J. Math. Anal. Appl. 325 (2007) 1216–1239 www.elsevier.com/locate/jmaa Patterns in parabolic problems with nonlinear boundary conditions Alexandre N. Carvalho a,∗,1, German Lozada-Cruz b,2 a Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Cx. Postal: 668, São Carlos, Brazil b Instituto de Biociências, Letras e Ciências Exatas - IBILCE/UNESP, 15054-000 São José do Rio Preto, Brazil Received 9 June 2005 Available online 24 March 2006 Submitted by P. Broadbridge Abstract We obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator in such domains. This information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and a uniform trace theorem. © 2006 Elsevier Inc. All rights reserved. Keywords: Semilinear parabolic problems; Nonlinear boundary conditions; Dumbbell domains; Stable nonconstant equilibria; Invariant manifolds 1. Introduction In this paper we study the existence of (asymptotically) stable nonconstant equilibrium solu- tions for semilinear parabolic problems of the form * Corresponding author. E-mail addresses: andcarva@icmc.usp.br (A.N. Carvalho), german@ibilce.unesp.br (G. Lozada-Cruz). 1 Research partially supported by CNPq # 305447/2005-0 and by FAPESP # 03/10042-0, Brazil. 2 Research supported by FAPESP # 00/01479-8, Brazil. 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.02.046 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1217 ⎧⎪⎪⎨ ⎪⎪⎩ ut = δ εN−1 �u + f (u) in Ωε, δ εN−1 ∂u ∂n = g(u) in ∂Ωε. (1.1) Here Ωε is a bounded smooth domain consisting of two fixed disconnected parts and a channel (ε-dependent) connecting them, ∂ ∂n is the outer normal derivative on the boundary ∂Ωε of Ωε , ε ∈ (0,1] is a parameter, δ > 0 is a constant and f,g are nonlinear functions satisfying certain growth and dissipativeness conditions that will be specified later. It is well known that stable nonconstant equilibrium solutions do not exist for parabolic prob- lems of the form⎧⎨ ⎩ ut = �u + f (u) in D, ∂u ∂n = 0 in ∂D, (1.2) when D is convex (see [10,21]). Therefore it is natural to seek for existence of stable nonconstant equilibrium solutions of (1.2) when D is not convex. Of course stable nonconstant equilibrium to (1.2) exists when D is disconnected and it becomes natural to investigate the existence of stable nonconstant equilibrium for (1.2) in domains consisting of two disconnected parts connected by a thin channel (dumbbell type domains). In [21], H. Matano shows an important result on existence of patterns for (1.1) and Ωε is a dumbbell type domain (as in Fig. 1). It also becomes important to detect which is the limiting problem when the channel shrinks to a one-dimensional domain and which dynamics of the limiting problem can be observed in the dumbbell type domains. In this direction are the works of Hale and Vegas [14], Vegas [24] and Jimbo [16–18]. In [11], N. Cònsul and J. Solà-Morales extended Matano’s results proving the existence of stable nonconstant equilibria for diffusion equations with nonlinear boundary conditions⎧⎨ ⎩ ut = �u in D, ∂u ∂n = kf (u) in ∂D. (1.3) Later, in [12], they obtain an abstract result on the stability of local minima of semilinear prob- lems of the form ut = Au + F(u) and apply this result to obtain stable nonconstant equilibrium to (1.1), (1.2) and to strongly damped semilinear wave equations with homogeneous Neumann boundary conditions. Jimbo and Morita in [19] studied the eigenvalue problem for the Laplacian with Neumann boundary conditions in a domain Ω ⊂ R N that consists of k fixed (disconnected) domains and thin channels joining them. The volume of each of thin channels is controlled by a small parame- ter ε > 0, and these channels shrink to a line segment as ε approaches zero (some of the channels Fig. 1. Domain Ωε . 1218 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 may be empty). They characterize the eigenvalues and eigenfunctions for the operator −� in this domain. Using the invariant manifold theory, they show in [23] (see also [22]) that the dynamics of (1.2) in such domains is equivalent to the dynamics of an explicitly given system of ordinary differential equations on an invariant manifold. Also, there are several works where eigenvalue problems for elliptic operators on varying domains or with varying diffusivity are studied (see, for example, [5–8,14,18,24]). Here, we use the invariant manifold theory and a uniform trace theorem to show that the dynamics of (1.1) is equivalent to the dynamics of a 2-dimensional ordinary differential equations on an invariant manifold. Patterns obtained by nonlinear boundary conditions are specially interesting for, in a control problem involving reaction–diffusion models the boundary is, in general, the only accessible part of the domain. The difference between our result and the results of Consul and Sóla-Morales [11,12], apart from the technique, is that the stable nonconstant equilibria obtained here are asymptotically stable while in [11,12] they are only stable. Let us now introduce the assumptions on the nonlinearities f,g. Suppose that f,g : R → R are twice continuously differentiable functions satisfying lim sup |u|→∞ f (u) u < 0 and lim sup |u|→∞ g(u) u < 0. In addition, assume some growth assumptions to ensure local well posedness of (1.1) (see [4]). Under these assumptions, problem (1.1) has a global attractor Aε , 0 � ε � ε0, and sup0�ε�ε0 supu∈Aε ‖u‖L∞(Ωε) < ∞ (see [4]). This enables us to cut f and g in such a way that the attractors Aε remain the same and in such a way that f,g together with its derivatives up to second order are bounded. Hereafter we assume (without loss of generality) that f,g are bounded functions with bounded derivatives up to second order. Next we specify the domain Ωε ⊂ R N (N � 2). It has a fixed part Ω and a parameter de- pendent part Rε , that is, Ωε = Ω ∪ Rε , where Ω = ΩL ∪ ΩR. We assume that ΩL, ΩR and Rε satisfy the following conditions: (I) ΩL, ΩR are bounded smooth domains in R N with disjoint closures. (II) There is an orthogonal system of coordinates x = (x1, x2, . . . , xN) = (x1, x ′) ∈ R N such that the following conditions hold for some positive constant ε0 > 0: ΩL ∩ { (x1, x ′) ∈ R N : x1 � 0, |x′| < ε0 } = { (0, x′) ∈ R N : |x′| < ε0 } , ΩR ∩ { (x1, x ′) ∈ R N : x1 � 1, |x′| < ε0 } = { (1, x′) ∈ R N : |x′| < ε0 } and Rε is expressed as follows Rε = { (x1, x ′) ∈ R N : 0 < x1 < 1, |x′| < εh(x1) } , where h ∈ C0([0,1]) ∩ C1((0,1)) and h(x1) � 0 for all x1 ∈ [0,1]. Define the line segment L = ⋂ ε∈(0,1] Rε = { (z,0, . . . ,0) ∈ R N : 0 � z � 1 } ∼= {z ∈ R: 0 � z � 1} whose endpoints are P0 = (0,0, . . . ,0) and P1 = (1,0, . . . ,0). A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1219 Without loss of generality we assume that |Ω| = 1 throughout the paper. If we denote by {λε n}∞n=1 and {ϕε n}∞n=1 the set of eigenvalues and the orthonormalized eigen- functions of the following eigenvalue problem⎧⎨ ⎩ −�u = λεu in Ωε = Ω ∪ Rε, ∂u ∂n = 0 in ∂Ωε, (1.4) then we have the following result. Theorem 1. The following holds: λε 1 = 0 and ϕε 1 = |Ωε |−1/2, ∀ε > 0, lim ε→0 λε 2 εN−1 = Co, ϕε 2 ε→0−−−→ ⎧⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ c0 1 := − √ |ΩR| |ΩL| , in ΩL, c0 2 := √ |ΩL| |ΩR| , in ΩR, in L2(Ω) and L2(∂Ω), sup ε>0 ∥∥ϕε 2 ∥∥ L∞(Ωε) < ∞, lim inf ε→0 λε 3 > 0, where Co = σN−1 ( 1 |ΩL| + 1 |ΩR| ) θ, θ = { 1∫ 0 dx1 hN−1(x1) }−1 and σN−1 is the Lebesgue measure of the unit ball in R N−1. A proof of this result (with stronger convergence properties for the eigenfunctions) is given in Appendix A. If u is a solution of problem (1.1), consider the following decomposition u(t, x) = u1(t)ϕ ε 1(x) + u2(t)ϕ ε 2(x) + ω(t, x), where u1, u2 and w are given by u1 = ∫ Ωε uϕε 1, u2 = ∫ Ωε uϕε 2, ω = u − u1ϕ ε 1 − u2ϕ ε 2 . This decomposition induces the following decomposition of (1.1) u̇1 = ∫ Ωε f (u)ϕε 1 + ∫ ∂Ωε γ ( g(u) ) γ ( ϕε 1 ) , u̇2 = −δ λε 2 εN−1 u2 + ∫ Ωε f (u)ϕε 2 + ∫ ∂Ωε γ ( g(u) ) γ ( ϕε 2 ) , ωt = δ εN−1 �ω + f (u) − [ ∫ f (u)ϕε 1 + ∫ γ ( g(u) ) γ ( ϕε 1 )] ϕε 1 Ωε ∂Ωε 1220 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 − [ ∫ Ωε f (u)ϕε 2 + ∫ ∂Ωε γ ( g(u) ) γ ( ϕε 2 )] ϕε 2, δ εN−1 ∂ω ∂n = g ( u1ϕ ε 1 + u2ϕ ε 2 + ω ) , (1.5) where γ (ϕ) denotes the trace of ϕ. In what follows, we give a heuristic argument to find the limiting ode that should contain the asymptotic behavior of (1.5). To that end, we introduce a very simple lemma from semigroup theory. Its proof follows immediately from the variation of constants formula. Lemma 2. Let X be a Banach space and A :D(A) ⊂ X → X be the generator of a strongly continuous semigroup of bounded linear operators {eAt : t � 0}. If f :X → X is a locally Lipschitz continuous map which satisfies supx∈X M‖f (x)‖X � N < ∞ and, for some ν > 0, ‖eAt‖L(X) � Me−νt and x(t, x0) is the global solution to ẋ = Ax + f (x), x(0) = x0, then ∥∥x(t, x0) ∥∥ X � Me−νt‖x0‖X + N ν . After this lemma we see that, asymptotically, the norm of the solution is proportional to 1 ν and, if ν is very large (compared to N ), all solutions are asymptotically small. Since the third eigenvalue λε 3 εN−1 goes to infinity when ε → 0, we guess that ω is not important in the asymptotic behavior (1.1) (for very small values of ε) and we have ⎧⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ u̇1 ∼ ∫ Ωε f ( u1ϕ ε 1 + u2ϕ ε 2 ) ϕε 1 + ∫ ∂Ωε g ( u1ϕ ε 1 + u2ϕ ε 2 ) ϕε 1, u̇2 ∼ −δ λε 2 εN−1 u2 + ∫ Ωε f ( u1ϕ ε 1 + u2ϕ ε 2 ) ϕε 2 + ∫ ∂Ωε g ( u1ϕ ε 1 + u2ϕ ε 2 ) ϕε 2 . According to Theorem 1, we should have the limit system given by⎧⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ u̇1 = ∫ Ω f (u1 + u2φ2) + ∫ ∂Ω g(u1 + u2φ2), u̇2 = −δ Cou2 + ∫ Ω f (u1 + u2φ2)φ2 + ∫ ∂Ω g(u1 + u2φ2)φ2, (1.6) where φ2 = c0 1χΩL + c0 2χΩR . Or, in terms of c0 1, c0 2 and Co given in Theorem 1, ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ u̇1 = ∣∣ΩL ∣∣f ( u1 + c0 1u2 )+ ∣∣ΩR ∣∣f ( u1 + c0 2u2 ) + ∣∣∂ΩL ∣∣g(u1 + c0 1u2 )+ ∣∣∂ΩR ∣∣g(u1 + c0 2u2 ) , u̇2 = −δCou2 + ∣∣ΩL ∣∣c0 1f ( u1 + c0 1u2 )+ ∣∣ΩR ∣∣c0 2f ( u1 + c0 2u2 ) + ∣∣∂ΩL ∣∣c0g ( u + c0u )+ ∣∣∂ΩR ∣∣c0g ( u + c0u ) . (1.7) 1 1 1 2 2 1 2 2 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1221 The variables u1 and u2 may not be the best choice of variables to study this problem. A better choice of variables is probably that which reflects the averages over ΩL and ΩR. To relate u1 and u2 with these averages we consider v1 = ∣∣ΩL ∣∣−1 ∫ ΩL u(x)dx and v2 = ∣∣ΩR ∣∣−1 ∫ ΩR u(x)dx. Thus { u1 = ∣∣ΩL ∣∣v1 + ∣∣ΩR ∣∣v2, u2 = −(∣∣ΩL ∣∣∣∣ΩR ∣∣)1/2 (v1 − v2) and { v1 = u1 + c0 1u2, v2 = u1 + c0 2u2. (1.8) With the variables (1.8), system (1.7) becomes⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ v̇1 = −δCo ∣∣ΩR ∣∣(v1 − v2) + f (v1) + |∂ΩL| |ΩL| g(v1), v̇2 = δCo ∣∣ΩL ∣∣(v1 − v2) + f (v2) + |∂ΩR| |ΩR| g(v2). (1.9) Now our aim is to show that the dynamics of (1.1) can be described by the dynamics of (1.9). With this in hand, we will try to produce asymptotically stable equilibrium solutions (v1, v2) for (1.9) with v1 = v2 and they should correspond to asymptotically stable equilibrium solutions to (1.1) which are nonconstant for v1 reflects the average in ΩL and v2 reflects the average in ΩR. This paper is organized as follows. In Section 2 we state the main results of the paper, in Section 3 we prove the main results and in Section 4 we give an example for which we can obtain stable nonconstant equilibria from the boundary nonlinearity. At the end of the paper we include three appendixes where we prove Theorem 1 (Appendix A), an invariant manifold theorem to take into account the dependence on ε (Appendix B) and a uniform (with respect to ε) trace theorem (Appendix C). 2. Main results In this section we will state the main results of this paper. The proofs will be given in Section 3. Before we proceed, we need to introduce some terminology. For ε > 0, let Xε = L2(Ωε) and Lε :D(Lε) ⊂ Xε → Xε be the operators defined by D(Lε) = { u ∈ H 2(Ωε): ∂u ∂n = 0 } , Lεu = δ εN−1 �u. It is well known that Lε is an unbounded, self adjoint, nonpositive definite operator which has compact resolvent. It follows that −Lε is a sectorial operator and, for ζ > 0 fixed, we can define the fractional powers (−Lε + ζ I )α and corresponding fractional power spaces Xα ε 1222 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 (Xα ε = D((−Lε + ζ I )α) endowed with the graph norm), α > 0. Xα ε is a Hilbert space with the inner product 〈ϕ,ψ〉α = ∫ Ωε (−Lε + ζ I )αϕ(−Lε + ζ I )αψ. Then, X1/2 ε = H 1(Ωε) (with norm ‖ψ‖2 X 1/2 ε = δ εN−1 ‖∇ψ‖2 L2(Ωε) +ζ‖ψ‖2 L2(Ωε) ) and X1 ε = D(Lε) with the graph norm. Also, let V = span [ ϕε 1, ϕε 2 ] , V ⊥ = { ϕ ∈ L2(Ωε): ∫ Ωε ϕψ dx = 0, for all ψ ∈ V } and V ⊥ α = V ⊥ ∩ Xα ε = {ϕ ∈ Xα ε : 〈ϕ,ψ〉α = 0, for all ψ ∈ V }. Define Aε :D(Aε) ⊂ V ⊥ → V ⊥ by D(Aε) = V ⊥ 1 and Aεω = Lεω for all ω ∈ D(Aε); that is, the restriction of Lε to V ⊥. We also denote by Aε its realization in X 1/2 ε . Also define Bε = [ 0 0 0 −δ λε 2 εN−1 ] . If ρ = sup0<ε�1 δ λε 2 εN−1 , β(ε) = λε 3 εN−1 and α � 0, the operators Aε and Bε satisfy∥∥eAεtw ∥∥ V ⊥ α � e−β(ε)t‖w‖V ⊥ α , t � 0,∥∥eAεtw ∥∥ V ⊥ α � t−αe−β(ε)t‖w‖V ⊥ , t > 0,∥∥eBεt z ∥∥ R2 � e−ρt‖z‖R2, t � 0. (2.1) Note that there is a constant c, independent of ε, such that ‖ψ‖H 2α(Ωε) � cεα(N−1)‖ψ‖V ⊥ α , 1 2 � α � 0, (2.2) for all ψ ∈ V ⊥ α . Also note that, according to (1.5), if ψ ∈ V ⊥ 1/2, then 〈wt,ψ〉 = − δ εN−1 〈∇w,∇ψ〉 + ∫ Ωε f ( u1ϕ ε 1 + v2ϕ ε 2 + w ) ψ + ∫ ∂Ωε g ( u1ϕ ε 1 + v2ϕ ε 2 + w ) ψ. Proceeding as in [9] we can now rewrite system (1.5) as{ ω̇ = Aεω + HΩ ε (y,ω) + H∂Ω ε (y,ω), ẏ = Bεy + Gε(y,ω), (2.3) where y = (u1, u2) ⊥, u = u1ϕ ε 1 + u2ϕ ε 2 + ω, Gε(y,ω) = ( Gε 0(y,ω),Gε 1(y,ω) )⊥ , Gε 0(y,ω) = ∫ f (u)ϕε 1 + ∫ g(u)ϕε 1, Ωε ∂Ωε A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1223 Gε 1(y,ω) = ∫ Ωε f (u)ϕε 2 + ∫ ∂Ωε g(u)ϕε 2, 〈 HΩ ε (y,ω),ψ 〉 V ⊥−α,V ⊥ α = ∫ Ωε f ( u1ϕ ε 1 + u2ϕ ε 2 + w ) ψ, 〈 H∂Ω ε (y,ω),ψ 〉 V ⊥−α,V ⊥ α = ∫ ∂Ωε g ( u1ϕ ε 1 + u2ϕ ε 2 + w ) ψ. We identify u = u1ϕ 1 ε + u2ϕ 2 ε + w ∈ X 1/2 ε with (y,w) ∈ R 2 × V ⊥ 1/2. This identification is an isomorphism which is bounded with bounded inverse uniformly with respect to ε. If f and g are bounded with bounded derivatives up to second order, (y0,w0) ∈ R 2 × V ⊥ 1/2, then there is a unique solution (y,w)(t, (y0,w0)) of (2.3) defined in [0,∞). We can now state the main results of this paper. Theorem 3. There is a continuously differentiable map σε : R2 → V ⊥ 1/2 such that Sε = { (y,ω): ω = σε(y), y ∈ R 2} is an exponentially attracting invariant manifold for (1.5). The flow on Sε is given by u(t, x) = (y(t), σε(y(t))), where y is the solution of ẏ = Bεy + Gε ( y,σε(y) ) . (2.4) Furthermore, σε → 0 in C1(R2,V ⊥ 1/2), as ε → 0. The following theorem tells us that the asymptotic dynamics of system (1.9) is equivalent to that of system (2.4). In particular, stable equilibria of (1.9) with v1 = v2 corresponds to stable nonconstant equilibria of (1.1). Theorem 4. Assume that system (1.9) is structurally stable. Then for small enough ε, the flow on the invariant manifold given by (2.4) is topologically equivalent to the flow (1.9). Remark 5. Making |ΩL| = |ΩR|, |∂ΩL| = |∂ΩR|, r := δCo|ΩL| = δCo|ΩR| and f (s) + |∂ΩL| |ΩL| g(s) = f (s) + |∂ΩR| |ΩR| g(s) := s − s3 we can verify the conditions of [1, Theorem V, p. 395] for r = 1 2 and r = 1 3 (see [6, p. 400]). Hence, in this case, (1.9) is structurally stable. For the general case, see [13] for conditions ensuring that (1.9) is structurally stable. 3. Proof of the main results In the proof of Theorem 3, we use the following uniform trace theorem whose proof is given in Appendix C. 1224 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 Theorem 6 (Uniform Trace Theorem). Let γ :V ⊥ α → L2(∂Ωε) be the bounded linear operator which associates to each w ∈ V ⊥ α its trace γ (w), α > 1 4 . Then ‖γ ‖L(V ⊥ α ,L2(∂Ωε)) � Gε(N−2)/2 � G (3.1) for some G > 0 independent of ε. Proof of Theorem 3. Note that ϕε 1(x) = |Ωε |−1/2 � 1, |∂Ωε | � |∂Ω| + |∂R1| and that m2 := supε>0 supx∈Ωε |ϕε 2(x)| < ∞ (the last estimate follows from [4, Lemma B.1] and from Theo- rem 1). From our assumptions on f,g we have that M1 := sup x∈R ∣∣f (x) ∣∣ < ∞, L1 := sup x∈R ∣∣f ′(x) ∣∣ < ∞, M2 := sup x∈R ∣∣g(x) ∣∣ < ∞, L2 := sup x∈R ∣∣g′(x) ∣∣ < ∞. With this we have∣∣Gε 0(y,ω) ∣∣ � ∫ Ωε ∣∣f (y,w) ∣∣∣∣ϕε 1 ∣∣+ ∫ ∂Ωε ∣∣g(y,w) ∣∣∣∣ϕε 1 ∣∣ � M1|Ωε |1/2 + M2 |∂Ωε | |Ωε |1/2 � M1 ( 1 + |R1| )1/2 + M2 (|∂Ω| + |∂R1| ) = N1, (3.2) and ∣∣Gε 1(y,ω) ∣∣ � M1 ∫ Ωε ∣∣ϕε 2 ∣∣+ M2 ∫ ∂Ωε ∣∣ϕε 2 ∣∣ � M1|Ωε |1/2 + M2m2|∂Ωε | � M1 ( 1 + |R1| )1/2 + M2m2 (|∂Ω| + |∂R1| ) = N2. (3.3) Hence, if NG = √ N2 1 + N2 2 ,∥∥Gε(y,ω) ∥∥ R2 � NG. To obtain global Lipschitz continuity, let y1 = (u1, u2), y2 = (v1, v2) ∈ R 2, ω1,ω2 ∈ V ⊥ 1/2. Then ∣∣Gε 0(y1,ω1) − Gε 0(y2,ω2) ∣∣ � L1 [ |u1 − v1| + |u2 − v2| ∫ Ωε ∣∣ϕε 1 ∣∣∣∣ϕε 2 ∣∣+ ∫ Ωε |ω1 − ω2| ∣∣ϕε 1 ∣∣] + L2 [ |u1 − v1| ∫ ∂Ωε ∣∣ϕε 1 ∣∣2 + |u2 − v2| ∫ ∂Ωε ∣∣ϕε 1 ∣∣∣∣ϕε 2 ∣∣+ ∫ ∂Ωε |ω1 − ω2| ] � L1 [√ 2‖y1 − y2‖R2 + cε(N−1)/2‖ω1 − ω2‖X 1/2 ε ] + L2 [ C1‖y1 − y2‖R2 + C2ε (N−2)/2‖ω1 − ω2‖X 1/2 ε ] , where C1 := √ 2(|∂Ω|+|∂R1|)max{1,m2}, C2 = c (|∂Ω|+|∂R1|)1/2G and we used Theorem 6 and (2.2). Hence there is a constant L1, independent of ε, such that∣∣Gε 0(y1,ω1) − Gε 0(y2,ω2) ∣∣ � L1 [‖y1 − y2‖R2 + ‖ω1 − ω2‖V ⊥ ] . 1/2 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1225 Similarly, there is a constant L2, independent of ε, such that∣∣Gε 1(y1,ω1) − Gε 1(y2,ω2) ∣∣ � L2 [‖y1 − y2‖R2 + ‖ω1 − ω2‖V ⊥ 1/2 ] . Thus if LG = √ L2 1 + L2 2,∥∥Gε(y1,ω1) − Gε(y2,ω2) ∥∥ R2 � LG [‖y1 − y2‖R2 + ‖ω1 − ω2‖V ⊥ 1/2 ] . It is easy to see that ∂ ∂u1 Gε 0(u1, u2,ω) = ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + w )( ϕε 1 )2 + ∫ ∂Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + w )( ϕε 1 )2 , ∂ ∂u2 Gε 0(u1, u2,ω) = ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 1ϕε 2 + ∫ ∂Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 1ϕε 2, ( ∂ ∂ω Gε 0(u1, u2,ω) ) ψ = ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 1ψ + ∫ ∂Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 1ψ (for all ψ ∈ V ⊥ 1/2) and, proceeding just as before, that they are Lipschitz continuous. Hence Gε 0 : R2 ×V ⊥ 1/2 → R is continuously differentiable. Similarly Gε 1 : R2 ×V ⊥ 1/2 → R is continuously differentiable. Now for HΩ ε : R2 × V ⊥ 1/2 → V ⊥−α , using (2.2), it is easy to see that∥∥HΩ ε (y,ω) ∥∥ V ⊥−α � NF εα(N−1) and that∥∥HΩ ε (y1,ω1) − HΩ ε (y2,ω2) ∥∥ V ⊥−α � εα(N−1)LΩ (‖y1 − y2‖R2 + ‖ω1 − ω2‖V ⊥ 1/2 ) . The derivatives of HΩ ε are given by〈 ∂ ∂u1 HΩ ε (u1, u2,ω)h,ψ 〉 V ⊥−α,V ⊥ α = h ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 1ψ, h ∈ R, 〈 ∂ ∂u2 HΩ ε (u1, u2,ω)k,ψ 〉 V ⊥−α,V ⊥ α = k ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 2ψ, k ∈ R, 〈 ∂ ∂ω HΩ ε (u1, u2,ω)ξ,ψ 〉 V ⊥−α,V ⊥ α = ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ξψ, ξ ∈ V ⊥ 1/2. It is not difficult to see that these derivatives are continuous. Just to give an idea of the techniques involved, we observe that the continuity derivative with respect to ω (after the use of Hölder’s inequality) requires that∥∥f ′(u1ϕ ε 1 + u2ϕ ε 2 + w1 )− f ′(v1ϕ ε 1 + v2ϕ ε 2 + w2 )∥∥ Lp(Ωε) → 0 as max{|u1 − v1|, |u2 − v2|,‖w1 − w2‖X 1/2 ε } ε→0−−−→ 0, for large values of p. This follows from the fact that it goes to zero when p = 2 and it is bounded when p = ∞. Finally, for H∂Ω ε : R2 × V ⊥ 1/2 → V ⊥−α it is easy to see that∥∥H∂Ω ε (y,ω) ∥∥ ⊥ � NF ε(N−2)/2 V−α 1226 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 and that∥∥H∂Ω ε (y1,ω1) − H∂Ω ε (y2,ω2) ∥∥ V ⊥−α � L∂Ωε(N−2)/2(‖y1 − y2‖R2 + ε(N−2)/2‖ω1 − ω2‖V ⊥ 1/2 ) . The derivatives of H∂Ω ε are given by〈 ∂ ∂u1 H∂Ω ε (u1, u2,ω)h,ψ 〉 V ⊥−α,V ⊥ α = h ∫ Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 1ψ, h ∈ R, 〈 ∂ ∂u2 H∂Ω ε (u1, u2,ω)k,ψ 〉 V ⊥−α,V ⊥ α = k ∫ Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ϕε 2ψ, k ∈ R, 〈 ∂ ∂ω H∂Ω ε (u1, u2,ω)ξ,ψ 〉 V ⊥−α,V ⊥ α = ∫ Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + w ) ξψ, ξ ∈ V ⊥ 1/2. It is easy to see that these derivatives are continuous. With this we see that Fε and Gε satisfy the hypotheses of Theorem B.2. It follows from (2.1) that the conditions on Aε and Bε of Theorem B.2 are also satisfied. The proof of Theorem 3 now follows from Theorem B.2. � Proof of Theorem 4. For suitably small ε, the flow in the invariant manifold is given by v(t, x) = u1(t)ϕ 0 1(x) + u2(t)ϕ 0 2(x) + σε(u1(t), u2(t))(x), where (u1(t), u2(t)) is a solution of{ u̇1 = Gε 0 ( u1, u2, σε(u1, u2) ) , u̇2 = −δu2 + Gε 1 ( u1, u2, σε(u1, u2) ) . (3.4) To obtain that the flow on the attractor of (1.6) (or equivalently (1.9)) is topologically equivalent to the flow on the attractor of (3.4), we only need to prove that the vector fields Xε(u1, u2) = ( X0(ε),X1(ε) ) and X0(u1, u2) = ( X0(0),X1(0) ) , where X0(ε) = ∫ Ωε f ( u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 1 + ∫ ∂Ωε g ( u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 1, X1(ε) = −δ λε 2 εN−1 u2 + ∫ Ωε f ( u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 2 + ∫ ∂Ωε g ( u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 2, X0(0) = ∫ Ω f (u1 + u2φ2) + ∫ ∂Ω g(u1 + u2φ2), X1(0) = −δ Cou2 + ∫ f (u1 + u2φ2)φ2 + ∫ g(u1 + u2φ2)φ2 Ω ∂Ω A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1227 are C1 close. This follows easily from the fact that σε approaches 0 in the C1(R2,V ⊥ 1/2) topology and the asymptotic properties of the eigenvalues and eigenfunctions of − δ εN−1 � as ε → 0. Just to give an idea of the techniques involved, let us prove that ∂u1X1(ε) converges to ∂u1X1(0) as ε → 0. Note that ∂u1X1(ε)(u1, u2) − ∂u1X1(0)(u1, u2) = −δ ( λε 2 εN−1 − Co ) u2 + ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 2ϕε 1 − ∫ Ω f ′(u1 + u2φ2)φ2 + ∫ ∂Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 2ϕε 1 − ∫ ∂Ω g′(u1 + u2φ2)φ2 + ∫ Ωε f ′(u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 2∂u1σε(u1, u2) + ∫ ∂Ωε g′(u1ϕ ε 1 + u2ϕ ε 2 + σε(u1, u2) ) ϕε 2∂u1σε(u1, u2). All but the last two lines in the above expression go to zero, uniformly in bounded subsets of R 2, because of the convergence properties of λε 1, λ ε 2, ϕε 1 and ϕε 2 and because ‖σε(u1, u2)‖V ⊥ 1/2 → 0. For the last two lines we only need to observe that ‖∂u1σε(u1, u2)‖V ⊥ 1/2 → 0 and use Uniform Trace Theorem (Theorem 6) to conclude that ‖∂u1σε(u1, u2)‖L2(∂Ωε) ε→0−−−→ 0. � 4. Patterns In this section we return to system (1.9). Now we want to obtain stable nonconstant equilib- rium solutions for (1.1). For this purpose we consider δ sufficiently small, g(u) = u − u3 and f (u) = 0. Thus system (1.9) has nine equilibrium points (see Fig. 2) which (for δ � 1) are ap- proximately equal to P1 = (0,0), P2 = (0,1), P3 = (0,−1), P4 = (1,0), P5 = (1,1), P6 = (1,−1), P7 = (−1,0), P8 = (−1,1), P9 = (−1,−1). All these equilibrium points are hyperbolic. P5, P6, P8, P9 are stable. We have seen that the dynamics of (1.9) is equivalent to the dynamics of (1.1). The stable equilibrium points of the form P = (v1, v2) with v1 = v2 correspond to stable nonconstant equilibrium for (1.1). Thus, if ε is suitably small, system (1.1) has asymptotically stable nonconstant equilibria. 1228 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 Fig. 2. Approximate phase portrait: δ � 1, g(u) = u − u3, f ≡ 0. Acknowledgment We express our gratitude to an anonymous referee whose comments greatly improved the manuscript. Appendix A. Convergence of eigenvalues and eigenfunctions In this appendix we show Theorem 1 concerning the eigenvalues λε n arranged in increasing order (counting multiplicity) and a complete system of orthonormalized eigenfunctions ϕε n asso- ciated to problem (1.4). Let φ1 = |Ω|−1/2 = 1 and φ2 = c0 1χΩL + c0 2 χ ΩR . For the results in this section we follow the ideas in [2,3]. Lemma A.1. lim sup ε→0 λε 2 εN−1 � σN−1 1∫ 0 hN−1(x1) ∣∣ξ ′(x1) ∣∣2 dx1 = Co, (A.1) where ξ(x1) is the solution of the boundary value problem{ (hN−1ξ ′(s))′ = 0, s ∈ (0,1), ξ(0) = c0 1, ξ(1) = c0 2, (A.2) where c0 1 and c0 2 are defined in Theorem 1. Proof. From the variational characterization, we know that λε 2 = inf {∫ Ωε |∇ϕ|2 dx∫ Ωε |ϕ|2 dx : ϕ ∈ H 1(Ωε), ϕ = 0, ∫ ϕ dx = 0 } . Ωε A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1229 Defining ψ̃(x1, x ′) by ψ̃(x1, x ′) = ⎧⎪⎨ ⎪⎩ ξ(x1) in Rε, c0 1 in ΩL, c0 2 in ΩR, and ψ(x1, x ′) = ψ̃(x1, x ′) − 1 |Ωε | ∫ Ωε ψ̃(x1, x ′) dx1 dx′, since ψ̃ ∈ H 1(Ωε), then ψ ∈ H 1(Ωε) and ∫ Ωε ψ dx1 dx′ = 0. So we have λε 2 � ∫ Ωε |∇ψ |2 dx∫ Ωε |ψ |2 dx . From the definition of ψ , we have∫ Ωε |∇ψ |2 dx = ∫ Rε |∇ψ̃ |2 dx = ∫ Rε ∣∣∇ξ(x1) ∣∣2 dx1 dx′ = σN−1 εN−1 1∫ 0 hN−1(x1) ∣∣ξ ′(x1) ∣∣2 dx1 dy′. Also, ∫ Ωε |ψ |2 dx = ∫ Ωε ∣∣ψ̃(x1, x ′) ∣∣2 dx1 dx′ − 1 |Ωε | ( ∫ Ωε ψ̃(x1, x ′) dx1 dx′ )2 and since∫ Ωε ψ̃(x1, x ′) dx1 dx′ = ∫ Rε ψ̃(x1, x ′) dx1 dx′ = O ( εN−1), we obtain∫ Ωε |ψ |2 dx = ∫ Ω ∣∣ψ̃(x1, x ′) ∣∣2 dx1 dx′ + ∫ Rε ∣∣ψ̃(x1, x ′) ∣∣2 dx1 dx′ + O ( ε2(N−1) ) = 1 + σN−1ε N−1 1∫ 0 hN−1(x1) ∣∣ξ(x1) ∣∣2 dx1 + O ( ε2(N−1) ) . That gives us∫ Ωε |∇ψ |2 dx∫ Ωε |ψ |2 dx = σN−1ε N−1 ∫ 1 0 hN−1(x1)|ξ ′(x1)|2 dx1 dy′ 1 + σN−1εN−1 ∫ 1 0 hN−1(x1)|ξ(x1)|2 dx1 + O(ε2(N−1)) and consequently λε 2 � σN−1ε N−1 1∫ hN−1(x1) ∣∣ξ ′(x1) ∣∣2 dx1 dy′ + O ( ε2(N−1) ) . 0 1230 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 Hence, lim sup ε→0 λε 2 σN−1εN−1 � 1∫ 0 hN−1(x1) ∣∣ξ ′(x1) ∣∣2 dx1. (A.3) Now note that the solution of (A.2) can be found explicitly by ξ(x1) = c0 1 + ( c0 2 − c0 1 ){ x1∫ 0 dt hN−1(t) }{ 1∫ 0 dx1 hN−1(x1) }−1 (A.4) and thus ∣∣ξ ′(x1) ∣∣2 = ( c0 2 − c0 1 )2 { 1 hN−1(x1) }2 { 1∫ 0 dx1 hN−1(x1) }−2 = ( 1 |ΩL| + 1 |ΩR| ){ 1 hN−1(x1) }2 { 1∫ 0 dx1 hN−1(x1) }−2 . Replacing it in the right-hand side of Eq. (A.8), Co appears. � Theorem A.2 (Convergence of eigenfunctions). Let n ∈ N and ϕε n be eigenfunctions for prob- lem (1.4), then (i) ϕε 1 → φ1 in Hk(Ω) as ε → 0, k � 1, (ii) supε ‖ϕε 2‖L∞(Ω) < ∞ and ϕε 2 → φ2 in H 1(Ω) as ε → 0. Proof. (i) Observe that ϕε 1 = |Ωε |−1/2 and φ1 = |Ω|−1/2 = 1 are the corresponding eigenfunc- tions associated to λε 1 = μ1 = 0. It is clear that ϕε 1 → φ1 in Hk(Ω) for all integer k � 0. (ii) Let λε 2 be the second eigenvalue of (1.4) and ϕε 2 be a corresponding normalized eigenfunc- tion. From Lemmas A.1 and [4, Lemma B.1], we have that ϕε 2 ∈ L∞(Ωε) and∥∥ϕε 2 ∥∥ L∞(Ωε) � C ∥∥ϕε 2 ∥∥ L2(Ωε) for some constant C = C(|Ωε |,N). This implies ∫ Rε |ϕε 2 |2 ε→0−−−→ 0 and ∫ Rε ϕε 2 ε→0−−−→ 0 and, consequently,∫ Ω ∣∣ϕε 2 ∣∣2 ε→0−−−→ 1 and ∫ Ω ϕε 2 ε→0−−−→ 0. Also, since∫ Ω ∣∣∇ϕε 2 ∣∣2 � ∫ Ωε ∣∣∇ϕε 2 ∣∣2 = λε 2, (A.5) we have that∫ ∣∣∇ϕε 2 ∣∣2 ε→0−−−→ 0. (A.6) Ω A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1231 Since H 1(Ω) is compactly embedded in L2(Ω), ϕε 2 has a convergent subsequence in L2(Ω), which we denote again by ϕε 2 . Let ϕ be its limit. Thus we have⎧⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ ϕε 2 ε→0−−−→ ϕ, strongly in L2(Ω), ∇ϕε 2 ε→0−−−→ 0, strongly in L2(Ω),∫ Ω ϕ = 0, ∫ Ω ϕ2 = 1. Hence, ∇ϕ = 0. From this we have that ϕ = c1χ ΩL + c2χΩR where c1 and c2 must satisfy{ c1 ∣∣ΩL ∣∣+ c2 ∣∣ΩR ∣∣ = 0, (c1) 2 ∣∣ΩL ∣∣+ (c2) 2 ∣∣ΩR ∣∣ = 1. (A.7) Thus, c1 = ± √ |ΩR| |ΩL| and c2 = ∓ √ |ΩL| |ΩR| . So, ∫ Ω ϕε 2φ2 = ∫ ΩL ϕε 2φ2 + ∫ ΩR ϕε 2φ2 ε→0−−−→ ∫ ΩL c1c 0 1 + ∫ ΩR c2c 0 2. From the first equation of (A.7) we see that c1 and c2 have opposite signs. We consider c1 with negative sign and c2 with positive sign. In this case c1 = c0 1 and c2 = c0 2. Thus we have∫ Ω ϕε 2φ2 ε→0−−−→ ( c0 1 )2∣∣ΩL ∣∣+ ( c0 2 )2∣∣ΩR ∣∣ = 1. Now we show ϕε 2 ε→0−−−→ φ2. In fact, ∥∥ϕε 2 − φ2 ∥∥2 L2(Ω) = ∥∥ϕε 2 ∥∥2 L2(Ω) + ‖φ2‖2 L2(Ω) − 2 ∫ Ω0 ϕε 2φ2 � ∥∥ϕε 2 ∥∥2 L2(Ωε) + 1 − 2 ∫ Ω ϕε 2φ2 � 2 − 2 ∫ Ω ϕε 2φ2 ε→0−−−→ 0. From this and (A.6) we conclude that ‖ϕε 2 − φ2‖H 1(Ω) ε→0−−−→ 0. � The exact rate of convergence of λε 2 to 0 is given by the following proposition. Proposition A.3. lim ε→0 λε 2 εN−1 = Co. (A.8) Proof. From Lemma A.1 it remains to show lim inf ε→0 λε 2 σN−1εN−1 � 1∫ hN−1(x1) ∣∣ξ ′(x1) ∣∣2 dx1. (A.9) 0 1232 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 For this, we consider the following family of functions ξε(x1, y ′) = ϕε 2(x1, εy ′), where (x1, y ′) ∈ R1 = {(x1, y ′): 0 � x1 � 1, |y′| < h(x1)}. If x′ = εy′, εN−1 ∫ R1 [( ∂ξε ∂x1 )2 + 1 ε2 |∇y′ξε |2 ] dx1 dy′ = ∫ Rε ∣∣∇ϕε 2 ∣∣2 dx1 dx′ � λε 2. (A.10) Using (A.3) and (A.10), we have sup ε>0 ∫ R1 [( ∂ξε ∂x1 )2 + 1 ε2 |∇y′ξε |2 ] dx1 dy′ < ∞. (A.11) It follows that the family of functions {ξε}ε>0 is uniformly bounded in H 1(R1). So we can find a subsequence {εn} with εn n→∞−−−−→ 0, and a function ξ0 ∈ H 1(R1) such that ξεn n→∞−−−−→ ξ0 weakly in H 1(R1) and strongly in Hs(R1) for all s < 1. Hence, from (A.11) we conclude that ξ0 is independent of x′. Let K be the convex and closed set of H 1(0,1) defined by K = { u ∈ H 1(0,1): u(0) = c0 1, u(1) = c0 2 } . Now we show that ξ0 ∈ K. In fact, since ξ0 ∈ H 1(R1), we conclude immediately that ξ0 ∈ H 1(0,1). To find the boundary value of ξ0 we use the fact that ϕε 2 ε→0−−−→ φ2 strongly in H 1(Ω). Thus, from the continuity of the trace operator, it follows that ξεn |∂R1∩Ω → ξ0|∂R1∩Ω in H 1/2(∂R1). Thus ξ0(0) = c0 1 and ξ0(1) = c0 2, which proves that ξ0 ∈ K. Note that, from (A.10), λε 2 � εN−1 ∫ R1 ( ∂ξε ∂x1 )2 dx1 dy′ (A.12) and that, if J :K → R is given by J (u) = 1∫ 0 hN−1(x1)u ′(x1) 2 dx1, then J (ξ) = minu∈K J (u). Hence J (ξ) � J (ξ0). From (A.12), it follows that lim inf n→∞ λ εn 2 εN−1 � lim inf εn→0 ∫ R1 ( ∂ξεn ∂x1 )2 dx1 dy′ � ∫ R1 ( ∂ξ0 ∂x1 )2 dx1 dy′ � σN−1 1∫ 0 hN−1(x1) ∣∣ξ ′(x1) ∣∣2 dx1. This gives (A.9), which along with (A.3) implies (A.8). This concludes the proof. � Finally, to conclude the proof of Theorem 1 we need to ensure that λε 3 is bounded away from zero. A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1233 Proposition A.4. lim inf ε→0 λε 3 > 0. Proof. Note that ϕε 3 satisfies∫ Ωε ϕε 3 = 0, ∫ Ωε ϕε 3ϕε 2 = 0, ∥∥ϕε 3 ∥∥ L2(Ωε) = 1. (A.13) Suppose that there is a sequence {εj }∞j=1 with εj j→∞−−−→ 0 such that limj→∞ λ εj 3 = 0. Then∫ Ω ∣∣∇ϕ εj 3 ∣∣2 � ∫ Ωεj ∣∣∇ϕ εj 3 ∣∣2 = λ εj 3 ∥∥ϕ εj 3 ∥∥2 = λ εj 3 j→∞−−−→ 0. Hence there are ϕ ∈ H 1(Ω) and a subsequence of {ϕεj 3 }, with ϕ εj 3 j→∞−−−→ ϕ weakly in H 1(Ω) and strongly in L2(Ω). Thus we have ϕ εj 3 s−L2(Ω)−−−−−→ ϕ, ∇ϕ εj 3 s−L2(Ω)−−−−−→ 0. It follows that ϕ = { c1, in ΩL, c2, in ΩR. (A.14) Now supj∈N λ εj 3 < ∞ and [4, Lemma B.1] imply that supj∈N ‖ϕεj 3 ‖L∞(Ωεj ) < ∞. This together with (A.13) imply that c1 and c2 satisfy⎧⎪⎨ ⎪⎩ c1 ∣∣ΩL ∣∣+ c2 ∣∣ΩR ∣∣ = 0, c1c 0 1 ∣∣ΩL ∣∣+ c2c 0 2 ∣∣ΩR ∣∣ = 0, (c1) 2 ∣∣ΩL ∣∣+ (c2) 2 ∣∣ΩR ∣∣ = 1. (A.15) Since the above system does not have a solution, we have a contradiction and the result is proved. � Appendix B. Invariant manifold theorem In this section, we state and give hints of the proof for the invariant manifold theorem used to prove Theorem 3. The proof is adapted from the results in Henry [15, Chapter 6]. Let ε ∈ (0,1], Xε and Yε be Banach spaces, Aε :D(Aε) ⊂ Xε → Xε be a sectorial operator and Bε ∈ L(Yε). Denote by Xα ε the fractional power spaces associated to Aε , α ∈ [0,1). Let Fε :Xα ε × Yε → Xε and Gε :Xα ε × Yε → Yε be Lipschitz continuous functions and consider the following system of weakly coupled semilinear differential equations in Xα ε × Yε{ ẋ = Aεx + Fε(x, y), ẏ = Bεy + Gε(x, y). (B.1) Definition B.1. A set Sε ⊂ Xα ε ×Yε is an invariant manifold for (B.1) if there exists σε :Yε → Xα ε such that Sε = {(x, y) ∈ Xα ε × Yε : x = σε(y)} and, for each (xε 0 , yε 0) ∈ Sε there exists a solution (xε(·), yε(·)) of (B.1), (xε(0), yε(0)) = (xε, yε), defined on R such that (xε(t), yε(t)) ∈ Sε , 0 0 1234 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 ∀t ∈ R. An invariant manifold Sε is exponentially attracting if there are positive constants ν and K such that∥∥xε(t) − σε ( yε(t) )∥∥ Xα ε � Ke−νt ∥∥xε 0 − σε ( yε 0 )∥∥ Xα ε , whenever (xε(t), yε(t)) is a solution to (B.1) with (xε(0), yε(0)) = (xε 0 , yε 0) ∈ Xα ε × Yε . Theorem B.2. Assume that∥∥Fε(x, y) − Fε(z,w) ∥∥ Xε � LF (ε) (‖x − z‖Xα ε + ‖y − w‖Yε ) ,∥∥Fε(x, y) ∥∥ Xε � NF (ε),∥∥Gε(x, y) − Gε(z,w) ∥∥ Yε � LG(ε) (‖x − z‖Xα ε + ‖y − w‖Yε ) ,∥∥Gε(x, y) ∥∥ Yε � NG(ε), for each (x, y), (z,w) in Xα ε × Yε where NF (ε) > 0, LF (ε) > 0, NG(ε) > 0 and LG(ε) > 0. Also, assume that there are positive constants MA, MB and ρ, independent of ε and β(ε) > 0 such that∥∥e−Aεtw ∥∥ Xα ε � M A e−β(ε)t‖w‖Xα ε , t � 0,∥∥e−Aεtw ∥∥ Xα ε � MAt−αe−β(ε)t‖w‖Xε , t > 0,∥∥e−Bεt z ∥∥ Yε � MBe−ρt‖z‖Yε , t � 0, for any w ∈ Xα ε and z ∈ Yε . If NF (ε) β(ε)α−1 ε→0−−−→ 0, LF (ε) β(ε)α−1 ε→0−−−→ 0 and LF (ε)LG(ε) β(ε)α−1 ε→0−−−→ 0, then, for small enough ε, there is an exponentially attracting invariant manifold Sε = { (x, y): x = σε(y), y ∈ Yε } for (B.1), where σε :Yε → Xα ε satisfies s(ε) = sup y∈Yε ∥∥σε(y) ∥∥ Xα ε ε→0−−−→ 0, sup {‖σε(y) − σε(z)‖Xα ε ‖y − z‖Yε : y, z ∈ Yε, y = z } ε→0−−−→ 0. If Fε , Gε are smooth, then σε is smooth and its derivative Dσε satisfies sup y∈Yε ∥∥Dσε(y) ∥∥ L(Yε,Xα ε ) ε→0−−−→ 0. Proof. The first and crucial step is to obtain the existence of the invariant manifold. Given D > 0 and � > 0, let σε :Yε → Xα ε be a function satisfying |||σε ||| := sup y∈Yε ∥∥σε(y) ∥∥ Xα ε � D, ∥∥σε(y) − σε(y ′) ∥∥ Xα ε � �‖y − y′‖Yε . (B.2) Denote by yε(t) = ψ(t, τ, η, σε) the solution of ẏε = −Bεyε + Gε ( σε(yε), yε ) , t < τ, yε(τ ) = η (B.3) A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1235 and define Θ(σε)(η) = τ∫ −∞ e−Aε(τ−s)Fε ( σε ( y(s) ) , y(s) ) ds. (B.4) Note that ∥∥Θ(σε)(η) ∥∥ Xα ε � τ∫ −∞ NF MA(τ − s)−αe−β(ε)(τ−s) ds � NF MAΓ (1 − α) β(ε)1−α . (B.5) Let ε0 > 0 be such that, for ε � ε0, supη∈Yε ‖Θ(σε)(η)‖Xα ε � D. Suppose that σε and σ ′ ε are functions satisfying (B.2), η,η′ ∈ Yε , yε(t) = ψ(t, τ, η, σε) and y′ ε(t) = ψ(t, τ, η′, σ ′ ε). Then yε(t) − y′ ε(t) = e−Bε(t−τ)(η − η′) + t∫ τ e−Bε(t−s) ( Gε ( σε(yε), yε )− Gε ( σ ′ ε ( y′ ε ) , y′ ε )) ds. Now, ∥∥yε(t) − y′ ε(t) ∥∥ Yε � MBeρ(τ−t)‖η − η′‖Yε + MB τ∫ t eρ(s−t) ∥∥Gε ( σε(yε), yε )− Gε ( σ ′(y′ ε ) , y′ ε )∥∥ Yε ds � MBeρ(τ−t)‖η − η′‖Yε + MBLg τ∫ t eρ(s−t) (∥∥σε(yε) − σ ′(y′ ε )∥∥ Xα ε + ∥∥yε − y′ ε ∥∥ Yε ) ds � MBeρ(τ−t)‖η − η′‖Yε + MBLg τ∫ t eρ(s−t) (∥∥σε(yε) − σ ′(yε) ∥∥ Xα ε + (1 + �) ∥∥yε − y′ ε ∥∥ Yε ) ds � MBeρ(τ−t)‖η − η′‖Yε + MBLg τ∫ t eρ(s−t) (|||σε − σ ′||| + (1 + �) ∥∥yε − y′ ε ∥∥ Yε ) ds � MBeρ(τ−t)‖η − η′‖Yε + MBLg(1 + �) τ∫ t eρ(s−t) ∥∥yε − y′ ε ∥∥ Yε ds + eρ(τ−t)MBLg ρ |||σε − σ ′|||. Let φ(t) = eρ(t−τ)‖yε(t) − y′ ε(t)‖Yε . Then φ(t) � MB [ ‖η − η′‖Yε + Lg ρ |||σε − σ ′|||Xα ε ] + MBLg(1 + �) τ∫ φ(s) ds. t 1236 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 By Gronwall’s lemma we have ∥∥yε(t) − y′ ε(t) ∥∥ Yε � [ MB‖η − η′‖Yε + LG ρ |||σε − σ ′|||Xα ε ] e[ρ+cΓ ](τ−t), where cΓ = MBLg(1 + �). Therefore ∥∥Θ(σε)(η) − Θ ( σ ′ ε ) (η′) ∥∥ Xα ε � MA τ∫ −∞ (τ − s)−αe−β(ε)(τ−s) ∥∥Fε ( σε(y), y )− Fε ( σ ′(y′), y′)∥∥ Xε ds � MALF τ∫ −∞ (τ − s)−αe−β(ε)(τ−s) (∥∥σε(y) − σ ′(y′) ∥∥ Xα ε + ‖y − y′‖Yε ) ds � MALF τ∫ −∞ (τ − s)−αe−β(ε)(τ−s) (|||σε − σ ′||| + (1 + �)‖y − y′‖Yε ) ds and consequently ∥∥Θ(σε)(η) − Θ ( σ ′ ε ) (η′) ∥∥ Xα ε � MALF τ∫ −∞ (τ − s)−αe−β(ε)(τ−s) ( 1 + LG(1 + �) ρ e[ρ+cΓ ](τ−s) ) ds|||σε − σ ′||| + MAMBLF (1 + �) τ∫ −∞ (τ − s)−αe−[β(ε)−ρ−cΓ ](τ−s) ds‖η − η′‖Yε = Iη(ε)‖η − η′‖Yε + Iσ (ε)|||σε − σ ′|||, where Iσ (ε) = MALF Γ (1 − α) β(ε)1−α + MALF LG(1 + �)Γ (1 − α) ρ(β(ε) − ρ − cΓ )1−α Iη(ε) = MAMBLF (1 + �)Γ (1 − α) (β(ε) − ρ − cΓ )1−α . It is easy to see that, given θ < 1, there exists ε0 > 0 such that, for ε � ε0, Iσ (ε) � θ , Iη(ε) � � and ∥∥Θ(σε)(η) − Θ(σ ′)(η′) ∥∥ Xα ε � Iη(ε)‖η − η′‖Yε + Iσ (ε) ∣∣∣∣∣∣σε − σ ′ ε ∣∣∣∣∣∣. (B.6) The inequalities (B.5) and (B.6) imply that Θ is a contraction from the class of functions that satisfy (B.2) into itself. Hence it has a unique fixed point σ ∗ ε = Θ(σ ∗ ε ) in this class. The rest of the proof follows in the same manner as in [15, Chapter 6]. � A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. Appl. 325 (2007) 1216–1239 1237 Appendix C. Proof of Theorem 6 We remark that this result is similar in nature to those presented in [20, Section 4.2]. The main differences being that, in our case, the cylinder is not straight (each section has a different radius), we consider the trace of less regular functions and we embed the trace space into fractional power spaces. For ω ∈ V ⊥ α , α > 1 4 , let γ (ω) be its trace. Our aim is to show that there is a constant G, independent of ε, such that∥∥γ (ω) ∥∥ L2(∂Ωε) � G‖ω‖Xα ε . For simplicity of notation we also denote by ω its trace γ (ω). Note that ‖ω‖L2(∂Ωε) � ( ∫ ∂Ω |ω|2 ds + ∫ ∂Rε\∂Ω |ω|2 ds )1/2 � ‖ω‖L2(∂Ω) + ‖ω‖L2(∂Rε\∂Ω). Since Ω is a fixed domain, it is easy to estimate the norm ‖ω‖L2(∂Ω) in terms of the norm ‖ω‖ X 1/2 ε . Hence we only need to estimate ‖w‖L2(∂Rε\∂Ω). Before we proceed, let us introduce some notation: Rε = { (x, y, z) ∈ R × R N−2 × R: 0 < x < 1, ∥∥(y, z) ∥∥ RN−1 < εh(x) } , Dε = { (x, y) ∈ R × R N−2: 0 < x < 1, |y| < εh(x) } , Θε :Dε → R +, x2 + |y|2 + Θε(x, y)2 = ε2h(x)2, Θε(x, y) = εΘ1 ( x, y ε ) , Γ ± ε := {( x, y,±Θε(x, y) ) : (x, y) ∈ Dε } , ∂Rε\∂Ω = Γ + ε ∪ Γ − ε . If w :Rε → R we denote by w̃ :R1 → R the function defined by w̃(x, y′, z′) = w(x, εy′, εz′). Now, for 0 < ε � 1, we have that ε(N−1)/2‖w̃‖L2(R1) = ‖w‖L2(Rε) , (C.1) ε(N−1)/2‖w̃‖H 1(R1) � ‖w‖H 1(Rε) � ε(N−1)/2ε−1‖w̃‖H 1(R1) , (C.2) ε(N−1)/2‖w̃‖Hs(R1) � ‖w‖Hs(Rε) � ε(N−1)/2ε−s‖w̃‖Hs(R1), 0 < s < 1, (C.3) where (C.3) is obtained from (C.1) and (C.2) by interpolation. Let I± = ‖w‖L2(Γ ± ε ) and I = I+ + I− = ‖w‖L2(∂Rε \∂Ω). Hence, for 1 2 < s � 1, ( I±)2 = ∫ Γ ± ε |ω|2 dΓε = ∫ Dε ∣∣ω( x, y,±Θε(x, y) )∣∣2√1 + ( ∂xΘε(x, y) )2 + ( ∂yΘε(x, y) )2 dy = ∫ Dε ∣∣∣∣ω ( x, y,±εΘ1 ( x, y ε ))∣∣∣∣ 2 √ 1 + ( ε∂xΘ1 ( x, y ε ))2 + ( ε∂yΘ1 ( x, y ε ))2 dy = εN−2 ∫ ∣∣ω( x, εy′,±εΘ1(x, y′) )∣∣2√1 + ( ε∂xΘ1(x, y′) )2 + ( ∂y′Θ1(x, y′) )2 dy′ D1 1238 A.N. Carvalho, G. Lozada-Cruz / J. Math. Anal. 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