B GIO DC V O TO TRNG I HC S PHM H NI * V MNH TI TNH IU KHIN C CA MT S LP PHNG TRèNH PARABOLIC LUN N TIN S TON HC H Ni - 2016 B GIO DC V O TO TRNG I HC S PHM H NI * V MNH TI TNH IU KHIN C CA MT S LP PHNG TRèNH PARABOLIC Chuyờn ngnh: Phng trỡnh vi phõn v tớch phõn Mó s: 62 46 01 03 LUN N TIN S TON HC NGI HNG DN KHOA HC PGS.TS Cung Th Anh H Ni - 2016 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi di s hng dn ca PGS.TS Cung Th Anh Cỏc kt qu c phỏt biu lun ỏn l hon ton trung thc v cha tng c cụng b bt c mt cụng trỡnh no khỏc Nghiờn cu sinh V Mnh Ti LI CM N Lun ỏn c hon thnh di s hng dn nghiờm khc, tn tỡnh, chu ỏo ca PGS.TS Cung Th Anh Tỏc gi xin by t lũng kớnh trng v bit n sõu sc PGS.TS Cung Th Anh, ngi Thy ó dn dt tỏc gi lm quen vi nghiờn cu khoa hc t nhng ngy sau tt nghip i hc Ngoi nhng ch dn v mt khoa hc, s ng viờn v lũng tin tng ca thy dnh cho tỏc gi luụn l ng lc ln giỳp tỏc gi say mờ nghiờn cu Tỏc gi xin trõn trng gi li cm n n Ban Giỏm hiu, Phũng sau i hc, Ban Ch nhim Khoa Toỏn-Tin, Trng i hc S phm H Ni, c bit l cỏc thy cụ giỏo B mụn Gii tớch, Khoa Toỏn-Tin, Trng i hc S phm H Ni ó luụn giỳp , ng vin, to mụi trng hc nghiờn cu thun li cho tỏc gi Tỏc gi xin c by t lũng bit n n Ban Giỏm hiu trng i hc Thy li, cỏc thy cụ v cỏc anh ch ng nghip cụng tỏc ti B mụn Toỏn, Khoa Cụng ngh Thụng tin, Trng i hc Thy li ó luụn to iu kin thun li, giỳp v ng viờn tỏc gi sut quỏ trỡnh hc v nghiờn cu Li cm n sau cựng, tỏc gi xin dnh cho gia ỡnh, nhng ngi luụn yờu thng, chia s, ng viờn tỏc gi vt qua khú khn hon thnh lun ỏn Mc lc Li cam oan Li cm n Mc lc Mt s kớ hiu dựng lun ỏn M U L DO CHN TI TNG QUAN VN NGHIấN CU MC CH, I TNG V PHM VI NGHIấN CU 12 PHNG PHP NGHIấN CU 13 KT QU CA LUN N 13 CU TRC CA LUN N 15 Chng MT S KIN THC CHUN B 16 1.1 MT S KHễNG GIAN HM 16 1.1.1 Mt s khụng gian hm 16 1.1.2 Khụng gian hm ph thuc thi gian 17 1.2 L THUYT IU KHIN C CA H TUYN TNH TRONG KHễNG GIAN Vễ HN CHIU 18 1.2.1 Mt s nh ngha 18 1.2.2 Phng phỏp nht Hilbert (HUM) 20 1.3 MT S BT NG THC THNG DNG 21 1.3.1 Mt s bt ng thc kiu Hardy 21 1.3.2 Mt s bt ng thc s cp 23 1.4 MT S KT QU THNG DNG 24 Chng TNH IU KHIN C V CA PHNG TRèNH PARABOLIC CHA TON T GRUSHIN 26 2.1 T BI TON V PHT BIU KT QU CHNH 26 2.2 MT S KT QU B TR 28 2.2.1 Tớnh t ỳng ca bi toỏn 28 2.2.2 Khai trin Fourier 29 2.2.3 Tc tỏn x 30 2.2.4 Bt ng thc Carleman 32 2.3 CHNG MINH KT QU CHNH 44 2.3.1 Lc chng minh nh lớ 2.1 44 2.3.2 Bt ng thc quan sỏt c 45 2.3.3 Chng minh tớnh khụng iu khin c nh lớ 2.1 49 Chng TNH IU KHIN C V KHI THI GIAN LN CA PHNG TRèNH PARABOLIC CHA TON T GRUSHIN VI TH V Kè D 55 3.1 T BI TON V PHT BIU KT QU CHNH 55 3.2 CHNG MINH KT QU CHNH 58 3.2.1 Khai trin Fourier v tc tỏn x 58 3.2.2 Tớnh quan sỏt c u ca bi toỏn liờn hp 62 3.3 CHNG MINH BT NG THC CARLEMAN 67 3.3.1 Mt s tớnh cht ca hm trng 67 3.3.2 Chng minh nh lớ 3.3 70 Chng TNH IU KHIN C V CA MT LP PHNG TRèNH PARABOLIC MT CHIU NA TUYN TNH SUY BIN VI TH V Kè D 86 4.1 T BI TON 86 4.2 TNH T NG CA BI TON 88 4.2.1 Khụng gian hm v toỏn t 88 4.2.2 Tớnh t ỳng ca bi toỏn 89 4.3 TNH IU KHIN C V 95 4.3.1 Tớnh iu khin c v ca bi toỏn tuyn tớnh húa 95 4.3.2 Tớnh iu khin c v ca bi toỏn na tuyn tớnh 103 KT LUN 110 KT QU T C 110 KIN NGH MT S VN NGHIấN CU TIP THEO 110 DANH MC CC CễNG TRèNH CễNG B CA LUN N 112 TI LIU THAM KHO 113 MT S K HIU THNG DNG TRONG LUN N C0 () khụng gian cỏc hm kh vi vụ hn cú giỏ compact ã chun L ( ì (0, T )) S01 () khụng gian Sobolev cú trng dựng nghiờn cu cỏc bi toỏn cha toỏn t Grushin (xem trang 28) Sà,0 () khụng gian Sobolev cú trng dựng nghiờn cu cỏc bi toỏn cha toỏn t Grushin vi th v kỡ d (xem trang 57) H,0 (0, 1) khụng gian Sobolev cú trng dựng nghiờn cu bi toỏn cha toỏn t suy bin mt chiu (xem trang 88) Gs toỏn t Grushin (xem trang 9) vect gradient toỏn t Laplace D2 ma trn Hessian div ã toỏn t divergence I toỏn t ng nht IN1 ma trn n v cp N1 0RN1 phn t RN1 hm c trng ca hi t yu ab tớch tensor gia hai vect a v b M U L DO CHN TI Trong khong hai thp k gn õy, tớnh iu khin c (bao gm tớnh iu khin c chớnh xỏc, tớnh iu khin c v 0, tớnh iu khin c xp x) ó c nghiờn cu i vi nhiu lp phng trỡnh o hm riờng tuyn tớnh v na tuyn tớnh Bi phng phỏp nht Hilbert HUM (Hilbert Uniqueness Method) xut bi J.-L Lions (xem [48, 49, 50]), tớnh iu khin c ca bi toỏn tuyn tớnh c qui v tớnh quan sỏt c ca bi toỏn liờn hp tng ng thit lp tớnh quan sỏt c ca bi toỏn liờn hp tng ng thụng qua cỏc bt ng thc quan sỏt, mt nhng cụng c hiu lc nht l cỏc c lng kiu Carleman ton cc Cũn tớnh iu khin c ca bi toỏn na tuyn tớnh c chng minh bng cỏch s dng tớnh iu khin c ca bi toỏn tuyn tớnh húa tng ng v phng phỏp im bt ng xut ln u tiờn bi Zuazua [68, 69] cho phng trỡnh truyn súng na tuyn tớnh Mt nhng lp phng trỡnh o hm riờng c nghiờn cu nhiu l lp phng trỡnh tin húa kiu parabolic, cha ng phng trỡnh truyn nhit c in, nhiu lp phng trỡnh parabolic xut hin húa hc, sinh hc v c hc cht lng Nghiờn cu tớnh iu khin c ca cỏc phng trỡnh parabolic ó thu hỳt s quan tõm ca nhiu nh toỏn hc khong hai thp niờn gn õy Sau nhng nghiờn cu tiờn phong ca Fursikov v Imanuvinov [37, 43], Lebeau v Robbiano [46] bng cụng c c lng Carleman, ó cú nhiu tin b vic tỡm hiu v cỏc tớnh cht iu khin c ca cỏc phng trỡnh parabolic khụng suy bin vi cỏc h s bin thiờn Cỏc kt qu ny cng c m rng cho cỏc bi toỏn parabolic na tuyn tớnh [29, 31, 32, 33, 34, 70, 71] Cỏc kt qu t c u da trờn cụng c chớnh l bt ng thc Carleman cho nghim ca bi toỏn liờn hp tng ng Cỏc bt ng thc Carleman c thit lp ny yờu cu phn chớnh ca phng trỡnh l toỏn t elliptic u, b chn v khụng cú th v kỡ d Bờn cnh ú, tớnh iu khin c ca cỏc phng trỡnh parabolic u khụng b chn cng ó c nghiờn cu [18, 38, 55] Cú th núi ngy lớ thuyt iu khin c i vi cỏc phng trỡnh parabolic u ó khỏ hon thin c trng hp tuyn tớnh v na tuyn tớnh Trong khong mt thp k tr li õy, tớnh iu khin c ca phng trỡnh parabolic suy bin, khụng cú hoc cú th v kỡ d, ó c nghiờn cu bi nhiu nh toỏn hc Nhng nghiờn cu ny c thỳc y bi nhiu bi toỏn vt lớ khỏc nh mụ hỡnh tng lp biờn [17], cỏc mụ hỡnh di truyn qun th cỏ, cỏc mụ hỡnh khớ hu Bydyko-Sellers, Tuy nhiờn, hu ht cỏc kt qu t c hin ti ch yu trng hp mt chiu (xem [2, 19, 20, 23, 24, 35, 36, 52, 53, 62] v cỏc ti liu trớch dn ú), mi ch cú rt ớt kt qu iu khin c trng hp nhiu chiu, ch yu l trng hp hai chiu i vi phng trỡnh parabolic cha toỏn t div(A(x)u) vi A(x) l ma trn vuụng cp hai i xng [25], phng trỡnh parabolic cha toỏn t Grushin [12], phng trỡnh Kolmogorov [11, 45], v mt lp phng trỡnh suy bin nhiu chiu vi s hng i lu [65, 66, 67] Ngoi ra, cỏc kt qu v tớnh iu khin c ca cỏc phng trỡnh suy bin/kỡ d na tuyn tớnh cũn rt ớt õy ang l nhng thi s thu hỳt c s quan tõm nghiờn cu ca nhiu nh toỏn hc v ngoi nc Chỳng tụi s chn nhng ny lm ti nghiờn cu lun ỏn tin s ca mỡnh 106 uv 2L2 (0,T ;H + uv 2C([0,T ];L2 (0,1)) ( ) 2 exp(C(, , , )(1 + T )(1 + L)) u0 L2 (0,1) + hL2 (ì(0,T )) (4.56) ,0 (0,1)) Thay (4.55) vo (4.56) ta cú hng s dng C(, , , ) khụng ph thuc T, L v u0 : uv 2L2 (0,T ;H (0,1)) + uv 2C([0,T ];L2 (0,1)) ,0 ) ( 2k1 + + L + T L + L ) u0 2L2 (0,1) exp C(, , , )(1 + T + T T Vy (v)2X R2 vi mi v BX , ( ( )) 2 2k1 u0 2L2 (0,1) vi R = exp C(, , , ) + T + T + + L + TL + L T Vy ta cú c (i) Ta chỳ ý rng (v) H (0, T ; L2 (0, 1)) L2 (0, T ; D(A)) (xem nh lớ 4.1) Do ú ta cú (ii) vỡ tớnh compact ca phộp nhỳng H (0, T ; L2 (0, 1)) L2 (0, T ; D(A)) C([0, T ]; L2 (0, 1)) L2 (0, T ; H,0 (0, 1)) Phộp nhỳng compact ny cng c s dng cho chng minh (iii) Tht vy, vi vk X cho vk v X, k , ta chng minh rng uvk uv X, k õy uvk v uv l cỏc nghim ca (4.53) liờn kt vi vk , hvk v v, hv tng ng Ta cú (vk ) = uvk l nghim ca (4.53) tng ng vi iu khin hvk m cho uvk (T ) = 0, tc l uvt k (x uvxk )x uvk + cvk (x, t)uvk = hvk , (x, t) QT , x uvk (0, t) = uvk (1, t) = t (0, T ), uvk (x, 0) = u0 , uvk (x, T ) = x (0, 1) (4.57) 107 Khi ú, t (4.8) v (4.55) ta cú (ly dóy nu cn thit): uvk uv H (0, T ; L2 (0, 1)) L2 (0, T ; D(A)) C([0, T ]; H,0 (0, 1)), hvk h L2 ( ì (0, T )) (4.58) Do phộp nhỳng sau l compact H (0, T ; L2 (0, 1)) L2 (0, T ; D(A)) C([0, T ]; L2 (0, 1)) L2 (0, T ; H,0 (0, 1)), suy uvk uv C([0, T ]; L2 (0, 1)) (4.59) M v k v C([0, T ]; L2 (0, 1)), nờn cựng vi tớnh liờn tc ca cvk v (4.59) ta cú cvk (x, t)uvk (x, t) cv (x, t)uv (x, t), hu khp (x, t) QT (4.60) Do {cvk uvk } b chn L2 (QT ), nờn cvk uvk L2 (QT ) (4.61) T (4.60) v (4.61) ta suy (x, t) = cv (x, t)uv (x, t), hu khp (x, t) QT Vy cvk uvk cv uv L2 (QT ) (4.62) T (4.58), (4.62) ta cú th ly gii hn (4.57) kt lun (uv , hv ) tha (4.53) v uv (ã, T ) = Vy uv = (v) Do ú l liờn tc Vy cỏc gi thit ca nh lớ Schauder tha i vi nh lớ c chng minh Bõy gi ta chng minh kt qu chớnh ca chng 108 nh lớ 4.5 Gi s T > v u0 L2 (0, 1) cho trc Vi cỏc gi thit (4.3) v (4.2), bi toỏn (4.1) iu khin c v 0, tc l tn ti h L2 ( ì (0, T )) cho bi toỏn (4.1) cú nghim u tha u(ã, T ) = Hn na, hm iu khin tha T h dxdt C T u20 dx, (4.63) vi C T cú dng nh nh lớ 4.4 Chng minh Bc u tiờn, ta xột bi toỏn vt (x vx )x v + f (x, t, v) = 0, (x, t) QT /2 , x v(0, t) = v(1, t) = 0, t (0, T /2), v(x, 0) = u0 , x (0, 1) (4.64) Bi nh lớ 4.1, bi toỏn (4.64) cú nghim v L2 (0, T /2; H,0 (0, 1)), ú, (0, 1) tn ti thi im t0 (0, T /2) cho v(t0 , ã) =: u1 H,0 Bc tip theo, ta xột bi toỏn wt (x wx )x w + f (x, t, w) = h1 , (x, t) (0, 1) ì (t0 , T ), x (4.65) w(0, t) = w(1, t) = 0, t (t0 , T ), w(x, t0 ) = u1 , x (0, 1) Bi nh lớ 4.4, bi toỏn (4.65) iu khin c v 0, tc l, tn ti iu khin h1 L2 ( ì (t0 , T )) cho w(ã, T ) = 0, v T h21 dxdt t0 C T t0 u21 dx, vi hng s dng C T t0 cú dng C T nhng thay T bi T t0 Bõy gi ta xỏc nh u v h bi v(t) vi mi t [0, t0 ], u := v h := w(t) vi mi t [t0 , T ], h1 vi mi t [0, t0 ], vi mi t [t0 , T ] 109 Khi ú u l nghim ca (4.1) v tha u(x, T ) = vi mi x (0, 1), h tha (4.63) Chỳ ý cui chng Kt qu chng ny l m rng kt qu v tớnh iu khin c v ca lp phng trỡnh parabolic mt chiu suy bin vi th v kỡ d ca Vancostenoble [62] t trng hp tuyn tớnh sang trng hp na tuyn tớnh KT LUN CHNG Trong chng ny, chỳng tụi ó nghiờn cu mt lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d trng hp di ti hn Kt qu chớnh t c l chng minh tớnh iu khin c v bng cỏch s dng c lng Carleman [62] chng minh c tớnh iu khin c v ca bi toỏn tuyn tớnh húa v sau ú dựng nh lớ im bt ng Schauder nhn c tớnh iu khin c v ca bi toỏn na tuyn tớnh 110 KT LUN KT QU T C Trong lun ỏn ny, chỳng tụi ó nghiờn cu tớnh iu khin c ca lp phng trỡnh parabolic cha toỏn t Grushin khụng cú/cú th v kỡ d trng hp nhiu chiu v lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d Cỏc kt qu chớnh t c l: i vi bi toỏn iu khin ca phng trỡnh parabolic cha toỏn t Grushin trng hp hỡnh hp nhiu chiu: Chng minh c tớnh iu khin c v ti mi thi im T > s (0, 1) (suy bin yu) Khi s = (suy bin mnh) ta chng minh c tớnh iu khin c v thi gian iu khin ln v tớnh khụng iu khin c v thi gian iu khin quỏ nh Chng minh c tớnh khụng iu khin c v s > (suy bin quỏ mnh) Chng minh c tớnh iu khin c v thi gian iu khin ln ca phng trỡnh parabolic cha toỏn t Grushin s = vi th v kỡ d à/|x|2 trng hp nhiu chiu Chng minh c tớnh iu khin c v ca mt lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d KIN NGH MT S VN NGHIấN CU TIP THEO Bờn cnh cỏc kt qu ó t c lun ỏn, mt s m cn tip tc nghiờn cu nh: 111 Nghiờn cu tớnh iu khin c v ca phng trỡnh parabolic cha toỏn t Grushin vi th v kỡ d s (0, 1) Nghiờn cu tớnh iu khin c v ca lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d trng hp ti hn Nghiờn cu tớnh iu khin c ca phng trỡnh parabolic suy bin/kỡ d iu khin nm trờn biờn (bi toỏn iu khin biờn) õy l rt khú, c trng hp mt chiu 112 DANH MC CC CễNG TRèNH CễNG B CA LUN N C T Anh and V M Toi (2013), Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains, Nonlinear Analysis: Theory, Methods and Applications, Vol 93, 181-196 (ISI) C T Anh and V M Toi (2015), Null controllability for semilinear degenerate/singular parabolic equations, Fixed Point Theory, Vol 16, 15-30 (ISI) C T Anh and V M Toi (2016), Null controllability in large time of a parabolic equation involving the Grushin operator with an inverse-square potential, Nonlinear Differential Equations and Applications, Vol 23, 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NGHIấN CU Mc ớch lun ỏn: Nghiờn cu tớnh iu khin c v 0 ca phng trỡnh parabolic cha toỏn t Grushin trong trng hp nhiu chiu, phng trỡnh parabolic cha toỏn t Grushin cú th v kỡ d trong trng hp nhiu chiu, phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d i tng nghiờn cu: Bi toỏn iu khin i vi lp phng trỡnh parabolic cha toỏn t Grushin... trỡnh tin húa kiu parabolic suy bin hoc cú th v kỡ d vn cũn nhiu vn m Núi riờng, nhng vn m m chỳng tụi quan tõm nghiờn cu trong lun ỏn ny bao gm: Tớnh iu khin c ca phng trỡnh parabolic suy bin cha toỏn t Grushin trong trng hp nhiu chiu Tớnh iu khin c ca phng trỡnh parabolic suy bin cha toỏn t Grushin vi th v kỡ d kiu Hardy à/|x|2 trong trng hp nhiu chiu Tớnh iu khin c ca phng trỡnh parabolic mt chiu... trng hp nhiu chiu v lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d Phm vi nghiờn cu: Ni dung 1: Bi toỏn iu khin c i vi phng trỡnh parabolic cha toỏn t Grushin trong min nhiu chiu 13 Ni dung 2: Bi toỏn iu khin c i vi phng trỡnh parabolic cha toỏn t Grushin vi th v kỡ d kiu Hardy trong min nhiu chiu Ni dung 3: Bi toỏn iu khin c i vi lp phng trỡnh parabolic mt chiu na tuyn tớnh suy... Mt lp phng trỡnh parabolic rt c quan tõm khỏc l lp phng trỡnh parabolic cha toỏn t Laplace vi th v kỡ d: Aà = à/|x|2 Cỏc kt qu v tớnh t ỳng ca bi toỏn cng nh dỏng iu tim cn nghim ca phng trỡnh parabolic cha t Aà ó c nghiờn cu bi nhiu nh toỏn hc (xem [8, 9, 16, 64] v cỏc ti liu trớch dn trong ú) Trong khi ú, tớnh iu khin c ca phng trỡnh parabolic cha toỏn t ny ó nhn c trong cỏc cụng trỡnh ca Vancostenoble-Zuazua... h parabolic na tuyn tớnh cha toỏn t ny ó c nghiờn cu gn õy trong c trng hp ụtụnụm v khụng ụtụnụm (xem, chng hn, [4, 5, 7]) Tớnh iu khin c ca phng trỡnh parabolic cha toỏn t Grushin c nghiờn cu u tiờn trong trng hp hai chiu bi Beauchard, Cannarsa v Guglielmi [12] Xem thờm kt qu gn õy trong [14] Tuy nhiờn, tớnh iu khin c ca lp phng trỡnh ny trong trng hp nhiu chiu vn cũn nhiu vn m Mt lp phng trỡnh parabolic. .. iu khin c v 0 khi thi gian iu khin ln ca phng trỡnh parabolic cha toỏn t Grushin khi s = 1 vi th v kỡ d à/|x|2 trong trng hp nhiu chiu Chng minh c tớnh iu khin c v 0 ca mt lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin cú th v kỡ d Cỏc kt qu ca lun ỏn l mi, cú ý ngha khoa hc, v gúp phn vo vic hon thin lớ thuyt iu khin c i vi lp phng trỡnh parabolic suy bin khụng cú/cú th v kỡ d Cỏc kt qu chớnh... Chng 2 trỡnh by cỏc kt qu tớnh iu khin c v 0 ca phng trỡnh parabolic cha toỏn t Grushin trong trng hp hỡnh hp nhiu chiu Chng 3 trỡnh by tớnh iu khin c v 0 khi thi gian ln ca phng trỡnh parabolic cha toỏn t Grushin khi s = 1 vi th v kỡ d kiu Hardy bờn trong min trong trng hp nhiu chiu Chng 4 trỡnh by tớnh iu khin c v 0 ca mt lp phng trỡnh parabolic mt chiu na tuyn tớnh suy bin vi th v kỡ d 16 Chng... trng hp kỡ d trờn biờn Gn õy, trong trng hp hai chiu, tớnh iu khin c xp x cho phng trỡnh parabolic cha toỏn t Grushin vi th v kỡ d à/|x|2 ó c nghiờn cu bi Morancey [56] nh tớnh cht thỏc trin duy nht ca toỏn t tng ng Hn na, trong [21], cỏc tỏc gi ó chng minh tớnh iu khin c v 0 khi thi gian ln cho phng trỡnh parabolic cha toỏn t Grushin vi th v kỡ d à/|x|2 khi s = 1 v min khụng gian l (0, 1) ì (0,... phng trỡnh parabolic tuyn tớnh thỡ tớnh iu khin c chớnh xỏc thng khụng t c do hiu ng trn ca nghim so vi d kin ban u Hn na tớnh iu khin c v 0 kộo theo tớnh iu khin c xp x ca h Do vy trong lun ỏn ny chỳng tụi ch tp trung vo vic nghiờn cu tớnh iu khin c v 0 ca nhng lp phng trỡnh trờn Ngoi ra, chỳng tụi cng ch xột bi toỏn khi iu khin cú giỏ bờn trong min Bi toỏn iu khin biờn i vi lp phng trỡnh parabolic. .. o thỡ iu khin c v 0 Nhn xột 1.2 Nu (1.1) l parabolic u thỡ Tớnh iu khin c chớnh xỏc ca h (1.1) khụng t c vỡ hiu ng trn ca nghim (nghim trn hn iu kin ban u) Tớnh iu khin c chớnh xỏc n qu o ca (1.1) tng ng vi tớnh iu khin c v 0 ca h (1.1) Tớnh iu khin c v 0 ca h (1.1) suy ra tớnh iu khin c xp x ca (1.1) 20 Do ú trong lớ thuyt iu khin c i vi cỏc phng trỡnh parabolic tuyn tớnh, ngi ta c bit quan tõm