Attractors for Parabolic Problems with Nonlinear Boundary Conditions

Transcription

1 JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 27, ARTICLE NO AY Conens Aracors for Parabolic Probles wih Nonlinear Boundary Condiions Alexandre N Carvalho Deparaeno de Maeaica do ICMSC, Uniersidade de Sao Paulo, 1356, Sao Carlos-SP, Brazil Sergio M Oliva Deparaeno de Maeaica Aplicada do IME, Uniersidade de Sao Paulo, Sao Paulo-SP, Brazil Anonio ˆ L Pereira Deparaeno de Maeaica do IME, Uniersidade de Sao Paulo, Sao Paulo-SP, Brazil and Anıbal Rodriguez-Bernal Deparaeno de Maeaica Aplicada, Uniersidad Copluense de Madrid, Madrid 284, Spain Subied by Jack K ale Received June 21, Inroducion and saeen of he resuls 2 Noaions and background resuls 3 Local well posedness 31 The case 1,, N 3, and growh condiions 32 The case s,, s 1, N 3, and non-criical growh condiions 33 The fracional power spaces case and no growh condiions 4 Exisence of global aracors 41 The fracional power spaces case 42 The energy space case 421 Gradien syses in he criical growh case 422 Non-gradien syses in he non-criical growh case Research parially suppored by CNPq, Process Parially suppored by projecs DGICYT, PB9-235, Spain, and EEC Gran SC1-CT X97 $25 Copyrigh 1997 by Acadeic Press All righs of reproducion in any for reserved

2 41 CARVALO ET AL 5 Alernaie dissipaieness condiions 51 Conracing recangles 52 Conracing piecewise sooh convex regions 6 Appendix 61 The firs eigenvalue of differenial operaors 62 La Salle s invariance principle 1997 Acadeic Press 1 INTRODUCTION AND STATEMENT OF TE RESULTS Le be a bounded sooh doain of N In his paper we sudy he long ie behavior of soluions of weakly coupled reacion diffusion syses wih dispersion of he for N u u Div a x u Ý Bj x uf u, on, x j j1 u g u, on, n a 11 where u u,, u, 1, ax diaga x,,a x 1 1, ai C 1, a x, x, 1i,un ² au,n,nis : i a he 1 ouward noral o, B diag b,,b j j j is coninuous on, 1jN,, and f f 1,, f :, g g 1,, g : Noe ha he coupling appears boh on he reacion er, f, and in he boundary condiions hrough he nonlinear radiaion er g The saring poin for he sudy of he long ie behavior of he soluions of 11 is having a good funcional fraework in which consrucing he soluions In his direcion one needs o have uniqueness and regulariy of soluions o define a well-behaved non-linear seigroup in a suiable Banach space Then, one ries o prove ha he dynaical syse is dissipaive and possesses a global aracor ha capures he asypoic behavior Therefore, one firs needs o choose in which space o work In aking such a decision here are several alernaive ways o proceed A firs approach is based on he fac ha for he scalar case 1 and wihou dispersion ers B j here is a naural energy associaed o he soluions, given by V a F G 2 2 where F and G are priiives of f and g, respecively which is dissipaed as ie increases Therefore, one ay hink of using L 2 as an underlying space and he properies of he second order ellipic operaor in his

3 ATTRACTORS FOR PARABOLIC PROBLEMS 411 space o prove he well posedness of 11 Once his is done, he use of he space 1 urns ou naurally in view of he energy V The proble hen becoes ha one us ipose soe resricions in he growh of he non-linear ers in order o ensure ha he inegrals above are finie These resricions becoe ore sringen as he diension N increases The resricions can be soewha relaxed by working in suiable sup-spaces of Sobolev spaces s, appearing as he fracional power spaces associaed o he second order ellipic operaor In soe ranges for N and s, no growh assupions are needed on he non-lineariies Alernaively, one ay assue ha here are invarian regions, as in 27, 38, and hen work wih bounded funcions u, again geing rid of growh assupions on f and g Anoher approach is using he properies of he ellipic operaor aking L p for p N as he underlying space, see 13 The reason for his is ha, in his case, he fracional power spaces becoe ebedded in space of coninuous funcions and hen no growh assupions are needed in f and g in order o have he non-linear seigroup well defined Once his firs sep is copleed saisfacorily he proble becoes proving he dissipaiveness of he flow induced by he equaions in such a space One has o obain esiaes on he soluion for he nor of he space in which he dynaical syse is se, ha allows one o prove he exisence of aracors, for exaple, by using soe of he general resuls in 16 A his poin i is coon ha he case wih he naural energy V described above does no give any inforaion abou he dissipaiveness in he righ nor We adop here an inerediae approach We sick o he L 2 seing, which possesses soe echnical advanages and ease, and hen consider eiher he case in which one works in he energy space, 1, wih he naural growh resricions on he non-lineariies, or work in a differen space, which is essenially s C, for soe s, and, since he funcions are bounded, again no growh assupions are needed In he firs case, we use he naural energy V o prove he exisence of he aracor, while in he second we have o use soe conracing regions o obain dissipaiveness in he proper nor In order o have dissipaiveness soe rearks are needed For his, we inerpre u as a hea disribuion in he body ence, we assue for he oen ha u In his siuaion, noe ha for ranges in which f is posiive we have absorpion of hea, while when f is negaive we have sources of hea The sae holds for g; when g is posiive we have a flow of hea hrough he boundary of ha exracs hea fro he body, while in he opposie case, hea is flowing inside When one of he ers is absen, soe resuls are available Firs, if g, hen i is well known ha if f has he wrong sign, soluions ay

4 412 CARVALO ET AL develop explosions in finie ie 12, 24, while for he good sign dissipaiveness follows 16, 4, 8 Also, noe ha soluions of he ode uuf u are consan in space soluions of 11, if g, and ha soluions of his equaion ay blow-up in finie ie Siilarly, when f, if g has he wrong sign, explosions ay occur in finie ie 25, 4, 26, while in he opposie case boundedness of soluions follows 4, 41 When boh f and g are non-zero we expec o have soe kind of copeiion beween boh echaniss and only if he righ sign is doinan can we expec o obain he desired dissipaiveness Therefore, in wha follows we will ipose soe growh andor sign assupions on he non-linear ers under which we will prove boh he well posedness and dissipaiveness of he seigroup We will resric ourselves o he case N 3, bu i will be clear fro he proofs ha everyhing works he sae for any diensions Only he growh condiions we ipose below are affeced by he value of N To be ore precise, we assue he following Concerning growh condiions, we assue ha f, g: are C 1 and C 2 funcions, respecively, saisfying: N 2 and for every, here exiss c such ha u 2 2 f u f c e e u, 12 for every u,,orn3 and 2 2 f u f L 1 u u 13 for all u, ; and N 2 and for every, here exiss c such ha 2 2 u g u g c e e u, u 2 2 Dg u Dg c e e u, 14 for every u,,orn3 and 1 1 g u g L 1 u u, Dg u Dg L 1 u u, 15 for all u, and soe 1 Observe ha, no growh assupions are ade for N 1 Also, observe ha hese assupions are saisfied in he scalar case, 1, if he following condiions hold f s li 2 s s e,, if N 2, f s 2 L 1s,if N3 16

5 ATTRACTORS FOR PARABOLIC PROBLEMS 413 and g s li 2,, if N 2, s s e 17 g s L 1s, 1, if N 3 Also noe ha a funcion saisfying 14 or 15 also saisfies 12 or 13, respecively ence, he condiions on he boundary non-linear er are sronger han he condiions on he reacion er Concerning sign assupions, we assue he following Le f f 1,, f :, g g 1,, g : be sooh funcions saisfying he following There exiss 1,, such ha i and here are consans c and d wih i i sifi s gi s c i, and d i, for all si i 18 s s i i uniforly in s, j i, where c and d are such ha he firs eigenvalue j i i of he proble 1 Div a B x c, on n a N Ý j j1 x j 19 d, on, for c c,,c and d d,,d 1 1, is posiive Observe ha eiher ci or di can be negaive This allows for one of he non-linear ers o have he wrong sign for large values of is arguen In such a case, he above hypohesis iplies ha he wrong er is copensaed by he oher; see Secion 6 Also, observe ha his hypohesis, which can be seen as a principle of linearized dissipaiveness a infiniy, jus allows for a sub-linear behavior of he non-lineariies in he region of he bad sign, excluding hen he cases of polynoial non-lineariies having he bad sign Finally, noe ha hese assupions have been previously used by enry 22, working for N 1 and 1, where he proves he syse is Morse-Sale See also 3

6 414 CARVALO ET AL Before we proceed any furher le us inroduce soe noaion Le XL 2, and A: D A XX be he operaor A diaga 1, A defined by D A D A D A wih 1 2 D A :, i ½ n ai 5 N i Ai Div ai x Ý bj x, x j where n a ², n : a i, i 1,, i We can define he fracional powers Ai and A of Ai and A, respecively, see 21, and he fracional power spaces X DA i i and X D A endowed wih he graph nor,, where X X, if In his case we can always view A as a secorial operaor wih copac resolven fro X 1 ino X which is posiive and self adjoin if B, for j 1,,N In fac, fro he diagonal srucure of A, we have j X X1 X These spaces provide a naural fraework for solving 11, see 21 I is worh noing a his poin ha for every i 1,,, Xi is a closed 2 subspace of he Sobolev space, for and in paricular 12 X 1 i, ie, he energy space Le us now review soe of he known resuls using his fraework For he case of no dispersion B j, linear boundary condiions, g, scalar equaions 1, and N 3, he exisence of a global aracor, A, in he energy space 1, for proble 11, under condiions 13 and he dissipaiveness assupion j1 f u u li inf u u is a well known resul 16 Also, if g, 1, and N 2, he exisence of a global aracor for 11 in he energy space 1, under he sae dissipaiveness assupion, has been esablished in 7, assuing he non-lineariy saisfies u f u ce, for soe 2 Therefore, condiion 12 slighly relaxes his assupion, for N 2, while 18 and 19 give a generalized dissipaiveness condiion Working ouside he energy space, ie, in X for 12 and for he case g in 15, he exisence of a local aracor for 11 is proved, which coincides wih he ebedding of he aracor for u u f u

7 ATTRACTORS FOR PARABOLIC PROBLEMS 415 ino he subspace of consan funcions of X, 34, under he addiional assupion ha he diffusion coefficien ax is large enough See also 18, 19, 6 for he case of linear boundary condiions, ie, guc x u, wih c x diagc x,,c x 1 owever, hese echniques can only be exended o global aracors if soe a priori bound, in X, on he size of an absorbing se could be obained and only if he diffusion coefficien is large see 7, 5 These a priori bounds are obained in 8, 4, always for he case g and Bj, also working in X, for 34 and N 3 and assuing f verifies ha here exiss 1,, such ha i and sifi s, for all s s i In 4 he auhor uses a echnique of conracing recangles o obain he desired esiaes in X and he exisence of aracors Noe ha in all hese aricles he auhors work in X for 1 34 and N 3 and ipose no growh condiions in he non-lineariies The reason for his is he ebedding X C, owever, for 34, he space X incorporaes he boundary condiion u n 13 a, and herefore, i canno be he righ space o work in when g owever, for N 1 and orforn2 and 34 12, we have X C, and X does no incorporae any boundary condiion Therefore, we work in his range of, for N 1, 2 and in his case again no growh condiion is iposed on f and g On he oher hand, as soon as N 3, X is never included in C,, when Therefore, o avoid growh condiions, in he case N 3, we will work in Y X C, and prove he well posedness of he proble in his space i i 2 NOTATIONS AND BACKGROUND RESULTS Concerning funcional spaces, we will use he sandard Sobolev spaces 1 1, p 12 11p, and W and he spaces of races and W p Also, we will denoe by s he dual space of s, eiher on or Noe ha his sybol is usually reserved o denoe he dual space of s owever, his noaion should produce no confusion The dualiy pairing beween hese spaces will be denoed ², : s, s In paricular, he scalar 2 produc in L will be denoed by ², : If here is no possible confusion, we will no indicae if he spaces or dualiy producs are referred o funcions on or When required, we will wrie ², : and ², : o differeniae boh cases The sybol will always represen he nor in L 2

8 416 CARVALO ET AL We will denoe by he race operaor defined on s, wih values s12 in, for s 12 Moreover, for a given funcion f s,we s12 will idenify is race, f, wih he linear for f 12 1, such ha for every 1 ² : def f, ² f,: f, 1,1 def s ha is, we use he ebedding L 1, Even ore, fro he race heore, he race operaor : W q 11 q, W q is bounded 14 We will also consider he noral derivaive operaor, relaive o he diffusion operaor Divaxu, defined as follows: if 4 def 1 2 uz z,div a x z L 12 hen u n and i is defined as a ; u, Div a x u au na 12,12 Div au 21 for every 1 Under hese condiions, and assuing, we inroduce he canoni- 1 1 cal isoeric isoorphis beween and is dual,, such ha for every u, 1 ² L u,: au u 22 1,1 Now ha we can hen wrie 21 as u, ; ² L u,: 1,1 ² Div a x u u, : n a 12, Also, we consider in he scalar produc ² : a u, au u L u, 24 1,1 which gives a nor equivalen o he usual one

9 ATTRACTORS FOR PARABOLIC PROBLEMS 417 In L 2 we define A : D A L 2 L 2 as he operaor defined by 5 u 2 D A ½u :, on n a 25 AuDiv au u, u D A, wih Then, we have he following well known resul PROPOSITION 21 The operaor A defined aboe is posiie, self adjoin, and has copac resolen in L 2 In paricular, i is a secorial operaor in 2 2 L and is fracional power spaces erify X, for and in paricular X 1 D A, X 12 1, X L 2, X 12 1, 1 def 1 where we hae se The resricion of L o L 2, ie, he resricion of L o he doain Du 1, such ha L u L 2 4, coincides wih A and D 1 D A Moreoer, L is a secorial operaor in wih doain 1 Noe ha all he above reains rue, excep for A o be posiive, 12 if Also, when, he nor in X 1 is given by au 2 u 2, which is precisely he nor in 1 given by he bilinear for 24 By sandard perurbaion resuls 21, Theore 148, we have PROPOSITION 22 The operaor, A, in L 2 gien by D A D A and N u Au AuÝ bj x x j is a secorial operaor wih copac resolen and he sae fracional power spaces as hose of A In paricular A can be exended in a unique way o an 1 1 operaor L fro o is dual, gien by he bilinear for j1 N u ² L u,: 1,1 auý bj x u 26 x j1 j 1 1 for eery u, Moreoer, L is a secorial operaor in wih doain 1

10 418 CARVALO ET AL Le us now define X L 2, and le A: D A XX be he operaor A diag A,, A defined by D A D A D A 1 1 wih 2 D A :, i ½ n ai 5 N i Ai Div ai x Ý bj x, x j where n a ², n : a i, i 1,, i Then A is a secorial operaor in X and fro he diagonal srucure of A, we have X X1 X In paricular, A generaes an analyic seigroup on X such ha i saisfies he esiaes j1 A e u Me u,, X X A e u Me u,, X X 27 for soe, M 1 In paricular, if Bj, j 1,, N, in order o have, can be aken as any posiive nuber On he oher hand, if dispersion is presen, has o be aken large enough for he seigroup o decay exponenially Fro he previous resuls, we can exend A o he operaor L diag L 1,, L, beween X, and is dual, X, which is also secorial The analyic seigroup generaed by L in X 12, e L,is A he unique exension of e o his space and is such ha 27 also holds As shown in 35, 36, for solving probles wih non-hoogeneous boundary condiions, i is naural o consider a special class of eleens 1 h defined as ² h,: 1,1 ² f,: ² g, : for every, where f L and g So, for shor, def h f g Throughou he paper and especially in he proofs, we will denoe by c i generic posiive consans, whose values are irrelevan for he resuls 3 LOCAL WELL POSEDNESS In his secion we consider he local well posedness of probles 11 To ha end, we will ake soe hypoheses on he non-lineariies f and g ha will allow us o ake use of general absrac resuls for parabolic evoluion equaions, described below

11 ATTRACTORS FOR PARABOLIC PROBLEMS 419 Assue A is a secorial operaor in a ilber space X We can define he fracional powers A of A, and he fracional power spaces X D A, endowed wih he graph nor,, where X X, for Even ore, A is secorial in X wih doain X 1, for any, see 2, 3, 21, 34 TEOREM 31 Wih he aboe noaions, assue h: X X is locally Lipschiz and bounded on bounded ses, where 1 Then, he absrac parabolic proble u Au h u u ux 31 has a unique locally defined soluion, gien by he Variaion of Consans Forula A As u e u e h u s ds, where e A denoes he analyic seigroup generaed by A Moreoer, u erifies 1 u C,T, X C,T, X, u C,T, X for eery 1 and he equaion is erified in X Een ore, eiher he soluion is defined for all or i blows up, in X nor, in finie ie We will apply his resul o he operaor L inroduced in he previous secion and for his, we consider non-linear appings of he for h u f u g u, 2 12 where, a leas, f : X L, and g : X,, for soe Noe ha h acs on es funcions r,, for r12, as ² hu, : ² f u,: ² g u, : Depending on he exra regulariy properies of g, o be ade precise below, we will ge h: X X for suiably chosen and Bu firs, we will show soe naural a priori requireens on he exponens and Recall ha for he absrac resul we need 1 On he one hand, since we wan o give accoun of non-hoogeneous ers on he boundary, ie, we consider he case g, ha iplies Oherwise, we can always ake Since, fro he resuls in 1 35, we are ineresed in reading he equaion in,, hen we

12 42 CARVALO ET AL need 12 Also, as shown below, for obaining energy esiaes on he soluion we are ineresed in having enough regulariy o have u, u 1,, for, and for his one needs, according o he soohing effec in Theore 31, 1 12 and hen 12 On he oher hand, if we wan he non-linear er gu o depend on he values of u on, hen we us have 14 in order o have he race of u well defined; oherwise we can ake In case we wan o have iniial daa a leas in 1,, we us require 12 Finally, noe ha in he case of non-zero ers on he boundary, here is anoher naural upper bound for and In fac, 34, since for 34 he space X incorporaes he boundary condiion u n a Also, fro he soohing resul in Theore 31, we us also have 1 34, ie, 14 Suarizing, if g, hen 34 14, 14 12, and 1 while if g, we have he sandard case, 1 Then we have TEOREM 32 Assue f, g,, and are as aboe and h: X X is locally Lipschiz and bounded on bounded ses Then, for eery u X, here exiss a unique, locally defined soluion of gien by which erifies u L u h u, u u 32 L Ls u e u e h u s ds 1 u C,T, X C,T, X, u C,T, X for eery 1 and N u u au Ý Bj x u x j1 j f u ² g u, : 33

13 ATTRACTORS FOR PARABOLIC PROBLEMS for eery, In paricular, we hae N u u Div a x u Ý Bj x uf u, on x j j1 34 u g u, n a on and eiher he soluion is defined for all or i blows up, in X nor, in finie ie In paricular, assue f : X L 2, g : X L 2, or een r, wih r 12 are locally Lipschiz non-linear funcions, for r2 34 Then here exiss such ha erifying all he aboe Proof Fro Theore 31, we have he exisence and regulariy pars of he saeen Since 12, he equaion u L u h u 1 holds as an equaliy in X and also in, Bu fro he soohing effec in Theore 31 and since 1 12, for, u 1, and u L 2, and hen fro he previous equaion we ge 33, by aking a es funcion 1, Furherore, we read he equaion as a diagonal ellipic syse L u u f u g u and, fro ellipic regulariy heory, we ge 34 Noe ha if g akes values in L 2, hen h f g is well s s2 defined acing on es funcions in, X for every s 12, s 2 herefore h X and we can ake 12 s214 Fi- r nally, if g akes values in,, wih r 12, hen h is well r12 r214 defined acing on es funcions in, X and herefore h X r 214 and we can ake 12 r21414 In order o apply Theore 31, we need r2 14 1, which leads o he condiion r2 34 Observe ha if he assupions of Theore 32 are verified, hen his resul allows one o define a non-linear seigroup S in X, such ha Su u, he unique soluion of 11 The nex wo subsecions are devoed o showing ha hese assupions are verified for a class of non-lineariies in soe X space

14 422 CARVALO ET AL 31 The Case 1,, N 3, and Growh Condiions Assue now ha f, g: are C 1 and C 2 funcions, respecively, saisfying 12 or 13 and 14 or 15, respecively We denoe by f and g he coposiion Neisky operaors defined by f and g, for funcions defined on, while we denoe by g he Neisky operaor defined by g for funcions defined on Observe ha if he races of u and g u are defined hen g u g u We will show below ha, under he growh assupions 12 15, he aps f and g are such ha h f g verifies he assupions of Theore 32 above in 1, For he case N 2, we will ake use of he following resul due o N S Trudinger 39, 29 LEMMA 31 u 1 There exis wo posiie consans and K such ha if 1 hen 2 u e 2 L K 35 Furherore, he consan is bounded aboe by2 Wih his resul we obain he following LEMMA 32 Assue f erifies 12, 13, hen he apping f : 1, L p, 1 is Lipschiz coninuous and bounded on bounded subses of,, for p, if N 1, for any 1 p, if N 2 and for p 2 if N 3 Proof The case N 1 is obvious, since for, in 1, such ha 1 r, we ge, f f c r c r 1 L, R 1 2, where we used ha he derivaive of f is bounded on bounded ses and he ebedding 1, L, We now prove he case N 2 Le r and, be funcions in 1, such ha 1 r and 1, R, R r Choose such ha for p 1, 2 pr 2, wih as in Lea 31 Then, fro p p u2 12, here exiss c such ha f u f c e e 2 p

15 ATTRACTORS FOR PARABOLIC PROBLEMS 423 p u, for every u, and hen p p L, R f f 2 2 p p x x p c e e x x dx ž / 2 2 2p x x 1ž / / 2p p x x 2p ž c e e dx x x dx 12 p 1, R c e e dx, 1 2p where we have used he ebedding, L, Bu, fro 2pu Lea 31 and he choice of, and verify e 2 2 L K, and his concludes he case N 2, since he inegral er above is bounded by a consan depending only on K The case N 3 is uch sipler and well known bu he proof is given for copleeness Fro 13, we have f u f 2 L 2 1u u for all u, and hen if r, 1, R r, 1 r, we have, R L, R L 1 ž / ž / f f L 1 2 c r 1 2, R, where we have used older s inequaliy and he ebedding 1, 6 L, Assue now g verifies 14, 15 As noed above, hen g also verifies 12 and 13, respecively, and hen g verifies Lea 32 owever, he funcion g has beer properies As observed above, if he race of g u is defined, we have g u g u Now, we have LEMMA 33 If g erifies 14, 15, he ap g : 1, W 1, q, is Lipschiz coninuous and bounded on bounded subses of 1, for any 1 q 2, if N 1, for any 1 q 2, if N 2 and for any 1 q 6 4, if N 3 12

16 424 CARVALO ET AL Proof The case N 1 follows as he case N 2 below, wih q 2 1 1, and p If N2, we firs show ha g :, W q, is a bounded ap, for any 1 q 2 Since 14 iplies ha Dg also verifies Lea 32, hen if u 1 r, we ge gu q,r L,R c q 2 q 1 r, q, Dg u L, R c2 r, q, and Dg u Dg L, R 2 c r, qu 1 3, R for any 1 q Observe ha now if 1 q 2 and p is chosen such ha 1q 1p 12, Dg u u Dg u u q 2 p 2 L, R L, R L, R rc2 r, p c4 r, q 1 1, Therefore, g is a bounded ap fro, ino W q,, 1 q 2 On he oher hand, if again we choose p such ha 1q 1p 12, we have q L, R Dg u u Dg Dg u u Dg u Dg Dg u q q L, R L, R 2 p u 2 L, R L, R 2 p 2 L, R L, R Dg u Dg c r, p u 2 rc r, p u 1 2 L, R 3, R 1 c r, p u 5, R Tha concludes he case N 2 The case N 3 is proved in he following way Fro he previous lea, g is a bounded ap fro 1, ino L q,, for any 1q2 Now we prove ha if u 1 r, hen Dg u u q, R L, R is bounded by a consan depending only on r, for soe q 1, 2 For his, noe ha q q L, R q q Dg u u Dg u u Mu q ž / ž / 2q 2 q2 1 2q 2q 2 L 1u u and since 1, L 6,, we have o ake p 12q 2q 6, ie, q 6 4 for he righ hand side o be bounded

17 ATTRACTORS FOR PARABOLIC PROBLEMS 425 For he lipchizness, we have fro L 61, R 2 Dg u Dg L 1u u 61 ž / ž / L 1u u c r u 61 1, 36 6, R where we have used older s inequaliy and, for he las inequaliy, he ebedding 1, L 6, Wih his 64 L, R Dg u u Dg Dg u u 64 L, R 64 L, R Dg u Dg Dg u 2 61 u 2 L, R L, R L, R L, R Dg u Dg c r u 2 c r u 1, 7 L, R 8, R where we have used he las inequaliy follows fro esiae 36 Then, we have LEMMA 34 If g erifies 14, 15, hen 1 12 i If n 1, g :,, is Lipschiz coninuous on bounded ses ii If N 2, g : 1, r, is Lipschiz coninuous on bounded ses for r 12 1 r iii If N 3 and 1, g :,, is Lipschiz coninuous on bounded ses, for 12 r 2 1 1, Proof Since, fro he lea above, g :, W q, is Lipschiz on bounded ses, for soe q, and using he race heore, i 1, q 11 q, q r suffices o show ha : W, W,, is bounded, for soe r and such q i The proof is obvious, since we can ake q 2 11q, ii If N 2 hen q 1, 2, and W q, r, if 32 2q r ence aking he larges possible value of q, we ge r 12

18 426 CARVALO ET AL 11q, iii If N 3, hen q q 6 4 and W q, r, if 2 3q r Taking again he larges possible value of q, we ge r 2 and we can ake r 12 iff p 1 Therefore, we ge TEOREM 33 Assuing he growh condiions and 1 if N3, 11 defines a local seigroup in 1, Proof According o Leas 32 and 34, h f g is locally Lip- 12 r schiz and bounded on bounded ses of X ino,, wih r 12 Therefore, Theore 32 applies wih The Case s,, s 1, N 3, and Non-criical Growh Condiions In his secion we will give a local exisence resul for iniial values in s,, for soe s 1, N 3, assuing he non-lineariies grow a lile slower han in Again no assupions are ade for N 1 This local exisence resul, apar fro is inrinsic ineres, will be of grea help when proving he exisence of aracors for 11 in Secion 4 More precisely, le f, g: be C 1 and C 2 funcions, respecively, such ha 18 holds Assue also ha f and g saisfy, insead of 12 15, f u f L 1 u u 37 for all u, and arbirary, if N 2 and 2if N3, and 1 1 g u g L 1 u u, Dg u Dg L 1 u u, 38 for all u, and soe 1, for N 2, 3 The following resuls are refineens of Leas 32 and 33 LEMMA 35 If f erifies 37, he ap f : s, L p, is Lipschiz coninuous on bounded subses for p if N 1 and 1 s 12, for any 1 p if 1 s 1 2p 1 and N 2, and for any 1p6 1 if 5, 1 s 3212p 1, and N 3 Proof The case N 1 is as in Lea 32, since s, L, for 1 s 12

19 ATTRACTORS FOR PARABOLIC PROBLEMS 427 Le R and u, s,, s 1, be such ha u s, R R and s R Then, R p p L, R f u f p p p L 1 u x x u x x dx p p p c 1u pq pq u pq, 1 L, R L, R L, R where 1q 1q 1 Therefore, if q can be chosen such ha pq and p q are such ha we can apply Sobolev inequaliies, we ge p p L, R f u f p p p c 1u s s u s 2 p c R u s 3, R, R, R, R and he resul is proved For N 2, fro Sobolev inclusions, s, L p, wih coninuous inclusion for any p such ha s 1 2p, ie, if 1p 1s 2 Therefore, he nuber q can be chosen as above iff s 1 2p 1 which is copaible wih s 1 for any p and Finally, for N 3, fro Sobolev inclusions, 2, L p, wih coninuous inclusion for any p such ha s 32 3p, ie, if 1p 3 2 s 6 and he condiion on q reduces o s 32 3p 1, which is copaible wih s 1 iff 1 p 6 1 and 5 Reark 31 Noe ha for he proof above, i suffices 5if N3, which is a weaker resricion on f han 37 Then, we can ake p 2if 1s 32 1 and 2 as in 37 LEMMA 36 If g erifies 38, he ap g : s, W 1, q, is Lipschiz coninuous and bounded on bounded subses of s, for 1q2 32s if N1 and 1 s 12, for 1 q 2 1 s 2 1 if N 2 and 1 s 1 2, and for 1 q 6 32s 2 2 if 1 s and N 3 Proof Noe ha fro he previous lea, g : s, L q, is Lipschiz coninuous on bounded subses for q if N 1 and 1 s 12, for any 1 q if 1 s 1 2q 2 and N 2, and for any 1 q 6 2 if 1s 3212q 2 and

20 428 CARVALO ET AL s N3 Now, if u R we have, R Dg u u Dg Dg u u q q L, R L, R q L, R 1q q1 q Dg u Dg 5ž / c 1u u 1q q q q 5ž / c 1 u u We deal wih each of he inegrals in he righ hand side of he above expression separaely For he firs er, we ge 6 L, R L, R q1 q q1 q1p q 1 u u c 1 u u pq, wih 1p 1p 1, while for he second er we apply older s inequaliy wih hree ers o ge 1u q q u q q q q q 7 L, R L, R c 1 u q q pq u qr L, R L, R wih 1p 1r 1 1 Therefore, if p in he firs case and p, r, in he second case can be chosen such ha for he exponens pq, q 1 p and q, pq, and qr, respecively, we can use he Sobolev inequaliies hen we ge and 8, R, R q1 q q1 q 1 u u c 1u s u s, 1u q q u q q q q q s, R s, R s, R s, R c 1u u 9 and we ge he resul As in Lea 35, s, L p,, for p if 1 s 12 and N 1, for any p such ha 1p 1 s 2 if N2, and for any p s 1, such ha 1p 3 2 s 6 if N3 Also,, W p,

21 ATTRACTORS FOR PARABOLIC PROBLEMS 429 for any p such ha 1p 3 2 s 2 if N1, for any p such ha 1p 2s 2if N2, and for any p such ha 1p 5 2 s 6 if N3 For N 1 we can ake in he esiaes above p 1 and r and p1, respecively, assued ha q s, which is copaible wih q 1 since 1 s 12 For N 2 i is easy o check ha p and p, r, can be chosen verifying all he above iff q 2 1 s 2 1 which is copaible wih q1 iff 1 s Analogously, for N 3, p and p, r, can be chosen verifying all he above iff q s 2 2 which is copaible wih q 1 iff s which in urn is copaible wih s 1 since 2 Then, we have, analogously o Lea 34 LEMMA 37 If g erifies 38, hen g : s, r, is Lipschiz coninuous on bounded ses, if i N1, 1 s 12, and 12 r 1 s 2 ii N 2, 1 s 1 2, and 12 r 1 s 2 12 iii N 3, 1 s , and 12 r 3 2 s s 1, Proof Since, fro he lea above, g :, W q, is Lipschiz on bounded ses, for soe q, and using he race heore, i 1, q 11 q, q r suffices o show ha : W, W,, 11 q, q r is bounded, for soe r and such q Also, W,, if 1 1q r and N 1, 32 2q r and N 2 and if 2 3q r if N 3 i According o he previous lea, if N 1 he ranges for s and q are 1 q q s and 1 s 12 Therefore, aking he larges value of q in he ebedding above, we ge 1 s 2 r 12 ii For N 2 he ranges are 1 q q 2 1 s 2 1 and 1 s 1 2 Proceeding as before we ge 1 s 2 12r, which is less han 12 iff 1 s 1 2 iii For N 3 he ranges are 1 q q s 2 2 and 1 s In his case we ge 3 2 s 2 2 2r, which is less han 12 iff 1 s

22 43 CARVALO ET AL Fro all his, we ge TEOREM 34 Assuing he growh condiions 37 38, Theore 32 applies wih s2 proided i N1 and 1 s 12 ii N 2 and 1 s ax 1, iii N 3 and 1 s 32 ax 1, Therefore, 11 defines a local seigroup in s, for s in he ranges aboe Proof Noe ha he resricions on s deerined by iply, according o Lea 35, ha f aps s, ino L 2, s2 Since X s,, fro he resuls above and he proof of Theore 32, we ge ha he non-linear er aps X s2 ino X for r214 Therefore, o apply he general resuls in Theore 32, we need s2 1, ie, s r 32 Fro he esiaes on r given in Lea 37 his condiion can be e if respecively i N1 and any 1 s 12 ii N 2 and s 1 which holds rue, since fro Lea 37, 1 s iii N 3 and s which holds rue, since fro Lea 37, 1 s The Fracional Power Spaces Case and No Growh Condiions Noe ha he basic resul Theore 32 iposes soe condiions on he non-lineariies, naely ha h f g aps X ino X, and his, in urn, iposes soe growh condiions on he appings f, g: Therefore, in order o obain an exisence resul wihou any growh condiion on he non-lineariies we will work in Y X C, for 34 14, endowed wih he nor u u u Y X Observe ha Theore 32 is, in principle, no direcly applicable in his conex owever, noe ha hen, if N 1 and 14 orif N2 and 3412 hen, fro he Sobolev inclusions, X C, and

23 ATTRACTORS FOR PARABOLIC PROBLEMS 431 hen Y X and Theore 32 can be applied In fac, we have LEMMA 38 Assue ha f: and g: are C 1 and C 2 funcions, respeciely If N 1 and or if N 2 and 34 12, hen and 1 f : X C,, g : X,, g : X 12, are Lipschiz coninuous on bounded ses of X Proof Le u, X such ha u R and X X R ence, we have f u f c R u c R u and he sae for g Moreover, 1 2 Dg u u Dg c R u uc R u 3 4 c R u 5 X 1 and hen g : X, is Lipschiz on bounded ses of X Using he race operaor, we ge he resul Reark 32 Noe ha if g is only of class C 1, hen he sae proof above for f, bu only wih poins in he boundary, gives he Lipschizness of g : X L, and Theore 32 can sill be applied In his case soluions will be less regular As a consequence of he above and Theore 32, we ge TEOREM 35 Wih he aboe assupions, 11 defines a local seigroup in X, if N 1 and or if N 2 and Now we coe back o he cases no covered by he previous resul, ie, N 2 and orn3 and Assuing f: and g: are C 1 and C 2 funcions, respecively, and in view of Theore 32, we will look for soluions of 11 which are funcions uc, T, Y, ha are soluions of he fixed poin proble wih h f L Ls u e u e h u s ds, g Firs noe ha his equaion is equivalen o A As Ls u e u e f u s ds e g u s ds 2 as soon as u, f u s L, X

24 432 CARVALO ET AL To prove he exisence resul we will ake use of he following resul ha has been proven in 36 PROPOSITION 31 Le X be a Banach space and A, D A a secorial operaor in X and consider proble ½ u u u Au g 39 p wih g L, T, X, u X, and 1 p, T Then, he soluion erifies a uc, T, X for all 1p Moreoer, if u X, hen uc, T, X b The apping p X L,T, X u, g u C,T, X is Lipschiz Moreoer, if Re A hen ha holds also for T In ha case u C,, X b Wih his, we can sae TEOREM 36 Wih he aboe assupions 11 defines a local seigroup in Y Moreoer, soluions are defined for all or hey blow-up, in he Y nor, in finie ie Proof Define, for u Y and u C, T, Y A As Ls Fu e u e f u s ds e g u s ds Firs, noe ha f u C, T, C, and g u C, T, C, and hen, in paricular, hu C, T, X, for any 12, 14 Therefore, fro Proposiion 31, we ge F u C,T, X C,T, X, for every 34 A Now, observe ha since u X C,, fro he fac ha e defines an analyic seigroup in X 21 and in C, 28, we ge A e u C, T, Y and i is a classical soluion of he linear hoogeneous proble boh in he equaion and in he boundary condiions As Now, we define F u e f u s ds and F u 1 2 Ls e g u s ds Then, since f u C, T, C,, he resuls in 28, cobined wih Proposiion 31, give us F u 1 2 C, T, C, for every 1

25 ATTRACTORS FOR PARABOLIC PROBLEMS 433 Finally, fro a classical resul by Pogorzelski 31, see also 1, since, 2 g u C, T, C,, we have F u C, T, 2, for every, 1 and oreover F u C sup g u, y 2,2, y 2 In paricular, F u C, T, C, Puing his inforaion ogeher we ge, in paricular, F u C, T, Y Therefore, i is legiiae o look for fixed poins of F in C, T, Y For his, we will prove ha for a given u Y, F is a conracion in a suiable closed subse of C, T, Y, for sall enough T Once his is done, classical coninuaion arguens conclude he proof More precisely, for a given u Y, we denoe V u C, T, Y, u u r,, T 4 Y, where r and T are o be chosen below Firs we prove ha F aps V ino iself For his, noe ha for a given r, T can be chosen sall enough such ha e u u r3 A coninuous seigroup in boh spaces Also, fro 21, and aking any 12, 14 such ha 1, A for all, T, for he nors of boh X and C,, since e is a ž / Ls 1 X e h u s ds C s ds c r since hus akes values in a bounded subse of X Consequenly, he expression above can be ade saller han r3, for all, T,ifT is sall enough On he oher hand, arguing now in C, As 1 F u e f u s ds As, 2 e f u s dsc r and again, his can be ade saller han r3, for all, T,ifT is, 2 sall enough Finally, since F u C, T, 2, for every, 1 and F u, x, we ge, for every x and, T, 2 F2 u, x 2 F2 u 2,2 C sup g u, y, y 2 c3 r,

26 434 CARVALO ET AL and again, his can be ade saller han r3, uniforly in x and, T,ifT is sall enough Consequenly, F V V, ift is sall enough Now, we prove ha F is a conracion in V For his, le u, V Then, F u F X C s h u s h s ds ž / 4,T C s ds c r sup u s s Therefore, for sall enough T, we ge for all, T F u F X 14 sup u s s Y,T On he oher hand, noe ha F u F F u F F u F and hen 2 As, F u F e f u s f s ds 1 1 c r sup u s s,t 5 18 sup u s s Y,T for sall enough T Finally, as before, for sall enough T F2 u F2 2 C sup g u, y g, y, y 2 c 6 r, sup u s s,t 18 sup u s s Y,T Puing all hese ogeher, we ge F u F V 12u V and he resul is proved Concerning regulariy, we have COROLLARY 31 erifies Wih he aboe noaions, for eery u Y, he soluion u C,T, Y C,T, X, u C,T, X, for any 34 and 33 and 34 hold also rue, while he soluion exiss

27 ATTRACTORS FOR PARABOLIC PROBLEMS 435 Proof Taking any 12, 14 such ha 1, ie, any 1, 14, as in 21, Lea 332, we prove u X is locally older Wih his, we ge hu X is also locally older, since, for sall h, f u h, x f u, x c1 u h, x u, x, for all x, g u h, x g u, x c2 u h, x u, x, for all x Inegraing he firs inequaliy in we ge f u h f u 2 c u h u, L, R X while inegraing he second one in, we ge g u h g u 2 c u h u 2 L, R L, R 2 1 c u h u X Now, 21, Theore 322 iplies ha u is a srong soluion in X Bu even ore, 21, Theore 352iplies u C, T, X, for any 1 Since is arbirary in 1, 14, we ge he resul 3 4 EXISTENCE OF GLOBAL ATTRACTORS In his secion we prove ha under he dissipaiveness assupion 18, 19, he syse 11 has a global aracor Noe ha fro he diagonal srucure of 19, i reduces o solving scalar eigenvalue probles Fro he resuls in 32, 33, 23, we have ha he firs eigenvalue of his proble is always real, where by firs we ean ha all ohers have greaer real par, see Secion 6 Also, he firs eigenvalue, 1, is given by he infiu of he firs eigenvalues of each scalar eigenvalue proble 41 The Fracional Power Spaces Case In his secion we firs prove ha soluions of 11 wih iniial daa in YX C, are globally defined and orbis of bounded subses of Y, under he seigroup deerined by 11, S, are also bounded in Y Moreover, we will show he seigroup is copac Second, we will show ha he seigroup is dissipaive in Y and for his, we will find conracing regions, siilar o he conracing recangles considered in 4 Wih hese and he resuls in 16, we will obain he exisence of a copac aracor in Y

28 436 CARVALO ET AL I is worh noing ha he whole idea for proving boundedness and dissipaiveness is ha esiaes on he sup nor are ransferred o esiaes in he Y nor, as shown below We firs sar wih he following iporan reark LEMMA 41 Wih he assupions of Secion 3 and in paricular wih he noaions of Theores 32, 35, and 36, le u be a local soluion of 11 in YX C, defined in a axial ie ineral, ax If u C, R c, for all, 1 ax, for soe c1, hen, ax and u X c 2, for all and soe c 2 Een ore, if a bounded se B Y is such ha S B, 4 is bounded in C,, hen S B, 4 is bounded in Y and S B, 14 is copac in Y Proof Firs noe ha by replacing A and f by A ai and f ai, we can always assue ha 27 holds wih Then, using he variaion of consans forula, we obain, for u u, s X u Mu e M e s h u s ds, for soe Therefore, for finie ax, we have s u X Mu e MK e s ds, 41 wih K sup hu 4, which is finie since u, C,R c a x 1 Therefore, since he nor in X, and hence he nor in Y, reains bounded in finie ie, hen he soluions us exis for Moreover, he righ hand side of 41 reains uniforly bounded in ie and he resul is proved Analogously, if u B, hen he righ hand side of 41 reains uniforly bounded in ie and u B and consequenly, SB, 4 is bounded in Y Even ore, fro he fac ha L has copac resolven, SB, 4is bounded in Y, and he variaion of consan forula, we 4 ge as in 16, Theore 422 ha SB,1 is copac in X On he oher hand, since, wih he noaions in Theore 36, S S S 1 2, wih A As 1 S u e u e f u s ds, Ls 2 S u e g u s ds and A has copac resolven in C,, again using 16, Theore 422, we ge S B,14 is copac in C, Finally, as in Theo- 1

29 ATTRACTORS FOR PARABOLIC PROBLEMS 437, 2 2 re 36 and using 31, 1, we have S u C,,, for every, 1 and S u C, B 2,2 and he righ hand side is unifor in and u B In paricular S 2 B, 4 1 is in a bounded se of C, and herefore i is copac in C, ence, SB,1 4is copac in Y To obain esiaes in C, for he soluion of 11, in wha follows we will use coparison resuls and he noion of invarian regions We sar by defining sub- and super-soluions for 11 super-soluion of 11 if, for 1 i, i saisfies 2 N DEFINITION 41 A C funcion u:, T is a N u i i u Div a u b x u x j Ý i i i j i j1 fi u 1,,u i1,u i,u i1,,u, on, u i gi u 1,,u i1,u i,u i1,,u, on n a ui u i, 42 where u u,,u is he unique soluion of 11 1 wih iniial value u Siilarly, we define a sub-soluion u by replacing he sign by he sign in 42 We hen have LEMMA 42 If u is he soluion of 11 wih iniial alue u and if u and u are super- and sub-soluions of 11 in he sense defined aboe, hen ui, x ui, x ui, x for eery 1 i, while he soluions exis Proof For each i 1,,, le fˆ, x, f i i u 1,, u i1,, u,,u and g, x, g u,,u,,u,,u i1 ˆi i 1 i1 i1 Therefore, ui and ui are super- and sub-soluions of N i Div a b x fˆ i Ý j i, x,, on, x j j1 g, x,, on n ai ˆi u i

30 438 CARVALO ET AL Fro he resuls in 3, we ge ha here exiss a soluion of he proble above, such ha u, x, x u, x i i Bu he unique soluion of his proble is ui and he resul is proved Assue now f, g saisfy 18 and 19 Le i and i be, respec- ively, he firs posiive eigenvalue and noralized eigenfuncion of each coponen of 19 and in x i x i For each,,, define 1 4 u Y : u x x, for all x 43 i i i Below we prove ha is an invarian aracing region LEMMA 43 Le be as in 18 Then, wih he noaion aboe, i le ii i and for eery, denoe i e, i i for, and,, 1 i Then, for any iniial daa in Y, such ha u x e x, i i i for all x, are respeciely a sub- and super-soluion of 11 for, and consequenly S e e i as, where e e i i In paricular, for eery Mi i, define ˆ i M sup 1 log M, and M e i i i i i i M i, hen for, S S M Mˆ Me ˆ as In paricular, is a posiiely inarian region for 11 i Proof Firs ake e Then, for,, x i i i i i e and consequenly, fro 18 i i i N i i i i i Ý j i i j1 x j Div a b x c N i i i i i Ý j i j1 x j Div a b x fi u 1,,u i1, i, u i1,,u i i di i gi u 1,,u i1, i,u i1,,u n n a a Thus, is a super-soluion The sae applies for wih all he inequaliies reversed The res is obvious

31 ATTRACTORS FOR PARABOLIC PROBLEMS 439 As a consequence, we have COROLLARY 41 All soluions of 11 wih iniial daa in Y exis for, and he seigroup S is well defined on Y for Moreoer, for eery bounded se B Y, SB, 4 is bounded in Y and S B, 14 is copac in Y Een ore, if is as in Lea 43, hen is an absorbing se for S ˆ Proof Clearly, for any bounded se B Y, here exiss ˆ such ha B ˆ and we can assue ha ii i Then, Lea 43 iplies ha SBS as B ˆ Therefore, fro Lea 41 we ge he resul Noe ha is no a bounded se in Y and herefore dissipaiveness does no follow fro he corollary owever, we have TEOREM 41 The proble 11 has a global aracor A in Y X C Furherore, A 44 for eery, such ha ii i, for i 1,, Proof Since we already have ha orbis of bounded subses of Y under S are bounded in Y and ha S is a copac seigroup, i jus reains o prove poin dissipaiveness o prove he exisence of a global aracor A for S; 4in Y, see 16, Theore 346 In fac we show below ha S is bounded dissipaive For his, noe ha fro he Corollary above, for any bounded se B Y here exiss such ha SB, for all Fro he variaion of consans forula, where as in Lea 41 we can assue wihou loss of generaliy ha 27 holds wih, we have for any u B and, S u Me L X 12 s MP e s ds, 45 where Ssu L, for every s and u B, and P Y sup f s g s 4, wih s, s M 4 s i i, wih M sup x i i Noe ha and he righ hand side above are indepen- den of u B, and hen leing we obain 12 s li sup S u MP e s ds, 46 X and since he righ hand side of 46 does no depend on u B Y, S is bounded dissipaive in Y Finally, since is absorbing, we ge A

32 44 CARVALO ET AL Wih his resul, he nex one follows fro 16 and fro he fac ha a consan equilibriu u for 11 ay only happen if u f u gu COROLLARY 42 The ellipic proble u Div au B x uf u, on x j N Ý j j1 u g u, n a on, has a leas one soluion which is non-riial wheneer I f and g hae no coon zeros 42 The Energy Space Case 421 Gradien Syses in he Criical Growh Case Now we consider 11 wihou dispersion B, ha is, u Div au u f u, on, u 47 g u, on, n a wih iniial values in 1,, where f and g saisfy and he dissipaiveness assupions 18, 19 Noe ha now he eigenvalue proble 19 reduces o Div a c, in n a j d, on, 48 and fro he diagonal and self adjoin srucure of his proble we have ha he firs eigenvalue, 1, is given by he infiu of he firs eigenvalues of each scalar eigenvalue proble, ie, he infiu of a c d i i i i inf 2 1 which are assued o be posiive for i 1,,

33 ATTRACTORS FOR PARABOLIC PROBLEMS 441 Noe ha he dissipaiveness assupion is e if fi s si li inf s i s i 49 gi s li inf si and for each i 1,, one of he inequaliies is sric Moreover, we assue ha here are scalar poenials F: and G: such ha F s f s and G sg s, for every s,a condiion ha is always saisfied for 1 Consider he energy funcional V: 1, defined by Ý i i 2 i1 2 V a F G 41 The nex resul ensures ha V is well defined LEMMA 44 Assue f: and F: are such ha F s f s i If f saisfies 12 or 13 hen for eery 1,, one has F L 1, and F 1c r, if 1 L r for soe coninuous increasing funcion c r ii If f saisfies 14 or 15 hen for eery 1,, one has F L 1, and F 1c r, if 1 L r for soe coninuous increasing funcion c r 1 Proof Since f is a gradien, we have F s f rs sdr, for all s and hen F s 1 f rss dr s If 12 holds rue, hen for every, f s c e 2 s f and s2 s herefore F s c 2 e f s Since s e 2 we ge F s c2f e s2 and we conclude using Lea 31, as in Lea 32 If 14 holds rue hen as before and as in Leas 33 and 34, we ge F 1 1, 1 L c F W cr If 13 holds rue, hen f s c1s 3 and herefore F sc1s 4 and again we ge he resul as in Lea 32 Finally, if 15 holds rue, hen proceeding as before, we ge F sc1s p for soe p 4 and, a he sae ie, if 1 12,, hen he race is in, L q,, for q 4 and we ge he resul s i

34 442 CARVALO ET AL TEOREM 42 Under he aboe assupions, we hae i V is a Liapuno funcion for 47 ii All soluions of 47 wih iniial daa in 1, are globally defined iii The proble 47 has a global aracor A in 1, Furherore 47 is a gradien syse and herefore, A W u E, where E is he se of equilibria of 47, ie, E 1, : L h 4 u and W E denoes he unsable se of E Moreoer, if each eleen of E is hyperbolic, hen E is finie and E A W u, where W u denoes he unsable anifold of he equilibriu poin Proof Fro he soohing effec in Theore 32, we have u 1, for, and hen aking u as a es funcion in 33, we ge d 2 V u u d for any soluion u of 47 and herefore V decreases along soluions of 47 Observe ha fro he dissipaiveness assupion, we have c d F s s s c, G s s c i 2 i 2 Ý i 1 Ý i 1 i1 i1 for all s and soe consan c Therefore, 1 and hen 1 c d V Ý a Ý Ý c i 2 i 2 i i i i 2 i1 i1 i i i i i 2 4 i1 4 i1 Ý Ý V a c This inequaliy iplies ha soluions are globally defined, since Vu Vu and hen he 1, nor of he soluion reains bounded since i

35 ATTRACTORS FOR PARABOLIC PROBLEMS 443 Fro Lea 44 we ge V C r, 1 if, R r, where C : is a coninuous increasing funcion Therefore, orbis of bounded ses are bounded in 1, Also, for each, S is copac, since A has copac resolven 16 Since V decreases along rajecories, fro La Salle s invariance principle, he -lii se of any rajecory lies inside he se 1,, 4 1 V, bu his is he se of equilibria of 47, ie, E, : Lh 4 Therefore, E aracs uniary ses in 1, under he seigroup S, 4 Nex, we prove ha E is bounded in 1, and hen he seigroup S, 4is poin dissipaive 16 Since he equilibriu poins of 47 saisfy Lu hu, ie, Div au u f u, on aking u as a es funcion in 411, we ge u g u, on 411 n a 2 2 Ý i i i1 a u u f u u g u u Fro he dissipaive condiion we ge f s ss 2 Ý c 2s 2 i1 i i 2 c3 and g s sýi1 di2 si c3 for all s and soe consan c 3 Therefore, hus c 2 i 2 i 2 Ý a u Ý u Ý u i i i i c4 i1 i1 2 i1 2 Ý i1 2 2 a u u i i Ý i i 2c4 i1 and hen he se E of equilibria is bounded in 1, The res follows fro he resuls in 16 Our nex goal is proving ha he aracor consruced above is a bounded se in C, and oreover ha he esiaes 43, 44, of d

36 444 CARVALO ET AL he previous secion sill hold rue For his, we firs prove he following lea LEMMA 45 Assue N 3 and g erifies 15, wih 1 Then, if u, wih 32 1 erifies 2 Div au u F L, u g u na on, s hen u,, where s in2, A B 4, wih A 2, B 31 2 Moreoer, u s 2 C F L, u In paricular u C Proof Noe ha we can resric ourselves o he case 1, since we only use he soohness and growh assupions on g Also, noe ha fro ellipic regulariy resuls, since F L 2 we ge u 2 loc, bu he regulariy of gu on deerines he overall regulariy of u on Therefore, we prove ha gu has a cerain degree of regulariy on and hen use he ellipic regulariy heory o ge he resul Noe ha fro 15 and he Sobolev inclusion for we ge ha Dg u L p for p On he oher hand, fro 1 he Sobolev inclusions for we ge u L q, 3 for q 65 2 Fro his we ge Dg u u L r, 3 for 1r 1p 1q and his leads o r r Also, fro 15 and he Sobolev inclusion for we ge gu p 1, r L for p Therefore, we ge guw 11r, r Now, fro he race heore, g u W s for s s 23r Fro ellipic regulariy heory, we ge u wih s in2, 72 3r 4 and we ge he resul Now, noe ha s C,if s32 ence, if j A B, wih A 2, B is greaer han 32 and he conclusion s n follows If no, we repea he previous arguen o ge u for n Bu noe ha he funcion j is onoonic and has a unique fixed poin a f 1, since 1 Therefore, here exis n such ha s n 32 for any 32 1 and he conclusion follows PROPOSITION 41 Le u be a soluion of 47, wih iniial daa in 1, Then u C,, C,

37 ATTRACTORS FOR PARABOLIC PROBLEMS 445 Moreoer, if N 1, 2 or if N 3 and u, 4 is bounded in L 2, for, hen 4 sup u C, R is bounded aboe by a consan depending on he bound on u in 1, and he bound on u in L 2,, if N 3 Proof Noe ha wihou loss of generaliy, we can assue ha 27 holds rue wih Fro Theore 42, u is coninuous and uniforly bounded in 1, for ence, if N 1, here is nohing 1 o prove since, C, If N2, we ge fro Theore 1 32, u C,, X and oreover, fro he variaion of consans forula, for any 1 12 u X M e u 12 X s Me s h u s X ds and he righ hand side is uniforly bounded for Bu since 12, can be chosen such ha and hen we ge he resul if N 2, since X C, Finally, if N 3, as seen in Lea 34 and Theore 33, in he arguen above we can ake r214 for 12 r 2 and hen we ge he bound in X for 3 4 and hen in, for Also, fro Theore 32, u C,, L 2, and reading he equaion as Div au u u f u u g u na and working on finie ie inervals, we ge, fro Lea 45, u C,, C, On he oher hand, if u, 4 is bounded in 2 L, for, again Lea 45 gives he resul We now prove he following absrac resul PROPOSITION 42 Assue A is a secorial operaor wih copac resol- en and assue he proble u Au h u, where h: X X is Lipschiz on bounded ses, wih 1, defines a globally defined seigroup, S, in X ha has a global aracor A

38 446 CARVALO ET AL Then A is a bounded subse of X for any 1 and oreoer, here exis a consan c, only depending on he bound of A in X, such ha for any soluion u lying on he aracor, one has for any 1 u c, for any 412 X Proof Noe ha A is a bounded subse of X and is invarian, ie, S A Afor all Le u, wih, be a soluion lying on he aracor Fro he variaion of consans forula, for any 1we s ge u X M e u X Me s hus X ds Taking 1, he righ hand side above is bounded by a consan independen of u and hen S 1 A A is a bounded se of X Noe ha if 412 is proved, hen we can ge he bound on he aracor for he case 1, since we have Au u hu and he righ hand side is bounded in X The proof of 412 is based on he proof of 21, Theore 352, Lea 351 and he invariance of he aracor Take u, wih, a soluion lying on he aracor and consider he inerval, 1, 1 for Proceeding as in 21, Theore 352, using he bound on X, 1, and ha he non-lineariy is Lipschiz on bounded ses, we ge ha js hus is bounded in X and saisfies, for s s h 1, j sh j s X hc1 s s 1 c sr j rh j r 2 X dr and fro here using Gronwall s lea as in 21, p 6, one ges jsh js X chs 1 hk s 3 where c 1, c 2, c3 are independen of u and Noe ha 1 K s ds is bounded by an absolue consan, and hen, 21, Lea 351 gives ha on,, for any 1 1 s 4 5 u s c s c sr K r dr Taking s 1 and changing variables r z, and using 1 1 one ges u c 6, for soe absolue consan c 6, for all and u on he aracor Therefore, we ge 1 TEOREM 43 The aracor of 47 in, is a bounded se in C, Furherore, for any u A one has u x x, for all x, i 1,, i i i wih, x as in 43 and 44