Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations
We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations.
Saved in:
| Date: | 2010 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2904 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508897880571904 |
|---|---|
| author | Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. |
| author_facet | Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. |
| author_sort | Aldashev, S. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:12Z |
| description | We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations. |
| first_indexed | 2026-03-24T02:32:31Z |
| format | Article |
| fulltext |
UDK 517.956
S. A. Aldaßev (Zap.-Kazax. ahrar.-texn. un-t, Ural\sk, Kazaxstan)
SUWESTVOVANYE SOBSTVENNÁX FUNKCYJ
SPEKTRAL|NOJ ZADAÇY TRYKOMY DLQ NEKOTORÁX
KLASSOV MNOHOMERNÁX SMEÍANNÁX
HYPERBOLO-PARABOLYÇESKYX URAVNENYJ
We show that there exists a denumerable set of eigen functions of the Tricomi spectral problem for
multidimensional mixed hyperbolic parabolic equations.
Pokazano, wo isnu[ zliçenna mnoΩyna vlasnyx funkcij spektral\no] zadaçi Trikomi dlq bahato-
vymirnyx mißanyx hiperbolo-paraboliçnyx rivnqn\.
Teoryq kraev¥x zadaç dlq hyperbolo-parabolyçeskyx uravnenyj na ploskosty
xoroßo yzuçena [1], a yx mnohomern¥e analohy, naskol\ko yzvestno avtoru,
yssledovan¥ malo [2].
Pust\ D — koneçnaq oblast\ evklydova prostranstva Em+1 toçek
( ), , ,x x tm1 … , ohranyçennaq v poluprostranstve t > 0 konusamy K x t0 : = ,
K x t1 1: = − , 0 1 2≤ ≤t / , a pry t < 0 cylyndryçeskoj poverxnost\g Γ =
= { }( , ) :x t x = 1 y ploskost\g t = t0 = const, hde x — dlyna vektora x =
= ( ), , ,x x xm1 2 … .
Oboznaçym çerez D+
y D−
çasty oblasty D, leΩawye sootvetstvenno v
poluprostranstvax t > 0 y t < 0 . Çasty konusov K0 , K1 , ohranyçyvagwyx
oblasty D+ , oboznaçym çerez S0 y S1
sootvetstvenno.
Pust\ S = { }( , ) : ,x t t x= < <0 0 1 , Γ0 = { }( , ) : ,x t t x= =0 1 .
V oblasty D rassmotrym smeßann¥e hyperbolo-parabolyçeskye uravnenyq
γ u =
∆ x tt i x
i
m
tu u a x t u b x t u c x t u t
i
− + + + >
=
∑ ( , ) ( , ) ( , ) ,
1
00
0
1
,
( , ) ( , ) , ,∆ x t i x
i
m
u u d x t u e x t u t
i
− + + <
=
∑
(1)
hde γ — dejstvytel\noe çyslo, ∆ x — operator Laplasa po peremenn¥m
x x xm1 2, , ,… , m ≥ 2.
Sleduq [1], v kaçestve mnohomernoj spektral\noj zadaçy Trykomy rassmot-
rym sledugwug zadaçu.
Zadaça''Tγγγγ . Najty reßenye uravnenyq (1) v oblasty D pry t ≠ 0 yz klassa
C D C D C D D( ) ( )\ ( )Γ0
1 2∩ ∩ ∪+ − , udovletvorqgwee kraev¥m uslovyqm
© S. A. ALDAÍEV, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 723
724 S. A. ALDAÍEV
u
S0
= 0, u Γ = 0. (2)
V dal\nejßem nam udobno perejty ot dekartov¥x koordynat x x tm1, , ,… k
sferyçeskym r tm, , , ,θ θ1 1… − , r ≥ 0 , 0 21≤ <θ π , 0 ≤ ≤θ πi , i = …2 3, ,
… −, m 1.
Pust\ { }, ( )Yn m
k θ — systema lynejno nezavysym¥x sferyçeskyx funkcyj po-
rqdka n, 1 ≤ ≤k kn ,
( )! ! ( )! ( )m n k n m n mn− = + − + −2 3 2 2 , θ θ θ= … −( , , )1 1m ,
W Sl
2 ( ) , l = …0 1, , , — prostranstva Soboleva.
Ymeet mesto sledugwaq lemma [3].
Lemma. Pust\ f r W Sl( , ) ( )θ ∈ 2 . Esly l m≥ − 1, to rqd
f r( , )θ =
n
n
k
n m
k
k
k
f r Y
n
=
∞
=
∑ ∑
0 1
( ) ( ), θ , (3)
a takΩe rqd¥, poluçenn¥e yz neho dyfferencyrovanyem porqdka p l m≤ − + 1,
sxodqtsq absolgtno y ravnomerno.
Çerez �a r tin
k ( , ) , a r tin
k ( , ) , �b r tn
k ( , ) , �c r tn
k ( , ) , �d r tin
k ( , ) , d r tin
k ( , ) , �e r tn
k ( , ) , ρn
k ,
τn
k r( ) , νn
k r( ) oboznaçym koπffycyent¥ rqda (3) sootvetstvenno funkcyj
a t ti ( , , ) ( )θ ρ θ , a
x
ri
i ρ , b r t( , , )θ ρ , c r t( , , )θ ρ , d r ti ( , , )θ ρ , d
x
ri
i ρ , e r t( , , )θ ρ ,
i m= …1, , , ρ θ( ) , τ θ θ( , ) ( , , )r u r= 0 , ν θ θ( , ) ( , , )r u rt= 0 , pryçem ρ θ( ) ∈
∈ C H∞( ) , H — edynyçnaq sfera v Em
.
Pust\ a x ti ( , ) , b x t( , ) , c x t W Dl( , ) ( )∈ +
2 , d x ti ( , ) , e x t W D C Dl( , ) ( ) ( )∈ ⊂− −
2 ,
l m≥ + 1, pry πtom d x e xi ( , ) ( , )0 0 0= = , 0 1< <x , i m= …1, , .
Tohda spravedlyva sledugwaq teorema.
Teorema. Zadaça T γ dlq kaΩdoho γ ymeet sçetnoe mnoΩestvo sobstven-
n¥x funkcyj.
Dokazatel\stvo. V sferyçeskyx koordynatax uravnenye (1) v oblasty D+
ymeet vyd
L u0( ) ≡ u
m
r
u
r
u u a r t urr r tt i x
i
m
i
+
−
− − +
=
∑1 1
2
1
δ θ( , , ) +
+ b r t u c r t ut( , , ) ( , , )θ θ+ = γ u , (4)
δ ≡ –
1
1
1
1
1
g j
m j
j j
m j
j
jj
m
sin
sin
− −
− −
=
− ∂
∂
∂
∂
θ θ
θ
θ∑∑ ,
g1 = 1, g j = ( )sin sinθ θ1 1
2… −j , j > 1.
Yzvestno [3], çto spektr operatora δ sostoyt yz sobstvenn¥x çysel λn =
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
SUWESTVOVANYE SOBSTVENNÁX FUNKCYJ SPEKTRAL|NOJ ZADAÇY … 725
= n n m( )+ − 2 , n = …0 1, , , kaΩdomu yz kotor¥x sootvetstvuet kn ortonor-
myrovann¥x sobstvenn¥x funkcyj Yn m
k
, ( )θ .
Pry t → − 0 na S poluçym funkcyonal\noe sootnoßenye meΩdu τ θ( , )r
y ν θ( , )r vyda
τ τ δτ γ τrr r
m
r r
+
−
− −
1 1
2
= ν θ( , )r . (5)
Reßenye zadaçy Tγ v oblasty D+
budem yskat\ v vyde
u r t( , , )θ =
n k
k
n
k
n m
k
n
u r t Y
=
∞
=
∑ ∑
0 1
( , ) ( ), θ , (6)
hde u r tn
k ( , ) — funkcyy, podleΩawye opredelenyg. Podstavyv (6) v (4), a za-
tem umnoΩyv poluçennoe v¥raΩenye na ρ θ( ) ≠ 0 y proyntehryrovav po edy-
nyçnoj sfere H, dlq un
k
poluçym [4, 5]
ρ ρ ρ0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
u u
m
r
a urr tt
i
m
i− +
−
+
=
∑ rr tb u c u u1
0
1
0
1
0
1
0
1
0
1
0
1+ + −� � γρ +
+
n k
k
n
k
nrr
k
n
k
ntt
k
n
k
i
mn
u u
m
r
a
=
∞
= =
∑ ∑ ∑− +
−
+
1 1 1
1
ρ ρ ρ iin
k
nr
k
n
k
nt
ku b u
+
�
+
+ � �c
r
a na un
k
n
n
k
in
k
in
k
i
m
n− + −( )
−
=
∑λ
ρ
2 1
1
kk
n
k
n
ku−
γρ = 0. (7)
Teper\ rassmotrym beskoneçnug systemu dyfferencyal\n¥x uravnenyj
ρ ρ ρ0
1
0
1
0
1
0
1
0
1
0
11
u u
m
r
urr tt r− +
−
= γρ0
1
0
1u , (8)
ρ ρ ρ λ
ρ
γρ1 1 1 1 1 1 1
1
2 1
1k
rr
k k
tt
k k
r
k
k
ku u
m
r
u
r
u− +
−
− − 11 1
k ku =
= –
1
1 1
0
1
0
1
0
1
0
1
0
1
0
1
k
a u b u c u
i
m
i r t
=
∑ + +
� � , n = 1, k = 1 1, k , (9)
ρ ρ ρ λ
ρ
γρn
k
nrr
k
n
k
ntt
k
n
k
nr
k
n
n
k
n
ku u
m
r
u
r
u− +
−
− −
1
2 nn
k
n
ku = –
1
1
1 1
1
1
k
a u
n i
m
in
k
n r
k
k
kn
=
− −
=
∑∑
−
+
+ � � �b u c a n an
k
n t
k
n
k
in
k
in
k
i
− − − − −
=
+ + − −( )1 1 1 2 11( )
11
1
m
n
ku∑
− , k = 1, kn , n = 2, 3, … .
(10)
Netrudno ubedyt\sq, çto esly { }un
k , k = 1, kn , n = 0, 1, … , — reßenye
system¥ uravnenyj (8) – (10), to ono qvlqetsq y reßenyem uravnenyq (7).
Zametym, çto kaΩdoe uravnenye system¥ (8) – (10) moΩno predstavyt\ v vyde
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
726 S. A. ALDAÍEV
u u
m
r
u
r
unrr
k
ntt
k
nr
k n
n
k− +
−
−
1
2
λ
= γu f r tn
k
n
k+ ( , ) , (11)
hde f r tn
k ( , ) opredelqgtsq yz pred¥duwyx uravnenyj πtoj system¥, pry πtom
f r t0
1( , ) ≡ 0.
Dalee, uçyt¥vaq ortohonal\nost\ sferyçeskyx funkcyj Yn m
k
, ( )θ [3], yz (5)
y yz pervoho kraevoho uslovyq (2) v sylu (6) ymeem
τ τ
λ
τ γ τnrr
k
nr
k n
n
k
n
km
r r
r+
−
− −
1
2 ( ) = νn
k r( ) , 0 < r < 1, (12)
u r rn
k ( , ) = 0, 0 ≤ r ≤ 1, k = 1, kn , n = 0, 1, … . (13)
V¥polnqq v (11) – (13) zamenu peremenn¥x u r tn
k ( , ) = r u r tm
n
k( )/ ( , )1 2−
y pola-
haq ξ = ( ) /r t+ 2 , η = ( ) /r t− 2 , sootvetstvenno poluçaem
L un
k ≡ u un
k n
n
k
ξη
λ
ξ η
+
+( )2
= γ ξ ηu fn
k
n
k+ ( , ) , (14)
τ
λ
ξ
τ γ τξηn
k n
n
k
n
k+ −
2
= ν ξn
k ( ) , 0 < ξ <
1
2
, (15)
un
k ( , )ξ 0 = 0, 0 ≤ ξ ≤
1
2
, (16)
fn
k ( , )ξ η = ( ) ( , )( )/ξ η ξ η ξ η+ + −−m
n
kf1 2 , τ ξn
k ( ) = ( ) ( )( )/2 21 2ξ τ ξm
n
k− ,
ν ξn
k ( ) = ( ) ( )( )/2 21 2ξ ν ξm
n
k− , λn =
(( ) ( ) )m m n− − −1 3 4
4
λ
,
k = 1, kn , n = 0, 1, … .
S yspol\zovanyem obweho reßenyq uravnenyq (14) (sm. [6]) v [5] pokazano,
çto reßenye zadaçy Koßy dlq uravnenyq (14) ymeet vyd
un
k ( , )ξ η =
1
2
1
2
τ η η η ξ η τ ξ ξ ξ ξ ηn
k
n
kR R( ) ( , ; , ) ( ) ( , ; , )+ +
+
1
2
1 1 1 1 1 1ν ξ ξ ξ ξ η τ ξ ξ η ξ ηn
k
n
kR
N
R( ) ( , ; , ) ( ) ( , ; ,−
∂
∂
)) ξ η
η
ξ
ξ
1 1
1=
∫ d +
+
1 2 0
1 1 1 1 1 1
/
( , ) ( , ) ( , ;
ξ η
γ ξ η ξ η ξ η ξ∫ ∫ +
u f Rn
k
n
k ,, )η ξ ηd d1 1, (17)
hde R( , ; , )ξ η ξ η1 1 = P zµ ( ) = Pµ
ξ η ξ η ξ η ξη
ξ η ξ η
( ) ( ) ( )
( ) ( )
1 1 1 1
1 1
2− − + +
+ +
— funkcyq
Rymana uravnenyq L un
k = 0 [7], P zµ ( ) — funkcyq LeΩandra, µ = n
m
+
− 3
2
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
SUWESTVOVANYE SOBSTVENNÁX FUNKCYJ SPEKTRAL|NOJ ZADAÇY … 727
∂
∂ =N ξ η1 1
=
∂
∂
∂
∂
+
∂
∂
∂
∂
⊥ ⊥
=
ξ
η
η
ξ
ξ η
1
1
1
1
1 1
N N
, a N ⊥
— normal\ k prqmoj ξ = η
v toçke ( , )ξ η1 1 , napravlennaq v storonu poluploskosty η ≤ ξ .
Yz uravnenyq (17) pry η = 0 s uçetom (16) poluçaem
0 =
τ ξ
ν ξ
ξ
ξ
ξ
τ ξ
µ
ξ
n
k
n
k n
k
P d
( )
( )
(
2
1
2
1
21
1
1
0
1+
−∫
))
ξ
ξ
ξ
ξµ
ξ
1
1
1
0
′
∫ P d ,
0 < ξ <
1
2
. (18)
Dalee, yz (15), (18) ymeem
0 =
τ ξ
τ
λ
ξ
τ ξ γ τ ξξ ξ
n
k
n
k n
n
k
n
k( )
( ) ( )
2
1
2 1 1
1
2 1 1+ + −
∫ P dµ
ξ ξ
ξ
ξ1
1
0
–
–
1
2
1
1
1
1
0
τ ξ
ξ
ξ
ξ
ξµ
ξ
n
k
P d
( ) ′
∫ , 0 < ξ <
1
2
. (19)
Reßenye uravnenyq (19) budem yskat\ v vyde
τ ξn
k ( ) = ξβ , 1 < β = const, 0 < ξ <
1
2
. (20)
Podstavlqq (20) v (19), poluçaem
1 2 1 1
2 1
1
0
+ + −
−∫λ β ξ ξ
ξ
ξ
ξβ
µ
ξ
n P d( ) = γ ξ
ξ
ξ
ξβ
µ
ξ
1
1
1
0
P d
∫ ,
0 < ξ <
1
2
. (21)
Yz formul¥ [8]
0
1
∫ z P z dzα
µ ( ) =
π α
α µ α µ
α2 1
1 2 2 2 2 3 2
1− − +
+ − + +
Γ
Γ Γ
( )
( ) ( )/ / / / /
, α > – 1,
hde Γ ( )z — hamma-funkcyq, sleduet, çto esly β = µ − 2s , s = 1, 2, … , to
0
1
2 1
1
ξ
β
µξ
ξ
ξ
ξ∫ −
P d =
0
1
1
1
ξ
β
µξ
ξ
ξ
ξ∫
P d = 0.
Sledovatel\no, ravenstvo (21) ymeet mesto dlq lgboho γ .
Dalee, podstavyv (20), (15) v (17), poluçym yntehral\noe uravnenye Vol\ter-
ra vtoroho roda
un
k ( , )ξ η = γ ξ η ξ η ξ η
ξ η
µ
1 2 0
1 1 1 1
/
( , ) ( ) ( , )∫ ∫ +u P z d d Fn
k
n
k , (22)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
728 S. A. ALDAÍEV
Fn
k ( , )ξ η =
1
2
1
2
1 1
2
1( ) ( ( ) )ξ η β β λ ξ γ ξ
ξβ β
η
ξ
β β
µ+ + − + −
∫ −
n P 11
2
1
+
+
ξη
ξ ξ η( )
–
–
ξ ξ η
ξ η
ξ ξη
ξ ξ η
β
µ
1
1
1
2
12
− −
+
′
+
+
( )
( ) ( )
P dd f P z d dn
kξ ξ η ξ η
ξ η
µ1
1 2 0
1 1 1 1+ ∫ ∫
/
( , ) ( ) . (23)
Takym obrazom, reßyv snaçala zadaçu (8), (13) ( )n = 0 , a zatem (9), (13)
( )n = 1 y t. d., najdem posledovatel\no vse u r tn
k ( , ) yz (22), k = 1, kn , n = 0,
1, … .
Ytak, pokazano, çto v oblasty D+
ρ θ γ( ) ( )L udH
H
0 −∫ = 0. (24)
Pust\ f r t( , , )θ = R r T t( ) ( ) ( )ρ θ , pryçem R r V( ) ∈ 0 plotno v L t t2 1(( , ))− ,
ρ θ( ) ( )∈ ∞C H plotno v L H2( ) , a T t V( ) ∈ 1 plotno v L2 0 1 2(( , ))/ . Tohda
f r t V( , , )θ ∈ , V = V H V0 1⊗ ⊗ , plotna v L D2( )+
[9].
Otsgda y yz (24) sleduet, çto
f r t L udD
D
( , , )( )θ γ0 − +
+
∫ = 0
y
L u0 = γ u ∀ ∈ +( , , )r t Dθ .
Uçyt¥vaq ocenky [3]
kn ≤ C nm−2 ,
∂
∂
p
j
p n m
kY
θ
θ, ( ) ≤ C nm p/2 1+ − ,
C = const, j = 1 1, m − , p = 0, 1, … ,
netrudno pokazat\, çto rqd
τ θ( , )r =
n k
k
l m
n m
k
n
n r Y
=
∞
=
− + −∑ ∑
1 1
1 2β θ( )/
, ( ) (25)
sxodytsq absolgtno y ravnomerno, esly l > 3 2m / , β = µ − 2s > ( ) /m − 1 2 .
Takym obrazom, funkcyq
u r t( , , )θ =
n k
k
l m
n
k
n m
k
n
n r u r t Y
=
∞
=
− −∑ ∑
1 1
1 2( )/
,( , ) ( )θ (26)
qvlqetsq reßenyem zadaçy (4), (2), (25) v oblasty D+ , hde funkcyy u r tn
k ( , ) ,
k = 1, kn , n = 1, 2, … , naxodqtsq yz (22) y prynadleΩat klassu C D( )+
∩
∩ C1 ( ) ( )D S C D+ +∪ ∩ 2
.
Sledovatel\no, m¥ pryßly v oblasty D−
k pervoj kraevoj zadaçe dlq
uravnenyq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
SUWESTVOVANYE SOBSTVENNÁX FUNKCYJ SPEKTRAL|NOJ ZADAÇY … 729
L u1 ≡ u
m
r
u
r
u u d r t u e rrr r t
i
m
i xi
+
−
− − + +
=
∑1 1
2
1
δ θ θ( , , ) ( , ,, )t u = γ u (27)
s uslovyqmy
u
S
= τ θ( , )r , u Γ = 0. (28)
Reßenye zadaçy (27), (28) budem yskat\ v vyde (6).
Podstavlqq (6) v (27), ymeem
ρ ρ ρ0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
u u
m
r
d urr t
i
m
i r− +
−
+
=
∑ 11
0
1
01
1
0
1
0
1+ −�e u uγρ +
+
n k
k
n
k
nrr
k
n
k
nt
k
n
k
i
m
i
n
u u
m
r
d
=
∞
= =
∑ ∑ ∑− +
−
+
1 1 1
1
ρ ρ ρ nn
k
nr
ku
+
+ � �e
r
d nd un
k
n
n
k
i
m
in
k
n
k
n
k− + −( )
=
−∑λ
ρ
2
1
1 −−
γρn
k
n
ku = 0. (29)
Teper\ rassmotrym beskoneçnug systemu dyfferencyal\n¥x uravnenyj
ρ ρ ρ0
1
0
1
0
1
0 0
1
0
11
u u
m
r
urr t
t
r− +
−
= γρ0
1
0
1u , (30)
ρ ρ ρ λ
ρ
γρ1 1 1 1 1 1 1
1
2 1 1
1k
rr
k k
t
k k
r
k
k
ku u
m
r
u
r
u− +
−
− − kk ku1 =
= –
1
1 1
0
1
0
1
0
1
0
1
k
d u e u
i
m
i r
=
∑ +
� , n = 1, k = 1 1, k , (31)
ρ ρ ρ λ
ρ
γρn
k
nrr
k
n
k
nt
k
n
k
nr
k
n
n
k
n
k
nu u
m
r
u
r
u− +
−
− −
1
2
kk
n
ku =
= –
1
1 1
1 1 1
1
1
k
d u e d
n k
k
i
m
in
k
n r
k
n
k
i
mn
= =
− − −
=
−
∑ ∑ ∑+ +� �
iin
k
in
k
n
kn d u− − −− −( )
2 1 11( ) ,
k = 1, kn , n = 2, 3, … . (32)
Netrudno ubedyt\sq, çto esly { }un
k
, k = 1, kn , n = 0, 1, … , — reßenye sys-
tem¥ uravnenyj (30) – (32), to ono qvlqetsq y reßenyem uravnenyq (29).
Zametym, çto kaΩdoe uravnenye system¥ (30) – (32) moΩno predstavyt\ v
vyde
u u
m
r
u
r
unrr
k
nt
k
nr
k n
n
k− +
−
−
1
2
λ
= γu g r tn
k
n
k+ ( , ) , (33)
hde g r tn
k ( , ) opredelqgtsq yz pred¥duwyx uravnenyj πtoj system¥, pryçem
g r t0
1 0( , ) ≡ .
V¥polnqq v (33) zamenu peremenn¥x u r t r u r tn
k m
n
k( , ) ( , )( )/= −1 2 , poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
730 S. A. ALDAÍEV
L un
k ≡ u u
r
u unrr
k
nt
k n
n
k
n
k− + −
λ
γ
2
= g r tn
k ( , ) , k = 1, kn , n = 0, 1, … , (34)
pry πtom kraevoe uslovye (28) prynymaet vyd
u r0
1 0( , ) = 0, u rn
k ( , )0 = n rl− β , u tn
k ( , )1 = 0, k = 1, kn , n = 0, 1, … ,(35)
g r tn
k ( , ) = r g r tm
n
k( )/ ( , )1 2− .
Reßenye zadaçy (34), (35) ywem v vyde
u r tn
k ( , ) = u r t u r tn
k
n
k
1 2( , ) ( , )+ , (36)
hde u r tn
k
1 ( , ) — reßenye zadaçy
L u n
k
1 = g r tn
k ( , ) , (37)
u rn
k
1 0( , ) = 0, u tn
k
1 1( , ) = 0, (38)
a u r tn
k
2 ( , ) — reßenye zadaçy
L u n
k
2 = 0, (39)
u r20
1 0( , ) = 0, u rn
k
2 0( , ) = n rl− β , u tn
k
2 1( , ) = 0. (40)
Reßenye ukazann¥x v¥ße zadaç, analohyçno [10], rassmotrym v vyde
u r tn
k ( , ) = R r T ts s
s
( ) ( )
=
∞
∑
1
, (41)
pry πtom pust\
g r tn
k ( , ) = a t R rs s
s
( ) ( )
=
∞
∑
1
, n rl− β = b R rs s
s
( )
=
∞
∑
1
. (42)
Podstavlqq (41) v (37), (38), s uçetom (42) poluçaem
R
r
R Rsrr
n
s s+ + −
λ
µ γ
2
( ) = 0, 0 < r < 1, (43)
Rs ( )0 = 0, Rs ( )1 = 0, (44)
T Tst s+ µ = – a ts ( ) , (45)
Ts ( )0 = 0. (46)
Ohranyçennoe reßenye zadaçy (43), (44) ymeet vyd [11]
R rs ( ) = r J rsν µ( ) , (47)
hde ν = n
m
+
− 2
2
, J zν( ) — funkcyq Besselq pervoho roda, µs — ee nuly,
µ = γ µ+ s
2 .
Reßenye zadaçy (45), (46) ymeet vyd
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
SUWESTVOVANYE SOBSTVENNÁX FUNKCYJ SPEKTRAL|NOJ ZADAÇY … 731
T ts ( ) = – a t ds s
t
( ) exp ( )( )[ ]ξ γ µ ξ ξ− + −∫ 2
0
. (48)
Podstavlqq (47) v (42), naxodym
r g r tn
k−1 2/ ( , ) =
s
s sa t J r
=
∞
∑
1
( ) ( )ν µ , 0 < r < 1, (49)
n rl− −β 1 2/ =
s
s sb J r
=
∞
∑
1
ν µ( ) , 0 < r < 1. (50)
Rqd¥ (49), (50) — razloΩenyq v rqd¥ Fur\e – Besselq [12], esly
a ts ( ) =
2
1
2
0
1
[ ]( )
( , ) ( )
J
g t J d
s
n
k
s
ν
ν
µ
ξ ξ µ ξ ξ
+
∫ , (51)
bs =
2
1
2
1 2
0
1
n
J
r J d
l
s
s
−
+
+∫[ ]( )
( )/
ν
β
ν
µ
µ ξ ξ , (52)
µs , s = …1 2, , , — poloΩytel\n¥e nuly funkcyj Besselq, raspoloΩenn¥e v
porqdke vozrastanyq.
Yz (47), (48) poluçym reßenye zadaçy (37), (38) v vyde
u r tn
k
1 ( , ) = –
s
s s s
t
r J r a t d
=
∞
∑ ∫ − + −
1
2
0
ν µ ξ γ µ ξ ξ( ) ( ) exp ( )( )[ ]
, (53)
hde a ts ( ) opredelqetsq yz (51).
Dalee, podstavlqq (41) v (39), (40), poluçaem uravnenye
T Tst s s+ +( )γ µ2 = 0,
reßenyem kotoroho qvlqetsq
T ts ( ) = exp ( )[ ]− +γ µs t2 . (54)
Yz (47), (54) s uçetom (42) poluçym reßenye zadaçy (39), (40):
u r tn
k
2 ( , ) = b r J r ts
s
s s
=
∞
∑ − +
1
2
ν µ γ µ( ) exp ( )[ ], (55)
hde bs naxodytsq yz (52).
Sledovatel\no, reßyv snaçala zadaçu (30), (35) ( n = 0 ) , a zatem (31), (35)
( n = 1 ) y t. d., najdem posledovatel\no vse u r tn
k ( , ) yz (36), hde u r tn
k
1 ( , ) ,
u r tn
k
2 ( , ) opredelqgtsq yz (53) y (55), pry πtom u r tj0
1 0( , ) ≡ , j = 0, 1.
Ytak, pokazano, çto v oblasty D−
ρ θ γ( ) ( )L udH
H
1 −∫ = 0. (56)
Pust\ g r t( , , )θ = R r T t( ) ( ) ( )ρ θ , pryçem R r V( ) ∈ 0 plotno v L2 0 1(( , )) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
732 S. A. ALDAÍEV
ρ θ( ) ( )∈ ∞C H plotno v L H2( ) , a T t V( ) ∈ 1 plotno v L t2 0 0(( , )) . Tohda
g r t V( , , )θ ∈ , V V H V= ⊗ ⊗0 1, plotna v L D2( )−
.
Otsgda y yz (56) sleduet, çto
g r t L udD
D
( , , ) ( )θ γ1 − −
−
∫ = 0
y
L u1 = γ u ∀ ∈ −( , , )r t Dθ .
Takym obrazom, funkcyq (26) qvlqetsq reßenyem zadaçy (27), (28) v oblasty
D−
, hde funkcyy u r tn
k ( , ) , k = 1, kn , n = 1, 2, … , opredelqgtsq yz (36) y
prynadleΩat klassu C D C D S C D( ) ( ) ( )\− − −Γ0
1 2∩ ∪ ∩ .
Sledovatel\no, zadaça Tγ dlq kaΩdoho γ ymeet sobstvenn¥e funkcyy vy-
da (26), pryçem v sylu (23) y (52) yx sçetnoe mnoΩestvo.
Teorema dokazana.
1. Naxußev A. M. Zadaçy so smewenyem dlq uravnenyq v çastn¥x proyzvodn¥x. – M.: Nauka,
2006. – 287 s.
2. Vrahov V. N. Kraev¥e zadaçy dlq neklassyçeskyx uravnenyj matematyçeskoj fyzyky. –
Novosybyrsk: Novosyb. hos. un-t, 1983. – 84 s.
3. Myxlyn S. H. Mnohomern¥e synhulqrn¥e yntehral¥ y yntehral\n¥e uravnenyq. – M.:
Fyzmathyz, 1962. – 254 s.
4. Aldaßev S. A. O zadaçax Darbu dlq odnoho klassa mnohomern¥x hyperbolyçeskyx uravne-
nyj // Dyfferenc. uravnenyq. – 1998. – 34, # 1. – S.S64 – 68.
5. Aldaßev S. A. Kraev¥e zadaçy dlq mnohomern¥x hyperbolyçeskyx y smeßann¥x
uravnenyj. – Almat¥: H¥l¥m, 1994. – 170 s.
6. Bycadze A. V. Uravnenyq smeßannoho typa. – M.: Yzd-vo AN SSSR, 1959. – 164 s.
7. Copson E. T. On the Riemann – Green function // J. Ration. Mech. and Anal. – 1958. – 1. –
P. 324 – 348.
8. Bejtmen H., ∏rdejy A. V¥sßye transcendentn¥e funkcyy. – M.: Nauka, 1973. – T.1. – 294 s.
9. Kolmohorov A. N., Fomyn S. V. ∏lement¥ teoryy funkcyj y funkcyonal\noho analyza. –
M.: Nauka, 1976. – 543 s.
10. Tyxonov A. N., Samarskyj A. A. Uravnenyq matematyçeskoj fyzyky. – M.: Nauka, 1977. –
659 s.
11. Kamke ∏. Spravoçnyk po ob¥knovenn¥m dyfferencyal\n¥m uravnenyqm. – M.: Nauka,
1965. – 703 s.
12. Bejtmen H., ∏rdejy A. V¥sßye transcendentn¥e funkcyy. – M.: Nauka, 1974. – T.2. – 295 s.
Poluçeno 26.08.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
|
| id | umjimathkievua-article-2904 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:32:31Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c3/73284ce8204e670a55ec379277bcd3c3.pdf |
| spelling | umjimathkievua-article-29042020-03-18T19:40:12Z Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations Существование собственных функций спек тральной задачи Трикоми для некоторых классов многомерных смешанных гиперболо-параболических уравнений Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations. Показано, що існує злічеппа множина власних функцій спектральної задачі Трікомі для багатовимірних мішаних гіперболо-параболічних рівнянь. Institute of Mathematics, NAS of Ukraine 2010-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2904 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 6 (2010); 723 – 732 Український математичний журнал; Том 62 № 6 (2010); 723 – 732 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2904/2559 https://umj.imath.kiev.ua/index.php/umj/article/view/2904/2560 Copyright (c) 2010 Aldashev S. A. |
| spellingShingle | Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| title | Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| title_alt | Существование собственных функций спек тральной задачи Трикоми для некоторых классов многомерных смешанных гиперболо-параболических уравнений |
| title_full | Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| title_fullStr | Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| title_full_unstemmed | Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| title_short | Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| title_sort | existence of eigenfunctions of the tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2904 |
| work_keys_str_mv | AT aldashevsa existenceofeigenfunctionsofthetricomispectralproblemforsomeclassesofmultidimensionalmixedhyperbolicparabolicequations AT aldaševsa existenceofeigenfunctionsofthetricomispectralproblemforsomeclassesofmultidimensionalmixedhyperbolicparabolicequations AT aldaševsa existenceofeigenfunctionsofthetricomispectralproblemforsomeclassesofmultidimensionalmixedhyperbolicparabolicequations AT aldashevsa suŝestvovaniesobstvennyhfunkcijspektralʹnojzadačitrikomidlânekotoryhklassovmnogomernyhsmešannyhgiperboloparaboličeskihuravnenij AT aldaševsa suŝestvovaniesobstvennyhfunkcijspektralʹnojzadačitrikomidlânekotoryhklassovmnogomernyhsmešannyhgiperboloparaboličeskihuravnenij AT aldaševsa suŝestvovaniesobstvennyhfunkcijspektralʹnojzadačitrikomidlânekotoryhklassovmnogomernyhsmešannyhgiperboloparaboličeskihuravnenij |