Bài toán biên đối với một số lớp phương trình truyền sóng trong miền không trơn

B GIO DC V O TO TRNG I HC S PHM H NI * NGUYN THANH TNG BI TON BIấN I VI MT S LP PHNG TRèNH TRUYN SểNG TRONG MIN KHễNG TRN LUN N TIN S TON HC H NI - 2017 B GIO DC V O TO TRNG I HC S PHM H NI * NGUYN THANH TNG BI TON BIấN I VI MT S LP PHNG TRèNH TRUYN SểNG TRONG MIN KHễNG TRN Chuyờn ngnh: Phng trỡnh vi phõn v tớch phõn Mó s: 62 46 01 03 LUN N TIN S TON HC TP TH HNG DN KHOA HC TS V Trng Lng GS.TSKH Nguyn Mnh Hựng H NI - 2017 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi di s hng dn ca TS V Trng Lng v GS TSKH Nguyn Mnh Hựng 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 Nguyn Thanh Tựng LI CM N Lun ỏn c hon thnh di s hng dn tn tỡnh, chu ỏo ca TS V Trng Lng v GS.TSKH Nguyn Mnh Hựng Ngoi nhng ch dn v mt khoa hc, cỏc thy cũn to ng lc ln giỳp tỏc gi t tin, say mờ v quyt tõm nghiờn cu Tỏc gi xin by t lũng kớnh trng v bit n sõu sc TS V Trng Lng v GS.TSKH Nguyn Mnh Hựng Tỏc gi cng xin c t lũng bit n ln lao ti cỏc thy cụ b mụn Gii tớch, c bit l PGS TS Trn ỡnh K v PGS TS Cung Th Anh ó tn tỡnh ch bo cho tỏc gi quỏ trỡnh nghiờn cu v hon thnh lun ỏn Tỏc gi cm n cỏc bn nghiờn cu sinh ó úng gúp nhiu ý kin quý bỏu cho lun ỏn ca tỏc gi Tỏc gi xin c by t cm n ti Ban Giỏm hiu, phũng Sau i hc, khoa Toỏn - Tin Trng i hc S phm H Ni ó to iu kin thun li tỏc gi hon thnh quỏ trỡnh hc tp, nghiờn cu ca mỡnh Tỏc gi xin c by t cm n n Ban Giỏm hiu Trng i hc Tõy Bc, cỏc thy cụ v cỏc anh ch ng nghip cụng tỏc ti khoa Toỏn-Lý-Tin, Trng TH, THCS & THPT Chu Vn An ó luụn to iu kin thun li, giỳp v ng viờn tỏc gi sut quỏ trỡnh hc v nghiờn cu Sau cựng, tỏc gi by t lũng bit n ti gia ỡnh, ngi thõn v gia ỡnh TS V Trng Lng - nhng ngi luụn yờu thng, chia s, ựm bc, ng viờn tỏc gi vt qua khú khn hon thnh quỏ trỡnh hc v nghiờn cu ca khúa hc NCS Tỏc gi Mc lc Li cam oan Li cm n Mc lc Mt s kớ hiu dựng lun ỏn M U Lch s v lớ chn ti Mc ớch, i tng v phm vi nghiờn cu 11 Phng phỏp nghiờn cu 12 Kt qu ca lun ỏn 12 Cu trỳc ca lun ỏn 13 Chng MT S KIN THC CHUN B 14 1.1 Khụng gian cỏc hm, hi t yu, cỏc nh lý nhỳng 14 1.1.1 Mt s khụng gian hm 14 1.1.2 Hi t yu 17 1.1.3 nh lý nhỳng Sobolev v nh lý nhỳng RellichKondrachov 18 1.2 Mt s bt ng thc 19 1.2.1 Bt ng thc Cauchy v bt ng thc Young 19 1.2.2 Bt ng thc Hăolder 19 1.2.3 Mt s phỏt biu ca bt ng thc Gronwall 20 1.2.4 Bt ng thc Gagliardo-Nirenberg 21 1.3 Mt s kin thc cn bn v lớ thuyt toỏn t 22 1.4 Mt s b nhỳng cú cnh, bi toỏn Dirichlet i vi phng trỡnh elliptic cp hai a din 25 1.4.1 Mt s b nhỳng cú cnh 25 1.4.2 Bi toỏn Dirichlet i vi phng trỡnh elliptic cp hai a din 27 1.5 Mt s b nhỳng v bi toỏn Dirichlet i vi phng trỡnh elliptic mnh nún cú cnh 27 1.5.1 Min nún cú cnh 27 1.5.2 Mt s b nhỳng 28 1.5.3 Bi toỏn Dirichlet i vi h elliptic mnh nún cú cnh 28 1.6 Mt s kin thc cn bn v lớ thuyt na nhúm cỏc toỏn t tuyn tớnh b chn 30 1.7 Mt s kin thc cn bn v o khụng compact v ỏnh x nộn 35 Chng BI TON BIấN BAN U I VI PHNG TRèNH HYPERBOLIC NA TUYN TNH TRONG TR KHễNG TRN 41 2.1 Thit lp bi toỏn 41 2.2 S tn ti v tớnh nht ca nghim yu a phng 44 2.3 S tn ti v tớnh nht ca nghim yu ton cc 59 Chng BI TON DIRICHLET-CAUCHY I VI PHNG TRèNH HYPERBOLIC NA TUYN TNH TRONG CC MIN A DIN 63 3.1 Bi toỏn Dirichlet-Cauchy i vi phng trỡnh hyperbolic na tuyn tớnh cú cnh 63 3.1.1 M u 63 3.1.2 Bi toỏn tuyn tớnh 65 3.1.3 Bi toỏn na tuyn tớnh 81 3.2 Bi toỏn Dirichlet-Cauchy i vi phng trỡnh hyperbolic na tuyn tớnh nún cú cnh 87 3.2.1 M u 87 3.2.2 Bi toỏn tuyn tớnh 88 3.2.3 Bi toỏn na tuyn tớnh 92 Chng PHNG TRèNH TRUYN SểNG NA TUYN TNH VI CU TRC TT DN 107 4.1 Thit lp bi toỏn 107 4.1.1 Vớ d m u 107 4.1.2 Bi toỏn 109 4.2 S tn ti nghim mm ca bi toỏn 111 4.3 S tn ti nghim mm phõn ró ca bi toỏn 118 KT LUN 126 Kt qu t c 126 Kin ngh mt s nghiờn cu tip theo 126 DANH MC CC CễNG TRèNH CễNG B CA LUN N 127 TI LIU THAM KHO 128 MT S K HIU THNG DNG TRONG LUN N Chỳng tụi s dng cỏc ký hiu N l cỏc s t nhiờn, R l s thc Vi mi x = (x1 , ã ã ã , xn ) Rn v a ch s = (1 , ã ã ã , n ) Nn , n ta ký hiu x = x ã ã ã x n v || = i , ! = !2 ! ã ã ã n ! i=1 Vi a ch s = (1 , ã ã ã , n ) N, ta ký hiu D := Dx = Dx11 ã ã ã Dxnn , := x = x11 ã ã ã xnn ! ku v = Cỏc v Thờm na, ta ký hiu utk = k t !( )! khụng gian cỏc hm c ký hiu nh sau: ký hiu (m liờn thụng) b chn Rn vi biờn ký hiu hp ca v QT ký hiu tớch Descartes ca v (0, T ), vi < T ST ký hiu tớch Descartes ca v (0, T ), vi < T K ký hiu nún R3 vi nh ti gc vi biờn K KT ký hiu tớch Descartes ca nún K vi (0, T ) d KT ký hiu tớch Descartes ca i vi (0, T ), ú i l cỏc i=1 mt nhn ca nún K C k () ký hiu khụng gian cỏc hm cú o hm liờn tc n cp k , k C0k () ký hiu khụng gian cỏc hm kh vi cp k cú giỏ compact , k Lp () ký hiu khụng gian Banach gm tt c cỏc hm kh tng cp p, p < theo ngha Lebesgue L () ký hiu khụng gian Banach gm tt c cỏc hm o c v b chn hu khp ni trờn Lp (0, T ; X) ký hiu khụng gian tt c cỏc hm kh tng t [0, T ] vo khụng gian Banach X vi p < L (0, T ; X) ký hiu khụng gian cỏc hm u xỏc nh trờn [0, T ] cú giỏ tr X ng thi vi mi t, u(t) o c v b chn hu khp ni trờn X H k () ký hiu khụng gian Hilbert bao gm tt c hm kh tng a phng u trờn cho D u tn ti thuc Lp () k () ký hiu khụng gian Hilbert bao gm cỏc hm u H k () H cho D u = trờn vi tt c || k k () H k () ký hiu khụng gian i ngu ca H Hm () ký hiu khụng gian Sobolev cú trng R gm cỏc hm v D () khụng gian C0 () c trang b tụ pụ compact cho r+||m D v L2 (), || m, r = |x| Val,p () ký hiu khụng gian Sobolev cú trng a R, l bao úng ca C0 ( \ l0 ), ú l vi biờn gm hai siờu phng , cú giao l a l0 v < p < , Hal () = Val,2 () l (K) ký hiu bao úng ca C0 (K \ S), S = {0} M1 ã ã ã Md , vi V, R, = (1 , ã ã ã , d ) Rd , Mi l cỏc cnh ca nún K M U Lch s v lớ chn ti Cỏc bi toỏn biờn tuyn tớnh i vi phng trỡnh, h phng trỡnh o hm riờng cỏc vi biờn trn [3] ó c cỏc nh toỏn hc nghiờn cu khỏ hon thin na u th k XX Cỏc bi toỏn biờn loi dng cỏc trn ó c nghiờn cu nh phộp phõn hoch n v a bi toỏn ang xột v bi toỏn ton khụng gian v na khụng gian [19, 24, 27] Cỏc bi toỏn biờn khụng dng cỏc hỡnh tr vi ỏy l cú biờn trn c nghiờn cu nh phộp bin i Laplace hoc phộp bin i Fourier a v bi toỏn dng vi tham bin trn T gia th k XX, bi toỏn biờn tng quỏt i vi phng trỡnh elliptic vi biờn k d ó c nghiờn cu, cỏc kt qu quan trng v tớnh t ỳng ca bi toỏn cng nh tớnh trn v tim cn ca nghim vi cỏc im nún trờn biờn ó nhn c [49, 50] Nh khoa hc V.A.Kondratiev ó gii quyt c mt s mang tớnh nguyờn lớ khc phc im kỡ d kiu nún ca bi toỏn biờn tng quỏt i vi phng trỡnh elliptic Tip theo, mt s nh toỏn hc khỏc ó da trờn cỏc phng phỏp ca V.A.Kondratiev nghiờn cu cỏc bi toỏn biờn i vi cỏc h dng cỏc vi cỏc im k d trờn biờn [15, 25, 26, 51, 47, 52, 53] Bi toỏn biờn tng quỏt i vi phng trỡnh elliptic a din ó c V Mazya, J Rossomann nghiờn cu v tớnh gii c cỏc khụng gian L2 Sobolev cú trng, khụng gian Hăolder cú trng cỏc nh din, nún cú cnh, kiu a din [60], nhng kt qu cn bn ca toỏn t pencil ó c ỏp dng vic khng nh tớnh gii c ca bi toỏn Nhng kt qu t c ca bi toỏn biờn tng quỏt i vi phng trỡnh elliptic cỏc cú im nún, cú im lựi, cú cnh, kiu a giỏc l c s quan trng cho cỏc 120 (tc l sup et un (t) X n) cho sup et F (un )(t) X > n Khi ú tR+ tR+ et F1 (un )(t) X et S2 (t) x0 L(X) D(A) + g(un ) D(A) et S2 (t) L(X) x0 D(A) + g ( un C ( )t x0 D(A) + g (R) e C x0 D(A) + g (R) ) (4.48) t et F2 (un )(t) X et S2 (t s)S1 (s) y0 L(X) + h(un ) X X + Ax0 C y0 + h ( un X ) X + Ag(un ) + Ax0 X ds X t + g ( un ) e(2 )s ds e(2 )t C y0 X + h (R) + Ax0 X + g (R) , (4.49) vi mi t 0, iu ny cú c l un BCR (n) õy C1 l hng s dng c lp vi un t et F3 (un )(t) X s et S2 (t s)S1 (s ) L(X) f (, un ( )) X d ds t s e2 (ts) e1 (s ) m( ) un ( ) C et 0 t s e2 (ts) nC et X d ds e1 (s ) m( )d ds, ta thy rng s s e1 (s ) m( )d = e1 (s )+(s ) es m( )d s = es e(1 )(s ) m( )d Kes , 121 iu ny cú c l nh Do ú t e t F3 (un )(t) X t nC Ke e (ts)s nKC ds (4.50) T (4.48)-(4.50), ta cú et F (un )(t) X C2 x D(A) + y0 X + Ax0 X nKC , + (1 + )g (R) + h (R) + vi un BCR (n) v t 0, õy C2 = max{C, C1 } T ú suy 1< sup et F (un )(t) n t0 X C2 n x0 D(A) + y0 X + Ax0 X KC + (1 + )g (R) + h (R) + Qua gii hn n ta thu c iu mõu thun vi (4.43) B 4.4 Gi s (A ), (G ), (H ), (F ) tha Khi ú ta luụn cú F (D) M g + h + g + (D), (4.51) vi tt c cỏc b chn D BCR () Chng minh Cho D M l b chn bt k Ta cú F (D) = F (D) + d F (D) (4.52) Trc ht, ta cú F (D) F1 (D) + F2 (D) + F3 (D) (4.53) Nh B 4.2, ta thu c cỏc ỏnh giỏ sau F1 (D) M g (D), (4.54) F2 (D) 4(h + g ) (D), (4.55) F3 (D) (D) (4.56) T (4.53)(4.56), ta cú F (D) M g + h + g + (D) (4.57) 122 Bõy gi ta ly D BCR () l mt b chn Khi ú vi tt c u D, ta cú et F (u)(t) X t iu ny kộo theo F (u)(t) X et , u D v vi t ln iu ny tng ng vi T ln, ta cú dT F (D) eT Do ú d F (D) = lim dT F (D) = T (4.58) T (4.52), (4.57) v (4.58), ta cú kt lun ca b nh lý 4.2 Gi s cỏc gi thit ca B 4.3 tha v l := M g + h + g ) + < (4.59) Khi ú bi toỏn (4.8), (4.9) cú ớt nht mt nghim mm xỏc nh trờn R+ tha et u(t) X = O(1) t + Chng minh Bi gi thit (4.59), toỏn t nghim F l nộn Thc vy, nu D BCR () l b chn tha (D) F (D) , ú theo B 4.4, ta thu c (D) F (D) l (D) Do ú (D) = 0, nờn D l compact tng i Theo B 4.3, F BCR () BCR () v F liờn tc p dng nh lý 1.15, ỏnh x F xỏc nh bi (4.24) cú im bt ng F ix(F ) BCR () l compact, khỏc iu ny chng t rng bi toỏn (4.8), (4.9) cú nghim u(t) biu din bi (4.16) tha lim u(t) = t minh cho s tn ti nghim mm phõn ró theo tc m ca bi toỏn tng quỏt, ta quay li xột bi toỏn (4.1)(4.4) Nh ó nờu Vớ d mc M u, ta xột biờn theo hai trng hp sau: Min Rn cú biờn trn Khi ú vi X = L2 (), A = x cú () H () sinh C0 na nhúm {T (t)}t0 gii tớch, xỏc nh H compact, n nh m, T (t) L(X) e1 t , vi > l giỏ tr riờng u tiờn ca A Do ú {T (t)}t0 l C0 na nhúm liờn tc chun Ta nhc li cỏc kớ hiu sau: u(t) = u(t, ã), f (t, v) = f (t, ã, v(ã)), n g(u) = k(ã, y)u(0, y)dy, Ci u(ti , ã) h(u) = i=1 123 Ta xột hm phi tuyn f, gi thit rng tn ti hm m L1loc (R+ ) cho |f (t, x, z)| m(t)|z|, (t, x, z) R+ ì ì R Khi ú ta cú f (t, v) X m(t) v X , v X Chỳ ý rng toỏn t G(v) = k(ã, y)v(y)dy l toỏn t kiu Hilbert-Schmidt trờn L2 (), nờn nú l toỏn t compact iu ny ỏp dng cho ỏnh x g xỏc nh bi g(u) = G u(0, ã) cng l toỏn t compact Tng t, ta cú Ag l ỏnh x compact Vỡ vy iu kin (G )(ii) l tha vi g = g = Thờm na, ta thy rng g(u) X k L2 (ì) u(0, ã) X k u(0, ã) L2 (ì) C Do ú (G )(i) tha Tip tc, xột cỏc iu kin i vi hm h, ta cú N h(u1 ) h(u2 ) X N Ci u1 (ti , ã) u2 (ti , ã) X i=1 N Khi ú h() Ci u1 u2 C i=1 Ci (), vi mi b chn D BC(R+ ; X) i=1 N iu ny cho thy gi thit (H )(ii) tha món, õy h = Ci Thờm i=1 na, ta cú ỏnh giỏ N h(u) X N Ci i=1 u(ti , ã) X Ci u , u BC(R+ ; X) i=1 iu ny cho thy gi thit (H )(i) l tha Bi cỏc kt qu thu c trờn, qua tớnh toỏn, ta cú M = 1, = 0, (2 ) , g (r) = k t N e(1 )(t ) m( )d, h (r) = K = sup t0 L2 (ì) r, Ci ) r i=1 p dng nh lý 4.2, bi toỏn (4.1)(4.4) cú ớt nht mt nghim u BC R+ ; L2 () tha et u(t) L2 () = O(1) t + xỏc nh 124 bi cỏc iu kin K < , k L2 (ì) N C i + k + (2 ) L2 (ì) + < i=1 Ta xột bi toỏn (4.1)-(4.4) trờn R2 b chn vi biờn trn tr im gc x = 0, cỏc hm g, h c xột nh trờn Tuy nhiờn i vi hm f, ta gi s nh sau: Cho f1 : R+ ì ì R R, : R+ ì R v f2 : ì R R cho (a) f1 l hm liờn tc tha f(t, x, 0) = v f1 (t, x, z1 ) f1 (t, x, z2 ) m1 (t)|z1 z2 | vi tt c x , z1 , z2 R, õy m1 L2 (R+ ) hoc m1 l hng s; (b) BC R+ , L2 () ; (c) f2 l hm liờn tc v |f2 (x, z)| l(x)|z| vi l L2 () Hm f : R+ ì X X tha f (x, t, v) = f1 (t, v)(x) + f2 (t, v)(x), vi f1 (t, v)(x) = f1 (t, x, v(x)); f2 (x, v(x))dx f2 (t, v)(x) = à(t, x) Xột f1 , ta cú f1 (s, v1 ) f1 (s, v2 ) m1 (s) v1 v2 iu ny a n f1 (s, D) m1 (s)(D) (4.60) vi tt c cỏc b chn D X Xột f2 , s dng bt ng thc Hăolder, ta cú f2 (t, v) X f2 x, v(x) dx à(t)|2X à(t) X |l(x)||v(x)|dx à(t) X l X v X Mt khỏc, vi bt k D X, ta thy rng f2 (s, D) {à(s, ã) : R} iu ny a n f2 (s, D) nm khụng gian mt chiu ca X 125 Do ú f2 (t, V ) = 0, (4.61) vi t R+ T (4.60) v (4.61), ta thu c f (s, D) f1 (s, D) + f2 (s, D) m1 (s)(D) vi mi b chn D X Nh vy, f (t) = m1 (t) Thờm na, ta cú f (ã, t, v) X f1 (t, v) X + f2 (t, v) Do ú m(t) = m1 (t) + à(t) X l X m1 (t) + à(t) X l X v X X Bi cỏc kt qu thu c vớ d ó nờu mc M u ca chng, lp lun tng t nh trng hp biờn trn, ta cng thu c kt qu M = 1, K , g (r) = k (2 ) m1 L2 (R) + 2L 1 ú L = max à(t) t0 X l , N L2 (ì) r, h (r) = Ci ) r, i=1 m1 L2 (R+ ) (21 ) X Tip theo, ta cú L + m1 L2 (R+ ) 1 (21 ) , p dng nh lý 4.2, bi toỏn (4.1)(4.4) cú ớt nht mt nghim u BC R+ ; L2 () tha et u(t) L2 () = O(1) t + xỏc nh bi cỏc iu kin K < , N Ci + k L2 (ì) i=1 + + < k L2 (ì) + (2 ) Kt lun chng Trong chng 4, chỳng tụi ó thit lp c s tn ti nghim mm phõn ró tc theo cp m ca bi toỏn (4.8), (4.9) v a c cỏc vớ d minh cho kt qu t c i vi phng trỡnh truyn súng na tuyn tớnh vi cu trỳc tt dn trờn trn bt k v cú im gúc R2 Kt qu t c tng quỏt hn so vi cựng kt qu [22, 21] trờn hai phng din, th nht l bi toỏn c xột vi iu kin biờn khụng a phng, th hai l bi toỏn c xột vi lp hm phi tuyn rng hn 126 KT LUN Kt qu t c S tn ti v tớnh nht nghim yu trờn [0, +) ca bi toỏn giỏ tr biờn ban u i vi phng trỡnh hyperbolic na tuyn tớnh cp cao cỏc tr khụng trn S tn ti nht v biu din tớnh chớnh quy ca nghim yu trờn [0, T ] (0 < T < +) theo bin thi gian khụng gian Sobolev cú trng ca bi toỏn Dirichlet-Cauchy i vi phng trỡnh hyperbolic na tuyn tớnh cp hai cú cnh, nún cú cnh S tn ti nghim mm phõn ró tc theo cp m phng trỡnh truyn súng na tuyn tớnh vi cu trỳc tt dn cựng iu kin ban u khụng a phng, xột vi lp hm phi tuyn cú giỏ tr khụng gian Banach liờn tc b chn Kin ngh mt s nghiờn cu tip theo Bờn cnh cỏc kt qu ó t c lun ỏn, mt s m cn tip tc nghiờn cu nh: Biu din tim cn ca nghim ca bi toỏn Dirichlet-Cauchy i vi phng trỡnh hyperbolic phi tuyn cỏc cú cnh S tn ti nht, tớnh chớnh quy, biu din tim cn ca nghim ca bi toỏn Dirichlet-Cauchy i vi phng trỡnh hyperbolic phi tuyn a din S tn ti nghim mm ca phng trỡnh truyn súng cú cu trỳc tt dn vi tr vụ hn cựng iu kin ban u khụng a phng 127 DANH MC CễNG TRèNH CễNG B CA LUN N Vu Trong Luong, Nguyen Thanh Tung, "The first initial boundary value problem for nonlinear hyperbolic equations of higher order in cylinders with singular points", International journal of evolution equations 2015, Volume 9, Number 2, pp 167-180 Vu Trong Luong, Nguyen 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NGUYỄN THANH TÙNG BÀI TOÁN BIÊN ĐỐI VỚI MỘT SỐ LỚP PHƯƠNG TRÌNH TRUYỀN SÓNG TRONG MIỀN KHÔNG TRƠN Chuyên ngành: Phương trình vi phân tích phân Mã số: 62 46 01 03 LUẬN ÁN TIẾN SĨ TOÁN HỌC TẬP THỂ... miền trơn Từ vấn đề nêu trên, định nghiên cứu toán biên số lớp phương trình truyền sóng miền không trơn Trong nghiên cứu tồn nghiệm yếu toàn cục, nghiệm yếu địa phương toán biên ban đầu phương trình. .. biến thời gian toán biên ban đầu thứ nhất, thứ hai trụ với đáy miền với biên xác định Đặc biệt, tính trơn nghiệm theo biến không gian toán biên hệ phương trình hyperbolic trụ với đáy miền chứa điểm