Directory UMM :Journals:Journal_of_mathematics:VMJ:

“„Š 517.5+517.9

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loc

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­¥à ¢¥­á⢮

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D . ‡¤¥áì jj | ¥¢ª«¨¤®¢  ­®à¬ . ‚ à ¡®â¥ [8]

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k

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ãà ¢­¥­¨¥ ¢¨¤ 

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£¤¥ 5

x u =(

@u

@x1 ;:::;

@u

@xn

), A=(A 1

;:::;A n

), ä㭪樨 A(x;t;;) ¨ B(x;t;;)

㤮¢«¥â¢®àïîâ ãá«®¢¨î Š à â¥®¤®à¨.

à¥¤¯®«®¦¨¬,ç⮯ਢá¥å(x;t)2Q

T

¨ 2R nf0g; 2R n

¤«ï­¥ª®â®à®£®

p2(1;1)áãé¥áâ¢ã¥âª®­áâ ­â >0¨¨§¬¥à¨¬ë¥ä㭪樨 f

1 ;f

2 ;f

3 ;g

2 ;g

3 ;h

3

â ª¨¥, çâ® ¢ë¯®«­¥­ë á«¥¤ãî騥 ãá«®¢¨ï:

jA(x;t;;)jjj p,1

!(x)+g

2 jj

p,1

!(x)+g

3

!(x); (4)

jB(x;t;;)jf

1 jj

p,1

!(x)+f

2 jj

p,1

!(x)+f

3

!(x); (5)

A(x;t;;) jj p

!(x),f

2 jj

p

!(x),h

3

!(x); (6)

¤«ï ¯®ç⨠¢á¥å x 2 ¨ ¢á¥å 2 R , 2 R n

, £¤¥ !(x) | p-¤®¯ãáâ¨¬ë© ¢¥á,

f

1 2L

p;Q

T ;

, f

2 ;f

3 ;h

3 2L

1;Q

T ;

, g

2 ;g

3 2L

p

p,1 ;Q

T ;

.

Ž¡®¡é¥­­ë¬à¥è¥­¨¥¬ ãà ¢­¥­¨ï(3)¯à¨ãá«®¢¨ïå(4){(6)­ §®¢¥¬

äã­ª-æ¨î u 2W 1;0

p;loc (Q

T

;),¥á«¨

ZZ

Q

T

f,u'

t

+A(x;t;u;5

x u)5

x

',B(x;t;u;5

x

u)'gdxdt =0

¤«ï ¢á¥å '2C 1

0 (Q

T ).

ˆá¯®«ì§ãïâ¥å­¨ªã,¯à¥¤«®¦¥­­ã.‘¥à¨­ë¬ [15],  ­ «®£¨ç­®

१ã«ì-â â ¬ ‹. Œ. . ‘ à ¨¢ë [4] ¤®ª §ë¢ ¥âáï

‹¥¬¬  2. ãáâì Q

T

| ®âªàë⮥ ¬­®¦¥á⢮ ¢ R n+1

¨ b2 R nf0g. ãáâì

u ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨ï (3) ¢Q

T

¨ ¯ãáâì | ¯à®¨§¢®«ì­ ïäã­ªæ¨ï

á ª®¬¯ ªâ­ë¬ ­®á¨â¥«¥¬ ¢ Q

T

â ª ï, çâ® 2L

1 (Q

T

) ¨ 2W 1;0

p (Q

T

;); p

1;

t 2L

1 (Q

T

). ’®£¤  ¨¬¥¥â ¬¥áâ® à ¢¥­á⢮

ZZ

Q

T

A(x;t;u;5

x u)5

x ( e

bu

),B(x;t;u;5

x u) e

bu

, e

bu

b t

p;loc

z 2W 1

p (Q

T

;)\C

0 (Q

T ) ¨ z

t 2L 1;Q T , ¯à¨ç¥¬ kuk 1;Q T +kzk 1;Q T

+k5

x zk p p;QT; +kz t k 1;Q T M:

’®£¤  áãé¥áâ¢ã¥â ¯®áâ®ï­­ ï C =C(M;f

1 ;f 2 ;f 3 ;g 2 ;g 3 ;h 3 ) â ª ï, çâ® ZZ QT jz 5 x uj p

ddtC:

C ”ã­ªæ¨ï =jzj p 2W 1;0 p (Q T

;)\C

0 (Q

T

). ˆá¯®«ì§ãï«¥¬¬ã 2, ¨¬¥¥¬

ZZ Q T ,e ,bu @jzj p @t

+A5

x (jzj

p

e bu

),B(jzj p

e bu

)

dxdt=0:

‘ ãç¥â®¬ ãá«®¢¨©(4){(6) ¯®«ã稬

I

3 pe

bkuk1

kg

3 k

p

p,1 ;QT;

k5

x zk

p;Q

T ;

;

I

4 e

bkuk

1

kzk

1

kz5

x uk

p,1

p;QT; kf

1 k

p;Q

T ;

;

I

5 e

bkuk

1

kuk p,1

1

kzk p

1 kf

2 k

1;Q

T ;

;

I

6 e

bkuk

1

kzk p

1 kf

3 k

1;Q

T ;

;

I

7 +I

8

C(M;f

2 ;h

3 );

I

9 e

bkuk

1

kzk p,1

1 kz

t k

1;Q

T :

Žæ¥­¨¬ ⥯¥àì «¥¢ãîç áâì

ZZ

Q

T j5

x uj

p

jzj p

e bu

ddte ,kuk1

ZZ

Q

T j5

x uj

p

jzj p

ddt:

‘®¡¨à ï ¢¬¥á⥠®æ¥­ª¨ «¥¢®© ¨ ¯à ¢®© ç á⥩ ­¥à ¢¥­á⢠ (7), ®ª®­ç â¥«ì­®

¨¬¥¥¬

kz5

x uk

p

p;Q

T ;

kz5

x uk

p,1

p;Q

T ;

C

3 (M;f

1 )+C

6 (M;f

2 ;f

3 ;g

2 ;g

3 ;h

3 ):

‹¥¬¬  ¤®ª § ­ . B

’¥®à¥¬  1. ãáâì Q

T

| ®âªàë⮥ ®£à ­¨ç¥­­®¥ ¬­®¦¥á⢮ ¢ R n +1

, e |

ª®¬¯ ªâ ¢ Q

T

­ã«¥¢®© (p;)-¥¬ª®áâ¨. …᫨ äã­ªæ¨ï u 2 W 1;0

p;loc (Q

T

;ne;)\

L

1 (Q

T

) 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (2) ­  ¬­®¦¥á⢥ Q

T

ne, â® áãé¥áâ¢ã¥â

¥¤¨­á⢥­­®¥ ¯à®¤®«¦¥­¨¥ u~ 2 W 1;0

loc ;p (Q

T

;) à¥è¥­¨ï u â ª®¥, çâ® u~ ¥áâì

à¥-襭¨¥ ãà ¢­¥­¨ï (3) ­  Q

T .

C ãáâì k

| ¯®á«¥¤®¢ â¥«ì­®áâì ä㭪権 ¨§ «¥¬¬ë 1. ãáâì äã­ªæ¨ï

2 C 1

0 (Q

T

) á ª®¬¯ ªâ­ë¬ ­®á¨â¥«¥¬ S Q

T

; 0 1; = 1 ¢

®ªà¥áâ-­®á⨠! ¬­®¦¥á⢠e. Ž¯à¥¤¥«¨¬ ¯®á«¥¤®¢ â¥«ì­®áâì ä㭪権 z k

= (1, k

).

Žç¥¢¨¤­® z k

2 C 1

0 (Q

T

ne). ‹¥£ª® ¢¨¤¥âì, çâ® k5

x z

k

k

p;QT;

C, £¤¥ C |

¯®áâ®ï­­ ï, ­¥ § ¢¨áïé ï ®â k. ’ ª ª ª u 2 W 1;0

p;loc (Q

T

;ne;), = 1 ­  !, ¨

¬¥à  ¬­®¦¥á⢠ e à ¢­  ­ã«î, â®

Z

! juj

p

j1, k

j p

ddt

®áâ ¥âáï ®£à ­¨ç¥­­ë¬ ¯à¨k !1. ’ ª ª ª k

!0 ¯. ¢., â®u 2L

p;Q

T ;;loc

.

„ «¥¥ ¯® «¥¬¬¥ 2 kz k

5

x uk

p;QT;

C, ¨ â. ª. z k

! ¯®ç⨠¢áî¤ã, â®

k 5

x uk

p;Q ;

C ¨ 5

x u2L

„ «¥¥  ­ «®£¨ç­ë¬ à áá㦤¥­¨¥¬ ¤®ª §ë¢ ¥¬, çâ®

x

ï¥âáï

áâà®-£¨¬ á㡣ࠤ¨¥­â®¬, â. ¥. ­ ©¤¥âáï ¯®á«¥¤®¢ â¥«ì­®áâì ä㭪権

' i

2 C 1

(

Q

T

)

â ª¨å, çâ® ¢ë¯®«­ïîâáï ãá«®¢¨ï (2).

’¥¯¥àì ¤®ª ¦¥¬, çâ®

u

ï¥âáï à¥è¥­¨¥¬ (3) ¢

Q

T

. ãáâì

'2 C 1

0

(

Q

T

),

¨¬¥¥¬

ZZ

Q

T

[

A5 x

((1

, k

)

'

)

,B

(1

, k

)

',u

(1

, k

)

'

)

t

]

dxdt

= 0

:

‘«¥¤®¢ â¥«ì­®,

ZZ

Q

T

(1

, k

)[

A5

x

',B',u'

t

] =

ZZ

Q

T

'

[

u

(

k

)

t

,A

(

k

)

x

]

dxdt:

‚¢¨¤ã ⮣®, çâ®

k

!

0 ¯. ¢., ¯¥à¢ë© ¨­â¥£à « áâ६¨âáï ª

ZZ

Q

T

(

A5 x

',B',u'

t

)

dxdt:

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

kuk

1 k'k

p k

jt k

p

+

k5

x

k

k

p;Q

T ;

k'k

1 ,

k5

x uk

p,1

p;Q

T ;

+

kuk p,1

1 kg

2 k

p

p,1 ;Q

T ;

+

kg

3 k

p

p,1 ;Q

T ;

:

’ ª ª ª

k k

t k

p

!

0

; k5 x

k

k

p;Q

T ;

!

0, â® ¯®á«¥¤­¥¥ ¢ëà ¦¥­¨¥ ¬®¦­® ᤥ« âì

ª ª 㣮¤­® ¬ «ë¬. ‘«¥¤®¢ â¥«ì­®,

ZZ

Q

T

(

A5 x

',B',u'

t

)

dxdt

= 0

:

’ ª¨¬ ®¡à §®¬

u

| ï¥âáï à¥è¥­¨¥¬ (3) ¢

Q

T

¨ ⥮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.

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

so-lutions 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

L

p

-⥮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¥-襭¨© 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

sat-isfying Hormander's condition and applications // Revista Matematica

Iberoamericana.|1992.|V. 8, No. 4.|P. 367{439.

13. Fabes E. B., Kenig C. E., and Serapioni R. R. The local regularity of solutions

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.