Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology
We consider problems with free boundaries for an unknown scalar function and its level surface. We study problems of uniqueness, monotonicity, spatial localization and the existence of solutions. We survey the constructive methods developed for solving specific problems and mathematical prediction i...
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| Дата: | 1992 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1992
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7796 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512757130985472 |
|---|---|
| author | Mitropolsky , Yu. A. Berezovsky, A. A. Plotnitsky, T. A. Митропольский, Ю. А. Березовский, А. А. Плотницкий, Т. А. |
| author_facet | Mitropolsky , Yu. A. Berezovsky, A. A. Plotnitsky, T. A. Митропольский, Ю. А. Березовский, А. А. Плотницкий, Т. А. |
| author_sort | Mitropolsky , Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | We consider problems with free boundaries for an unknown scalar function and its level surface. We study problems of uniqueness, monotonicity, spatial localization and the existence of solutions. We survey the constructive methods developed for solving specific problems and mathematical prediction in the corresponding areas of the natural sciences. |
| first_indexed | 2026-03-24T03:33:51Z |
| format | Article |
| fulltext |
YL!K 517.946.9
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Ha )lBe nono6JJaCTH Ql (t) 1-l Q2 (t). O6e CBOCOL!HbJe noBepXHOCTH r (t): (D (x, t) =
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U (X, f), <D* (f) H f (t) TaKHe, l.JTO
~u-k (u)u, =-f(u) B Q(t)={u<c=const }, t>O,
u (x, 0) = ,P (x) B Q (0), ( l)
vu + a (u) (u - cp) = 0 Ha S (t) = i)Q (t) n iJD, t > 0,
[U] = 0, u = u· ' I Vil , v<D*) = p<D; Ha <D* (t) = Q l (t) n Q2 (t), t > 0,
u = C, yU = 0 Ha r (t) = i)Q (t) n D, t > 0.
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pa3 pbIB n e p Boro p oJla B TOl.JKe u (/) = u * (t) = const; / (u) - 3 aLlaHHaH cj)yHK
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TI051BJIHI0!iicl5JC5J K aK noBepXHOCTb yp0BHH U (X, f) = u* ; r (t) HJIH (J) (t) -
noBe p xHOCTb u (x, t) - c = O; p = con s t - 3a)la HHaH KOHCTa HTa; 3 HaK [ ]
0 3 Hal.JaeT CKal.J0K CTOHI.ll.ett no.n HHM cpyHKUHH npH n e p exo.ne 4 e pe3 yKa3aHHble
noBepXHOCTH.
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KOHeLIHOh CKOp0CTblO . B1-1e Q (t) U (x, t) = C H no3TOMY np0ilOJJ)l{eHI-1oe n o He
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C JO . A. MHTP0n0JJ bCKHl'I , A. A. 6EPE30BCKHFI, T. A. ITJIOTHHUl(HH, 1992
ISSN 0041-6053. Y,cp_ MaT. :HCYPH., 1992, T. 44, M I 6Z
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I<0rop1.,1x y1<a.tKeM yc.~oBHH I 1, 2]
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6- u =0 - f (u) B Q = {u < c},
vu + ex (u) (ll - (p) = 0 Ha S = c)Q n dD,
u = C, vu = 0 Ha r = 8Q n D.
(3)
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ypoB HH 6y.11.eM npe,!LCTaBJIHTb B BH)Le (4, 5]
z = z (x, y, u, t), (4)
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(5)
I 2/
Ul/11 = - z [Zyy -(z" z,Jul,
u
no3BOJimoT nepecjJOpMyJrnpoBaTb 3anaqy (I) AJJH u B 3a,1LaLiy co CBo6oAHbIMH
rpaHHUaMH LlJIH 2 [6-9]
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U::;#=-u te
(6)
z(x,y,u . 0)=Z0 (x,y.u B Qu(0\
.!E:__ + ex (u) (u - cp) = 0, u = !!:. (x, y, t), t > 0 ,
z.,
[2] = 0, U = u•. t ::::,.. 0 .
68 ISSN 0041-6053. Jl,cp. ,1ia1. [)/(;yp11.., 1992, r. 44, M 1
r,ne
- 1- = 0, 11 = C, f>O ,
Zu
Z = Z8 (x, y , t) , U = ~ (x, y , t) , t > 0,
Az = L.\,z - ( I ; z l2
) , L.\, 2 = v,v,z,
Zu . u
L.\,z = Zn +211!/• vz = {- Zx, - Zy, l}.
(7)
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sanmow,aH noBepXHOCTb s (t) : 2 - 28 (x , y, t) = O; 20 (x, y, u) onpe.ue.rrneTCH
113 ypaBI-1e1-11151 [ l 0, 11]
'P (x, y, 2 0)- u = 0 (8)
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DJI0CK0CTH U = u*, TaK H B He06XOllHMOCTH onpeneJJHTb CBOcom1y10 rp a m,uy
r4 (t) : ll - u (x, y, t) = 0. EcJIH B 3cl}latJe (I) l·Ja llJJe)KHT onpe,ueJJHTb noBe px -
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l{Oflla Ha S (t) 3a}laH0 ycJJ0BHe JlHpHXJ!e U = cp Ha S (t), CT3 H0B HTC51 H3BeCT I-IOH
H rpa1rnua fu (f): U - /I (X, y, f) = U - cp (X, y, f) = 0. JlJJH 2 = 2 (X, y,
u, t) npH 3T0M noJJytJaeM -;-leJJHHeHHYIO 3 3,llatJ y Ha con pm1<e 1rne B H3BeCTJ-IOI°l 06-
JiaCTH Qu (t) : {x, y E .0u, cp (x, y, t) ~ u ~ c}.
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fi! !± /J iiQu ff» u
- lvzl2 Vuldu- ss V 1vz121cdxdy,
211 !!iJ Zu y
(9)
r.11.e $ - cetJeH He o6JJaCTH Q,, (t) TIJI0CKOCTblO ll = C, a fl)u - npOH3B0JlbHOH
DJJOCK0CTblO; .25,, C fl); b = min!!. (x, If, t): -; _ opT BHellJHeH HOpMaJJH K 3aMK-
X , 1,
HYTOMY KOHTYPY oQu, orpmlWIHBaI0lUeMy fl)u; v(x,y,u,t)'H z(x,y,u,t)
npoH3BOJ!bHble, 11B3)1{}!bl 11errpepb1BH0 11mpcpepe1-m11pyeMbJe cj:JyHKUHH;
-
y',2 = {- Zx, - 2y}-
noJl3f351 B (9) v = 21 - 2 2 H z = Zi, a 3aTeM z = 2 2 , noJJytJaeM .UBa paaeH
CTBa, 113 KOTOpblX ITOCJJe Bbll.JHTaHHH BTOporo H3 nepaoro CJJe,llyeT COOTHOIIJem,e
< C - ➔ s s dxdy \ (A21 - A22) (2 1 - 22) du= 5 du ~ (V, (2 1 - 22), n) (21 - 22) dl -
fi) ~ h ~
- H dxdy f [1 Vt (21 - 22) 12 - ( I V/112 _ I v;2 12 \)(Z1u - Z2u)]du-
@ ~ lu 2u
(10)
ISSN 0041-6053. YKp. AtaT. [HCljpH., 1992, T. 44, M 1 69
TTpeo6pa3yeM rIO,Ubll-ITerpaJi bHOe Bb!paw;eHHe BTOporo HHTerpaJia npaBOll
-.acTH (10):
( 11 )
B pe3y.TibTaTe OKOHqaTeJlbHO IlOJiyqaeM
C C O (2 2
) S dxdy S (A21 - A2 2 )(21 - 22) du= ~ du f (2 1 - 22 ) 10~ 2) dl =
9' " h a!filu
(12)
TTycTb 21 = 21 (x, y , ll, t) H 22 = 22 (x, y, u, t) - !!Ba pa3JIHqHblX peweHHH
aa,n.atJH (6). l(JIH T3KHX peweHHI1 Bbll10Jil·l5ll0TC51 paBeHCTBa
a a
~ (2 1 - 2,) = - a (u) (u - cp) ~ (2 1 - 2,), u = u, (1 3) uu - uu -
BhITeKaIOlll.He H3 KpaeBblX ycJIOBHH 33,UatJH (6). C HX yqeTOM cpopMyJ13 (12) y rr
pomaeTCH K BH.UY
C C
S S dxdy S (A21 -A22)(21 - 22) du = - Sa (u) (u - cp) du x
9' ~ b
- 2 (v21, V22) 1 du. (14)
BocrrOJib30B3BW HCb paseHCTBOM
(u - cp) :u (2 1 - 22 , 2 = [(21 - 22)2 (u - cp)Ju -(21 - 22 ) 2 , (15)
tta 11,n.eM
(u - cp)(21 - 22) (21 - 22)u = ; {[(u - cp)(21 - 22f lu - (21 - 22 ) 2}. (16)
B cJiyqae KyCO'lHO-IlOCTOHH HOrr 33BHCHMOCTH CX (u) = al + (a2 - a1) 'Y] (u -
- u*) , r ,ue 'Y] (u) - cj)yHKUHH XeBHCaiina, IlOCJie HHTerpnpoBaHHH no u C y q e
TOM KpaeBbIX ycJIOBHH cj:>opMyJia f pHHa (14) OKQHq aTeJi bHO rrpeo6pa3yeTCH K
BH.llY
C 1 C
S S dxdy S (A21 - A22 ) (21 - 22 ) du= - 2 Sa (u) du f (z 1 - 22 ) 2 dl -
@ ~ b ~
7U ISSN 0041-6053. Ji,cp. Mar. 'JtCypn., 1992, r. 44, M l
TaK KaK 21 H 22 - MOHOTOHHbie cpyHKUHH no nepeMeHHOH u, TO 2iu122u > 0,
Z2u l zlU > 0. ;:ho HO3BOJIHeT, BOCnOJlb3OB3BlllHCb HepaBeHCTBOM lOHra 2xy ~
~a2x2 + a-2y2, noJ1yq11Tb o u eHKy
(18)
B CHJIY 3TOro HepaBeHCTBa npaBaH qaCTb (17) HenoJJO)KHTeJihHa, qTo H ycTaHaB
JH!BaeT MOHOTOHHOCTb onepaTopa A Ha p ellle HHHX 3a,naq11 (6). MoHOTOHHOCTb
o ncpaTopa A no3BOJIHeT llOK333Tb TeopeMy e,UHHCTBeHI-IOCTH p ellleHH51 3TOH 3a
na0rn. H eTpyllHO BH.lleTh, 'ITO YCJIOBH5l conpmKeHH5l B (6) MO)KHQ Bl<JIIO'-IHTb
C HOMOI.11.b!O lleJibTa-cpyHKUHH 1lttpa1<a {j (u - u*) B .nmpcpepeHUH3JlbHOe
y paBHe1rne
Az = k (u) Zt + f (u) Zu + pz16 (u - u*) B Qu (t). (19)
11CKJIIO'-laH Az1 H Az2, C noMOI.11.blO (19) npeo6pa3 yeM npaBylO '-13CTb (17) K BH.llY
C C
5 5 dxdy 5 (Az 1 - Az 2 ) ( 2 1 - 2 2 ) du - S .\ dxdy S {[k (u) + pf, (u - u*)] x
Pl! ~ Pl! ~
X (z1 - 2 2 ) + f (u) (21 - 2 2)u}(z1 - 22 )du =-½-55 {P (z1 - 22 )~1u• +
§ii
C
+ 5 [k(u) (21 - 22)~- t' (u) (21 - 2 2)2] du} dxdy.
!:
(20)
E cJIH f (u) - MOHOTOHHO y6bIB3IOIJ.{aH cpymnurn (f' (u) < 0), TO H3 (20) CJie.ny
e T, 'ITO JJeBaH l/3CTb (17) HeOTpttuaTeJJ b Ha. lleHCTBHTeJlbHO, TaK KaK nptt t = 0
B CHJIY H3'-13JlbHbIX ycJIOBHH 33.llal/H (6) (21 - 22) 2 = 0 H BeJIHl/HHa (21 - 22) 2 >
> 0 npH t > 0, TO (21 - 22)~ > 0. npaBa5l 43CTb (17) B CliJIY MOHOTOHHOCTH
onep aTopa A HenoJIO)KHTeJibHa. BbIX0.11 113 3TOfO npOTHBope'-11151 BO3MO)KeH TOJ!b·
KO B cJJyqae, Kor;i,a o6e OHH paBHbl HYJIIO, LJTO BO3MO)Kl·IO J!Hlllb npH paBeHCTBe
21 = 22, x , y, u E Qu (t), t > 0. :ho H noKa3bIBaeT Teop eMy enm-1cTBeHHOCTH
pelllemrn 3a1ta'l11 (6) .
5. Mo Ho To H Hoc Th p e ill e H H H (oJlHH npocToH npHMep). Pac-
CMoTpHM npocTeHlllYIO 3a}la'ly, HBJl5110llI,YIOC51 '-13CTHbIM cJiyqaeM 3a}laqtt (6)
z,+(+)+t(u)2u=O, 0<u<l, t>0,
u fl
z(u . 0) =20 (u), 0<u<l,
z(0,t)=0, z,,( l.t) = oo , t>0.
(21)
nycTb k > I - He4eTHOe l/HCJIO. YMI-IO)KHM (21) Ha ( 1/z~)u . HecJIO)KHblH no}l
C4eT npHBOllHT K TO}!{,UeCTBy
(:; t+ k
1 1 (2!~1}+(k~kl)2 ( ~-~ • ):+ k!:_I (2~~1 - z;~,)= 0-
Zu
I
CT pennoJio)KHM, 'ITO f (0) = 0 11 BBe.neM o6o3Ha01eH11e .\ a (u) du = (a).
0
TerpttpyeM (22) OT 0 }10 1. Y 0IHTbl835l (2 1 ), noJJyl/aeM
I I 4h ( I )2 h f u
k- 1 <zk~I )t + k- 1 · < z( k-t- l )/Z ) = k- 1 ( zk-1 ).
u u u u
JSSN 0041-6053. Y,w Mar. 3!Cypn ., 1992, r. 44, M l
(22)
TTpottH-
(23)
71
I
06rotta11aH II a IILpto. I) = (aP) P = II a II, 0Tc10,ua rroJiytJaeM HepaBeHCTBO
II L II~ k; l fu II+. II , (f u = f u (u), U E (0, 1)), (24)
cepmo oueHoK
(25)
H BKJIIOtJeHHe
-k+-,-E Loo (0, t; L2 (0, l))n L2 (0, t; wJ (0, 1)). (26)
- 2
Zu
- ll + l~
n ocKOJibKY 1-13 ycJIOBHH f u < 00 CJie,uyeT --k- f u < 00, TO, rrepexo.UH K rrpe-
.ueJiy rr ptt k -+- oo, rroJiytJaeM
_l_ EC ((0, t) x (0, 1 )). (27)
Zu
EcJIH B HatJaJibHb!H MOMeHT z = Zo (u) MOHOTOHI!a5J cpyHI<UHH TaKaH, tJTO Zu >
> a > 0, TO TaKOH )Ke OHa OCTaHeTC5J tt rrpH t > 0 B cJiytJae a = a (t). TaKI IM
o6pa3oM, perneHHe 3a,ua1rn (21) np1-1
Zou> a> 0, f u < oo, f (0) = 0 (28)
MOHOTOHHO H, CJie,UoBaTeJibHO, e,UHHCTBeHHO.
CJie.I.I.yeT 3aMeTHTb, tJTO BKJIIO'leHHe (26) MO)l{HO IIOJiy<IHTb npH 6oJiee C'Jla-
6b!X npe,urroJJO)KeHH51X OTHOCHTeJlbllO 0,JYHKU11H t, eCJIH nocJie,UHee CJiaraeMoe
(22) rrpe.11.crnBHTb B BH,Ue
/I 1 ( ,)-, ). ~ ,,2f i(
2
~ ). ,~, (29)
z7
H BOCIIOJib30BaTbCH ttepaBeHCTBOM feJib.LJ.epa.
6. CT p O C T p a H C T B e H H a 51 JI O K a JI H 3 a U H 51 (,uocTaTOtJHbie
ycJIOBHH). YMHO)l{HB (21) Ha zk, IIOJIYtJHM TO)l{,UeCTBO
2~+1 + (__!__) - kzk-1 + (--t- zk+I) - ~ zk+I = 0. (30)
k + 1 Zu u k + 1 u k + 1
CTpoHHTerp1-1pyeM ero OT 0 .11.0 u ~ 1, o603HatJHB a (u) tJepe3 a. l1crroJib3YH
ycJIOBHH (21), rroJiyqaeM
k+I - k --k+I -
~ + -2- - k (zk- 1) + E.__ - ___b_ (z"+1)=0 (3 1) k+l z k+l k+l '
u.
OTK)'.LJ.a CJie,uyeT HepaBeHCTBO
(32)
H COOTBeTCTBYIOIUaH oueHKa
2fut
k+l k2 +k llzi12 ~llzoll 2e ---- (33)
fv
2ho 3HatJHT, tJTO II z II< oo rrpH t > 0, H, cJie,uoBaTeJibHO,
zk ti"+'
-=- + k + I < oo.
Zu
(34)
ISSN 0041-6053. JIKp. MaT. ~!/PH., 1992, T. 44, Ni J
EcJinf>O,TOOTc1onacJienyeT,t1TOZ<=npHBCexu<l, a np» f(l)>O z
6y,rr_eT orpaHttt!eHa H npH U= I, T. e. JIOK3.'lbHO no t 6yi 1.eT HM eTb MeCTO n po
CTpaHCTBeJIIl351 JIOK3Jl l1 3 a1:1rn . 4T06bl JJOK3JIH33UH51 6bI Jia pa B HOMe p1-1ofi 110 t,
Heo6XO,ll_IIMO, lJT06b! B (33) fu 6bIJIO OT])HUaTeJII,HblM. '.:ho YCJIOBHe H a t llOCTH·
raeTCH npH llOCT3TOlJHO 60JlbWHX k, ecm-1 f,, < 0 B OKp eCTHOCTH u = I, noc
KOJibKY 2 - MOHOTOHHO B03 pacTaIOUJ:3fl no U cpym<UHfl II U MO,IUIO BbI6paTb
llOCTclT01IHO 6JIH3KHM K I. T aKHM o6pa30M, npH Bb!TIOJJHeHim yc.'JOBHfr
20 ,, > a > O, f>0, f,, < =, fu<0, (u > ui), /( 1) >0, f(0) = 0
(35)
p e w e rme 3 3,ll_3lJH (21) MOHOTOHHO, ellHHCTBeHHO 11 npocTpaHCTBC' IIHO JJOK3JIH -
30B3HO .
7. Cy Ill, e CT BOB a H H e p e ill e H H 51. 06oc110BaHHe oueHOK 6y,rr_eM
npoBOllHTb T])3 ,7J,HUHOHHO- MeTonm1 KOH('llJib! X pa3 HOCTe tt no t. TTo 3 31{3HHOMY
z = z., (u) Ha CJJOe !_ = 0 nOCT])OHl\l p eweHHe Ha CJJOe t - !. = t,,.t KaK perne 1me
«rrp116JIH)Ke 1-rnoi1» 3an:atrn
z ' l ) 2 Tt + (-2- + f (u) Zu = !),,t ,
u u
P acCMOT])HM scnoMoraTeJil, I-JYIO 33.'lalJy
I1p» s = z rroJJytJaeM cTau1-10HapHoe peme1-rne
( - 1-) + f (u) z., = 0,
z., u
f = - qq,, * q = - Jf 2 Sf (u) du,
EcJIH npellTIOJIO)KHTb, lJTO
u
z = 1 ⇒ z 5 du
U q C = q (U)
0
q(l) = 0,
u
(36)
(37),
(38),
(39)
TO Zc (u) - MOHOTOI-11-Iafl, orpaHHlJeHHafl cpyHKUHH, T. e . pewe1me CT3UHOHa pHoi:i
3311341-1 rrpocTpaI-ICTBeI·IHO JIOK 3 JIH30B3HO. I1ycTb
(40)
To r.r1.a 11JI5l y = 1/zu rroJiy<1aeM sa,1:iatJy Kow»
/(u) s-: Yu+y =- M y(l)=O, (41 )
OTK y n:a c.nen.ye -r, lJTO yseJmtJeHHe (JioKaJibHO no u) s o6ycnoBJJHBaeT yseJiw1e
~rn e lj . nocr<OJlbKY f (u) > 0, H, CJiellOBaTeJibHO, yMeHbllleI-me 2 . TaKHM o6pa -
3 0:11, oe U~HHe (37) He rrp eBbI\UaeT Zc (u) npH s > 2 H HeTIOllBH)KH351 TOlJKa co-
OTBeTCTBl-15! ~ - 2 flBJJfleTCH peweHHeM 33ll3lJJ,I (36) 2 ~ 2 ~ Zc , l103BOJl5110WHM
Bbl l!HCJI HTb z H a l10CJien:y1oll1eM CJioe, H T. 11- I1pn6JIH)KeHHOe perue1-me 6y,u.e-r
rJI 3ll.KHM H yt > 0 YllOBJieTBOpHeT HepaBeHCTBY
(42)
ISSN 0041-6053. YKp. MaT. J1CypH., 1992, T. 44, M J
13KHM o6pa30 M, Mb! ITOJIYtIHJUI CJie,!I.yIOlll.HH pe3yJibTaT: eCJI H Bbll10JIHeHbl yc
.l!0BIUI
I
q EC1 (0, l), qu< O, q(l)=O, j q~~) < oo ,
0
(43)
I l - > - , Z0 (0) = --- = 0, f = - qqu,
Zou q Zou (l)
1'0 33,!I.atia (21) HMeeT e)lHHCTBeI-11-l0e M0I-I0T0HH0e pewem-ie, YJl0BJleTBopmomee
ttepaBeHCTBaM
I
S du
z0 (u ) ~z (u,t) ~zc (u)= q(u) · (44)
u
I1ccJie)loBa HHIO KOHKpeTHb!X 33,!I.34 c o CB060llI-lblMH rpaHHll3 MH ,[lJH! HeJrn
'IICl1HblX 3BOJIIOUHOJ-1Hb!X ypaB1-1e 1rn11, B03HHKaI01UHX np11 pewe1-1 1111 8KTY3Jlbllb!X
npo6JieM MeTaJJ.11yprn11, 0l!O.rnrHH H 3K0JI0fHH, ll0CB511UeHbl pa60Tbl l1 2-29],
B I{OTOpblX p accM0Tp eHbl MaTeM3TH4ec 1rne M0)leJJH TeTIJJ0Bbl X H nmptj)y3HOHI-lblX
ll poueccoB 3JieKTpOI-IH0-.,1ylJeBO l1 IT.TJaBKH MeTaJJJJOB , Kp HO,IleCTp y 1rnHH 6H0JJO
r HqecKHX TKa1-1eii H nepe p acnpe,!I.eJJemrn 1rn1-1ueHTpaum1 KHc.n opona H ceposono
pona no rny6111-1 e 4 epHoro M0pH. D.syMepHb!e CT3UHOHap Hbre 33.l:W 'IH co CB0-
0O)lHOH rpaHHUeH CBe)leHbl K p ewen mo 1-1em-11-1 ei":HblX HHTerpaJJbHb!X ypam-I eHHH
MHH HM3JlbHOH pa3MepH0CTH. l1x pewem-1e ocymecTBJJeHO MeTO)lOM KOHe4HblX
:1J1eMe11ToB c ucno.'!b30BaHHeM Ma.r1h1x 3BM. llJJn p erneHHH o,zu-10M ep1-1b1x necTa
UHor-iapHblX 33)l3lJ co CB060.lll-lblMH rpal·lHU3MH npHMe H e Hbl cneUH3a1 bHbl e BapH
-8UH0HHhJe MeT0)lhl, MeTO)l PoTe C IT0CJJe)ly1onlHM nepeXOll0M K I !eJJ HHeHHblM l'!H
TerpaJibHblM ypaBHeHHHM BoJibTe ppa , He51B1-lble pa3 H0CTHble cxeMbl.
B npo6JieMaX c neJ.I,3JJ eKTpOMeTaJIJiyp rHH, CB5!33HHblX C pactJeTaMM TeITJIO
BbIX no.'!eH B aBT0TlffJie llpH BaKyy1111-1oi't 3.TJeI<Tpo1rno-Jiy 'l eBOH r apI·IHC3)1<HOH
nJiaBKe MeTaJ!JIOB H CI1.'13B0B , T04Hb!e, npH6JIH)KeHHble 31-1 3JIHTH4eCIUi e H l!HC
.1ermo- a 1-1aJIHTH4eCr<He pemeHH 51 O)lHO- H llByMepI-Ibl X 33,[l8l! TI03BOJIHJIM ycnrno
BllT b npaKTH4 eCKH B3)K Hble )1JI51 nporH03 Hp0B8HH51 npoCTb!e (j)yHKUHOHcrn l,llble
33 BHCHM0CTH, 0ITHC 3 T b llH H8 MHKY n pou ec ca !lJiaBJie mrn 14, 6-10, 23, 241.
BecbMa aKTyaJibHOH )1JI51 cos p eMem-1011 Men11uHHb1 HBJ1r1eTO1 npo6JJeMa pac
•qeTa H nporH03a B03)1€HCTBH51 HH3 KHX TeMne p aTyp Ha )h ! IBYI0 0HOJIOrI1 4eCKYIO
TKaHb . 8 43CTHOCTH, l]Jl 51 x11pypr1rn Heo6X01!HMO npo r H03Hf)0B3 I-JHe pe3 y Ji bTa
T0B 3a111o p a)KHBa HH H H ll€CTpyKUHH. 8 OTJIHLl!1e OT c n eU3JieKTpo111eTaJIJiypnrn
3)leCb B0 3 HHK310T 3 8,[l3lJH C HCT0lJHHK8MH , cymecTBeHH0 3 8BHC5lli_lJlMII OT TeM n e
p aTypb!. 3To rrplIBO)lHT K T3Kl-lM !{84eCTBeHI➔0 J-IOBblM p e3y .'lbTclT8M, 1(81{ CTa6H
JIH3 3UH51 pemeHHI! I< npe)leJJ bl-l0MY CT 8 UHOHap 1-10My COCT051HHI0 H p eaJibHO 1-1a6-
JIIOl]aeMa5l npoc-r paHCTBeHI-1351 JIOK8JIH33UH51 - TellJJ0B0e B03My!lleHI1e
noKaJJbHO .
TTo.TJy ll e HHhie pewe mrn 0llHOMepHblX HeCTaUI-I0HapHbI X 381!alJ TI03BOJJ 5l!OT
llf)OC4HT3Tb [)51)1 T HHH 4 Hbl X CJiyqaeB 3aMop a)I01B3Hll51 0HOTKalletr , C LIOCTclTOlJHOH
T04HOCT blO onpe,lle.1 HTb 1-13CTHbi e 11 O0 il!He 3 aJ(0H0MepH0CTH npouecca, COCTaBHTb
ceprrn 1-lOM0rpaMM, 11eo6XOlUIMh!X llJl5! HCIT0JJh3 0B8HH51 B npaKTH4eCKOH M e)lHUH
He, 1-1aMeT HT b )laJibHeiiurne 1-1 a npanJre H1I H 1:cc.1e)loBa HH'H l 12, 13, 15, 17, 21,
25-281.
TTpo6JieMa pa3 p a60TKH MeT0)lOB H c p e)lCTB npe.U0TBpmuemrn B03M O}KH0ll
HeraTHBHOH rrepecTpOHKH 3KOJ!OfH'I.J€CKOH CHCTeMhl l.Jep11oro M0pH, K8!{ H Jlpy
rHX BHYT PHM8Tep11K0Bb]X B0)lOE'M0B 11 0 TJk'.'l h l :b lX pai\ol!Oil MHJJOBOfO 0I<eaHa ,
C0)l€p)K81UH X paCTBopeHI-Ib!H B B0)le cepoB0Ll0P0ll, ('l3$ i'.i ,Jlli'I C OllC l !KOll 11 npor-
1!030M npocTp aHCTBeHHO -Bp eMeH HOH H3MeH'IHB0CTll pac TBope ll Hb!X l(OHUe HTpa
UHII KHCJiopo)la 11 cep0BO)l0po)la , o6ycJIOBJieHHbl X H3M e HeJHJeM npupo,!I.Hb! X H
aHTponoreHHbl X cpaKTOpOB . TTpocTeHmHe n mpcpy3 HOHHbi e , XHMHl! E'CKHe H 61-10-
010fH4eCKHe npe)lcTaBJieHH51 0 )lHH3MHKe npouecca rrpHBO)l5lT 3)leCb K 3 3)lalJaM
nma CTecj)al-13 JJ.JIJI CHCT€Mbl HeJIHHeHHb!X .3BOJJI0UH0H HblX ypaBJl eHHll )1Hcj:cj)y 3 HH
C peaKUHeii.
14 ISSN 0041-6053. YKp . . Mar. ,icypn., 1992, T. 44, .M I
P e3yJibTaTbl paclleTOB KOHKpeTHblX BapHaHTOB }la.)l{e rrpH HeITOJII-lblX BXO}l
Hb!X }lal-!Hb!X xopowo cor,1acy10TC51 C 3KCrrepHMeHTaJi bI-IO H3MepeHHblMH p acrrpe
.lleJieHH51MH KOI-1Ue1npaUHH no r Jiy6tt1-1e. AHaJII13 rro.1ytieHHbIX pewem111 ITOKa-
3bIBaeT , l!To ocHOBHy10 poJib B rrpouecce rrepep acrrpe,n:eJiemrn ttrpaeT K03Cp<pH
u u eHT Typ6yJieHTHOH ,n:mjxj)y3 HH (5-7, 14, 18, 20, 22, 291.
Y Ka3al-!Hb!e rrpHJIO)KeH!rn ITO)llJepKHBaJOT a1<TyaJibI-IOCTb HCCJie)l.oBaHH.ll
3a)l.al! co CB060)l.I-lb!MH rpaHHUaMH (1) H (6) .
I. MapTttHCOH JI. K. 0 KOHeqHQH CK0p0CTH pacnpoCTpatteHlrn TenJJOBblX B03MymettHfl a
cpe.uax C n0CT0l!HHblM K03cjlcjmm1eHTOM TenJIonp0B0.!lH0CTH // )KypH. Bbl'IHCJIHT. MaTeMa
THKH H MaT. cjlH3HKH.- 1976.-16, N2 5.-C. 1233- 1241.
2. Ka1ia1111-1uKoa A. C. 0 xap3KTepe p3cnpocTp3Hemrn B03Myll.le,mii B 33.!l3'I3X ue,1HHeiiuofl ren
JIOnposol(H0CTH c nornoll.leIrneM // T aM J-Ke.- 1974.- 14, NQ 4. - C. 89 1-905.
3. </Jpuo,\ta/-1. A. B3pi-iaUHOl!Hble 11JJHIIUHllbl H 33Jl3 4H co CB060llHblMH rpamIUaMH.- M . : H ayKa,
1990 .- 536 C.
4 . Bepeaoac,wii, A. A ., D Aom1-1.ui1Kw1 T . A., Jleo1-1.mbea 10. B . O6parnb1e npeo6pa3osamrn B 33.
/la 'l3X Kj)HCT3Jl.1H33UHH // O6paTHble 3all,1'1H 13 3311_a·rnx l(!Jl1CT3JIJIH33 UHH H QJH3HKH M0pll.
K 11cs, 198:3. - C. 3-9.- (TTpenp1rnr / A H YCCP . I-1H-r MaTeMaTHKH; 83.54).
5 . S epe308CKULi. A. A., E o?cyc.rnacKUU C. r. /V\aTeMaTf!l!eCK35! MOJ\eJJb cjlop MHJJOB3 HH 5! cjlH3H'-leCKHX
H XHMH'ICCK HX noJief! rJiy6HHHblX B0!l 4 ep1-IOro M0JJ 5! / / O6paTI-Ibie npeo6pa30Ba 1rn11 B 33.lf.343X
"PHCTa.~.11-1 3auIrn H cpH3HKe M0JJ5!.- K11es, 1983.- C. 10-15 .- (TTpenpHHT / AH YCCP .
l-11-1-T MaTeMaTHl(H; 83 .54).
6. Mumpono.1bCKui110 . A .. Sepe30/3cKut1 A. A. 3a.ua,m CTecjlaHa c npe11_eJ1bHbIM cTau11ouap Hb1M
C0CT0HHHCM B cneu3JieKTJJ0MeTa.~Jiypnm, Kj)H0XHpyprnH H cjlH3HKe M0 JJ SI / / M 3T. cjJH 3HKa H
11em 11-1eiiH. MeX3 HHK3.- 1987.- B brn . 7.- C. 50-60.
7. EepeaoacKuii A. A . l(J13cc11,1ec"He H cneu11aJibI-Ibre nocTaH0BKH 331(3 4 CTecpaua // H ecTaUHO·
11 apHb1e 3alla '-IH CTecjl3I-I3.- KHeB, 1988. - C. 3-20.- (IlpenpHHT / AI-I YCCP. I-111-T M3Te·
M3THI<11; 88 .49).
8. MumponoAbCKuil JO. A., E epeaoacKut1 A . A. 3311_a •m CTecjl3H3 13 MeTaJJJJyprna , KpaoxHpyprnH
H cp11 3HKC MOJJH.- Kaes, 1989.- 42 c. - (Ilpenpmn / AH YCCP. I1H -T MaTeManma: 89. 11).
9. Eepe3oacKui1 A. A. TTpoeKUHOHI-IO-ceTO'lHbiii MeT0ll pe·,, e1mS1 0.!lH0MepnbIX 3a.!l3 '1 C-recp311a //
M a-r. cj)IBHKa H 11eJ1aueirn . MeX3 HH K3 .- 1985.- Bbm. 4.- C. 71-713.
10 . Bepe30tlCKW1 A. A. JleKUHH no neJJHI-IeHH bIM KJJ3CBblM 3a;(a43M MaTeMaTH'JeCKOII cpH3HKH:
B 2-x •1. - K11 es : Hayre llYMKa, 1976.- LJ. 1. - 452 c. ; G. 2.- 292 c.
II. E epe30/3CKUU A. A. llByMepHbJe M3TeMaTHqecKHe M0.!lCJI H KJJH 0.!leCTpyKUHH 6HOTK3HH //
M 3TeMaTH'leCKOe M0l(eJIHJ)0BaHHe cjlH3H'-!eCKHX npoueccoB.- K11eB : I1I-I-T M3TCM3THKH
AH YCCP, 1989. - C. 14- 38.
12. 5epe3nl3CKui1 •A. A. O.!lHOMepHaH JJ0K3JlbHaSI 33.!!3 '13 CTecjl3H3 nJIOCKOn3p3JJJJeJJbHOii Kpao.ue
c-rp y KUHH 6HOJIOrH'leCK0H TK3Hll / / 33.!l3'1H TenJJonp0B0.!lI-IOCTH C llO.!lBHlKHblMH rp3HHUa
MH.-KHeB, 1985.- C. 3-8.- (TTpenpmn / AH YCCP. l-11-1--r MaTeM3THKH ; 85.2).
1:, . E ep e30/JCKUU A. A. OpocTpaHCTBel-IH3H JI0K3JIH33UHH KJJHOI.l03l(CHCTBH5! Ha 61-IOJIOrH'leCKHe
TK3HH // TTpocTp311crnem1311 JJ0K3JJH33UH5! u 33.!l343X CTecjl3I-I3.- Kaes, 1987. - C . 3-12.
(Ilpe npHin I AI-I YCCP. l-111 -T MaTeMaTHKH; 87. 60) .
14 . Bepe3oacKui1 A. A., EoeycAaacKafl E. C. 331!34H rnna CTecjl3H3 B npo6JJeMe JIHH3MHKH ceposo
ilOPOllHOii 30Hbl 4 epHOro M0JJ51 / / T3M lKe.- C. 15-22.
15. 5epeaoacKut1 A. A. OilllOMepHble M3TCM3Tf!ll eCKHe M0.!leJIH Kj)H0.!leCTPYKUHH 6HOJJOnP1ecKOH
TK3HH / / llmpcjlepeHUH3JlbHbie YJJ3 BHei-IH5! C '13CTHblMH np0H3B0.!lHblMH B npHKJla.!lHb!X 33.!l3-
'l3X .- Kne11: l-11-1-T MaTeMaTHKH AI-I YCCP, 1986.- C. 3-18.
16. B,·pe:1nBcKui1 A. A. 1lsyxcpa,H1,1e ocecHMMe-rpn'IHb!e CT3UH0I-I3pHb1e 33.!l34H CTecpa11a // I-leKo•
T0JJble B0l!JJOCbl 3CHM11T0TH'leCK HX MeT0JI.OB neJIHI-1ei'mo"1 MCX31-!HKH .- KaeB : l1H•T MaTeMa
THKH AH YCCP, 1986.- C . 26-33.
17 . B ep e30/3cKut1 A. A . O.!lHOMep11a51 JI0K3JlbHaH 3311_3 43 CTecjl3H3 nJJOCKO-n3paJJJ1e.11,1-wii KJJHO
/lec-rpyl(UHH 6HOJIOrH 4eC KOll TK3HH / I 3a.lf.3 '1H TenJJonpoB0.!lH0CTH C 110/lBHlKHb!MH rpaiiHU3-
MH.- K11eu, 1985.- C. 3-8.- (TTpenpHHT / AI-I YCCP . I1u-T M3TeM3THKH, 85.5).
18. BeoeaoacKuil A. A., E oe11cAaBcKaJ1 E. C. M a-reMaTH'IecKa>I MO.!lCJJb /\HI-I3MHKH 30I-IbI cywec-rso
sa 11H5! KHcJI0p0ll3 n cepoBoJlopona 13 Gep1-IOM Mope// TaM i Ke. - C. 9- 17.
19 . B epeaoac,wu A.. A., Eoe1.JCAD.8cKaJ1 l; . C. M3TeMan14ecKoe npor1103apoB3HHe /\HI-I3MHKH ce po
BO!lopo.!lHOii 30Hbl 4 epi-ioro MOJJSI // Mop. rnapoq:>H3. )Kyp 11. - 1986.- N2 6.- C. 32-39.
20. Mame.,1tamu'1 eCKOe M0.!leJJnpoBaHHe !1HH3MHKH cepOBOJ\Oj)0.ll-HOfi 30Hbl Gepuoro MOPS!/
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| id | umjimathkievua-article-7796 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:51Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f0/aa104c122923a2314d2db3c3a3a58cf0.pdf |
| spelling | umjimathkievua-article-77962023-09-14T12:03:56Z Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology Задачи со свободными границами для нелинейного эволюционного уравнения в проблемах металлургии, медицины, экологии Mitropolsky , Yu. A. Berezovsky, A. A. Plotnitsky, T. A. Митропольский, Ю. А. Березовский, А. А. Плотницкий, Т. А. We consider problems with free boundaries for an unknown scalar function and its level surface. We study problems of uniqueness, monotonicity, spatial localization and the existence of solutions. We survey the constructive methods developed for solving specific problems and mathematical prediction in the corresponding areas of the natural sciences. Рассмотрены задачи со свободными границами для искомой скалярной функции и поверхности ее уровня. Исследованы вопросы единственности, монотонности, пространственной локализации и существования решений. Дан обзор разработанных конструктивных методов решения конкретных задач и математического прогнозирования в соответствующих областях естествознания. Розглянуто задачі з вільними границями для шуканої скалярної функції та поверхні її рівня. Досліджені питання єдиності, монотонності, просторової локалізації та існування розв’язків. Зроблено огляд розроблених конструктивних методів розв’язку конкретних задач та математичного прогнозування в відповідних областях природознавства. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7796 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 67-76 Український математичний журнал; Том 44 № 1 (1992); 67-76 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7796/9464 Copyright (c) 1992 Yu. A. Mitropolsky, A. A. Berezovsky, T. A. Plotnitsky |
| spellingShingle | Mitropolsky , Yu. A. Berezovsky, A. A. Plotnitsky, T. A. Митропольский, Ю. А. Березовский, А. А. Плотницкий, Т. А. Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| title | Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| title_alt | Задачи со свободными границами для нелинейного эволюционного уравнения в проблемах металлургии, медицины, экологии |
| title_full | Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| title_fullStr | Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| title_full_unstemmed | Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| title_short | Problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| title_sort | problems with free boundaries for the nonlinear evolution equation in problems of metallurgy, medicine, ecology |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7796 |
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