I HC QUC GIA THNH PH H CH MINH TRNG I HC KHOA HC T NHIN

FG

CAO QUC DUY

PHNG TRNH SNG KIRCHHOFF

MT CHIU VI IU KIN BIN NEUMANN KHNG THUN NHT

MT PHN BIN

LUN VN THC S TON HC

Chuyn Nghnh: Ton gii tch M S : 60. 46. 01

Ngi hng dn khoa hc: TS NGUYN THNH LONG i Hc Khoa Hc T Nhin Tp.HCM

Thnh ph H Ch Minh Nm 2009

Lun vn c hon thnh ti Trng i hc Khoa hc T nhin Tp.H Ch Minh

Ngi hng dn khoa hc: TS Nguyn Thnh Long i hc Khoa hc T nhin Tp. HCM

Ngi nhn xt 1: TS Trnh Anh Ngc i hc Khoa hc T nhin Tp. HCM

Ngi nhn xt 2: TS L Th Phng Ngc Cao ng S phm Nha Trang

Hc vin cao hc: Cao Quc Duy

Lun vn s c bo v ti hi ng chm lun vn ti trng i hc Khoa hc T nhin Tp. H Ch Minh, vo lc gi ngy thng nm 2009.

C th tm hiu lun vn ti phng Sau i Hc, th vin trng i hc Khoa hc T nhin Tp. H Ch Minh.

Thnh Ph H Ch Minh Nm 2009

1

LI CM N

Li u tin, ti xin by t lng knh trng v bit n su sc nht n

Thy hng dn, TS Nguyn Thnh Long. Thy truyn t cho ti nhiu kin

thc qu bu v tn tnh hng dn ti trong sut kha hc v nht l trong vic

hon thnh lun vn ny.

Xin trn trng cm n Thy Trnh Anh Ngc v C L Th Phng Ngc

c cn thn lun vn ca ti v cho nhiu nhn xt qu bu lun vn c

hon chnh hn.

Xin chn thnh cm n Qu Thy C trong v ngoi khoa Ton Tin hc

trng i hc Khoa hc T nhin Tp. H Ch Minh tn tnh ging dy v

truyn t kin thc cho ti trong sut thi gian ti hc tp ti trng.

Xin chn thnh cm n Phng Sau i hc to iu kin thun li cho

ti hon thnh kha hc v lm cc th tc bo v lun vn.

Xin chn thnh cm n Ban Gim hiu, Khoa Khoa hc C bn trng

i hc B Ra Vng Tu to nhiu iu kin thun li v mt cng tc

ti hon thnh lun vn.

Xin cm n cc bn hc vin lp Cao hc Gii tch K16 v cc anh ch

trong nhm seminar do Thy hng dn t chc ng vin v nhit tnh gip

ti trong sut thi gian ti hc tp v lm lun vn.

Sau cng, xin gi n gia nh ti tt c nhng tnh cm thn thng, ni

to iu kin v gip ti vt qua nhiu kh khn trong cuc sng ti tip

tc con ng hc vn.

2

MC LC

Li cm n ...1

Mc lc .2

CHNG 0. TNG QUAN V BI TON....3

CHNG 1. CC CNG C CHUN B ..7

1.1. Cc k hiu v khng gian hm .7 1.2. Mt s cng c thng s dng .7

CHNG 2. S TN TI V DUY NHT NGHIM....10

2.1. Gii thiu ...10 2.2. Thut gii xp x tuyn tnh .11

2.3. S tn ti v duy nht nghim .25

CHNG 3. NGHIN CU KHAI TRIN TIM CN NGHIM YU

CA BI TON NHIU THEO MT THAM S B.33

CHNG 4. MINH HA BNG MT BI TON C TH 51

KT LUN 53

TI LIU THAM KHO 54

3

CHNG 0

TNG QUAN V BI TON

Trong lun vn ny, chng ti xt bi ton gi tr bin v ban u cho

phng trnh sng phi tuyn c h s cha tch phn thuc dng di y

( )2( ) ( , , ), 0 1, 0 ,tt x xxu u t u f x t u x t T = < < < < (0.1) (0, ) ( ), (1, ) 0,xu t g t u t= = (0.2) 0 1( ,0) ( ), ( ,0) ( )tu x u x u x u x= = . (0.3) trong 0 1, , , ,f g u u l cc hm cho trc tha cc iu kin m ta s ch ra sau.

Trong phng trnh (0.1), s hng phi tuyn ( )2( )xu t l hm ph thuc vo tch phn

12 2

0

( ) ( , )x xu t u x t dx= . (0.4) Phng trnh (0.1) c ngun gc t phng trnh m t dao ng phi

tuyn ca mt dy n hi (Kirchhoff [6])

2

00

( , )2

L

tt xxEh uhu P y t dy u

L y = + , (0.5)

y, u l vng, l khi lng ring, h l thit din, L l chiu di si dy trng thi ban u, E l mun Young v 0P l lc cng trng thi ban

u.

4

Trong [7], Carrier thit lp m hnh di dng

20 10

( , )L

tt xxu P P u y t dy u = + , (0.6)

trong 0P , 1P l cc hng s dng.

Bi ton (0.1) (0.4) c nhiu ngha trong vt l v c hc v c

nhiu nh ton hc quan tm nghin cu trong thi gian gn y.

Phng trnh (0.1) vi cc dng khc nhau ca v f v cc iu kin bin khc nhau c kho st bi nhiu tc gi, chng hn:

Trong [9], Nguyn Thnh Long v Trn Ngc Dim kho st phng

trnh (0.1) vi 1 , ( , , , , )x tf x t u u u v iu kin bin hn hp thun nht. Trong [10], Nguyn Thnh Long, Nguyn Cng Tm v Nguyn Th Tho

Trc kho st phng trnh (0.1) vi 1 , ( , , , , )x tf x t u u u v iu kin bin hn hp khng thun nht.

Khi 0f = v ( )2xu = l hm ph thuc vo 2xu vi iu kin bin hn hp hay Cauchy cng c nghin cu bi nhiu tc gi: Ebihara, Mederios,

Minranda [15], Pohozaev [16].

Trong [12 13], Nguyn Thnh Long v cc tc gi nghin cu

phng trnh (0.1) vi cc s hng phi tuyn v f c dng tng qut hn ( )2, xt u hoc ( )22, , xt u u v ( )22, , , , , ,x t xf x t u u u u u vi cc iu kin bin khc nhau.

Trong phn u ca lun vn ny, chng ti s lin kt vi phng trnh

(0.1) mt dy qui np tuyn tnh b chn trong mt khng gian hm thch hp. S

tn ti v duy nht nghim yu ca bi ton (0.1) (0.4) c chng minh da

5

phng php xp x Galerkin v k thut compact yu trong cc khng gian hm

Sobolev.

Trong phn hai ca lun vn, chng ti nghin cu khai trin tim cn

nghim yu ca bi ton nhiu theo mt tham s b , trong s hng nhiu l cc s hng phi tuyn trn phng trnh cng dng v biu thc ca iu kin

u nh bi ton sau

( )

( )

( ) ( )

2

1

0 011

1 11

2 2

1

( ) ( , , ), 0 1, 0 ,

(0, ) ( ), (1, ) 0,

( ,0) ( ) ( ) ,

( ,0) ( ) ( ) ,

( ) 1 ( ) ,

( , , ) ( , , ) ( , , ),

tt x xx

xN

kk

kN

kt k

k

x x

u u t u F x t u x t T

u t g t u t

u x u x u xQ

u x u x u x

u t u t

F x t u f x t u f x t u

+

=+

=

= < < <

6

Chng 3, chng ti nghin cu khai trin tim cn ca bi ton ( )Q theo mt tham s b .

Chng 4, chng ti cho mt v d c th minh ha v khai trin tim

cn ca bi ton ( )Q . K n l phn kt lun v sau cng l danh mc cc ti liu tham kho.

7

CHNG I

CC CNG C CHUN B.

1.1. Cc k hiu v khng gian hm

Chng ti b qua nh ngha cc khng gian hm thng dng v cho

tin li, ta k hiu:

( ) ( )0,1 , Q 0,T , 0T T = = > , ( ) ( ) ( ),2 , ,, ,p p m m m m p m pL L H H W W W= = = = ,

, , ln lt ch chun v tch v hng trn 2L , X

ch chun trn khng gian Banach X.

Ta k hiu ( )0, ; , 1pL T X p l khng gian Banach cc hm ( ): 0,u T X o c sao cho

( ) ( )1

0, ;0

( ) , 1T pP

L T X Xpu u t dt p = < + < ,

v

( ) ( )0, ; 0 sup ( ) ,L T X Xt Tu ess u t p = = . Ta vit

( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )t tt x xxu t u t u t u t u t u t u t u t u t= = = = ln lt thay cho 2 2

2 2( , ), ( , ), ( , ), ( , ), ( , )u u u uu x t x t x t x t x tt t x x

theo th t.

1.2. Mt s cng c thng s dng

8

Cho ba khng gian Banach 0 1, ,X X X sao cho 0 X X 1X vi cc

php nhng lin tc sao cho

0 1,X X phn x, (1.1)

php nhng 0X X compact. (1.2)

Vi 0 , 1 , 0, 1iT p i< < = ta t

( ) ( )0 10 10, ; : 0, ;p pdvW v L T X v L T Xdt = = .

Trn W ta trang b chun sau

( ) ( )0, ; 0, ;0 10 1W L T X L T Xp pv v v= + . Khi , W l khng gian Banach v ( )0 0, ;pW L T X . Ta cng c kt qu sau lin quan n php nhng compact.

B 1.1. [Lions [2], trang 57]. Vi cc gi thit (1.1), (1.2) v

1 , 0, 1ip i< < = th php nhng W ( )0 0, ;pL T X l compact. Sau cng chng ti trnh by mt s kt qu v l thuyt ph.

Ta thnh lp cc gi thit sau:

Cho V v H l hai khng gian Hilbert tha cc iu kin sau:

(i) Php nhng t V H l compact, (1.3)

(ii) V tr mt trong H . (1.4)

Cho :a V V \ l dng song tuyn tnh i xng, lin tc trn V V v cng bc trn V . Chnh xc hn, ta gi a l:

j/ dng song tuyn tnh nu ( , )u a u v tuyn tnh trn V vi mi v V v ( , )v a u v tuyn tnh trn V vi mi u V .

jj/ i xng nu ( , ) ( , ), ,a u v a v u u v V= .

9

jjj/ lin tc nu M sao cho ( , ) , ,V V

a u v M u v u v V . jv/ cng bc nu 0 > sao cho 2( , ) ,

Va u u u u V .

Khi ta c kt qu sau m chng minh c th tm thy trong [3 , trang

87, nh l 7.7].

B 1.2. Di cc gi thit (1.3) v (1.4) . Khi tn ti mt c s trc

chun { }1j j

w= ca H gm cc hm ring jw tng ng vi cc gi tr ring j

sao cho 1 20 ... ..., limj jj

< = + v

( , ) , , , 1, 2...j j ja w v w v v V j= = Hn na, dy { } { }j j jw w = cng l mt c s trc chun ca V i

vi tch v hng ( , ).a Ta cng c b nh gi sau m chng minh khng my kh khn:

B 1.3. Cho dy s thc { }m tha 0 10, 0 ,m m = + 1,m trong 0 1, 0 < > l cc hng s cho trc.

Khi , 11m

m .

10

CHNG II

S TN TI V DUY NHT NGHIM

2.1. Gii thiu

Trong chng ny, chng ti xt bi ton gi tr bin v ban u cho

phng trnh sng phi tuyn c h s cha tch phn thuc dng di y

( )2( ) ( , , ), 0 1, 0 ,tt x xxu u t u f x t u x t T = < < < < (2.1) (0, ) ( ), (1, ) 0,xu t g t u t= = (2.2)

0 1( ,0) ( ), ( ,0) ( )tu x u x u x u x= = . (2.3) trong 0 1, , , ,f g u u l cc hm cho trc tha cc iu kin m ta s ch ra sau. Trong phng trnh (2.1), s hng phi tuyn ( )2( )xu t l hm ph thuc vo tch phn

12 2

0

( ) ( , )x xu t u x t dx= . Bng cch i n hm, ta a bi ton khng thun nht (2.1) (2.3) v

bi ton vi iu kin bin thun nht nh sau.

Vi [ ]0,1x v 0t , ta t ( , ) ( 1) ( )x t x g t = , (2.4)

( , ) ( , ) ( , )v x t u x t x t= , (2.5) ( ), , ( , , ) ( , ) ( , , ) ( 1) ( )ttf x t v f x t v x t f x t v x g t= + = + , (2.6)

0 0 1 1( ) ( ) ( ,0), ( ) ( ) ( ,0)tv x u x x v x u x x= = , (2.7) cng vi iu kin tng thch

0 0(0) (0,0) (0), (1,0) (1) 0x xg u u u u= = = = . (2.8)

11

Khi bi ton (2.1) (2.3) tng ng vi bi ton gi tr bin ban u

sau y

( )2( ) ( ) ( , , ), 0 1, 0 ,tt x xxv v t g t v f x t v x t T + = < < <

12

Ta cng ch rng iu kin (H4) khng nht thit

( )1 [0,1]f C + \ \ . Cho trc 0M > v * 0T > , ta t:

( )*

0 0( , , )

, sup ( , , )x t v A

K K M f f x t v

= = , (2.15)

( ) ( )*

1 1 1 3( , , )

, sup ( , , )x t v A

K K M f D f D f x t v

= = + , (2.16)

trong ( ) { }* ** * , ( , , ) : 0 1, 0 , .A A M T x t v x t T v M= = Cho trc * 0.T > Vi mi 0M > v *(0, ],T T ta t

( ) ( ) ( ){ 2, 0, ; : 0, ;tW M T v L T V H v L T V = v ( )2 ,tt Tv L Q vi ( ) ( ) ( ) }2 20, ; 0, ;, , ,Tt ttL T V H L T V L Qv v v M (2.17)

( ) ( ) ( ){ }21 , , : 0, ;ttW M T v W M T v L T L= , (2.18) trong ( )0,TQ T= .

( ) ( ) ( )0 * 0 **1 1 0, 0,, C T C TM M T g g g = = + , (2.19) ( )

( )210 0

0, sup ( )

z M MK K M z

+= = , (2.20)

( )( )21

1 10

, sup ( )z M M

K K M z +

= = . (2.21)

Trong lun vn ny, chng ti t

{ }1 : (1) 0V v H v= = (2.22) v s dng dng song tuyn tnh sau

1

0

( , ) ,a u v u vdx u v V= . (2.23)

13

Trn V ta s dng chun ( , )V

v a v v v= = . Khi , ta c cc b sau:

B 2.1. Php nhng t V ( )0C l compact v [ ]( )0 0,1 xC Vv v v = ,

1 1

12 xH V H

v v v v = , vi mi v V .

B 2.2. Dng song tuyn tnh i xng ( , )a xc nh bi (2.23) lin tc trn V V v cng bc trn V .

Cc b 2.1 v 2.2 l cc kt qu quen thuc m chng minh ca n c

th tm thy trong nhiu ti liu lin quan n l thuyt v khng gian Sobolev,

chng hn trong [1].

Ch thch:

Trn V , ba chun sau l tng ng 1 , , ( , )H Vv v v a v v = . V l khng gian con ng ca 1H , do n cng l khng gian Hilbert

i vi tch v hng ca 1H .

B 2.3. Tn ti mt c s trc chun Hilbert { }1j j

w= ca

2L gm cc

hm ring jw tng ng vi cc gi tr ring j sao cho 1 20 ... ..., limj j

j

< = + ,

( , ) , , 1, 2...j j ja w v w v v V j= = Hn na, dy { } { }j j jw w = cng l mt c s trc chun ca V i

vi tch v hng ( , ).a

14

Hn na, cc jw tha bi ton bin di y

( )

( )

trong 0,1 ,

(0) (1) 0,

[0,1] .

j j j

j j

j

w ww w

w V C

= = =

B 2.3 c suy ra t b 1.2 vi 2H L= , , ( , )V a c xc nh bi (2.22)-(2.23).

Ta chn s hng u tin 0 0v . Gi s

( )1 1 ,mv W M T . (2.24) Ta lin kt bi ton (2.9) (2.11) vi bi ton bin phn sau:

Tm ( )1 ,mv W M T tha bi ton bin phn tuyn tnh sau y ( ), ( ) ( ), ( ),m m m mv t w t v t w F t w w V+ = , (2.25)

0 1(0) , (0)m mv v v v= = , (2.26) trong

( )21 1( ) ( ) ( ) , ( ) ( , , ( ))m m m mt v t g t F t f x t v t = + = . (2.27) S tn ti ca cc mv c cho bi nh l sau y.

nh l 2.1. Gi s cc gi thit (H1) (H4) ng. Khi tn ti mt

hng s 0M > ph thuc vo 0 1, , ,g u u v 0T > ph thuc vo 0 1, , , ,f g u u sao cho, vi 0 0v = , tn ti mt dy qui np tuyn tnh

{ } ( )1 ,mv W M T c xc nh bi (2.25) (2.27). Chng minh nh l 2.1 Bao gm cc bc di y.

Bc 1. Xp x Galerkin

15

Gi { }jw l c s trc chun ca V nh trong b 2.3, j j jw w = . t

( ) ( )

1( ) ( )

kk k

m mj jj

v t c t w=

= (2.28) trong ( ) ( )kmjc t tha h phng trnh vi phn tuyn tnh di y

( ) ( )( ), ( ) ( ), ( ), , 1k km j m m j m jv t w t v t w F t w j k+ = , (2.29) vi

( ) ( )0 1(0) , (0)

k km k m kv v v v= = , (2.30)

trong

0 0kv v trong 2V H mnh, (2.31) 1 1kv v trong V mnh. (2.32)

B 2.4. Gi s ( )1 1 ,mv W M T , h (2.28) (2.29) c nghim duy nht ( ) ( )kmv t trn [ ]0, .T

Chng minh b 2.4: B qua cc ch s m v k trong biu thc ca ( ) ( )kmv t , 0kv , 1kv h (2.28) (2.29) vit li nh sau

0 1

( ) ( ) ( ) ( ), , 1 ,

(0) : , , (0) : , .

j j m j j m j

j j j j j jV V

c t t c t F t w j k

c v w c v w

+ = = = = = (2.33)

vi 1

( ) ( )k

j jj

v t c t w=

= . Vit li h (2.33) di dng phng trnh tch phn

0 0

( ) ( ) ( ) ( )t

j j j m jc t h t d s c s ds

= , (2.34) vi

16

0 0

( ) ( ), , 1t

j j j j m jh t t d F s w ds j k

= + + . (2.35) Ta vit li h ny theo dng phng trnh im bt ng

( ) [ ]( ),c t H c t= (2.36) trong

( )1( ) ( ),..., ( ) ,kc t c t c t= ( )1[ ]( ) [ ]( ),..., [ ]( ) ,kH c t H c t H c t=

v

( )0 0

[ ]( ) ( ) ( ) .t

j j j m jH c t h t d s c s ds

= (2.37) Ta t

[ ]( )0 0, ; kY C T= \ , th Y l khng gian Banach vi chun 1 10 1

max ( ) , ( ) ( ) .k

jY t T jc c t c t c t

== =

D thy : .H Y Y Ta s nghim li rng vi 0n thch hp th 0 0

1: ( ) :n nH H H Y Y= l nh x co. Tht vy, t (2.34), (2.35), (2.37) v (2.20), vi mi ,c d Y v mi

[ ]0,t T , ta suy ra

0 0

[ ]( ) [ ]( ) ( ) ( ) ( )t

j j j m j jH c t H d t d s c s d s ds

20

0 0

( ) ( ) ,t

k j jK d c s d s ds

suy ra

17

201 1

0 0

[ ]( ) [ ]( ) ( ) ( )t

kH c t H d t K d c s d s ds

212k Yt c d . (2.38) vi

20k k K = . (2.39)

T (2.38) v (2.39) ta suy ra

21[ ] [ ]2kY Y

H c H d t c d , v bng qui np ta c

11[ ] [ ]( 1)!

n n nk YY

H c H d t c dn

+ +

11 , .( 1)!

nk Y

T c d nn

+ + (2.40)

Ta chn 0n sao cho

0 1

0

1 1.( 1)!

nk Tn

+

18

vi 2 2( ) ( ) ( )( ) ( ) ( ) ( )k k km m m mX t v t t v t= + , (2.43)

2 2( ) ( ) ( )( ) ( ) ( ) ( )k k km m m mY t v t t v t= + . (2.44) Khi , ta c b sau

B 2.5. Ta c

( )2 2( ) ( ) ( ) ( ) ( )0

( ) ( ) ( )

0 0 0

( ) (0) (0) ( ) ( ) ( )

2 ( ), ( ) 2 ( ), ( ) ( )

tk k k k k

m m m m m m

t t tk k k

m m m m m

S t X Y s v s v s ds

F s v s ds F s v s ds v s ds

= + + +

+ + +

( ) ( )1 2 3 4(0) (0)

k km mX Y I I I I= + + + + + . (2.45)

Chng minh b 2.5. Nhn (2.29) vi ( ) ( )kmjc t , sau ly tng theo j , v t (2.43) ta c

( ) ( ) ( ) ( ) ( )

2( )

1 ( ) ( ), ( ) ( ) ( ), ( )2

1 ( ) ( )2

k k k k km m m m m m

km m

d X t v t v t t v t v tdt

t v t

= +

+

2( ) ( )1( ), ( ) ( ) ( )2

k km m m mF t v t t v t= + . (2.46)

Trong (2.29) thay 1j jj

w w= , sau n gin j , ta c

( )( ) ( )( ), ( ) ( ), ( ),k km j m m j m jv t w t v t w F t w + = (2.47) Ta cng c cc kt qu sau

1( ) ( )

0

( ), ( , ) ( )k km j m jv t w v x t w x dx =

19

1( ) ( )

0( , ) ( ) ( ),

xk km j m jx

v x t w x v t w== = +

( ) ( ),km jv t w= . (2.48)

( ) ( )1( ) ( )0

( ), ( , ) ( )k km j m jv t w v x t w x dx = 1( ) ( )

0( , ) ( ) ( ),

xk km j m jx

v x t w x v t w== = +

( )( ) ( )( )

(1, ) 1 (0, ) (0)

( ),

k km j j m j j

km j

v t w v t w

v t w

= ++

( ) ( ),km jv t w= . (2.49) 1

0

( ), ( , ) ( )m j m jF t w F x t w x dx = 1

0( , ) ( ) ( ),

(1, ) (1) (0, ) (0) ( ),

x

m j m jx

m j m j m j

F x t w x F t w

F t w F t w F t w

== = +

= + +

( ),m jF t w= . (2.50) T (2.47) (2.50) ta suy ra

( ) ( )( ), ( ) ( ), ( ),k km j m m j m jv t w t v t w F t w + = (2.51) Nhn (2.51) vi ( ) ( )kmjc t , ri ly tng theo j , ta c

( ) ( ) ( ) ( ) ( )( ), ( ) ( ) ( ), ( ) ( ), ( )k k k k km m m m m m mv t v t t v t v t F t v t + = (2.52) T (2.52) v (2.44) ta suy ra

2( ) ( ) ( )1 1( ) ( ), ( ) ( ) ( )2 2

k k km m m m m

d Y t F t v t t v tdt

= + . (2.53)

20

T (2.46) v (2.53) ly tch phn ( ) ( )( ) ( )k km md dX s Y sds ds

+ theo t ri cng

vi 2( )

0

( )t

kmv s ds ta c (2.45).

B 2.5 c chng minh. Sau y l nh gi cc tch phn trong v phi ca (2.45):

Tch phn th nht:

T (2.27), (2.24), (2.19) v (2.21) ta c

( )21 1 11 1 1

( ) ( ) ( ) 2 ( ) ( ), ( ) ( )

2 ( ) ( ) ( ) ( )

m m m m

m m

t v t g t v t g t v t g t

K v t g t v t g t

= + + + + +

( )21 12K M M + . (2.54) Suy ra t (2.54) v (2.45) rng

( )( ) ( )

2 2( ) ( )1

0

2 22 ( ) ( )1 1

0

( ) ( ) ( )

2 ( ) ( )

tk k

m m m

tk k

m m

I s v s v s ds

K M M v s v s ds

= +

+ +

( )2 ( )1 10 0

2( )

tk

mK M M

S s ds+ . (2.55)

Tch phn th hai:

T (2.27), (2.24) v (2.15) ta c

[ ]1 122 2 21 0 00 0

( ) ( , , ( )) , 0,m mF t f x t v t dx K dx K t T , (2.56) Suy ra t (2.56) v (2.45) rng

21

( ) ( )2

0 0

2( ) 2 ( )0 0

0 0

2 ( ), ( ) 2 ( ) ( )

2 ( ) ( )

t tk k

m m m m

t tk k

m m

I F s v s ds F s v s ds

K v s ds TK v s ds

=

+

2 ( )0

0

( )t

kmTK X s ds + . (2.57)

Tch phn th ba:

T (2.27), (2.24) v (2.16) suy ra

( )( )1 3 1 1( ) , , ( , ) 1 ( , )m m mF s D f D f x s v x s v x sx + + , suy ra

( )21 1 1 1( ) 1 ( , ) 2 1m m mV VF s K v x s K vx + + ( )212 1K M + . (2.58)

T (2.58) v (2.44) (2.45) ta suy ra rng

( ) ( )3

0 0

2 ( ), ( ) 2 ( ) ( )t t

k km m m mV

I F s v s ds F s v s ds=

( ) ( )2 22 ( ) 2 2 ( )10 0

( ) ( ) 2 1 ( )t t

k km m mV

F s v s ds TK M v s ds + + +

( )2 2 ( )10

2 1 ( )t

kmTK M Y s ds + + . (2.59)

Tch phn th t:

T (2.29) suy ra ( ) ( )( ), ( ) ( ), ( ), , 1k km j m m j m jv t w t v t w F t w j k = . (2.60)

22

Nhn (2.60) vi ( ) ( )kmjc t , ri ly tng theo j , ta c ( ) ( ) ( ) ( ) ( )( ), ( ) ( ) ( ), ( ) ( ), ( )k k k k km m m m m m mv t v t t v t v t F t v t = ,

suy ra ( ) ( )( ) ( ) ( ) ( )k km m m mv t t v t F t + . (2.61)

Suy ra t (2.27), (2.20), (2.61) v (2.56) rng

( )2 22 2( ) ( )( ) 2 ( ) ( ) ( )k km m m mv t t v t F t + ( )22 ( ) 20 02 ( )kmK v t K + . (2.62)

T (2.62), (2.44) v (2.45) suy ra rng 22( ) ( ) 20

4 000 0

2( ) ( ) 2t t

k km m

KI v s ds Y s ds TK= + . (2.63) Tip theo, ta nh gi s hng ( ) ( )(0) (0)k km mX Y+ : T (2.43), (2.44) v (2.30) ta c

( )( )

2 22( ) ( ) ( ) ( )1

2 22( ) ( )1

(0) (0) (0) (0) (0) (0)

(0) (0) (0) (0)

k k k km m m m m

k km m m

X Y v v g v

v v g v

+ = + + + + +

( )( )2 2 2 21 1 0 0 0(0)k k k kv v v g v v= + + + + . (2.64) T (2.64), (2.31) v (2.32), ta suy ra tn ti mt hng s 0M > ph thuc

vo 0 1, , ,g u u sao cho 2

( ) ( )(0) (0)2

k km m

MX Y+ vi mi , .m k (2.65) T (2.42) (2.45), (2.55), (2.57), (2.59), (2.63) v (2.65), ta suy ra

23

( )

( )

22( ) ( ) 2 ( )1 1

00 0 0

22 2 ( ) ( ) 20

1 000 0

2( ) ( ) ( )

2

22 1 ( ) ( ) 2

t tk k k

m m m

t tk k

m m

K M MMS t S s ds TK X s ds

KTK M Y s ds Y s ds TK

+ + + +

+ + + + +

( ) ( )2 ( )1 20

, ( ) ,2

tk

mM C M T C M S s ds + + (2.66)

trong

( ) ( )( ) ( )

2 2 21 0 1

2 21 1 0

20

, 3 2 1 ,

2 21.

C M T TK TK M

K M M KC M

= + + + += + (2.67)

Khi , t (2.67), ta lun lun chn c hng s 0T > sao cho

( ) ( )( )( ) ( )( )

22

1 2

1 2

, exp ,2

2 , exp 1.T

M C M T TC M M

k D M T TD M

+ =

24

( )km mv v trong ( )20, ;L T V H yu*, (2.72) ( )km mv v trong ( )0, ;L T V yu*, (2.73) ( )km mv v trong ( )2 TL Q yu, (2.74)

v

( ), .mv W M T (2.75) S dng b 1.1 v tnh compact ca Lions [3], ta c php nhng

( ) ( )2 2 20, ; : 0, ;T dvW v L T H v L T Vdt = = ( )2 0, ;L T V l compact. M { } ( )( ) ,kmv W M T nn b chn trong ,TW do c th trch ra mt dy con ca dy { }( )kmv vn k hiu l { }( )kmv sao cho

( )km mv v trong ( )2 0, ;L T V mnh. (2.76)

Ta c t (2.76) rng

( )

0 0

( ) ( ) ( ), ( ) ( ) ( ),T T

km m j m m jt t v t w dt t t v t w dt

( )( )0

( ) ( ) ( ) ( ) ,T

km m m jt t v t v t w dt =

( )0

0

( ) ( ) ( )T

km m jK t v t v t w dt

( ) ( ) ( )2 2( )0 0, 0, ; 0, 0,km m jL T L T VK v v w D T . (2.77) T (2.72) (2.75) v (2.77), qua gii hn trong (2.29) (2.30) ta c mv

tha (2.25) (2.27) trong ( )2 0,L T yu. Mt khc, t (2.24), (2.25) v (2.75) ta suy ra

25

( ) ( )20, ;m m m mv t v F L T L = + , tc l ( )1 ,mv W M T . nh l 2.1 c chng minh hon ton.

2.3. S tn ti v duy nht nghim

nh l 2.2. Gi s (H1) (H4) ng. Khi tn ti mt hng s 0M > v 0T > sao cho bi ton (2.9) (2.11) c mt nghim yu duy nht

( )1 ,v W M T . Mt khc, dy qui np tuyn tnh { }mv xc nh bi (2.25) (2.27) hi t

mnh ti nghim v ca bi ton (2.9) (2.11) trong khng gian hm

( ) ( ){ }21( ) 0, ; : 0, ;W T v L T V v L T L = Hn na, ta c nh gi sau

( ) ( )20, ; 0, ; ,mm m TL T L L T Vv v v v Ck m + , (2.78) trong 1Tk < v C l hng s ch ph thuc vo 0 1, , , , ,T g f u u v .Tk

Chng minh nh l 2.2. Nhc li rng, trong phn ny chng ti dng

chun trn V l ( , ).a Ch rng ( )1W T l khng gian Banach vi chun

( ) ( ) ( )21 0, ; 0, ; .W T L T L L T Vv v v = + Chng minh tn ti nghim: Trc ht, ta s chng minh dy { }mv

Cauchy trong ( )1W T . t 1 ,m m mw v v+= khi mw tha bi ton bin phn sau

( )1 1( ), ( ) ( ), ( ) ( ) ( ),m m m m m mw t w t w t w t t v t w + ++ + 1( ) ( ), ,m mF t F t w w V+= , (2.79)

(0) (0) 0m mw w= = . (2.80)

26

T (2.79) (2.80) vi ch

( ) ( ) ( )( ) ( ) ( )( )

1

0

1

0

, ,

, ,

, ,

m m

xm mx

m

v t w v x t w x dx

v x t w x v t w

v t w

==

=

= =

ta suy ra

1( ), ( ) ( ),m m mw t w t w t w ++ ( )1 1( ) ( ) ( ), ( ) ( ), ,m m m m mt t v t w F t F t w w V + += + . (2.81)

Trong (2.81) ly ,mw w= sau khi ly tch phn theo ,t ta c

( )1 121 10 0

( ) ( ) ( ) 2 ( ) ( ) ( ), ( )m m m m m m mp t s w s ds s s v s w s ds + += + 1

10

2 ( ) ( ), ( ) ,m m mF s F s w s ds++ (2.82) vi

2 21( ) ( ) ( ) ( )m m m mp t w t t w t += + . (2.83)

T gi thit (H3), (2.21) v (2.27) ta suy ra

( ) ( )2 21 1( ) ( ) ( ) ( ) ( ) ( )m m m mt t v t g t v t g t + = + +

( )212 2

10

sup ( ) ( ) ( ) ( ) ( )m mz M M

z v t g t v t g t +

+ +

( )1 1 12 ( )mK M M w t + ( ) ( )11 1 12 .m W TK M M w + (2.84)

T gi thit (H4), (2.16), (2.27) v b 2.1 ta cng suy ra

1 1( ) ( ) ( , , ( )) ( , , ( ))m m m mF t F t f x s v t f x s v t+ =

27

3 10sup ( , , ) ( ) ( )m m

v MD f x t v v t v t

( )11 12 .m W TK w (2.85) T gi thit (H3) v (2.76), (2.84) (2.85) v (2.54) ta suy ra

( )2 2 2 200

1( ) ( ) 1 ( ) ( )m m m mw t w t w t w t + + +

0

11 ( )mp t +

( )( )( ) ( )1

221 1

0 0

1 1 1 10 0

11 2 ( )

11 2 2 2 ( )

t

m

t

m mW T

K M M w s ds

K M M M K w w s ds

+ + + + + +

( ) ( ) ( ) ( )12 2 21 1 20

, ( ) ( ) ,t

m m mW TD M T w D M w s w s ds + + (2.86)

vi

( ) ( )( )( ) ( ) ( )( )

1 1 1 10

22 1 1 1 1 1

0

1, 1 2 2 ,

11 2 2 2 .

D M T K M M M K T

D M K M M K M M M K

= + + + = + + + + +

T (2.86) v bt ng thc Gronwall ta suy ra

( ) ( ) ( )( )12 2 21 1 2( ) ( ) , exp .m m m W Tw t w t D M T w TD M+ do

( )11( ) ( ) ,m m T m W Tw t w t k w + (2.87) vi

28

( ) ( )( )1 22 , exp .Tk D M T TD M= T (2.87) ta suy ra

( ) ( )1 11m T mW T W Tw k w , v do

( ) ( )11 1 0 , , .1mT

m p m W TW TT

kv v v v m pk+

(2.88)

T (2.88) v ( )22.68 ta suy ra { }mv l dy Cauchy trong ( )1W T v do , tn ti mt v trong ( )1W T sao cho

mv v trong ( )1W T mnh. (2.89) Cng v { } ( )1 ,mv W M T nn t dy { }mv ta trch ra c mt dy con

vn k hiu l { }mv sao cho mv v trong ( )20, ;L T V H yu*, (2.90) mv v trong ( )0, ;L T V yu*, (2.91) mv v trong ( )2 TL Q yu, (2.92)

v

( ), .v W M T (2.93) T (2.20) (2.21) v nh gi tng t nh trong (2.84) ta suy ra

( )20 0

( ) ( ), ( ) ( ) ( ) ( ), ( )T T

m mt v t w t dt v t g t v t w t dt +

( )0

( ) ( ) ( ) , ( )T

m mt v t v t w t dt

29

( )( )20

( ) ( ) ( ) ( ), ( )T

m t v t g t v t w t dt + +

( )0

0

1 1 10

( ) ( ) ( )

2 ( ) ( ) ( )

T

m

T

m

K v t v t w t dt

K M M M v t v t w t dt

+ +

( ) ( ) ( )( ) ( )11 10 1 1 1 0, ;2 ,m m L T VW T W TK v v K M M M v v w + + ( )1 0, ; .w L T V (2.94)

T (2.89) v (2.94) ta suy

( ) ( ) ( ) ( ) ( )( ) ( ) ( )20 0

, , ,T T

m mt v t w t dt v t g t v t w t dt + ( )1 0, ;w L T V . (2.95)

T (2.16) v b 2.1 ta suy ra

( )

( ) ( )0 0

0

( ), ( ) , , ( ) , ( )

, , ( ) , , ( ) , ( )

T T

m

T

m

F t w t dt f x t v t w t dt

f x t v t f x t v t w t dt

1 10

( ) ( ) ( )T

mK v t v t w t dt ( ) ( ) ( )11 11 1 0, ;2 , 0, ; .m L T VW TK v v w w L T V (2.96)

T (2.89) v (2.96) ta suy ra

( )10 0

( ), ( ) ( , , ( )), ( ) , 0, ;T T

mF t w t dt f x t v t w t dt w L T V . (2.97)

30

Khi , ta qua gii hn trong (2.25) (2.27) khi m+ ta thu c t (2.90) (2.93) v (2.95), (2.97) vi ch ( ) ( )2 10, ; 0, ;L T V L T V , rng tn ti

( ),v W M T tha phng trnh ( )2( ), ( ) ( ) ( ), ( , , ( )), , .v t w v t g t v t w f x t v t w w V+ + = (2.98)

v tha iu kin u

0 1(0) , (0) .v v v v= = (2.99) Mt khc, ta cng c t (2.98) v (2.93) rng

( ) ( )2 2( , , ( )) 0, ;v v g v f x t v t L T L = + + . (2.100) Vy, ta thu c ( )1 ,v W M T . (2.88) cho p + , ta thu c (2.78). S tn ti nghim c chng minh hon tt.

Chng minh s duy nht nghim: Gi s 1 2,v v l hai nghim yu ca

bi ton (2.9) (2.11) sao cho

( )1 , , 1, 2ii iv W M T i = (2.101) vi ,i iM T l M v T tng ng vi , 1, 2.iv i =

Khi { }1 2 1 2( ) ( ) ( ), 0 min ,w t v t v t t T T T= = tha phng trnh bin phn sau

( )1

1 2 2 1 2

( ), ( ) ( ),

( ) ( ) ( ), ( ) ( ), , ,

w t t w t

t t v t F t F t V

+ = +

(2.102)

v tha iu kin u

(0) (0) 0w w= = , (2.103) trong

( )( ) ( ) ( )i it v t g t = + , ( ) ( , , ( )), 1, 2i iF t f x t v t i= = . (2.104)

31

Trong (2.102), ly w = v lp lun tng t nh trong phn chng minh tn ti nghim, ta thy

( )21 1 2 20 0

( ) ( ) ( ) 2 ( ) ( ) ( ), ( )t t

Z t s w s ds s s v s w s ds = + 1 2

0

2 ( ) ( ), ( ) ,t

F s F s w s ds+ (2.105) vi

2 21( ) ( ) ( ) ( )Z t w t t w t= + . (2.106)

t { }1 2max ,M M M= , nh gi tng t nh trong (2.54), (2.84), (2.85), ta c

( ) ( )2 211 1 1 1 1( ) 2 2t K M M K M M + + , (2.107) 1 2( ) ( )t t ( )1 12 ( )K M M w t+ , (2.108)

11 2 1 1 2 1 1( ) ( ) ( ) ( ) ( ) 2 ( )HF t F t K v t v t K w t K w t . (2.109) T (2.105) (2.106), (2.107) (2.109) v gi thit (H3), ta suy ra

2 2

0

1( ) ( ) 1 ( )w t w t Z t + +

( )

( ) ( )22

1 10 0

2 21 1 1

0 0

11 2 ( )

11 2 2 ( ) ( )

t

t

K M M w s ds

K M M M K w s w s ds

+ + + + + + +

( ) ( )2 20

, ( ) ( )t

D M T w s w s ds + . (2.110)

32

vi ( ) ( ) ( )( )21 1 1 1 10

1, 1 2 2 2D M T K M M K M M M K = + + + + +

T (2.110) v bt ng thc Gronwall suy ra

2 2( ) ( ) 0w t w t+ , ngha l 1 2v v . S duy nht nghim c chng minh hon tt v nh l 2.2 c chng

minh hon ton.

33

CHNG III

NGHIN CU KHAI TRIN TIM CN NGHIM YU

CA BI TON THEO MT THAM S B

Trong chng ny, ta lun gi thit (H1) (H2), (H4) ng, ngoi ra ta

cn thnh lp cc gi thit b sung:

(H5) ( )1 , ( ) 0 0,C z z+ \ ((H5) thay cho (H3)) (H6) 1f tha gi thit (H4),

(H7) 0 1, , 1,..., 1k ku u k N= + tha gi thit (H2).

Ta xt bi ton nhiu di y, trong l tham s b, 1

( )

( )

( ) ( )

2

1

0 011

1 11

2 2

1

( ) ( , , ), 0 1, 0 ,

(0, ) ( ), (1, ) 0,

( ,0) ( ) ( ) ,

( ,0) ( ) ( ) ,

( ) 1 ( ) ,

( , , ) ( , , ) ( , , ).

tt x xx

xN

kk

kN

kt k

k

x x

u u t u F x t u x t T

u t g t u t

u x u x u xQ

u x u x u x

u t u t

F x t u f x t u f x t u

+

=+

=

= < < <

34

( )( )( )2

0

1

1 ( ) ( ) ( , , ), 0 1, 0 ,

(0, ) 0, (1, ) 0,( ,0) ( ),( ,0) ( ),

tt xx

x

t

v v t g t v F x t v x t T

v t v tQv x v xv x v x

+ + = < < <

35

l 2.2, rng gii hn 0v trong cc khng gian hm thch hp ca h { }v khi 0 l nghim yu duy nht ca bi ton ( )0Q tng ng vi 0 = tha

( )0 1 ,v W M T (3.3) Do , u v = + (tng ng 0 0u v= + ) l nghim yu duy nht ca bi ton ( )Q (tng ng ( )0Q vi 0 = ), hn na ta c nh l sau: nh l 3.1. Gi s (H1) (H2), (H4), (H6) (H7) ng. Khi tn ti

cc hng s 0M > v 0T > sao cho vi mi , 1 , bi ton ( )Q c duy nht mt nghim yu ( )1 ,u W M T v tha nh gi tim cn ( ) ( )20 00, ; 0, ;L T V L T Lu u u u C + , (3.4)

trong C l hng s ch ph thuc vo ( )0 1, , , ,T M K M f ( )1 , ,K M f ( )0 , ,K M 0 1, , 1,2,..., .k ku u k N=

Chng minh nh l 3.1

t 0 0w u u v v = = th w tha bi ton bin phn sau

( )2 01

01

1

11

( ), ( ),

( ) ( ), ( ) ( ), , ,

(0) (0) ,

(0) (0) .

Nk

kkN

kk

k

w t w t

u t u t f t f t V

w u

w u

+

=+

=

+ = + = =

(3.5)

trong

1( ) ( , , ( )) ( , , ( ))f t f x t v t f x t v t = + (3.6) Trong ( )1 3.5 , ly w = , ta c

36

( ), ( ) ( ), ( )w t w t w t w t+ = ( )2 0( ) ( ), ( ) ( ) ( ), ( )u t u t w t f t f t w t + . (3.7)

Mt khc, ta c

( ) 2 ( ), ( ) 2 ( ), ( ) ,d t w t w t w t w tdt = + (3.8)

vi

2 2( ) ( ) ( ) .t w t w t = + (3.9) T (3.7) v (3.8) ta suy ra

( )2( ) 2 ( ) ( ), ( )d t u t u t w tdt = 02 ( ) ( ), ( )f t f t w t+ . (3.10)

Tch phn (3.10) theo t vi ch ( )2 21 1

1 01 1

0 ,N N

k kk k

k ku u + +

= == + nh

gi tng t nh trong (2.86) ta cng thu c bt ng thc sau

( )2 21 1

1 01 1

20

0 0

( )

2 ( ) ( ), ( ) 2 ( ) ( ), ( )

N Nk k

k kk k

t t

t u u

u s u s w s ds f s f s w s ds

+ +

= == +

+ +

( ) ( )1 2 2 21 0 01 0

1 2 ( )tN

k kk

N u u K M w s ds +=

+ + + ( ) ( )( )1 0 1

0

2 2 , ( ) , ( )t

K M f w s K M f w s ds+ + ( ) ( )1 2 2 21 0

1

1N

k kk

N u u +=

+ +

37

( )( ) ( )220 1 00

, ( )t

K M f K M w s ds+ + + ( ) ( )2 21

0

2 , ( ) ( )t

K M f w s w s ds+ + 21 2

0

( )t

E T E s ds + , (3.11) vi

( ) ( ) ( ) ( )( ) ( ) ( )

12 2

1 0 1 0 1 01

2 0 1 0 1

, , 1

, , 2 ,

N

k kk

E K M f K M M N u u

E K M f K M M K M f

+

=

= + + + + = + +

(3.12)

Tip theo, do (3.11) v bt ng thc Gronwall ta thu c

( )21 2( ) expt E T E T vi mi [ ]0,t T , do

( ) ( )20 00, ; 0, ;L T V L T Lu u u u C + , (3.13)

trong ( )1 22 expC E T E T= l hng s ch ph thuc vo , , ,T M ( )0 1, ,K M f ( ) ( )1 0, , ,K M f K M , 0 1, , 1,2,..., .k ku u k N=

nh l 3.1 c chng minh hon ton. Trong phn tip theo chng ti nghin cu khai trin tim cn ca u n

cp 1N + theo , vi 1 . Ta s dng k hiu ( )21 1[ ] ( , , ), [ ] ( , , ), [ ] ( )xf u f x t u f u f x t u u u t = = = ,

[ ] 1 [ ]u u = + . By gi ta gi thit thm rng

(H8) ( ), ( ) 0 0,NC z z + \

38

(H9) ( ) ( )1 1[0,1] , [0,1]N Nf C f C+ + + \ \ \ \ v tha (H9i) 3 (1, , ) 0, 1,...,

kD f t u k N= = , vi mi 0t v mi u\ , (H9ii) 3 1(1, , ) 0, 1,..., 1

kD f t u k N= = , vi mi 0t v mi u\ . Gi 0u l nghim ca bi ton ( )0 ,Q ( )1 2 1, ,..., ,Nu u u W M T (vi 0M > v 0T > thch hp) l nghim ca cc bi ton ( ) ( ) ( )1 2, ,..., NQ Q Q tng ng s xc nh sau y:

( )1 1 1 1

1 1 1

1 01 1 11

[ ], 0 1, 0 ,(0, ) 0, (1, ) 0,

( ,0) ( ), ( ,0) ( ),

u u F u x t TQ u t u t

u x u x u x u x

= < < <

39

vi cc 0 1[ ] [ , , ,..., ]p p pf f u u u = , 0 1[ ] [ , , ,... ]p p pu u u = c xc nh bi cng thc hi qui

( )1 30

1[ ] [ ]p

p k p kk

f p k D f up

=

= , (3.19) ( )1 1

0 0

1[ ] 2 [ ] , .p p k

p k z j p k jk j

p k j u up

= =

= (3.20) Ta cng ch rng cc [ ]p f v [ ]p l cc hm bc nht theo cc ,pu

.pu Nu t ( )1 ,u W M T l nghim yu duy nht ca bi ton ( ),Q th 0

N ppp

w u u u h == l nghim yu ca bi ton ( )

10 1

11 1

[ ] [ ] [ ] [ ] [ ]( , ), 0 1, 0 ,

(0, ) (1, ) 0,( ,0) ,

( ,0) ,

NN

NN

w w h w F w h F h w h h hE x t x t T

w t w tw x uw x u

++

++

+ = + + + + < < <

40

( )0

1[ ] [ ] , 0 1.!

p

p p h p Np =

= (3.25)

Chng minh b 3.1:

D dng thy rng

0 00[ ] [ ] [ ]f f h f u == = = ( )0, , .f x t u (3.26)

Vi 1p = , ta c

( )1 3 3 0 1 0 3 100 0

[ ] [ ] [ ] ( ) [ ] [ ] .f f h D f h h D f u u D f u

== = = = = = (3.27)

Gi s tnh c [ ]k f v 3[ ]k D f vi 0, 1,..., 1k p= t cc cng thc (3.26) (3.27) v (3.23) th:

Ta c

( ) ( ) ( ) ( )1 1 1 310

[ ] [ ] [ ] ,p p k p kp

kpp p k p k

kf h f h C D f h h

=

= = suy ra

[ ] ( ) ( ) ( )1 1 300 0 0

1 1[ ] [ ]! !

p k p kpk

p pp k p kk

f f h C D f h hp p

== = =

= = ( )1 1 3

0

1 ! [ ] !!

pkp k p k

kk C D f p k u

p

== ( )1 3

0

1 [ ] ,p

k p kk

p k D f up

=

= tc l (3.23) ng. Chng minh tng t (3.24) cng ng.

Tip theo, ta cng d thy

( )0 00[ ] [ ] [ ]h u == = , (3.28) Vi 1p = , ta c

( )10

[ ] [ ] .h

== (3.29)

41

nhng

( ) ( ) ( ) ( )2 2 2[ ] ,zh h h h = = (3.30) v

1,

Np

pp

h u =

= ( )2 2 ,h h h = suy ra

( )2 0 10

2 ,h u u =

= . (3.31)

T (3.29) (3.31) suy ra

( )21 0 0 1 0 0 1[ ] 2 , 2 [ ] ,z zu u u u u = = . (3.32) Gi s ta tnh c [ ]k v [ ]k z vi 0, 1,..., 1k p= t cc cng thc

(3.28), (3.32) v (3.25) th:

Ta c

[ ]( ) ( ) ( ) ( )1 1 2110

[ ] [ ]p p k p kp

kp zp p k p k

kh h C h h

=

= = . (3.33) Mt khc, ta c

( ) 1 12 110

2 , 2 ,m m j m jm

jmm m j m j

jh h h C h h

=

= = ,

0

! , 0i

ii h i u i N =

= ,

suy ra

( ) 12 100 0 0

2 ,m j m jm

jmm j m j

jh C h h

== = =

= ( )1 1

02 ! , !

mj

m j m jj

C j u m j u

=

=

42

( )1 10

2 ! ! ,m

jm j m j

jj m j C u u

== . (3.34)

T (3.25), (3.33) v (3.34) ta suy ra

( ) ( )1 2100 0

1 1[ ] [ ] ! [ ]! !

p p kpk

p p k zp p kk

h k C hp p

== =

= = ( )1 11 1

0 0

2 ! [ ] ! ! ,!

p p kk jp k z p k j p k j

k jk C j p k j C u u

p

= ==

( )1 10 0

12 [ ] , ,p p k

k z j p k jk j

p k j u up

= ==

tc l (3.25) ng.

Vy, b 3.1 c chng minh. B 3.2. Gi s (H1) (H2) v (H8) (H9) ng. Khi , tn ti mt

hng s K c lp vi sao cho ( )2

1

0, ,

N

L T LE K + , (3.35)

trong K ch ph thuc vo , ,M T N v cc hng s

( )*

30 1,0 ,0 0

, sup [ ], 1, 2,..., 1,i

ji

x t T u M jK M f D f u i N

== = +

( )*

1 3 10 1,0 ,0 0

, sup [ ], 1, 2,..., ,i

ji

x t T u M jK M f D f u i N

== =

( ) ( )*0 1,0 ,0 0

, sup [ ], 1, 2,...,i

ji z

x t T u M jK M u i N

== = .

Chng minh b 3.2:

Trng hp 1N = , chng minh b 3.1 khng kh khn nn chng ti b qua v chng minh vi 2.N

43

Bng cch khai trin MacLaurin xung quanh im 0 = ca hm [ ]f h v 1[ ]f h n cp 1N + v cp N ta c

10 1 1 1

1[ ] [ ] [ ] [ , , ], 0 1

Np N

p Np

f h f u f R f + +=

= + <

44

Tng t, ta cng Khai trin MacLaurin xung quanh im 0 = ca hm [ ]h n cp N ta c

1

01

[ ] [ ] [ ] [ , , ], 0 1N

p Np N

ph u R

= = + <

45

1( , ) [ ] [ ] [ ] [ ]

Np

o p pp

E x t F h f u h h F u =

= + 1 1 1

1 1[ ] [ ] [ ] [ ]

pNp

p p i p i p pp i

f f u F u = =

= + + ( )1 (1) (1)1 1 2[ , , , , ] [ , , , ]N N NR f f R h ++ +

( )1 (1) (1)1 1 2[ , , , , ] [ , , , ] .N N NR f f R h += + (3.46) Bi tnh b chn ca cc hm , , 0, 1,...,i iu u i N = trong khng gian

( )10, ;L T H , chng ti thu c t (3.38), (3.39), (3.41), (3.43), (3.45) v (3.46) rng

( )21

0, ,,N

L T LE K + (3.47)

trong K l hng s ch ph thuc vo , ,M T N v cc hng s ( ),iK M f , 1, 2,..., 1i N= + , ( )1,iK M f , 1, 2,...,i N= , ( ),iK M , 1, 2,..., .i N=

Vy, b 3.2 c chng minh.

By gi ta xt dy hm { }mw c nh ngha nh sau

( )0

1 1 1

10 1

11 1

0,[ ] [ ] [ ] [ ] [ ]

( , ), 0 1, 0 , (3.48)(0, ) (1, ) 0,

( ,0) ,

( ,0) , 1.

m m m m m

m mN

m NN

m N

ww w h w F w h F h w h h h

E x t x t Tw t w t

w x uw x u m

++

++

+ = + + + + < < <

46

1 1

1 11

1 0 1

11 1 1

[ ] ( , ), 0 1, 0 ,(0, ) (0, ) 0,

( ,0) ,

( ,0) .

NN

NN

w h w E x t x t Tw t w t

w x uw x u

++

++

= < < <

47

( )2 14 N + ( ) ( )( )2 211 1 10, ; 0, ;8 N L T L L T LK T w w ++ + ( ) ( )( )2 21 1 1

0

4t

w s w s ds+ + , (3.53) vi 2 21 1 0 0 1(1 )N Nu K u + += + + . (3.54)

T (3.53) v bt ng thc Gronwall ta suy ra

( ) ( )( )21 1w t w t+ ( ) ( ) ( )( )2 212 1 1 1 1 10, ; 0, ;4 8 exp(4 ), 0NN L T L L T LK T w w T t T ++ + + < c lp vi m v ,

sao cho

( ) ( )2 21

0, ; 0, ;,Nm m TL T L L T Lw w C m ++ (3.56)

Tht vy, bng cch ly tch v hng hai v ca ( )13.48 vi mw , sau khi ly tch phn theo t , ta c

( )

2 2,

22 1,

0 0

( ) ( ) ( )

( ) ( ) 2 [ ] [ ], ( )

m m m

t tN

m m m m

w t t w t

s w s ds F w h F h w s ds

++ =

+ +

( )10 0

2 [ ] [ ] , ( ) 2 ( , ), ( )t t

m m mw h h h w s ds E x t w s ds + + +

48

( ) 22 1 ,0 0

( ) ( ) 2 [ ] [ ] ( )t t

Nm m m ms w s ds F w h F h w s ds + + +

110 0

2 [ ] [ ] ( ) 2 ( ) ,t t

Nm m mw h h h w s ds K w s ds ++ + + (3.57)

vi

( )2, 1 1 1( ) 1 [ ] 1 ( ) ( ) ,2m m mt w h w t h t = + + = + + ( nh) (3.58)

, 1 1 1

1 1 1

( ) 2 [ ] ( ) ( ), ( ) ( )

2 [ ] ( ) ( ) ( ) ( )

m z m m m

z m m m

t w h w t h t w t h t

w h w t h t w t h t

= + + + + + +

( ) 21 12 2 ,K N M + (3.59) v

( ) ( )2 21 1[ ] [ ]m mw h h w h h + = + ( )( )1 12 1 1 mK N M w + + , (3.60)

( )1 1 1 1 11 1 1 1

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]m m m

m m

F w h F h f w h f h f w h f h

f w h f h f w h f h

+ = + + + + + +

( )*

3 3 1 1( , , )

sup ( , , ) mx t u A

D f D f x t u w +

( ) ( )( )1 1 1 12 , , .mK M f K M f w + (3.61) T (3.57) (3.61) ta suy ra tip

( )2 2 2 2,( ) ( ) 2 ( ) ( ) ( )m m m m mw t w t w t t w t+ + ( ) ( )22 22 1 11

0 0

2 2 ( ) 2 2 ( )t t

N Nm mw s ds K T w s ds + + + + +

49

( )2 22 10

2 ( ) ( )t

m mw s w s ds + + ( ) ( ) ( ) ( )( )2 22 2 22 1 1 2 1 10, ; 0, ;2 2N N m mL T L L T LT K w w + + + + +

( ) ( )22 22 1 110 0

2 2 ( ) 2 2 ( )t t

N Nm mw s ds K T w s ds + + + + +

( )2 22 10

2 ( ) ( )t

m mw s w s ds + + ( ) ( )2 21 2

0

2 1 ( ) ( ) ,t

m mw s w s ds + + + + (3.62) vi ( ) ( )( ) ( )( )2 1 1 1 12 , , 2 1 1 .K M f K M f K N M = + + + +

Nu ta t

( ) ( )2 22 2

0, ; 0, ;,m m mL T L L T Lw w = + (3.63)

v 1 21 = + + , (3.64) th t (3.62), (3.63) (3.64) v bt ng thc Gronwall ta c

( ) ( )22 1 1 2 12 2 exp(2 )N Nm mT K T + + + + . (3.65) Ta vit li (3.65) di dng

1m m + , (3.66) vi

( ) ( )2

22 1 1

2 exp(2 )

2 exp(2 )N NT T

K T T

+ += = +

(3.67)

50

T 1(3.67) ta c th chn 0T > thch hp sao cho 0 1 < . Khi , p dng b 1.3 cho dy { }m xc nh nh (3.66) (3.67)vi ch 0 0 = , ta c

( ) ( )2 22 2

0, ; 0, ;,

1m mL T L L T Lw w m + . (3.68)

T (3.67) v (3.68) ta suy ra

( ) ( )2 21

0, ; 0, ;,Nm m TL T L L T Lw w C m ++ , (3.69)

vi

12 2

2

exp(2 )41 2 exp(2 )T

K T TCT T

+=

l hng s c lp vi m v .

Mt khc, dy qui np tuyn tnh { }mw xc nh bi (3.48) hi t mnh trong khng gian hm ( )1W T ti nghim w ca bi ton (3.21). Suy ra, cho m+ trong (3.69), ta c ( ) ( )2 2

1

0, ; 0, ;

NTL T L L T L

w w C ++ , tc l

( ) ( )2

1

0 00, ; 0, ;

N NNp p

p p Tp pL T L L T V

u u u u C

+

= = + . (3.70)

Tm li, ta c nh l sau :

nh l 3.2. Gi s (H1) (H3) v (H5) (H9) ng. Khi tn ti mt

hng s 0M > v 0T > thch hp sao cho vi mi , 1 , bi ton ( )Q c duy nht mt nghim yu ( )1 ,u W M T tha nh gi tim cn n cp 1N + nh trong (3.70), cc hm 0 1, ,..., Nu u u l nghim yu ca cc bi ton

( )0 ,Q ( )1 ,..Q ( )..., NQ tng ng.

51

CHNG IV

MINH HA BNG MT BI TON C TH.

Trong chng ny, chng ti xt mt v d c th v khai trin tim cn

cho bi ton ( )Q ng vi 510, , ( ) , 3.f f u z z N = = = Gi 0 1 2 3, , , ,u u u u u ln lt l nghim ca cc bi ton sau

( )

( )2 54

0 01

4

1 11

1 ( ) , 0 1, 0 ,

(0, ) ( ), (1, ) 0,

( ,0) ( ) ( ) ,

( ,0) ( ) ( ) .

tt x xx

x

kk

k

kt k

k

u u t u u x t T

u t g t u t

Q u x u x u x

u x u x u x

=

=

+ = < < <

52

Trong , cc hm , 1, 2, 3pF p = c tnh tng minh nh sau 251 1 0 0 0[ ] ,F u u u u= + 242 2 0 1 0 1 0 1 0[ ] 5 , ,F u u u u u u u u= + +

24 3

3 3 0 2 0 1 0 2

20 1 1 0 1 0 1 0

[ ] 5 10

, 2 , .

F u u u u u u u

u u u u u u u u

= + + + + +

Nu t 30

ppp

w u u u h == th w l nghim yu ca bi ton ( ) ( )2 2 25 5

404

414

1 ( ) ( ) ( )

( , ), 0 1, 0 ,(0, ) (1, ) 0,

( ,0) ,

( ,0) .

w w t h t w w h h w h h h

E x t x t Tw t w t

w x uw x u

+ + = + + + + < < <

53

KT LUN

Qua lun vn ny, tc gi bc u lm quen vi cng vic nghin cu

khoa hc mt cch nghim tc v c h thng. Tc gi cng hc tp c cch

c ti liu v tho lun trong cc nhm sinh hot hc thut. Tc gi bc u

tm hiu c mt s cng c ca gii tch hm phi tuyn m tc gi p dng

trong lun vn nh phng php xp x Galerkin, phng php compact yu v

cc k thut nh gi tin nghim qua gii hn trong cc khng gian hm

Sobolev. Tc gi cng c lm quen vi phng php khai trin tim cn

nghim. y l nhng kin thc v cng qu bu m tc gi hc c trong qu

trnh lm lun vn di s s dn dt ca Thy hng dn. Tuy nhin, do nng

lc v s hiu bit hn ch ca bn thn nn khng trnh khi nhng sai st, tc

gi rt mong hc hi t s ng gp v ch bo ca Qu Thy C trong v ngoi

hi ng. Tc gi xin chn thnh cm n.

54

TI LIU THAM KHO

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