h, τ ) ma[mo
G t h f h d
t
2 2
0
0( ) ′ ( ) − ( )∫ , , , ,τ µ τ τ τ ≤ C20
. (20)
Z umov teoremy ta ocinok (13) ta (20) vyplyva[
v ( 0, t ) ≥ G t f d
t
2 1
0
0 0 0( ) ( ) − ′( )( )∫ , , , ,τ τ µ τ τ –
– C C
V
t
d
t
21 22
1 2
0
0 1
− ( ) +
( ) − ( )
( − ) +
∫ τ τ τ
θ θ τ
τ
β γ γ/
. (21)
Dlq ocinky perßoho intehrala u (21) pidstavymo funkcig Hrina (6) ta vydilymo
z rqdu dodanok, wo vidpovida[ n = 0:
G t f d
f
t
d
t t
2 1
0
1
0
0 0 0 1 0( ) ( ) − ′( ) =
π
( ) − ′( )
( ) − ( )
( )∫ ∫, , , ,
,τ τ µ τ τ τ µ τ
θ θ τ
τ +
+ 2 0 1
2 2
10
π
( ) − ′( )
( ) − ( )
−
( ) − ( )
=
∞
∑∫ f
t
n h
t
d
n
t
,
exp
τ µ τ
θ θ τ θ θ τ
τ ≥
≥ 1 0 1
0
π
( ) − ′( )
( ) − ( )∫ f
t
d
t
, τ µ τ
θ θ τ
τ .
Todi z (21) znaxodymo
v ( 0, t ) ≥ 1 0 1
0
21 22
1 2
0
0 1
π
( ) − ′( )
( ) − ( )
− ( ) +
( ) − ( )∫ ∫
( − ) +f
t
d C C
V
t
d
t t
,
–
/τ µ τ
θ θ τ
τ τ τ τ
θ θ τ
τ
β γ γ
. (22)
Z toho, wo pry t → + 0 osoblyvist\ druhoho intehrala v (22) menßa za osobly-
vist\ perßoho, vyplyva[, wo isnugt\ taki znaçennq t1
, 0 < t1 ≤ T, ta q, 0 < q <
< 1, wo
C
V
t
d C q
f
t
d
t t
22
1 2
0
21
1
0
0 1 1 0τ τ τ
θ θ τ
τ τ µ τ
θ θ τ
τ
β γ γ( − ) + ( ) +
( ) − ( )
+ ≤
π
( ) − ′( )
( ) − ( )∫ ∫
/ ,
, t ∈ [ 0, t1 ].
Todi z (22) oderΩu[mo
v ( 0, t ) ≥ ( − )
π
( ) − ′( )
( ) − ( )∫1 1 0 1
0
q
f
t
d
t
, τ µ τ
θ θ τ
τ . (23)
Pidstavlqgçy (23) v rivnqnnq (11), otrymu[mo
a ( t ) ≤
π ( )
( − ) ( ) − ′( )
( ) − ( )∫
µ
τ µ τ
θ θ τ
τ
3
1
0
1
0
t
q
f
t
d
t
,
.
Vykorystovugçy (17), ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ … 1491
a0 ( t ) ≤
π ( ) ( )
( − ) + ( ) − ′( )
−+ +∫
µ
β τ µ τ
τ
τβ
β β
3
1
1 1
0
1 1
0
t a t
q t
f
t
d
t
max
,
.
Vvedemo poznaçennq
H ( t ) ≡
π ( )
+ ( ) − ′( )
−+ +∫
µ
β τ µ τ
τ
τβ
β β
3
1
1 1
0
1
0
t
t
f
t
d
t
,
. (24)
Z umov teoremy vyplyva[, wo funkciq H ( t ) [ dodatnog na ( 0, T ] ta naleΩyt\
klasu C ( 0, T ].
Dovedemo isnuvannq hranyci lim
t
H t
→ +
( )
0
. Dlq c\oho vykorysta[mo teoremu
pro seredn[ ta zaminu zminnyx z = τ
t
:
lim lim
,
t t tH t
t
t f t t d
t
→ + → +
+ +
( ) = π ( )
+ ( ) − ′( )
−
( ) ∫
0 0
3
1 1 1
0
1 0
µ
β µ τ
τ
β
β β
=
= π
+
( )
( ) − ′( )
−
→ + ( + )
+
( ) ∫
β
µ
µβ
β
1
0
1
0
3
1 2
1 1
0
1lim
,/t
t
t f t t dz
z
=
= π
+ ( ) − ′( )( )
M
f Iβ µ1 0 0 01 1,
> 0.
Tut
I1 = dz
z1 1
0
1
− +∫ β
, M = lim /t
t
t→ + ( + )
( )
0
3
1 2
µ
β ,
t — deqke çyslo z [ 0, T ]. ProdovΩugçy ocinku funkci] a ( t ) z vykorystannqm
oznaçennq funkci] H ( t ), otrymu[mo
a0 ( t ) ≤ 1
1 −
( ) ( )
q
H t a tmax abo amax ( t ) ≤ 1
1 −
( ) ( )
q
H t a tmax max ,
de Hmax ( t ) = max
0≤ ≤
( )
τ
τ
t
H . Zvidsy ma[mo
amax ( t ) ≤ 1
1 2
2
( − )
( )
q
H tmax , t ∈ [ 0, t1 ]. (25)
Ostatoçno dlq a ( t ) oderΩu[mo
a ( t ) ≤ A t1
β, de A1 = 1
1 2
2
( − )
( )
q
H Tmax > 0, t ∈ [ 0, t1 ]. (26)
Dlq ocinky a ( t ) znyzu vykorysta[mo oznaçennq funkci] V ( t ) ta nastupnu
ocinku v ( 0, t ):
v ( 0, t ) ≤ C
f
t
d C
V
t
d
t t
23
1
0
24
1 2
0
1 0 0
+
π
( ) − ′( )
( ) − ( )
+ ( )
( ) − ( )∫ ∫
( − ) +, /τ µ τ
θ θ τ
τ τ τ
θ θ τ
τ
β γ
+
+ C
d
t
t
25
0
1τ τ
θ θ τ
γ
( ) − ( )∫ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1492 M. I. IVANÇOV, N. V. SALDINA
Pidstavymo cej vyraz u rivnqnnq (11), vykorystavßy oznaçennq funkci] amin ( t ):
a ( t ) ≥ µ β τ µ τ
τ
τ
β β3 23
1
1 1
0
1 0( )
+ +
π ( )
( ) − ′( )
−+ +∫t C
a t
f
t
d
t
min
,
+
+
C
a t
V
t
d
C
a t
d
t
t t
24
1 2
1 1
0
25
1 1
1
0
1 10 1β τ τ
τ
τ β τ τ
τ
β γ
β β
γ
β β
+
( )
( )
−
+ +
( ) −
( − ) +
+ + + +
−
∫ ∫
min
/
min
.
Podilymo obydvi çastyny nerivnosti na tβ i zvedemo ]] do vyhlqdu
a0 ( t ) ≥ a t
C a t t
t
t
t
f
t
d
t
min
min ,( )
( )
( )
+ +
π ( )
( ) − ′( )
−+ +∫23
3 3
1
1 1
0
1 0β β
β βµ
β
µ
τ µ τ
τ
τ +
+ C t
t
V
t
d C t
t
d
t
t t
26
3
1 2
1 1
0
27
3
1 1
1
0
0 1β β γ
β β
β γ
β βµ
τ τ
τ
τ
µ
τ τ
τ( )
( )
−
+
( ) −
( − ) +
+ + + +
−
∫ ∫
/
.
ProdovΩymo ocinku a0 ( t ), vykorystavßy isnuvannq dodatno] hranyci
lim /t
t
t→ + ( + )
( )
0
3
1 2
µ
β ta oznaçennq funkci] H ( t ):
a0 ( t ) ≥ a t C a t t
H t
C t
V
t
d
t
min min
/ /
/
( )
( ) +
( )
+ ( )
−
( − ) ( − )
( − ) +
+ +∫28
1 2
29
1 2
1 2
1 1
0
1 0β β
β γ
β β
τ τ
τ
τ +
+ C t
d
t
a t H t C t
t
30
1 2
1 1
1
0
31
1 21
1( − )
+ +
−
( − )
−
≥ ( ) ( )
+∫β
γ
β β
βτ τ
τ
/
min
/ +
+ C t
V
t
d C t
d
t
t t
32
1 2
1 2
1 1
0
33
1 2
1 1
1
0
0 1( − )
( − ) +
+ +
( − )
+ +
−
( )
−
+
−
∫ ∫β
β γ
β β
β
γ
β β
τ τ
τ
τ τ τ
τ
/
/
/
. (27)
Z ohlqdu na povedinku funkci] V ( t ) robymo vysnovok pro te, wo isnu[ take zna-
çennq t2
, 0 < t2 ≤ T, wo
C t C t
V
t
d C t
d
t
t t
31
1 2
32
1 2
1 2
1 1
0
33
1 2
1 1
0
0 1( − ) ( − )
( − ) +
+ +
( − )
+ +
+ ( )
−
+
−∫ ∫β β
β γ
β β
β
γ
β β
τ τ
τ
τ τ τ
τ
/ /
/
/ ≤
≤ C t C t C t31
1 2
34 35
0 1( − ) + +β γ γ/ ≤ q, t ∈ [ 0, t2 ].
Todi z (27) ma[mo
amin ( t ) ≥
a t H t
q
min( ) ( )
+1
abo amin ( t ) ≥
H t
q
min
2
21
( )
( + )
, t ∈ [ 0, t2 ], (28)
de
Hmin ( t ) = min
0≤ ≤
( )
τ
τ
t
H .
Ostatoçno dlq a ( t ) otrymu[mo
a ( t ) ≥ A t0
β , t ∈ [ 0, t2 ], A0 = 1
1 2
2
( + )
( )
q
H Tmin > 0. (29)
Magçy ocinky zverxu ta znyzu dlq funkci] a ( t ), prodovΩu[mo ocinku funk-
ci] V ( t ). Dlq c\oho pidstavymo u (18) ocinku amin ( t ) z (28) ta vraxu[mo (19):
V ( t ) ≤ C
C
t
C
t
V
t
d
C
t
d
t
t t
12
36
1 2
37
2
1 2
0
38
2
0
0 1
+ + ( )
−
+
−( − )
( − ) +
∫ ∫β β
β γ
β
γτ τ
τ
τ τ τ
τ/ /
/
/ . (30)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ … 1493
PomnoΩymo obydvi çastyny na t( − )β 1 2/
ta vvedemo novu funkcig w ( t ) ≡
≡ V t t( ) ( − )β 1 2/
. Todi otryma[mo
w ( t ) ≤ C
C
t
w
t
d
C
t
d
t
t t
39
37
0
38
0
0 1
+ ( )
−
+
−∫ ∫τ τ
τ
τ τ τ
τ
γ γ
. (31)
Poklademo t = σ, domnoΩymo obydvi çastyny nerivnosti na
1
t − σ
i zintehru[-
mo vid 0 do t:
w d
t
C t C d
t
w
d C d
t
d
t t t( )
−
≤ +
( − )
( )
−
+
( − ) −∫ ∫ ∫ ∫ ∫σ σ
σ
σ
σ σ
τ τ
σ τ
τ σ
σ σ
τ τ
σ τ
γσ γσ
0
40 37
0 0
38
0 0
0 1
.
Peretvorymo druhyj ta tretij dodanky nerivnosti. Zminggçy porqdok intehru-
vannq ta vraxovugçy zaminu zminnyx z =
σ τ
σ
−
−t
, ma[mo
w d
t
C t C w d C t
t t( )
−
≤ + π ( ) +∫ ∫ − +σ σ
σ
τ τ τγ γ
0
40 37
1 2
0
41
1 20 1/ /
. (32)
Podamo (31) u vyhlqdi
w ( t ) ≤ C C t
w
t
d C t
t
39 37
1 2
0
42
0 1+ ( )
−
+− ∫γ γτ
τ
τ/
.
Vykorystovugçy ocinku (32), otrymu[mo
w ( t ) ≤ C C t w d
t
43 44
1 2 1 2
0
0 0+ ( )− −∫γ γτ τ τ/ /
abo
w t t C t C w d
t
( ) ≤ + ( )− − −∫1 2
43
1 2
44
1 2
0
0 0 0/ / /γ γ γτ τ τ . (33)
U vypadku, koly γ0 ≥ 1
2
, dana nerivnist\ rozv’qzu[t\sq za dopomohog lemy Hro-
nuolla [9, s. 188], i, qk naslidok, funkciq w ( t ) obmeΩena deqkog stalog, qka
zaleΩyt\ vid vyxidnyx danyx. Nexaj γ0 < 1
2
. Poznaçymo pravu çastynu neriv-
nosti (33) çerez χ ( t ). Todi
′( ) − ( ) ≤ −
− − −χ χ γγ γt C t t C t44
2 1
43 0
1 20 01
2
/
.
DomnoΩymo obydvi çastyny nerivnosti na exp −
−∫C d
t
44
2 1
0
0τ τγ
ta zintehru[mo
vid 0 do t. V rezul\tati otryma[mo
χ ( t ) ≤ C t
C
t43
1 2 44
0
20 0
2
/ exp−
γ γ
γ
.
Pidstavlqgçy otrymanu ocinku v (33), oderΩu[mo
w ( t ) ≤ C
C
t43
44
0
2
2
0exp
γ
γ
≤ C45 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1494 M. I. IVANÇOV, N. V. SALDINA
Todi ma[mo
V ( t ) ≤
C
t
45
1 2( − )β / , t ∈ [ 0, t2 ] , abo | v ( x, t ) | ≤
C
t
45
1 2( − )β / , ( x, t ) ∈ [ 0, h ] × ( 0, t2 ],
(34)
de C45 — stala, wo vyznaça[t\sq vyxidnymy danymy.
3. Dovedennq isnuvannq rozv’qzku. Vvedemo novu funkcig ˜ ,v( )x t ≡
≡ v( ) ( − )x t t, /β 1 2
i zapyßemo systemu rivnqn\ (7), (8), (11) u vyhlqdi
u ( x, t ) = u0 ( x, t ) +
+ G x t b c u d d
ht
1 1 2
00
( ) ( ) ( ) + ( ) ( )
( − )∫∫ , , , ,
˜ ,
, ,/ξ τ ξ τ ξ τ
τ
ξ τ ξ τ ξ τβ
v
, ( x, t ) ∈ Qt0
, (35)
̃ ,v( )x t = v0 ( x, t )t( − )β 1 2/ +
+
t G x t b c u d dx
ht
( − )
( − )( ) ( ) ( ) + ( ) ( )
∫∫β
βξ τ ξ τ ξ τ
τ
ξ τ ξ τ ξ τ1 2
1 1 2
00
/
/, , , ,
˜ ,
, ,
v
, ( x, t ) ∈ Qt0
,
(36)
a ( t ) =
µ β
3
1 2
0
( )
( )
( − )t t
t
/
˜ ,v
, t ∈ [ 0, t0 ], t0 = min { t1 , t2 }. (37)
Podamo systemu rivnqn\ (35) – (37) v operatornij formi
ω = P ω, (38)
de ω = ( )u a, ˜,v , P = ( P1 , P2 , P3 ), operatory P1 , P2 , P3 vyznaçagt\sq pravymy
çastynamy rivnqn\ (35) – (37). Vyznaçymo mnoΩynu N = { ( u ( x, t ), ̃ ,v( )x t , a ( t ) ) ∈
∈ C Qt( )
0
× C Qt( )
0
× C [ 0, t0 ] : | u ( x, t ) | ≤ U, | ̃ ,v( )x t | ≤ C45 , A0 ≤ a t
t
( )
β ≤ A1 }. Zhid-
no z ocinkamy (12), (34), (26), (29) operator P perevodyt\ mnoΩynu N v sebe.
PokaΩemo, wo operator P [ cilkom neperervnym na N . Zhidno z teoremog
Arcela – Askoli dlq c\oho slid vstanovyty, wo dlq dovil\noho ε > 0 isnu[ take
δ > 0, wo
| Pi ( x2 , t2 ) – Pi ( x1 , t1 ) | < ε, i = 1, 2,
| P3 ( t2 ) – P3 ( t1 ) | < ε, ∀( u ( x, t ), ̃ ,v( )x t , a ( t ) ) ∈ N ,
qkwo | t2 – t1 | < δ, | x2 – x1 | < δ, de ( x1 , t1 ), ( x2 , t2 ) ∈ Qt0
. Dovedennq kompakt-
nosti pokaΩemo na prykladi odnoho z dodankiv, wo vxodyt\ do intehral\noho
operatora P:
K = t f G t d
t
2
1 2
1 2 2
0
0 0 0
2
( − ) ( )( ) − ′( ) ( )∫β τ µ τ τ τ/ , , , , –
– t f t G t d
t
1
1 2
1 2 1
0
0 0 0
1
( − ) ( )( ) − ′( ) ( )∫β τ µ τ τ/ , , , , .
Prypustymo, wo ti
, i = 1, 2, [ dosyt\ malymy. Rozhlqnemo intehral
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ … 1495
ˆ , , , ,
,/
/
K t f G t d t f
t
d
t t
= ( ) − ′( ) ( ) =
π
( ) − ′( )
( ) − ( )
( − )
( − )
( )∫ ∫β
β
τ µ τ τ τ τ µ τ
θ θ τ
τ1 2
1 2
0
1 2
1
0
0 0 0
0
+
+ 2
0 1
2 2
10
1 2
f
t
n h
t
d K K
n
t ( ) − ′( )
( ) − ( )
−
( ) − ( )
= +
=
∞
∑∫ ,
exp ˆ ˆτ µ τ
θ θ τ θ θ τ
τ .
Vykorysta[mo poznaçennq (17) ta oznaçennq funkci] θ ( t ). Todi dlq druhoho do-
danka otryma[mo ocinku
ˆ max ,
/
min ,
K
t
a t
f t t
tt T
t
2
1 2
0
1 1 1
0
2 1
0 1≤ +
π ( )
( ) − ′( )
−
( − )
∈[ ] + +
( ) ∫β µ
τ
β
β β
×
× exp
max
− ( + )
( )( − )
+ +
=
∞
∑ n h
a t t
d
n
2 2
1 1
1
1β
τ
τβ β .
Z oznaçennq mnoΩyny N robymo vysnovok pro obmeΩenist\ pidintehral\noho
vyrazu K̂2 . OtΩe, K̂2 prqmu[ do nulq pry t → + 0. Rozhlqnemo perßyj doda-
nok K̂1 , vykorystovugçy teoremu pro seredn[ ta zaminu zminnyx z = τ
t
:
ˆ ,
/
K
t
a t
f t t d
t
t
1
1 2
0
1 1 1
0
1
0= +
π ( )
( ) − ′( )
−
( − )
+ +
( ) ∫
β
β β
β µ τ
τ
=
=
( )( ) − ′( ) +
π ( ) − +∫f t t
a t
dz
z
t
0 1
1
1
0
1
0
, µ β
β
,
de t ∈ [ 0, T ]. Poznaçymo lim ˆ
t
K
→ + 0
1 = κ0
. Todi, povertagçys\ do K, otrymu[mo
K ≤ t f G t d
t
2
1 2
1 2 2
0
00 0 0
2
( − ) ( )( ) − ′( ) ( ) −∫β τ µ τ τ τ κ/ , , , , +
+ t f G t d
t
1
1 2
1 2 1
0
00 0 0
1
( − ) ( )( ) − ′( ) ( ) −∫β τ µ τ τ τ κ/ , , , , .
MoΩna vkazaty take znaçennq t*
, wo pry 0 < ti < t*
, i = 1, 2, budut\ vykonuva-
tys\ nerivnosti
t f G t di i
ti
( − ) ( )( ) − ′( ) ( ) − <∫β τ µ τ τ τ κ ε1 2
1 2
0
00 0 0
2
/ , , , , .
OtΩe, K < ε pry 0 < ti < t*
, i = 1, 2. Vypadok ti > t*
, i = 1, 2, ta inßi inteh-
ral\ni operatory, wo vxodqt\ v P1 , P2 , doslidΩugt\sq analohiçno do vypadku
syl\noho vyrodΩennq dlq rivnqnnq teploprovidnosti [10]. Operator P [ cil-
kom neperervnym na N .
Za teoremog Íaudera isnu[ rozv’qzok systemy (35) – (37), qkyj ma[ potribnu
hladkist\. Isnuvannq rozv’qzku zadaçi (1) – (4) dovedeno.
3. Dovedennq [dynosti rozv’qzku. Prypustymo, wo isnugt\ dva rozv’qzky
( ai ( t ), ui ( x, t ), v i ( x, t ) ), i = 1, 2, systemy (7), (8), (11). Poznaçymo riznycg cyx
rozv’qzkiv çerez a ( t ) ≡ a1 ( t ) – a2 ( t ), u ( x, t ) ≡ u1 ( x, t ) – u2 ( x, t ), v ( x, t ) ≡ v1 ( x, t ) –
– v2 ( x, t ). Dlq cyx funkcij otryma[mo systemu rivnqn\
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1496 M. I. IVANÇOV, N. V. SALDINA
u ( x, t ) = u0 ( x, t ) +
G x t b c u d d
ht
1
1
00
( ) ( ) ( ) + ( ) ( )( )∫∫ , , , , , , ,ξ τ ξ τ ξ τ ξ τ ξ τ ξ τv +
+
00
1
1
1
2
2
ht
G x t G x t b∫∫ ( )(( ) − ( ) ( ) ( ), , , , , , , ,ξ τ ξ τ ξ τ ξ τv +
+ c u d d( ) ( ))ξ τ ξ τ ξ τ, ,2 , ( x, t ) ∈ QT , (39)
v ( x, t ) = v0 ( x, t ) +
G x t b c u d dx
ht
1
1
00
( ) ( ) ( ) + ( ) ( )( )∫∫ , , , , , , ,ξ τ ξ τ ξ τ ξ τ ξ τ ξ τv +
+
00
1
1
1
2
2
ht
x xG x t G x t b∫∫ ( )(( ) − ( ) ( ) ( ), , , , , , , ,ξ τ ξ τ ξ τ ξ τv +
+ c u d d( ) ( ))ξ τ ξ τ ξ τ, ,2 , ( x, t ) ∈ QT , (40)
a ( t ) =
− ( ) ( ) ( )
( )
a t a t
t
t1 2
3
0v ,
µ
, t ∈ [ 0, T ], (41)
de G x ti
1( ), , ,ξ τ , i = 1, 2, — funkci] Hrina perßyx krajovyx zadaç dlq rivnqn\
uit = ai ( t ) uixx , u0 ( x, t ) = u01 ( x, t ) – u02 ( x, t ), v0 ( x, t ) = v01 ( x, t ) – v02 ( x, t ).
Dlq dovedennq [dynosti rozv’qzku ocinymo a ( t ), vyxodqçy z rivnqnnq (41).
Spoçatku ocinymo velyçyny | u ( x, t ) |, | v ( x, t ) |, | u0 ( x, t ) |, | v0 ( x, t ) |, vid qkyx
zaleΩyt\ ocinka | v ( 0, t ) |. Dlq prykladu rozhlqnemo odyn iz dodankiv, wo vxo-
dyt\ do v0 ( x, t ). Poznaçymo
R2 =
0
1
1 1
2
1 1
0 2
4
t
n
f
t
x nh
t∫ ∑( ) − ′( )
π ( ) − ( )
− ( + )
( ) − ( )
( ) ( )= ∞
∞,
exp
–
τ µ τ
θ θ τ θ θ τ
–
–
f
t
x nh
t
d
n
( ) − ′( )
π ( ) − ( )
− ( + )
( ) − ( )
( ) ( )= ∞
∞
∑0 2
4
1
2 2
2
2 2
,
exp
–
τ µ τ
θ θ τ θ θ τ
τ .
Peretvorymo R2 , vydilqgçy z rqdiv dodanky, wo vidpovidagt\ n = 0:
R2 ≤ 1 0
4
1
0
1
2
1 10 1 1π
( ) − ′( )
−
( ) − ( )
( ) − ( )[ ]
( )
( )∫max , exp
,T
t
f t t x
t t
µ
θ θ τ θ θ τ
–
– 1 1
42 2 2 20
2
1 1θ θ τ
τ
θ θ τ θ θ τ( ) − ( )
+
( ) − ( )
−
( ) − ( )
∫ ( )t
d
t
x
t
t
exp –
– exp exp
–
−
( ) − ( )
+
( ) − ( )
− ( + )
( ) − ( )
( ) ( )∫ ∑
= ∞
≠
∞
x
t
d
t
x nh
t
t
n
n
2
2 2 1 10
2
1 1
0
4
1 2
4θ θ τ
τ
θ θ τ θ θ τ
–
– 1 2
42 2
2
2 2
0
46 2
1
3
θ θ τ θ θ τ( ) − ( )
− ( + )
( ) − ( )
=
( )= ∞
≠
∞
=
∑ ∑t
x nh
t
C R
n
n
i
i
exp
–
.
Podamo R23 u vyhlqdi
R23 =
θ θ τ
θ θ τ
τ
2 2
1 1
1 2
4
2
0
0 ( )− ( )
( )− ( )
= ∞
≠
∞
∫ ∑∫ ∂
∂
− ( + )
t
t
n
n
t
z z
x nh
z
dz dexp
–
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ … 1497
Vraxovugçy obmeΩenist\ pidintehral\noho vyrazu, otrymu[mo ocinku
R23 ≤ C t t d C d a a d
t t t
47 1 1 2 2
0
47
0
1 2θ θ τ θ θ τ τ τ σ σ σ
τ
( ) − ( ) − ( ) + ( ) ≤ ( ) − ( )∫ ∫ ∫ ≤
≤ C a t t48
2˜max( ) +β , de ˜ maxmaxa t a
t
( ) ≡ ( )
≤ ≤0
0
τ
τ , a0 ( τ ) ≡ a t
t
( )
β .
Dlq ocinky R22 vykorysta[mo nerivnosti | ex – ey | ≤ | x – y | max { ex, ey
} ta (26):
R22 ≤ x
t t t
t2
2 2 1 1 2 20
4
1 1 1
θ θ τ θ θ τ θ θ τ( ) − ( ) ( ) − ( )
−
( ) − ( )∫ ×
× exp −
( − )
+ +
x
C t
d
2
49
1 1β βτ
τ =
= x t t
t t
x
C t
d
t2
2 2 1 1
2 2
3 2
1 10
2
49
1 14
θ θ τ θ θ τ
θ θ τ θ θ τ τ
τβ β
( ) − ( ) − ( ) + ( )
( ) − ( ) ( ) − ( )
−
( − )
( ) ( )∫ + +/ exp .
Vraxovugçy ocinky (28), ma[mo
θ θ τ θ θ τ σ σ σ τ
β
β
τ
β β
1 1 2 2 0
1 1
1
( ) − ( ) − ( ) + ( ) ≤ ( ) ≤ −
+
( )∫
+ +
t t a d
t
a t
t
˜max ,
(42)
θ θ τ σ σ σ τ
β
β
τ
β β
i i i
t
t a d
H t
q
t( ) − ( ) = ( ) ≥ ( )
( + )
−
+∫
+ +
0
2
2
1 1
1 1
min , i = 1, 2.
Beruçy do uvahy nerivnist\ x ep qx− 2
≤ Cp, q
, x ≥ 0, p ≥ 0, q > 0, ta ocinky (42),
otrymu[mo
R22 ≤ C a t d
t
C a t
t
t
50 1 1
51
1 2
0
˜
˜
max
max
/( )
−
≤ ( )
+ + ( − )∫ τ
τβ β β .
Vyraz R21 podamo u vyhlqdi
R21 =
0
2
1 1
2 2 1 1
2 2 1 14
t
x
t
t t
t t
d∫ −
( ) − ( )
( ) − ( ) − ( ) − ( )
( ) − ( ) ( ) − ( )( ) ( )( )
exp
θ θ τ
θ θ τ θ θ τ
θ θ τ θ θ τ
τ ≤
≤
0
2 2 1 1
2 2 1 1 2 2 1 1
t
t t
t t t t
d∫ ( ) − ( ) − ( ) + ( )
( ) − ( ) ( ) − ( ) ( ) − ( ) + ( ) − ( )( )( )( )
θ θ τ θ θ τ
θ θ τ θ θ τ θ θ τ θ θ τ
τ .
Beruçy do uvahy (42), dlq R21 oderΩu[mo
R21 ≤ C a t d
t
C a t
t
t
52 1 1
53
1 2
0
˜
˜
max
max
/( )
−
≤ ( )
+ + ( − )∫ τ
τβ β β .
Ostatoçno otrymu[mo ocinku R2 ≤
C a t
t
54
1 2
˜max
/
( )
( − )β . Nastupnyj dodanok, wo vxodyt\
do v0 ( x, t ), ocing[t\sq analohiçno do poperedn\oho:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1498 M. I. IVANÇOV, N. V. SALDINA
R3 =
0
2
1 1
2
1 1
2 1
4
t
n
f h
t
x h n
t∫ ∑′ ( ) − ( )
π ( ) − ( )
− + ( − )
( ) − ( )
( )
( )
( )= ∞
∞µ τ τ
θ θ τ θ θ τ
,
exp
–
–
–
′ ( ) − ( )
π ( ) − ( )
− + ( − )
( ) − ( )
≤ ( )
( )
( )
( )= ∞
∞
( − )∑µ τ τ
θ θ τ θ θ τ
τ β
2
2 2
2
2 2
55
1 2
2 1
4
f h
t
x h n
t
d
C a t
tn
,
exp
˜
–
max
/ . (43)
Dlq dvox inßyx dodankiv u vyrazi v0 ( x, t ) magt\ misce ocinky
R1 = ( )( ) − ( ) ′( ) ≤ ( )∫ G x t G x t d C a t
h
2
1
2
2
0
560 0, , , , , , ˜maxξ ξ ϕ ξ ξ ,
(44)
R4 =
00
2
1
2
2
57
ht
G x t G x t f d d C ta t∫∫ ( )( ) − ( ) ( ) ≤ ( ), , , , , , , ˜maxξ τ ξ τ ξ τ ξ τξ .
OtΩe, ostatoçno ma[mo | v0 ( x, t ) | ≤
C a t
t
58
1 2
˜max
/
( )
( − )β .
Vyraz | u0 ( x, t ) | ocing[t\sq analohiçno, i dlq n\oho spravdΩu[t\sq ocinka
| u0 ( x, t ) | ≤ C a t59 ˜max( ).
Poznaçymo U ( t ) = max
,x h∈[ ]0
| u ( x, t ) |, V ( t ) = max
,x h∈[ ]0
| v ( x, t ) |. Todi z (39), (40), vy-
korystovugçy ocinky v0 ( x, t ), u0 ( x, t ) ta umovy teoremy, ma[mo
V ( t ) ≤
C a t
t
C
t
V U
t
d
t
60
1 2
61
2
1 2
0
0 1˜max
/ /
/( ) + ( ) + ( )
−( − )
( − ) +
∫β β
β γ γτ τ τ τ
τ
τ , (45)
U ( t ) ≤ C a t C V U d
t
62 63
1 2
0
0 1˜max
/( ) + ( ) + ( )( )( − ) +∫ τ τ τ τ τβ γ γ
. (46)
Rozv’qzugçy nerivnist\ (46) wodo U ( t ), otrymu[mo
U ( t ) ≤ C a t C V d
t
64 65
1 2
0
0˜max
/( ) + ( )( − ) +∫ τ τ τβ γ
. (47)
Pidstavlqgçy (47) u (45), pryxodymo do nerivnosti
V ( t ) ≤
C a t
t
C
t
V
t
d
t
66
1 2
61
2
1 2
0
0˜max
/ /
/( ) + ( )
−( − )
( − ) +
∫β β
β γτ τ
τ
τ +
+
C
t
d
t
V d
t
67
2
0
1 2
0
1
0
β
γ
β γ
ττ τ
τ
σ σ σ/
/
−
( )∫ ∫ ( − ) +
. (48)
Rozv’qzugçy (48) analohiçno do (30) i vraxovugçy (47), otrymu[mo
V ( t ) ≤
C a t
t
68
1 2
˜max
/
( )
( − )β ta U ( t ) ≤ C a t69 ˜max( ), t ∈ [ 0, t0 ]. (49)
Takym çynom, my otrymaly ocinky | u ( x, t ) |, | v ( x, t ) |. Vyraz | v ( 0, t ) | ociny-
mo okremo. Rozhlqdagçy (43) pry x = 0, robymo vysnovok, wo pidintehral\nyj
vyraz R3 [ obmeΩenym, tomu R3 ≤ C a t70 ˜max( ) . Povedinka dodankiv R1
, R4 ne
zming[t\sq, otΩe, zalyßagt\sq pravyl\nymy ocinky (44). Vyraz R2 ocinymo
toçniße. Pidstavlqgçy x = 0 v R2 ta vydilqgçy z rqdiv dodanky, wo vidpovi-
dagt\ n = 0, ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
OBERNENA ZADAÇA DLQ PARABOLIÇNOHO RIVNQNNQ … 1499
′ =
π ( ) − ( )
−
( ) − ( )
( ) − ′( )∫ ( )R
t t
f d
t
2
1 1 2 20
1
1 1 1 0
θ θ τ θ θ τ
τ µ τ τ, +
+ 2 0 0
0
1
1 1
2 2
1 11
1
2 2π
( ) − ′( )
( ) − ( )
−
( ) − ( )
− ( ) − ′( )
( ) − ( )∫ ∑
=
∞t
n
f
t
n h
t
f
t
,
exp
,τ µ τ
θ θ τ θ θ τ
τ µ τ
θ θ τ
×
× exp −
( ) − ( )
= ′ + ′
=
∞
∑ n h
t
d R R
n
2 2
2 21
21 22θ θ τ
τ .
Pidintehral\nyj vyraz ′R22 ne ma[ osoblyvosti, tomu ′R22 ≤ C a t71˜max( ) . Dlq
′R21 vykorysta[mo ocinky (42) ta oznaçennq funkci] H ( t ):
′R21 = 1 0
0
1 1 2 2 1
1 1 2 2 1 1 2 2π
( ) − ( ) − ( ) + ( ) ( ) − ′( )
( ) − ( ) ( ) − ( ) ( ) − ( ) + ( ) − ( )∫ ( )
( )( )( )
t
t t f
t t t t
d
θ θ τ θ θ τ τ µ τ
θ θ τ θ θ τ θ θ τ θ θ τ
τ,
≤
≤
β τ µ τ
τ
τ µ
β β β
+ ( + ) ( )
π ( )
( ) − ′( )
−
≤ ( + ) ( ) ( )
( )+ +∫1 1
2
0 1
2
3
3
1
1 1
0
3
3
4
q a t
H t
f
t
d
q t a t
H t t
t˜ , ˜max
min
max
min
.
Ostatoçno
| v0 ( 0, t ) | ≤
( + ) ( ) ( )
( )
+ ( )1
2
3
3
4 72
q t a t
H t t
C a t
µ
β
˜
˜max
min
max . (50)
Pidstavlqgçy (50) u (40) ta vraxovugçy (49), otrymu[mo
| v ( 0, t ) | ≤
( + ) ( ) ( )
( )
+ ( ) + ( ) +
( − )
1
2
3
3
4 73 74 1 2
0 1q t a t
H t t
C a t C a t
t t
t
µ
β
γ γ
β
˜
˜ ˜max
min
max max / . (51)
Beruçy do uvahy (25), z (41) znaxodymo
| a0 ( t ) | ≤
H t
q
t
t
tmax ,
4
4
31
0
( )
( − ) ( )
( )
β
µ
v .
ProdovΩymo ocinku a0 ( t ), pidstavyvßy (51):
| a0 ( t ) | ≤
( + ) ( )
( − ) ( )
+ + ( + )
( )( − )1
2 1
3 4
4 4 75
1 2
76
0 1
q H t
q H t
C t C t t a tmax
min
/
max˜β γ γ
abo
˜ ˜max
max
min
/
maxa t
q H t
q H t
C t C t t a t( ) ≤ ( + ) ( )
( − ) ( )
+ + ( + )
( )( − )1
2 1
4 4
4 4 75
1 2
76
0 1β γ γ
.
Z toho, wo lim max
t
H t
→ +
( )
0
= lim min
t
H t
→ +
( )
0
, vyplyva[, wo dlq zadanoho q, 0 < q <
< 1, isnu[ take çyslo t*
, 0 < t* ≤ T , wo
H t
H t
max
min
4
4
( )
( )
≤ 1 + q , C t75
1 2( − )β / +
+ C t t76
0 1( + )γ γ ≤ q, t ∈ [ 0, t* ]. Zafiksu[mo çyslo q tak, wob 0 < q <
2 1
2 1
5
5
−
+
.
Otryma[mo
( + ) ( )
( − ) ( )
+ + ( + ) ≤ ( + )
( − )
( + ) + <( − )1
2 1
1
2 1
1 1
4 4
4 4 75
1 2
76
4
4
0 1
q H t
q H t
C t C t t
q
q
q qmax
min
/β γ γ
.
Todi ˜maxa t( ) ≤ 0 na promiΩku [ 0, t* ], wo nemoΩlyvo. OtΩe, ˜maxa t( ) ≡ 0 pry
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1500 M. I. IVANÇOV, N. V. SALDINA
t ∈ [ 0, t0 ], de t0 = min { t1 , t2 , t* }. Zvidsy a ( t ) ≡ 0, t ∈ [ 0, t0 ] i u ( x, t ) ≡ 0,
v ( x, t ) ≡ 0, x ∈ [ 0, h ], t ∈ [ 0, t0 ].
Teoremu dovedeno.
ZauvaΩennq. Dovedenu teoremu isnuvannq ta [dynosti rozv’qzku zadaçi (1) –
(4) moΩna vykorystaty dlq doslidΩennq analohiçno] zadaçi z inßymy krajovy-
my umovamy ta umovamy perevyznaçennq. Dijsno, rozhlqnemo zadaçu (1) z poçat-
kovog umovog (2), krajovymy umovamy ux ( 0, t ) = µ1 ( t ), ux ( h, t ) = µ2 ( t ) ta dodat-
kovog umovog u ( 0, t ) = µ3 ( t ). Todi zaminamy ux ( x, t ) = v ( x, t ) ta u ( x, t ) = µ3 ( t ) +
+
v( )∫ η η, t d
x
0
dana zadaça zvodyt\sq do tako]:
vt =
a t b x t b x t c x t c x t t t dxx x x x
x
( ) + ( ) + ( ) + ( ) + ( ) ( ) + ( )
( ) ∫v v v v, , , , ,µ η η3
0
+
+ f x tx( ), , ( x, t ) ∈ QT ,
v ( x, 0 ) = ϕ′ ( x ), x ∈ [ 0, h ],
v ( 0, t ) = µ1 ( t ), v ( h, t ) = µ2 ( t ), t ∈ [ 0, T ],
a ( t ) vx ( 0, t ) = µ4 ( t ), t ∈ [ 0, T ],
de µ4 ( t ) ≡ ′ ( )µ3 t – b ( 0, t ) µ1 ( t ) – c ( 0, t ) µ3 ( t ) – f ( 0, t ).
Lehko baçyty, wo naqvnist\ intehral\noho dodanka v rivnqnni ne vplyva[ na
doslidΩennq dano] zadaçi analohiçno do zadaçi (1) – (4).
1. Ivançov M. I., Saldina N. V. Obernena zadaça dlq rivnqnnq teploprovidnosti z vyrodΩennqm
// Ukr. mat. Ωurn. – 2005. – 57, # 11. – S. 1563 – 1570.
2. Jones B. F. The determination of a coefficient in a parabolic differential equation. Part I // J. Math.
and Mech. – 1962. – 11, # 6. – P. 907 – 918.
3. HadΩyev M. M. Obratnaq zadaça dlq v¥roΩdagwehosq πllyptyçeskoho uravnenyq // Pry-
menenye metodov funkcyon. analyza v uravnenyqx mat. fyzyky. – Novosybyrsk, 1987. – S. 66
– 71.
4. Eldesbaev T. O nekotor¥x obratn¥x zadaçax dlq v¥roΩdagwyxsq hyperbolyçeskyx urav-
nenyj // Dyfferenc. uravnenyq. – 1976. – 11, # 3. – S. 502 – 510.
5. Eldesbaev T. Ob odnoj obratnoj zadaçe dlq v¥roΩdagwehosq hyperbolyçeskoho uravne-
nyq vtoroho porqdka // Yzv. AN KazSSR. Ser. fyz.-mat. – 1987. – # 3. – S. 27 – 29.
6. Saldina N. Obernena zadaça dlq paraboliçnoho rivnqnnq z vyrodΩennqm // Visn. L\viv. un-
tu. Ser. mex.-mat. – 2005. – Vyp. 64. – S. 245 – 257.
7. Lad¥Ωenskaq O. A., Solonnykov V. A., Ural\ceva N. N. Lynejn¥e y kvazylynejn¥e uravne-
nyq parabolyçeskoho typa. – M.: Nauka, 1967. – 736 s.
8. Ivanchov M. Inverse problems for equations of parabolic type. – Lviv: VNTL Publ., 2003. – 238 p.
9. Bekkenbax ∏., Bellman R. Neravenstva. – M.: Myr, 1965. – 276 s.
10. Ivanchov M., Saldina N. An inverse problem for strongly degenerate neat equation // J. Inv. Ill-
Posed Problems. – 2006. – 14, # 5. – P. 465 – 480.
OderΩano 06.03.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
|
| id | umjimathkievua-article-3549 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:44:38Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/90/49aae789bbca0ba116c77e56870a2290.pdf |
| spelling | umjimathkievua-article-35492020-03-18T19:57:29Z Inverse problem for a parabolic equation with strong power degeneration Обернена задача для параболічного рівняння з сильним степеневим виродженням Ivanchov, N. I. Saldina, N. V. Іванчов, М. І. Салдіна, Н. В. We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. Розглянуто обернену задачу визначення залежного від часу коефіцієнта при старшій похідній у повному параболічному рівнянні, який дорівнює нулю у початковий момент часу. Встановлено умови існування та єдиності класичного розв'язку вказаної задачі. Institute of Mathematics, NAS of Ukraine 2006-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3549 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 11 (2006); 1487–1500 Український математичний журнал; Том 58 № 11 (2006); 1487–1500 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3549/3833 https://umj.imath.kiev.ua/index.php/umj/article/view/3549/3834 Copyright (c) 2006 Ivanchov N. I.; Saldina N. V. |
| spellingShingle | Ivanchov, N. I. Saldina, N. V. Іванчов, М. І. Салдіна, Н. В. Inverse problem for a parabolic equation with strong power degeneration |
| title | Inverse problem for a parabolic equation with strong power degeneration |
| title_alt | Обернена задача для параболічного рівняння з сильним степеневим виродженням |
| title_full | Inverse problem for a parabolic equation with strong power degeneration |
| title_fullStr | Inverse problem for a parabolic equation with strong power degeneration |
| title_full_unstemmed | Inverse problem for a parabolic equation with strong power degeneration |
| title_short | Inverse problem for a parabolic equation with strong power degeneration |
| title_sort | inverse problem for a parabolic equation with strong power degeneration |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3549 |
| work_keys_str_mv | AT ivanchovni inverseproblemforaparabolicequationwithstrongpowerdegeneration AT saldinanv inverseproblemforaparabolicequationwithstrongpowerdegeneration AT ívančovmí inverseproblemforaparabolicequationwithstrongpowerdegeneration AT saldínanv inverseproblemforaparabolicequationwithstrongpowerdegeneration AT ivanchovni obernenazadačadlâparabolíčnogorívnânnâzsilʹnimstepenevimvirodžennâm AT saldinanv obernenazadačadlâparabolíčnogorívnânnâzsilʹnimstepenevimvirodžennâm AT ívančovmí obernenazadačadlâparabolíčnogorívnânnâzsilʹnimstepenevimvirodžennâm AT saldínanv obernenazadačadlâparabolíčnogorívnânnâzsilʹnimstepenevimvirodžennâm |