The Dynamics of a One-Dimensional Parabolic Problem versus the Dynamics of Its Discretization

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1 Joural of Differeial Equaios 168, 6792 (2000) doi: jdeq , available olie a hp: o The Dyamics of a Oe-Dimesioal Parabolic Problem versus he Dyamics of Is Discreizaio Simoe M. Bruschi 1 Deparameo de Maema ica, IGCE-UNESP, Cx. Posal 178, Rio Claro, SP, Brazil sbruschims.rc.uesp.br Alexadre N. Carvalho 2 Deparameo de Maema ica, ICMC-USP, Cx. Posal 668, Sa~ o Carlos, SP, Brazil adcarvaicmc.sc.usp.br ad Jose G. Ruas-Filho 3 Deparameo de Maema ica, ICMC-USP, Cx. Posal 668, Sa~ o Carlos, SP, Brazil jgrfilhoicmc.sc.usp.br Received November 2, 1998; revised May 24, 1999 dedicaed o professor jack hale o he occasio of his 70h birhday I his paper we prove ha he spaial discreizaio of a oe dimesioal sysem of parabolic equaios, wih suiably small sep size, coais exacly he same asympoic dyamics as he coiuous problem Academic Press Key Words: parabolic equaios; spaial discreizaio; opological equivalece of aracors. INTRODUCTION I order o keep he preseaio simple we will cosider he scalar case ad laer poi ou ha he chages eeded whe cosiderig sysems of oe dimesioal parabolic equaios. Cosider he followig oe dimesioal scalar parabolic problem u =au xx + f(u), 0<x<1, >0 u x (0)=u x (1)=0, >0, (0.1) 1 Research parially suppored by FAPESP Gra Research parially suppored by CNPq Gra Research parially suppored by CNPq Gra Copyrigh 2000 by Academic Press All righs of reproducio i ay form reserved.

2 68 BRUSCHI, CARVALHO, AND RUAS-FILHO where a>0 ad f: R R is a C 2 fucio saisfyig he dissipaiveess codiio f(u) u<0, u >!, for some!>0. Also, cosider he semi-implici discreizaio of (0.1) wih p equally spaced seps U4 =alu+f(u), (0.2) where L is he p_p 2_ marix give by }}} }}} }}} L= p b b b.. b b b, (0.3) }}} }}} }}} f(u)=( f(u 1 ),..., f(u p )) ad U=(u 1,}}},u p ) Uder he above assumpios o f we have he exisece of a global aracor A for (0.1) ad a global aracor A p for (0.2). The aim of his work is o show ha he asympoic dyamics of he wo equaios above are opologically equivale for a sufficiely large p; ha is, for sufficiely small sep size. I order o illusrae he differeces ha may arise bewee he dyamics of (0.1) ad (0.2) we cosider he case p=2 i (0.2); ha is if we wrie, x 1 =14, x 2 =34 ad deoe by u 1 ()=u(x 1, ) ad u 2 ()=u(x 2, ), he we have (already wih he boudary codiios icorporaed) he followig equaio: u* 1=4a(u 1 u 2 )+ f(u 1 ), u* 2=4a(u 1 u 2 )+ f(u 2 ). (0.4) Take f(u)=uu 3. We observe ha for ay value of a he equaio (0.4) has a mos ie equilibrium pois whereas he problem (0.1) for small values of a may have ay umber of equilibrium pois (see, [CI]). Besides his, for 4a<13 we have he exisece of equilibrium pois for (0.4) which are sable ad of he form U=(u 1, u 2 ) where u 1 {u 2. If he dyamics of (0.4) were equivale o he dyamics of (0.1) he equilibrium poi U would correspod o a sable, ocosa equilibrium poi for (0.1); ha is, a paer. I is well kow (see [Ch, CH]) ha paers do o exis

3 A ONE-DIMENSIONAL PARABOLIC PROBLEM 69 for he problem (0.1). Tha way eve for values of a o so small he dyamics of he discreized equaio may differ sigificaly from ha of he coiuous problem. This has bee poied ou previously i [Ro]. Of course a similar reasoig could be carried ou for larger values of p he advaage of p=2 is he possibiliy of compuig all he equilibrium pois of (0.4) which gives a complee picure of is aracor. I has bee show i ([CP]) ha, for ay p give, here is a fucio a( } ) such ha he dyamics of (0.2) is equivale o he dyamics of he problem u =(a(x) u x ) x + f(u), 0<x<1, >0 u x (0)=u x (1)=0, >0 (0.5) Afer havig preseed he problems ha may arise whe comparig he dyamics of (0.1) ad (0.2) we are ready o sae he mai resul of his paper. Theorem 0.1. For p large eough, here is a homeomorphism H: A A p which maps orbis oo orbis preservig ime direcio. The proof of his resul requires us o embed he discree problem io he seig of he coiuous problem. Sice he coiuous problem is ifiie dimesioal he firs ask is o reduce i o a problem o a fiie dimesioal space. Tha is accomplished hrough he ivaria maifold heorem. Uforuaely if we cosider he coiuous problem o a fixed fiie dimesioal ivaria maifold of dimesio ad cosider he discreizaio wih sepsize 1, we are o able o prove ha he vecor fields of he coiuous ad discree problem wih same dimesio are close (due o he fac ha he eigevalues of he L ad of he 1-d Neuma Laplacia are o uiformly close). Keepig he coiuous problem o a fixed maifold, he proximiy of he vecor fields (o he par ha cocers L ad he projeced oe dimesioal Neuma Laplacia) will come whe he sep size is very small ad herefore he dimesio of he discree problem will ow exceed he dimesio for he coiuous problem. We could ow projec he discree problem oo a ivaria maifold wih same dimesio as ha of he coiuous problem. Tha akes care of he covergece of he par of he vecor field comig from he projecio of he Laplacia ad he projecio of L bu he we eed o sudy he covergece of he olieariies projeced o he ivaria maifolds. For ha we eed he covergece of he ivaria maifolds which leads o echical complicaios. Our approach is o allow he dimesios of boh ivaria maifolds o icrease i such a way ha he ivaria maifolds are boh very fla ad

4 70 BRUSCHI, CARVALHO, AND RUAS-FILHO herefore very close o oe aoher i he C 1 opology. The sice we already kow ha he aracors are all coaided i submaifolds of fixed dimesio we have ha vecor fields are also C 1 close i hese submaifolds ad he opological equivalece is a cosequece of he srucural sabliliy for he coiuous problem. Le us ow cosider he slighly more geeral siuaio (0.5) wih a beig a sricly posiive C 1 ((0, 1), R) C([0, 1], R) fucio. Wih a chage of variables (0.5) ca be covered io u s =u!! +a~ (!) f(u), 0<!<L, s>0 u! (0)=u! (1)=0, s>0, (0.6) where s=(a(x)) 1,!= x 0 1 a(s) ds ad a~ (!)=a(x(!)), L= a(s) ds. This leads us o sudy oly he case (0.1) possibly wih f also depedig o he space variable. All he resuls proved here are for he case whe he olieariy f depeds oly upo he ukow u. More geeral siuaios like he case whe he fucio f also deped o he space variable ad o he dispersio ca be obaied i a similar fashio. The assumpios required for hese more geeral siuaios ad he Dirichle boudary codiio case ca be foud i Secio 5. There has bee several works i he lieraure where par of he resuls preseed here have bee aouced. Amog hem we cie [Ha, FR]. To our kowledgeme here is o rigorous proof of such resuls i he lieraure. This paper is orgaized as follows. I Secio 1 we prese he discreized problem obaiig uiform bouds for he aracors A p ad he exisece of a expoeially aracig ivaria maifold for i. I Secio 2 prese he coiuous problem obaiig uiform bouds for he aracor A ad he exisece of a expoeially aracig ivaria maifold for i. I Secio 3 i is proved ha a cerai se of eigevalues ad eigefucios of he discree problem coverges uiformly o he eigevalues ad eigefucios of he coiuous problem ad use hese facs o compare he vecor fields of he discree ad coiuous problem o he ivaria maifolds. I Secio 5 we make several commes o possible exesios of he resuls. Fially, i he Appedix, we prove a heorem o exisece of expoeially aracig ivaria maifolds ha deals wih chagig spaces ad dimesios.

5 A ONE-DIMENSIONAL PARABOLIC PROBLEM DISCRETIZATION Firsly we discuss he spaial discreizaio of (0.1), for ha cosider he pois x j = j12 p, j=1,..., p ad deoe u j ()=u(x j, ). The, we have u* 1= p 2 (u 2 u 1 )+ f(u 1 ), u* j= p 2 (u j1 2u j +u j+1 )+ f(u j ), j=2,..., p1 (1.1) u* p= p 2 (u p1 u p )+ f(u p ) Observe ha he boudary codiios have chaged o u 1 =u 0, u p+1 =u p ad have bee icorporaed o he liear operaor L. Deoig U=(u 1,..., u p ) ad rewriig he above equaio i a marix form, we obai U4 =LU+f(U), (1.2) where L is a p_p marix give by (0.2) ad f(u)=( f(u 1 ),..., f(u p )). We observe ha he sysem (1.2) is geerically MorseSmale (see [FO]). By he codiios imposed o f, he above problem has a global aracor A p ha saisfies A p /R p!, (1.3) where R p! =[v # R p ; v i!, 1i p], see [CDR]. Sice we are ieresed o sudyig he soluios of he above problem i he aracor oly we may cu he olieariy i such a way ha f is bouded wih bouded firs ad secod derivaives. Theorem 1.1. The eigevalues of L are give by * p k =4p2 si 2 k? 2p ad he associaed eigevecors are w p =(cos k?x k 1,..., cos k?x p ) for k=0,..., p1. Besides ha, lim p * p=* k k, where * k =(k?) 2 is he (k+1)h eigevecor of he operaor 2 wih homogeeous Neuma boudary codiio. To be able o compare he dyamics of he discree problem wih he dyamics of he coiuous problem we mus assig o R p a orm which is compaible wih he orm adoped for he coiuous problem. Tha leads us o defie i R p he ier produc: (x, y)= 1 p x p i=1 i y i, which is iheried from he L 2 ier produc ad will be referred as discreized L 2 ier produc. Normalizig w p k accordig o his ier produc we obai: p = w p k?x k 1,..., cos k?x p ) k w p =(cos. k 1 p p : cos 2 k?x i i=1

6 72 BRUSCHI, CARVALHO, AND RUAS-FILHO If we wrie p : cos k?x i / Ii (x) p(x)= i=1 k ; 1 p p : cos 2 k?x i i=1 where we deoe by I i he ierval [ i1 i p, p], we obai ha k(x)# p L (0, 1) ad p (x)- 2 cos k?x k 0 whe p. Cosider a base of eigevecors p k, 0k p1, i R p. This basis is orhoormal wih respec o he ier produc previously described. We cosider he discreized equaio i his ew coordiaes, ha is, if we wrie: v 1 =(U, p ),..., v 0 p=(u, p ) ad v=(v p1 1,..., v p ) we obai: v* =L v+f(v), (1.4) where L is he p_p marix give by L =diag(* p,..., * p 0 p1 ) ad F(v)= (F 1 (v),..., F p (v)) wih each F j (v) give by F j (v)=(f(u), p j1 )= : p k=1 1 p p j1 k f( p 0k v 1+}}}+ p ( p1) k v p), (1.5) where p deoes he k-h coordiae of p jk j. We deoe he marix of chage of basis by Z; i is give by z kj = p ad he marix ( j1) k Z1 is give by (1p) Z. Now, we cosider a discreizaio wih p= 3 pois ad cosider he followig decomposiio of R 3 =R R 3 where R =spa[ 3 0,..., ] 3 1 ad R 3 =spa[ 3,..., 3 1], wih his decomposiio we obai he 3 followig weakly coupled sysem v* +B v = g~ (v, w ) w* +A w = f (v, w ), (1.6) where B is he _ diagoal marix give by B =diag(* 3 0,..., ), A *3 1 is he ( 3 )_( 3 ) diagoal marix give by A =diag(* 3,..., * 3 1), 3 g~ (v, w )=(F 1 (v, w ),..., F (v, w )) ad f (v, w )=(F (+1) (v, w ),..., F 3(v, w )). For he weakly coupled sysem (1.8) we show ha here exiss a expoeially aracig -dimesioal ivaria maifold; ha is, he followig holds:

7 A ONE-DIMENSIONAL PARABOLIC PROBLEM 73 Theorem 1.2. Le f be wice coiuously differeiable, bouded wih bouded firs ad secod derivaives; he, he problem (1.6) for sufficiely large, possess a ivaria maifold M =[(v, w )#R 3 w =_~ (v )], which is expoeially aracig, where _~ is a smooh fucio, _~ : R R 3 ad he flux o M is give by u()=v ()+_~ (v ()) where v () is soluio of v* +B v = g~ (v, _~ (v )). (1.7) To prove his heorem we use he followig resul. This resul is also used o prove he proximiy of he vecor fields of he coiuous ad discree problems afer hey are projeced o heir ivaria maifolds. Lemma 1.3. Le ad X, Y be a sequece Baach spaces, A : D(A )/ X X be a sequece of secorial operaors ad B : D(B )/Y Y be a sequece of geeraors of C 0 -groups of bouded liear operaors. Suppose ha f : X : _Y : X ad g : X : _Y : Y are a sequece of fucios saisfyig: f (x, y)f (z, w) X L f (xz X :+yw Y :), f (x, y) X N f, for every (x, y), (z, w) i X : _Y : ad g (x, y)g (z, w) Y L g (xz X :+yw Y :), g (x, y) Y N g, for every (x, y), (z, w) i X : _Y :. Assume ha e A w X :M a e ;() w X :, 0 e A w X :M a : e ;() w X, >0, e B z Y :=e B () z Y :M b e \() z Y :, e B z Y :M b () : e \() z Y, <0, 0, for ay w # X : ad z # Y, where ;()\() + as. Cosider he weakly coupled sysem { x* =A x+ f (x, y), y* =B y+ g (x, y). (1.8)

8 74 BRUSCHI, CARVALHO, AND RUAS-FILHO The, for large eough, here is a expoeially aracig ivaria maifold for (1.8) where _ : Y : X : saisfies S=[(x, y) :x=_ ( y), y # Y : ], s()= sup _ ( y) X :, [y# Y : ] _ ( y)_ (z) X :l() yz Y :, wih s(), l() 0 whe. If f, g are smooh; he, _ is smooh ad is derivaive D_ saisfy sup D_ ( y) L(Y :, X )l(). : y # Y The proof of his resul ca be foud i he appedix. Proof of Theorem 1.2. Makig :=0 i he previous lemma we have: Y _X where Y =R ad X =R 3, g~ : Y _X Y ad f : Y _ X X. We make he followig disicio relaively o he several orms used here, whe he idex of he orm is X or Y we are usig he base of eigevecors ad ha way he orm is give by }=( i x 2 i )12, whe he idex of he orm is R k we are usig he caoical basis ad he orm give by he L 2 discreized ier produc. Firsly we compue he eeded esimaes o f ad g~ g~ (v, w ) Y = \ : i= (F i (v, w )) f(z(v, w ) ) R 3= \ 3 : = \ : 3 i=1 \ : 3 i=1 Similarly, we obai he esimae i= ( f([z(v, w 3 ) ] i )) 12 1 f 2 3 = f (f i(z(v 3, w ) )) f (v, w ) X f. (1.9)

9 A ONE-DIMENSIONAL PARABOLIC PROBLEM 75 For he Lipschiz cosas we have g~ (v, w )g~ (z, u ) Y = \ : i= (F i (v, w )F i (z, u )) f(z(v, w ) )f(z(z, u ) ) R 3 = \ 2+ 3 : 1 ( f([z(v, w 3 ) ] i )f([z(z, u ) ] i )) i=1 \ : 1 3 L2([Z(v f, w ) ] i [Z(z, u ) ] i ) i=1 = \ : 1 3 L2([Z(v f z, w u ) ] i ) i=1 =L f Z((v z ), (w u )) R 3-2 L f (v z Y +w u X ), where L f is he Lipschiz cosa of he fucio f. I he same way we obai he esimaive for he Lipschiz cosa of f. All he cosas are uiform i. The cosas ;() ad \() are: ;()=* 3 12 ad \()=* 3 1. Tha gives us ha ;() 2 ad \()(1) 2 as ad his gives us ha ;()\() as. 2. THE CONTINUOUS PROBLEM We ow ur o he problem (0.1). Le X=L 2 (0, 1), we defie f e : H 1 (0, 1)/X X by f e (,)(x)= f(,(x)) ad we defie: A: D(A)/L 2 (0, 1) L 2 (0, 1) D(A)=H 2 N (0, 1)=[, # H 2 (0, 1):,$(0)=,$(1)=0] A,=,". Tha way, we rewrie he problem (0.1) as: d d u+au= f e (u), (2.1) u(0)=u 0

10 76 BRUSCHI, CARVALHO, AND RUAS-FILHO By he codiios imposed o f we obai ha f e is Lipschiz coiuous i bouded subses of H 1 (0, 1). The, he above problem has a global aracor A ha saisfies sup u # A sup u(x)!. (2.2) x #[0,1] The above boud allow us o cu (wihou chagig he aracor) he olieariy f i such a way ha i becomes bouded ad has bouded firs ad secod derivaive. Also afer cuig he olieariy we may pose he problem i L 2 (0, 1) keepig he same aracor. Here afer we assume ha f is bouded ad has firs ad secod derivaive we also assume ha he problem is posed i L 2 (0, 1). Le * 0 <* 1 <* 2 < } } } be he sequece of eigevalues of A, where * k =(k?) 2 ad, 0,, 1,, 2,... a correspodig sequece of ormalized eigefucios,, k (x)=- 2 cos(k?x). Now cosider he followig decomposiio of X=WW = where W=spa[, 0,, 1,...,, 1 ] W = =[, # X: (,, w)=0, \w # W], (2.3) where ( }, }) is he ier produc of L 2 (0, 1). The, u # L 2 (0, 1) ca be wrie as where u=v 1, 0 +v 2, 1 +}}}+v, 1 +w, v i = 1 0 w=u : u(x), i1 (x) dx, i=1 v i, i1 i=1,..., ad Le u be a soluio of (2.1); he, for each, we ca wrie u(, x)=v 1 (), 0 (x)+v 2 (), 1 (x)+ }}} +v (), 1 (x)+w(, x) (2.4) where A deoes A D(A) W =. v* i=* i1 v i +(f(u),, i1 ) 1 w +A w= f(u) : (f(u),, i ), i, i=0

11 A ONE-DIMENSIONAL PARABOLIC PROBLEM 77 Wriig v=(v 1, v 2,..., v ), u=(v, w) ad B a _ diagoal marix B =diag(* 0, * 1,..., * 1 ) we obai he followig sysem v* +B v= g (v, w) w +A w= f (v, w), (2.5) where g (v, w)=((f(v, w),, 0 ),..., (f(v, w),, 1 )) ad f (v, w)= f(v, w) 1 i=0 (f(v, w),, i), i. Theorem 2.1. Le f # C 2 (R, R) be bouded wih bouded firs ad secod derivaives; he, for sufficiely large here exiss a expoeially aracig, smooh ivaria maifold S for (2.5). The flux o S is give by: u(, x)=(v(), _ (v())) where v is soluio of v* +B v= g (v, _ (v)) (2.6) Proof. Le L f e be Lipschiz cosa of f e ad N f e= f. Take L f =L g =L f e, N f =N g =N f e, ;()=*, \()=* 1 ad observe ha ;()\()=? 2 (2+1). The heorem follows form Lemma UNIFORM SPECTRAL CONVERGENCE To compare he asympoic dyamics of he discreized problem wih he asympoic dyamics of he coiuous problem we projec he firs o he ivaria maifold _~ ad he secod o he ivaria maifold _, oly afer ha we are able o compare heir asympoic dyamics. This is accomplished comparig he vecor fields (ow wih same fiie dimesio). To compare he vecor fields we eed o obai a way of comparig B ad B. Tha is achieved if we prove he uiform (wih respec o ) covergece of he eigevalues ad eigefucios of B o he eigevalues ad eigefucios of B as. Aoher way o compare he vecor fields would be projecig boh problems o fixed ivaria maifolds of same dimesio ad he o sudy he covergece of he vecor fields. Tha would ivolve sudyig he covergece of he ivaria maifolds ad would lead o uecessary echical complicaios. This approach has he clear propery ha he eigevalues ad eigefucios (a fixed umber) coverge uiformly. Here we exploi he fac ha for large values of he ivaria maifolds have a very small C 1 orm ad herefore we ca simply eglec hem; o he oher had oe eeds o be careful i order o guaraee he uiform covergece of eigevalues ad eigefucios. Tha is he reaso why we

12 78 BRUSCHI, CARVALHO, AND RUAS-FILHO make he cu bewee he h ad (+1)h eigevalue wih he discree problem havig 3 eigevalues. If we cosider he marix L wih p= 3 we have 3 simple eigevalues ad orhoormal eigefucios for L. Of course hese eigevalues ad eigefucios do o coverge uiformly o he firs 3 eigevalues ad eigefucios of he Neuma Laplacia as. I is also clear ha ay fiie subse of eigevalues of L coverge uiformly o he correspodig eigevalues of he Neuma Laplacia as ad i fac more is rue. The firs eigevalues ad eigefucios of he 3 _ 3 marix L will coverge uiformly o he firs eigevalues ad eigefucios of he Neuma Laplacia as. Tha is wha we prove i Subsecios 3.1 ad Uiform Covergece of Eigevalues The eigevalues of he operaors B ad B are respecively * 3 ad * k k, wih k=0,..., 1. I his case we have ha k? * 2 k k =(k?) }\si 3 2 k? 2 3 * }. Usig he power series expasio of he fucio si we obai * 3 * k k (k?) } 3!\ 2 k? o 2 \\k? 2 +}. Tha way we have ha for k, k=0,..., 1 * 3 * k k (?) } 3!\ 2? o \\ } =o \ 1 + so * 3 k * k 0 for all k=0,..., 1 uiformly, as Uiform Covergece of Eigefucios We will show ha 3 (x)- 2 cos(k?x) k = for sufficiely large ad for all k=0,..., 1. Firs we cosider cos(k?x)cos(k?x j ) for x #[ j1 j, 3 3]. For x #[ j1 j, ] we have ha 3 3 cos(k?x)cos(k?x j ) k?

13 A ONE-DIMENSIONAL PARABOLIC PROBLEM 79 For all k=0,..., 1 ad for all j=1,..., 3 we have cos(k?x)cos(k?x j )? 2 2 ad hece, 3 k (x)- 2 cos(k?x) c 2, for some cosa c. 4. COMPARISON OF THE VECTOR FIELDS Now we show he proximiy of he vecor fields. Deoe by g~ i ad g i he ih coordiae fucio of g ~ ad g respecively. The, we have: g j (v, 0)g~ j (v, 0) = } 3 : k= f ( j1) k \ : 3 (l1) k l+ v l=1 f \ : l=1 v l - 2 cos(l1)?x + - 2cos(j1)?x dx } } 3 : k3 ( 3 f k=1 (k1) 3 ( j1) k \ : 3 (l1) k l+ v l=1 f \ : l=1 v l - 2cos(l1)?x + - 2cos(j1)?x + dx } } 3 : k3 f \ : 3 k=1 (k1) 3 (l1) k l+ v (3 l=1 + } 3 : k3 3\ k=1 k1 f \ : l=1 3 : k=1 k3 (k1) : k3 k=1 (k1) 3 f \ : f l=1 f \ : l=1 3 (l1) k v l+ ( j1) k - 2 cos( j1)?x) dx } v l - 2 cos(l1)?x cos(( j1)?x) dx } } f \ : l=1 } f \ : 3 v (l1) k l+} (3 ( j1) k - 2 cos( j1)?x) dx l=1 3 (l1) k v l+ v l - 2 cos(l1)?x - 2 cos(( j1)?x) dx +} c 2+L c - 2 f : 2 l=1 v l.

14 80 BRUSCHI, CARVALHO, AND RUAS-FILHO Therefore, g (v, _ (v))g~ (v, _~ (v)) Y g (v, _ (v))g (v, 0) Y +g (v, 0)g~ (v, 0) Y +g~ (v, 0)g~ (v, _~ (v)) Y L f _ (v)+l f _~ (v)+ 1 - \ f c +L f!c - 2 +, where! is as i (1.3). Sice, by Lemma 1.3 _ 0ad_~ 0as he g (v, _ (v))g~ (v, _~ (v)) 0 as. Similarly, usig he fac ha f $ is globally Lipschiz, D_ 0 ad D_~ 0 as, we show ha he fucios g (v, _ (v)) ad g~ (v, _~ (v)) are C 1 close. So, we have he followig heorem: Theorem 4.1. Le f # C 2 (R, R) be a bouded fucio wih firs ad secod derivaives. Assume ha he flow o A is srucurally sable. The, for large eough he flow of (2.5) o he aracor A ad he flow of (1.6) o A are opologically equivale. Proof. We firs oe ha, from [He2], we have ha (0.1) is geerically MorseSmale ad herefore, our assumpio o srucural sabiliy is o a srog resricio o he class of maps f uder cosideraio. So, we have ha (0.1) is A-srucurally sable. We also have ha he vecor field of (2.5) is a C 1 small perurbaio of he (1.6) ad he heorem is proved. 5. FURTHER COMMENTS Though we have chose o prese he resuls i he simples formulaio hey ca be exeded o much more geeral siuaios. The proofs ca be easily adaped for he case whe f also depeds upo he space variable x. Aoher simple exesio is ha for which f depeds o x, u ad u x. The laer ca be doe if he fucio f(x, u, u x ) is a locally Lipschiz fucio ha saisfies f(x, u, 0) u<0, u >!>0

15 A ONE-DIMENSIONAL PARABOLIC PROBLEM 81 ad f(x, u, p)f(x, u, q) L f pq, \x, u. I his case we use Lemma 1.3 for :=12 ad he compariso of he vecor fields eed addiioal care bu i ca all be accomplished wihou sigifica chage. We poi ou he mai differeces. Firs oe ha he coiuous problem has a global aracor A saisfyig (2.2) ad addiioally here is a cosa C such ha sup u # A sup s #[0,1] u x (s) C. Tha esures ha we may assume ha he olieariy f is globally bouded wih globally bouded parial derivaives of firs ad secod order. The discreized equaios i his case are u* 1= p 2 (u 2 u 1 )+ f(x 1, u 1,0), u* j= p 2 (u j1 2u j +u j+1 ) + f(x j, u j,(p2)(u j+1 u j1 )), j=2, }}}p1 u* p= p 2 (u p1 u p )+ f(x p, u p,0). (5.1) I is easy o check ha if p>l he each recagle of he form [', '] p wih '>! is ivaria ad he resuls of [CDR] esure ha he aracor for he discreized problem is coaied i i he recagle [!,!] p for ay p. To efficiely hadle he depedece of he olieariies o he spaial derivaive we eed o chage from he L 2 seig o he H 1 seig whe obaiig he ivaria maifolds for he coiuous ad discree problems. The coiuous problem ca be projeced o he ivaria maifold obaied from Lemma 1.3 wih :=12. The discree problem eeds more aeio. The firs remark is ha, eve hough i is a fiie dimesioal problem, i should be reaed as is ifiie dimesioal couerpar. Deoe by U=(u 1,..., u p ) ad cosider f p :(R p, ( }, }) 12 ) (R p, ( }, })) defied by f p (U)=( f(x 1, u 1,0), f(x 2, u 2,(p2)(u 3 u 1 )),..., f(x p, u p,0)), where (u, v) 12 =(u, v)+(lu, v) ad ( }, }) is he discree L 2 ier produc. The, we apply Lemma 1.10 for :=12. Afer projecig boh problems o he ivaria maifolds we use he L 2 orm o sudy heir proximiy. The discree ad coiuous spaial derivaive require ha we cosider p= 4 isead of p= 3. As for he spliig of he specrum, i is sill doe bewee he h ad (+1) h eigevalue. Afer hese addiioal cosideraios he proofs will follow as i he case wihou dispersio.

16 82 BRUSCHI, CARVALHO, AND RUAS-FILHO The case of Dirichle boudary codiio ca be reaed i a a compleely similar way. I his case he marix L has o be replaced by he marix 2_ }}} }}} }}} L D = p b b b.. b b b. (0.3) }}} }}} }}} The eigevalues ad eigefucios of L D are give by * k p =4p2 si 2 k? 2( p+1),, k = k? 2k? pk? p si, si,..., si \ p+1 p+1 p+1+, k=1,..., p (see [Sm], for example). Afer ormalizaio of he eigefucios hey ca be used o prove he resuls for he Dirichle boudary codiio case followig he Neuma case sep by sep. Fially we observe ha if we cosider a sysem of oe-dimesioal parabolic equaios of he form (0.1), he same resuls will hold as log as he sysem is srucurally sable. The dissipaiveess assumpios o he fucio f: R R ca be of he form f i (u) u i <0, u i >!, 1i. There are oher dissipaiveess assumpios ha will also work. They are eeded o guaraee ha he aracor for he sysem of parabolic equaios remai bouded i he uiform opology. Tha allow us o cu he olieariy i such a way ha i has bouded firs ad secod derivaives. For he dissipaiveess codiios above he uiform bouds o he aracors are prove i [ACR]. 6. APPENDIX This secio is devoed o he proof of Lemma 1.3. This resul is reproduced from classical ivaria maifold resuls as i [He2]. Is proof is adaped o ecompass he possibiliy ha he space (icludig space

17 A ONE-DIMENSIONAL PARABOLIC PROBLEM 83 dimesio) chages accordig o a parameer ad o rack he depedece of he ivaria maifold upo he parameer. I he case for which we apply he absrac ivaria maifold resul coaied i Lemma 1.3, he parameer is a aural umber. I meas ha we are spliig he phase space io he space geeraed by he firs eigefucios of he problem ad is orhogoal compleme. Afer projecig he hea equaio oo hese spaces we produce he pair of equaios ha appear i he saeme of he lemma. Before we ca sar he proof of he Lemma 1.3 we eed o esablish a geeralized versio of Growall's lemma. Tha requires ha we sudy he covergece of he series E ; (z)= : k=0 z ;k 1(k;+1). I is o hard o see ha E ; is a eire fucio ad followig [E] we may obai ha here is a cosa c such ha E ; (z)ce z. Lemma 6.1 [Geeralized Growall's Lemma]. Le <r,,: [, r] R + be a coiuous fucio, a: [, r] R + be a iegrable fucio, b>0 ad 0<;1. Assume ha The, wih,()a()+b r (s) ;1,(s) ds, r. (6.1),() : k=0 (B k a)() (6.2) B k a()= r (b1(;)) k (s) k;1 a(s) ds. (6.3) 1(k;) Furhermore, if a()#a=cos he we have ha,()ae ; ((b1(;)) 1; (r))ace (b1(;))1; (r), (6.4) if a()=c 0 r (s) : e \(sr) ds, \>0, he we have ha,() c 0 b [E ;((b1(;)) 1; (r))1] c 0 b [ce(b1(;))1; (r) 1] (6.5)

18 84 BRUSCHI, CARVALHO, AND RUAS-FILHO ad fially, if : [, r] R + is a coiuous fucio ad a()= c 0 r (s): e \s (s) ds, \>0 he we have ha,()c 0 c1(;) r (s) ;1 e \s e (b1(;))1; (s) (s) ds. (6.6) The proof of his resul ca be easily adaped from similar resuls coaied i [He2]. Le X ad Y be Baach spaces, A: D(A)/X X be a secorial operaor such ha Re _(A)>0 ad B: D(B)/Y Y be he geeraor of a C 0 -group of bouded liear operaors [S(), 0] o Y. Le [T() 0] be he aaliic semigroup of bouded liear operaors geeraed by A ad deoe by (A) : he : fracioal power of A ad X : =D((A) : ) edowed wih he graph orm. Defiiio 6.1. Le f: X : _Y : X, g: X : _Y : Y be locally Lipschiz coiuous fucios. A se S/X : _Y : is a ivaria maifold for a differeial equaio x* =Ax+ f(x, y) y* =By+ g(x, y), if here exiss _: Y : X : such ha S=[(x, y)#x : _Y : : x=_( y)] ad, for ay (x 0, y 0 )#S, here exiss a soluio (x(}), y( } )) of he differeial equaio o R such ha (x(), y()) # S \ # R. A ivaria maifold S is said expoeially aracig if here are posiive cosas # ad K such ha x()_( y()) X :Ke # x(0)_( y(0)) X :, wheever (x(), y()) is a soluio of he differeial equaio. Proof of Lemma 1.3. The firs sep is o prove he exisece of he ivaria maifold. For D>0, 2>0 give, le _ : Y : X : saisfyig _ := sup _ ( y) X :D, _ ( y)_ ( y$) X :2 y y$ Y :. y # Y : (6.7) Le y()=(, {, ', _ ) be he soluio of dy d =B y+ g (_ ( y), y), for <{, y({)=', (6.8)

19 A ONE-DIMENSIONAL PARABOLIC PROBLEM 85 ad defie G(_ )(')= { e A ({s) f (_ ( y(s)), y(s)) ds. (6.9) Noe ha G(_ )( } ) X : { N f M a ({s) : e ;()({s) ds. (6.10) Le 0 be such ha, for 0, G(_ )( } ) X :D. Nex, suppose ha _ ad _$ are fucios saisfyig (6.7), ', '$#Y : ad deoe y()= (, {, ', _ ), y$()=(, {, '$, _$ ). The, y()y$()=e B '+ ({) { e B (s) g (_ ( y), y) ds Ad e B '$ ({) { e B (s) g (_$ ( y$), y$) ds. y ()y$ () Y : M b e \()({) ''$ Y :+M b { (s) : e \()(s) _g (_ ( y ), y )g (_$ ( y$ ), y$ ) Y ds M b e \()({) ''$ Y : +M b L g { (s) : e \()(s) _(_ ( y )_$ ( y$ ) X :+y y$ Y :) ds M b e \()({) ''$ Y : +M b L g { (s) : e \()(s) _(_ ( y$ )_$ ( y$ ) X :+(1+2) y y$ Y :) ds M b e \()({) ''$ Y : +M b L g { (s) : e \()(s) ((1+2) _y y$ Y :+ $ X :) ds M b e \()({) ''$ Y : +M b L g (1+2) { (s) : e \()(s) y y$ Y : ds +M b L g $ X : { (s) : e \()(s) ds.

20 86 BRUSCHI, CARVALHO, AND RUAS-FILHO Le,()=e \() ({) y ()y$ () Y :. The,,()M b [''$ Y :+L g { (s) : e \()(s{) ds $ X :] +M b L g (1+2) { (s) :,(s) ds. By Geeralized Growall's Lemma y ()y$ () Y :[c 1 ''$ Y :+c 2 $ X :] e [\()+c 1 ]({). where c 1 =(M b L g (1+2) 1(1:)) 1(1:). Thus, G(_ )(')G(_$ )('$) X : M a { ({s) : e ;()({s) f (_ ( y), y)f (_$ ( y$), y$) X ds M a { ({s) : e ;()({s) _(L f _ ( y)_$ ( y$) X :+L f y y$ Y :) ds M a { ({s) : e ;()({s) L f ((1+2) y y$ Y :+ $ ) ds M a { ({s) : e ;()({s) L f (1+c 2 (1+2) e [\()+c 1 ]({s) ) ds _ $ +c 1 M a L f (1+2) { ({s) : e [;()\()c 1 ]({s) ds ''$ Y :. Le I _ ()=M a L f { ({s) : e ;()({s) (1+c 2 (1+2) e [\()+c 1 ]({s) ) ds

21 A ONE-DIMENSIONAL PARABOLIC PROBLEM 87 ad I ' ()=c 1 M a L f (1+2) { ({s) : e [;()\()c 1 ]({s) ds. I is easy o see ha, give %<1, here exiss a 0 such ha, for 0, I _ ()% ad I ' ()2 ad G(_ )(')G(_$ )('$) X :I ' () ''$ Y :+I _ () $. (6.11) The iequaliies (6.10) ad (6.11) imply ha G is a coracio map from he class of fucios ha saisfy (6.7) io iself. Therefore, i has a uique fixed poi _*=G(_ *) i his class. I remais o prove ha S=[( y, _*( y)) : y # Y : ] is a ivaria maifold for (1.10). Le (x 0, y 0 )#S, x 0 =_ *( y 0 ). Deoe by y*() he soluio of he followig iiial value problem dy d =B y+ g (_ *( y), y), y(0)=y 0. This defies a curve (_*( y *()), y *()) # S, of # R. Bu he oly soluio x* =A x+ f (_ *( y *()), y *()), which remais bouded as is x*()= e A (s) f(_ *( y *(s), y *(s)) ds=_ *( y *()). Therefore, (_ *( y *()), y *()) is a soluio of (1.10) hrough (x 0, y 0 ) ad he ivariace is proved. From (6.10) i is clear ha s() 0 as ad from (6.11) ha l() 0as. The ex sep is o prove ha, for large eough, he ivaria maifold S is expoeially aracig. Specifically, if (x (), y ()) is a soluio of (1.10), here are posiive cosas # ad K such ha x ()_ *( y ()) X :Ke # x ( )_ *( y ( )) X :. Le!()=x ()_ *( y ()) ad y *(s, ), s be he soluio of dy * ds =B y *+g (_ *( y *), y *), s, y *(, )=y (), s=.

22 88 BRUSCHI, CARVALHO, AND RUAS-FILHO The, y*(s, )y (s) Y : =e B (s) y *(, )+ s e B (s%) g (_ *( y *(%, )), y *(%, )) d% e B (s) y () s e B (%s) g (x (%), y (%)) d% Y : M b (%s) : e \()(%s) s _g (_ *( y*(%, )), y *(%, ))g (x (%), y (%)) Y d% M b L g (%s) : e \()(%s) s _(_ *( y*(%, ))x (%) X :+y *(%, )y (%) Y :) d% M b L g (%s) : e \()(%s) s _(_ *( y (%))x (%) X :+(1+2)y *(%, )y (%) Y :) d% M b L g (%s) : e \()(%s) s _((1+2) y*(%, )y (%) Y :+!(%) X :) d%. Therefore, z(s)m b L g (1+2) (%s) : z(%) d% s +M b L g (%s) : e \() %!(%) X : d%, s where z(s)=e \() s y *(s, )y () Y :. By Geeralized Growall's Lemma, y*(s, )y (s) Y :c 3 (%s) : e [\()+c 1 ](%s)!(%) X : d%, s s.

23 A ONE-DIMENSIONAL PARABOLIC PROBLEM 89 Le s. I wha follows esimaes for y *(s, )y *(s, ) Y : are obaied. y *(s, )y *(s, ) Y : e B (s ) [ y*( 0, )y ( )] Y : + s e B (s%) [ g (_ *( y *(%, )), y *(%, )) g (_ *( y *(%, )), y *(%, ))] d% Y : c 3 M b e \()( s) (% ) : e [\()+c 1](%)!(%) X : d% +M b L g (1+2) (%s) : e \()(%s) y*(%, )y *(%, ) Y : d%, s ad by Geeralized Growall's Lemma y*(s, )y*(s, ) Y :c 4 (% ) : e [\()+c 1 ](%s)!(%) X :. Nex, he esimaes above are used o esimae!() X :.!()e A ( )!( ) =x ()_ *( y ())e A ( ) (x ( )_ *( y ( ))) = = e A (s) f (x (s), y (s)) ds_ *( y ())+e A ( ) _ *( y ( )) e A (s) f (x (s), y (s)) ds e A (s) f (_*( y *(s, ), y *(s, )) ds +e A ( ) e A ( s) f (_*( y *(s, ), y *(s, )) ds = e A (s) [ f (x (s), y (s))f (_ *( y *(s, ), y *(s, ))] ds e A (s) [ f (_ *( y *(s, ), y *(s, )) f (_ *( y *(s, ), y *(s, ))] ds

24 90 BRUSCHI, CARVALHO, AND RUAS-FILHO ad z()=!()e A ( )!( ) X : M a L f (s) : e ;()(s) (x (s)_*( y *(s, )) X : +y (s)y *(s, ) Y :) ds +M a L f (1+2) (s) : e ;()(s) _y*(s, )y *(s, ) Y : ds M a L f (s) : e ;()(s)!(s) X : ds +c 3 M a L f (1+2) (s) : e ;()(s) _ (%s) : e [\()+c 1 ](%s)!(%) X : d% ds s +c 4 M a L f (1+2) (s) : e ;()(s) _ (% ) : e [\()+c 1](%s)!(%) X : d% ds M a L f (s) : e ;()(s)!(s) X : ds +c 5 (%) : e ;()(%)!(%) X : _ % (%s) : e [;()(\()+c 1 )](%s) ds d% +c 6 (% ) : e [\()+c 1](%)!(%) X : _ (s) : e [;()[\()+c 1 ]](s) ds d% c 5 1(1:) 1: _M al f + [;()\()c 1 ] (s) : e ;()(s)!(s) X : ds c + 6 1(1:) [;()\()c 1 ] (% 1: 0 ) : e [\()+c 1 ](%)!(%) X : d%.

25 A ONE-DIMENSIONAL PARABOLIC PROBLEM 91 Thus,!() X :M a e ;()( )!( ) X : c +[M a L f + 5 1(1:) [;()\()c 1 ] 1:] _ (s) : e ;()(s)!(s) X : ds c 6 1(1:) + [;()\()c 1 ] (s 1: 0 ) : e [\()+c 1](s)!(s) X : ds ad, if w()=sup[!(s) X :, s], he where e ;()( )!() X :M a!( ) X :+#() e ;()( ) w() #()= 1(1:) ;() [M c 5 1(1:) c al 1: f + [;()\()c 1 ] 1:]+ 6 1(1:) K [;()\()c 1 ] 1: where K=sup '0 ( ' 0 u: e [\()+c 1 ](u') du). Choose 0 >0 such ha #() 1 2 for every 0. Therefore ad e ;()( )!() X :e ;()( ) w()m a!( ) X :+#() e ;()( ) w()!() X :2M a!( ) X : e ;()( ). The smoohess of _ * is proved i he same way as i [He2] ad he esimae for he derivaive follows from he esimae for is Lipschiz cosa. This cocludes he proof. REFERENCES [ACR] J. Arriea, A. N. Carvalho, ad A. Rodriguez-Beral, Aracors for parabolic problems wih oliear boudary codiios. Uiform bouds, Comm. Parial Differeial Equaios 25 (2000), 137. [C] A. N. Carvalho, Reacio-diffusio problems wih oliear boudary codiios i cell issues, Resehas 3 (1997), [CDR] A. N. Carvalho, T. Dloko, ad H. M. Rodrigues, Upper semicoiuiy of aracors ad sychroizaio, J. Mah. Aal. Appl. 220 (1998), [CP] A. N. Carvalho ad A. L. Pereira, A scalar parabolic equaio whose asympoic behavior is dicaed by a sysem of ordiary differeial equaios, J. Differeial Equaios 112 (1994),

26 92 BRUSCHI, CARVALHO, AND RUAS-FILHO [CH] R. G. Case ad C. J. Hollad, Isabiliy resuls for reacio diffusio equaios wih Neuma boudary codiios, J. Differeial Equaios 27 (1978), [Ch] N. Chafee, Asympoic behavior for soluios of a oe-dimesioal parabolic equaio wih homogeeous Neuma boudary codiios, J. Differeial Equaios 18 (1975), [CI] N. Chafee ad E. F. Ifae, A bifurcaio problem for a oliear parial differeial equaio of parabolic ype, Appl. Aal. 4 (1974), [E] M. A. Evgrafov, ``Asympoic Esimaes ad Eire Fucios,'' Gordo ad Breach, New York, [FR] B. Fiedler ad C. Rocha, Orbi equivalece of global aracors of semiliear parabolic differeial equaios, Tras. Amer. Mah. Soc. 352 (2000), [FO] G. Fusco ad W. M. Oliva, Jacobi marices ad rasversaliy, Proc. Royal Soc. Ediburgh Sec. A 109 (1998), [Ha] J. Hale, Numerical ad Dyamics, ``Chaoic Numerics,'' Coemporary Mahemaics, Vol. 172, pp. 130, Amer. Mah. Soc., Providece, RI, [He1] D. Hery, Some ifiie-dimesioal MorseSmale sysems defied by parabolic parial differeial equaios, J. Differeial Equaios 59 (1985). [He2] D. Hery, ``Geomeric Theory of Semiliear Parabolic Equaios,'' Lecure Noes i Mahemaics, Vol. 840, Spriger-Verlag, Berli, [Ro] C. Rocha, Bifurcaios i discreized reacio-diffusio equaios, Resehas IME-USP 1 (1994), [Sm] G. D. Smih, ``Numerical Soluio of Parial Differeial EquaiosFiie Differece Mehods,'' Oxford Uiversiy Press, 1978.