{'fraction_non_alphanumeric': 0.08894040483998643, 'fraction_numerical': 0.02834633796976893, 'mean_word_length': 3.19903450459006, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 86, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions

We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a freeboundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data does not identically vanish, then for sufficiently large interspecific competition, all non-negative solutions of the competition-diffusion system converge to stationary states as time tends to infinity. Such dynamics are much simpler than those found for the corresponding system with either homogeneous Neumann or homogeneous Dirichlet boundary conditions.


Introduction
In this paper, we show that, under certain conditions, the competition-diffusion system with inhomogeneous Dirichlet boundary conditions has simple long-time dynamics for large positive values of the competition parameter k.
Here Ω ⊂ R N is smooth and bounded, f and g are positive on (0, 1) and negative elsewhere, and α > 0. Such reaction-diffusion systems are well-known in the modelling of competition between two species of population densities u(x, t) and v(x, t), and we refer to the introduction of [6] for a brief review. These models can be used to study the dynamics of the spatial segregation between the competing species. The parameters k and α may be thought of as representing the interspecific competition rate and the competitive advantage of v over u respectively. Zero flux (that is, zero Neumann) are the most commonly imposed boundary conditions. But when the two species have quite different preferences for environmental conditions, then competition occurs mainly in a region Ω where their habitats overlap and this gives rise to boundary conditions (2) on ∂Ω [22]. More precisely, we prove that if αm 1 − m 2 is not identically zero on ∂Ω and all stationary solutions of the limit problem are non-degenerate (see Definition 4.1), then for k sufficiently large, all non-negative solutions of (1) approach stationary states as t → ∞. Two remarks on our hypotheses and results should be made at the outset. First, provided we suppose that αm 1 − m 2 is not identically zero on ∂Ω, our system (1) with inhomogeneous Dirichlet boundary conditions has much simpler dynamics than the corresponding system with zero Dirichlet or Neumann boundary conditions. In [14], it is observed that such systems may have solutions that are small (O(1/k)) for all time. To ensure that for large k these solutions converge to a stationary solution of the k−dependent system as t → ∞, it is necessary to impose a condition of there being no "circuits" of positive heteroclinic orbits of an associated limit system (see [14, Assumption C3] and [9]) plus a condition on a linear limit problem ([14, Assumption C1]). No such additional assumptions are needed here to show simple dynamics. Compare Theorem 4.4 with [14,Thm 5].
Second, the condition that all solutions of the stationary limit problem (3) are nondegenerate does not always hold for our boundary conditions -not even in one space dimension. This contrasts with the case of zero Neumann or zero Dirichlet boundary conditions, in which non-degeneracy does hold in one space-dimension -see a remark in [14, p 472]. But some genericity results can be shown for our inhomogeneous Dirichlet case, and we discuss these, together with the possible failure of non-degeneracy in one dimension, in Section 6.
Our methods owe much to [14], which treats (1) with zero Neumann boundary conditions. The idea is first to use a blow-up method to show that for each δ > 0, one of u or/and v must be small at each (x, t) ∈ Ω × [δ, ∞) for sufficiently large k (Section 2). This results in the linear combination w = αu − v satisfying the scalar equation where O k (1) L 2 (Ω) → 0 as k → ∞ uniformly in t ∈ [δ, ∞). Note that here we can only estimate the L 2 -norm of O k (1), rather than the L ∞ -norm, as in [14], if the given boundary data m 1 , m 2 is not assumed to be segregated on ∂Ω. But this L 2 -estimate is sufficient to study the long-time behaviour of (1). The Lyapunov function for (4) with O k (1) = 0 can then be used (Section 3) to show that w must lie close to solutions of (3) for k, t large, under the condition that solutions of (3) are isolated in L 2 (Ω). Section 4 then shows that if these stationary solutions are in fact all non-degenerate, then solutions of (1) must approach stationary states of (1) as t → ∞. Note that the non-degeneracy required in Section 4 does imply the isolatedness used in Section 3, even though the function h in (3) being only locally Lipschitz at its zero set means that the inverse function theorem cannot be applied directly to the operator w → ∆w + h(w) (see, for example, remark (ii) at the end of Section 6). That there is a (locally) unique stationary solution of (1) close to (α −1 w + , −w − ) for w a non-degenerate solution of (3) is shown in Section 5 using index-theory arguments similar to those in [11,10]. Our inhomogeneous boundary values here necessitate careful modification of various arguments in [14,11,10], particularly the blow-up argument in Section 2, and also in the bounds and index arguments used to prove local uniqueness in Section 5. Section 6 is devoted to non-degeneracy of stationary solutions of (3), as mentioned above. We use an approach from [24,8] to show that all stationary solutions of the limit problem are non-degenerate for generic boundary data by applying the version of Sard's Theorem from [25] to a suitable map. Our function h defined in (3) is locally Lipschitz but not in general continuously differentiable; [8] extends the work of [24] to deal with such non-smooth functions, and we use the ideas from [8] here.
This paper follows on from the related work [6], in which a spatial segregation limit is derived for the generalisation of (1) in which the diffusion coefficients of u and v are allowed to differ. It is shown there that for each T > 0, u and v converge in L 2 (Ω×(0, T )) as k → ∞, where in the limit, uv = 0 almost everywhere and w = αu − v is the solution of a limiting free boundary problem. Here, our assumption that the diffusion coefficients of u, v are in fact the same enables us to form the equation (4) which plays a key rôle in the rest of our analysis. It also allows us to establish the key estimates in Section 1 uniformly in t, which enables us to use the Lyapunov-function argument in Section 3. (Note that the argument in Section 1 yields estimates uniform in the unbounded time-interval t ∈ [δ, ∞) for each δ > 0 and we exploit this in the energy argument that is given in Section 3. But estimates uniform on bounded time intervals would in fact be sufficient to obtain the result in Section 3 using a slightly different argument exploiting [18,Theorem 3.4.1] -see [14].) where it is supposed throughout that (a) f and g are continuously differentiable functions on [0, ∞) such that f (0) = g(0) = 0 and f (s) < 0, g(s) < 0 for all s > 1; Some of our results will need the following stronger hypothesis on the limiting boundary behaviour of (u, v); (b5) let Γ 1 , Γ 2 be closed smooth sub-manifolds-with-boundary of ∂Ω, with non-empty interior in ∂Ω, and such that ∂Ω = Γ 1 ∪ Γ 2 . Then m 1 and m 2 in (b3) are such that m i ≡ 0 on Γ j where j = i.
We note the following basic consequence of (b2) and (b3).
By a solution of problem (P k ) we will mean a pair (u, v) such that u, v ∈ C(Q) ∩ C 2,1 (Ω × [t 0 , ∞)) for any t 0 > 0. We will say that (u, v) is a solution of problem (P k, We begin with some standard preliminaries on a priori bounds and global well-posedness for the problem (P k ).
Since L i (0) = 0, i = 1, 2, it follows from the maximum principle that u k , v k ≥ 0. One can then check that L i (M) ≥ 0, i = 1, 2, which completes the proof of Lemma 2.2. 2 Proof. By [17,Thms 9.15 and 9.19], there exist h 1 , h 2 ∈ C ∞ (Ω) ∩ W 2,p (Ω) such that for i ∈ 1, 2, ∆h i = 0 in Ω and h i = m k i on ∂Ω (in the sense of trace). Defining U := u − h 1 , V := v − h 2 allows application of [21,Prop 7.3.2] to the corresponding system for U and V with homogeneous boundary conditions to yield the existence of a unique solution (u k , v k ) of (P k,T ) for some T > 0. That we can take T = ∞ follows from the a priori bounds of Lemma 2.2 and the last part of [21,Prop 7.3.2]. 2 Given the solution (u k , v k ) of (P k ), define Then w k satisfies the equation Note that the explicitly k−dependent terms in (P k ) cancel on forming the equation for w k . Together with Lemma 2.2 and (b2), this gives k-independent bounds for w k which are crucial in the following. Now fix β > 0 and ξ ∈ (0, 1 2 ) and define The following lemma is crucial.
Proof. We adapt the blow-up argument used in the proof of [14,Thm 1], and will prove parts (i) and (ii) in parallel. The first step is to use a contradiction argument to obtain a limiting system for part (i) that is defined on R N × R, and then to use a similar argument for part (ii) to obtain the same limiting equations but defined on either R N × R or on H × R for a half-space H. The second step will be to show that for both possible limit problems, one component must vanish identically, which will lead to a contradiction.
Consider what happens to the system (9) as j → ∞. Note first that since t j ≥ t 0 for each j and k j → ∞, [−k j t j , ∞) tends to R as j → ∞, in the sense that given a compact interval I ⊂ R, there exists j 0 such that I ⊂ [−k j t j , ∞) for all j ≥ j 0 . This will enable us to obtain limiting problems defined for all t ∈ R, which will be vital to conclude our contradiction argument.
Thus given an arbitrary compact subset K of R N , K ⊂ Ω j for j sufficiently large, and hence given and is uniformly bounded away from ∂Ω j × {−k j t j } for j sufficiently large. And since 0 ≤ U k j , V k j ≤ M for all j, it follows from the interior estimates of [20, p 342] that U k j , V k j are bounded independently of j in W 2,1 p (K ×[−T, T ]) for every p ∈ [1, ∞) and thus in C 1+λ, 1+λ 2 (K ×[−T, T ]) for every λ ∈ (0, 1) (see [20, p 5] for the definition of the parabolic space W 2,1 p (K × [−T, T ])). So there is a subsequence of U k j , V k j that converges strongly in C 1+λ, 1+λ 2 (K × [−T, T ]) for each λ ∈ (0, 1).
as j → ∞, passing to the limit in the weak form of (9) yields a weak solution U, V ∈ C 1+λ, 1+λ That in fact U, V ∈ C 2+λ,1+ λ 2 (K × [−T, T ]) and is a classical solution of (10) on int(K × [−T, T ]) then follows immediately from [20, p 224]. And thus by a diagonalisation argument, a subsequence of U k j , V k j converges uniformly on compact subsets of R N × R to a solution U, V of (10) with 0 ≤ U, V ≤ M and U(0, 0) ≥ ǫ 0 , V (0, 0) ≥ ǫ 0 . Now consider part (ii), for which condition (b5) is assumed to hold. Proceeding by contradiction as for part (i), the argument above leading to the system (9) follows through with the single change that now x j ∈ Ω instead of x j ∈ Λ k j . This leads to there being two possible types of limit problem that arise from letting j → ∞ in (9) in this case, depending on the behaviour of the sequence {x j } ∞ j=1 . If dist(0, ∂Ω j ) → ∞ for a subsequence as j → ∞, then exactly as above, we obtain a solution (U, V ) of (10) on R N × R. The second possible type of limit problem arises if {dist(0, ∂Ω j )} ∞ j=1 is bounded. In this case, there is a subsequence (not re-labelled) for which Ω j approaches a half-space H as j → ∞ in the sense of the definition below. There exists a subsequence of {x j } ∞ j=1 which we denote again by {x j } ∞ j=1 and a point x 0 such that Since by the rescaling, and since by hypothesis dist(0, ∂Ω j ) is bounded independently of j, (11) implies that x 0 ∈ ∂Ω. Furthermore, it turns out that ∂H is parallel to the tangent plane to ∂Ω at x 0 . The precise sense of the convergence of the sequence {Ω j } ∞ j=1 is as follows.
Definition. We say that Ω j approaches a half-space H as j → ∞ if : (i) let K ⊂ H be an arbitrary compact set contained in H; then there exists j 0 , depending on K, such that K ⊂ Ω j for all j ≥ j 0 ; (ii) similarly let K ′ ⊂ int(R N \ H) be an arbitrary compact set; then there exists j ′ 0 depending on K ′ such that K ′ ⊂ R N \ Ω j for all j ≥ j ′ 0 .
Note that since 0 ∈ Ω j for each j, 0 ∈ H and it follows as above that as {k j } → ∞, (U k j , V k j ) converges uniformly on compact subsets of H × R to a function pair (U, V ), which is continuous on H ×R and solves (10) on H ×R. The limiting pair (U, V ) is in fact uniformly continuous on H × [−T, T ] for each T > 0. This follows from the fact that for some λ > 0, independently of j for each T > 0, which can be proved in the following steps: (i) straightening the boundary of Ω j locally (as done, for instance, in [17, p97-98]) leads to transforming (9) into a more complicated system for a transformed pair (Ũ k j ,Ṽ k j ) which can be shown to be bounded in the C λ, λ 2 -norm of its flat domain independently of j for some λ > 0 using [20, p204]; (ii) reversing the straightening of the boundary then gives the required uniform Hölder bound on (U k j , V k j ) because the C 2,µ norm of the function which defines the boundary at a point of ∂Ω j is bounded from above independently of j by the C 2,µ norm of ∂Ω.
We now claim that U = m 1 (x 0 ) and V = m 2 (x 0 ) on ∂H × R, where m 1 , m 2 are as in (b3).
So we have a solution U, V of (10) on H × R with 0 ≤ U, V ≤ M. Moreover, by condition (b5), at least one of U, V is identically zero on ∂H. And as in the first possible limit problem above, U(0, 0) ≥ ǫ 0 and V (0, 0) ≥ ǫ 0 .
Thus a contradiction approach to proving both parts (i) and (ii) of Lemma 2.4 leads to a solution of the limit equations (10) on either R N × R or on H × R for a half-space H ⊂ R N .
In what follows we complete the proof by showing that for both possible limit problems, at least one of U, V must be identically zero, which is inconsistent with U(0, 0) ≥ ǫ 0 and V (0, 0) ≥ ǫ 0 . We focus on the details of the case where U, V are defined on H × R; the case when U, V are defined on R N × R is slightly simpler and is treated in [14].
Note first that U and V are constant on ∂H × R. We can suppose that H = {x : x N > 0} without loss of generality, since ∆ is invariant under rotation and translation of the spatial domain. Now extend η : It follows immediately from (10) that on {x N > 0}×R,η is pointwise classically differentiable up to second order in space and first order in time andη t = ∆η. And the extension construction (12) gives that the same holds in {x N < 0} × R, sinceη t = ∆η is autonomous and all spatial derivatives are of even order. Now let φ ∈ C ∞ 0 (R N × R) be supported in a ball B in R N ×R, and note that the outward unit normals ν,ν to {x N > 0} ×R and {x N < 0} ×R respectively are the (N + 1)-vectors ν = (0, . . . , 0, −1, 0) andν = (0, . . . , 0, 1, 0). Then for each i = 1, . . . , N, Green's Theorem gives that since ν = −ν and so the boundary terms cancel. Thusη has weak first order spatial (and likewise, first order time and second order spatial) derivatives that equal the pointwise classical derivatives away from ∂H × R. Soη is a weak solution ofη t = ∆η on any bounded subdomain of R N × R. And since η is continuous on H × R (since U and V are) we have thatη is continuous on R N × R, and hence by [20, p223, Thm 12.1],η is a classical solution ofη t = ∆η on R N × R. So the fact that a bounded solution ofη t = ∆η on R N × R must be constant [3] implies thatη ≡η|

Consider the case when
We will show that U ≡ 0. Note that, unlike in the homogeneous Neumann boundary condition case considered in [14], here it is necessary to consider (13) on H × R rather than on R N × R because our extended functionη is odd rather than even.
Since 0 ≤ U ≤ M and U is constant on ∂H ×R, well-known local estimates [20] imply that U is bounded in C 2+λ,1+ λ 2 uniformly in H × R for each λ ∈ (0, 1). Define z(t) = sup x∈H U(x, t). To see that z is Lipschitz and thus differentiable almost everywhere ([16, p 81]), take s, t ∈ R and let a n ∈ H be such that U(a n , t) ≥ z(t) − 1/n. Then Now fixt ∈ R. Because U ≥ 0 on H ×R, U = 0 on ∂H ×R and U is uniformly continuous on H × {t}, U(·,t) either attains its supremum over x ∈ H at somex ∈ intH or there exists a sequence x n with dist(x n , ∂H) ≥ δ > 0 for every n and U(x n ,t) → sup x∈H U(x, t) as n → ∞. Suppose that U(·,t) attains its supremum atx ∈ intH. It follows from the definition of z that for h > 0, Hence, since ∆U(x,t) ≤ 0 and m 2 (x 0 ) ≥ 0, Now the local estimates [20] clearly imply that a subsequence of U(· + x n , ·) converges uniformly on compact sets of either (13). Note that in both cases, 0 belongs to the interior of the domain ofŨ , that and also thatŨ (0,t) = sup xŨ (x,t).
Next definez(t) = sup xŨ (x, t), t ∈ R. Then sinceŨ satisfies (13) andŨ(·,t) attains its supremum in the interior of the domain ofŨ, the argument given above in the analysis of z applies toz to give It follows that for every t ∈ R, limsup h→0 2 , and hence on the set of full measure on which z is differentiable, If (14) can be integrated to obtain that for any t 0 < t ∈ R, Since t 0 ∈ R was arbitrary, we can let t 0 → −∞ in (15) to find that z(t) = 0 for every t ∈ R.
Recall the definition of w k from (5). Lemma 2.4 yields the following convergence result, which gives the convergence properties that will be used in subsequent sections to analyse the long-time behaviour of solutions of problem (P k ) for large k. and and sup t≥t 0 (ii) If, in addition, m k 1 , m k 2 satisfy the supplementary condition (b5), then (16) and (19) hold with the norm · L 2 (Ω) replaced by the norm · L ∞ (Ω) .
(Here w + := max{w, 0}, w − := min{w, 0} and thus w = w + + w − .) Proof. Fix t 0 > 0 and let ǫ > 0. Lemma 2.4 implies that there exists k 0 such that for each Lemma 2.4 ensures that one of these three possibilities must arise for each ( and hence for R : we have where K f , K g are the Lipschitz constants of f, g respectively restricted to the interval (20) and (21) establishes (16) and (19).
If, in addition, condition (b5) holds, then it follows from Lemma 2.4 that the above argument holds with Λ k replaced by Ω throughout, from which the last statement of Lemma 2.5 is immediate. 2 Remark We conclude this opening section by noting a relation with a special case of [6]. In [6], a spatial segregation limit is derived for the generalisation of problem (P k ) in which the diffusion coefficients of u and v are allowed to differ. It is shown that u k → u and v k → v in L 2 (Ω × (0, T )) for every T > 0, where uv = 0 almost everywhere in Ω × (0, T ) and w = αu − v is the unique weak solution of a limiting free boundary problem (see [6,Section 3] for details).
Now if the diffusion coefficients are in fact the same, then for each 0 < t 0 < T , 3 Long-time behaviour (1) : closeness to stationary solutions of the limit problem In this section we will show that for sufficiently large k, solutions of (P k ) are close to stationary solutions of a certain limit problem for sufficiently large time. The appropriate notion of limit-problem stationary solutions is as follows. Recall the definition of h from Lemma 2.5 and note from (b3) that m 1 , m 2 ∈ W 2,p (Ω). We will say that w ∈ W 2,p (Ω) is a solution of (S) if which immediately implies that w ∈ C 2 (K) for all compact sets K ⊂ Ω and that w ∈ C 1,λ (Ω). Note that the results in this and the following section hold whether or not the supplementary condition (b5) holds.
We first collect some standard regularity, boundedness and compactness results for solutions (u k , v k ) of (P k ) and w k that will be useful in this and the following sections.
and there existsM (independent of k) such that for each t ≥ t 0 , (note that since p > N, these estimates clearly also hold with W 2,p (Ω) replaced by Proof. We use the semiflow framework of [18,Chp 3] in the space X = L p (Ω) × L p (Ω) with the domain W 2,p 0 (Ω) × W 2,p 0 (Ω) for the system with homogeneous boundary conditions discussed in the proof of Lemma 2.3 (respectively X = L p (Ω), domain W 2,p 0 (Ω), for the corresponding homogeneous equation for the linear combination w k ).
Part (a) follows from [18, Thm 3.5.2] and the fact that given β < 1 sufficiently close to 1, the fractional power space X β ⊂ C 1,λ (Ω) for some λ > 0. That for such a λ > 0  (6) give the independence of k of the bound on w k t (·, t) C 1,λ (Ω) ). Part (b) then follows using Lemma 2.2 again, together with [17,Lem 9.17]. For Part (c), note from the bounds on u k t , v k t , w k t just observed, together with Part (b) and condition (a) on f and g, that given t 0 > 0, there are compact (18), (19) from Lemma 2.5, in addition to Part (b) and condition (a). Thus ∆u k (·, t), ∆v k (·, t) ∈Λ ′ t 0 ,k and ∆w k t (·, t) ∈ Λ ′ t 0 and Part (c) follows since ∆ maps W 2,p (Ω) bijectively to L p (Ω). Part (d) is then immediate from (c) and [18,Thm 4.3.3]. 2 The main result of this section is the following.
Theorem 3.2 Suppose that solutions of (S) are isolated in L 2 (Ω). Then given ǫ > 0 and for all t sufficiently large (where how large t needs to be depends on k).
Proof. First some preliminary remarks.
(i) We will show that there exists a solutionw of (S) such that w k = αu k − v k is close tow in L 2 (Ω) for large time. (24) will follow from this together with (16) from Lemma 2.5.
(ii) Denote the set of all solutions of (S) by S. Then S is a compact subset of L 2 (Ω) since S is bounded in W 1,2 (Ω) and thus any sequence in S has a subsequence that converges weakly in W 1,2 (Ω) and L 2 (∂Ω), and strongly in L 2 (Ω), to a limit w which is a solution of the weak form of (S), and hence, by regularity, of (S). Hence S is a finite set, by the assumption that the solutions of (S) are isolated in L 2 (Ω). In the rest of the proof, let (iii) It follows from Lemma 3.1 (c) that for each t 0 > 0, w k (·, t) lies in compact subsets of W 1,2 (Ω) and C(Ω) independently of k and of t ≥ t 0 . Since a continuous bijection on a compact set is a homeomorphism, this implies that the L ∞ -, L 2 -and W 1,2 -norms generate equivalent metrics on the set {w k (·, t) : k ∈ N and t ≥ t 0 }.
(iv) Fix η > 0 and t 0 > 0. Then there exist δ > 0 and k 0 such that if for k ≥ k 0 and t ≥ t 0 , For if not, there exist sequences k j and t j ≥ t 0 such that Then by Lemma 3.1 (c), there exists w ∈ W 1,2 (Ω) such that a subsequence w k j → w in both W 1,2 (Ω) and L 2 (∂Ω). Hence, using (b3) and regularity theory, w is a solution of (S). But this contradicts that w k j (·, t k j ) − w i L 2 (Ω) ≥ η for each 1 ≤ i ≤ r, and (25) follows.
Our approach, which follows closely that in [14], is to show that the natural energy for the limit problem evaluated at w k (·, t) decreases at a certain rate when w k (·, t) lies outside L 2 -neighbourhoods of the elements of S. Choose and fix t 0 > 0. For w ∈ W 1,2 (Ω) ∩ L ∞ (Ω), define an energy where H is a primitive of h. Note first that, by Lemma 3.1 (b), w k (·, t) lies in bounded sets in W 1,2 (Ω) and L ∞ (Ω) independently of k and t ≥ t 0 , and hence E(w k (·, t)) is bounded independently of k and of t ≥ t 0 . Now fix ǫ > 0. By Lemma 2.5, where sup t≥t 0 R(u k , v k ) L 2 (Ω) → 0 as k → ∞ for eacht 0 > 0. And by Lemma 3.1 (a), w k is sufficiently smooth that for t ≥ t 0 , E(w k (·, t)) is differentiable with respect to t, and (27), So if t ≥ t 0 and w k (·, t) − w i L 2 (Ω) ≥ ǫ/4 for every i ∈ {1, . . . , r}, then this together with (25) with η = ǫ/4 gives the existence of δ 1 > 0 and k 0 ∈ N (larger than above if necessary) such that d dt since there exists δ > 0 such that ∆w k + h(w k ) L 2 (Ω) ≥ δ and then k 0 can be chosen so that ∆w k + h(w k ) L 2 (Ω) − R(u k (·, t), v k (·, t)) L 2 (Ω) ≥ δ/2 for k ≥ k 0 . Denote by B R (w) the ball in L 2 (Ω), centre w, radius R. We would like to show that there exists β > 0 such that for k sufficiently large, E(w k (·, t)) is a decreasing function of t outside ∪ r i=1 B βǫ (w i ) and the drop in E(w k (·, t)) when w k (·, t) moves from inside B βǫ (w i ) at some t to the boundary of B ǫ (w i ) (at some later timet) is larger than the possible range of E(w k (·, t)) when w k (·, t) − w i L 2 (Ω) ≤ βǫ. This implies that if w k (·, t) moves from inside B βǫ (w i ) to ∂B ǫ (w i ), then w k (·, t) cannot re-enter B βǫ (w i ) at any later time. Recall remark (iii) and note that it follows that E(w k (·, t)) is close to E(w i ) when w k (·, t) is close to w i in L 2 (Ω) because E(·) is continuous as a function of w ∈ W 1,2 (Ω) ∩ {w ∈ L ∞ (Ω) : w L ∞ (Ω) ≤ M}.
To prove this, we first show that there exists T > 0 such that if t ≥ t 0 and w k (·, t) ∈ B ǫ/4 (w i ) and w k (·, t +t) ∈ B ǫ (w i ),t > 0, thent ≥ T . To see this, recall from Lemma 3.1 (b) that w k (·, t) lies in a bounded set in W 2,2 (Ω) for all k and all t ≥ t 0 , so there exists M 1 > 0 such that ∆w k (·, t) + h(w k (·, t)) L 2 (Ω) ≤ M 1 for all such k and t. And Hence Now this together with (29) implies that in going from B ǫ/4 (w i ) to ∂B ǫ (w i ), E(w k (·, t)) drops by at least T ǫ δ 1 . And we can choose β > 0 (≤ 1 4 ) so that for t ≥ t 0 , Now we can apply (25) with η = βǫ together with (28) to obtaink ≥ k 0 and δ 2 > 0 such that for t ≥ t 0 , when w k (·, t) − w i L 2 (Ω) ≥ βǫ for each i ∈ {1, . . . , r}. Thus for k ≥k, E(w k (·, t)) decreases when w k (·, t) lies in B ǫ (w i ) \ B βǫ (w i ), for some i, and the drop is at least T ǫ δ 1 as w k (·, t) moves from inside B βǫ (w i ) to ∂B ǫ (w i ), since β ≤ 1 4 . It follows using (30) that if w k leaves B βǫ (w i ) and moves out to ∂B ǫ (w i ), it cannot re-enter B βǫ (w i ) at a later time.
Now if w k (·, t) ∈ ∪ r i=1 B βǫ (w i ) for all t sufficiently large, then E(w k (·, t)) would decrease at at least rate −δ 2 for all large time, which would contradict the fact that E(w k (·, t)) is bounded below independently of t ≥ t 0 . Hence there is a sequence of times t n → ∞ for which w k (·, t n ) ∈ ∪ r i=1 B βǫ (w i ), and since there are a finite number of w i , there exists i 0 and a subsequence ( Long-time behaviour (2) : convergence to stationary solutions of (P k ) Note first that here all solutions of (S) are not identically equal to zero, since it is supposed in (b3) that αm 1 − m 2 is not identically zero on ∂Ω. Now observe that (as in [11,14]), it follows from [4] that such solutionsw of (S) only take the value zero on a set of measure zero. Hence h ′ (w(x)) exists for almost every x ∈ Ω. This enables us to make the following definition.
Definition 4.1 A solutionw ∈ W 2,p (Ω) of (S) is said to be non-degenerate if the only solution w ∈ W 2,p (Ω) of the linearised equation is identically equal to zero.
Lemma 4.2 Suppose that a solutionw of (S) is non-degenerate. Then given M > 0, there exist ǫ, k 0 > 0 such that for each k ≥ k 0 , a solution (u k , v k ) of (P k ) that is defined for all t ∈ R and 0 ≤ u k , v k ≤ M and satisfies must have u k t and v k t identically zero on Ω × R. (Note that for this lemma we do not assume that u k (·, 0), v k (·, 0) are given by (b4).) Proof. Our proof follows that in [14,Thm 3] which establishes the corresponding result with zero Neumann boundary conditions. Much of the argument is un-changed and we give an outline here, giving most detail in a blow-up argument where the main differences with [14] lie.
Suppose that the result is false for some M > 0. Then there exist sequences k j → ∞, ǫ j → 0 and solutions (u k j , v k j ) of (P k j ) that are defined for all t ∈ R, 0 ≤ u k j , v k j ≤ M, w k j (·, t) −w L 2 (Ω) ≤ ǫ j for all t ∈ R for the solutionw of (S) but (u is not identically zero on Ω × R. First consider (P k ) for fixed k. Let (u k , v k ) be a solution defined for all t ∈ R with 0 ≤ u k , v k ≤ M and (u k t , v k t ) not identically zero (so (u k , v k ) is a non-stationary solution of (P k )). Now since 0 ≤ u k , v k ≤ M for all t ∈ R, standard parabolic estimates [20] yield that (u k t , v k with the boundary condition (h, l)(x, t) = 0 for (x, t) ∈ ∂Ω × R since (u k , v k ) satisfies time-independent Dirichlet boundary conditions (by (b3)). Since (u k t , v k t ) is not identically zero, we can multiply (u k t , v k t ) by a constant to obtain a solution of (33), called (h k , l k ), say, that satisfies Now a Kato-inequality argument gives that h k and l k are bounded in L ∞ (Ω × R) independently of k; since the proof is identical to that in [14] modulo replacing the zero Neumann boundary conditions for (h k , l k ) in [14] by zero Dirichlet conditions here, we omit the details.
We now use a blow-up argument to deduce that one of h k j and l k j is uniformly small away from the set wherew = 0 if j is large. More precisely, given a compact subset Λ of (intΩ) \ {x :w(x) = 0} and an ǫ 0 > 0, we prove that there exists j 0 > 0 such that Suppose that (35) is false. Then there exist x j ∈ Λ and t j ∈ R such that |h k j (x j , t j )| ≥ ǫ 0 and |l k j (x j , t j )| ≥ ǫ 0 for a sequence of j ′ s tending to infinity (not re-labelled). Without loss, we can assume, by a shift in time, that t j = 0 for every j (note that the u k j and v k j in (33) must also be shifted in time). Now thanks to the uniform-in-k bounds on h k j , l k j obtained above, we can rescale and blow-up (33) much as in the proof of Lemma 2.4. Note that since x j ∈ Λ and Λ is compactly contained in (intΩ) \ {x :w(x) = 0}, any limit point of the x j cannot lie on ∂Ω, and hence rescaling always yields a limit system defined on R N × R (rather than H × R for a half-space H ⊂ R N ). Note also that since w k j (·, t) lies in a compact subset of C(Ω) independently of j and t (since w k j (·, t) lies in a bounded set in C γ (Ω) for some γ > 0, independent of j, t), the fact that sup t∈R w k j (·, t) −w L 2 (Ω) → 0 as j → ∞ implies that sup t∈R w k j (·, t) −w L ∞ (Ω) → 0 as j → ∞. This is because given a compact subset of L ∞ (Ω), the L ∞ -and L 2 -norms generate equivalent metrics on this set due to the fact that a continuous bijection on a compact set is a homeomorphism. Moreover, since Λ ⊂⊂ Ω, it follows as in the proof of Lemma 2.5 (i) that as j → ∞ for each fixed T ∈ R andΛ a compact subset of Ω. So takingΛ with Λ ⊂⊂ Λ ⊂⊂ Ω, we have the existence of j 0 such that x ′ √ k j + x j ∈Λ for j ≥ j 0 for all x ′ ∈ K ⊂ R N compact (where j 0 is independent of x ′ for a given K), and hence the uniform convergence in (36) gives the existence ofx ∈ Λ such that for a subsequence as j → ∞, uniformly in (x ′ , t ′ ) ∈ K ×[−T, T ] for every T > 0 and K ⊂ R N compact. We thus obtain an L ∞ -solution (h,l) on R N × R of such that |h(0, 0)| ≥ ǫ 0 and |l(0, 0)| ≥ ǫ 0 . Now note that sincex ∈ Λ and Λ is compactly contained in Ω 0 \ {x :w(x) = 0}, exactly one ofw + (x) andw − (x) is non-zero. Supposew + (x) = 0. Thenw + (x) > 0 and in the latter case, replacel by −l (which still satisfies (38)) and | −l(0, 0)| > 0. Now as in the proof of Lemma 2.4, define Arguing as in the proof of Lemma 2.4 then gives thaṫ Now since sup (x,t)∈R N ×Rl (x, t) > 0, there exists t with z(t) > 0, so z(s) ≥ z(t) > 0 for all s ≤ t, since (39) implies that z is non-increasing when it is non-negative. Hence for any and so since t 0 < t was arbitrary, we can let t 0 → −∞ in (40) to find that z(t) ≤ 0, which contradicts the above. Similarly, ifw − (x) = 0, the equation forh yields a contradiction. Hence the claim (35) is true. It remains to establish thatŵ k j := αh k j − l k j is uniformly small on Ω × R if j is large. The argument given for the corresponding result in [14,Thm 3] applies almost un-changed and we omit the details. Note that the requirement thatw be a non-degenerate solution of (S) is needed here. The idea is that via a contradiction argument, a non-trivial bounded solution of the linearisation of the parabolic equation satisfied by the limitŵ as k j → ∞ ofŵ k j is obtained. Since this solution is non-trivial, there exists a timet such that w(·,t) ≡ 0, and by the non-degeneracy assumption, there must be a non-zero real eigenvalue λ of the linearisation of (S) such that the L 2 -inner product ofŵ(·,t) with a corresponding eigenfunction φ is non-zero. But then z(t) :=<ŵ(·, t), φ > can be shown to satisfyż = λz, and thus cannot be bounded, which is a contradiction. Note that havingŵ satisfy zero Dirichlet rather than zero Neumann conditions causes no difficulties, and that the fact that as j → ∞ for each T > 0 and each Λ ⊂⊂ Ω, is enough to pass to the limit in the various weak forms of equations obtained by multiplying by smooth functions of compact support in Ω × R (see [14,Thm 3]).
To conclude, suppose that Λ is a compact subset of Ω \ {x :w(x) = 0}. Sinceŵ k j = αh k j − l k j converges uniformly to zero on Ω × R and since, by (35), given ǫ > 0 there exists j 0 such that j ≥ j 0 implies |h k j (x, t)| < ǫ or |l k j (x, t)| < ǫ for each x ∈ Λ, t ∈ R (by the blow-up argument above), it follows that l k j and h k j each converge uniformly to zero on Λ × R as j → ∞. Hence givenǫ > 0, there existsĵ, independent oft, such that for all j ≥ĵ, Now h k j L ∞ (Ω×R) is bounded independently of j and |Ω \ Λ| can be made arbitrarily small by a suitable choice of Λ, since {w(x) = 0} and ∂Ω each have zero n-dimensional measure. So there existsj such that for all t ∈ R, j ≥j implies that h k j L 2 (Ω×[t,t+1]) ≤ 1/8. A similar estimate for l k j L 2 (Ω×[t,t+1]) can be established, giving a contradiction with the normalisation (34) for (h k j , l k j ) ′ . The result follows. 2
Proof. Let ǫ, k 0 be as in Lemma 4.2, and for (fixed) k ≥ k 0 , let (u k , v k ) satisfy the hypotheses above (that such (u k , v k ) exist if there are non-degenerate solutions of (S) follows from Theorem 3.2). Let Γ denote the ω-limit set of (u k 0 , v k 0 ) in W 2,p (Ω) × W 2,p (Ω). Recall Lemma 3.1 (d) and note that the fact that Γ is invariant implies that it consists of the union of trajectories of (P k ) that are defined for all t ∈ R. Now it follows from (41) that each (γ u , γ v ) ∈ Γ satisfies αγ u − γ v −w L 2 (Ω) ≤ ǫ. And by the characterisation of the omega-limit set, given (γ u ,γ v ) ∈ Γ, there exists a solution (η u , η v ) of (P k ), defined for all t ∈ R, such that (i) (η u (·, t), η v (·, t)) ∈ Γ for every t ∈ R, and (ii) (γ u ,γ v ) = (η u (·,t), η v (·,t)) for somet ∈ R.
Since dist((u k , v k )(·, t), Γ) → 0 as t → ∞, by Lemma 3.1 (d), it remains to show that for k sufficiently large, the elements of Γ are isolated in W 2,p (Ω) × W 2,p (Ω). Consider F : Then since W 2,p (Ω) ֒→ C 0,λ (Ω) for some λ > 0, F ∈ C 1 (W 2,p (Ω) × W 2,p (Ω), L p (Ω) × L p (Ω)). Moreover, an argument the same as part of the proof of [11,Thm 1.2] gives that for k sufficiently large, the Fréchet derivative is also surjective and has bounded inverse follows from the Fredholm Alternative. The isolatedness of elements of Γ is then a consequence of the Inverse Function Theorem (see [2,Thm 1.2], for example) and the result follows. 2 We conclude with a result on simple dynamics for (P k ) and some remarks.

Theorem 4.4
Suppose that all the solutions of (S) are non-degenerate. Then there exists k 0 > 0 such that if k ≥ k 0 , there exists a non-negative stationary solution (ũ k ,ṽ k ) of (P k ) such that u k (·, t) →ũ k (·) and v k (·, t) →ṽ k (·) in W 2,p (Ω) and in C 1,λ ′ (Ω) for all λ ′ ∈ (0, λ) as t → ∞. Note that k 0 is dependent on the boundary data m k 1 | ∂Ω , m k 2 | ∂Ω but is independent of the choice of initial data for (P k ).
Proof. Note first that there exists M > 0 such that for any k and any initial data (u k , there exists T (dependent on k and (u k 0 , v k 0 )) such that 0 ≤ u k (·, t), v k (·, t) ≤ M for all t ≥ T . This follows from the fact that z(t) := sup x∈Ω u k (x, t) (similarly, sup x∈Ω v k (x, t)) decreases at a certain rate for t for which z(t) > sup k∈N, x∈Ω {m k 1 (x), m k 2 (x), 2} =: M; indeed one can check that z t ≤ f (z) so that z lies below the solution of the ordinary differential equation U t = f (U) together with the same initial condition sup x∈Ω u 0 (x). Thus z(t) < M for somet; then, since z(t) is decreasing whenever z(t) > sup k∈N, x∈Ω {m k 1 (x), m k 2 (x), 1}, z(t) cannot increase above M for any t >t.
The result now follows immediately from Theorem 3.2 and Theorem 4.3 applied with this value of M. 2 5 On the local existence and uniqueness of stationary solutions of (P k ) close to a non-degenerate solution of (S) for large k We first prove the following result on the existence (and total degree) of positive stationary solutions of (P k ) near (α −1 w + 0 , −w − 0 ) in L p (Ω) × L p (Ω). Note that we assume here that the boundary conditions m k 1 , m k 2 are in fact independent of k and write m k 1 = m 1 , m k 2 = m 2 .
Theorem 5.1 Suppose w 0 is an isolated (in L p (Ω)) solution of (S) which changes sign and has non-zero index. Suppose further that the boundary conditions m k 1 , m k 2 in (P k ) are independent of k. Then there exist k 0 and δ 1 > 0 such that for k ≥ k 0 , (P k ) has a positive stationary solution (u, v) in the δ 1 -neighbourhood in L p (Ω) × L p (Ω) of (α −1 w + 0 , −w − 0 ). Here p is as in condition (b1)and by the index of w 0 we mean the fixed point index in Ω, (42) y = αm 1 − m 2 on ∂Ω.
(Note that we use the notation B 2 here for ease of reference with [10].) Proof. This is an analogue of [10, Thm 3.3], which establishes a similar result for a system with homogeneous Dirichlet boundary conditions. Some key parts of the proof differ from that of [10, Thm 3.3] and so we include a proof here, giving most detail where the differences lie. Consider the homotopy where t ∈ [0, 1]. We first note that positive solutions (u, v) are bounded in L ∞ (Ω) independently of t ∈ [0, 1] and k. Indeed, it follows from (a) that there exists c > 0, independent of (u, v), k and t, such that −∆u ≤ c, −∆v ≤ c. Now let y 1 , y 2 ∈ W 2,p (Ω) be such that −∆y i = c in Ω and y i = m i on ∂Ω, (i = 1, 2). Then the maximum principle gives that u ≤ y 1 , v ≤ y 2 . Since u, v ≥ 0, it follows that there exists a constant M 0 > 0 such that for any non-negative solution (u, v) of (43), Now, as in [10, Thm 3.3], let f 1 (u, v, t) and f 2 (u, v, t) denote the right-hand-sides of the equations for u and v respectively in (43), and define u M = min{u, M}, v M = min{v, M}. Next definef has the same non-negative solution set as (43). Indeed every nonnegative solution pair (u, v) of (45) is such that u, v ≤ M 0 so that (u, v) satisfies (43). Now choose δ > 0 small enough that w 0 is the only solution of (S) in the δ−neighbourhood N δ (w 0 ) of w 0 in L p (Ω). Then choose δ 1 > 0 so that Here Next we prove the following result.
Lemma 5.2 For the above choice of δ 1 , there exists k 0 such that (45) has no non-negative Proof. Because of our boundary conditions, more care is needed here than in the corresponding proof in [10, Lemma 3.1]. Suppose, for contradiction, that there are k n → ∞, t n ∈ [0, 1] such that (45) has a non-negative solution (u n , v n ) ∈ ∂N δ 1 (α −1 w + 0 , −w − 0 ). Then by (44), (u n , v n ) is a solution of (43) for k = k n , t = t n . Setting w n = αu n − v n gives and b n is bounded in L ∞ (Ω) independently of n, again using (44). Now let ψ ∈ W 2,p (Ω) be such that ∆ψ = 0 in Ω and ψ = αm 1 − m 2 on ∂Ω, and set W n = w n − ψ. Then −∆W n = b n , and there is a constant K > 0 such that Hence w n is bounded in W 1,2 (Ω) and thus, taking a subsequence if necessary, there exists w ∈ W 1,2 (Ω) such that w n ⇀ w in W 1,2 (Ω) and L 2 (∂Ω), and w n → w ∈ L 2 (Ω).
We now adapt an idea from [6]. Let φ ∈ W 1,2 0 (Ω) satisfy −∆φ = λφ in Ω, φ = 0 on ∂Ω with λ > 0 and φ > 0 in Ω. Then multiplication of (43) by u n φ, integration by parts and the fact that u n , v n ≥ 0 give and hence there exists K 1 > 0, independent of n, such that Similarly, there exists K 2 , K 3 > 0 such that and (See [6] for details of similar arguments for the parabolic system (P k ).) It follows from (50), (51) and (44) that {u n } ∞ n=1 , {v n } ∞ n=1 are each bounded in W 1,2 (Ω ′ ) and hence relatively compact in L 2 (Ω ′ ) for each Ω ′ ⊂⊂ Ω. Using (44) again, it follows that there are subsequences of u n , v n (not relabelled) and u, v ∈ L ∞ (Ω) such that u n → u, v n → v in L p (Ω) and a.e. in Ω, and by (52), which, together with (48), gives that u = α −1 w + and v = −w − . Note that (53) also gives that (u, v) ∈ ∂N δ 1 (α −1 w + 0 , −w − 0 ). Now take a subsequence if necessary to ensure t n → t ∈ [0, 1]. It follows from (47), (48) and (53) that for φ ∈ C ∞ 0 (Ω), Thus w is a solution of (S). By (46), (48) and (53), w = αu − v ∈ N δ (w 0 ), and thus w = w 0 , so u = α −1 w + 0 and v = −w − 0 . This contradicts that (u, v) ∈ ∂N δ 1 (α −1 w + 0 , −w − 0 ), and completes the proof of Lemma 5.2. 2 Next we return to the proof of Theorem 5.1. Now given k ≥ k 0 , choose M k > 0 sufficiently large thatf for any u, v ≥ 0 and t ∈ [0, 1], and also Then A t is completely continuous (that is, A t is continuous and compact) and maps the natural positive cone P in L p (Ω) × L p (Ω) into itself. Moreover, by Lemma 5.2 and the homotopy invariance of the degree (see, for example, [1, p201]), for k ≥ k 0 , Note that (u, v) = A 0 (u, v) if and only if (u, v) solves (45) with t = 0, and by (44), such As in [10, Thm 3.3], we will show that for k sufficiently large, (60) has a unique non-negative is a non-negative solution of (60), thenw 0 := αu − v is a solution of (S). And hencew 0 = w 0 , by the choice of δ 1 . It follows that αu − v = w 0 , and, since u, v ≥ 0, that Now observe that for any k > 0, the equation has a unique solution u k satisfying u k ≥ α −1 w + 0 . Indeed, α −1 w + 0 is a lower solution of (62). This is because Kato's inequality (see, for example, [19,13]) gives that in the sense of distributions, Then because |w 0 | + w 0 = 2w + 0 , adding (63) and (23) gives that in the sense of distributions, and thus since α −1 w 0 = m 1 − α −1 m 2 ≤ m 1 on ∂Ω and hence α −1 w + 0 ≤ m 1 on ∂Ω, it follows that α −1 w + 0 is a lower solution of (62). And any large positive constant is an upper solution of (62), so there exists a solution u k ≥ α −1 w + 0 . Uniqueness follows from the fact that the right-hand-side of (62) is non-increasing in u when u ≥ α −1 w + 0 . (Note that this argument differs from that used in [10,Thm 3.3] since here u ≡ 0 is no longer necessarily a lower solution if αm 1 − m 2 is large on ∂Ω. ) Next, we prove that u k → α −1 w + 0 in L p (Ω) as k → ∞. This argument is the same as that in [10,Thm 3.3]; we give it here for completeness. Note that u k ≥ α −1 w + 0 and u k (αu k − w 0 ) ≥ 0. This second inequality gives that u k 1 is an upper solution of (62) if k 1 ≤ k and hence u k 1 ≥ u k if k 1 ≤ k. Now let lim k→∞ u k (x) = u(x), x ∈ Ω. Then for any φ ∈ C ∞ 0 (Ω), is a non-negative solution of (60), then w 0 = αu − v solves (S) and w 0 ∈ N δ (w 0 ) so w 0 = w 0 . Thus and hence u = u k and v = αu − w 0 = v k . Thus for k sufficiently large, (u k , v k ) is the unique non-negative solution of (60) in Let Since A 0 maps P into C, it follows from two applications of the commutativity of the fixed point index ( [15, p 214], [23]) that whereC = {(ũ,ṽ) ∈ C 1 0 (Ω) × C 1 0 (Ω) :ũ + h 1 ≥ 0,ṽ + h 2 ≥ 0}, h i , i = 1, 2 satisfy ∆h i = 0 in Ω and h i = m i on ∂Ω, and for (ũ,ṽ) ∈ C 1 0 (Ω) × C 1 0 (Ω), where the inverse (−∆ + M k ) −1 is taken under zero Dirichlet boundary conditions. Now the strong maximum principle gives that u k > 0, v k > 0 in Ω, and that the outward (Note that our inhomogeneous boundary conditions necessitate a slightly different argument here from that in [10,Thm 3.3].) We now use the homeomorphism h(u, v) = (u, αu − v) in C 1 0 (Ω) × C 1 0 (Ω), as in [10,Thm 3.3]. Note that h −1 = h, and that for (ũ,w) ∈ C 1 0 (Ω) × C 1 0 (Ω), So the commutativity of the fixed point index and the product formula give that Hence, by (66) and (69), where B 2 , K are as defined in the statement of the theorem. It remains to show that index C 1 0 (Ω) (B, u k − h 1 ) = 1. First note that α −1 w + 0 and u 0 (the solution of (62) with k = 0) are lower and upper solutions for (62) respectively (neither of which are solutions). Just as in the system (58) we add to both sides of (62) a term of the form M k u with M k large enough so thatB maps the set C * = {ũ ∈ C 1 0 (Ω) : α −1 w + 0 − h 1 ≤ũ ≤ u 0 − h 1 } into itself (in fact, into intC * , by the strong maximum principle). Also, by the uniqueness for (62) discussed above, u k − h 1 is the only solution ofũ =Bũ in C * . Also,B(C * ) is a bounded set in C 1 0 (Ω); choose a large ball B R in C 1 0 (Ω) such thatB(C * ) ⊂ B R . Let S = C * ∩ B R . Then S is a bounded convex set in C 1 0 (Ω) andB maps S into itself. Moreover,B has a unique fixed point in intS. Thus It follows from (59), (64), (71), (72) and the hypotheses of Theorem 5.1 that from which the result follows by the existence property of degree. 2 Theorem 5.3 Suppose w 0 is a non-degenerate solution of (S). Then there exists k 0 > 0 such that for any k ≥ k 0 , (P k ) has a unique positive stationary solution (u, v) near Proof. We mimic the proof of [11,Thm 1.2] which treats the corresponding problem with homogeneous rather than inhomogeneous Dirichlet boundary conditions. That in turn draws on [10,Thm 3.3], of which our analogue is Theorem 5.1 above.
First, it follows exactly as in the proof of [11, Thm 1.2] that if δ 1 > 0 is sufficiently small, then for large enough k, the linearisation of the stationary system (b1)). The next step is to show that for each such k, the value of index P (A 1,k , (u k , v k )) at any positive solution (u k , v k ) of (P k (S)) in N δ 1 (α −1 w + 0 , −w − 0 ) is the same, where P is (as above) the natural positive cone in L p (Ω) × L p (Ω) and A 1,k is defined in (57) (using (55) and (56)) in the proof of Theorem 5.1. This, together with Theorem 5.1, will give the existence of a unique positive solution of (P k (S)) in N δ 1 (α −1 w + 0 , −w − 0 ), since Theorem 5.1 gives the existence of such solutions, and the proof of Theorem 5.1 gives that if δ 1 > 0 is sufficiently small, then for large enough k, (see (73)). Note that index K (B 2 , w 0 ) is +1 or −1 since w 0 is a non-degenerate fixed point of B 2 (B 2 is defined in (42) and in (70)). Now for (u k , v k ) as above, it follows from the commutativity of the fixed point index that just as in (66), for positive solutions of (P k (S)) where C is as defined in (65). And note (as in [11]), that in C, small neighbourhoods of solutions are uniformly close, and hence the truncations in the definition of A 1,k do not affect A 1,k near fixed points. Thus we can delete the truncations and work with the mapÂ 1,k defined to be A 1,k | C without the truncations. As in [11], the reason for doing this is thatÂ 1,k is differentiable. Now (see also (66) and (67)) where has an analogous effect to that in (67) Further, it follows from the proof of Theorem 5.1 that the system . Note that solutions of (75) are fixed points of the operator A 0,k from the proof of Theorem 5.1. And it follows as in the proof of [11, Thm 1.2] that there exists δ 1 > 0 such that if k is sufficiently large, then since the linearisationsÂ ′ 1,k ,Â ′ 0,k act in the space C 1 0 (Ω) × C 1 0 (Ω), with homogeneous Dirichlet boundary conditions, as in [11]. Since we prove that Problem (60) has a unique solution and since the indices of non-degenerate fixed points of nonlinear maps are equal to those of the corresponding linearised maps, it follows that solutions (u k , v k ) of (P k (S)) in N δ 1 (α −1 w + 0 , −w − 0 ) all have the same index γ, where γ = ±1. Now let N be the number of stationary solutions of Problem P k . We deduce from (74) that N ×γ = index K (B 2 , w 0 ) = ±1, which in turn implies that N = 1. 2

On the non-degeneracy condition
Here we discuss the key non-degeneracy condition (31). Consider solutions w ∈ W 2,p (Ω) of the stationary limit problem (S). Note first that, in contrast to the case when w satisfies homogeneous Neumann boundary conditions (see a remark in [14, p 472]), (31) no longer always holds in one space dimension. To see this, let Ω be an interval, say Ω = (0, 1). If m 1 = m 2 = 0 on ∂Ω, it follows from, for example, [27], that there may be many solutions of (S). Now suppose that (αm 1 − m 2 )(0) > 1 and (αm 1 − m 2 )(1) < −1. Then a solution w of (S) must be decreasing in x ∈ (0, 1). This is because the form of h (see (18) and (a)) forbids a local maximum (resp. minimum) of w(x) at x 0 if w(x 0 ) > 1 (resp. < −1) and the fact that (S) has only even order derivatives together with uniqueness for initial-value problems gives that w must be symmetric about any critical point.
where H is a primitive of h, and note that γ w is independent of x for a given w. Hence if w,ŵ are two solutions of (S) with w(0) =ŵ(0) = (αm 1 − m 2 )(0) > 1 and |w ′ (0)| ≥ |ŵ ′ (0)|, then |w ′ (x)| ≥ |ŵ ′ (x)| at any x at which w(x) =ŵ(x). So if (αm 1 − m 2 )(1) < −1, this, together with the fact that w,ŵ are both decreasing, yields that w ≡ŵ, and thus (S) has at most one solution in this case. This shows that (31) cannot hold for every m 1 , m 2 ≥ 0, because if it did, the number of solutions of (S) would be preserved as m 1 , m 2 varied, by the inverse function theorem applied to a suitable projection (in fact, to the mapping π V | F −1 (0) defined in the proof of Theorem 6.1).
However, it is possible to prove some results on non-degeneracy holding for all solutions of (S) for a generic set of boundary data. We use ideas from [24] and [8]. [24] prove that if h ∈ C 1 and h(0) = 0 then for generic φ, the equation ∆w + h(w) = 0 in Ω, w = φ on ∂Ω, has only non-degenerate solutions. [8] extends their main ideas to h with possible discontinuities in h ′ in the context of generic domain (rather than boundary data) dependence. Note that h defined in (18) is locally Lipschitz but is not in general C 1 . Theorem 6.1 (a) There is a dense subset A of {y ∈ W 2,p (Ω) : ∆y = 0 in Ω} such that if φ ∈ A and non-negative m 1 , m 2 ∈ W 2,p (Ω) are such that αm 1 − m 2 = φ on ∂Ω, then every solution of (S) is non-degenerate.
Note first that u+v is not identically zero for any (u, v) ∈ U ×V . This, together with [4], gives that for any solution (u, v) of F (u, v) = 0, u + v only takes the value zero on a set of measure zero. It follows, arguing as in [10, p 468-469] and [7, p 248], that F is strictly differentiable at (u, v) whenever F (u, v) = 0 (see [5, p 48] for the definition of strictly differentiable). This is the key property that allows h to be only locally Lipschitz -see [8]. Next note two technical properties of F . Firstly, F (·, v) : u → F (u, v) is Fredholm of index zero. The mapping u → ∆u is an isomorphism from X onto Z, and u → h ′ (u 0 +v 0 )u is a linear compact operator from X into Z for each (u 0 , v 0 ) ∈ U ×V (note that u 0 +v 0 = 0 a.e., so h ′ (u 0 , v 0 ) makes sense, and that h ′ (u 0 + v 0 ) ∈ L ∞ (Ω), by (a) and (18) ). Secondly, F is proper, in the sense that the set of u ∈ U such that F (u, v) = 0 with v belonging to a compact set in Y is relatively compact in Y . This is because if v n → v in W 2,p (Ω) and F (u n , v n ) = 0, then since {u n +v n } ∞ n=1 is bounded in W 2,p (Ω), {u n } ∞ n=1 is bounded in W 2,p (Ω) and has a subsequence u n k → u in C(Ω) (since p > N/2). Since h is continuous, it follows that h(u n k + v n k ) converges in L p (Ω), so {∆(u n + v n )} ∞ n=1 is relatively compact in L p (Ω), and thus {u n + v n } ∞ n=1 , and so {u n } ∞ n=1 , is relatively compact in W 2,p (Ω).
Most of the properties noted for (a) follow in the same way here. To see that zero is a regular value of F , note that if F (u 0 , v 0 ) = 0, u ∈ U = W 2,p 0 (Ω) satisfies ∆u+h ′ (u 0 +v 0 +ψ)u = 0 a.e and Ω h ′ (u 0 + v 0 + ψ)yu dx = 0 for all y ∈ Y, which yields that ∂u/∂ν = 0 on Γ 2 . This is enough to deduce that u ≡ 0 in Ω -again, see [8, p 144]. The remainder of the proof is the same as for (a). 2 Remarks. (i) Note that if N = 1, the boundary data is necessarily constant on each component of ∂Ω. In particular, (b) implies that if Ω = (0, 1) and we fix m 1 (0) > 0, m 1 (1) = 0 and m 2 (0) = 0, then there is a dense set of positive constant values for m 2 (1) for which every solution of (S) is non-degenerate. When N > 1 and 0, 1 are replaced by Γ 1 , Γ 2 respectively, our method does not give a corresponding result for boundary data constant on each of the components Γ 1 , Γ 2 of ∂Ω since in that case we can no longer deduce from (76) that ∂u/∂ν ≡ 0 on Γ 2 .
(ii) Application of the inverse function theorem to π V | F −1 (0) at a non-degenerate solution u 0 + v 0 of (S) yields isolatedness in W 2,p (Ω) of solutions of (S) equal to v 0 on ∂Ω, from which isolatedness in L 2 (Ω) follows (because solutions of (S) belong to a bounded set in L ∞ (Ω) for a given v 0 , so non-isolatedness in L 2 (Ω) would imply non-isolatedness in L p (Ω), and since solutions of (S) satisfy (23) and h is locally Lipschitz, non-isolatedness in L p (Ω) would imply non-isolatedness in W 2,p (Ω)).