B GIO DC V O TO TRNG I HC S PHM H NI * DNG TRNG LUYN V MT S PHNG TRèNH ELLIPTIC V HYPERBOLIC PHI TUYN SUY BIN LUN N TIN S TON HC H NI - 2017 B GIO DC V O TO TRNG I HC S PHM H NI * DNG TRNG LUYN V MT S PHNG TRèNH ELLIPTIC V HYPERBOLIC PHI TUYN SUY BIN 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: GS.TSKH Nguyn Minh Trớ H NI - 2017 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi Cỏc kt qu ny c lm di s hng dn ca GS TSKH Nguyn Minh Trớ Cỏc kt qu lun ỏn vit chung vi thy hng dn u ó c s nht trớ ca thy hng dn a vo lun ỏn Cỏc kt qu lun ỏn l trung thc v cha tng c cụng b cỏc cụng trỡnh ca cỏc tỏc gi khỏc Nghiờn cu sinh: Dng Trng Luyn LI CM N Lun ỏn c thc hin v hon thnh ti B mụn Gii tớch, Khoa Toỏn - Tin, Trng i hc S phm H Ni, di s hng dn ca GS TSKH Nguyn Minh Trớ Thy ó dn dt tỏc gi lm quen vi nghiờn cu khoa hc tỏc gi cũn l hc viờn cao 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 giỳp tỏc gi tin tng v say mờ nghiờn cu khoa hc Vi tm lũng tri õn sõu sc, tỏc gi xin by t lũng bit n chõn thnh v sõu sc nht i vi thy 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 giỏo, cụ giỏo B mụn Gii tớch, Khoa Toỏn - Tin, Trng i hc S phm H Ni, cỏc thy giỏo, cụ giỏo Phũng Phng trỡnh vi phõn, Vin Toỏn hc, ó luụn giỳp , ng vin, to mụi trng hc nghiờn cu thun li cho tỏc gi Tỏc gi xin cm n Ban Giỏm hiu, cỏc anh ch em Khoa T nhiờn, Trng i hc Hoa L ó to mi iu kin thun li giỳp tỏc gi quỏ trỡnh hc nghiờn cu v hon thnh lun ỏn Tỏc gi xin trõn trng cm n qu NAFOSTED ó ti tr cho tỏc gi sut quỏ trỡnh hc nghiờn cu sinh Li cm n sau cựng, xin dnh cho gia ỡnh ca tỏc gi, nhng ngi ó dnh cho tỏc gi tỡnh yờu thng trn vn, tng ngy chia s, ng viờn tỏc gi vt qua mi khú khn hon thnh lun ỏn Mc lc Trang Li cam oan Li cm n Mc lc Mt s quy c v kớ hiu M u Tng quan 10 Chng1 MT S KIN THC CHUN B 17 1.1 1.2 Toỏn t v mt s khụng gian hm 17 1.1.1 Toỏn t 17 1.1.2 Mt s khụng gian hm 19 1.1.3 Mt s tớnh cht 20 Tp hỳt ton cc v tớnh cht 23 1.2.1 Mt s nh ngha 23 1.2.2 Mt s tớnh cht 26 Chng2 S TN TI NGHIM V TNH CHNH QUY CA NGHIM CA BI TON BIấN I VI PHNG TRèNH ELLIPTIC SUY BIN 2.1 28 Mt s nh lớ v s tn ti nghim yu 28 2.1.1 nh lớ v s tn ti nghim yu 29 2.1.2 nh lớ v s tn ti nghim yu khụng õm 41 2.2 Tớnh chớnh quy ca nghim ca bi toỏn biờn elliptic suy bin 44 Chng3 TP HT TON CC I VI PHNG TRèNH HYPERBOLIC TT DN CHA TON T ELLIPTIC SUY BIN MNH TRONG MIN B CHN 52 3.1 S tn ti v nht ca nghim tớch phõn 53 3.1.1 t bi toỏn 53 3.1.2 S tn ti v nht ca nghim tớch phõn 54 3.2 () ì L2 () S tn ti hỳt ton cc S(k ,k2 ),0 3.3 ỏnh giỏ s chiu fractal ca hỳt ton cc 61 69 Chng4 TP HT TON CC I VI PHNG TRèNH HYPERBOLIC TT DN CHA TON T GRUSHIN TRấN TON KHễNG GIAN 4.1 4.2 82 S tn ti nht ca nghim tớch phõn 83 4.1.1 t bi toỏn 83 4.1.2 S tn ti v nht ca nghim tớch phõn 84 S tn ti hỳt ton cc Sk2 (RN ) ì L2 (RN ) 86 Kt lun v kin ngh 111 Cỏc kt qu t c 111 Kin ngh mt s nghiờn cu tip theo 111 Danh mc cụng trỡnh khoa hc ca tỏc gi liờn quan n lun ỏn 113 Ti liu tham kho 113 MT S QUY C V K HIU Trong ton b lun ỏn, ta thng nht mt s kớ hiu nh sau: RN khụng gian vect thc N chiu R+ cỏc s thc khụng õm R+ cỏc s thc dng |x| chun Euclid ca phn t x khụng gian RN C k () khụng gian cỏc hm kh vi liờn tc n cp k C0 () khụng gian cỏc hm kh vi vụ hn cú giỏ compact Lp () khụng gian cỏc hm ly tha bc p kh tớch Lebesgue H ã, ã khụng gian i ngu ca khụng gian Banach H i ngu gia H v H (ã, ã)H tớch vụ hng khụng gian H Id ỏnh x ng nht hi t yu phộp nhỳng liờn tc phộp nhỳng compact Vol() o Lebesgue ca khụng gian RN x y z Toỏn t Laplace theo bin x RN1 : x = Toỏn t Laplace theo bin y RN2 : y = Toỏn t Laplace theo bin z RN3 : z = N1 i=1 N2 j=1 N3 l=1 x2i yj2 zl2 M U Lớ chn ti Lớ thuyt phng trỡnh vi phõn o hm riờng c nghiờn cu u tiờn cỏc cụng trỡnh ca J DAlembert (1717-1783), L Euler (17071783), D Bernoulli (1700-1782), J Lagrange (1736-1813), P Laplace (1749-1827), S Poisson (1781-1840) v J Fourier (1768-1830), nh l mt cụng c chớnh mụ t c hc cng nh mụ hỡnh gii tớch ca Vt lớ Vo gia th k XIX vi s xut hin cỏc cụng trỡnh ca Riemann, lớ thuyt phng trỡnh vi phõn o hm riờng ó chng t l mt cụng c thit yu ca nhiu ngnh toỏn hc Cui th k XIX, H Poincarộ ó ch mi quan h bin chng gia lớ thuyt phng trỡnh vi phõn o hm riờng v cỏc ngnh toỏn hc khỏc Sang th k XX, lớ thuyt phng trỡnh vi phõn o hm riờng phỏt trin vụ cựng mnh m nh cú cụng c gii tớch hm, c bit l t xut hin lớ thuyt hm suy rng S L Sobolev v L Schwartz xõy dng Nghiờn cu cỏc phng trỡnh, h phng trỡnh elliptic tng quỏt v phng trỡnh hyperbolic ó úng vai trũ rt quan trng lớ thuyt phng trỡnh vi phõn Hin cỏc kt qu theo hng ny ó tng i hon chnh Cựng vi s phỏt trin khụng ngng ca toỏn hc cng nh khoa hc cụng ngh nhiu bi toỏn liờn quan ti trn ca nghim ca cỏc phng trỡnh, h phng trỡnh khụng elliptic v phng trỡnh hyperbolic tt dn suy bin ó xut hin Cú mt s lp phng trỡnh, ú cú lp phng trỡnh elliptic suy bin v phng trỡnh hyperbolic tt dn suy bin, mt khớa cnh no ú cng cú mt s tớnh cht ging vi phng trỡnh elliptic v phng trỡnh hyperbolic tt dn cha toỏn t Tuy nhiờn cỏc kt qu t c cho cỏc phng trỡnh elliptic v hyperbolic tt dn suy bin cũn ớt, cha y Vi cỏc lớ nờu trờn chỳng tụi ó chn ti nghiờn cu cho lun ỏn ca mỡnh l V mt s phng trỡnh elliptic v hyperbolic phi tuyn suy bin Mc ớch nghiờn cu Ni dung : Nghiờn cu bi toỏn biờn elliptic suy bin cha toỏn t vi cỏc ni dung sau: - Nghiờn cu s tn ti nghim yu ca bi toỏn; - Tớnh chớnh quy ca nghim yu Ni dung : Nghiờn cu phng trỡnh hyperbolic tt dn cha toỏn t elliptic suy bin mnh b chn vi cỏc ni dung sau: - Nghiờn cu s tn ti v nht nghim tớch phõn; - Nghiờn cu s tn ti hỳt ton cc; - ỏnh giỏ s chiu fractal ca hỳt ton cc Ni dung : Nghiờn cu phng trỡnh hyperbolic tt dn cha toỏn t Grushin ton khụng gian vi cỏc ni dung sau: - Nghiờn cu s tn ti v nht nghim tớch phõn; - Nghiờn cu s tn ti hỳt ton cc i tng v phm vi nghiờn cu i tng nghiờn cu ca lun ỏn l xột bi toỏn biờn v bi toỏn biờn giỏ tr ban u cú cha toỏn t elliptic suy bin N := j=1 xj j2 xj Phng phỏp nghiờn cu nghiờn cu s tn ti nghim yu ca bi toỏn chỳng tụi s dng phng phỏp bin phõn nghiờn cu tớnh chớnh quy ca nghim chỳng tụi s dng nh lớ nhỳng kiu Sobolev v mt s bt ng thc nghiờn cu s tn ti nht ca nghim tớch phõn chỳng tụi s dng phng phỏp na nhúm (xem [53, 59]) nghiờn cu dỏng iu tim cn ca nghim, chỳng tụi s dng cỏc cụng c v phng phỏp ca lớ thuyt h ng lc vụ hn chiu (xem [13,14,21,22,52,58,62,74]), núi riờng l phng phỏp phng trỡnh nng lng v phng phỏp ỏnh giỏ phn uụi ca nghim chng minh s chiu fractal ca hỳt ton cc l b chn chỳng tụi s dng phng phỏp qu o (xem [51, 55]) Cỏc kt qu t c v ý ngha ca ti Lun ỏn ó t c nhng kt qu chớnh sau õy: i vi bi toỏn biờn elliptic suy bin a v chng minh c s tn ti nghim yu ca bi toỏn vi mt s iu kin ca s hng phi tuyn.V cng chng minh c tớnh chớnh quy ca nghim õy l ni dung ca Chng i vi phng trỡnh hyperbolic tt dn cha toỏn t elliptic suy bin mnh b chn: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc v ỏnh giỏ c s chiu fractal ca hỳt õy l ni dung ca Chng i vi phng trỡnh hyperbolic tt dn cha toỏn t Grushin ton khụng gian: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc õy l ni dung ca Chng Cỏc kt qu ca lun ỏn l mi, cú ý ngha khoa hc, v gúp phn Vy hm f (X, u) tha iu kin (4.2) Ta cú f (X, 0) = L2 (R2 ), nờn f (X, u) tha iu kin (4.3) Vi iu kin (4.4), ta chn C1 = 14 , g1 (X) = 4(|X|14 +1) L1 (R2 ) C2 |X|4 +1 Tht vy, ta tỡm C1 v g2 (X) = C2 F (X, u) C1 f (X, u)u + |X|4 + u(1u2 ) u2 u4 |X|4 +1 C1 |X|4 +1 + C2 |X|4 +1 |u| 1, u2 u4 14 + C2 |u| 1, |X| +1 C1 u u |u| 1, u2 u4 uC1 u2 + C2 u2 u4 C1 u u2 + C2 C1 u4 + C1 u2 C2 tha |u| 1, C1 ta chn C1 = |u| 1, 4 u4 + C1 u2 C2 + |u| 1, ú C2 tha |u| 1, 41 u2 C2 u2 C2 + |u| 1, nờn ta chn C2 = 41 Khi ú ta cú 1 F (X, u) f (X, u)u , X R2 , u R 4 4(|X| + 1) Ta cú |X|4 + 1 |X|4 + u2 u4 0, X R2 , |u| 1, u2 u4 0, X R2 , |u| 109 Nờn F (X, u(X)) 0, vi mi u S1/2 (R2 ) Vy hm f (X, u) tha iu kin (4.5) Do ú f (X, u) tha cỏc iu kin (4.2)(4.5) Nờn ỏp dng nh lớ 4.2.7, ta cú Vớ d 4.2.8 tn ti mt hỳt ton cc 2 2 (R2 )ìL2 (R2 ) S AS1/2 1/2 (R ) ì L (R ) i vi na nhúm S(t) sinh bi Vớ d 4.2.8 KT LUN CHNG Trong chng ny, chỳng tụi nghiờn cu bi toỏn hyperbolic tt dn cha toỏn t Grushin ton khụng gian, vi s hng phi tuyn tha mt s tớnh cht Cỏc kt qu t c bao gm: 1) Chng minh c s tn ti nht ca nghim tớch phõn ca bi toỏn (nh lớ 4.1.5 ) 2) Chng minh c s tn ti hỳt ton cc ca bi toỏn (nh lớ 4.2.7) Cỏc kt qu ca Chng l s m rng cỏc kt qu tng ng trc ú i vi phng trỡnh hyperbolic tt dn cha toỏn t elliptic ton khụng gian [27, 33] cho toỏn t Grushin v hm phi tuyn Phng phỏp c s dng õy l phng phỏp c lng uụi nghim cho hỡnh cu suy bin tng ng vi toỏn t Grushin 110 KT LUN V KIN NGH Cỏc kt qu t c Trong lun ỏn ny, chỳng tụi nghiờn cu s tn ti nghim, tớnh chớnh quy ca nghim ca bi toỏn biờn cú cha phng trỡnh elliptic suy bin b chn cú biờn trn v s tn ti nghim ton cc, hỳt ton cc ca bi toỏn biờn giỏ tr ban u i vi phng trỡnh hyperbolic tt dn cú cha toỏn t elliptic suy bin Cỏc kt qu chớnh t c lun ỏn bao gm: i vi bi toỏn biờn elliptic suy bin a v chng minh c s tn ti nghim yu ca bi toỏn vi mt s iu kin ca s hng phi tuyn, v tớnh chớnh quy ca nghim i vi phng trỡnh hyperbolic tt dn cha toỏn t elliptic suy bin mnh b chn: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc liờn thụng compact khụng gian S(k () ì L2 () ,k2 ),0 v chng minh c s chiu fractal ca hỳt l hu hn i vi phng trỡnh hyperbolic tt dn cha toỏn t Grushin ton khụng gian: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc khụng gian Sk2 (RN ) ì L2 (RN ) Kin ngh mt s nghiờn cu tip theo Bờn cnh cỏc kt qu t c lun ỏn, mt s m liờn quan cn c tip tc nghiờn cu nh: 111 Nghiờn cu iu kin tn ti nghim ca cỏc bi toỏn biờn núi trờn khụng b chn, b chn vi iu kin biờn Dirichlet khụng thun nht Nghiờn cu s tn ti hỳt hm phi tuyn ph thuc thi gian nh hỳt lựi, hỳt u v cỏc tớnh cht ca hỳt Nghiờn cu s tn ti nghim v dỏng iu tim cn nghim ca cỏc phng trỡnh hyperbolic suy bin vi cỏc iu kin biờn khỏc nhau, chng hn iu kin biờn khụng thun nht, iu kin biờn Neumann, iu kin biờn hn hp, iu kin biờn phi tuyn, lm c iu ny cn xõy dng c khụng gian cú trng tng ng, nh lớ nhỳng kiu Sobolev Nghiờn cu cỏc mụ hỡnh ng dng thc t 112 DANH MC CễNG TRèNH KHOA HC CA TC GI LIấN QUAN N LUN N [1] D T Luyen and N M Tri (2012), On boundary value problem for semilinear degenerate elliptic differential equations AIP Conf Proc 1450, 1317 [2] D T Luyen and N M Tri (2015), Existence of solutions to boundary value problems for semilinear differential equations Mathematical Notes, Vol 97, No 1, 7384 [3] D T Luyen and N M Tri (2016), Large - time behavior of solutions to damped hyperbolic equation involving strongly degenerate elliptic differential operators Siberian Mathematical Journal, Vol 57, No 4, 632649 [4] D T Luyen and N M Tri (2016), Global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator Annales Polonici Mathematici, Vol 117, No 2, 141162 113 Ti liu tham kho [1] C T Anh (2010), Pullback attractors for non-autonomous parabolic equations involving Grushin operator Electron J Differential Equations, no 11, 114 [2] C T Anh (2014), Global attractor for a semilinear strongly degenerate parabolic equation on RN NoDEA Nonlinear Differential Equations Appl 21, no 5, 663678 [3] C T Anh; P Q Hung; T D Ke; T T Phong (2008), Global attractor for a semilinear parabolic equation involving the Grushin operator Electron J Differential Equations, no 32, 111 [4] C T Anh; T D Ke (2009), Existence and continuity of global attractors for a degenerate semilinear parabolic equation Electron J Differential Equations, no 61, 113 [5] C T Anh; B K My (2016), Existence of solutions to Laplace equations without the Ambrosetti-Rabinowitz condition Complex Var Elliptic Equ 61, no 1, 137150 [6] C T Anh; B K My (2016), Liouville-type theorems for elliptic inequalities involving the Laplace operator Complex Var Elliptic Equ 61, no 7, 10021013 [7] C T Anh; N V Quang (2011), Uniform attractors for a nonautonomous parabolic equation involving Grushin operator Acta Math Vietnam 36, no 1, 1933 114 [8] C T Anh; V M Toi (2013), Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains Nonlinear Anal 93, 181196 [9] C T Anh; V M Toi (2016), Null controllability in large time of a parabolic equation involving the Grushin operator with an inversesquare potential NoDEA Nonlinear Differential Equations Appl 23, no 2, Art 20, 26 pp [10] J M Arrieta; A N Carvalho; J K Hale (1992), A damped hyperbolic equation with critical exponent Comm Partial Differential Equations 17, no 56, 841866 [11] A V Babin; M I Vishik (1983), Regular attractors of semigroups and evolution equations J Math Pures Appl (9) 62, no 4, 441491 [12] A V Babin; M I Vishik (1990), Attractors of partial differential evolution equations in an unbounded domain Proc Roy Soc Edinburgh Sect A116, no 34, 221243 [13] A V Babin; M I Vishik (1992), Attractors of Evolution Equations Translated and revised from the 1989 Russian original by Babin Studies in Mathematics and its Applications, 25 North-Holland Publishing Co., Amsterdam x+532 pp [14] J M Ball (2004), Global attractor for damped semilinear wave equations Partial differential equations and applications Discrete Contin Dyn Syst 10, no 12, 3152 [15] K Beauchard; P Cannarsa; R Guglielmi (2014), Null controllability of Grushin-type operators in dimension two J Eur Math Soc (JEMS) 16, no 1, 67101 115 [16] T Bieske (2002), Viscosity solutions on Grushin type planes Illinois J Math 46, no 3, 893911 [17] H Brezis; L Nirenberg (1983), Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Comm Pure Appl Math 36, no 4, 437477 [18] F E Browder (1956), On the regularity properties of solutions of elliptic differential equations Comm Pure Appl Math 9, 351361 [19] C Budd; M C Knaap; L A Pelrtier (1991), Asymptotic behaviour of solutions of Elliptic equations with critical exponents and neumann boundary conditions Proc Roy Soc Edinburgh Sect A117, 225250 [20] L Capogna; D Danielli; N Garofalo (1993), An embedding theorem and the Harnack inequality for nonlinear subelliptic equations Comm Partial Differential Equations 18, no 910, 17651794 [21] A Carvalho; J A Langa; J C Robinson (2013), Attractors for Infinite Dimensional Non-Autonomous Dynamical Systems Applied Mathematical Sciences, 182 Springer, New York xxxvi+409 pp [22] V V Chepyzhov; M I Vishik (2002), Attractors for Equations of Mathematical Physics American Mathematical Society Colloquium Publications, 49 American Mathematical Society, Providence, RI xii+363 pp [23] I D Chueshov (1999), Introduction to the theory of infinite - dimensional dissipative systems [University Lectures in Contemporary Mathematics] AKTA, Kharkiv 436 pp 116 [24] N M Chuong; T D Ke; N V Thanh; N M Tri (1999), Non existence theorem for boundary value problem for some classes of semilinear degenerate elliplic operators Proceeding of the conference on PDEs and their applications, Hanoi Dec, 185190 [25] N M Chuong; T D Ke (2004), Existence of solutions for a nonlinear degenerate elliptic system Electron J Differential Equations, no 93, 15 pp [26] N M Chuong; T D Ke (2005), Existence results for a semilinear parametric problem with Grushin type operator Electron J Differential Equations, no 107, 12 pp [27] Fall Djiby (2005), Longtime dynamics of hyperbolic evolutionary equations in ubounded domains and lattice systems Graduate Theses and Dissertations, University of South Florida, http://scholarcommons.usf.edu/etd/2875 [28] E Feireisl (1994), Attractors for semilinear damped wave equations on R3 Nonlinear Anal 23, no 2, 187195 [29] A Friedman (1958), On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations J Math Mech 7, 4359 [30] D Gilbarg; N Trudinger (2001), Elliptic partial differential equations of second order Reprint of the 1998 edition Classics in Mathematics Springer-Verlag, Berlin xiv+517 pp [31] V V Grushin (1970), A certain class of hypoelliptic operators (Russian) Mat Sb.(N.S.) 83 (125), 456473 117 [32] V V Grushin (1971), A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold (Russian) Mat Sb (N.S.) 84 (126), 163195 [33] H B Xiao (2009), Global attractors for semilinear damped wave equations with critical exponent on RN Chaos Solitons Fractals 41, no 5, 25462552 [34] J K Hale (1985), Asymptotic behaviour and dynamics in infinite dimensions Nonlinear differential equations (Granada, 1984), 142, Res Notes in Math., 132, Pitman, Boston, MA [35] J K Hale (1988), Asymptotic behavior of dissipative systems Mathematical Surveys and Monographs, 25 American Mathematical Society, Providence, RI x+198 pp [36] J K Hale; G Raugel (1993), Attractors for dissipative evolutionary equations International Conference on Differential Equations, Vol 1, (Barcelona, 1991), 322, World Sci Publ., River Edge, NJ, [37] A Haraux (1985), Two remarks on hyperbolic dissipative problems (French summary) Nonlinear partial differential equations and their applications Collốge de France seminar, Vol VII (Paris, 19831984), 6, 161179, Res Notes in Math., 122, Pitman, Boston, MA [38] B Helffer; J Nourrigat (1985), Hypoellipticitộ maximale pour des opộrateurs polynụmes de champs de vecteurs (French) [Maximal hypoellipticity for polynomial operators of vector fields] Progress in Mathematics, 58 Birkhăauser Boston, Inc., Boston, MA x+278pp [39] L Hăormander (1966), Psedo-differential operators and non-elliptic boundary problems Ann of Math.(2) 83, 129209 118 [40] D S Jerison; J M Lee (1987), The Yamabe problem on CR manifolds J Differential Geom 25, no 2, 167197 [41] N I Karachalios; N M Stavrakakis (1999), Existence of a global attractor for semilinear dissipative wave equations on RN J Differential Equations 157, no 1, 183205 [42] T T Khanh; N M Tri (2010), On the analyticity of solutions to semilinear differential equations degenerated on a submanifold J Differential Equations 249, no 10, 24402475 [43] A Kh Khanmamedov (2005), Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain Appl Math Lett 18, no 7, 827832 [44] A E Kogoj; E Lanconelli (2012), On semilinear Laplace equation Nonlinear Anal 75, no 12, 46374649 [45] A E Kogoj; S Sonner (2013), Attractors for a class of semi-linear degenerate parabolic equations J Evol Equ 13, no 3, 675691 [46] A E Kogoj; S Sonner (2014), Attractors met Xelliptic operators J Math Anal Appl 420, no 1, 407434 [47] A E Kogoj; S Sonner (2016), Hardy type inequalities for Laplacians Complex Var Elliptic Equ 61, no 3, 422442 [48] E Lanconelli; A E Kogoj (2000), Xelliptic operators and Xcontrol distances Contributions in honor of the memory of Ennio De Giorgi (Italian) Ricerche Mat 49, suppl., 223243 [49] D Li; C Sun (2016), Attractors for a class of semi-linear degenerate 119 parabolic equations with critical exponent J Evol Equ 16, no 4, 9971015 [50] D T Luyen (2017), Two nontrivial solutions of boundary value problems for semilinear differential equations, Math Notes 101, no 5, 815823 [51] J Mỏlek; D Prazỏk (2002), Large time behavior via the method of trajectories J Differential Equations 181, no 2, 243279 [52] A Miranville; S Zelik (2008), Attractors for dissipative partial differential equations in bounded and unbounded domains Handbook of differential equations: evolutionary equations Vol IV, 103200, Handb Differ Equ., Elsevier/North-Holland, Amsterdam [53] A Pazy (1983), Semigroups of linear operators and applications to partial differential equations Applied Mathematical Sciences, 44 Springer-Verlag, New York viii+279 pp [54] S I Pohozaev (1965), On the eigenfunctions of the equation u + f (u) = (Russian) Dokl Akad Nauk SSSR 165, 3336 [55] D Prazỏk (2002), On finite fractal dimension of the global attractor for the wave equation with nonlinear damping J Dynam Differential Equations 14, no 4, 763776 [56] D T Quyet; L T Thuy; N X Tu (2016), Semilinear strongly degenerate parabolic equation with a new class of nonlinearities Vietnam J Math doi:10.1007/s10013-016-0228-5 120 [57] G Raugel (2002), Global Attractors in Partial Differential Equations Handbook of dynamical systems, Vol 2, 885982, NorthHolland, Amsterdam [58] J C Robinson (2001), Infinite-Dimensional Dynamical Systems An introduction to dissipative parabolic PDEs and the theory of global attractors Cambridge Texts in Applied Mathematics Cambridge University Press, Cambridge, xviii+461 pp [59] G R Sell; Y You (2002), Dynamics of Evolutionary Equations Applied Mathematical Sciences, 143 Springer-Verlag, New York xiv+670 pp [60] M Struwe (1996), Variational Methods Applications to nonlinear partial differential equations and Hamiltonian systems Second edition Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34 Springer-Verlag, Berlin xvi+272 pp [61] D Tartakoff; L Zanghirati (2005), Local real analyticity of solutions for sums of squares of non-linear vector fields J Differential Equations, 213, no 2, 341351 [62] R Temam (1988), Infinite-dimensional dynamical systems in mechanics and physics Applied Mathematical Sciences, 68 SpringerVerlag, New York xvi+500 pp [63] R Temam (1995), Navier-Stokes equations and nonlinear functional analysis Second edition CBMS-NSF Regional Conference Series in Applied Mathematics, 66 Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA xiv+141 pp 121 [64] N T C Thuy; N M Tri (2002), Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators Russ J Math Phys 9, no 3, 365370 [65] P T Thuy; N M Tri (2012), Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations NoDEA Nonlinear Differential Equations Appl 19, no 3, 279298 [66] P T Thuy; N M Tri (2013), Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators NoDEA Nonlinear Differential Equations Appl 20, no 3, 12131224 [67] F Treves, Symplectic geometry and analytic hypo-ellipticity Differential equations: La Pietra 1996 (Florence), 201219, Proc Sympos Pure Math., 65, Amer Math Soc., Providence, RI [68] N M Tri (1998), On the Grushin equation Mat Zametki 63, no 1, 95105; translation in Math Notes 63, no 12, 8493 [69] N M Tri (1998), Critical Sobolev exponent for degenerate elliptic operators Acta Math Vietnam 23, no 1, 8394 [70] N M Tri (1999), Semilinear perturbation of powers of the Mizohata operators Comm Partial Differential Equations 24, no 12, 325354 [71] N M Tri (2008), Semilinear hypoelliptic differential operators with multiple characteristics Trans Amer Math Soc 360, no 7, 38753907 122 [72] N M Tri (2010), Semilinear Degenerate Elliptic Differential Equations, Local and global theories Lambert Academic Publishing, 271p [73] N M Tri (2014), Recent Progress in the Theory of Semilinear Equations Involving Degenerate Elliptic Differential Operators Publishing House for Science and Technology of the Vietnam Academy of Science and Technology, 380p [74] B Wang (1999), Attractors for reaction-diffusion equation in unbounded domains Phys D 128, no 1, 4152 [75] C J Xu (1992), Regularity for quasilinear second-order subelliptic equations Comm Pure Appl Math 45, no 1, 7796 123 ... trỡnh elliptic suy bin phi tuyn, phng trỡnh parabolic suy 10 bin, v phng trỡnh hyperbolic suy bin Cỏc kt qu ó t c l: s tn ti v khụng tn ti nghim ca bi toỏn biờn cho phng trỡnh elliptic suy bin,... trỡnh, h phng trỡnh khụng elliptic v phng trỡnh hyperbolic tt dn suy bin ó xut hin Cú mt s lp phng trỡnh, ú cú lp phng trỡnh elliptic suy bin v phng trỡnh hyperbolic tt dn suy bin, mt khớa cnh no... biờn i vi phng 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