DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number, February 01, Pages S (011) Artcle electroncally publshed on September 15, 011 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG Abstract. In ths paper, we prove that the soluton of the Lotka-Volterra competng speces system wth strong competton converges to a statonary pont under some natural condtons. We also study the movng boundary problem of the sngular lmt equaton, whch plays an mportant role n our proof. Contents 1. Introducton 961. The sngular lmt: Statonary case The sngular lmt: Parabolc case Asymptotcs n strong competton Approxmate Clean Up Lemma Systems wth zeroth order terms Proof of the Approxmate Clean Up Lemma A lnearzed verson A boundary verson 100 Acknowledgement 1004 References Introducton The Lotka-Volterra model of competng speces descrbes the competton of a number of speces n a fxed doman. Its general form s as follows: u t u = f (u ) u b j u j, n Ω (0, + ), u =0(or u n =0), on Ω (0, + ), u = φ, on Ω {0}, Receved by the edtors Aprl 8, 010 and, n revsed form, September 3, Mathematcs Subject Classfcaton. Prmary 35B40, 35R35, 35K57, 9D5. Key words and phrases. Competng speces, reacton-dffuson system, free boundary problem. Ths work was supported by the Australan Research Councl and the Natonal Natural Scence Foundaton of Chna ( , ). 961 c 011 Amercan Mathematcal Socety

2 96 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG where b j > 0 are constants and 1, j M, wth M the number of the speces, and Ω s a bounded doman n R n (n 1) wth smooth boundary. The ntra-speces constrant term f (u ) s usually the logstc model: f (t) =a t t, t R. Usually we consder a homogeneous Drchlet u = 0 or Neumann boundary condton u n = 0, and we mpose a consstent condton on the ntal values φ 0. We only study nonnegatve solutons, that s, u 0 for all. The study of ths reacton-dffuson system has a long hstory and there exst a great many works on the subject. However, most of these works are concerned wth the case of two speces. As far as we know, study n the case of many competng speces has been very lmted. In the 1990s Dancer and Du studed three speces of competton systems and receved very nterestng exstence results. It s beleved that generally ths system has complcated dynamcs (see [14], [15]), even n the ordnary dfferental equaton cases (see [7]). In recent years, people have shown much nterest n strongly competng systems wth many speces, that s, the system (or ts ellptc case) u t u = a u u κu b j u j, where κ s suffcently large. Cont, Terracn and Verzn [9], [10], Caffarell, Karakhanyan and Ln [] [3], etc., establshed the regularty of the sngular lmt (and the partal regularty of ts free boundary) as κ + and the unform regularty for all κ>0. Cont, Terracn and Verzn found that n the sngular lmt of the statonary problem, speces are spatally segregated and satsfy a remarkable system of dfferental nequaltes, and these two condtons are also satsfed by the soluton of a varatonal problem. Although t s not fully establshed, t s very possble that ths sngular lmt has a varatonal structure. That s, the soluton of a correspondng ellptc problem (as the equatons do not contan the terms f (u )) s the harmonc map from the doman Ω nto a metrc space Σ wth nonpostve curvature, whch has been studed by many authors snce the work of [0]. Here the metrc space Σ s defned as follows: Σ:={(u 1,u,,u M ) R M : u 0,u u j =0for j}. Ths harmonc map s the crtcal pont (n the weak sense as n [9]) of the functonal M u, =1 Ω defned n the class of functons u =(u 1,u,,u M ) (H 1 (Ω)) M satsfyng u 0 and u u j = 0, a.e.; see [9]. In our orgnal problem and the correspondng ellptc problem, a term F (u) needs to be added to the above formulaton: M ( 1 u F (u )), =1 Ω Ω where F (u) = u 0 f (s)ds. The soluton of the parabolc system of our sngular lmt could also be seen as a weak soluton of the heat flow for the harmonc map (gnorng the zeroth order terms as above).

3 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 963 So although our orgnal problem n case κ fnte has no varatonal structure, ts sngular lmt ndeed has. Thus t s natural to conjecture (ths s probably due toe.n.dancer)thatforκ large, ths system has smple dynamcs, n partcular, has no perodc soluton. Ths s also related to an open problem of Wemng N concernng the exstence of a perodc soluton for fxed κ (the report gven at the Pacfc Insttute for the Mathematcal Scence, Unversty of Brtsh Columba, 001.7). Ths problem has been studed by Dancer and Zhang [18] for the case of two speces. They proved that, under some natural assumptons on nondegeneracy, for κ large and two speces, n any dmenson, any soluton of the ntal value problem wll converge to a statonary soluton as tme t tends to nfnty. By usng some deas from the recent work of Caffarell, Karakhanyan and Ln [], we obtan new results concernng ths problem, wth the number of speces greater than, n dmenson 1. Under some natural assumptons on nondegeneracy smlar to those n [18], for κ large, any soluton of the ntal value problem wll converge to a statonary soluton as tme t tends to nfnty. The problem we study n ths paper s the followng ntal value problem wth a homogeneous Drchlet (or Neumann) boundary value condton n 1 dmenson: (1.1) u t u x = a u u κu b j u j n [0, 1] (0, + ), u (0) = u (1) = 0 (or u x x=0,1 =0), u 0 n [0, 1] (0, + ), u = φ on [0, 1] {0}. Here =1,,M, φ are gven Lpschtz contnuous functons on [0, 1] whch satsfy φ 0, and φ φ j =0for j. In the paper, we wll smply take b j 1 for smplcty. If there s any need to modfy the argument arsng from b j 1, we wll ndcate t n the remark. From the classcal theory of reacton-dffuson systems (see, for example, [3]), we know that the soluton u,κ of (1.1) exsts globally and s unque. Moreover, from the regularty results of [] (see Theorem 1 and the Remark n the paper; ths can also be proven usng the blow up method of [10]), α (0, 1), u,κ are unformly bounded for κ n C α ([0, 1] (0, + )). Here the dstance functon s the parabolc dstance n [0, 1] (0, + ), defned as dst((t 1,x 1 ), (t,x )) := t 1 t 1 + x1 x. So we can say (up to a subsequence) that they converge unformly n any compact subset of [0, 1] (0, + ) tou. More precsely, gven any sequence u,κn, κ n +, we can extract a subsequence whch converges to u n the above sense; u s also Hölder contnuous. Smlar to the ellptc case treated by S. Terracn et al. n [9]

4 964 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG and [10], t s easly seen that u satsfy (1, j M) (1.) u t u x a u u n (0, 1) (0, + ); ( t x )(u u j ) a u u j u j u (a j)n(0, 1) (0, + ); u (0) = u (1) = 0 (or u x x=0,1 =0); u = φ on [0, 1] {0}; u 0; u u j =0for j n [0, 1] (0, + ). The above dfferental nequaltes are always understood n the weak sense (and n ths paper, ths knd of dfferental nequaltes are always understood n the weak sense), that s, e.g., ϕ(x, t) 0 smooth wth compact support n (0, 1) (0, + ), for the frst nequalty we have (, 1 M) + 1 ϕ 0 0 t u + u ϕ x x (a u u )ϕdxdt 0. In [], they also prove that the lmt u are locally Lpschtz contnuous n the parabolc dstance n [0, 1] (0, + ) (see ther Theorem ). In fact, u are Lpschtz contnuous n the parabolc dstance snce they are unformly bounded n our case. Here we need to note that, although our equatons look a lttle dfferent from those n [], ther method can stll be appled to our case, snce the term f (u )weadd can be seen as a small perturbaton after a rescalng. For the detals, please see Secton 4.1 below. Frst we study the dynamcs of (1.): we show that after a fnte perod of tme the set where u s postve has only fntely many components and that the number of these s bounded. Ths s true even f at tme zero the set where u s postve has nfntely many components. After a perod of tme no extncton can happen any more. Fnally, the u converge to a statonary state as tme t goes to nfnty, that s, f a speces s suffcently small compared wth the other speces, t wll become extnct n the future after a fxed fnte tme. Imposng some assumptons, for system (1.1) wth strong competton (large κ), we can prove that for any ntal value, the soluton wll eventually converge to a statonary state (see Theorem 4.1). The proof follows the dea of [18], but n our case we need some dstnctly nontrval techncal changes. Note that, dfferent from the case wth only two speces, whch can be transferred nto a sngle equaton by subtractng the two equatons, f the number of speces M 3, there may exst nteractons between all the speces, and we can t transfer our equatons nto a sngle equaton. We fnd that, n 1 dmenson, due to the topologcal restrcton, n most areas, locally t appears that there are only two speces competng, that s, other speces are very small compared to these two domnatng speces. Ths s essentally the Approxmate Clean Up Lemma, whch orgnates from the Clean Up Lemma n [] and s vald n any dmenson. We also need a lnearzaton verson and a boundary verson of the Approxmate Clean Up Lemma. The proof of these varous forms of the Clean Up Lemma follows from the powerful teraton scheme of []. In the paper [18], for the solutons of the lnearzed equaton of (1.1), the authors used the Kato nequalty to derve a

5 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 965 unform bound from the L bound. Ths causes no restrcton n speces, but f the number of speces s more, unless we assume the symmetry b j = b j, the Kato nequalty can t be appled drectly. Here we can avod ths by our lnearzaton verson of the Approxmate Clean Up Lemma. Possbly n one dmenson we may fnd other more drect methods to attack ths problem, but we stll thnk that our treatment may be useful n hgher dmensons. At last we want to say somethng about our assumptons. In order to prove the convergence to a statonary pont, we mpose some nondegeneracy condtons (see Secton 5). Some of these condtons are verfed n one dmenson. However, these condtons seem to be too restrctve n a hgher dmenson. Anyway, f we add these condtons n a hgher dmenson, most of the arguments n ths paper can go through f we bypass the regularty of the free boundares near the fxed boundary. It seems that even f we do not mpose these condtons artfcally, we can stll prove a weak verson of the conjecture above (see Corollary 4.3 for the proof n the case of one dmenson). More precsely, we show that for κ large any perodc soluton must stay near some statonary soluton of the sngular lmt system. That s, t looks lke a statonary soluton whch slowly changes n tme. We wll pursue ths problem n the future. The paper s organzed as follows. In Secton we gve a prelmnary analyss of the statonary case of (1.), establshng that there are only fntely many solutons. In Secton 3 we study the system (1.) and show that the soluton converges to the statonary state (see Theorem 3.1). In the process of the proof we also descrbe the extncton phenomena. In Secton 4 we establsh an Approxmate Clean Up Lemma and ts lnearzaton verson and boundary verson. In Secton 5 we prove that, under some assumptons, the soluton of (1.1) converges to a statonary state.. The sngular lmt: Statonary case In ths secton we study the statonary case of dfferental nequaltes (1.): d u dx a u u n (0, 1); (.1) d dx (u u j ) a u u j u j u (a j)n(0, 1); u 0, u (0) = u (1) = 0; u u j = 0 n [0, 1], for j(1, j M). The man result of ths secton s Theorem.4. The Neumann boundary value problem can be treated smlarly. We know that the above problem can have a soluton wth some dentcally zero components, thus t may have multple solutons, but f the set where u s postve has several components, we treat these as dfferent speces (because we are n the statonary case). Here the number of speces may become nfnte. Below we exclude ths case. We have the followng theorem. (Note thatlemma.belowensuresthatthesetwhereu s postve has only fntely many components.) Theorem.1. M N, there exsts at most one soluton (u 1,u,,u M ) of (.1) (up to permutaton) such that each u s not dentcally zero and each of ts supports s an nterval.

6 966 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG Frst we need a well-known lemma. For later use and completeness we gve the proof here. Lemma.. For each L> (.) There s no postve soluton f L π, there exsts a unque postve soluton u of a d u dx = au u n (0,L), u(0) = u(l) =0. π a. Proof. The exstence can be easly proven by, for example, the method of sub- and sup-solutons. In fact, we have the followng conservaton quantty: (.3) ( du dx ) + au 3 u3 c, for some constant c. If we have two postve solutons u and v, thenontheopensetd := {u >v} (f notempty)wehave whch s a contradcton. 0 = = = > 0, D D D D u ν v v ν u uv vu (au u )v +(av v )u uv(u v) Corollary.3. The constant c defned n (.3), seen as a functon of L, sncreasng n L. Proof. Assume L 1 >L > π a wth u 1 and u the soluton of (.) n [0,L 1 ]and [0,L ], respectvely. Because 0=u (L ) <u 1 (L ), the same method as n the prevous lemma gves that n [0,L ], u 1 >u. Ths mples du 1 dx (0) > du dx (0). Note that here we have the strct nequalty because otherwse we wll have u 1 u. Because c (L )=( du dx (0)), our clam follows.

7 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 967 Now we gve the proof of Theorem.1. Proof. Snce each support {u > 0} s an nterval, and for j, {u > 0} and {u j > 0} are dsjont (snce u u j = 0), we can assume wth (by renumberng f necessary) {u > 0} =(α,β ), 0 α 1 <β 1 α <β α m <β m 1. Because on the set (β 1,α +1 ), where all u j 0forj, u satsfes the equaton (by combnng the frst and second equaton of (.1)) (.4) d u dx = a u u. Moreover, u 0andu does not vansh dentcally on ths nterval, so we must have u > 0n(β 1,α +1 ). Thus α +1 = β and α 1 =0,β m =1, that s, the set where all u vansh contans at most fntely many ponts. In (α,β ), there exsts a constant c > 0 such that (.5) ( du dx ) + a u 3 u3 = c. Because u (α )=u (β )=0,then (.6) ( du dx (α )) = c =( du dx (β )). In (α,β +1 ), from the second nequaltes n (.1) d dx (u u +1 ) a u u (a +1 u +1 u +1) n(α,β +1 ), we see that d dx (u +1 u ) a +1 u +1 u +1 (a u u )n(α,β +1 ), (.7) d (u u +1 ) dx = a u u a +1 u +1 + u +1. Ths mples u u +1 C 1 (α,β +1 ). Then we have ( du dx (β )) =( du +1 dx (β )). So all of the c n (.5) are the same, whch we denote by c. Now from Corollary.3 we know that each β α s unquely determned by c (because t s a strctly ncreasng functon of c). Then there exsts at most one c such that (β α )=1. Theorem.4. There are only fntely many solutons of (.1).

8 968 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG Proof. It s well known that f u>0 n a bounded doman Ω wth boundary value 0andf u = au u, then a>λ 1, where λ 1 s the frst egenvalue of Ω wth a Drchlet boundary condton. Now n our case each connected component of u, whch s an nterval, has length π a. So the total number of connected components of each possble soluton of (.1) has an upper bound, whch depends on those a,1 M. By Theorem.1 we can fnsh the proof. 3. The sngular lmt: Parabolc case In ths secton, we study the dynamcs of the sngular lmt equaton (1.). The man result s the followng Theorem 3.1. Wth a smlar dea, we also gve a descrpton of the connectng orbt (Proposton 3.1) and a unqueness result of the ntal value problem (Proposton 3.14). Theorem 3.1. Let s defne m(t ):=#{connected components of {u > 0} {t = T }}. Then we have: (1) m(t ) s a nonncreasng functon of T. () There exst at most countable ponts T 1 >T > wth T k 0, whch are all of the places where the value of m(t ) jumps. (3) For t (T +1,T ) (here we defne T 0 =+ ), we can rearrange the ndces (where we can throw away those speces vanshng dentcally here) such that f we defne u := u 1 u + u 3 u 4 +, t satsfes the followng equaton: (3.1) ( t )u = f, x where f := a 1 u 1 u 1 (a u u )+ s a Lpschtz contnuous functon. (4) In [0, 1] (T 1, + ) we have lm t + u (x, t)=v (x), wherev :=(v 1,v, ) s a soluton of (.1). Moreover, ether v 0 or v s nontrval and changes sgn exactly m(t 1 ) tmes. Frst, we show a knd of regularzng property of (1.). Ths s a specal case of the Dmenson Reducton Prncple (see, for example, [6]). However, t s not wrtten explctly n the lterature, so we wll gve the arguments here for completeness. Proposton 3.. For each t>0, for the free boundary F(u) := {u > 0} n the problem of (1.), F(u) {t} contans only fntely many ponts. Proof. Assume for some t>0 that there exst nfntely many ponts n F(u) {t}. Take a x 0 F(u) {t} (0, 1) such that there exst x k x 0 n F(u) and

9 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 969 lm k x k = x 0. Wthout loss of generalty, we can assume x 0 =0andt =0. (We can change the doman by translaton.) Take r k = x k x 0, whch we can assume to be a postve number. Now defne the blow up sequence u k (x, t) := 1 u(r k x, r L kt), k where L k = 1 C rk C rk u and C rk =[ r k,r k ] [ rk,r k ] s the parabolc cylnder. As n [] (or Theorem.1 n [6]), we know there exsts a subsequence (stll denoted by u k ) convergng to v(x, t) locally unformly on R 1 (, + ), wth v nontrval and nonnegatve. Moreover, there exsts a postve nteger d, whchs the frequency of (0, 0) (for the defnton and ts property, please see Secton 9 n []), such that n R 1 (, + ), v t v x 0, ( (3.) t x )(v v j ) 0, v v j =0for j, v (λx, λ t)=λ d v (x, t). Because (1, 0) F(u k ), we also have v (1, 0) = 0 for all, or equvalently (1, 0) F(v). Then from the homogenety of v we have R 1 + {0} F(v). Ths contradcts the dmenson estmate of the free boundary. (From Secton 10 of [] and Theorem.3 n [6], we know that the Hausdorff dmenson of the free boundary restrcted to tme t = 0 s 0.) If x 0 les on the fxed boundary, the blow up lmt v s defned on R 1 + (, + ), wth the boundary condton v =0, on {0} (, + ). Ths can be treated smlarly. Now we have the followng lemma: Lemma 3.3. T 1 >T > 0, any connected component of {u > 0} {t<t 1 } must ntersect wth {t = T }. Proof. We know that {u > 0} s an open subset of [0, 1] [0, + ), so t s locally path connected and the path connected components concde wth the connected components. Assume there exsts a connected component A, wth {t = T } A =. Now consder the restrcton û of u on A. Frst, t s stll contnuous. So we have û t û x a û û n {û > 0}, (3.3) û 0 n [0, 1] [T,T 1 ], û =0 on p ([0, 1] [T,T 1 ]),

10 970 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG where p ([0, 1] [T,T 1 ]) s the parabolc boundary. Now we must have û 0. In fact, take a postve constant C>a and defne We have n the open set {v >0} v(x, t) :=e Ct û (x, t). v t v x e Ct ( Cû + a û û ) < 0. If û s not dentcally zero, then max [0,1] [T,T 1 ] v s attaned at some pont (x 0,t 0 ) where û (x 0,t 0 ) > 0. We know û s postve n a neghborhood of (x 0,t 0 ) because û s contnuous. Then v s smooth n ths neghborhood, and we can apply the maxmum prncple to get a contradcton. Remark 3.4. From the proof, we see that ths lemma remans true n hgher dmensons and wth a more general nonlnearty. Wthout loss of generalty, we can assume that as t = T > 0, u satsfy the condton that {u > 0} {t = T } s an nterval for each. Later t wll be clear that, as tme evolves, two connected components may merge nto one but a sngle connected component can t splt nto two. Corollary 3.5. T >T, {u > 0} {t = T } s an nterval f t s not empty. Proof. If T > 0, such that {u > 0} {t = T } contans two dsjont ntervals (α 1,β 1 ) (α,β )where α 1 <β 1 <α <β and (β 1,α ) {u > 0} =. From Lemma 3.3 we know there exst two contnuous paths γ 1 (t) andγ (t) whch start from a pont n {u > 0} {t = T } and end n two ponts lyng n (α 1,β 1 ) {T } and (α,β ) {T }, respectvely. There exsts some j such that ((β 1,α ) {T }) {u j > 0}. However, ths set s contaned n a connected component whch can t be connected to {t = T } through a contnuous path, contradctng Lemma 3.3, and we conclude the proof of the corollary. Now the frst and the second parts of Theorem 3.1 are easly seen. The ponts T k are exactly where some components of u become extnct. Assume n (T +1,T )thatu 1,u,,u m are not dentcally zero, where m M. Wth the help of Lemma 3.3, we can show that any two connected components of {u > 0} and {u j > 0} wth j can t nterweave, that s, f at t 1 (T +1,T ), {u > 0} les above {u j > 0}, then at another t (T +1,T ), {u > 0} can t le below {u j > 0}. So we can rearrange the ndex properly so that {u > 0} s adjacent to {u 1 > 0} and {u +1 > 0} (here we treat the same speces wth

11 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 971 dfferent domans as dfferent speces; ths causes no confuson n (T +1,T )). Now the thrd part of Theorem 3.1 s proved. Now n ths tme nterval we have Lemma 3.6. t (T +1,T ),thereexst 0=α 0 (t) α 1 (t) <α m (t) =1 such that {u j > 0} {t} =(α j 1 (t),α j (t)). Moreover, all the α j (t) are contnuous functons of t. Proof. From Corollary 3.5 we know that for each t>0 there exst 0=α 0 (t) <β 1 (t) α 1 (t) <β (t) <β m (t) =1 such that {u j > 0} {t} =(α j 1 (t),β j (t)). From the unque contnuaton property of the lnear parabolc equaton u (3.4) t u x = Vu, where V := (a 1 u 1 ) (a u )+ s an L functon, we know that, t, theset{x : u(x, t) =0} can t contan an open set. From ths, we easly see β j (t) =α j (t). Now {(α j(t),t):t (T +1,T )} s the nodal set of u, and the contnuty of α j (t) s easly seen by ts local unqueness. Remark 3.7. In the proof we need the fact that u s not dentcally zero n [0, 1] (T +1,T ). Ths can be guaranteed by the unque contnuaton property of the lnear parabolc equaton (3.4) (see [6]; note here we have zero boundary condton). Note that although the form of the equaton changes when crossng those extncton tme T,fu 0n{t >T },thenu 0att = T and we can apply the backward unqueness n (T +1,T ] agan. These curves are n fact regular. That s, Proposton 3.8. t (T +1,T ),at(α j (t),t), we have (here u s defned as n Theorem 3.1) u x 0. In partcular, α j (t) are C 1 n t. Proof. Note that snce u satsfes (3.4) and that both V and u are bounded, standard parabolc estmates mply that u s C 1 n the space varable x. So the formulaton n the proposton makes sense. We need the characterzaton of the sngular ponts of the free boundares (see Secton 9 of [] or [6]). Assumng we are at (0, 0),wehaveapostventegernumber d (the frequency) such that the blow up sequences (3.5) u λ (x, t) =λ d u(λ t, λx)

12 97 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG converge to v(x, t) asλ 0, whch s a polynomal of the form (the Hermtan polynomal) (3.6) [ d ] k=0 d! k!(d k)! xd k t k. Moreover, f u u x =0,thend. In other words, x 0 s equvalent to d =1. Now, {v(x, t) =0} s composed of d nonsngular curves C whch have the form t = c x, 1 d. Here c are the zeros of the correspondng Hermte polynomals, and nonsngular means at these ponts v x 0 (ths s because Hermte polynomals have smple zeros). Now because u λ converges to v n C 1, by the nverse functon theorem we know for λ small near each C we have a nonsngular nodal curve of u λ. In partcular, there exst two ponts x 1,λ x,λ such that u λ ( 1,x 1,λ )=u λ ( 1,x,λ )=0, f λ small enough. Comng back to u, ths means u( λ,λx 1,λ )=u( λ,λx,λ )=0. Ths s a contradcton, because from the prevous lemma we have near (0, 0) that the nodal curve of u λ (that s, of u) s a sngle contnuous curve, whch mples there exsts a unque α( λ ) such that u( λ,α( λ )) = 0. Remark 3.9. Smlar results to Proposton 3.8 already appeared n [6]. See also Secton10n[]. Wth ths regularty property of the free boundary we can gve an energy dentty. Defne the energy (3.7) E(t) := 1 [0,1] u x 1 a u u3. We have Proposton For t (T +1,T ) (3.8) d dt E(t) = [0,1] u dt. In partcular, E(t) s decreasng n t.

13 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 973 Proof. We have (3.9) = d dt β (t) α (t) β (t) α (t) ( u x 1 u x 1 a u u3 u x t a u u t + u u t )dt [ 1 ( u x ) 1 u u3 ](α (t),t) dα (t) dt +[ 1 ( u x ) 1 u u3 ](β (t),t) dβ (t) dt β (t) u u = t [ x + a u u ] α (t) + u x (β (t),t) u t (β (t),t) u x (α (t),t) u t (α (t),t) 1 u x (α (t),t) dα (t) + 1 dt u x (β (t),t) dβ (t). dt By dfferentatng u(α (t),t) = 0 and notng the defnton of u we have (3.10) So we get u t (α (t),t)+ u x (α (t),t) dα (t) =0. dt u u (3.11) x t (β (t),t)= u x (β (t),t) dβ (t) dt and u u (3.1) x t (α (t),t)= u x (α (t),t) dα (t). dt Note that we have α (t) =β 1 (t). Summng (3.9) up n, wegettheresult. The fourth part of Theorem 3.1 can be proved usng the same method n [18]. The man dea s that f the soluton s very close to a statonary soluton for some tme, then t ether stays near ths statonary soluton all the tme or t leaves away and never comes back. Here we wll establsh a property whch, roughly speakng, can be sad to be no extncton at nfnty. From the dscusson above, we know for a soluton of (1.) there exsts a T 1 > 0, such that for all t>t 1, no extncton occurs. But at ths stage we can t exclude the possblty that some u 0ast + whle remanng postve n any fnte tme. Now let s consder ths problem: Proposton For each, u (x, t) converges to u (x). If (u 1 (x),u (x), ) are not dentcally zero, then each u (x) s not dentcally zero. Proof. From the energy dentty (3.8), we know that (3.13) lm t + Because + 1 t 0 u t =0. (3.14) ( t x ) u u = ˆV t t,

14 974 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG where ˆV := (a 1 u 1 ) (a u )+ s unformly bounded, by the standard parabolc estmate we get (3.15) lm So we have at tme t sup t + x u (x, t) =0. t (3.16) sup u x x + a u u ɛ(t), where lm ɛ(t) = 0. From the statement about the Lpschtz regularty of solutons of (1.), u t + x are unformly bounded. We can multply (3.16) wth t to get (3.17) sup u u x x x + a u u x u u x Cɛ(t), for some constant C ndependent of t. Integratng ths n x we get, for x (α (t),β (t)), (3.18) [ 1 ( u x ) + 1 a u 1 3 u3 ](x, t) [ 1 ( u x ) + 1 a u 1 3 u3 ](α (t),t) Cɛ(t), where α (t) can be replaced by α +1 (t). From ths, f we defne C(t) :=[ 1 ( u x ) + 1 a u 1 3 u3 ](α (t),t) for some fxed, then there exsts a constant C>0such that for all (smlar to the ellptc case consdered n Secton ), (3.19) [ 1 ( u x ) + 1 a u 1 3 u3 ](x, t) C(t) Cɛ(t). In partcular, f C(t) s unformly bounded from below by a fxed postve constant, then at tme t, f we are at the maxmal pont of u (x, t), because we have u x =0, (3.0) [ 1 a u 1 3 u3 ](x, t) C(t) Cɛ(t). We can solve ths equaton to obtan max u (x, t) C (t) Cɛ(t), x where C (t) are bounded from below because they only depend on C(t) anda.so no u (x, t) can converge to 0 for any t +. On the other hand, f lm nf C(t) = 0, then we have for a sequence t k + t + that lm u(x, t k) = 0. Moreover, from the standard parabolc estmate ths convergence can be taken n C 1 topology, and we have for ths k + sequence lm t k + Ω 1 u x 1 a u u3 =0.

15 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 975 From the energy decreasng property (3.8), we then have 1 lm t + Ω u x 1 a u u3 =0. Inthscase,wemusthaveforallthat lm u (x, t) =0. t + Ths s because any nontrval soluton of the statonary equaton (.1) has energy < 0 strctly, whch can be seen by an ntegraton by parts. The above method can be used to descrbe the entre soluton, whch s defned for all tme as t (, + ). Frst, from the energy decreasng property we have ether lm E(t) =+ t or lm E(t) < +. t If the latter happens, we n fact have the lmt as t exsts, that s, Proposton 3.1. For a soluton u of (1.) defned n [0, 1] (, + ), f lm E(t) < +, t then there exsts a statonary soluton v of (1.) (that s, a soluton of (.1)) such that lm u (t) =v. t Proof. From the energy dentty we have 0 (3.1) u [0,1] dt < +. In partcular, t (3.) lm u t [0,1] dt =0. Because u (t) are unformly Lpschtz contnuous functons on [0, 1] (note here we have a control of the energy; see []), for any sequence t k there exsts a subsequence convergng to a Lpschtz contnuous functon v for all. We clam that (v )sasolutonof(.1).infact,defne u,k (x, t) =u (x, t + t k ). It s a soluton of (1.) defned on [0, 1] (, + ) and unformly Lpschtz contnuous, so t converges locally unformly to a soluton v of (1.). Also we have u,k (x, 0) v (x), so v (x, 0) = v (x). Moreover, we have E(u k,t)=e(u, t + t k ). So for t>0, (3.3) E(u k, t) E(u k,t)+ε(k, t),

16 976 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG where lm ε(k, t) = 0 for any fxed t. By the unform Lpschtz contnuty, t, k + lm E(u k,t)=e(v, t). k + Passng to the lmt n the nequalty (3.3) and notng the energy decreasng property (3.8) (ths can be appled to the current case), we get Usng the energy dentty (3.8), we have t (3.4) E(v, t) =E(v, t). t [0,1] v t =0. Ths means v t 0. That s, v s the statonary soluton of (1.). We know that the statonary soluton of (1.) s a fnte set (Theorem.4), but from standard theory we know that the ω-lmt set of u(x, t) ast s a connected set, so t must be a sngle pont. In other words, f there exst two lmt ponts, then we can prove that the lmt set contans a path connectng these two ponts, contradctng our prevous clam. The complete argument s as follows: Take small neghborhoods V 1 and V of the two lmtng ponts v 1, v, and neghborhoods V of all other statonary solutons v, whch are dsjont. We know there exst t k,1 and t k, whch converge to such that u(t k,1 ) V 1 and u(t k, ) V. From connectedness of the orbt u(t) we know there exsts t k lyng between t k,1 and t k, such that u(t k )snotn V. But from the unform Lpschtz contnuty, u(t k ) converge to a lmt n C[0, 1], whch s not a statonary pont. Ths s a contradcton. Remark For any soluton w of (1.) whch exsts on [0, 1] (, + ), assume that there exst u and v, whch are solutons of (.1) such that lm w (x, t) =u (x), lm w (x, t) =v (x). t t + Then we have two cases: (1) u 0andv s nontrval. () Both u and v are nontrval, wth the energy of u strctly larger than that of v. Now we consder the unqueness of the ntal value problem (1.) n a specal case: Proposton For the system (1.), f the total number of connected components of the ntal value φ s fnte, then there exsts a unque soluton.

17 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 977 Proof. The exstence s obvous, that s, the soluton can be constructed as the lmt of the solutons of system (1.1) (κ + ). We just need to prove the local unqueness. Assume there exst two local solutons of (1.), u and v. By Theorem 3.1, we know there exsts some ɛ>0, such that n [0,ɛ), the sum n of the number of connected components of the support of the component u (or v )equalsthatofφ = (φ 1,φ,,φ M ), and the free boundares are nonsngular curves (see Proposton 3.8). Let d(u, v) = u v. Usng the Kato nequalty we have h = sgn(h) h, a.e., h sgn(h) h, a.e. ( t )d(u, v) x sgn(u v )( t x )(u v ) sgn(u v )[a (u + v )](u v ) Cd(u, v). Here sgn(u v ) s the sgnature of u v and C s a constant dependng only on a and φ. The second nequalty s vald by the followng argument: by the regularty of the free boundares, n fact we have (3.5) ( t x )u = a u u + u ν δ α (t), α (t) where α (t) s the boundary of {u > 0} at tme t (t conssts of fntely many ponts) and ν s the outward unt normal vector to {u > 0} (although n 1 dmenson ths s trval, we keep ths notaton for clearness), and δ s the Drac measure supported on these ponts. Summng these terms n we fnd that n the nteror these δ measures cancel each other out and we are left wth the regular part, whle those at boundary ponts 0 (or 1) have the form sgn(u v ) (u v ) δ 0. ν If near x =0,u v > 0, then (u v ) ν (0) 0 and vce versa. So ths s a nonpostve term, and n the nequalty we can throw t away. Now d(u, v) 0att = 0. Then the maxmum prncple gves for all t [0,ɛ), that s, d(u, v) 0 u v.

18 978 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG 4. Asymptotcs n strong competton In ths secton, we study the asymptotc behavor of (1.1) for κ large. From the analyss n Secton 3, we know that the sngular lmt as κ + s a gradent system (havng a varatonal structure), wth ts soluton convergng to the statonary state as t +. So the natural queston arses: does (1.1) for κ large also behave lke a gradent system? In ths secton we wll show that wth a few assumptons, the answer s yes: for κ large the dynamcs of (1.1) s smple. For smplcty, we assume the coeffcents b j =1, j. Wthout ths assumpton, our proof s stll vald wth mnor changes, due to the specal property of dmenson 1. In the followng, we wll pont ths out whenever necessary. In the proof of the man result of ths secton, especally n Theorem 4.4, we need some techncal results. These results gve the unform a pror estmate and the convergence of the solutons of the lnearzed equatons of (1.1) as κ +. The precse statement and proof of these results are found n the next secton. We need to mpose some assumptons on the statonary equaton of (.1), exactly as n [18]. From Secton, we know any soluton of (.1) can be arranged properly so that f we defne u := u 1 u +, then u satsfes (4.1) d u = f(x, u) on[0, 1],u(0) = u(1) = 0, dx where f(x, u) =a u ( 1) +1 u, for x {x [0, 1] u (x) > 0}. If b j 1, n the above defnton of u, u has the form u 1 c 1 u + c u, where c 1,c, are constants dependng only on b j. The followng statement and proof can be changed accordngly. Any soluton u of (4.1) s nondegenerate, that s, any soluton v of (4.) v = g(x, u)v a.e. on [0, 1],v(0) = v(1) = 0 wth boundary value 0, s dentcally zero. Here g(x, u) =f (x, u) =a ( 1) +1 u, for x {x [0, 1] u (x) > 0}. In fact, we know that f(x, u) s dfferentable n u except at u =0,andf u(x, u) < f(x,u) u,u 0,f(x, 0) = 0. (Note that nontrval solutons of (4.1) only vansh on a set of measure zero so that the lnearzaton (4.) makes sense.) If z s a soluton of (4.1) wth k nteror zeroes, then z s a soluton of the lnearzed equaton (4.) and t must have at least k+1 zeroes n (0, 1), one between each two zeroes of z n [0, 1]. Suppose that v satsfes (4.). By the Sturm comparson theorem (comparng wth z ) v hasatleastk zeroes n (0, 1) (one between any two zeroes of z ). Now z f(x, z) z = 0 a.e. on [0, 1],z(0) = z(1) = 0 z and z has k nteror zeroes. Hence 0 must be the (k + 1)th egenvalue of the egenvalue problem h f(x, z) h = λh a.e. on [0, 1],h(0) = h(1) = 0. z

19 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 979 Snce f u(x, u) < f(x,u) u as u 0 a.e. on (0, 1), t follows by egenvalue comparson that the (k + 1)th egenvalue of v g(x, u)v = λv a.e. on [0, 1],v(0) = v(1) = 0 s postve. Thus f 0 s an egenvalue, t must be the lth egenvalue where l<k+1. Hence the correspondng egenfuncton has less than l 1 nteror zeroes, and so less than k nteror zeroes. Ths contradcts the frst part of the proof, and hence 0 s not an egenvalue,.e., (4.) has no nontrval solutons. We assume: (1) The system of dfferental nequaltes (4.3) d u dx a u n [0, 1], d dx (u u j ) a u a j u j n [0, 1], u (0) = u (1) = 0, u 0,u u j =0for j, n [0, 1], (1, j M) has no nontrval solutons. (Assume (u ) s a nontrval soluton (we stll consder a functon on dsjont supports as two dstnct functons). Smlar to Secton, there exst α m (1 m k for some postve nteger k) such that 0=α 0 <α 1 <α < <α k =1, and there exst constants c m > 0,d m = u m (x) = α m 1 α m α m 1 π (1 m k) such that { π cm sn( α m α m 1 x + d m ), f x (α m 1,α m ), 0, otherwse, and f j m for all m, u j 0. Moveover, m, c m+1 am+1 and α m α m 1 =. Ths last condton mples π a m = c m am (4.4) k m=1 π am =1. Ths s very restrctve. So we can assume {a,=1,,,m} such that there s only trval soluton of (4.3).) () Nonconstant (n tme) bounded postve solutons of the followng system ( =1,,,M), u 1 t u 1 x u t u x = a 1u 1 u 1 u j n [0, 1] (, + ), j 1 = a u u u j n [0, 1] (, + ), j u M u M t x = a M u M u M u j n [0, 1] (, + ), j M u (0,t)=u (1,t)=0fort (, + ),

20 980 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG approach dstnct statonary solutons as t ±. The postve statonary solutons are hyperbolc, and there are no crcuts of heteroclnc postve orbts. (Ths condton has been studed n [18], where they construct an example (wth two equatons) satsfyng ths condton.) (3) a >π ( =1,,,M, π s the frst egenvalue of w = λw, w(0) = w(1) = 0). (Assumpton 3 mples that the equaton ( =1,,,M) d u dx = a u n (0, 1), (4.5) u(0) = u(1) = 0 has no postve solutons.) Now we state our man result Theorem 4.1. Under the above assumptons (1) (3), forκ large, any soluton of (1.1) converges to a statonary pont as t +. Frst, we need to establsh a theorem whch s the analogue to Theorem n [18]. However, ther proof can t be drectly appled here, so we need some mprovements. Theorem 4.. ɛ >0, thereexstsκ(ɛ) such that f κ>κ(ɛ), then there s a statonary soluton w of (.1) such that u (x, t) w (x) ɛ for all x [0, 1] f t s large. Proof. Take the space X := C α ([0, 1], R M )forsomefxedα (0, 1). For the soluton u κ defne A κ := {u κ (t), t [0, + )}. t Because u κ (t) are unformly bounded n C β ([0, 1], R M ) for any β (0, 1) wth bounds ndependent of κ and t (see [] for example; ths can also be proved by the blow up method n [10]), we know that A κ s a pre-compact set of X. Sowecan κ defne the lmt set A := {u X : v κ A κ, such that v κ u n X}. Clam 1. A s a compact, closed set composed of the connectng orbts of the lmt equaton (1.) wth correspondng statonary ponts. Here we add a remark: because we are n the stuaton of an ntal value problem, the connectng orbt may connect the ntal value to a statonary pont of (1.). However, ths only causes mnor changes n the followng proof, because the local compactness and unform convergence stll hold. The dscusson below can be modfed slghtly to deal wth ths case; so we wll omt t. The compactness s from the fact that v κ C β C(β) andv κ u n C[0, 1], whch mples u C β C(β), u A. The second clam that A s nvarant can be shown as follows. Aassume u κ = u κ (t κ ) u. Frst we have for j lm sup u,κ u j,κ =0, κ [0,1] [0,+ )

21 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 981 so u u j 0for j. Next defne v κ (t) =u κ (t κ + t). Then the standard method shows that v κ locally converges (n the sense of [5]) to a soluton v(t) of(1.)wthv(0) = u. If t κ s bounded, then v s defned on [ T 0, + ) forsomet 0 > 0 (notng that for any T>0, u κ (t) converges unformly on [0,T]). Herewemusthavev( T 0 )=ϕ, the ntal value. If t κ +, thenv s defned on (, + ). We need to prove that v s a connectng orbt. In vew of Proposton 3.1, we just need to prove that lm E(v, t) < +, t where the energy E s defned before Proposton Ths can be guaranteed by the unform Lpschtz contnuty of u κ (t) for all κ>0andt>0, whch mples C, such that for any t (, + ), the Lpschtz constant of v(t) s bounded by C. However, f we are n the case b j b j, we don t know whether the unform Lpschtz contnuty s true, so we need to proceed as follows. In vew of the monotoncty of E(t) n the lmt case, we just need to prove that t+1 lm t t E(v, τ)dτ < +. Ths can be seen by takng the lmt n t+1 E(u t κn,τ)dτ. Frst we know that as a measure, κ n u,κn u j,κn converge weakly to a Radon measure ν,whchs supported on {v > 0}. Wth the convergence of u κn to v n C([T 1,T ] [0, 1]) (T 1 T gven), we get T κ n u,κ n u j,κn dxdt lm n + T 1 [0,1] T = lm n + T 1 T = v dν =0. T 1 [0,1] [0,1] κ n v u,κn u j,κn dxdt By multplyng the equaton wth u,κn,weget 1 1 T (4.6) [0,1] u,κ n (T ) [0,1] u,κ n (T 1 )+ u,κn T 1 [0,1] T = a u,κ n 1 3 u3,κ n κ n u,κ n u j,κn. T 1 [0,1] By takng the lmt and notng that other terms converge, we get T T (4.7) lm u,κn = v. n + T 1 [0,1] T 1 [0,1]

22 98 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG Moreover, from (4.6) we see that the above quanttes are unformly bounded dependng on T T 1 and sup u only. Thus t+1 1 v x 1 a v v3 t [0,1] [0,1] are unformly bounded ndependent of t. Ths ends the verfcaton of Clam 1. Now assume all the solutons of (.1) are {w 0,w 1, } (by Theorem.4; ths s a fnte set) wth w 0 = 0 (that s, all of the components are dentcally zero). In the space X, take small open neghborhoods V for each w, and take a neghborhood U of A. We know that for κ large, A κ U (from compactness). We can also take an open neghborhood V 1 of the ntal value ϕ, andϕ s denoted by w 1. Clam. There exst two unversal constants T>T (dependng on our choce of the open neghborhoods V only) such that for any connectng orbt of (1.), ts tme lyng outsde V s smaller than T and greater than T. Ths can be proved by a compactness argument (Note that all of the connectng orbts form a compact set and the statonary ponts are fnte.) Clam 3. ɛ >0, κ 0, such that f κ>κ 0, then there exst some connectng orbts v 1 (t),v (t), of (1.), wth ether u κ (t) V or for some t κ sup u κ (t) v (t t κ ) ɛ. [ T, T ] Here T s the constant n Clam. Ths roughly says that outsde V, u κ (t) can t be rotatng too much around the manfold composed of connectng orbts: t almost goes down wth a connectng orbt drectly. Later we wll show that the number of these connectng orbts are fnte, but at ths stage we can t exclude that there s an nfnte number of them. However, at least we can choose a countable number of these connectng orbts (each occupy a perod of tme T ). The proof of Clam 3 s easy: f u κ (t κ ) v A wth v not n V,thenfrom compactness, on [ T, T ] u κ (t + t κ ) v(t), where v(0) = v. Clam 4. δ >0, dependng only on our choce of V, such that for any connectng orbt v, f for every t [ T, T ], v(t) les outsde V,thenwehave E(v, T ) E(v, T ) δ. Thscanbeprovenbycompactness. Clam 5. κ 0, such that for κ κ 0, f for every t [ T V,thenwehavesomeh>0 ndependent of κ, such that T +h T E(u κ,t)dt T +h T, T E(u κ,t)dt δ. ], u κ(t) les outsde

23 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 983 Assume ths s wrong. Then there exsts a sequence κ n + such that u κn s the soluton of (1.1) wth the fxed ntal value φ, andu κn ([t κn T,t κ n + T ]) le outsde V, but (4.8) tκ n + T +h t κ n + T E(u κ,t)dt tκ n T +h t κ n T E(u κ,t)dt δ. If we assume the unform Lpschtz estmate, then for all t [ T, T ]andn, u κn (t κn + t) are unform Lpschtz contnuous. So after takng a subsequence, t converges to a soluton v(t) of (1.), whch s defned on [ T, T ], and v(t) le outsde V ( V s an open set), and (4.9) T +h T E(v, t)dt T +h T E(v, t)dt δ. Ths contradcts Clam 4. In fact, n the above n order to guarantee that the quantty n (4.8) converge to those n (4.9), we do not need ths strong condton on unform Lpschtz contnuty; see the last part of the proof of Clam 1. Thus our Clam 5 follows. Clam 6. If we choose V small enough, then h >0, κ 0 large enough, such that for any soluton u κ of (1.1) wth κ>κ 0,fu κ ([T 1 h, T 1 ]) and u κ ([T,T + h]) le n V (here T >T 1 ), then T +h T E(u κ,t)dt T1 T 1 h E(u κ,t)dt + δ 4. By (4.6), the gradent terms n the above ntegral can be transformed nto those terms contanng only u,κ. However, u,κ are unformly Hölder contnuous wth respect to the parabolc dstance, so f we choose V and h small enough (dependng on δ only), then for any t [0,h], sup u κ (T 1 h + t) u κ (T + t) δ Ω 16. At last we note that the last term n (4.6) converges to 0 as κ + (usng compactness, we can prove that ths convergence s unform, that s, ɛ >0, κ 0, such that κ >κ 0,thstermssmallerthanɛ>0) and Clam 6 follows. Now we have the followng pcture. If u κ (t 0 )snotn V because t s close to some connectng orbt v(t), from Clam 1, we know that after a tme T, u κ wll enter some V. Moreover, ts energy decays by an amount at least δ. If u κ (t 0 ) V, then for any t>t 0, ether we have u κ (t) V (stayng nsde and not gettng out), and then n vew of our arbtrary choce of V, we conclude; or there exsts a t 1 >t 0 such that u κ (t 1 ) gets outsde of V. Then after a tme T,twllgetntosomeV agan and ts energy decays by an amount at least δ. However, we know the energy E(u κ,t) s bounded, so these procedures can happen at most fntely many tmes, and after some tme t wll stay n some V and never get outsde of t. That s, there exsts a t > 0 such that for any t>t,wehave u κ (t) V for some. In vew of the arbtrary choce of V, we conclude.

24 984 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG The above theorem can be used to prove that: Corollary 4.3. For κ large, any perodc soluton of (1.1) must stay near some statonary pont of the sngular lmt system (1.) n C α [0, 1] for any α (0, 1). Proof. We just need to note that for a perodc soluton, f for large tme t t stays n an open neghborhood V of some w (the same notaton as n the prevous theorem), then for all tme t tstaysnthsopenneghborhood. Now we need to study the structure of orbts for κ large, near a nontrval statonary pont of the lmt equaton. Theorem 4.4. Assume that w s a nontrval statonary soluton of (.1). Then there exsts ɛ, κ 0 > 0 such that f u,κ s a soluton of (1.1) for κ > κ 0 wth u,κ w ɛ for all large t, thenu,κ (t) v unformly on [0, 1] as t +, where v s a nonnegatve nontrval statonary soluton of (1.1) wth v near w (n L ([0, 1]). Proof. By usng omega lmt sets, we see that t suffces to prove that the only solutons u κ of (1.1) defned for all t and satsfyng u,κ w ɛ for all t are the constant solutons (statonary n tme). Suppose ths s false. We frst consder fxed κ. If u,κ s a nonstatonary soluton of (1.1) whch s bounded for all t, standard local parabolc estmates mply that u,κ t are unformly bounded (and at least one s nontrval). By dfferentatng the equaton n tme t we get on [0, 1] R (4.10) ( t x ) u,κ =(a u,κ ) u,κ κ u,κ t t t We can rescale u,κ t (4.11) (4.1) u j,κ κu,κ to v,κ so that (after a translaton n tme) t+1 1 sup v,κ 10 =1, t t v 10,κ 1. u j,κ. t Wth ths ntegral bound, by the proof of Proposton 5.15 (n the nteror) and Proposton 5.18 (near the boundary), we get (4.13) sup v,κ C, [0,1] (,+ ) for some constant C ndependent of κ. Now let κ +. We can prove n some weak sense specfed below that u,κ (x, t) u (x, t),v,κ (x, t) v (x, t), where (1) u (x, t) =w (x).thssbecausefrstwehavefor t (, + ) u (t) w ɛ, and then by Proposton 3.1 and the fact that w s solated we have lm u (x, t) =w (x). t ±

25 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 985 Now we can use the energy dentty (Proposton 3.10) to conclude that u t 0. () v (x, t) 0 outsde {w > 0}. Ths s an easy consequence of Lemma 5.7, Proposton 5.15 (the case near the free boundary) and Proposton 5.18 (dealng wth the boundary pont). (3) In {w > 0} (, + ) (4.14) ( t x )v = a v w v. Ths s because n the equaton of v,κ,forj, u j,κ and v j,κ converge to 0 rapdly n any compact subset of ths doman. Thus we can take the lmt. (4) Near the regular part of {w > 0} {w j > 0} (here all of the free boundares are regular: they are lnes), we have (4.15) ( t x )(v v j )=a v w v a j v j +w j v j. Ths can be proved by Corollary (5) (4.16) (4.17) v Ths can be proved by the combnaton of (4.1) and (4.13), whch mples that for a small δ>0, v,κ 10 1 w δ} 4. [0,1] [0,1]\{ Then because n [0, 1] [0, 1] \{ w δ}, v,κ converge to v unformly, we can take the lmt to get (4.18) [0,1] [0,1]\{ v 10 1 w δ} 4. In our stuaton, we can rearrange v so that f we defne v := v 1 v +, then v satsfes ( t )v = g(x, w)v, x wth g(x, w) := ( 1) +1 (a w ), for x {x [0, 1] w (x) > 0}. Usng the expanson wth egenfunctons of the operator d dx +g(x, w), from the nondegeneracy of (4.1) and the boundedness of v, wecanshow v(x, t) 0, whch contradcts (4.16). Remark 4.5. In the proof of the unform boundedness, (4.13), from the bound on the L 10 norm, (4.1), we can also use the Kato nequalty, as n [18]. That s, by the equatons satsfed by v,κ,weget (4.19) ( t x ) v,κ (a u,κ ) v,κ κ v,κ u j,κ + κu,κ v j,κ.

26 986 E. N. DANCER, KELEI WANG, AND ZHITAO ZHANG Summng over, the last two terms are canceled, and we get (4.0) ( t x ) v,κ (a u,κ ) v,κ. Then standard parabolc estmates gve our bound on sup v,κ. Unfortunately, t seems mpossble to extend ths method to the settng (4.1) ( t x )v,κ =(a u,κ )v,κ κv,κ b j u j,κ + κu,κ b j v j,κ, where b j need not equal b j. On the other hand, our proof can be carred out n ths case. Corollary 4.6. Any nontrval perodc soluton of (1.1) for κ large must stay near 0. In partcular, ts sup-norm s small. Proof. Frst, the method of Theorem 4. gves that for any soluton u κ of (1.1), whchsdefnedon(, + ), there exsts a T>0and a statonary pont w of (1.) such that for any t>t, u κ (t) les near w. Inourstuatonofaperodc soluton, t mples that for all t (, + ), u κ (t) les near w. NowTheorem4.4 says that f w 0,u κ must be statonary. At last we consder the case for those solutons stayng near 0. Theorem 4.7. Wth the assumptons (1), (), and (3), thereexstɛ, κ 0 > 0 such that f u,κ s a soluton of (1.1) for κ>κ 0 wth u,κ ɛ for all large t, then u,κ (t) converges unformly on [0, 1] as t +. Proof. As n Theorem 4.4, we only need to consder solutons defned on all tme. We can assume (by translaton n tme) t+1 1 (4.) sup u,κ = ɛ(κ), t t (4.3) u,κ 1 ɛ(κ). We defne v,κ = 1 ɛ(κ) u,κ, whch satsfes (4.4) v,κ t v,κ x 0 0 = a v,κ ɛ(κ)v,κ κɛ(κ)v,κ v j,κ on [0, 1] R. Takng the lmt we get three cases: (1) κɛ(κ) 0; () κɛ(κ) + ; (3) κɛ(κ) λ for some postve constant λ. We can prove n all of these cases that v,κ are unformly C α contnuous wth respect to the parabolc dstances. (The cases (1) and (3) are easy, and we n fact have hgher unform regularty. Case () can be proved usng the same method of [] or smply by the blow up method n [10].) So we can say as κ + that v,κ v

27 DYNAMICS OF STRONGLY COMPETING SYSTEMS WITH MANY SPECIES 987 locally unformly, and they satsfy the lmt equaton weakly. Moreover, by takng the lmt n (4.) we have t+1 1 (4.5) sup v =1, t t (4.6) v Ths mples global Lpschtz contnuty of v wth respect to the parabolc dstance. We study these cases separately: Case 1. Here we have the lmt equaton v (4.7) t v x = a v on [0, 1] R. We can easly prove, usng the energy dentty, that as t ±, v (x, t) converge to the nontrval statonary solutons. Then by the unqueness of the soluton of (4.5) (because they are lnear ODEs) and the energy dentty agan, we know that v (x, t) =v(x). Ths s a nontrval soluton of (4.5), and we get a contradcton to assumpton (3). Case. Here the lmt equaton s v t v x a v on [0, 1] R, (4.8) ( t x )(v v j ) a v a j v j on [0, 1] R, where v have dsjont support. Here we can prove an energy dentty exactly as n Proposton 3.10, and then the same method n Case 1 gves a contradcton to our assumpton (1). Case 3. Here we have the lmt equaton (4.9) v t v x for some λ>0. By defnng ˆv = λv,weget (4.30) ˆv t ˆv x = a v λv v j on [0, 1] R, = a ˆv ˆv ˆv on [0, 1] R. So ths case can be treated smlar to [18] (cf. the last part of Theorem 4 on page 484) to get a contradcton to assumpton (). Ths completes the proof. Remark 4.8. We wll pursue the hgher dmensonal case n a forthcomng paper. The dffculty n hgher dmenson s, generally, we do not know f the solutons of the ellptc sngular system are fnte. We also do not know f there are fntely many crtcal values of the correspondng functonal.