Inverse problem for the heat equation with degeneration
We consider the inverse problem of determining the time-dependent thermal diffusivity that 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.
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| Дата: | 2005 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3708 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509839101263872 |
|---|---|
| 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 | 2020-03-18T20:02:37Z |
| description | We consider the inverse problem of determining the time-dependent thermal diffusivity that 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 | 2026-03-24T02:47:28Z |
| format | Article |
| fulltext |
UDK 517.95
M.�I.�Ivançov, N.�V.�Saldina (L\viv. nac. un-t)
OBERNENA ZADAÇA DLQ RIVNQNNQ
TEPLOPROVIDNOSTI Z VYRODÛENNQM
We consider an inverse problem of the determination of time-dependent heat conduction coefficient
which vanishes at the initial time. We establish conditions for the existence and uniqueness of the
classical solution of the problem considered.
Rozhlqnuto obernenu zadaçu vyznaçennq zaleΩnoho vid çasu koefici[nta temperaturoprovid-
nosti, qkyj dorivng[ nulg u poçatkovyj moment çasu. Vstanovleno umovy isnuvannq ta [dynosti
klasyçnoho rozv’qzku vkazano] zadaçi.
Do paraboliçnyx rivnqn\ z vyrodΩennqm zvodyt\sq rqd praktyçno vaΩlyvyx
zadaç, sered qkyx zadaça pro oprisnennq mors\kyx vod, rux ridyn i haziv u porys-
tomu seredovywi, qvywa v plazmi [1, 2] ta in. Prqmi zadaçi dlq paraboliçnyx
rivnqn\ z vyrodΩennqm doslidΩuvalys\ bahat\ma avtoramy [3 – 11]. Prote ober-
nenym zadaçam dlq c\oho praktyçno vaΩlyvoho klasu zadaç prydilqlos\ malo
uvahy. Sered ostannix moΩna bulo b vidznaçyty zadaçu vyznaçennq vil\noho
çlena v eliptyçnomu rivnqnni z vyrodΩennqm [12].
U danij roboti doslidΩu[t\sq obernena zadaça dlq rivnqnnq teploprovid-
nosti, v qkomu nevidomyj zaleΩnyj vid çasu koefici[nt temperaturoprovidnosti
prqmu[ do nulq pry t → 0 za stepenevym zakonom. Rozhlqnuto vypadok slab-
koho vyrodΩennq, dlq qkoho znajdeno umovy isnuvannq ta [dynosti hladkoho
(klasyçnoho) rozv’qzku.
1. Formulgvannq zadaçi ta osnovni rezul\taty. V oblasti QT ≡ { ( x , t ) :
0 < x < h , 0 < t < T } rozhlqda[mo rivnqnnq teploprovidnosti
ut = a ( t ) uxx + f ( x , t ) (1)
z nevidomym koefici[ntom a ( t ) > 0, t ∈ ( 0 , T ] , z 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)
Pid rozv’qzkom zadaçi (1) – (4) rozumi[mo paru funkcij ( a ( t ) , u ( x , t ) ) z kla-
su C [ 0 , T ] × C2,
1
( QT ) ∩ C QT
1 0, ( ) , a ( t ) > 0, t ∈ ( 0 , T ] , takyx, wo isnu[ hranycq
lim ( )
t
a t
t→+0 β > 0, de 0 < β < 1 — zadane çyslo, i zadovol\nqgt\sq rivnqnnq (1) ta
umovy (2) – (4).
Otrymani umovy isnuvannq ta [dynosti rozv’qzku zadaçi (1) – (4) sformul\o-
vano u nastupnyx teoremax.
Teorema 1. Nexaj vykonugt\sq umovy:
1) ϕ ∈ C1
[ 0 , h ] , µi ∈ C1
[ 0 , T ] , i = 1, 2, µ3 ∈ C [ 0 , T ] , f ∈ C QT
1 0, ( ) ;
2) ϕ′ ( x ) > 0, x ∈ [ 0 , h ] , ′µ1( )t ≤ 0, ′µ2( )t ≥ 0, t ∈ [ 0 , T ] , µ3 ( t ) > 0, t ∈ ( 0 , T ] ,
isnu[ hranycq lim
( )
t
t
t→+0
3µ
β > 0, f ( x , t ) ≥ 0, ( x , t ) ∈ QT ;
3) ϕ ( 0 ) = µ1 ( 0 ) , ϕ ( h ) = µ2 ( 0 ) .
Todi isnu[ rozv’qzok zadaçi (1) – (4).
© M.7I.7IVANÇOV, N.7V.7SALDINA, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11 1563
1564 M.7I.7IVANÇOV, N.7V.7SALDINA
Teorema 2. Nexaj vykonugt\sq umovy:
1) ϕ ∈ C2
[ 0 , h ] , µi ∈ C1
[ 0 , T ] , i = 1, 2, f ∈ C QT
1 0, ( ) ;
2)
µ
β
3( )t
t
≠ 0, t ∈ ( 0 , T ] .
Todi rozv’qzok zadaçi (1) – (4) [ [dynym.
2. Dovedennq teoremy isnuvannq rozv’qzku. Poznaçymo çerez G x tk ( , , , )ξ τ ,
k = 1, 2, funkci] Hrina perßo] ta druho] krajovo] zadaçi dlq rivnqnnq (1), qki
magt\ vyhlqd
Gk ( x , t , ξ , τ ) = 1
2
2
4
2
π θ θ τ
ξ
θ θ τ( ) ( )( ) ( )
exp
( )
( ) ( )t
x nh
tn−
− − +
−
=−∞
∞
∑ +
+ ( ) exp
( )
( ) ( )( )
− − + +
−
1
2
4
2
k x nh
t
ξ
θ θ τ
, (5)
de θ ( t ) = a d
t
( )τ τ
0∫ . Tymçasovo vvaΩagçy funkcig a ( t ) vidomog, rozv’qzok za-
daçi (1) – (3) podamo u vyhlqdi
u ( 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( , , , ) ( ) ( ) ( , , , ) ( , )∫ ∫∫+ . (6)
Dlq pidstanovky v umovu perevyznaçennq prodyferencig[mo cej vyraz.
Vykorystovugçy vlastyvosti funkci] Hrina
G x t G x tx1 2( , , , ) ( , , , )ξ τ ξ τξ= − , G x t
G x t
a2
2
ξξ
τξ τ ξ τ
τ
( , , , )
( , , , )
( )
= − , (7)
intehrugçy çastynamy i beruçy do uvahy umovy uzhodΩenosti, znaxodymo
ux ( x , t ) = G x t d G x t d
h t
2
0
2 1
0
0 0( , , , ) ( ) ( , , , ) ( )ξ ϕ ξ ξ τ µ τ τ′ − ′∫ ∫ +
+ G x t h d G x t f d d
t
x
ht
2 2
0
1
00
( , , , ) ( ) ( , , , ) ( , )τ µ τ τ ξ τ ξ τ ξ τ′ +∫ ∫∫ .
Pidstavyvßy cej vyraz v umovu perevyznaçennq (4), otryma[mo rivnqnnq wodo
a ( t ) :
a ( t ) = µ ξ ϕ ξ ξ τ µ τ τ3 2
0
2 1
0
0 0 0 0( ) ( , , , ) ( ) ( , , , ) ( )t G t d G t d
h t
′ − ′
∫ ∫ +
+ G t h d G t f d d
t
x
ht
2 2
0
1
00
1
0 0( , , , ) ( ) ( , , , ) ( , )τ µ τ τ ξ τ ξ τ ξ τ′ +
∫ ∫∫
−
, t ∈ [ 0 , T ] . (8)
OtΩe, obernenu zadaçu (1) – (4) zvedeno do rivnqnnq (8). Dlq dovedennq isnu-
vannq rozv’qzku rivnqnnq (8) zastosu[mo do n\oho teoremu Íaudera pro neruxo-
mu toçku cilkom neperervnoho operatora.
Vstanovymo ocinky rozv’qzkiv rivnqnnq (8). Vraxovugçy umovy teoremy ta
rivnist\
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
OBERNENA ZADAÇA DLQ RIVNQNNQ TEPLOPROVIDNOSTI Z VYRODÛENNQM 1565
G x t d
h
2
0
0( , , , )ξ ξ∫ = 1, (9)
qku lehko pereviryty, vyxodqçy z oznaçennq (5) funkci] Hrina, ma[mo
G t d
h
2
0
0 0( , , , ) ( )ξ ϕ ξ ξ′∫ ≥ min ( ) ( , , , ) min ( )
[ , ] [ , ]0 2
0
0
0 0
h
h
h
x G t d x′ = ′∫ϕ ξ ξ ϕ .
Todi otryma[mo ocinku dlq a ( t ) zverxu:
a ( t ) ≤
µ
ϕ
3
0
( )
min ( )
[ , ]
t
x
h
′
≤ A1 t
β, (10)
de A1 — dodatna stala.
Dlq ocinky funkci] a ( t ) znyzu spoçatku ocinymo koΩnyj dodanok znamen-
nyka (8). Vraxovugçy (9), znaxodymo
G t d
h
2
0
0 0( , , , ) ( )ξ ϕ ξ ξ′∫ ≤ max ( )
[ , ]0 h
x′ϕ .
Z obmeΩenosti funkci] G t h2 0( , , , )τ ma[mo
G t h d
t
2 2
0
0( , , , ) ( )τ µ τ τ′∫ ≤ C1 .
Nerivnist\ dlq funkci] Hrina [13]
G2 ( 0 , t , 0 , τ ) ≤ 1
π θ θ τ( )( ) ( )t −
+ C2
da[ moΩlyvist\ vstanovyty ocinku
G t d
t
2 1
0
0 0( , , , ) ( )τ µ τ τ′∫ ≤ C d
t
t
3
0
τ
θ θ τ( ) ( )−∫ + C4
z vidomymy stalymy C3 , C4 > 0. Vykorystavßy vlastyvosti funkci] Hrina, oci-
nymo ostannij intehral z (8):
G t f d dx
ht
1
00
0( , , , ) ( , )ξ τ ξ τ ξ τ∫∫ ≤ max ( , ) ( , , , )
Q
ht
T
f x t G t d d−( )∫∫ 2
00
0ξ ξ τ ξ τ =
= max ( , ) ( , , , ) ( , , , )
Q
t
T
f x t G t G t h d2 2
0
0 0 0τ τ τ−( )∫ ≤ C d
t
t
5
0
τ
θ θ τ( ) ( )−∫ .
OtΩe, funkciq a ( t ) obmeΩena znyzu vyrazom
a ( t ) ≥
µ
τ
θ θ τ
3
6 7
0
( )
( ) ( )
t
C C d
t
t
+
−∫
. (11)
Vvedemo poznaçennq
a0 ( t ) = a t
t
( )
β , min ( )
[ , ]0 0T
a t = amin , µ0 ( t ) =
µ
β
3( )t
t
. (12)
Zhidno z cymy poznaçennqmy z (11) otryma[mo
a0 ( t ) t
β ≥
µ
τ
σ σ σ
β
β
τ
0
6 7
00
( )
( )
t t
C C d
a d
t
t
+
∫
∫
≥
µ
β τ
τ
β
β β
0
6 7 1 1
0
1
( )
min
t t
C C
a
d
t
t
+ +
−+ +∫
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1566 M.7I.7IVANÇOV, N.7V.7SALDINA
Vykonavßy v intehrali zaminu zminnyx z = τ
t
, oderΩymo
d
t
t τ
τβ β+ +−∫ 1 1
0
= t dz
z
1
2
1
0
1
1
−
+−∫
β
β
≤ C8 , t ∈ [ 0 , T ] .
Povertagçys\ do ocinky a ( t ) znyzu, pryxodymo do nerivnosti
a0 ( t ) ≥
C
C
C
a
9
6
10+
min
.
Nerivnist\ spravdΩu[t\sq dlq vsix t ∈ [ 0 , T ] , vidpovidno dlq amin otrymu[mo
nerivnist\
C6 amin + C10 amin – C9 ≥ 0,
z qko] znaxodymo
amin ≥
2
4
9
10
2
6 9 10
2
C
C C C C+ +
≡ A0 > 0.
OtΩe, dlq funkci] a ( t ) vykonu[t\sq nerivnist\
a ( t ) ≥ A0 t
β, t ∈ [ 0 , T ] , (13)
de A0 — vidoma stala.
Dovedemo isnuvannq hranyci lim ( )
t
a t
t→+0 β > 0. Dlq c\oho skorysta[mosq formu-
lamy (12), qki pidstavymo v umovu perevyznaçennq
a0 ( t ) =
µ0
0
( )
( , )
t
u tx
.
Isnuvannq potribno] hranyci zaleΩyt\ vid isnuvannq lim ( , )
t xu t
→+0
0 . Z umov teore-
my, vlastyvostej intehrala Puassona ta vstanovlenyx vywe ocinok vyplyva[
isnuvannq hranyci
lim ( , , , ) ( )
t
h
G t d
→+
′∫0 2
0
0 0ξ ϕ ξ ξ = ϕ′ ( 0 ) .
OtΩe,
lim ( )
t
a t
t→+0 β = lim ( )
t
a t
→+0 0 = lim
( )
( , )t
x
t
u t→+0
0
0
µ
=
µ
ϕ
0
0
( )
( )
t
′
> 0.
Vyznaçymo mnoΩynu N = a t C T A a t
t
A( ) [ , ] : ( )∈ ≤ ≤{ }0 0 1β . Rozhlqnemo (8) qk
operatorne rivnqnnq a ( t ) = P a ( t ) wodo a ( t ) z operatorom P, qkyj vyznaça-
[t\sq rivnistg P a ( t ) =
µ3
0
( )
( , )
t
u tx
i, zhidno z ocinkamy (10), (13), vidobraΩa[ mno-
Ωynu N v sebe. PokaΩemo, wo operator P [ cilkom neperervnym na N .
Zhidno z teoremog Arcela, vstanovymo odnostajnu neperervnist\ mnoΩyny N .
Zadamo dovil\ne ε > 0 i vyznaçymo isnuvannq takoho δ > 0, wo
| P a ( t2 ) – P a ( t1 ) | < ε pry | t2 – t1 | < δ . (14)
ZvaΩagçy na te, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
OBERNENA ZADAÇA DLQ RIVNQNNQ TEPLOPROVIDNOSTI Z VYRODÛENNQM 1567
| P a ( t2 ) – P a ( t1 ) | =
µ µ3 2
2
3 1
10 0
( )
( , )
( )
( , )
t
u t
t
u tx x
− ≤
≤
µ µ µ3 2 3 1
2
3 1 2 1
2 10
0 0
0 0
( ) ( )
( , )
( ) ( , ) ( , )
( , ) ( , )
t t
u t
t u t u t
u t u tx
x x
x x
− + −( )
,
i beruçy do uvahy ocinku u t xx h
( , ) min ( )
[ , ]
0
0
≥ ′ϕ ≡ M1 > 0, nerivnist\ (14) dovodymo
tak samo, qk i v nevyrodΩenomu vypadku [13].
OtΩe, umovy teoremy Íaudera dlq rivnqnnq (8) vykonugt\sq, i tomu isnu[
rozv’qzok a = a ( t ) rivnqnnq (8) z klasu C [ 0 , T ] , qkyj zadovol\nq[ ocinky (10),
(13) i dlq qkoho isnu[ dodatna hranycq lim ( )
t
a t
t→+0 β . Pidstavlqgçy joho v rivnqn-
nq (1), za formulog (6) znaxodymo funkcig u = u ( x , t ) , qka, qk ce vyplyva[ z
dovedennq teoremy, ma[ potribnu hladkist\.
Teoremu 1 dovedeno.
ZauvaΩennq 1. Teorema 1 poßyrg[t\sq i na vypadok β = 1, u çomu lehko
perekonatys\, povtoryvßy ]] dovedennq. Krim c\oho, teorema 1 zalyßa[t\sq
spravedlyvog, qkwo zamist\ analohiçnyx umov teoremy prypustyty, wo ϕ′ ( x ) ≥
≥ 0, x ∈ [ 0 , h ] , ′µ1( )t < 0, t ∈ [ 0 , T ] . Ocinka znamennyka rivnqnnq (8) matyme
vyhlqd
ux ( 0 , t ) ≥ − ′ ≥ − ′( )
−∫ ∫G t d t d
t
t
T
t
2 1
0
0 1
0
0 0 1( , , , ) ( ) min ( )
( ) ( )[ , ]
τ µ τ τ
π
µ τ
θ θ τ
≥
≥ C d
t
C
t
11 2 2
0
11 2
τ
τ
π
−
=∫ > 0.
Inßi mirkuvannq z dovedennq teoremy 1 zalyßagt\sq bez zmin.
3. Dovedennq teoremy [dynosti rozv’qzku. Prypustymo, wo isnugt\ dva
rozv’qzky ( ai ( t ) , ui ( x , t )) , i = 1, 2, zadaçi (1) – (4). Dlq riznyci rozv’qzkiv a ( t ) ≡
≡ a1 ( t ) – a2 ( t ) , u ( x , t ) ≡ u1 ( x , t ) – u2 ( x , t ) otryma[mo zadaçu
ut = a1 ( t ) uxx + a ( t ) u2xx , ( x , t ) ∈ QT , (15)
u ( x , 0 ) = 0, x ∈ [ 0 , h ] , (16)
u ( 0 , t ) = u ( h , t ) = 0, t ∈ [ 0 , T ] , (17)
a1 ( t ) ux ( 0 , t ) = – a ( t ) u2x ( 0 , t ) , t ∈ [ 0 , T ] . (18)
Za dopomohog funkci] Hrina G x t1
1( )( , , , )ξ τ rozv’qzok zadaçi (15) – (17) poda-
mo u vyhlqdi
u ( x , t ) = G x t a u d d
ht
1
1
2
00
( )( , , , ) ( ) ( , )ξ τ τ ξ τ ξ τξξ∫∫ . (19)
Pidstavlqgçy (19) v umovu (18), otrymu[mo intehral\ne rivnqnnq wodo a ( t ) :
a ( t ) u2x ( 0 , t ) = − ∫∫a t G t a u d dx
ht
1 1
1
2
00
0( ) ( , , , ) ( ) ( , )( ) ξ τ τ ξ τ ξ τξξ , t ∈ [ 0 , T ] . (20)
Podamo a ( t ) u vyhlqdi (12). Todi rivnqnnq (20) perepyßet\sq tak:
a0 ( t ) = − ∫∫a t
t u t
G t a u d d
x
x
ht
1
2
1
1
0 2
000
0
( )
( , )
( , , , ) ( ) ( , )( )
β
β
ξξξ τ τ τ ξ τ ξ τ , (21)
abo
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1568 M.7I.7IVANÇOV, N.7V.7SALDINA
a0 ( t ) = K t a d
t
( , ) ( )τ τ τ0
0
∫ , t ∈ [ 0 , T ] , (22)
de
K ( t , τ ) = − ∫a t
t u t
G t u d
x
x
h
1
2
1
1
2
00
0
( )
( , )
( , , , ) ( , )( )τ ξ τ ξ τ ξ
β
β ξξ .
Vstanovymo ocinku dlq qdra K ( t , τ ) . Dlq c\oho zapyßemo rozv’qzok u2 ( x , t )
zadaçi (1) – (3) u vyhlqdi (6), vykorystavßy funkcig Hrina G1
2( ) dlq rivnqnnq
ut = a2 ( t ) uxx . Vzqvßy do uvahy vlastyvosti funkci] Hrina (7) ta prointehruvav-
ßy çastynamy, znajdemo
u2xx ( x , t ) = G x t d G x t f d
h t
1
2
0
1
2
1
0
0 0 0( ) ( )( , , , ) ( ) ( , , , ) ( ) ( , )ξ ϕ ξ ξ τ µ τ τ τξ′′ + ′ −( )∫ ∫ +
+ G x t h f h d G x t f d d
t ht
1
2
2
0
1
2
00
ξ ξ ξτ τ µ τ τ ξ τ ξ τ ξ τ( ) ( )( , , , ) ( , ) ( ) ( , , , ) ( , )− ′( ) −∫ ∫∫ . (23)
Ocinymo koΩnyj dodanok formuly (23). Vraxovugçy (9), znaxodymo
G x t d
h
1
2
0
0( )( , , , ) ( )ξ ϕ ξ ξ′′∫ ≤ max ( )
[ , ]0 h
x′′ϕ .
Vykorystovugçy zobraΩennq funkci] Hrina (5), otrymu[mo
G x t f d
t
1
2
1
0
0 0ξ τ µ τ τ τ( )( , , , ) ( ) ( , )′ −( )∫ ≤ 1
2
0
0
1π
µmax ( ) ( , )
[ , ]T
t f t′ − ×
× 1 2
2
42 2
3 2
0
2
2 2θ θ τ θ θ τ
τ
( ) ( )
( ) exp
( )
( ) ( )/t
x nh
x nh
t
d
t
n−( )
+ − +
−( )
∫ ∑
=−∞
∞
.
Vydilymo dodanok, wo vidpovida[ n = 0, i rozhlqnemo intehral
I1 ≡ x
t
x
t
d
t
θ θ τ θ θ τ
τ
2 2
3 2
0
2
2 24( ) ( )
exp
( ) ( )/−( )
−
−( )
∫ .
Z oznaçennq funkci] θ ( t ) ta vlastyvostej funkci] a2 ( t ) otrymu[mo
I1 ≤ C x
t
x
C t
d
t
1 1 1 3 2
0
2
2
1 1β β β β
τ τ
τ
+ + + +
−( )
−
−( )
∫ / exp .
V ostann\omu intehrali vykona[mo zaminu zminnyx z = x
C t2
1 1( )β βτ+ +−
. Todi
I1 ≤ C
z dz
t x
C z
x
C t
3
2
1
2
2
2
1
2
1
exp( )−
−
+ +
∞
+
∫
β
β
β
β
=
=
C
t
z z dz
z x
C t
z x
C t
x
C t
3
2
1 2
2
1
1
2
1
1
2
1
β
β
β
β
β
β
β
β
β
β
+
+
+
+
+
∞ −
−
+
+
∫ exp( )
≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
OBERNENA ZADAÇA DLQ RIVNQNNQ TEPLOPROVIDNOSTI Z VYRODÛENNQM 1569
≤
C
t
z z dz
z x
C t
x
C t
3
1 2
2
1
1
2
1
β
β
β
β
β
β
β
+
+
+
∞ −
−
+
∫ exp( )
.
Zaminog z – x
C t2
1β+
= y zvedemo ocinku I1 do vyhlqdu
I1 ≤
C
t
y y x
C t
y x
C t
dy3 1
2
1
1
2
1
2
0
β
β
β
β
β
β
β
−
+
+
+
+
∞
+
− +
∫ exp .
Vykorysta[mo nerivnist\ x ep qx− 2
≤ Cp q, , x ≥ 0, p ≥ 0, q > 0. Todi
I1 ≤
C
t
y y x
C t
dy
C
t
y
y
dy4 1
2
1
2
0
4 1
2
0
1
2 2β
β
β
β β
β
β
−
+
+
∞ −
+
∞
− +
≤ −
∫ ∫exp exp ≤
C
t
5
β .
Analohiçno do poperedn\oho ma[mo
G x t h f h d
t
1
2
2
0
ξ τ τ µ τ τ( )( , , , ) ( , ) ( )− ′( )∫ ≤
C
t
6
β .
Rozhlqnemo intehral
G x t f d d
ht
1
2
00
ξ ξξ τ ξ τ ξ τ( )( , , , ) ( , )∫∫ ≤ 1
4
1
2 2
3 2
00
π θ θ τ
max ( , )
( ) ( ) /Q
x
ht
T
f x t
t −( )∫∫ ×
× x nh
x nh
t
x nh
n
− + − − +
−( )
+ + +
=−∞
∞
∑ ξ ξ
θ θ τ
ξ2
2
4
2
2
2 2
exp
( )
( ) ( )
×
× exp
( )
( ) ( )
− + +
−( )
x nh
t
d d
ξ
θ θ τ
ξ τ2
4
2
2 2
≡ I2, 1 + I2, 2 .
Zaminog z =
x nh
t
− +
−
ξ
θ θ τ
2
2 2 2( ) ( )
peretvorymo I2, 1 :
I2, 1 ≤ C d
t
z z dz
t
n x n h
t
x nh
t
7
2 20 2 1
2
2
2
2
2 2
2 2τ
θ θ τ
θ θ τ
θ θ τ
( ) ( )
exp
( )
( ) ( )
( ) ( )
−
−( )∫∑ ∫
=−∞
∞
+ −
−
+
−
≤
≤ C d
t
z z dz C d
t
C t
t t
7
2 20
2
8
2 20
9
1
2τ
θ θ τ
τ
θ θ τ
β
( ) ( )
exp
( ) ( )−
−( ) ≤
−
≤∫ ∫ ∫
−∞
∞ −
.
Analohiçno ocing[mo I2, 2 . Ostatoçno otrymu[mo ocinku
| u2xx ( x , t ) | ≤
C
t
10
β , ( x , t ) ∈ [ 0 , h ] × ( 0 , T ] . (24)
Podamo a1 ( t ) u vyhlqdi a1 ( t ) = a01 ( t ) t
β. Z umovy perevyznaçennq (4) ta pry-
puwen\ teoremy ma[mo
u2x ( 0 , t ) =
µ3
2
( )
( )
t
a t
≠ 0, t ∈ [ 0 , T ] .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1570 M.7I.7IVANÇOV, N.7V.7SALDINA
Vraxovugçy ce, a takoΩ ocinku (24), vstanovlg[mo ocinku qdra intehral\noho
rivnqnnq (22)
| K ( t , τ ) | ≤
C
t
11
1 1β βτ+ +−
.
Todi
K t d
t
( , )τ τ
0
∫ ≤ C t d
12
1
2
1
0
1
1
−
+−∫
β
β
σ
σ
, (25)
tobto qdro K ( t , τ ) [ intehrovnym. Za vlastyvostqmy intehral\nyx rivnqn\
Vol\terra druhoho rodu rivnqnnq (22) ma[ lyße tryvial\nyj rozv’qzok a0 ( t ) ≡
≡ 0. Zvidsy a ( t ) ≡ 0, t ∈ [ 0 , T ] , i u ( x , t ) ≡ 0, ( x , t ) ∈ QT , qk rozv’qzok odno-
ridno] zadaçi (15) – (17).
Teoremu 2 dovedeno.
ZauvaΩennq. 2. Zhidno z ocinkog (25), dovedennq teoremy 2 ne7poßyrg[t\-
sq na vypadok β = 1.
3. Qkwo v teoremi 1 prypustyty, wo ϕ ∈ C2
[ 0 , h ] , to, povtorggçy doveden-
nq teoremy 1 ta vykorystovugçy vyhlqd druho] poxidno] uxx ( x , t ) z dovedennq
teoremy 2, lehko perekonatys\, wo zadaça (1) – (4) ma[ rozv’qzok ( a ( t ) , u ( x , t ) ) ,
komponenta a ( t ) ma[ taki Ω vlastyvosti, wo j u teoremi 1, a druha komponenta
u ( x , t ) naleΩyt\ do klasu C2,1
( [ 0 , h ] × ( 0 , T ] ) , pryçomu funkci] u t ( x , t ) ,
t
β
uxx ( x , t ) [ neperervnymy v QT .
1. Podhaev(A.(H. O kraev¥x zadaçax dlq nekotor¥x kvazylynejn¥x parabolyçeskyx uravnenyj
s neklassyçeskym v¥roΩdenyem // Syb. mat. Ωurn. – 1987. – 28, #72. – S.7129 – 139.
2. Caffarelli L. A., Friedman A. Continuity of the density of a gas flow in a porous medium // Trans.
Amer. Math. Soc. – 1979. – 252. – P. 99 – 113.
3. Hlußko(V.(P. V¥roΩdagwyesq lynejn¥e dyfferencyal\n¥e uravnenyq // Dyfferenc.
uravnenyq. – 1968. – 4,7#711. – S.71957 – 1966.
4. Kalaßnykov(A.(S. O rastuwyx reßenyqx lynejn¥x uravnenyj vtoroho porqdka s neotryca-
tel\noj xarakterystyçeskoj formoj // Mat. zametky. – 1968. – 3, 7#72. – S.7171 – 178.
5. Kalaßnykov(A.(S. Zadaça bez naçal\n¥x uslovyj v klassax rastuwyx reßenyj dlq nekoto-
r¥x lynejn¥x v¥roΩdagwyxsq parabolyçeskyx system vtoroho porqdka. I // Vestn. Mosk.
un-ta. Ser. mat., mex. – 1971. – #72. – S.742 – 48.
6. Kalaßnykov(A.(S. Zadaça bez naçal\n¥x uslovyj v klassax rastuwyx reßenyj dlq nekoto-
r¥x lynejn¥x v¥roΩdagwyxsq parabolyçeskyx system vtoroho porqdka. II // Tam Ωe. –
#73. – S.73 – 9.
7. Olejnyk(O.(A., Radkevyç(E.(V. Uravnenyq vtoroho porqdka s neotrycatel\noj xarakterys-
tyçeskoj formoj // Ytohy nauky y texnyky. „Mat. analyz. 1969” / VYNYTY. – 1971. –
S.77 – 252.
8. DΩuraev(T.(D. O kraev¥x zadaçax dlq lynejn¥x parabolyçeskyx uravnenyj, v¥roΩdag-
wyxsq na hranyce oblasty // Mat. zametky. – 1972. – 12, #75. – S.7643 – 652.
9. Hlußko(V.(P. O razreßymosty smeßann¥x zadaç dlq parabolyçeskyx uravnenyj vtoroho
porqdka s v¥roΩdenyem // Dokl. AN SSSR. – 1972. – 207, #72. – S.7266 – 269.
10. Hlußak(A.(V., Ímulevyç(S.(D. O nekotor¥x korrektn¥x zadaçax dlq parabolyçeskyx urav-
nenyj v¥sokoho porqdka, v¥roΩdagwyxsq po vremennoj peremennoj // Dyfferenc. urav-
nenyq. – 1986. – 22, #76. – S.71065 – 1068.
11. Voznqk(O.(I., Ivasyßen(S.(D. Zadaça Koßi dlq paraboliçnyx system z vyrodΩennqm na
poçatkovij hiperplowyni // Dopov. NAN Ukra]ny. – 1994. – #76. – S.77 – 11.
12. HadΩyev(M.(M. Obratnaq zadaça dlq v¥roΩdagwehosq πllyptyçeskoho uravnenyq //
Prymenenye metodov funkcyon. analyza v uravnenyqx mat. fyzyky. – Novosybyrsk, 1987. –
S.766 – 71.
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OderΩano 15.07.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
|
| id | umjimathkievua-article-3708 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:47:28Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/63/4e5df45ee0dc80be279588205cd7ea63.pdf |
| spelling | umjimathkievua-article-37082020-03-18T20:02:37Z Inverse problem for the heat equation with degeneration Обернена задача для рівняння теплопровідності з виродженням Ivanchov, N. I. Saldina, N. V. Іванчов, М. І. Салдіна, Н. В. We consider the inverse problem of determining the time-dependent thermal diffusivity that 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 температуропровідності, який дорівнює нулю у початковий момент часу. Встановлено умови існування та єдиності класичного розв'язку вказаної задачі. Institute of Mathematics, NAS of Ukraine 2005-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3708 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 11 (2005); 1563–1570 Український математичний журнал; Том 57 № 11 (2005); 1563–1570 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3708/4141 https://umj.imath.kiev.ua/index.php/umj/article/view/3708/4142 Copyright (c) 2005 Ivanchov N. I.; Saldina N. V. |
| spellingShingle | Ivanchov, N. I. Saldina, N. V. Іванчов, М. І. Салдіна, Н. В. Inverse problem for the heat equation with degeneration |
| title | Inverse problem for the heat equation with degeneration |
| title_alt | Обернена задача для рівняння теплопровідності з виродженням |
| title_full | Inverse problem for the heat equation with degeneration |
| title_fullStr | Inverse problem for the heat equation with degeneration |
| title_full_unstemmed | Inverse problem for the heat equation with degeneration |
| title_short | Inverse problem for the heat equation with degeneration |
| title_sort | inverse problem for the heat equation with degeneration |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3708 |
| work_keys_str_mv | AT ivanchovni inverseproblemfortheheatequationwithdegeneration AT saldinanv inverseproblemfortheheatequationwithdegeneration AT ívančovmí inverseproblemfortheheatequationwithdegeneration AT saldínanv inverseproblemfortheheatequationwithdegeneration AT ivanchovni obernenazadačadlârívnânnâteploprovídnostízvirodžennâm AT saldinanv obernenazadačadlârívnânnâteploprovídnostízvirodžennâm AT ívančovmí obernenazadačadlârívnânnâteploprovídnostízvirodžennâm AT saldínanv obernenazadačadlârívnânnâteploprovídnostízvirodžennâm |