Bài toán ổn định hoá đối với phương trình vi phân cấp phân số nửa tuyến tính

B GIO DC V O TO TRNG I HC S PHM H NI NG TH THO BI TON N NH HểA I VI PHNG TRèNH VI PHN CP PHN S NA TUYN TNH Chuyờn ngnh: Toỏn gii tớch Mó s: 60 46 01 02 LUN VN THC S TON HC Ngi hng dn khoa hc: PGS TS Trn ỡnh K H NI, 2015 LI CM N Em xin by t lũng bit n sõu sc ti PGS.TS Trn ỡnh K ó tn tỡnh hng dn em quỏ trỡnh thc hin lun ny Em xin chõn thnh cm n Ban Giỏm Hiu, Phũng sau i hc, cựng ton th cỏc thy giỏo, cụ giỏo Khoa Toỏn Trng i Hc S Phm H Ni 2, ó ng viờn giỳp v to iu kin thun li em cú iu kin tt nht sut quỏ trỡnh hc tp, thc hin ti v nghiờn cu khoa hc Do thi gian v kin thc cú hn nờn lun khụng trỏnh nhng hn ch v thiu sút nht nh Em xin chõn thnh cm n ó nhn c nhng ý kin úng gúp ca cỏc thy giỏo, cụ giỏo v cỏc bn hc viờn H Ni, ngy 08 thỏng 07 nm 2015 Tỏc gi ng Th Tho LI CAM OAN Tụi xin cam oan, di s hng dn ca PGS.TS Trn ỡnh K, lun tt nghip "Bi toỏn n nh húa i vi phng trỡnh vi phõn cp phõn s na tuyn tớnh" c hon thnh bi s nhn thc ca chớnh bn thõn tỏc gi v khụng trựng vi bt k lun no khỏc Trong quỏ trỡnh lm lun vn, tụi ó k tha nhng thnh tu ca cỏc nh khoa hc vi s trõn trng v bit n H Ni, ngy 08 thỏng 07 nm 2015 Tỏc gi ng Th Tho Mc lc M u 1 Kin thc chun b 1.1 n nh hoỏ cho h iu khin cp phõn s tuyn tớnh 1.2 Khụng gian pha 10 1.3 o khụng compact v ỏnh x a tr nộn 11 Tớnh gii c ca h iu khin phn hi 15 Tớnh n nh hoỏ c yu ca h iu khin 22 3.1 Tớnh n nh húa c yu 22 3.2 p dng 31 Ti liu tham kho 34 ii M u Lý chn ti Gi s X v U l khụng gian Banach Xột h iu khin dng D0 x(t) Ax(t) + Bu(t) + F (t, xt ), t > 0, (0.1) x(t) = (t), t 0, (0.2) ú D0 , (0, 1), l o hm cp phõn s theo ngha Caputo, A sinh C0 -na nhúm trờn X , B : U X l toỏn t tuyn tớnh b chn, B vi B l khụng gian pha s c gii thiu phn sau, F : [0, ) ì B P(X) l ỏnh x a tr, v xt ký hiu trng thỏi lch s ca h v c xỏc nh bi xt (s) = x(t + s) vi s (, 0] Phng trỡnh vi phõn cp phõn s ó c chng minh l mt cụng c hu hiu mụ t nhiu hin tng vt lý nh cỏc dũng chy mụi trng vt liu xp v cỏc quỏ trỡnh dao ng (xem [14, 25, 28]) Cỏc khỏi nim v kt qu c bn v phng trỡnh vi phõn cp phõn s cú th tỡm thy cỏc cun sỏch chuyờn kho [20, 24, 29] Vic nghiờn cu cỏc bao hm thc vi phõn cp phõn s c thỳc y bi nhiu bi toỏn Mt s ú n t lý thuyt phng trỡnh vi phõn vi phn phi tuyn khụng liờn tc (xem [10]) C th, hm a tr F (0.1) l hm chớnh quy hoỏ ca hm n tr khụng liờn tc f theo cỏch F (t, y) l bao li úng ca (y) = {z : yn y, z = lim f (t, yn )} n Theo cỏch ny, bao hm thc vi phõn nhn c c xem xột nh l mụ hỡnh xp x cho phng trỡnh vi phõn vi chỳ ý F l ỏnh x a tr na liờn tc trờn Tớnh gii c ca (0.1)-(0.2) c nghiờn cu bi nhiu tỏc gi di cỏc gi thit khỏc v na nhúm etA v s hng phi tuyn F (xem [5, 21, 23, 32]) Bờn cnh ú, cõu hi v tớnh iu khin c ca (0.1)-(0.2) cng thu hỳt s quan tõm ca nhiu nh khoa hc (xem [26, 27, 32]) Tuy nhiờn, cỏc kt qu v tớnh n nh húa cho cỏc h vi phõn cp phõn s dng (0.1)-(0.2) cha c bit n Vi mong mun tỡm hiu sõu hn v lý thuyt n nh húa i vi phng trỡnh vi phõn cp phõn s, tụi chn "Bi toỏn n nh húa i vi phng trỡnh vi phõn cp phõn s na tuyn tớnh" cho ti nghiờn cu ca lun Vỡ tớnh nht nghim ca h (0.1)-(0.2) khụng c m bo nờn lý thuyt n nh Lyapunov khụng ỏp dng c, c trng hp = Do ú ta s dng khỏi nim n nh hoỏ yu nh sau nh ngha 0.1 H (0.1)-(0.2) c gi l n nh hoỏ c yu bi iu khin phn hi nu tn ti toỏn t tuyn tớnh D : U X cho vi u(t) = Dx(t), nghim S() = v tho n nh: vi bt kỡ > 0, tn ti > cho ||B < thỡ supt>0 |xt |B < , vi mi x S(); Hỳt yu: vi mi B, tn ti nghim ton cc x S() cho |xt |B t + Ta s chng minh kt qu n nh hoỏ yu cho (0.1)-(0.2) bng cỏch s dng kt qu n nh hoỏ cho cỏc h tuyn tớnh v lý thuyt im bt ng cho cỏc ỏnh x a tr nộn Khú khn bi toỏn ny bao gm hai yu t Trc ht l s xut hin ca tr vụ hn cựng vi khụng gian pha nghiờn cu tớnh hỳt yu ta cn tỡm cỏc iu kin hp lý cho khụng gian pha ca h Khú khn tip theo l thiu tiờu chun compact khụng gian BC(0, ; X), khụng gian cỏc hm liờn tc, b chn trờn (0, ) v ly giỏ tr X vt qua khú khn ny, ta xõy dng mt o khụng compact (MNC) chớnh qui trờn mt khụng gian ca BC(0, ; X), l khụng gian nghim cho bi toỏn ang xột Mc ớch nghiờn cu Nghiờn cu tớnh n nh húa c theo ngha yu ca h (0.1)-(0.2) da trờn tớnh n nh húa c ca h tuyn tớnh tng ng Nhim v nghiờn cu Tỡm hiu v lý thuyt na nhúm; Tỡm hiu v gii tớch bc phõn s; Nghiờn cu iu kin n nh húa ca h tuyn tớnh; Nghiờn cu tớnh n nh húa theo ngha yu ca h phi tuyn i tng v phm vi nghiờn cu i tng nghiờu cu: H iu khin mụ t bi bao hm thc vi phõn cp phõn s Phm vi nghiờn cu: iu kin n nh húa theo ngha yu ca h (0.1)- (0.2) D kin úng gúp mi Chng minh chi tit cỏc kt qu cụng trỡnh [19] Phng phỏp nghiờn cu Lun s dng mt s phng phỏp v cụng c ca gii tớch bao gm: Gii tớch a tr, gii tớch bc phõn s, o khụng compact; Lý thuyt na nhúm; Lý thuyt im bt ng cho ỏnh x nộn Lun c trỡnh by ba chng Chng nhc li mt s khỏi nim v kt qu liờn quan ti lý thuyt na nhúm v gii tớch bc phõn s Thờm vo ú, chỳng tụi s trỡnh by mt kt qu c bn cho tớnh n nh hoỏ ca h tuyn tớnh tng ng vi (0.1)-(0.2) Trong Chng 2, chỳng tụi trỡnh by kt qu v s tn ti nghim ton cc trờn (, T ], vi T > Kt qu tn ti nghim õy khụng yờu cu v tớnh compact ca na nhúm etA hoc tớnh Lipschitz ca F Nu etA khụng compact, ta s yờu cu tớnh chớnh qui ca F qua o khụng compact Hausdorff (xem [5, 21, 23, 32] cho trng hp s dng gi thit compact hoc Lipschitz) Chng dnh cho vic chng minh kt qu chớnh v tớnh n nh hoỏ c yu vi gi gi thit rng phn tuyn tớnh tng ng l n nh hoỏ c Phn cui chng ny chỳng tụi trỡnh by mt vớ d minh ho cỏc kt qu t c Chng Kin thc chun b 1.1 n nh hoỏ cho h iu khin cp phõn s tuyn tớnh Gi s A l toỏn t tuyn tớnh vi D(A) l phn t sinh ca mt C0 -na nhúm {etA }t0 trờn X Ký hiu (A) l ph ca A, s(A) l bỏn kớnh ph ca A, (A) l tng trng m ca A, l tng trng ct yu ca A, ess (A) ln etA t t ú s(A) = sup{Re() : (A)}; (A) = lim ln etA t t lim ess , õy ã L(X) v ã ess L(X) ; ess (A) = tng ng l ký hiu ca chun v chun ct yu ca toỏn t tuyn tớnh trờn X (xem sỏch chuyờn kho ca Engel v Nagel [9]) Ta s nhc li mt s kt qu liờn quan ti tớnh cht ca cỏc i lng trờn Chng minh chi tit cỏc kt qu ny cú th xem [9] B 1.1 Gi s A l phn t sinh ca C0 -na nhúm trờn khụng gian Banach X Khi ú (i) (A) = max{s(A), ess (A)}; (ii) Nu na nhúm {etA } liờn tc theo chun, ngha l, hm t etA liờn tc vi t > thỡ s(A) = (A); (iii) ess (A) = ess (A + K), ú K : X X l toỏn t compact; (iv) Vi mi > ess (A) (A) { C : Re() } l b chn Nhc li rng vi mi > ess (A) c nh ( c gi l im tỏch ph) thỡ (A) cú th c vit di dng (A) = u (A) s (A) (xem [9, nh lý V.3.7]), ú u (A) = (A) { C : Re() } s (A) = (A) { C : Re() < } Chỳng ta cú th thy rng u (A) l b chn v tỏch s (A) bi mt ng cong úng C Khi ú theo kt qu ca Kato [17, nh lý 6.17], ta thu c biu din tng ng ca khụng gian trng thỏi v toỏn t A nh sau X = Xu Xs , A = Au As , õy Xu = P X, Xs = (I P )X , P l toỏn t chiu P = 2i (I A)1 d C v Au = A Xu , As = A Xs Thc t, Xu v Xs l hai khụng gian bt bin di tỏc ng ca toỏn t A, ngha l, AXu Xu , AXs Xs Hn na, (Au ) = u (A), (As ) = s (A) v Au l toỏn t b chn trờn Xu Ký hiu L1 (0, T ; X) l khụng gian cỏc hm trờn [0, T ] ly giỏ tr khụng gian X v kh tớch theo ngha Bochner nh ngha 1.1 Tớch phõn cp phõn s > ca hm f L1 (0, T ; X) c xỏc nh bi I0 f (t) = () t (t s)1 f (s)ds, ú l hm Gamma nh ngha 1.2 o hm cp phõn s (0, 1) theo ngha Caputo ca f C ([0, T ]; X) cho bi D0 f (t) t = (1 ) (t s) f (s)ds Xột bi toỏn Cauchy D0 x(t) = Ax(t) + f (t), t > 0, (0, 1], (1.1) x(0) = x0 (1.2) nh ngha 1.3 Mt hm x C([0, T ]; X) c gi l nghim tớch phõn ca (1.1)-(1.2) trờn khong [0, T ] v ch t x(t) = S (t)x0 + (t s)1 P (t s)f (s)ds, t [0, T ], (1.3) A (1.4) ú ()et S (t) = ()et P (t) = d, A (1.5) d, () = (1)n1 n1 n=1 (n + 1) sin n, (0, ) n! (1.6) Cụng thc ca (1.3) v ca cỏc toỏn t S (ã), P (ã) cú th tỡm thy bi bỏo [33] Chỳng ta gi S (ã) v P (ã) l toỏn t gii thc cp phõn s sinh bi toỏn t A Trong (1.1)-(1.2), ta t f (t) = Bu(t), ú chỳng ta bn v h iu khin tuyn tớnh D0 x(t) = Ax(t) + Bu(t), t > 0, (1.7) x(0) = x0 (1.8) ngn gn, chỳng ta s dng ký hiu(A, B) ch h (1.7)-(1.8) v (A, B) ch h (A, B)1 Nhc li rng, h (A, B) n nh hoỏ m bi phn hi nu cú toỏn t tuyn tớnh D : U X cho toỏn t A = A + BD sinh na nhúm n nh m {S(t)}t0 , ngha l, S(t) L(X) M eat , vi M 1, a > Trong phn cũn li ca lun vn, ta núi toỏn t tuyn tớnh b chn D : U X l toỏn t phn hi nu A = A + BD sinh C0 -na nhúm nh ngha 1.4 Ta núi rng h (A, B) l n nh hoỏ c bi iu khin phn hi nu tn ti toỏn t phn hi D : U X cho nghim ca bi toỏn Mnh 3.1 ([1]) MNC xỏc nh bi (3.4) l chớnh quy Chỳng ta nghiờn cu tớnh hỳt yu ca h (0.1)-(0.2) nh sau u tiờn, ta xõy dng mt li úng b chn ca BC0 v bt bin di tỏc ng ca toỏn t nghim , cú dng BR () = {y BC0 : y BC Rv t y(t) , t 0}, ú < < min(, ), R v l cỏc s thc dng Khi ú ta s ch rng : BR () P(BR ()) l ỏnh x a tr vi giỏ tr li compact, v nộn theo MNC lm c iu ny, ta thay th cỏc gi thit (A), (B) v (F) bi gi thit mnh hn (A*) Gi thit (A) c tho v (A, B) n nh hoỏ c bi iu khin phn hi (B*) Khụng gian pha B tho (B1)-(B4) vi K BC(R+ ) v M tho t M (t) = O(1) t +, õy BC(R+ ) l khụng gian cỏc hm liờn tc b chn trờn R+ (F*) Hm phi tuyn a tr F tho (F) vi m, k L (R+ ) Do gi thit (A*), ta cú th gi s rng tn ti toỏn t phn hi D cho A = A + BD sinh C0 -na nhúm S(ã) tho S(t) L(X) M eat , t 0, õy M 1, a > Do (B*) chỳng ta cú K = sup K(t) < +, M = sup M (t) < +, t0 t0 M = sup t M (t) < + t0 Do (F*) chỳng ta cú m = ess suptR+ m(t) < +, k = ess suptR+ k(t) < + 23 B 3.2 Gi s (A*), (B*) v (F*) tho Khi ú (BCR ()) BCR () vi R, > 0, l t = sup t0 (t s)1 P (t s) L(X) m(s)M (s) ds < + (3.5) (t s)1 P (t s) L(X) m(s)K(s) ds cho hỡnh cu BR = {x BC0 : x R} BC bt bin di tỏc ng ca toỏn t , ngha l, (BR ) BR Gi s ngc li vi mi n N tn ti xn BC0 vi xn BC n v zn (xn ) cho zn BC > n Theo cụng thc ca toỏn t , chỳng ta tỡm c fn PFp (xn ) cho t (t s)1 P (t s)fn (s) ds, t > zn (t) = S (t)(0) + Do ú t zn (t) S (t) L(X) (t s)1 P (t s) (0) + L(X) m(s)|xn []s |B ds (3.9) Li cú |xn []s |B K(s) sup xn (s) + M (s)||B r[0,s] nK(s) + M (s)||B (3.10) Khi ú, kt hp (3.9) vi (3.10), ta cú ỏnh giỏ sau zn (t) S (t) L(X) t (0) (t s)1 P (t s) + L(X) m(s) (nK(s) + M (s)||B ) ds (3.11) t (t s)1 P (t s) M (0) + n L(X) m(s)K(s) ds + ||B (t), (3.12) 24 ú t (t s)1 P (t s) (t) = L(X) m(s)M (s) ds Do ú t (3.12) ta cú zn BC M (0) + ||B + n n 1< Ly gii hn n bt ng thc trờn ta thu c mõu thun vi (3.6) Bc Ta ch s tn ti ca > cho BR () = BR {x BC0 : t x(t) , t 0}, bt bin di tỏc ng ca toỏn t Gi s ngc li: tn ti yn BR (n) v zn (yn ) cho sup t |zn (t) > n Ly fn PFp (yn ) cho t0 t (t s)1 P (t s)fn (s)ds, zn (t) = S (t)(0) + ú t zn (t) S (t) L(X) (0) + t (t s)1 P (t s) + t L(X) (0) , L(X) m(s)|ys |B ds = I1 (t) + I2 (t) + I3 (t), õy I1 (t) = S (t) t (t s)1 P (t s) I2 (t) = L(X) m(s)|ys |B ds, t (t s)1 P (t s) I3 (t) = L(X) m(s)|ys |B ds t S dng B 1.2, ta cú t I1 (t) = t S (t) L(X) (0) M t (0) as t + T õy v t t I1 (t) t 0, ta cú sup t I1 (t) < + t0 25 (3.13) Chỳng ta li cú t tiờn (B3) vi = c lng |ys |B K(s) sup y(r) + M (s)||B r[0,s] RK + M ||B (3.14) Liờn quan ti I2 (t) ta cú t (t s)1 P (t s) t I2 (t) Ct L(X) m(s)ds t t Ct [(1 )t] P (t s) t (1) C(1 ) t L(X) m(s)ds , t > 0, ú C = RK + M ||B Khi ú t I2 (t) t + Mt khỏc CM m t t I2 (t) () t (t s)1 ds = t CM m (1 ) t+ ( + 1) t Do ú sup t I2 (t) < + (3.15) t0 c lng I3 (t) u tiờn, s dng tiờn (B3) vi = s v bt ng thc (3.14) ta cú |ys |B K(s s) sup y(r) + M (s s)|ys |B r[s,s] K sup y(r) + CM (s(1 )) r[s,s] Vỡ th s |ys |B K s (s) sup r y(r) + C(1 ) [(1 )s] M ((1 )s) r[s,s] K n + C(1 ) M Th c lng trờn vo t I3 (t), ta cú t (t s)1 s P (t s) t I3 (t) = t L(X) m(s)s |ys |B ds t t (t s)1 P (t s) t K L(X) m(s)[K n + C (1 ) M 26 n + C(1 ) M ]ds (3.16) Kt hp (3.13), (3.15) v (3.16), ta cú 1< 1 sup t |z(t) n t0 n sup t I1 (t) + sup t I2 (t) + sup t I3 (t) t0 t0 K + t0 D , n (3.17) ú D = sup t I1 (t) + sup t I2 (t) + C (1 ) M < + t0 t0 Qua gii hn bt ng thc (3.17) n , ta thu c mõu thun vi (3.8) Do ú ta cú iu phi chng minh Bõy gi, chỳng ta khng nh rng luụn tn ti s L nh ngha ca T (1.29) cho t (t s)1 P (t s) := sup t>0 L(ts) k(s) ds L(X) e < (3.18) Tht vy, ta cú t (t s) P (t s) L(ts) L(X) e M k k(s) ds () t (t s)1 eL(ts) ds t M k r1 eLr dr = () M k L + L Ta chng minh tớnh cht -nộn ca toỏn t b sau B 3.3 Gi s (A*) (B*) v (F*) tho Nu cỏc iu kin (3.5)-(3.8) cng tho thỡ l -nộn trờn BR () Chng minh Chỳng ta ch rng vi bt kỡ BR (), thỡ (()) ã () u tiờn chỳng ta chỳng minh rng d (()) = t z (), thỡ z BR () Theo cụng thc ca BR () ta cú t z(t) , t Vi T > 0, ta cú z(t) T , t T 27 Vỡ z () tu ý nờn dT (()) = sup sup z(t) T z() tT Cho T +, ta cú d (()) = Trong phn sau, ta ch rng (()) ã () Gi s T > l s dng c nh tu ý Vỡ T () l b chn C([0, T ]; X) nờn PFp () l b chn kh tớch Lp (0, T ; X) T iu ny v Mnh 1.5, ta cú T (()) = S (ã)(0) + Q PFp (T ()) l liờn tc ng bc C([0, T ]; X) Do ú (3.19) modT (T (())) = S dng Mnh 1.7, chỳng ta cú vi mi t [0, T ] thỡ t (()(t)) (t s)1 P (t s)PFp ()(s) ds t (t s)1 P (t s)PFp ()(s) ds (3.20) Nu na nhúm S(ã) l compact thỡ (()(t)) = 0, vỡ (P (t s)PFp ()(s)) = vi s (0, t) Nu trỏi li, s dng (F)(4) vi chỳ ý l vi mi r 0, [](r) = {(r)} l mt im, chỳng ta cú ỏnh giỏ (P (t s)PFp ()(s)) P (t s) p L(X) (PF ()(s)) P (t s) L(X) k(s) sup ([](s + )) P (t s) L(X) k(s) sup ((r)) r[0,s] Th (3.21) vo (3.20), chỳng ta cú t (t s)1 P (t s) (()(t)) L(X) k(s) sup ((r)) ds r[0,s] 28 (3.21) T bt ng thc ny ta cú t eLt (()(t)) (t s)1 P (t s) L(X) e L(ts) k(s) sup eLr ((r)) ds r[0,s] t (t s)1 P (t s) L(ts) k(s) ds L(X) e T (T ()) T ú, ta cú t T (T (())) (t s)1 P (t s) sup t>0 L(X) e L(ts) k(s) ds T (T ()) (3.22) Kt hp cỏc bt ng thc (3.19) v (3.22) ta i n (()) ã (), vi t (t s)1 P (t s) =4 L(ts) k(s) ds L(X) e < Do ú (()) = (()) + d (()) = (()) ã () ã () T õy ta cú iu phi chng minh Kt qu chớnh ca phn ny c phỏt biu nh sau nh lớ 3.4 Di cỏc gi thit ca B 3.3, bi toỏn (2.1)-(2.2) cú ớt nht mt nghim tớch phõn (, +) tho x(t) = O(t ) t Chng minh Theo B 3.2, chỳng ta xột toỏn t nghim : BR () Kv(BR ()) vi R, > Lp lun tng t nh chng minh ca B 2.2, chỳng ta cú l toỏn t úng Hn na, theo B 3.3, l toỏn t nộn theo MNC Vỡ vy kt lun ca nh lý 3.4 c suy t nh lý 1.10 Bõy gi ta phỏt biu tớnh n nh hoỏ yu ca h (0.1)-(0.2) Nú ch n nh tim cn yu ca h (2.1)-(2.2) di cỏc gi thit (A*), (B*) v (F*) 29 Theo nh ngha ca nghim v (F)(3), ta cú t x(t) S (t) L(X) (0) + (t s)1 P (t s) L(X) m(s)|xs |B ds (t s)1 P (t s) L(X) m(s)K(s) t S (t) L(X) (0) + x BC ds t (t s)1 P (t s) + L(X) m(s)M (s)||B ds M ||B + x BC + ||B T bt ng thc ny ta cú (1 ) x BC (M + )||B S dng tiờn (B3) vi = ta c |xt |B K sup x(s) + M ||B s[0,t] K x BC + M ||B K (M + ) + M ||B , t (3.23) Do ú ta ó kim tra c phỏt biu th nht nh ngha 0.1 kim tra phỏt biu th hai, s dng tiờn (B3) vi = 2t chỳng ta cú |xt |B K sup x(s) + M t st K t t t |x t |B 2 sup s x(s) + t st t t M t |x t |B |x t |B K sup s x(s) + M t st (3.24) n õy, nu x l nghim ca h (0.1)-(0.2), theo nh lý 3.4, chỳng ta cú sup s x(s) sup s x(s) < + t st s0 Do ú t cỏc bt ng thc (3.23)-(3.24) ta cú |xt |B t + 30 3.2 p dng Gi s l mt b chn RN vi biờn trn Xột h iu khin xỏc nh nh sau t2 Z(t, x) = x Z(t, x) + Z(t, x) + b(x)u(t) + c(t, x)v(t), t > 0, (3.25) v(t) (, y) f1 (Z(t + , y)), f2 (Z(t + , y)) dyd, (3.26) Z(t, x) = 0, x , t > 0, (3.27) Z(t, x) = (t, x), x , t 0, (3.28) ú t2 l o hm Caputo bc 12 , x l toỏn t Laplace theo bin x, l mt s dng v u l thnh phn iu khin, b L2 () Trong (3.26), ta hiu on [r1 , r2 ] = { r1 + (1 )r2 : [0, 1]}, vi r1 , r2 R Xột toỏn t A = + I vi xỏc nh D(A) = H () H01 () Khi ú A sinh mt C0 -na nhúm compact S(t) = et et trờn khụng gian X = L2 () Hn na, nu > , giỏ tr riờng nh nht ca toỏn t A0 = thỡ S(ã) khụng cú tớnh cht n nh m Gi s {i } i=1 l cỏc giỏ tr riờng ca A0 v {ei }i=1 l cỏc vộc-t riờng tng ng Ta gi s > , = i , i t m = max{i : i > 0} Xột h iu khin tuyn tớnh dZ (t, x) = x Z(t, x) + Z(t, x) + b(x)u(t), dt Z(t, x) = 0, x , t > 0, Z(0, x) = Z0 (x), x Ta bit rng (xem [30]) h ny n nh húa c nu b(x)ei (x)dx = 0, i = 1, 2, , m (3.29) Bõy gi ta t cỏc gi thit sau : (, 0] ì R l hm liờn tc tha |(, x)| C eh , (, x) (, 0] ì , (3.30) vi C > v h (0, 1) f1 , f2 : R R l cỏc hm liờn tc cho |fi (z)| Cf |z|, z R, vi Cf > 0, i = 1, 31 (3.31) c L (R+ ; L2 ()) t Bu(t)(x) = b(x)u(t), F (t, w)(x) = c(t, x) (, y) f1 (w(, y)), f2 (w(, y)) dyd, vi w B = CL2g , g() = eh v na chun B xỏc nh bi r |w|B = sup w() X eh w() + r0 X d , ú r = h1 ln(1 h) Khi ú ta thy h (3.25)-(3.28) cú dng (0.1)-(0.2) Rừ rng cp (A, B) l n nh húa c nu iu kin (3.29) c tha Vy ta cú (A*) Vi hm phi tuyn a tr F ta cú ỏnh giỏ F (t, w) X C2 Cf2 c(t, ã) X C2 Cf2 c(t, ã) h e h |w(, y)|dyd X h |w(, y)|dy e C2 Cf2 à() c(t, ã) h X = C2 Cf2 à() c(t, ã) h X C2 Cf2 à() c(t, ã) h C2 Cf2 à() c(t, ã) h d eh w(, ã) X d r eh w(, ã) + r w C([r,0];X) X d r X eh w(, ã) + X d 2 X |w|B , nh vo bt ng thc Hăolder Do ú F (t, w) X C Cf h v F tha (F*) vi m(t) = C Cf h compact ca 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