Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc21706/doc.pdf Дата изменения: Thu Mar 20 15:32:28 2008 Дата индексирования: Mon Oct 1 21:39:52 2012 Кодировка:

. ., . .
(, )
. p(t ) + q(t ) cos x 2 , p (t ) q(t ) . .

(

)

. , , . - . .. , .. , .. . ( ). ( ) . . [1]; , .

74


. ., . . -- -- 2007, . 2, . 74­82 Nikolsky I. M., Kurkina . S. -- MCE -- 2007, v. 2, p. 74­82

, , , (. [2]).

. :
ut = (uu x ) x + (u - u0 )(u - u1 ) u( x,0) = f ( x ) u0

(1) (2)

u1 > u0 > 0 . (0, T ) в R , T ­ . f ( x ) . (1)-(2). u (x, t ) , : u (x, t ) C
1, 2 tx

((0, T )

в R ) C ([0, T ) в R ) .

(1). , x . u ( x, t ) = p(t ) + q(t ) cos 2
12 2 p = 2 q + p + u0u1 - (u0 + u1 ) p q = 3 pq - (u0 + u1 ) q 2

(3)

(3) u1 > 2u0 ( u0 = 10 , u1 = 25 ) .1. , . u1 2u0 ­ (u0 , 0) (u1 , 0) . .
75


3. Part 3. Mathematical theory

p(0) = p0 > u1 p (t ) () T0 , t* . t* =
1 1 1 ln - C u -u (u1 - u0 ) 1 0

,

C=

(

u1 - z0 z0 - u0 )(u1 - u0

)

.

.1. (3)

, p0 > u0 + u1 q(t ) , p () q() . p (t ) / q(t ) t t t T0 . - . - . (.[3]). :
(u ( x, t ), A) = A u ( x, t ) u ( x, t ) - u ( x, t

)

.

, p0 > u0 + u1 p () / q(t ) 1, t

76


. ., . . -- -- 2007, . 2, . 74­82 Nikolsky I. M., Kurkina . S. -- MCE -- 2007, v. 2, p. 74­82

4 u ( x, t ), - s (x ) 0 t T0 - , 3 s ( x ) = 2 3
x 1 + cos 2

.

. f ( x ) (1)-(2) u0 . , , , , .

.2.

, , sup f ( x ) < u1 , sup u(t , x ) u0 (, u(t , x ) u0 t , x ).
x x

, u(t , x ) - U (t ) : sup f ( x ) U (0) u1 . . 2
x

(1)-(2). , .
77


3. Part 3. Mathematical theory

vt = (vv

xx

)

+ v 2 - (u0 + u1 ) v

(4)

. , f ( x ) v ( x,0) x , (1) u( x, t ) (. . 3). [4].

.3. ,

.4.

. Matlab 7.0. (400 ­ 1800 x, t). x. u0 10, u1 = 22 . . (1)-(2), . . 3 (.. u (t , x ) = u0 (a, b ) t (0, T ) , T - u (t , x ) ).

78


. ., . . -- -- 2007, . 2, . 74­82 Nikolsky I. M., Kurkina . S. -- MCE -- 2007, v. 2, p. 74­82


ut = u u

(

xx

)

+u



(5)

= + 1 (., , [3]). .4 . u ( x, tn ) tn
g (x ) = 1 max u ( x, t
x n

)

-u

(u ( x, t
0

n

)

- u0 ) .

, . .4 , . , , g s ( x ) = cos2 x 2 , u s (x, t )

(

)

ut = u u

(

xx

)

+ u2 ,

(.. (5) = 1 ). - . , , , g s ( x ) . 3 S-, .. wL . LS-, ( mes wL = 0 ), , HS-, . , ,

79


3. Part 3. Mathematical theory

, LS HS-.

. 5. HS ­

. 6. HS ­



ut = (uu ut = (uu

xx xx

) )

+u +u

1.8 2.5

- (u0 + u1 )u + u0u1 - (u0 + u1 )u + u0u1

(6) (7)

. 7. LS ­

. 8. LS ­
80


. ., . . -- -- 2007, . 2, . 74­82 Nikolsky I. M., Kurkina . S. -- MCE -- 2007, v. 2, p. 74­82

(6) , (.. mes wL ) (.5). HS . .7 (7). .8 . , , .. LS-. 1. ., .., .., .., .. : , , , , , . // . . " . ". . 28, 1987. . 95­205. 2. .., .., . . // . 2001. . 27, 4. . 310320. 3. . / .. [ .]; . .. -- .: , 1987. -- 480 c. 4. .., .. . // : . .. . 2006.

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3. Part 3. Mathematical theory

THE BEHAVIOR OF SOLUTIONS OF A NONLINEAR PARABOLIC EQUATION Nikolsky I. M., Kurkina . S.
(Russia, Moscow)
We study extinct and unbounded solutions of a nonlinear heat equation with special source. A family of analytic solutions of the form p(t ) + q(t ) cos x 2 , where p(t ) and q(t ) satisfy some dynamic system, has been built. Numerical and analytic investigation of the Cauchy problem for our equation has been performed.

(

)

82