Directory UMM :Journals:Journal_of_mathematics:VMJ:

‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
ˆî«ì{ᥭâï¡àì, 2000, ’®¬ 2, ‚ë¯ã᪠3

“„Š

517.5 + 517.9

ˆ‘Š‹ž—ˆ’…‹œ›… ŒŽ†…‘’‚€ „‹Ÿ …˜…ˆ‰
Š‚€‡ˆ‹ˆ…‰›• “€‚…ˆ‰ €€Ž‹ˆ—…‘ŠŽƒŽ ’ˆ€
‚ ‚…‘Ž‚›• Ž‘’€‘’‚€• ‘ŽŽ‹…‚€

Œ. ‘. €«¡®à®¢ 

ˆáá«¥¤ã¥âáï ¢®¯à®á ®¡ ãáâà ­¨¬®á⨠®á®¡¥­­®á⥩ ¤«ï ®£à ­¨ç¥­­ëå à¥è¥­¨© í¢®«î樮­­ëå ãà ¢­¥­¨© ¯ à ¡®«¨ç¥áª®£® ⨯ .

ãáâì QT =
 (0; T ) | ®£à ­¨ç¥­­ ï 樫¨­¤à¨ç¥áª ï ®¡« áâì, «¥¦ é ï
¢ ¥¢ª«¨¤®¢®¬ ¯à®áâà ­á⢥ R n+1 â®ç¥ª (x; t) = (x1; : : : ; xn; t).
à¥¤¯®«®¦¨¬, çâ® P (x; t; D) | ¤¨ää¥à¥­æ¨ «ì­ë© ®¯¥à â®à ¢ ç áâ­ëå
¯à®¨§¢®¤­ëå, ®¯à¥¤¥«¥­­ë© ­  ®âªàë⮬ ¬­®¦¥á⢥ QT  R n+1 ,   e | ª®¬¯ ªâ ¢ QT . ãáâì F (QT ne) ¨ F (QT ) | ­¥ª®â®àë¥ ª« ááë ä㭪権 ­  ¬­®¦¥á⢠å QT ne ¨ QT ᮮ⢥âá⢥­­®. Œ­®¦¥á⢮ e ­ §ë¢ ¥âáï ãáâà ­¨¬ë¬
¤«ï ¤¨ää¥à¥­æ¨ «ì­®£® ãà ¢­¥­¨ï P (x; t; D)f = 0 ®â­®á¨â¥«ì­® ä㭪樮­ «ì­ëå ª« áᮢ F (QT ne) ¨ F (QT ), ¥á«¨ ¤«ï ¢á类£® à¥è¥­¨ï f 2 F (QT ne)

ãà ¢­¥­¨ï P (x; t; D)f = 0 ­  QT ne áãé¥áâ¢ã¥â à¥è¥­¨¥ f~ 2 F (QT ) ãà ¢­¥­¨ï
P (x; t; D)f = 0 ­  QT â ª®¥, çâ® f~jQT ne = f . ¥è¥­¨¥ f~ ­ §ë¢ ¥âáï ¯à®¤®«¦¥­¨¥¬ à¥è¥­¨ï f ãà ¢­¥­¨ï P (x; D)f = 0 á QT ne ­  QT .
‡ ¤ ç  á®á⮨⠢ ­ å®¦¤¥­¨¨ ä㭪樮­ «ì­®-£¥®¬¥âà¨ç¥áª¨å å à ªâ¥à¨á⨪ ¬­®¦¥á⢠ e, ª®â®àë¥ £ à ­â¨àãîâ, çâ® ¬­®¦¥á⢮ e ãáâà ­¨¬®.
“áâà ­¨¬®áâì ª®¬¯ ªâ®¢ ¤«ï à¥è¥­¨© ¯ à ¡®«¨ç¥áª¨å ãà ¢­¥­¨© å®à®è®
¨§ã祭  ¢ ª« áᥠ®£à ­¨ç¥­­ëå £¥«ì¤¥à®¢ëå ä㭪権 ¨ ¢ ­¥ª®â®àëå ¯à®áâà ­áâ¢ å ‘®¡®«¥¢  (á¬., ­ ¯à¨¬¥à, [1{6]). “á«®¢¨ï ãáâà ­¨¬®á⨠®¯¨á뢠îâáï
¢ § ¢¨á¨¬®á⨠®â ¢¨¤  ãà ¢­¥­¨ï, ¢ â¥à¬¨­ å à ¢¥­á⢠ ­ã«î ᮮ⢥âáâ¢ãîé¨å ¥¬ª®á⥩ ¨ ¬¥à • ã᤮àä .
‚ ­ áâ®ï饩 áâ âì¥ ¨áá«¥¤ã¥âáï ¢®¯à®á ®¡ ãáâà ­¨¬®á⨠®á®¡¥­­®á⥩ ¤«ï
®£à ­¨ç¥­­ëå à¥è¥­¨© í¢®«î樮­­ëå ãà ¢­¥­¨© ¯ à ¡®«¨ç¥áª®£® ⨯ 

@u , divA(x; t; u; 5 u) + B (x; t; u; 5 u) = 0;
x
x
@t
£¤¥ A ¨ B | ­¥ª®â®àë¥ ä㭪樨, 㤮¢«¥â¢®àïî騥 ®¯à¥¤¥«¥­­ë¬ ãá«®¢¨ï¬
(á¬. x2).

c 2000 €«¡®à®¢  Œ. ‘.

3{4

Œ. ‘. €«¡®à®¢ 

„ ­­ ï à ¡®â  á®á⮨⠨§ ¤¢ãå ¯ à £à ä®¢. ‚ x1 ®¯à¥¤¥«¥­ë ¢¥á®¢ë¥
¯à®áâà ­á⢠ ‘®¡®«¥¢ . ‚ x2 ä®à¬ã«¨àãîâáï ¤®áâ â®ç­ë¥ ãá«®¢¨ï ãáâà ­¨¬®á⨠®á®¡¥­­®á⥩ ¤«ï ®£à ­¨ç¥­­ëå à¥è¥­¨© 㯮¬ï­ã⮣® ¢ëè¥ ª« áá  ¯ à ¡®«¨ç¥áª¨å ãà ¢­¥­¨©.
1. ”㭪樮­ «ì­®¥ ¯à®áâà ­á⢮

ãáâì ! | «®ª «ì­® ¨­â¥£à¨à㥬 ï ­¥®âà¨æ â¥«ì­ ï äã­ªæ¨ï
­ 
. ˆ§R
¬¥à¨¬®¬ã ¬­®¦¥áâ¢ã E ᮮ⢥âáâ¢ã¥â ¢¥á®¢ ï ¬¥à  (E ) = E !(x)dx. ’®£¤ 
d(x) = ! (x)dx, £¤¥ dx | ¬¥à  ‹¥¡¥£ . ƒ®¢®àïâ, çâ® ! ¥áâì p-¤®¯ãáâ¨¬ë© ¢¥á
(1 < p < 1), ¥á«¨ ¢ë¯®«­¥­ë á«¥¤ãî騥 ãá«®¢¨ï:
W 1. ! (x)dx ¨ dx ¢§ ¨¬­®  ¡á®«îâ­® ­¥¯à¥à뢭ë;
W 2. 0 < ! < 1 ¯®ç⨠¢áî¤ã ¢
¨ áãé¥áâ¢ã¥â ª®­áâ ­â  C â ª ï, çâ®
(B (x; 2r))  C(B (x; r)) ¤«ï «î¡®£® è à  B (x; r), £¤¥ 2B 
(ãá«®¢¨¥ 㤢®¥­¨ï);
W 3. ¥á«¨ 'i 2 C 1 (
) | ¯®á«¥¤®¢ â¥«ì­®áâì ä㭪権 â ª ï, çâ®
Z

Z
p
j'i j d ! 0 ¨ j 5x 'i , vjpd ! 0






¯à¨ i ! 1, £¤¥ v | ¢¥ªâ®à­®§­ ç­ ï äã­ªæ¨ï ¨§ Lp (
; ; Rm ), â® v = 0;
W 4. áãé¥áâ¢ãîâ ª®­áâ ­âë k > 1; r0 ; c3 > 0 â ª¨¥, çâ®
 R Z

1
 p1
1 Z j'jkp d kp  c r
p
3 (B ) j 5x 'j d
(B )

B

B

¤«ï «î¡®© ä㭪樨 ' 2 C01 (B ) ¨ è à  B = B (x; r) 
; x 2
 ; 0 < r < r0 ;
W 5. áãé¥áâ¢ãîâ ª®­áâ ­âë r0 ; c4 > 0 â ª¨¥, çâ®
Z
Z
B

j' , 'B jpd  c4 rp j 5x 'jp d
B

¤«ï «î¡®© ä㭪樨 ' 2 C 1 (BR) ¨ ¯à®¨§¢®«ì­®£® è à  B = B (x; r) 
; x 2

 ; 0 < r < r0 . ‡¤¥áì 'B = (1B) B 'd.
ˆ§ ãá«®¢¨ï W 4 ­  ¢¥á®¢ãî äã­ªæ¨î á ¯®¬®éìî ­¥à ¢¥­á⢠ ƒñ«ì¤¥à 

¯®«ãç ¥¬ ¢¥á®¢®¥ ­¥à ¢¥­á⢮ ã ­ª à¥: ¥á«¨ D 
| ®£à ­¨ç¥­­®¥ ®âªàë⮥
¬­®¦¥á⢮, diamD < 2r0 ¨ ' 2 C01 (D), â®
Z
Z
p
p
(1)
j'j d  C (diamD) j 5x 'jp d
D

D

ˆáª«îç¨â¥«ì­ë¥ ¬­®¦¥á⢠ ¤«ï à¥è¥­¨© ª¢ §¨«¨­¥©­ëå ãà ¢­¥­¨©

3{5

(á¬. [9]).
Ž¯à¥¤¥«¨¬ Lp (
; ) ª ª ª« áá ä㭪権



Z
 p1
p
Lp (
; ) = f : kf jLp (
; )k =
jf j d < 1 :



ƒ®¢®àïâ, çâ® äã­ªæ¨ï u ¯à¨­ ¤«¥¦¨â ª« ááã Wp1 (D; ), ¥á«¨ ¨ ⮫쪮 ¥á«¨
u 2 Lp (D; ) ¨ ­ ¤©¥âáï ¢¥ªâ®à­®§­ ç­ ï äã­ªæ¨ï v 2 Lp (D; ; Rn ) â ª ï, çâ®
¤«ï ­¥ª®â®à®© ¯®á«¥¤®¢ â¥«ì­®á⨠'k 2 C 1 (D) ('k 2 C01 (D))
Z
Z
p
j'k , uj d ! 0 ¨ j 5x 'k , vjp d ! 0
D

D

¯à¨ k ! 1. ‚¥ªâ®à-äã­ªæ¨î v ­ §ë¢ îâ á㡣ࠤ¨¥­â®¬ ä㭪樨 u ¢
Wp1 (D; ) ¨ ®¡®§­ ç îâ ᨬ¢®«®¬ 5x u.

Ÿá­®, çâ® ¯à®áâà ­á⢠ Wp1(
; ) ¨ W 1p (
; ) | ¡ ­ å®¢ë ¯à®áâà ­á⢠ ®â­®á¨â¥«ì­® ­®à¬ë kukWp1(
;) = kukLp(
;) + k 5x ukLp (
;) . Šà®¬¥ ⮣® ¯à®áâ
à ­á⢠ Wp1(
; ) ¨ W 1p (
; ) à¥ä«¥ªá¨¢­ë.
”ã­ªæ¨ï u ¯à¨­ ¤«¥¦¨â ¯à®áâà ­áâ¢ã Wp;1 loc (
; ), ¥á«¨ ®­  ¯à¨­ ¤«¥¦¨â
¯à®áâà ­áâ¢ã Wp1 (D; ) ¤«ï ¢á类© ®£à ­¨ç¥­­®© ®¡« á⨠D, § ¬ëª ­¨¥ ª®â®à®© ᮤ¥à¦¨âáï ¢
.
à®áâà ­á⢮ L1p (

; ) ®¯à¥¤¥«¨¬ ª ª ¯®¯®«­¥­¨¥ ¯à®áâà ­á⢠ C01 (
) ¯®
­®à¬¥
kukL 1 (
;) = k 5x ukLp (
; ):
p

”ã­ªæ¨ï u ¯à¨­ ¤«¥¦¨â ¯à®áâà ­áâ¢ã Liploc (
), ¥á«¨ ¤«ï ­¥¥ ¢ë¯®«­ï¥âáï
­¥à ¢¥­á⢮
ju(x) , u(y)j  jx , yj
¤«ï «î¡®© ¯®¤®¡« á⨠D; D 
. ‡¤¥áì j
j | ¥¢ª«¨¤®¢  ­®à¬ . ‚ à ¡®â¥ [8]
¤®ª § ­®, çâ® ¯à®áâà ­á⢮ Wp;1 lip (
; ) = Wp1 (
) \ Liploc (
) ¯«®â­® ¢ Wp1(
; ).
Šà®¬¥ ⮣® Wp;1 lip (

; ) ¥áâì ¢¥ªâ®à­ ï à¥è¥âª .
Žâ¬¥â¨¬, çâ® ¯à¨¢¥¤¥­­ë¥ ¢ëè¥ ¢¥á®¢ë¥ ¯à®áâà ­á⢠ ¤¨ää¥à¥­æ¨à㥬ëå ä㭪権 å®à®è® ¨§ãç¥­ë ¢ à ¡®â å ‘. Š. ‚®¤®¯ìï­®¢  [7{11] ¨ ¤à㣨å
 ¢â®à®¢. (á¬., ­ ¯à¨¬¥à, [12{14]).
ãáâì t | ¯¥à¥¬¥­­ ï ¢à¥¬¥­¨. Œë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® t 2 (0; T ),
£¤¥ T ª®­¥ç­® ¨«¨ à ¢­® +1. ãáâì B | ¡ ­ å®¢® ¯à®áâà ­á⢮. Ž¡®§­ ç¨¬
ç¥à¥§ Lp (0; T ; B ); p  1 ¯à®áâà ­á⢮ (ª« áᮢ) ä㭪権 f á® §­ ç¥­¨ï¬¨ ¢ B ,

3{6

Œ. ‘. €«¡®à®¢ 

¨§¬¥à¨¬ëå ­  (0; T ) (®â­®á¨â¥«ì­® ¬¥àë ‹¥¡¥£  dt ­  (0; T )). à®áâà ­á⢮
Lp (0; T ; B ) ï¥âáï ¡ ­ å®¢ë¬ ¯à®áâà ­á⢮¬ ¯® ®â­®è¥­¨î ª ­®à¬¥
8 RT
< f 0 kf (t)kpB dtg 1 ¯à¨ 1  p  1;
kf ; Lp(0; T ; B )k = : ess sup kf (t)kB
¯à¨ p = 1:
p

t2(0;T )



Ž¡®§­ ç¨¬ Wp1;0(QT ; ) = Lp (0; T ; Wp1(
; )) ¨ ¯ãáâì W 1p;0(QT ; ) | § ¬ëª ­¨¥
C01 (QT ) = fu 2 C 1 (Q T ) : u = 0 ­  ,T g ¯® ­®à¬¥ k
kW 1 0 (QT ; ), £¤¥
p

kukW 1 0(Q
p

;

T

;)

kukp;q;Q

‚ ¤ «ì­¥©è¥¬ kukp;p;Q

T

= kukpp;p;Q

T

; +

;

k 5x ukpp;p;Q

T

= ku; Lp(0; T ; Lq (
; )k:
; ¡ã¤¥¬ ®¡®§­ ç âì ª ª kukp;p;Q
1
ZZ
p
juj dxdt = kukp;Q ;
T

; ;

;

T

;

= kukp;Q

T

;

¨

p

T

QT

Lp (0; T ; Lq (
; )) = Lp;q;Q ; ;
Lp (0; T ; Lp(
; )) = Lp;Q ; ;
T

T

Lp (0; T ; Lq (
)) = Lp;q;Q ;
T

kuk1;
= vrai maxjuj:
ã¤¥¬ £®¢®à¨âì, çâ® äã­ªæ¨ï u 2 Lp;Q ; ¨¬¥¥â áâண¨© á㡣ࠤ¨¥­â v, ¥á«¨
­ ©¤¥âáï ¯®á«¥¤®¢ â¥«ì­®áâì 'k 2 C 1 (QT ) â ª ï, çâ®
T

ZZ

j'k , j

u p ddt

!0 ¨

QT

ZZ

j 5x 'k , vjp ddt ! 0:

QT

„«ï «î¡®£® ª®¬¯ ªâ­®£® ¬­®¦¥á⢠ e 2 QT ®¯à¥¤¥«¨¬ ª« áá ä㭪権
V (e; QT ) = fu 2 C01 (QT ) : u  1 ¢ ®ªà¥áâ­®á⨠e; 0  u  1 ¢ QT g

¨ ­ §®¢¥¬ (p; )-¥¬ª®áâìî ª®¬¯ ªâ  e ç¨á«®
ZZ

ZZ
p
j 5x 'j ddt + j't jdxdt :
cap(e) = '2Vinf
(e;Q )
T

QT

QT

(2)

ˆáª«îç¨â¥«ì­ë¥ ¬­®¦¥á⢠ ¤«ï à¥è¥­¨© ª¢ §¨«¨­¥©­ëå ãà ¢­¥­¨©

3{7

…¬ª®áâì ª ª äã­ªæ¨ï ¬­®¦¥á⢠ ¥áâ¥á⢥­­ë¬ ®¡à §®¬ à á¯à®áâà ­ï¥âáï
­  ¯à®¨§¢®«ì­ë¥ ¬­®¦¥á⢠.
’¥à¬¨­ ýª¢ §¨¢áî¤ãþ ®§­ ç ¥â, çâ® ¤ ­­®¥ ᢮©á⢮ ¢ë¯®«­ï¥âáï ¢áî¤ã,
§  ¨áª«î祭¨¥¬ ¬­®¦¥á⢠, ¨¬¥î饣® ¥¬ª®áâì ­®«ì.
‹¥¬¬  1.
e
(p; )
(1 < p <
1)
f k g
 k 2 C01 (QT ); 0 
 k  1;  k = 1
e;  k ! 0
QT k 5x  k kp;QT ; !
k
0; kt k1;QT ! 0
C ’ ª ª ª ¥¬ª®áâì cap(e) = 0, â® áãé¥áâ¢ã¥â ¯®á«¥¤®¢ â¥«ì­®áâì f'k g 2
V (e; QT ) â ª ï, çâ® k 5x 'k kp;QT ; ! 0 ¨ k'kt k1;QT ! 0. ˆá¯®«ì§ãï â¥å­¨ªã
„¦. ‘¥àਭ  [15], à áᬮâਬ äã­ªæ¨î k = max(min('k ; 1); 0). ’®£¤  k |
äã­ªæ¨ï á ª®¬¯ ªâ­ë¬ ­®á¨â¥«¥¬ ¢ QT ; 0  k  1; k = 1 ¢ ®ªà¥áâ­®á⨠e
¨ k 5x k kp;QT ; ! 0; kkt k1;QT ! 0.
ãáâìR  2 C01 (Rn+1 ) | äã­ªæ¨ï á ª®¬¯ ªâ­ë¬ ­®á¨â¥«¥¬ â ª ï, çâ®
(x; t)dxdt = 1. „«ï " 2 (0; 1) ¯ãáâì " ®¡®§­ ç ¥â äã­ªæ¨î
  0;
Rn+1
x ! "n1+1 ; ( x" ; "t ). „«ï ¤®áâ â®ç­® ¬ «ëå " ᢥà⪠  k = " k 2 C01 (QT ). ’ ª
ª ª k5x  k kp;QT ;  k5x k kp;QT ; ¨ ktk k1;QT  kkt k1;QT , â® k5x  k kp;QT ; ! 0
¨ ktk k1;QT ! 0.
ãáâì  k;t(x) =  k (x; t). ’®£¤   k;t 2 C01 (
); supp  x;t 
; QT =
(0; T ).
ˆá¯®«ì§ãï ¢¥á®¢®¥ ­¥à ¢¥­á⢮ ã ­ª à¥ (1), ¨¬¥¥¬
Z
Z
k;t
p
j (x)j d  C j 5x  k;tjp d
ãáâì

| ª®¬¯ ªâ­®¥ ¬­®¦¥á⢮

. ’®£¤  áãé¥áâ¢ã¥â ¯®á«¥¤®¢ â¥«ì­®áâì
¢ ®ªà¥áâ­®áâ¨

-¥¬ª®á⨠­®«ì

â ª ï, çâ®

¯. ¢.

¢

¨

.

D

D

¤«ï «î¡®£® ®âªàë⮣® ®£à ­¨ç¥­­®£® ¬­®¦¥á⢠ D 
.
‘ ¯®¬®éìî ­¥à ¢¥­á⢠ ƒ¥«ì¤¥à  ®ç¥¢¨¤­®
Z
p
Z
k;t
1
,
p
j jd  C j 5x  k;tjp d:
j(D)j
D

„ «¥¥ ¯®«ã稬

j(D)j1,p jT j1,p
RT R

ZT Z
0

D

D

 p1

j k jddt

C

ZT Z
0

j 5x  k jpddt:

D

Žç¥¢¨¤­®, j k jddt ! 0 ¤«ï «î¡®£® ®âªàë⮣® ®£à ­¨ç¥­­®£® ¬­®¦¥áâ0D
¢  D ¨§
. ‘«¥¤®¢ â¥«ì­®, ­ ©¤¥âáï ¯®¤¯®á«¥¤®¢ â¥«ì­®áâì f k g â ª ï, çâ®
 k 2 C01 (QT ); 0   k  1;  k = 1 ¢ ®ªà¥áâ­®á⨠e;  k ! 0 ¯. ¢. ¢ QT ¨
k 5x  k kp;QT ; ! 0; ktk k1;QT ! 0. B

3{8

Œ. ‘. €«¡®à®¢ 

2. “áâà ­¥­¨¥ ®á®¡¥­­®á⥩

ãáâì
| ®£à ­¨ç¥­­ ï ®¡« áâì ¢ R n ; n  1; QT =
 (0; T ] ¤«ï ­¥ª®â®à®£® 䨪á¨à®¢ ­­®£® T > 0 ¨ ST = @
 (0; T ]; ,T = ST [ (
 ft = 0g) (,T |
¯ à ¡®«¨ç¥áª ï £à ­¨æ  樫¨­¤à  QT ).
 áᬮâਬ ¢ QT ãà ¢­¥­¨¥ ¢¨¤ 

@u , divA(x; t; u; 5 u) + B (x; t; u; 5 u) = 0;
(3)
x
x
@t
@u ; : : :; @u ), A = (A1 ; : : :; An ), ä㭪樨 A(x; t; ;  ) ¨ B (x; t; ;  )
£¤¥ 5x u = ( @x
@xn
1

㤮¢«¥â¢®àïîâ ãá«®¢¨î Š à â¥®¤®à¨.
à¥¤¯®«®¦¨¬, çâ® ¯à¨ ¢á¥å (x; t) 2 QT ¨  2 Rnf0g;  2 R n ¤«ï ­¥ª®â®à®£®
p 2 (1; 1) áãé¥áâ¢ã¥â ª®­áâ ­â   > 0 ¨ ¨§¬¥à¨¬ë¥ ä㭪樨 f1; f2; f3; g2; g3; h3
â ª¨¥, çâ® ¢ë¯®«­¥­ë á«¥¤ãî騥 ãá«®¢¨ï:

jA(x; t; ;  )j  j jp,1!(x) + g2jjp,1!(x) + g3!(x);
(4)
jB (x; t; ;  )j  f1 j jp,1!(x) + f2 jjp,1!(x) + f3!(x);
(5)
A(x; t; ;  )
  j jp!(x) , f2jjp !(x) , h3 !(x);
(6)
¤«ï ¯®ç⨠¢á¥å x 2
¨ ¢á¥å  2 R,  2 R n , £¤¥ ! (x) | p-¤®¯ãáâ¨¬ë© ¢¥á,
f1 2 Lp;QT ; , f2; f3; h3 2 L1;QT ; , g2; g3 2 L p,p 1 ;QT ; .

Ž¡®¡é¥­­ë¬ à¥è¥­¨¥¬ ãà ¢­¥­¨ï (3) ¯à¨ ãá«®¢¨ïå (4){(6) ­ §®¢¥¬ äã­ª0 (Q ; ), ¥á«¨
æ¨î u 2 Wp;1;loc
T
ZZ

f,u't + A(x; t; u; 5xu) 5x ' , B (x; t; u; 5xu)'gdxdt = 0

QT

¤«ï ¢á¥å ' 2 C01 (QT ).
ˆá¯®«ì§ãï â¥å­¨ªã, ¯à¥¤«®¦¥­­ãî „¦. ‘¥à¨­ë¬ [15],  ­ «®£¨ç­® १ã«ìâ â ¬ ‹. Œ. . ‘ à ¨¢ë [4] ¤®ª §ë¢ ¥âáï
n+1 ¨ b 2 Rnf0g. ãáâì
‹¥¬¬  2. ãáâì QT | ®âªàë⮥ ¬­®¦¥á⢮ ¢ R
u ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨ï (3) ¢ QT ¨ ¯ãáâì | ¯à®¨§¢®«ì­ ï äã­ªæ¨ï
á ª®¬¯ ªâ­ë¬ ­®á¨â¥«¥¬ ¢ QT â ª ï, çâ®  2 L1 (QT ) ¨
2 Wp1;0(QT ; ); p 
1; t 2 L1 (QT ). ’®£¤  ¨¬¥¥â ¬¥áâ® à ¢¥­á⢮
ZZ 

QT

A(x; t; u; 5xu) 5x (

ebu)

, B (x; t; u; 5xu)

ebu

bu
, eb



t

dxdt = 0:

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‹¥¬¬  3.
u
(3) u 2 Wp;loc (QT ; ) \ L1 (
)
1
z 2 Wp (QT ; ) \ C0 (QT ) zt 2 L1;QT
ãáâì

| à¥è¥­¨¥ ãà ¢­¥­¨ï

¨

,

,

, ¯à¨ç¥¬

kuk1;QT + kzk1;QT + k 5x zkpp;QT ; + kzt k1;QT  M:
C = C (M; f1; f2; f3; g2; g3; h3 )

’®£¤  áãé¥áâ¢ã¥â ¯®áâ®ï­­ ï

â ª ï, çâ®

ZZ

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QT

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ZZ 

p
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@t

QT

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QT



ZZ

u p,1

j 5x j

QT

+
+

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ZZ

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p )ebu ddt +

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QT

u p,1

f2j j jzj

p ebu ddt +

QT

ZZ

ZZ

g2jujp,1 5x (jzjp )ebu ddt

QT

ZZ

f1 j 5x ujp,1 zp ebu ddt

QT

f3 jzj

p ebu ddt +

QT

+

ZZ
QT

f2 j j j z j
up

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ZZ

h3 jzjp ebu ddt

QT

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QT

9
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Ik :
b @t
k=1

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k 5x zkp;QT ; ;
p,1
k 5 z k
p,1
I2  pebkuk1
kuk1
x p;QT ;
kz k1
kg2 k p,p 1 ;QT ; ;

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kg3 k p,p 1 ;QT ;
k 5x z kp;QT ; ;

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kf k
I4  ebkuk1
kz k1
kz 5x ukpp;Q
1 p;QT ; ;
T ;
p,1
kz kp kf k
I5  ebkuk1
kuk1
1 2 1;QT ; ;
I6  ebkuk1
kz kp1
kf3 k1;QT ; ;
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p,1 kz k
I9  ebkuk1 kz k1
t 1;QT :
Žæ¥­¨¬ ⥯¥àì «¥¢ãî ç áâì

ZZ

j 5x j jzj
up

p ebu ddt

 e,kuk1

ZZ

j 5x ujp jzjp ddt:

QT

QT

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1
6
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T ; 3
‹¥¬¬  ¤®ª § ­ .

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1;0
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L1 (QT ) 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (2) ­  ¬­®¦¥á⢥ QT ne, â® áãé¥áâ¢ã¥â
1;0
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~ 2 Wloc;p (QT ; ) à¥è¥­¨ï u â ª®¥, çâ® u
~ ¥áâì à¥è¥­¨¥ ãà ¢­¥­¨ï (3) ­  QT .
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2 C01 (QT ) á ª®¬¯ ªâ­ë¬ ­®á¨â¥«¥¬ S  QT ; 0   1; = 1 ¢ ®ªà¥áâ­®á⨠! ¬­®¦¥á⢠ e. Ž¯à¥¤¥«¨¬ ¯®á«¥¤®¢ â¥«ì­®áâì ä㭪権 z k = (1 ,  k ).
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1;0
¯®áâ®ï­­ ï, ­¥ § ¢¨áïé ï ®â k . ’ ª ª ª u 2 Wp;loc (QT ; ne; ),
= 1 ­  ! , ¨
¬¥à  ¬­®¦¥á⢠ e à ¢­  ­ã«î, â®
’¥®à¥¬  1.

ãáâì

Z

jujpj1 ,  k jp ddt

!

k ! 1. ’ ª ª ª  k ! 0 ¯. ¢., â® u 2 Lp;QT ;;loc .
„ «¥¥ ¯® «¥¬¬¥ 2 kz k 5x ukp;QT ;  C , ¨ â. ª. z k !
¯®ç⨠¢áî¤ã, â®
5x ukp;QT ;  C ¨ 5x u 2 Lp;QT ;;loc .

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k

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â ª¨å, çâ® ¢ë¯®«­ïîâáï ãá«®¢¨ï (2).
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¨¬¥¥¬
ZZ

[A 5x ((1 ,  k )') , B (1 ,  k )' , u(1 ,  k )')t]dxdt = 0:

QT

‘«¥¤®¢ â¥«ì­®,
ZZ

(1 ,

 k )[A

5x ' , B' , u't ] =

QT

ZZ

'[u( k )t , A( k )x ]dxdt:

QT

‚¢¨¤ã ⮣®, çâ®  k ! 0 ¯. ¢., ¯¥à¢ë© ¨­â¥£à « áâ६¨âáï ª
ZZ

(A 5x ' , B' , u't )dxdt:

QT

‚â®à®© ¨­â¥£à « ¢ ᨫã ãá«®¢¨© (4){(6) ­¥ ¯à¥¢®á室¨â ¢¥«¨ç¨­ë
,

,1
kuk1
k'kp
kjt kp + k 5x  k kp;QT ;
k'k1 k 5x ukpp;Q
T ;



p,1
kg k p
p ;Q ; :
+kuk1
2 p,1 ;QT ; + kg3 k p,
1 T
’ ª ª ª ktk kp ! 0; k5x  k kp;QT ; ! 0, â® ¯®á«¥¤­¥¥ ¢ëà ¦¥­¨¥ ¬®¦­® ᤥ« âì
ª ª 㣮¤­® ¬ «ë¬. ‘«¥¤®¢ â¥«ì­®,
ZZ

(A 5x ' , B' , u't )dxdt = 0:

QT

’ ª¨¬ ®¡à §®¬ u | ï¥âáï à¥è¥­¨¥¬ (3) ¢ QT ¨ ⥮६  ¤®ª § ­ . B

‹¨â¥à âãà 
1. Aronson D. G. Removable singularities for linear parabolic equations // Arch.
Rational Mech. Anal.|1964.|V. 17.|P. 79{84.
2. Edmunds D. E., Peletier L. A. Removable singulirities of solutions to quasilinear parabolic equations // J. London Math. Soc.|1970.|V. 2, No. 2.|
P. 273{283.

3{12

Œ. ‘. €«¡®à®¢ 

3. Gariepy R., Ziemer W. P. Removable sets for parabolic equations // J. London
Math. Soc.|1981.|V. 21, No. 2.|P. 311{318.
4. Saraiva L. M. R. Removable singularities and quasilinear parabolic equations //
Proc. London Math. Soc.|1984.|V. 48, No. 3.|P. 385{400.
5. Šà ©§¥à ‚., Œî««¥à . “áâà ­¨¬ë¥ ¬­®¦¥á⢠ ¤«ï ãà ¢­¥­¨ï ⥯«®¯à®¢®¤­®á⨠// ‚¥áâ. Œƒ“.|1973.|ü 3.|C. 26{32.
6. Ziemer W. P. Regularity at the boundary and removable singularities for solutions of quasilinear parabolic equations // Proc. Center for Math. Anal.
Australia Nat. Univ.|1982.|V. 1.|P. 17{25.
7. ‚®¤®¯ìï­®¢ ‘. Š.  §à殮­­ë¥ ¬­®¦¥á⢠ ¢ ¢¥á®¢®© ⥮ਨ ¯®â¥­æ¨ « 
¨ ¢ë஦¤ î騥áï í««¨¯â¨ç¥áª¨¥ ãà ¢­¥­¨ï // ‘¨¡. ¬ â. ¦ãà­.|1995.|
T. 36, ü 1.|C. 28{36.
8. ‚®¤®¯ìï­®¢ ‘. Š. ‚¥á®¢ë¥ ¯à®áâà ­á⢠ ‘®¡®«¥¢  ¨ £à ­¨ç­®¥ ¯®¢¥¤¥­¨¥ à¥è¥­¨© ¢ë஦¤ îé¨åáï £¨¯®í««¨¯â¨ç¥áª¨å ãà ¢­¥­¨© // ‘¨¡. ¬ â.
¦ãà­.|1995.|T. 36, ü 2.|C. 278{300.
9. ‚®¤®¯ìï­®¢ ‘. Š. ‚¥á®¢ ï Lp -⥮à¨ï ¯®â¥­æ¨ «  ­  ®¤­®à®¤­ëå £à㯯 å //
‘¨¡. ¬ â. ¦ãà­.|1992.|T. 33, ü 2.|C. 29{48.
10. ‚®¤®¯ìï­®¢ ‘. Š., Œ àª¨­  ˆ. ƒ. ˆáª«îç¨â¥«ì­ë¥ ¬­®¦¥á⢠ ¤«ï à¥è¥­¨© áã¡í««¨¯â¨ç¥áª¨å ãà ¢­¥­¨© // ‘¨¡. ¬ â. ¦ãà­.|1995.|T. 36,
ü 4.|C. 805{818.
11. ‚®¤®¯ìï­®¢ ‘. Š., —¥à­¨ª®¢ ‚. Œ. à®áâà ­á⢠ ‘®¡®«¥¢  ¨ £¨¯®í««¨¯â¨ç¥áª¨¥ ãà ¢­¥­¨ï // ‹¨­¥©­ë¥ ®¯¥à â®àë, ᮣ« á®¢ ­­ë¥ á ¯®à浪®¬.|
®¢®á¨¡¨àáª: ˆ§¤-¢® ˆ­-â  ¬ â-ª¨ ‘Ž €,|1995.|C. 3{64.
12. Lu G. Weigthed Poincare and Sobolev inequalities for vector elds satisfying Hormander's condition and applications // Revista Matematica
Iberoamericana.|1992.|V. 8, No. 4.|P. 367{439.
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of degenerate elliptic equations // Comm. in P. D. E.|1982.|V. 7, No. 1.|
P. 77{116.
14. Jerison D. The Poincare inequality for vector elds satisfying Hormander's
condition // Duke Math. J.|1986.|V. 53, No. 2,|P. 503{523.
15. Serrin J. Introduction to dierentiation theory.|University of Minnesota,
1965.
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