Reaction-Diusion Systems with. 1-Homogeneous Non-linearity. Matthias Buger. Mathematisches Institut der Justus-Liebig-Universitat Gieen

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1 Reaction-Dision Systems ith 1-Homogeneos Non-linearity Matthias Bger Mathematisches Institt der Jsts-Liebig-Uniersitat Gieen Arndtstrae 2, D Gieen, Germany Abstract We describe the dynamics of a system of to reaction-dision eqations ith 1-homogeneos non-linearity We sho that either an order-presering property holds and can be sed in order to determine the limiting behaior in some (inariant) sets or the long time behaior of all soltions can be described by looking at one scalar reaction-dision eqation only AMS No 35K57 Key ords: Reaction-dision eqations, order-presering, redction to one eqation, oscillation nmber 1 Introdction The dynamics of soltions of one reaction-dision eqation d dt y = y xx + g(t; x; y) (1) ith Dirichlet or other bondary conditions on an interal (0; L), L > 0, has been examined by many athors [10], [11], [12], [15], [18] Far less is knon for a system of to reaction-dision eqations d dt = xx xx + f t; x; ; x 2 (0; L); t > 0: (2) In this paper e examine system (2) ith a 1-homogeneos non-linearity f(; ) = h 1 (; ), hich means that h 1 (; ) = h 1 (; ) for all real Frthermore, e assme that h 1 is C 1 in IR 2 n f(0; 0)g It trns ot that there are to dierent cases, depending on hether the set E := ( 0 2 IR :? sin 0 cos 0 1 h 1 cos 0 sin 0 = 0 )

2 2 MBger is empty or not Case 1 E 6= ; Then E has innitely many elements and e can take ; 2 E, <, sch that 62 E for all < < We rite the initial data ( 0 ; 0 ) in polar coordinates 0 0 = r 0 cos ' 0 sin ' 0 : (3) If r 0 is non-negatie and ' 0 has its ales in the interal (; ), then e sho that the soltion (; ) starting at ( 0 ; 0 ) is attened ot and tends to a planar soltion r 0 cos ' 0 sin ' 0 here ' 0 is a constant and eqals either or We call a state ( 0 ; 0 ) planar, if it can be ritten in the form (4) ith a constant, ie x-independent angle ' 0 We note that ( 0 ; 0 ) is planar if and only if 0 and 0 are linearly dependent A soltion (; ) is called planar if (; )(t) is planar for all t This means that the angle fnction ' might depend on t bt not on the space ariable x It is easy to obsere that (; )(t) is a planar soltion if and only if the initial ale is planar This notation is motiated by the fact that for a planar soltion (; ) the cre (4) t : [0; 1] 3 x 7 (x; (t; x); (t; x)) 2 [0; 1] IR 2 lies in a plane P t (hich depends on t), as one can see in the folloing gre,-axis 6 P t 0 1 Figre 1: cre t for a planar soltion (; ) The proof of the fact that the anglar fnction '(t; ) associated ith (; )(t) conerges either to or to is based on the folloing order-presering property: If (? ;? )(t) is a planar soltion ith '? (0) '(0; ), then e get '? (t) '(t; ) for all t 0 Similarly, e get a fnction ' + Hence, the soltion (; ) lies beteen a loer and an pper planar soltion Then it only remains to sho that the angle fnctions ' (t) of the planar soltions both conerge either to or to

3 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 3 6? 2 ẏ graph of (0; L) 3 x 7 0(x) 0 (x) 2 IR 2 2? '+ O Figre 2: a sitation in hich '?, ' + and ' conerge to We note that in the case E = 2 Z the non-linearity h 1 can be ritten in the form h 1 (; ) = 1(; ) : 2 (; ) Then the maximm priciple yields that soltions (; ) ith initial ale ( 0 ; 0 ), 0 > 0; 0 > 0, stay positie for all t > 0, and or reslt follos from ell knon order-presering properties [9], [14] We note that in this paper e se a dierent ordering relation hich has the adantage that it does also ork if E 6= 2 Z Therefore, or reslt in case 1 is a kind of generaliation of the ell knon orderpresering reslts for 1-homogeneos non-linearities Case 2 E = ; Another concept, hoeer, is needed in the case E = ; In this case, loosely speaking, the part of the non-linearity hich indces a rotation on IR 2 arond the origin does neer anish Hence, e cannot expect the existence of any stationary soltion besides ero, bt e shold be able to obsere periodic (or more diclt) motion heneer ero is nstable The description of the dynamics in this case seems to be really complicated, bt e sho that a major part can be done by looking at planar soltions hich are obtained from a scalar (time-periodic) reaction-dision eqation instead of a system of to eqations This method can be looked at as a generaliation of the approach sed in [3], [4], [5] for the model system d = xx +? + F (5) dt xx As a main reslt e obtain that each bonded soltion of (2) ith 1-homogeneos non-linearity h 1 tends either to a stationary or periodic (planar) soltion, ie e get a Poincare-Bendixson reslt Remarks on both cases We note that the fact that the dision rates coincide in both eqations is a crcial '? ū

4 4 MBger point Nmerical stdies sho that the dynamics can be totally dierent from the cases mentioned aboe for dierent dision rates, in general The restrictions on the non-linearity, hoeer, can be eakened It trns ot that all of or reslts hold for a non-linearity f of the form f(t; x; ; ) = h 1 (; ) + F (t; x; ; ) here F : [0; 1) [0; L] IR 2 IR is C 1 We ill ork ith the more general system d = xx + h dt 1 (; ) + F (t; x; ; ) (6) xx for the rest of this paper 2 Notation and preliminaries Let IR 2 := IR 2 n f(0; 0)g For eery (; ) 2 IR 2 the set f(; ); (?; )g is a base of IR 2 and, ths, h 1 (; ) 2 IR 2 can be ritten in the form h 1 = f 0 +? here f 0 (; ); g 0 (; ) 2 IR are niqely determined Since h 1 2 C 1 (IR 2 ; IR 2 ) and (7) implies that f 0 g 0 = = ? h 1 h 1 g 0 ; ; (7) e get f 0 ; g 0 2 C 1 (IR 2 ; IR) Since for all 2 IR = h 1 f 0 = + f 0? g 0 = h 1? + g 0 ; e obtain that f 0 = f 0 ; g 0 = g 0 : Therefore, it is scient to dene f 0 ; g 0 on S 1 here S 1 is dened by S 1 := f(x; y) 2 IR 2 : x 2 + y 2 = 1g :

5 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 5 In particlar, this implies that f 0 ; g 0 and their deriaties are bonded (on S 1 and, hence, on IR 2 ) For 0 2 IR e introdce the soltion (; 0 ) : IR IR of the ODE d dt = g 0(cos ; sin ) ; (0) = 0 : (8) Since g 0 is C 1 on S 1 (and has, ths, bonded deriaties), is ell dened Lemma 1 The fnction (; 0 ) is either constant or strictly monotone Proof If g 0 (cos 0 ; sin 0 ) = 0, then (t; 0 ) = 0 for all t We assme that g 0 (cos 0 ; sin 0 ) 6= 0, log e only deal ith the case g 0 (cos 0 ; sin 0 ) > 0 If as not strictly increasing, then there old be some t 0 2 IR ith d dt (t 0; 0 ) = 0 hich old imply that g 0 (cos (t 0 ; 0 ); sin (t 0 ; 0 )) = 0 and, ths, old be constant, contradicting the assmption Lemma 2 If g 0 (cos ; sin ) = 0 for some 2 [0; 2), ie 2 E, then (; 0 ) is bonded for all 0 ; if there is no sch, ie E = ;, then (; 0 ) is nbonded Proof 1 If there is as aboe, then eery non-constant fnction (; 0 ) satises (t; 0 ) 6= mod 2 for all t Since (; 0 ) is continos, (IR; 0 ) is contained in an interal of length 2; in particlar, (; 0 ) is bonded 2 If (; 0 ) is bonded, then the limit := lim t1 (t; 0 ) 2 IR exists by Lemma 1 Ths, there is a seqence t n % 1 sch that d dt (t n; 0 ) 0 (n 1) This implies that g 0 (cos ; sin ) = 0 : 3 Main reslts 31 The case E = ; An element ( 0 ; 0 ) of C 1 ([0; L]; IR 2 ) is called planar, if 0 and 0 are linearly dependent oer IR, ie if there are r 0 2 C 1 ([0; L]; IR) and 0 2 IR sch that 0 0 = cos 0 sin 0 The set of all planar elements of C 1 ([0; L]; IR 2 ) is denoted by P r 0 :

6 6 MBger Theorem 1 Let (; ) 2 C 1 ([0; 1)[0; L]; IR 2 ) be a (classical) soltion of system (6) ith Dirichlet or Nemann bondary conditions If E = ;, then e get dist C 1f(; )(t; ); Pg 0 (t 1) : This means that in the case E = ; the -limit set of any (bonded) soltion of (6) is contained in P By Theorem 1, the crcial part of the dynamics of (6) takes place in the set P of planar elements The folloing theorems deal ith the dynamics on P Theorem 2 The set P is positiely inariant, ie eery soltion (; ) of (6) hich starts in P is a planar soltion, ie (; )(t; ) 2 P for all t 0 If ( 0 ; 0 ) = (cos 0 ; sin 0 )r 0 ith 0 2 IR, then (; ) has the form (t; ) = cos (t; 0) sin (t; 0 ) r(t; ) (9) here r : [0; 1) [0; L] IR satises ths scalar eqation " d dt r = r cos (t; xx + r f 0 ) 0 sin (t; 0 ) + F t; x; cos (t; 0 ) sin (t; 0 ) r # (10) ith initial ale r 0 We note that the assertion of Theorem 2 is also tre in the case E 6= ;, bt the dynamics on P is important only if e hae the reslt of Theorem 1 hich ensres that all soltions nally tend to planar ones Theorem 3 Let (; ) be a (bonded) planar soltion and choose 0 ; r 0 ; r as in Theorem 2 If E = ;, then (; 0 ) mod 2 is periodic and the period p is gien by p = inff > 0 : j(; 0)j = 2g : In particlar, p depends only on g 0 If, in addition, (6) is atonomos or F is periodic in t ith a period p F sch that p F =p is rational, then the -limit set associated ith (; ) consists of periodic planar elements only, ie if e take an initial ale ( 0 ; 0 ) 2 (; ), then (; ) is a periodic planar soltion of (6) 32 The case E 6= ; Gien ; 2 IR, <, e call I ; := f( 0 ; 0 ) 2 C 1 ([0; L]; IR 2 ) : ( 0 ; 0 ) = r 0 (cos ' 0 ; sin ' 0 ) ith r 0 ; ' 0 2 C([0; L]; IR) sch that r 0 > 0 and < ' 0 < in (0; L)g the anglar space beteen and If, in addition, ; 2 E and

7 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 7 62 E for all < <, then e call I ; an anglar space associated ith E We note that E 6= ; implies that E contains innitely many elements We sho that an anglar space associated ith E is positiely inariant nder the semio generated by (6) and all soltions ith initial ale in this anglar space tend to planar soltions ith (constant) angle or Theorem 4 Let I ; be an anglar space associated ith E and (; ) a (bonded) soltion ith initial ale (; )(0) 2 I ; Then e get: (a) (; )(t; ) 2 I ; for all t 0 (b) There are continos fnctions ' : (0; 1) C([0; L]; IR) and r : (0; 1) C 1 ([0; L]; IR) sch that = r cos ' sin ' (c) If g 0 (cos ; sin ) > 0 for < <, then '(t; ) conerges to in C([0; L]; IR), in case g 0 (cos ; sin ) < 0 to 4 Interpretation of the reslts If e hae E = ;, then the rotation on the ales of (; ) drien by g 0 does neer stop formally this means that (; 0 ) is strictly monotone and nbonded In this case, or main reslt, Theorem 1, ensres that a major part of the dynamics of (6) can be described by looking only at the dynamics on P, becase e kno that all soltions of (6) tend to P for t 1 Then Theorems 2 and 3 sho that the dynamics on P can be described by soling rst the ODE (8) and then looking at the scalar reaction-dision eqation (10) hich has been stdied in [6], [7] and [8] This is obiosly mch easier than dealing ith the fll system (6); in particlar e can se oscillation nmber reslts in order to attack eqation (10) hich are, in general, not aailable for systems Neertheless, e note that the global attractor of (6) (if it exists) contains, in general, more elements than jst the nion of ( 0 ; 0 ) for all points ( 0 ; 0 ) We can een sho that in some cases the global attractor ill contain non-planar elements The reslts [5] on nstable directions for periodic soltions of the model system (5) cold, for example, be sed in order to proe the existence of non-planar heteroclinic soltions If e hae E 6= ; and 2 E, then (t; ) = for all t and the set f( 0 ; 0 ) := r 0 (cos ; sin ) : r 0 2 C 1 ([0; L]; IR)g is inariant nder the semio indced by (6) In this case Theorem 4 describes the dynamics of all soltions starting in an anglar space I ; beteen to conseqtie elements of E, ie in a space hich is bonded by inariant sets :

8 8 MBger Theorem 4 says that all soltions hich start in the anglar space I ; ill be attened ot and conerge to the bondary of this anglar space (see Figre 2) In the proof of Theorem 4 e ill see that the crcial point is that? '(t 0 ; ) + for some t 0 yields (t? t 0 ;? ) '(t; ) (t? t 0 ; + ) for all t t 0 This (eak) order-presering property together ith the fact that (; 0 ) conerges for any 0 is the main idea of the proof of Theorem 4 We note that otside the anglar spaces een non-planar eqilibria may exist We can, for example, take L = and h 1 (; ) = (; =2) hich has the non-planar eqilibrim (sin x; sin(2x)) (and E = 2 Z) 5 Constrction of an appropriate transformation Let (; ) 2 C 1 ([0; 1) [0; L]; IR 2 ) be a soltion of (6) ith Dirichlet or Nemann bondary conditions For ' 2 IR e consider cos '? sin ' M(') := : sin ' cos ' Gien 0 2 IR e introdce (t; ) := M(?(t; 0 )) (t; ) : An elementary comptation shos that (; ) satises d = xx +? " g dt xx 0 M((t; 0 ))? d # dt (t; 0) + " F t; x; M((t; 0 )) # +f 0 M((t; 0 )) : (11) 6 Oscillation nmber reslts The concept of oscillation nmbers [1], [13], [16] (sometimes also called lap nmbers or Matano nmbers) deals ith soltions y : [0; 1)[0; L] IR of a linear eqation d dt y = y xx + q(t; x)y (12) here q : [0; 1) [0; L] IR is bonded on [0; T ] [0; L] for all T > 0 We dene the nmber of sign changes (the oscillation nmber) by Z(t) := spfk 2 IN : 9 0 < x 1 < : : : < x k < L :

9 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 9 y(t; x j )y(t; x j+1 ) < j k? 1g : It is remarkable that Z(t) is nite for all t > 0, no matter ho many eros the initial state y(0; ) has, as Angenent [1] proed The most important reslt abot oscillation nmbers is that Z : (0; 1) IN is a non-increasing fnction Precisely, the ale of Z(t) decreases heneer y(t; ) has a mltiple ero, ie if y(t 0 ; ) has a mltiple ero, then Z(t 1 ) > Z(t 2 ) for all t 1 < t 0 < t 2 These oscillation nmber reslts play an important role in the examination of a scalar reaction-dision eqation bt can in general not be applied to systems of reaction-dision eqations In order to se these reslts for the fnction dened in the last section, e hae to sho that satises some eqation of the form (12) Since soles " d dt = xx + g 0 M((t; 0 )) " + F t; x; M((t; 0 ))? d # dt (t; 0) + f 0 M((t; 0 )) # by (11), it only remains to sho that q 1 := " g 0 M((t; 0 ))? d dt (t; 0) # is bonded on [0; T ] [0; L] for all T > 0 If q 1 is bonded, then ill sole eqation (12) ith q := q 1 + F t; x; M((t; 0 )) + f 0 M((t; 0 )) and e ill be able to apply oscillation nmber reslts to We assme that q 1 is not bonded Then there is T > 0 and a seqence (t n ; x n ) in [0; T ] [0; L] sch that jq 1 (t n ; x n )j 1 (n 1) and (t n ; x n ) 6= 0 for all n 2 IN We may assme that (t n ; x n ) conerges to (t 0 ; x 0 ) 2 [0; T ] [0; 1] Since g 0 and, by denition of, also d are bonded, this implies that dt (t n ; x n ) 0 (n 1) : (t n ; x n ) Hence, ( n ; n ) dened by n := n 1 q 2 (t n ; x n ) + 2 (t n ; n ) (t n ; x n ) (t n ; x n ) fll n 0 (n 1) This means that j n j 1 (n 1) and log e may assme that n 1 (n 1) Since the map [0; 2] S 1 3 ('; (; )) 7 g 0 M(') 2 IR

10 10 MBger is (niformly) continosly dierentiable, it is Lipschit-continos in (; ) 2 S 1 niformly for all ' 2 [0; 2] ith Lipschit-constant L g Hence, e get g (t 0 M((t n ; 0 )) n ; x n )? d (t n ; x n ) dt (t n; 0 ) = g 0 M((t n ; 0 )) L n g? 1 n 0 = L g q( n? 1) 2? 2 n : n n? g 0 M((t n ; 0 )) 1 0 We dene n 2 IR by ( n ; n ) = (cos n ; sin n ) Then e get tan n = (t n ; x n )=(t n ; 0 ) and n 0 (n 1) Frthermore, it follos that q ( n? 1) 2? n 2 = 2 n sin = O(tan n ) = O (t n; x n ) (n 1) 2 (t n ; x n ) and, ths, q 1 (t n ; x n ) is bonded for n 1 contradicting or assmption 7 Proof of Theorem 1 Step 1 Assme that the assertion is false and constrct a non-planar element in the -limit set nder this assmption By standard argments [12], [17] e kno that the soltions of (6) bild a (local) semio on some Sobole space; in particlar, the orbit? := f(; )(t) : t 0g of the bonded soltion (; ) is a precompact sbset of the space X := D( ), 2 (0; 1) (in the notation of Henry [12]) Taking < 1 sciently large and sing the Sobole embedding theorem, e obtain that? lies precompact in C 1 ([0; L]; IR 2 ) This implies that, if the assertion of or theorem does not hold, then there is ( 0 0; 0) 0 2 C 1 ([0; L]; IR 2 ) n P and a seqence (t n ), t n % 1, sch that jj(; )(t n ; )? ( 0 0; 0 0)jj C 1 0 (n 1) : This means that ( 0 ; ) is contained in the -limit set associated ith (; ) Let ( 0 ; 0 ) be the soltion of (6) ith initial ale ( 0 ; 0 0 ) We note that 0 (0 ; 0 ) is dened for all t 2 IR since ( 0 0; 0) 0 is an element of the -limit set Step 2 Denition of 0 We ant to take 0 2 IR sch that 0 (0; ) dened sing the formla 0 0 (t; ) := M(?(t; 0 )) 0 0 (t; )

11 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 11 has a mltiple ero If 0 or 0 0 hae a mltiple ero, then e can take 0 0 = =2 or 0 = 0 If neither 0 0 nor 0 0 hae mltiple eros, then e proceed as follos: Using the reslt of the last section, e can apply oscillation nmber reslts to 0 for eery 0 Since ( 0 ; 0 ) are also dened for negatie t, 0 soles an eqation of the form (12) for all t 2 IR Then Angenent's reslt [1] ensres that the nmber of eros of 0 (t; ) is nite for all t; in particlar for t = 0 Applying this reslt for 0 = =2 and 0 = 0, e obtain that 0 as ell as hae only nitely many eros (hich are all simple by assmption) Hence, there is a continos fnction ' 0 0 : [0; L] IR hich satises tan ' 0 0 (x) = 0 0(x) 0 0(x) for all x 2 [0; L] ith 0 0 (x) 6= 0 This reslt is an easy conseqence of de l'hospital's formla, and it can also be fond in [2,p696,Lemma 2] We set 0 := maxf' 0 0(x) : x 2 [0; L]g and take x 0 2 [0; L] sch that ' 0 0(x 0 ) = 0 Then e get 0 (0; x 0 ) = 0 sing the denition of 0 It remains to 0 (0; x 0 ) = 0 Case 1 0 (0; x 0 ) = 0 Then 0 (0; x 0 ) = 0 implies that 0 (x 0 0) = 0 (x 0 0) = 0 and, ths, Hence, it follos that tan 0 (x 0 0(x 0 0 (0; x 0 (0; x 0 ) = tan('0 (x 0 0)? 0 ) = 0 : In particlar, 0 (0; x 0 ) = 0 Case 2 0 (0; x 0 ) 6= 0 Let U 0 be a neighborhood of x 0 in [0; L] in hich 0 (0; ) has no eros Since ' 0 0(x) 0 for all x 2 [0; L], e get 0 tan(' 0 0(x)? 0 ) = 0 (0; x) 0 (0; x) for all x 2 U 0 Since 0 (0; ) has no sign change in U 0, it follos that 0 (0; ) mst not hae a sign change in U 0, too Hence x 0 mst be a mltiple ero of 0 (0; ) Step 3 Applying oscillation nmber reslts to 0 As shon in Section 6, e can apply oscillation nmber reslts to 0 Let Z 0 : IR IN be the corresponding oscillation nmber We note that e can dene Z 0 on the hole real axis becase 0 is dened for all t 2 IR (and soles a linear eqation of

12 12 MBger the form (12) on each interal [?; 1), 0) Since 0 (0; ) has a mltiple ero by Step 2, e get Z 0 (t? ) > Z 0 (t + ) for all t? < 0 < t + We note that 0 (t; ) can only hae mltiple eros for contably many t 2 IR since Z 0 has its ales in IN, is non-increasing and decreases each time 0 (t; ) has a mltiple ero Ths, e can choose t? < 0 < t + in a ay that 0 (t ; ) hae no mltiple eros Step 4 Denition of (; ) By assmption, E is empty and, ths, (; 0 ) is nbonded by Lemma 2 Therefore, (; 0 ) has to be periodic; e denote its period by p Wlog e may assme that t n mod p 2 [0; p) has the same ale for all n; otherise e take a sbseqence (t nk ) sch that m k := t nk mod p conerges in [0; p], denote its limit by and set k := t nk? (m k? ) Then e get (; )( k ) ( 0 0; 0), 0 and e can se ( k ) instead of (t n ) Wlog e may assme that = 0; otherise replace (; ) by (^; ^) := (; )(?) hich does also sole an eqation of the form (6) Then e introdce (; ) by (t; ) := M(?(t; 0 )) (t; ) for t 0 We note that (t n ) as chosen in a ay that t n is an integer mltiple of p for all n Ths, e get (t n ; 0 ) = 0 and (t n ; ) = M(?(t n ; 0 )) { } =M (? 0 ) Frthermore, an analogos comptation shos that (t n + t ; ) 0 0 (t n ; ) M(? 0 ) = (t ; ) in C 1 ([0; L]; IR) (0; ) (0; ) : (13) Since 0 (t + ; ) and 0 (t? ; ) hae no mltiple eros, it follos that the nmber of eros of (t n + t ; ) and 0 (t ; ) coincide from some index n on Since Z 0 (t? ) > Z 0 (t + ), the nmber Z(t) of eros of (t; ) accmlates at to dierent ales Z 0 (t ) Since oscillation nmber reslts hold for by Section 6, Z has to be non-increasing hich is a contradiction Remark We note that the assmption E = ; is sed in Step 4 only It ensres that (; 0 ) is nbonded and, ths, periodic Otherise (13) does not hold and e cannot conclde that ( 0 ; 0 )(0) is a limit point of some fnction (; ) dened as aboe

13 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 13 8 Proof of Theorems 2 and 3 Sppose that ( 0 ; 0 ), 0 and r 0 are gien as in Theorem 2 First dene r by (10) and then (; ) by (9) An elementary comptation shos that (; ) is a soltion of (6) ith initial ale ( 0 ; 0 ) Using the niqeness of the soltions of (6) (see, for example, [12]), the assertion of Theorem 2 follos By Lemma 2, (; 0 ) is nbonded In particlar, (; 0 ) is not constant and Lemma 1 implies that it is strictly monotone Wlog assme that (; 0 ) is strictly increasing Since (t; 0 ) is not bonded for t 1, there is a niqely determined p > 0 sch that (p ; 0 ) = : Ths, (; 0 ) is periodic ith period p Frthermore, there is t 0 2 [ 0 ; 0 + p ) sch that (t 0 ; 0 ) = 2k ith some integer k Then it follos that (t 0 + p j; 0 ) = 2(k + j) for all integers j In particlar, e get (t 0? p k; 0 ) = 0 and, ths, (p ; 0) = (p ; (t 0? p k; 0 )) = (t 0? p k + p ; 0 ) = (t 0? p (k? 1); 0 ) = 2 hich shos that p is gien by the expression mentioned in Theorem 3 We assme that F is t-periodic ith period p F here p =p F is rational This means that there is p > 0 ach that p=p as ell as p=p F are positie integers (in the atonomos case, one can jst take p = p ) In particlar, eqation (10) depends periodically on t (ith period p) Let ( 0 ; 0 ) be an element of (; ) and (; ) be the corresponding soltion of (6) Then (; ) is planar by Theorem 1 By Theorem 2, there is 0 2 IR and a fnction r sch that (9) and (10) hold Since ( 0 ; 0 ) = r(0)(cos 0 ; sin 0 ) is an element of (; ), it is a chain-recrrent point Then an elementary analysis like it as done in [5,Theorem 2] shos that r(0) is a also chain-reccrent point for eqation (10) Then [8] ensres that r is a p-periodic soltion of (10) and, since (; 0 ) is p-periodic, (; ) is a p-periodic soltion of (6) 9 Proof of Theorem 4 1 We ill proe the assertion only in the case =?=2 The general case can easily be redced to this sitation setting 0 0 (t) := M(?? =2) (t) : Then ( 0 ; 0 ) soles an eqation similar to (6) replacing F; f 0 ; g 0 by f 0 0 := f 0 M( + =2) ;

14 14 MBger F 0 t; x; g 0 0 := g 0 M( + =2) := F t; x; M( + =2) ; : Hence, e hae 0 =?=2, 0 =?? =2 and the assertion follos sing ' = ' =2 2 We note that =?=2 (Step 1) implies =2 2 E sing the fact that g 0 (0; 1) = g 0 (0;?1) (0-homogenicity) This gies 2 (?=2; =2] Hence, (; )(0) 2 I ; implies 0 = r 0 {} 0 cos ' {} 0 0 : 2(?=2;=2) Frthermore, e get g 0 (0;?1) = g 0 (cos ; sin ) = 0 Since the restriction of g 0 on S 1 is continosly dierentiable, then fnction ^g 0 gien by ^g 0 : S 1 3 (cos ; sin ) 7 g 0(cos ; sin ) cos sin 2 IR is ell dened and continos Setting ^g 0 (; ) := g 0(;) for all (; ) 2 IR 2, ^g 0 is 0-homogeneos, continos and bonded; in particlar, it is contained in L 1 (IR 2 ) Since satises d dt = xx + (F + f 0 )? g 0 = xx + (F + f 0? ^g 0 ) and (0; ) = 0 0, maximm principle argments imply that, for all t > 0, (t; x) > 0 for x 2 (0; L) and x (t; 0); x (t; 1) 6= 0 Ths, there is a continos fnction q : (0; 1) [0; L] IR sch that q(t; x) = (t; x) ; x 2 (0; L) : (t; x) (Note that q(t; 0); q(t; 1) can be dened sing de'l Hospital's formla) We dene ( ) ' : (0; 1) 3 t 7 [0; L] 3 x 7 arctan q(t; x) 2 (?=2; =2) 2 C([0; L]; IR) and r by r(t; ) := (t; )= cos '(t; ) Then, (b) follos We note that the positiity of and the fact that '(t; ) has its ales in (?=2; =2) implies that r(t; ) has only positie ales in (0; L) for all t > 0 3 We take t 0 > 0 and set? := min [0;L] '(t 0 ; ), + := max [0;L] '(t 0 ; ) Note that ' is continos by Step 2 Let = be dened as in Section 5 here the constant 0 shold hae the ale (?t 0 ; ) Then e get (t 0 ; ) = r(t 0 ; ) sin('(t 0 ; )? ) : Using the fact that r(t 0 ; ) is positie by Step 2 and '(t 0 ; )?? 2 [0; ), it follos that? (t 0 ; ) 0 We take x? 2 [0; L] sch that '(t 0 ; x? ) =? Then? (t 0 ; )

15 Reaction-Dision Systems ith 1-Homogeneos Non-linearity 15 has a mltiple ero at x = x? By Section 5, e can apply oscillation nmber reslts to? Hence,? (t; ) is either the ero fnction or strictly positie for all t > t 0 In particlar, e get Analogosly, e get min '(t; ) (t? t 0 ;? ) for all t > t 0 [0;L] max '(t; ) (t? t 0 ; + ) for all t > t 0 [0;L] 4 The soltions of (6) form a (local) semio on the space C 1 ([0; L]; IR 2 ) Hence, (; )(0) 2 I ; implies that there is > 0 sch that < '(t; ) < for all 0 t < Ths, e hae (; )(t) 2 I ; for t 2 [0; ) Take t 0 := =2 and dene as in Step 3 Then e get <? + < and, by denition of the fnction (; ), < (t;? ) (t; + ) < for all t 0 Ths, Step 3 implies that (; )(t) 2 I ; for all t =2 and, sing the reslt mentioned aboe, for all t 0 This proes (a) 5 Take t 0 > 0 and dene as in Step 3 Then e get (; )(t 0 ) 2 I ; by Step 4 and, ths, <? + < : If g 0 (cos 0 ; sin 0 ) > 0 for < 0 <, then Lemmata 1 and 2 yield (t; ) (t 1) Since (t? t 0 ;? ) min [0;L] '(t; ) max '(t; ) (t? t 0 ; + ) for all t > t 0 [0;L] by Step 3, it follos that '(t; ) conerges to in C([0; L]; IR) Analogosly, e get conergence to in the case g 0 (cos 0 ; sin 0 ) < 0 for < 0 < This proes (c) Acknoledgement The athor thanks Prof B Fiedler, Freie Uniersitat Berlin, for helpfl sggestions hich led to the examination of the problem dealt ith in this paper References [1] S Angenent, The ero set of a soltion of a parabolic eqation, J reine ange Math 390 (1988), 79{96 [2] M Bger, Torsion nmbers, a tool for the examination of symmetric reaction-dision systems related to oscillation nmbers, Disc Cont Dyn Sys 4 (1998), 691{708

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