2 JUNCHENG WEI where p; q; r > 0; s and qr > s(p? ). (For backgrounds and recent progress, please see [2], [3], [0], [4], [5], [8], [9], [2], [27], [2

Transcription

1 ON A NONLOCAL EIGENVALUE POBLEM AND ITS APPLICATIONS TO POINT-CONDENSATIONS IN EACTION-DIFFUSION SYSTEMS JUNCHENG WEI Abstract. We consider a nonlocal eigenvalue problem which arises in the study of stability of point-condensation solutions in the Gierer-Meinhardt system and generalized Gray-Scott system. We give some sucient conditions for stability and instability. The conditions are new and can be applied to the study of stability of single point-condensation solutions.. Introduction In the study of many reaction-diusion systems, it is observed numerically that when one of the diusion coecients is very small, there are solutions that concentrate at a nite number of points (we call these solutions point-condensation solutions). point-condensation phenomenon. Such a phenomenon is referred as We mention two reaction-diusion systems which give rise to point-condensations. One is the following generalized Gray-Scott system from chemical-reactor theory 8 < V t = D V V? bv + U q V p ; x 2 U t = D U U? U s V r + a(? U); x @ = 0 (.) where p > ; q > 0; s 0; r > 0; s(p? ) qr; b > a > 0 and N smooth bounded domain. (See [4], [5], [6], [2], [3], [7], [20], [22], [23], [24], [25], [26], [29], etc.) Another one is the following so-called Gierer-Meinhardt system from pattern formation 8 < A t = 2 A? A + A p H?q ; x 2 H t = D H H? H + A r H?(s?) ; x is = 0 (.2) 99 Mathematics Subject Classication. Primary 35B40, 35B45; Secondary 35J40. Key words and phrases. Non-local Eigenvalue Problem, Condensation Patterns, Stability, Instability, Metastability, eaction-diusion Systems.

2 2 JUNCHENG WEI where p; q; r > 0; s and qr > s(p? ). (For backgrounds and recent progress, please see [2], [3], [0], [4], [5], [8], [9], [2], [27], [28], etc. Note that here we have changed H?s in the original Gierer-Meinhardt system to H?(s?) to be consistent with (.).) To study the stability and instability of point-condensation solutions of the above two systems, one can decompose the eigenvalue problem into two problems: small eigenvalue problem (which is caused by the translational invariance of the equations) and large eigenvalue problem (which corresponds to the coupling eect between chemicals). (See Lemma A of [[29], page 599].) It can be shown that the the study of the large eigenvalues is equivalent to the study of the following nonlocal eigenvalue problem,?? + pw p?? (p? ) N w p = ; 2 H 2 ( N ); N where (.3) := qr s(p? ) > ; (.4) 2 C; 6= 0; (y) = (jyj); and w is the unique solution of the following problem w? w + wp = 0; w > 0 in N ; w(0) = max y2 N w(y); w(y)! 0 as jyj! : (.5) (See [28], [29] for details on the derivation of (.3).) By [28] and [29], if problem (.3) admits an eigenvalue with positive real part, then all single point-condensation solutions are unstable, while if all eigenvalues of problem (.3) have negative real part, then all single pointcondensation solutions are either stable or metastable. (Here we say that a solution is metastable if the eigenvalues of the associated linearized operator either are exponetially small or have strictly negative real parts.) Therefore it is vital to study problem (.3). For problem (.3), it is known that when = 0, there exists an eigenvalue = > 0. (See Theorem 2. of [6] and Lemma 2. of [29].) An important property of (.3) is that nonlocal term can push the eigenvalues of problem

3 POINT CONDENSATIONS 3 (.3) to become negative so that the point-condensation solutions of the Gray-Scott system or Gierer-Meinhardt system become stable or metastable. Note that when r 6= p +, problem (.3) is not self-adjoint. Thus problem (.3) may admit complex eigenvalues. In [29] and [28], the eigenvalues of problem (.3) in the following two cases r = 2; or r = p + are studied and the following results are proved. Theorem: () If (p; q; r; s) satises and (A) (B) r = 2; < p + 4 N where ( N +2) N?2 + = N +2 N?2 qr s(p? ) > ; or r = p + ; < p < ( N + 2 N? 2 ) +; when N 3 and ( N +2 N?2 ) + = + when N = ; 2. Then e() <?c < 0 for some c > 0, where 6= 0 is an eigenvalue of problem (.3). (2) If (p; q; r; s) satises (A) and (C) r = 2; p > + 4 N and < + c 0; for some c 0 > 0. Then problem (.3) has an eigenvalue > 0. The case when r 62 f2; p + g was left open. The purpose of this note is to give partial answers in the rest of cases. Note that when r = 2 or p +, problem (.3) has a nice structure: when r = p +, problem (.3) is a self-adjoint operator and when r = 2,? = w. In other cases, we have to work directly with problem (.3), which is dicult to solve. Let L 0 =?+pw p?. It is known that L 0 is an invertible operator from H 2 r ( N ) to L 2 r( N ). (Here H 2 r ( N ) (or L 2 r( N )) denotes the set of radially symmetric functions in H 2 ( N ) (or L 2 ( N )).) We denote the inverse of L 0 as L? 0. Throughout this paper, we assume that > ; r > ; r 6= 2; p +. Let w p+ 0 := q N < : (.6) w 2p w 2 N N Our rst result is on stability.

4 4 JUNCHENG WEI Theorem.. Assume that and + N (L? 0 wr? )? > 0; (.7) p < < + p ; (.8) + 0? 0 where 0 < is given by (.6). Then for any nonzero eigenvalue of problem (.3), we have e() <?c < 0 for some c > 0. emarks: (). When r = 2 or p +, condition (.7) is equivalent to assumption (B). (See [[29], page 600].) By continuity, (.7) is true for < p < p(r) when r is near 2 or p +. (2). The number 0 can be computed and thus condition (.8) can be veried. We remark that in [29] when r = 2 or p +, we don't have any condition on except that >. Theorem. suggests that plays a role on stability when r 6= 2; p +. In fact, from the proof of Theorem. in Section 3, we see that under the condition (.7), either Theorem. is true for all >, or problem (.3) has Hopf bifurcations, namely eigenvalues of purely imaginary, at some >. Our next result is on instability. Theorem.2. If (p; q; r; s) satises + 2r N < p < ( N + 2 N? 2 ) + and < + c 0 ; (.9) for some c 0 > 0. Then problem (.3) has a real eigenvalue > 0. We remark that the linear stability analysis for another scalar non-local problem has previously been conducted by Freitas [7], [8] and [9]. In those papers, he considered the linear operator of non-local problem as a perturbation of a local operator. (Similar approach has been used in [].) Our approach here is not perturbation type. Instead, we work directly with the non-local problem. The organization of this paper is as follows: In Section 2, we prove Theorem.. Section 3 contains the proof of Theorem.2. Finally in Section 4, we discuss some applications to Gray-Scott systems.

5 POINT CONDENSATIONS 5 Acknowledgments: This research is supported by an Earmarked esearch Grant from GC of Hong Kong. The author would like to thank the referees for carefully reading the manuscript and many valuable suggestions. 2. Proof of Theorem. In this section, we prove Theorem.. The approach is similar to Section 2 of [29], though it is more complicated. We assume that r > ; r 6= 2; p + and (.7) and (.8) hold. Let w be the unique solution of (.5) and set Let L :=? + pw p?? (p? ) L 0 =? + pw p? ; N? N w p ; 2 H 2 ( N ): X 0 := kernel(l 0 ) = jj = ; :::; j It is well-known that L 0 admits the following set of eigenvalues > 0; 2 = ::: = N + = 0; N +2 < 0; ::: (2.) where the eigenfunction corresponding to is of constant sign. See, e.g. Theorem 2. in [6]. We denote the eigenfunction corresponding to by 0 (we normalize it so that k 0 k L 2 ( N ) =.) We observe also that L 0 w = (p? )w p and hence L? 0 wp = w. p? Since L is not self-adjoint, we introduce a new operator as follows: L := L 0? (p? ) N? N w p? (p? ) N w p N wr? w p+? +(p? ) N N (? ; 2 H 2 ( N ): ) 2 N We have the following important lemma. Lemma 2.. () L is selfadjoint. (2) The kernel of L consists j ; j = ; :::; N. (3) There exists a positive constant a > 0 such that?(l ; ) (2.2)

6 6 JUNCHENG WEI 2(p? )? w p := (jrj 2 + 2? pw p? 2 ) + N N N w r N w p+?(p? ) N ( ) (? ) 2 2 N N a (; X ) 2 for all 2 H ( N ), where X := j = ; :::; Ng and (u; v) means the inner product in L ( N ), i.e., N (uv). Proof: The rst statement follows easily by direct verication. For (2), it is easy to see that X kernel(l ). On the other hand, if 2 Kernel(L ), then we have where c () = (p?) Hence which implies that c () = (p?)c () N w p N?(p?)? c ()L? L 0 = c ()? + c 2 ()w p N w p+ N? ( N ) 2 ; c 2 () = (p?) 0 wr?? c 2 () p? w 2 kernel(l 0): N? N : w p L? N 0 wr? w p+? L??(p?)c w r () N N 0 wr? ( ) 2 N N = c ()? c ()(p? ) N w p+ N? L? 0 wr? ( N ) 2 : By our assumption (.7), N L? 0 wr?? > 0, this implies that c () = 0; 2 X. It remains to prove (3). Suppose (3) is not true. Since w is exponentially small at innity, L has compact resolvent in H 2 ( N ). By () and (2), there exists (; ) such that (i) is real and positive, j ; j = ; : : : ; N, and (iii) L =. We show that this is impossible. From (ii) and (iii), we have w p? (L 0? ) = (p? ) N wr? + (p? ) N w p (2.3) N N

7 POINT CONDENSATIONS 7? w p+?(p? ) N N (? : ) 2 N We rst show that ( w p ) 2 + (? ) 2 6= 0: (2.4) N N Suppose this is not the case. Then > 0 is an eigenvalue of L 0. By (2.), = and = c 0 for some contant c 6= 0. Since 0 has constant sign, this contradicts with the fact that? w. Therefore 6= ; 0, and hence L 0? is invertible in X? 0. So (2.3) implies w p = (p? ) N w (L r 0? )?? N +(p?) N? N (L 0?)? w p?(p?) N? N w p+ ( N ) 2 (L 0?)?? : Thus we have obtained the following two equations [ p? 0?)? w p (p? ) w p+ )?? N N ( N ) 2 0?)?? )??] N N? + p? N 0? )?? )? w p = 0; N N N (2.5) [ p? 0?)? w p )w p (p? ) w p+? N N ( N ) 2 0?)?? )w p ] N N + [ p? 0? )?? )w p? ] w p = 0: (2.6) N N N Problems (2.5) and (2.6) have a nonzero solution (by (2.4)) if and only if N? the following holds [ p? (p? ) w p+ 0?)? w p )?? N N ( N ) 2 0?)?? )??] N N [ p? 0? )?? )w p? ] N N? p? 0? )?? )? N N [ p? 0?)? w p )w p (p? ) w p+? N N ( N ) 2 0?)?? )w p ] = 0 N N

8 8 JUNCHENG WEI which is equivalent to [ p? 0?)? w p (p? ) w p+ )??] 2 + N N ( N ) 2 0?)?? )? N N? p? 0? )?? )? p? N N 0? )? w p )w p = 0: N N Since p? 0? )? w p )?? = N N 0? )? w)? N N and (p? ) 2 (L 0? )? w p w p = (p? ) (L 0? )? w p L 0 w N N = (p? ) w p+ + (L 0? )? (L 0 w)w N N = (p? ) w p+ + w (L 0? )? ww; N N N we have ( (L 0? )? w? ) 2? (L 0? )? ww (L 0? )??? N N N (2.7)? w 2 (L 0? )??? = 0: N N By (.7), L? N 0 wr?? > 0. This implies that and N (L 0? )??? > 0; for < N (L 0? )??? < 0; for > : In fact, we note that the function h() = N (L 0? )??? is an increasing function and h(0) > 0. Hence h() > 0 for <. Note also that lim!+ h() = 0, which implies that h() < 0 for <. Thus the following function () :=? N 0? )? w)? (2.8) N 0? )?? )? is well-dened for 6=.

9 By the denition of, we have POINT CONDENSATIONS 9 0? )? (w + ()? ))? = 0 (2.9) N Set '() = 0? )? (w + ()? ))(w + ()? ) +? w 2 : N N Then a simple computation shows that equation (2.7) is equivalent to the following We now claim that (2.0) is impossible. '() = 0: (2.0) We rst observe that for near 0, say 2 (0; 0 ), we have '() > 0. Let 0 0 be the rst point for which '( 0 ) = 0. Then necessarily we have ' 0 ( 0 ) 0: Let us now compute ' 0 (): ' 0 () = 0? )?2 (w + ()? ))(w + ()? )??2 w 2 N N + 0 () 0? )?? )(w + ()? ) N = 0? )?2 (w + ()? ))(w + ()? )??2 w 2 : (by 2:9) N N Denote 0 = (L 0? 0 )? (w + ( 0 )? ). We have (L 0? 0 ) 0 = w + ( 0 )? : At 0, we have 2 0 0? 0 )? (w + ( 0 )? )) 2 w 2 : N N That is w 2 : (2.) N On the other hand, that '( 0 ) = 0 implies that N w 2 =? 0 =? 0 N 0w ( 2 0 0(w + ( 0 )? ) N 2 0) 2 ( N N w 2 ) 2

10 0 JUNCHENG WEI which implies that w 2 : (2.2) N Combining (2.) and (2.2), we have that 0 = cw for some constant c and thus c((p?)w p? 0 w) = w+( 0 )? which is impossible since r 6= 2; p+. Hence for 2 (0; ); '() > 0. (In fact, we have proved that if ' 0 () 0, then '() > 0.) Next we consider the case >. Observe that (L 0?)?? blows up as! but 0. Thus as!, ()!? Since 0? w? N 0 w N 0 w N 0.? N 0?? ; (L 0? )? (w? (L 0?)?? k(l 0?)?? k L 2 ( N )! N 0 w N 0?? ) exists. It is easy to see that as!, (L 0? )? (w + ()? )! (L 0? )? (w? N 0 w N 0?? ). Thus lim! '() exists and is nite. Moreover lim! '() 0. We claim that lim! + '() > 0. In fact, suppose not. We have lim! + '() = 0, which implies that for a subsequence i! +, lim i! + ' 0 () 0. The previous arguments leading to (2.2) show that lim i! + '() > 0. A contradiction! Following the previous arguments for >, we have that '() > 0 for >. (3) of Lemma 2. is thus proved. We now nish the proof of Theorem.. Suppose that L = 0 and 0 6= 0. Let 0 = + i I and = + i I. Since 0 6= 0, it follows that? kernel(l 0 ). Then we obtain two equations? L 0? (p? ) N w p =? I I ; (2.3) N? L 0 I? (p? ) N I w p = I + I : (2.4) N Multiplying (2.3) by and (2.4) by I and adding them together, we obtain? N ( I)

11 POINT CONDENSATIONS w r? N = (jr j )? p w p? 2 + (p? ) N N w r? + (jr I j I)? p w p? 2 I + (p? ) N I N N Multiplying (2.3) by w and (2.4) by w we obtain (p?) w p??(p?) N w N p+ = N N (p?) w p? I? (p?) N I w N p+ = N N Hence we have (p? )? w p + (p? )? I N N N w p+ = (p? ) N ((? ) 2 + (? I ) 2 ) N N N + (? w +? I w I ) N N N N + I ( Therefore we have N? I + ( I)+(?2) [ N N +(p? )(? ) 2 +(? 2) I ( N w? N??(L ; )? (L I ; I ) w p+ ( N N w r? N I N? w N? N w N + N N w p N w p N I : N N w? I N w I + I N w I ): N w p I N? I? N ) 2 + (? N I ) 2 N N? w N I ) N N w I ; N w w N I ] N =?(L ; )? (L I ; I ) + J + J 2 = 0 (2.5) where J = ( I)+(?2) [ N N and J 2 = (p? )(? ) 2 N w p+ N ( N? w N + N N? I? N ) 2 + (? N I ) 2 N w N I ] N

12 2 JUNCHENG WEI w r? N I w r? N +(? 2) I ( w N? N w N I ): N We rst note that Theorem. holds when = 2. In fact, suppose on the contrary 0. Then by (2.5) and (3) of Lemma 2. we have?(l ; ) =?(L I ; I ) = 0; N? = which implies that = 0; I = 0: This is impossible. N? I = 0 Next we set ( ; 2 ) to be the largest interval containing 2 such that Theorem. holds at 2 ( ; 2 ). Certainly < 2 < 2. We now show that and 2 +? 0. In fact, suppose not. Without loss of generality, we may assume that > By the denition of, we know that at =, there exists an eigenvalue = + i I 2 C of problem (.3) such that = 0. In this case, we have J = 0: (2.6) It remains to estimate J 2. Observe that multiplying (2.4) by and (2.3) by I and subtracting them together, we have I ( I) = (p? ) ( N N N (? ) N w p I? N (? I ) N w p ) By Schwarz inequality, we obtain s q s (p? ) w 2p j I j N I ( (? )) 2 + ( (? I )) 2 N N N N N and j? I w?? w I j N N N s N s ( (? )) 2 + ( N (? I )) 2 N ( w ) 2 + ( N s s s w 2 N ( (? )) 2 + ( N (? I )) 2 N Thus J 2 (p? )(? ) 2 N w p+ N ( N 2 + N w I ) 2 N? ) 2 + ( N? I ) 2 N N 2 I :

13 POINT CONDENSATIONS 3 q (p? ) w 2p w 2?j? 2j N N (? N ) 2 + (? N I ) 2 : w r N N Since + p +0 < < 2, it is easy to see that which implies that j? 2j s N w 2p N w 2 < (? ) 2 N w p+ J 2 0: (2.7) By (2.5), (2.6), (2.7), and (3) of Lemma 2., we have which implies that?(l ; ) =?(L I ; I ) = 0; J = 0; J 2 = 0 = 0; I = 0: This is impossible. Therefore we have proved that ( + p +0 ; + p?0 ) ( ; 2 ). In other words, if satises (.8), then Theorem. holds. The proof of Theorem. is thus completed. emark: The proof of Theorem. shows that if either > or 2 < +, then there is a Hopf bifurcation at either or 2. and 3. Proof of Theorem.2 In this section, we prove Theorem.2. Note that N (L? 0 w)wr? = L? 0 wp = w p? ; L? 0 w = p? w + xrw; (3.) 2 N ( p? w + 2 xrw)wr? = ( p?? N ) : 2r N (3.2) Assume that + 2r N < p < ( N +2 N?2 ) +; < < + c 0, where c 0 is to be determined later.

14 4 JUNCHENG WEI In this case, we consider the following function h 2 () := 0? )? w p )?? : N (p? ) N Note that for suciently small, we have? (L 0? )? w p =? L? 0 wp +? L?2 0 wp + O( 2 ) N N N = p? p? ( p?? N ) 2r N + Hence h 2 () = ( p?? (p? ) ) N + N + O( 2 ): p? ( p?? N ) + O( 2 ): 2r N Since (p? )?? 2rN? < 0 and (p? )?? ((p? ))? > 0, it is easy to show that there is an > 0 suciently small such that h 2 ( ) = 0, provided that? p? (p?) (? N ) =?? N p? p? 2r p? 2r is suciently small. We now take c 0 to be so small. Now we put = (L 0? )? w p : Since N w = ( (p? ))? N, it is easy to check that L = : Hence L has an positive eigenvalue. Moreover the eigenfunction corresponding to can be chosen to be radial.? 4. Applications to eaction-diffusion Systems In this section, we apply results of Theorem. and Theorem.2 to study the stability and instability of point-condensation solutions of Gray-Scott system. We shall follow the notations used in [29]. It would be helpful if the reader has a copy of [29] at hand. We say (p; q; r; s) satises assumption (D) if (D) r > ; r 6= 2; p + ; L? 0 wr?? > 0; and N

15 + POINT CONDENSATIONS 5 p < < + p : + 0? 0 We say (p; q; r; s) satises assumption (E) if (E) r > ; + 2r N < p < ( N + 2 N? 2 ) +; < < + c 0 where c 0 is the small number in Theorem.2. Let (V B ; ; U B ;) be the solutions constructed in Theorem. of [29]. Then we have Theorem 4.. For << ; <<, (V B ; ; U B ;) is linearly unstable if either P 0 is a nondegenerate critical point of H(P ) such that G B (P 0 ) contains one positive eigenvalue or (p; q; r; s) satises assumption (E). (V B ; ; U B ;) is linearly stable if P 0 is a nondegenerate critical point of H(P ) such that G B (P 0 ) contains no positive eigenvalue and (p; q; r; s) satises assumption (D). (V B ; ; U B ;) is metastable if P 0 is a nondegenerate critical point of H(P ) such that G B (P 0 ) contains at least one positive eigenvalue and (p; q; r; s) satises assumption (D). Similarly, let (V I ;;N ; U I ;;N) and (V I ;;D ; U I ;;D) be the solutions constructed in Theorems.2 and.3 of [29], respectively. Then we have Theorem 4.2. Let P 0 be a nondegenerate peak point. Then for << ; <<, (V I ;;N ; U I ;;N) is always linearly unstable. (V I ;;N ; U I ;;N) is metastable if (p; q; r; s) satises assumption (D). Theorem 4.3. Let P 0 be a nondegenerate peak point. Then for << ; <<, (V I ;;D ; U I ;;D) is linearly unstable if (p; q; r; s) satises assumption (E). (V I ;;D ; U I ;;D ) is linearly stable if (p; q; r; s) satises assumption (D). Finally, we remark that the results of [28] can be extended accordingly. eferences [] A. Bose and G. Kriegsman, Stability of localized structures in non-local reactiondiusion equations, to appear. [2] M. del Pino, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system, Indiana Univ. Math. J. 43 (994), pp

16 6 JUNCHENG WEI [3] M. del Pino, P. Felmer and M. Kowalczyk, Boundary spikes in the Gierer- Meinhardt system, preprint (999). [4] A. Doelman, A. Gardner and T.J. Kaper, Stability of singular patterns in the -D Gray-Scott model: A matched asymptotic approach, Physica D. 22 (998), -36. [5] A. Doelman, A. Gardner and T.J. Kaper, A stability index analysis of -D patterns of the Gray-Scott model, Technical eport, Center for BioDynamics, Boston University, submitted. [6] A. Doelman, T. Kaper, and P. A. egeling, Pattern formation in the onedimensional Gray-Scott model, Nonlinearity 0 (997) [7] P. Freitas, Bifurcation and stability of stationary solutions on nonlocal scalar reaction diusion equations, J. Dyn. Di. eqn. 6(994), [8] P. Freitas, A non-local Sturm-Liouville eigenvalue problem, Proc. oy. Soc. Edin. 24 A (994), [9] P. Freitas, Stability of stationary solutions for a scalar nonlocal reaction diusion equation, Q. J. Mech. Appl. Math. 48(995), [0] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 2(972), [] P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci. 38(983), [2] P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B! 3B; B! C, Chem. Eng. Sci. 39(984), [3] J.K.Hale, L.A. Peletier and W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, Technical eport, Mathematical Institute, University of Leiden, submitted, 998. [4] D. Iron and M. J. Ward, A metastable spike solution for a non-local reactiondiusion model, SIAM J. Appl. Math., to appear. [5] D. Iron, M. J. Ward and J. Wei, On the eect of strong coupling on the stability of multiple spike solutions, preprint (999). [6] C.-S. Lin and W.-M. Ni, On the diusion coecient of a semilinear Neumann problem, Calculus of variations and partial dierential equations (Trento, 986) 60{74, Lecture Notes in Math., 340, Springer, Berlin-New York, 988. [7] C.B. Muratov, V.V. Osipov, Spike autosolitions in Gray-Scott model, Los Alamose-print, patt-sol/980400, submitted. [8] W.-M. Ni and I. Takagi, Point-condensation generated by a reaction-diusion system in axially symmetric domains, Japan J. Industrial Appl. Math. 2 (995), [9] W.-M. Ni, I. Takagi and E. Yanagida, submitted. [20] Y. Nishiura and D. Ueyama, A skeleton structure of self-replicating dynamics, preprint, Laboratory of Nonlinear Studies, Hokkaido University, submitted. [2] W.-M. Ni, Diusion, cross-diusion, and their spike-layer steady states, Notices of Amer. Math. Soc. 45 (998), 9-8. [22] Y.Nishiura, Global structure of bifurcation solutions of some reaction-diusion systems, SIAM J. Math. Anal. 3(982), [23] V. Petrov, S.K. Scott, K. Showalter, Excitability, wave reection, and wave splitting in a cubic autocatalysis reaction-diusion system Phi. Trans. oy. Soc. Lond., Series A 347(994),

17 POINT CONDENSATIONS 7 [24] J.E. Pearson, Complex patterns in a simple system, Science 26, pp [25] J. eynolds, J. Pearson and S. Ponce-Dawson, Dynamics of self-replicating spots in reaction-diusion systems, Phy. ev. E 56 ()(997), [26] J. eynolds, J. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction diusion systems, Phy. ev. Lett. 72 (994), [27] J. Wei and M. Winter, On the two dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal., to appear. [28] J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness, spectrum estimates and stability analysis, Euro. J. Appl. Math., to appear. [29] J. Wei, Existence, stability and metastability of point condensation patterns generated by Gray-Scott system, Nonlinearity 3(999), Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong address: wei@math.cuhk.edu.hk