CONLEY INDEX CONTINUATION AND THIN DOMAIN PROBLEMS

MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI

IMUJ PREPRINT 2000/14

Abstract. Given ϵ > 0 and a bounded Lipschitz domain Ω in ℝ^{M} × ℝ^{N} let Ω_{ϵ} := { (x, ϵy) | (x, y) ∈ Ω } be the ϵ-squeezed domain. Consider the reaction-diffusion equation

(Ẽ_{ϵ})  u_{t} = Δu + f(u)

on Ω_{ϵ} with Neumann boundary condition. Here f is an appropriate nonlinearity such that (Ẽ_{ϵ}) generates a (local) semiflow π̃_{ϵ} on H^{1}(Ω_{ϵ}). It was proved by Prizzi and Rybakowski (J. Diff. Equations, to appear), generalizing some previous results of Hale and Raugel, that there are a closed subspace H_{s}^{1}(Ω) of H^{1}(Ω), a closed subspace L_{s}^{2}(Ω) of L^{2}(Ω) and a sectorial operator A_{0} on L_{s}^{2}(Ω) such that the semiflow π_{0} defined on H_{s}^{1}(Ω) by the abstract equation

u̇ + A_{0}u = f̂(u)

is the limit of the semiflows π̃_{ϵ} as ϵ → 0^{+}.

In this paper we prove a singular Conley index continuation principle stating that every isolated invariant set K_{0} of π_{0} can be continued to a nearby family K̃_{ϵ} of isolated invariant sets of π̃_{ϵ} with the same Conley index. We present various applications of this result to problems like connection lifting or resonance bifurcation.

1. Introduction

Let M and N be positive integers and Ω be an arbitrary nonempty smooth bounded domain in ℝ^{M} × ℝ^{N}. Write (x, y) for a generic point of ℝ^{M} × ℝ^{N}. Given ϵ > 0 squeeze Ω by the factor ϵ in the y-direction to obtain the squeezed domain Ω_{ϵ}. More precisely, let T_{ϵ}: ℝ^{M} × ℝ^{N} → ℝ^{M} × ℝ^{N}, (x, y) ↦ (x, ϵy), be the squeezing operator and define

The research of the first author was partially supported by the grant FAPESP-Brasil 99/02612-4.

Ω_{ϵ} := { (x, ϵy) | (x, y) ∈ Ω }. Consider the following reaction-diffusion equation on Ω_{ϵ}:

(Ẽ_{ϵ})  \begin{matrix}u_{t} = Δu + f(u), & t > 0, (x, y) ∈ Ω_{ϵ}, \\ ∂_{ν_{ϵ}}u = 0, & t > 0, (x, y) ∈ ∂Ω_{ϵ}, \end{matrix}

where ν_{ϵ} is the exterior normal vector field on ∂Ω_{ϵ} and f: ℝ → ℝ is a C^{1}-nonlinearity of appropriate polynomial growth which ensures that (Ẽ_{ϵ}) generates a (local) semiflow π̃_{ϵ} on H^{1}(Ω_{ϵ}).

If f is dissipative in the sense that

lim sup_{|s|→∞} f(s)/s ≤ −δ_{0} for some δ_{0} > 0

then the semiflow π̃_{ϵ} possesses a global attractor Ã_{ϵ}.

As ϵ → 0 the thin domain Ω_{ϵ} degenerates to a one-dimensional interval. One may therefore ask what happens in the limit to the family (π̃_{ϵ})_{ϵ>0} of semiflows and to the family (Ã_{ϵ})_{ϵ>0} of attractors. Is there a limit semiflow and a corresponding limit attractor?

This problem was first considered by Hale and Raugel in [21] for the case when N = 1, U is a bounded domain in ℝ^{M} and the domain Ω is the ordinate set of a smooth positive function g defined on cl U, i.e.

Ω = { (x, y) | x ∈ U and 0 < y < g(x) }.

The authors prove that, in this case, there exists a limit semiflow π̃_{0}, which is defined by the Neumann boundary value problem

(HR_{0})  \begin{matrix}u_{t} = (1/g) ∑_{i=1}^{M}(gu_{x_{i}})_{x_{i}} + f(u), & t > 0, x ∈ U \\ ∂_{ν}u = 0, & t > 0, x ∈ ∂U. \end{matrix}

Moreover, if f is dissipative, then π̃_{0} has a global attractor Ã_{0} and, in some sense, the family (Ã_{ϵ})_{ϵ≥0} is upper-semicontinuous at ϵ = 0.

Hale and Raugel also prove that one can modify the nonlinearity f in such a way that each modified semiflow π̃′_{ϵ} possesses an invariant C^{1}-manifold M̃_{ϵ} of some fixed dimension ν which includes the attractor Ã_{ϵ} of the original semiflow π̃_{ϵ}. The semiflows π̃_{ϵ} and π̃′_{ϵ} coincide on the attractor Ã_{ϵ} and, as ϵ → 0, the reduced flow on M̃_{ϵ} converges in the C^{1}-sense to the reduced flow on M̃_{0}.

If the domain Ω is not the ordinate set of some function (e.g. if Ω has holes or different horizontal branches) then an equation of the form (HR_{0}) cannot be a limiting equation for the family (Ẽ_{ϵ}), ϵ > 0. Nevertheless, as it was proved by Prizzi and Rybakowski in [33] the family π̃_{ϵ}, ϵ > 0 still has a limit semiflow. Moreover, if f is dissipative then there exists a limit global attractor and the upper-semicontinuity result continues to hold.

In order to describe the main results of [33] we first transfer the family (Ẽ_{ϵ}) to boundary value problems on the fixed domain Ω. More explicitly, we use the linear isomorphism Φ_{ϵ}: H^{1}(Ω_{ϵ}) → H^{1}(Ω), u ↦ u ∘ T_{ϵ}, to transform problem (Ẽ_{ϵ}) to the equivalent problem

(E_{ϵ})  \begin{matrix}u_{t} = Δ_{x}u + (1/ϵ^{2})Δ_{y}u + f(u), & t > 0, (x, y) ∈ Ω \\ ν_{1} · ∇_{x}u + (1/ϵ^{2})ν_{2} · ∇_{y}u = 0, & t > 0, (x, y) ∈ ∂Ω. \end{matrix}

Here, ν = (ν_{1}, ν_{2}) is the exterior normal vector field on ∂Ω.

Note that equation (E_{ϵ}) can be written in the abstract form

u̇ + A_{ϵ}u = f̂(u)

where f̂: H^{1}(Ω) → L^{2}(Ω) is the Nemitski operator generated by the function f and A_{ϵ} is the linear operator defined by

A_{ϵ}u = −Δ_{x}u − \frac{1}{ϵ^{2}} Δ_{y}u ∈ L^{2}(Ω) for u ∈ H^{2}(Ω) with ν_{1} · ∇_{x}u + \frac{1}{ϵ^{2}} ν_{2} · ∇_{y}u = 0 on ∂Ω.

Equation (E_{ϵ}) defines a semiflow π_{ϵ} on H^{1}(Ω) which is conjugated to π̃_{ϵ} via the isomorphism Φ_{ϵ}. Therefore, whatever property we prove about the transformed semiflow π_{ϵ} translates into a corresponding property about the original semiflow π̃_{ϵ}, and vice versa. In particular, if f is dissipative then π_{ϵ} has the global attractor A_{ϵ} := Φ_{ϵ}(Ã_{ϵ}), consisting of the orbits of all full bounded solutions of (E_{ϵ}).

The operator A_{ϵ} is, in the usual way, induced by the following bilinear form

a_{ϵ}(u, v) := ∫_{Ω}(∇_{x}u · ∇_{x}v + \frac{1}{ϵ^{2}} ∇_{y}u · ∇_{y}v) dx dy, u, v ∈ H^{1}(Ω).

Notice that, for every fixed ϵ > 0 and u ∈ H^{1}(Ω), the formula

|u|_{ϵ} = (a_{ϵ}(u, u) + |u|_{L^{2}(Ω)}^{2})^{1/2}

defines a norm on H^{1}(Ω) which is equivalent to | · |_{H^{1}(Ω)}. However, |u|_{ϵ} → ∞ as ϵ → 0^{+} whenever ∇_{y}u ≠ 0 in L^{2}(Ω).

In fact, we see that for u ∈ H^{1}(Ω)

lim_{ϵ→0^{+}} a_{ϵ}(u, u) = \begin{cases}∫_{Ω} ∇_{x}u · ∇_{x}u dx dy, & if ∇_{y}u = 0 \\ +∞, & otherwise. \end{cases}

Thus the family a_{ϵ}(u, u), ϵ > 0, of real numbers has a finite limit (as ϵ → 0) if and only if u ∈ H_{s}^{1}(Ω), where we define

H_{s}^{1}(Ω) := { u ∈ H^{1}(Ω) | ∇_{y}u = 0 }.

This is a closed linear subspace of H^{1}(Ω).

The corresponding limit bilinear form is given by the formula:

a_{0}(u, v) := ∫_{Ω} ∇_{x}u · ∇_{x}vdx dy, u, v ∈ H_{s}^{1}(Ω).

The form a_{0} uniquely determines a densely defined selfadjoint linear operator

A_{0} : D(A_{0}) ⊂ H_{s}^{1}(Ω) → L_{s}^{2}(Ω)

by the usual formula

a_{0}(u, v) = ⟨A_{0}u, v⟩_{L^{2}(Ω)}, for u ∈ D(A_{0}) and v ∈ H_{s}^{1}(Ω).

Here, L_{s}^{2}(Ω) is the closure of H_{s}^{1}(Ω) in the L^{2}-norm, so L_{s}^{2}(Ω) is a closed linear subspace of L^{2}(Ω).

It follows that the Nemitski operator f̂ maps the space H_{s}^{1}(Ω) into L_{s}^{2}(Ω). Consequently the abstract parabolic equation

(E_{0})  u̇ = −A_{0}u + f̂(u)

defines a semiflow π_{0} on the space H_{s}^{1}(Ω). This is the limit semiflow of the family π_{ϵ}. In the ordinate set case considered by Hale and Raugel, equation (E_{0}) reduces to their equation (HR_{0}) stated above. For a large class of two-dimensional domains (called nicely decomposed) equation (E_{0}) is equivalent to a parabolic equation defined on a finite onedimensional graph. For general higher dimensional domains and M ≥ 2 it does not seem possible to give a more explicit expression for equation (E_{0}). Nevertheless, many qualitative properties of the limit semiflow π_{0} may be deduced from the abstract equation (E_{0}).

For example, if f is dissipative then the semiflow π_{0} has a global attractor A_{0} and the family of attractors (A_{ϵ})_{ϵ≥0} is upper-semicontinuous at ϵ = 0. More precisely, we have the following result: Upper semicontinuity of attractors. (see Theorem 5.10 in [33]) The family of attractors (A_{ϵ})_{ϵ∈[0,1]} is upper-semicontinuous at ϵ = 0 with respect to the family | · |_{ϵ} of norms, i.e.

lim_{ϵ→0^{+}} sup_{u∈A_{ϵ}} inf_{v∈A_{0}} |u − v|_{ϵ} = 0.

Notice that each attractor A_{ϵ} for ϵ ≥ 0 is a compact isolated π_{ϵ}-invariant set with Conley index h(π_{ϵ}, A_{ϵ}) ≡ Σ^{0}. Therefore the above theorem implies that the invariant set A_{0} can be continued to a family of invariant sets with the same Conley index.

We will prove in this paper that the latter statement holds true for an arbitrary isolated invariant set of the limit semiflow: Conley index continuation theorem. Let K_{0} ⊂ H_{s}^{1}(Ω) be a compact isolated invariant set of the limit semiflow π_{0}. Then there is an ϵ_{0} > 0 and for each ϵ ∈ ]0, ϵ_{0}] there is a compact isolated invariant set K_{ϵ} ⊂ H^{1}(Ω) of the semiflow π_{ϵ} such that h(π_{ϵ}, K_{ϵ}) ≡ h(π_{0}, K_{0}) and the family (K_{ϵ})_{ϵ∈[0,ϵ_{0}]} is upper-semicontinuous at ϵ = 0 with respect to the family | · |_{ϵ} of norms, i.e.

lim_{ϵ→0^{+}} sup_{u∈K_{ϵ}} inf_{v∈K_{0}} |u − v|_{ϵ} = 0.

Note that in this theorem the function f need not be dissipative and it may depend not only on u but also on ϵ, x, y and ∇u. For a precise statement see Theorem 3.5 and its Corollary 3.7 in Section 3 below.

The reader should notice that our Conley index continuation Theorem 3.5 is not a direct consequence of the classical continuation principle in Conley index theory. In fact, we are dealing here with a singular perturbation problem and the limiting semiflow (unlike in the regular perturbation case) lives in the Banach space H_{s}^{1}(Ω) which is different from the Banach space H^{1}(Ω) on which the approximating semiflows are defined. Thus, although the proof of Theorem 3.5 follows the lines of the classical Conley continuation principle for admissible semiflows (as given in [37, 39]) several new ideas are required to cope with the present, more difficult situation (see Section 3 and the Appendix for details).

For the proof of Theorem 3.5 we need some convergence results, as ϵ → 0, of the semiflows π_{ϵ} and the linear semigroups generated by the operators A_{ϵ}. These results, presented in Section 2, strengthen and extend the convergence results previously obtained in [33].

In section 4 we present various applications of Theorem 3.5. First we show in Subsection 4.1 that, under appropriate assumptions, an orbit σ_{0} connecting equilibria u′_{0} and u′′_{0} of the limit semiflow π_{0} can be continued, for all ϵ > 0 small enough, to a ‘nearby’ family σ_{ϵ} of orbits connecting ‘nearby’ equilibria u′_{ϵ} and u′′_{ϵ} of π_{ϵ}.

This is then applied to a specific connection problem in parabolic equations on an interval, considered by Fiedler and Rocha [14].

We then prove that, generically in the domain, the family of the j-th eigenvalues λ_{ϵ,j} of the operators A_{ϵ} is strictly monotone decreasing in ϵ > 0, for j ≥ 2 (Subsection 4.2). Using this, together with Theorem 3.5 and the Conley index product formula we obtain, under suitable resonance hypotheses for the limit semiflow π_{0}, the existence of nontrivial equilibria of π_{ϵ} bifurcating from the trivial equilibrium 0 of π_{0} (Subsection 4.3). Finally, we show that these resonance hypotheses are satisfied for very simple two-dimensional domains (Subsection 4.3).

After the completion of this work we became aware of the preprint [25] of Q. Huang, in which the author proves an analogue of our Theorem 3.5 for ordinate set domains with g ≡ 1, i.e. domains Ω of the type Ω = U ×]0, 1[, where U is a bounded domain in ℝ^{M} with M = 1 or 2. The corresponding squeezed domain Ω_{ϵ} is then of the form Ω_{ϵ} = U ×]0, ϵ[. (See Theorem 6.1 in [25].) In the case M = 1 such domains are just rectangles.

The author obtains his result as an application of an abstract Conley index continuation theorem (see Theorem A in [25]). One could use that theorem, together with the convergence results established in Section 2 of the present paper and some additional estimates to give an alternative proof of our Theorem 3.5.

On the other hand, our proof of Theorem 3.5 uses only some general properties of the family π_{ϵ} of semiflows and the family Q_{ϵ} of projectors (cf. Section 2 and the Appendix below). By abstracting these properties one can establish a Conley index continuation result which is more general than Theorem A of [25]. In fact, we can replace the compactness condition (C) of [25] by a weaker admissibility assumption of the type introduced in [37]. Details are given in the paper [7]. Let us only remark here that this more general abstract result, unlike the results of [25], can be applied to singular perturbation problems for semiflows generated by damped wave equations (cf. [20, 22, 8]), neutral functional differential equations or infinite delay equations. The solution operators of such semiflows are not compact and yet the admissibility condition is satisfied.

Let us also remark that it is possible to formulate a general continuation principle for the Conley connection matrix in singular perturbation problems. See the forthcoming paper [9] for details.

For more results on the squeezed domain problems see the paper [34], which, among other things, contains an existence theorem for inertial manifolds containing the attractors A_{ϵ} of π_{ϵ}, improving and generalizing to arbitrary domains the results of Hale and Raugel mentioned above. Some other papers on thin domain problems are contained in the References, although the list is far from being complete.

The Reference-section also lists a few papers on the theory and applications of various versions of the Conley index.

2. Squeezed domains

In this section we will prove various convergence results for linear semigroups and nonlinear semiflows generated by reaction-diffusion equations on squeezed domains. These results, of interest in their own right, play an essential role in establishing the Conley index continuation principle of Section 3.

Let us first recall some definitions and basic results about squeezed domains proved in [33, 34], referring the reader to those papers and to the paper [21] of Hale and Raugel for more details on the subject. In this paper all vector spaces are over the real numbers.

Definition 2.1. Let V be a vector space and a: V × V → ℝ and b: V × V → ℝ be symmetric bilinear forms on V. If λ ∈ ℝ, u ∈ V \ {0} satisfy

a(u, v) = λb(u, v) for all v ∈ V

then we say that λ is an eigenvalue of the pair (a, b) and u is an eigenvector of the pair (a, b), corresponding to λ. The dimension of the span of all eigenvectors of (a, b) corresponding to λ is called the multiplicity of λ. If the set of eigenvalues of (a, b) is countably infinite, contains a smallest element and if each eigenvalue has finite multiplicity then the repeated sequence of the eigenvalues of (a, b) is the uniquely determined nondecreasing sequence (λ_{n})_{n∈ℕ} which contains exactly the eigenvalues of (a, b) and the number of occurrences of each eigenvalue in this sequence is equal to its multiplicity.

The following proposition is well-known:

Proposition 2.2. Let V, H be two infinite dimensional Hilbert spaces. Suppose V ⊂ H with compact inclusion, and V is dense in H. Let ∥ · ∥ and | · | denote the norms of V and H respectively, and ⟨·, ·⟩ be the inner product of H and b be the restriction of this inner product to V × V. Let a: V × V → ℝ be a symmetric bilinear form on V. Assume that there are constants d, C, α ∈ ℝ, α > 0, such that, for all u, v ∈ V,

|a(u, v)| ≤ C∥u∥∥v∥
a(u, u) ≥ α∥u∥^{2} − d|u|^{2}.

Then the set of eigenvalues of (a, b) is countably infinite, it has a smallest element and each eigenvalue has finite multiplicity. Let (λ_{n})_{n∈ℕ} be the repeated sequence of the eigenvalues of (a, b). Then the following properties are satisfied:

(1) There exists an H-orthogonal sequence (u_{n})_{n∈ℕ} such that for every k ∈ ℕ, u_{k} is an eigenvector of (a, b) corresponding to λ_{k}.

(2) Whenever (u_{n})_{n∈ℕ} is an H-orthogonal sequence such that for every k ∈ ℕ, u_{k} is an eigenvector of (a, b) corresponding to λ_{k}, then (u_{n})_{n∈ℕ} is H-complete. Moreover, the space V is characterized in the following way:

V = { v ∈ H | ∑_{n=1}^{∞} λ_{n}|⟨v, u_{n}⟩|^{2}/|u_{n}|^{2} < ∞ }.

(3) Let D be the set of all u ∈ V such that there is a w = w_{u} ∈ H with the property that

a(u, v) = ⟨w, v⟩

for all v ∈ V. Then w_{u} is uniquely determined by u, the set D is a dense linear subspace both of V and of H, and the map

A: D → H, u ↦ w_{u}

is a linear selfadjoint operator on (H, ⟨·, ·⟩) with compact resolvent. We call A the linear operator generated by the pair (a, ⟨·, ·⟩). A pair (λ, u) is an eigenvalueeigenvector pair of A if and only if (λ, u) is an eigenvalue-eigenvector pair of (a, b). □

Let Ω be an arbitrary nonempty bounded domain in ℝ^{M} ×ℝ^{N} with Lipschitz boundary and let ϵ > 0 be arbitrary. Write (x, y) for a generic point of ℝ^{M} × ℝ^{N}. As in the Introduction, let T_{ϵ}: ℝ^{M} × ℝ^{N} → ℝ^{M} × ℝ^{N}, (x, y) ↦ (x, ϵy), be the squeezing operator and Ω_{ϵ} := T_{ϵ}(Ω) be the squeezed domain. Define the symmetric bilinear forms

ã_{ϵ}: H^{1}(Ω_{ϵ}) × H^{1}(Ω_{ϵ}) → ℝ

by

ã_{ϵ}(u, v) := ∫_{Ω_{ϵ}} ∇u · ∇v dx dy

and

a_{ϵ}: H^{1}(Ω) × H^{1}(Ω) → ℝ

by

a_{ϵ}(u, v) := ∫_{Ω} (∇_{x}u · ∇_{x}v + \frac{1}{ϵ^{2}} ∇_{y}u · ∇_{y}v) dx dy.

Note that the assignment

Φ_{ϵ}: u ↦ u ∘ T_{ϵ}

restricts to linear isomorphisms L^{2}(Ω_{ϵ}) → L^{2}(Ω) and H^{1}(Ω_{ϵ}) → H^{1}(Ω).

Let b̃_{ϵ} be the restriction of ⟨·, ·⟩_{L^{2}(Ω_{ϵ})} to H^{1}(Ω_{ϵ}) × H^{1}(Ω_{ϵ}) and let b be the restriction of ⟨·, ·⟩_{L^{2}(Ω)} to H^{1}(Ω) × H^{1}(Ω).

It follows that a pair (λ, u) is an eigenvalue-eigenvector pair of (ã_{ϵ}, b̃_{ϵ}) if and only if the pair (λ, Φ_{ϵ}u) is an eigenvalue-eigenvector pair of (a_{ϵ}, b). The linear operators Ã_{ϵ} (resp. A_{ϵ}) defined by (ã_{ϵ}, b̃_{ϵ}) (resp. (a_{ϵ}, b)) satisfy the following properties:

(1) D(A_{ϵ}) = Φ_{ϵ}(D(Ã_{ϵ}));

(2) A_{ϵ}(Φ_{ϵ}u) = Φ_{ϵ}(Ã_{ϵ}u) for u ∈ D(Ã_{ϵ}).

Proposition 2.2 implies that there exists a nondecreasing sequence (λ_{ϵ,j}, w_{ϵ,j})_{j∈ℕ} of eigenvalue-eigenvector pairs of (a_{ϵ}, b) such that (w_{ϵ,j})_{j∈ℕ} is a complete orthonormal system on L^{2}(Ω). Moreover, λ_{ϵ,1} ≡ 0 and λ_{ϵ,j} > 0 for all ϵ > 0 and j ≥ 2. For convenience we set λ_{ϵ,0} := −∞.

Now define the “limit” space H_{s}^{1}(Ω) by

H_{s}^{1}(Ω) = { u ∈ H^{1}(Ω) | ∇_{y}u = 0 }.

Note that H_{s}^{1}(Ω) is a closed linear subspace of H^{1}(Ω). It can be proved that H_{s}^{1}(Ω) has infinite dimension.

Let us also define the space L_{s}^{2}(Ω) to be the closure of the set H_{s}^{1}(Ω) in L^{2}(Ω). It follows that L_{s}^{2}(Ω) is a Hilbert space under the scalar product of L^{2}(Ω).

Now let a_{0} : H_{s}^{1}(Ω) × H_{s}^{1}(Ω) → ℝ be the “limit” bilinear form defined by

a_{0}(u, v) := ∫_{Ω} ∇u · ∇v dx dy = ∫_{Ω} ∇_{x}u · ∇_{x}v dx dy.

The bilinear form a_{0} on H_{s}^{1}(Ω) together with the scalar product

⟨u, v⟩_{L_{s}^{2}(Ω)} := ∫_{Ω} uv dx dy on L_{s}^{2}(Ω)

satisfy the hypotheses of Proposition 2.2. Let b_{0} be the restriction of ⟨·, ·⟩_{L^{2}(Ω)} to H_{s}^{1}(Ω)× H_{s}^{1}(Ω).

Proposition 2.2 implies that there exists a nondecreasing sequence (λ_{0,j}, w_{0,j})_{j∈ℕ} of eigenvalue-eigenvector pairs of (a_{0}, b_{0}) such that (w_{0,j})_{j∈ℕ} is a complete orthonormal system on L_{s}^{2}(Ω). Moreover, λ_{0,1} ≡ 0 and λ_{0,j} > 0 for all j ≥ 2. For convenience we set λ_{0,0} := −∞.

Denote by A_{0} the operator defined by the pair (a_{0}, b_{0}).

Let us now recall the concept of a semiflow:

Let X be a topological space, let D be an open subset of [0, ∞[ × X and π: D → X be a continuous map. We write xπt := π(t, x) for (t, x) ∈ D. The map π is called a local semiflow on X if the following properties are satisfied:

(1) For every x ∈ X there is a number ω_{x} = ω_{x}^{π} ∈ ]0, ∞] such that (t, x) ∈ D if and only if 0 ≤ t < ω_{x}.

(2) xπ0 = x for all x ∈ X.

(3) If (t, x) ∈ D and (s, xπt) ∈ D then (t + s, x) ∈ D and

xπ(t + s) = (xπt)πs.

If ω_{x} = ∞ for every x ∈ X, then π is called a global semiflow on X.

Let J be an arbitrary interval in ℝ. A map σ: J → X is called a solution of π if for all t ∈ J and s ∈ [0, ∞[ for which t + s ∈ J , it follows that σ(t)πs is defined and σ(t)πs = σ(t + s). If 0 ∈ J and σ(0) = x, we say that σ is a solution through x. If J = ℝ (respectively J = ]−∞, 0]), then σ is called a full solution (respectively full left solution) relative to π.

Let π and π′ be local semiflows on the topological space X and Y be an arbitrary subset of X. We say that π and π′ coincide on Y if whenever J is an interval in ℝ and σ: J → Y is a map then σ is a solution of π if and only if σ is a solution of π′.

Now let X and X′ be topological spaces and let π (resp. π′) be a local semiflow on X (resp. on X′). We say that the map Φ: X → X′ conjugates π with π′ if Φ is a homeomorphism and σ: J → X is a solution of π if and only if Φ∘σ: J → X′ is a solution of π′. Example 2.3. Let X be a Banach space and A be a sectorial operator in X generating the family X^{β}, β ≥ 0, of fractional power spaces. Fix an α ∈ [0, 1[ and suppose U is an open set in X^{α} and f: U → X is a locally Lipschitzian map. The equation

u̇ = −Au + f(u)

defines, in the usual way, a local semiflow π_{A,f} on X^{α}. (see [24] or [39]). If f is globally Lipschitzian on U, then π_{A,f} is a global semiflow.

Example 2.4. Let A and U be as in the Example 2.3. Moreover, let f: U → X and f′: U → X be locally Lipschitzian maps. If Y is a subset of U and f(u) = f′(u) on Y, then π_{A,f} coincides with π_{A,f′} on Y. This easily follows from the definition of the solution of parabolic equations (see [24] or [39]).

For every ϵ > 0 the operator A_{ϵ} (resp. Ã_{ϵ}) is sectorial on X = L^{2}(Ω) (resp. on X = L^{2}(Ω_{ϵ})) and the corresponding fractional power space X^{α} with α = 1/2 satisfies X^{α} = H^{1}(Ω) (resp. X^{α} = H^{1}(Ω_{ϵ})). If f_{ϵ}: H^{1}(Ω) → L^{2}(Ω) (resp. f̃_{ϵ}: H^{1}(Ω_{ϵ}) → L^{2}(Ω_{ϵ})) is a locally Lipschitzian map we thus obtain the corresponding local semiflow π_{ϵ,f_{ϵ}} := π_{A_{ϵ},f_{ϵ}} (resp. π̃_{ϵ,f̃_{ϵ}} := π_{Ã_{ϵ},f̃_{ϵ}}) on H^{1}(Ω) (resp. on H^{1}(Ω_{ϵ})). Note that, given a locally Lipschitzian map f̃_{ϵ}: H^{1}(Ω_{ϵ}) → L^{2}(Ω_{ϵ}), the map Φ_{ϵ}(u) = u ∘ T_{ϵ} introduced above conjugates the semiflow π̃_{ϵ,f̃_{ϵ}} with the semiflow π_{ϵ,f_{ϵ}} where f_{ϵ} := Φ_{ϵ} ∘ f̃_{ϵ} ∘ Φ_{ϵ}^{−1}.

Now let g_{ϵ}: Ω_{ϵ} × ℝ × ℝ^{M} × ℝ^{N} → ℝ be a given function and f̃_{ϵ} := ĝ_{ϵ} be the Nemitski operator generated by g_{ϵ}, i.e. for u: Ω_{ϵ} → ℝ set

f̃_{ϵ}(u)(x, y) := g_{ϵ}((x, y), u(x, y), ∇_{x}u(x, y), ∇_{y}u(x, y)), for (x, y) ∈ Ω_{ϵ}.

Suppose f̃_{ϵ} restricts to a locally Lipschitzian map from H^{1}(Ω_{ϵ}) to L^{2}(Ω_{ϵ}). Then the map f_{ϵ} := Φ_{ϵ} ∘ f̃_{ϵ} ∘ Φ_{ϵ}^{−1} is clearly given by (1)

f_{ϵ}(u)(x, y) := g_{ϵ}((x, ϵy), u(x, y), ∇_{x}u(x, y), \frac{1}{ϵ}∇_{y}u(x, y)), for u ∈ H^{1}(Ω) and (x, y) ∈ Ω.

Now note that the “limit” operator A_{0} is sectorial on X = L_{s}^{2}(Ω) and the corresponding fractional power space X^{α} with α = 1/2 satisfies X^{α} = H_{s}^{1}(Ω). If f_{0} : H_{s}^{1}(Ω) → L_{s}^{2}(Ω) is a locally Lipschitzian map we thus obtain the corresponding local semiflow π_{0,f_{0}} := π_{A_{0},f_{0}} on H_{s}^{1}(Ω). Again, if f_{0} is globally Lipschitzian, then π_{0,f_{0}} is a global semiflow.

From now on, unless otherwise specified, we will work in a fixed bounded domain Ω and will often write | · |_{H^{1}} for | · |_{H^{1}(Ω)} and | · |_{L^{2}} for | · |_{L^{2}(Ω)}. Moreover, we will write ⟨u, v⟩ to denote the scalar product in L^{2}(Ω).

For every ϵ > 0 one can define on H^{1}(Ω) the norm

|u|_{ϵ} := (a_{ϵ}(u, u) + |u|_{L^{2}(Ω)}^{2})^{1/2} .

This norm is equivalent to |·|_{H^{1}(Ω)} for every fixed ϵ > 0, but |u|_{ϵ} → ∞ as ϵ → 0^{+} whenever ∇_{y}u ≠ 0 in L^{2}(Ω). Note that |u|_{ϵ} = |u|_{H^{1}(Ω)}, whenever ϵ > 0 and u ∈ H_{s}^{1}(Ω). It is easily seen that for every u ∈ H^{1}(Ω)

|u|_{ϵ}^{2} = ∑_{j=1}^{∞}(λ_{ϵ,j} + 1)|⟨u, w_{ϵ,j}⟩_{L^{2}(Ω)}|^{2}.

Moreover, for every u ∈ H_{s}^{1}(Ω)

|u|_{H^{1}(Ω)} = ∑_{j=1}^{∞}(λ_{0,j} + 1)|⟨u, w_{0,j}⟩_{L^{2}(Ω)}|^{2}.

The following concept will play a crucial in the convergence results established below: Definition 2.5. Given ϵ_{0} with 0 < ϵ_{0} ≤ 1 we say that the family (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} of maps satisfies hypothesis (A1) if the following properties are satisfied:

(1) f_{ϵ}: H^{1}(Ω) → L^{2}(Ω) for every ϵ ∈ ]0, ϵ_{0}] and f_{0} : H_{s}^{1}(Ω) → L_{s}^{2}(Ω).

(2) lim_{ϵ→0^{+}} |f_{ϵ}(u) − f_{0}(u)|_{L^{2}} = 0 for every u ∈ H_{s}^{1}(Ω).

(3) For every M ∈ [0, ∞[ there is an L ∈ [0, ∞[ such that

|f_{ϵ}(u) − f_{ϵ}(v)|_{L^{2}} ≤ L|u − v|_{ϵ}

for ϵ ∈ ]0, ϵ_{0}] and u, v ∈ H^{1}(Ω) satisfying |u|_{ϵ}, |v|_{ϵ} ≤ M. Moreover,

|f_{0}(u) − f_{0}(v)|_{L^{2}} ≤ L|u − v|_{H^{1}}

for u, v ∈ H_{s}^{1}(Ω) satisfying |u|_{H^{1}}, |v|_{H^{1}} ≤ M.

We say that (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfies hypothesis (A2) if the following properties are satisfied:

(1) f_{ϵ}: H^{1}(Ω) → L^{2}(Ω) for every ϵ ∈ ]0, ϵ_{0}] and f_{0} : H_{s}^{1}(Ω) → L_{s}^{2}(Ω).

(2) lim_{ϵ→0^{+}} |f_{ϵ}(u) − f_{0}(u)|_{L^{2}} = 0 for every u ∈ H_{s}^{1}(Ω).

(3) There is an L ∈ [0, ∞[ such that

|f_{ϵ}(u) − f_{ϵ}(v)|_{L^{2}} ≤ L|u − v|_{ϵ}

for ϵ ∈ ]0, ϵ_{0}] and all u, v ∈ H^{1}(Ω). Moreover,

|f_{0}(u) − f_{0}(v)|_{L^{2}} ≤ L|u − v|_{H^{1}}

for all u, v ∈ H_{s}^{1}(Ω).

The following proposition shows how we can obtain, in applications, families of maps satisfying hypothesis (A1): Proposition 2.6. Suppose the domain Ω satisfies the following condition:

(2)  L_{s}^{2}(Ω) = { u ∈ L^{2}(Ω) | ∇_{y}u = 0 }.

Let G: ℝ × ℝ^{M} × ℝ^{N} × ℝ × ℝ^{M} × ℝ^{N} → ℝ, (ϵ, x, y, ξ, η, ζ) ↦ G(ϵ, x, y, ξ, η, ζ), be a C^{1}- function for which there are constants β, γ and C ∈ [0, ∞[ such that for all (ϵ, x, y, ξ, η, ζ) ∈ ℝ × ℝ^{M} × ℝ^{N} × ℝ × ℝ^{M} × ℝ^{N} the following estimates are satisfied:

(1) |∂_{ϵ}G(ϵ, x, y, ξ, η, ζ)| ≤ C(1 + |ξ|^{β});

(2) |∇_{y}G(ϵ, x, y, ξ, η, ζ)| ≤ C(1 + |ξ|^{β});

(3) |∂_{ξ}G(ϵ, x, y, ξ, η, ζ)| ≤ C(1 + |ξ|^{γ});

(4) |∇_{η}G(ϵ, x, y, ξ, η, ζ)| + |∇_{ζ}G(ϵ, x, y, ξ, η, ζ)| ≤ C.

If n := M + N > 2 then we also assume that β ≤ 2^{∗}/2 and γ ≤ (2^{∗}/2) − 1, where 2^{∗} := 2n/(n − 2).

For ϵ > 0 define the function g_{ϵ}: Ω_{ϵ} × ℝ × ℝ^{M} × ℝ^{N} → ℝ by

g_{ϵ}((x, y), ξ, η, ζ) = G(ϵ, x, y, ξ, η, ζ) for ((x, y), ξ, η, ζ) ∈ Ω_{ϵ} × ℝ × ℝ^{M} × ℝ^{N}.

Then the Nemitski operator f̃_{ϵ} := ĝ_{ϵ} defined by the function g_{ϵ} is a well-defined map from H^{1}(Ω_{ϵ}) to L^{2}(Ω_{ϵ}). Define f_{ϵ}: H^{1}(Ω) → L^{2}(Ω) by f_{ϵ} = Φ_{ϵ} ∘ f̃_{ϵ} ∘ Φ_{ϵ}^{−1}. More specifically, f_{ϵ} is given by (1).

Furthermore, for u ∈ H_{s}^{1}(Ω) define f_{0}(u): Ω → ℝ by

f_{0}(u)(x, y) := G(0, x, 0, u(x, y), ∇_{x}u(x, y), 0) for (x, y) ∈ Ω.

Then f_{0}(u) ∈ L_{s}^{2}(Ω) and the family (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfies hypothesis (A1).

Proof. Since H^{1}(Ω) is continuously contained in L^{σ}(Ω) where σ ∈ [1, ∞[ for n = 2 and σ ∈ [1, 2^{∗}] for n > 2, it follows by an application of the mean-value theorem and Hölder inequality, using estimates 1 - 4 above, that f_{ϵ}(H^{1}(Ω)) ⊂ L^{2}(Ω) for ϵ > 0 and conditions 2 and 3 in the definition of hypothesis (A1) are satisfied. Let u ∈ H_{s}^{1}(Ω) be arbitrary and v := f_{0}(u). Then an application of Theorem 2.5 in [33] implies that ∇_{y}v = 0. Now condition (2) implies that v ∈ L_{s}^{2}(Ω). This shows that hypothesis (A1) is satisfied. The proof is complete. □

Remark 2.7. 1. Condition (2) is satisfied for domains Ω with connected vertical sections, i.e. domains such that the set

Ω_{x} := { y | (x, y) ∈ Ω }

is connected for all x ∈ ℝ^{M}. In fact in such cases functions u ∈ L^{2}(Ω) with ∇_{y}u = 0 are actually functions depending only on the variables x. By appropriately regularizing these functions with respect to these variables we can prove that u ∈ L_{s}^{2}(Ω). Obvious details are omitted. In particular, condition (2) is satisfied for domains Ω considered by Hale and Raugel. Condition (2) is also satisfied for nicely decomposed domains. This follows from results in [34].

2. If the function G is independent of the variables η and ζ, then condition (2) is superfluous. In fact, the proof that f_{0}(H_{s}^{1}(Ω)) ⊂ L_{s}^{2}(Ω) is, in that case, accomplished by a simple modification of the proof of Theorem 5.3 in [33].

3. Without changing the proof we may relax the growth conditions 1 - 3 in Proposition 2.6. In fact, we may allow factors involving certain positive powers of |η| and |ζ|. We do not give a precise statement here since our main interest is in functions G which do not depend on the variables η and ζ, so that the resulting semiflows π_{ϵ,f_{ϵ}} are gradient-like.

Condition (A1) is stronger than condition (A2). However, in many cases, one can modify a given family of maps satisfying (A1) in such a way as to obtain a family satisfying (A2) and so that both families coincide on a given bounded set. This is made more precise in the following result, which is obtained by a simple calculation, using Example 2.4: Proposition 2.8. Let 0 < ϵ_{0} ≤ 1 and (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfy hypothesis (A1). Let Y be the open ball in H^{1}(Ω) at zero with radius r > 0. Choose a smooth function h: ℝ → ℝ such that

h(s) = \begin{cases}1 & if |s| < r, \\ 0 & if |s| > 2r. \end{cases}

For every ϵ ∈ ]0, ϵ_{0}] define the map f′_{ϵ}: H^{1}(Ω) → L^{2}(Ω) by

f′_{ϵ}(u) = h(|u|_{ϵ})f_{ϵ}(u).

Moreover, define f′_{0} : H^{1}(Ω) → L^{2}(Ω) by

f′_{0}(u) = h(|u|_{H^{1}})f_{0}(u).

Then the family (f′_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfies hypothesis (A2). Moreover, f_{ϵ}(u) = f′_{ϵ}(u) for ϵ ∈ ]0, ϵ_{0}] and u ∈ Y. Besides, f_{0}(u) = f′_{0}(u) for u ∈ Y ∩ H_{s}^{1}(Ω). Consequently, the local semiflows π_{ϵ,f_{ϵ}} and π_{ϵ,f′_{ϵ}} coincide on Y_{ϵ} := { u ∈ H^{1}(Ω) | |u|_{ϵ} < r } for ϵ ∈ ]0, ϵ_{0}] and on Y ∩ H_{s}^{1}(Ω) for ϵ = 0. □

We can now state our first convergence result:

Theorem 2.9. Let (ϵ_{n})_{n∈ℕ} and (t_{n})_{n∈ℕ} be arbitrary sequences of positive numbers convergent to zero. Let u_{0} ∈ H_{s}^{1}(Ω) and (u_{n})_{n∈ℕ} be a sequence in H^{1}(Ω) such that

|u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Then

|e^{−t_{n}A_{ϵn}} u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Proof. Given v ∈ H^{1}(Ω), t > 0 and ϵ > 0 we have

|e^{−tA_{ϵ}}v|_{ϵ}^{2} = ∑_{j=1}^{∞}(λ_{ϵ,j} + 1)|⟨e^{−tA_{ϵ}}v, w_{ϵ,j}⟩|^{2} = ∑_{j=1}^{∞}(e^{−tλ_{ϵ,j}})^{2}(λ_{ϵ,j} + 1)|⟨v, w_{ϵ,j}⟩|^{2}.

Since λ_{ϵ,j} ≥ 0, it follows that

|e^{−tA_{ϵ}}v|_{ϵ} ≤ |v|_{ϵ}.

Hence we obtain

|e^{−t_{n}A_{ϵn}} u_{n} − u_{0}|_{ϵ_{n}} ≤ |e^{−t_{n}A_{ϵn}} (u_{n} − u_{0})|_{ϵ_{n}} + |e^{−t_{n}A_{ϵn}} u_{0} − u_{0}|_{ϵ_{n}}
≤ |u_{n} − u_{0}|_{ϵ_{n}} + |e^{−t_{n}A_{ϵn}} u_{0} − u_{0}|_{ϵ_{n}}.

Thus we only have to prove that

|e^{−t_{n}A_{ϵn}} u_{0} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Let δ > 0 be arbitrary. It follows that

|u_{0}|_{H^{1}}^{2} = ∑_{j=1}^{∞}(λ_{0,j} + 1)|⟨u_{0}, w_{0,j}⟩|^{2} < ∞.

Hence there is a k ∈ ℕ such that

d_{0} := ∑_{j=k}^{∞}(λ_{0,j} + 1)|⟨u_{0}, w_{0,j}⟩|^{2} < δ.

Let P_{n}: L^{2}(Ω) → L^{2}(Ω) be the L^{2}-orthogonal projection onto the span of the vectors {w_{ϵ_{n},1}, …, w_{ϵ_{n},k−1}}. Then

|u_{0}|_{H^{1}}^{2} = |u_{0}|_{ϵ}^{2} = ∑_{j=1}^{∞}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2} = c_{n} + d_{n},

where

c_{n} := ∑_{j=1}^{k−1}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2}

and

d_{n} := ∑_{j=k}^{∞}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2}.

Theorem 3.3 in [33] implies that for an appropriate choice of the eigenvectors (w_{0,j})_{j∈ℕ}

c_{n} → ∑_{j=1}^{k−1}(λ_{0_{n},j} + 1)|⟨u_{0}, w_{0,j}⟩|^{2} =: c_{0} as n → ∞.

Consequently, for n → ∞,

d_{n} → |u_{0}|_{H^{1}}^{2} − c_{0} = d_{0} < δ.

Thus there is an n_{0} such that 0 ≤ d_{n} < δ for n ≥ n_{0}. Now

|e^{−t_{n}A_{ϵn}} u_{0} − u_{0}|_{ϵ_{n}}^{2} = ∑_{j=1}^{∞}(e^{−t_{n}λ_{ϵn,j}} − 1)^{2}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2} = c′_{n} + d′_{n},

where

c′_{n} := ∑_{j=1}^{k−1}(e^{−t_{n}λ_{ϵn,j}} − 1)^{2}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2}

and

d′_{n} := ∑_{j=k}^{∞}(e^{−t_{n}λ_{ϵn,j}} − 1)^{2}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2}.

Now Theorem 3.3 in [33] implies that t_{n}·λ_{ϵ_{n},j} → 0·λ_{0,j} as n → ∞, so, again by Theorem 3.3 in [33],

c′_{n} → 0 as n → ∞,

so there is an n_{1} ≥ n_{0} such that

c′_{n} < δ for n ≥ n_{1}.

Furthermore, (e^{−t_{n}λ_{ϵn,j}} − 1)^{2} ≤ 1 so

d′_{n} ≤ ∑_{j=k}^{∞}(λ_{ϵ_{n},j} + 1)|⟨u_{0}, w_{ϵ_{n},j}⟩|^{2} = d_{n} < δ.

Therefore

|e^{−t_{n}A_{ϵn}} u_{0} − u_{0}|_{ϵ_{n}}^{2} < 2δ for n ≥ n_{1}.

Since δ is arbitrary, the theorem is proved. □

Remark 2.10. Let (ϵ_{n})_{n∈ℕ} and (t_{n})_{n∈ℕ} be arbitrary sequences of positive numbers convergent to zero. Let u_{0} ∈ L_{s}^{2}(Ω) and (u_{n})_{n∈ℕ} be a sequence in L^{2}(Ω) converging to u_{0} in L^{2}(Ω). Then

|e^{−t_{n}A_{ϵn}} u_{n} − u_{0}|_{L^{2}} → 0 as n → ∞.

The proof of this assertion is completely analogous to (and easier than) the proof of Theorem 2.9.

For the rest of this section let ϵ_{0} > 0 and (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfy hypothesis (A2). Write π_{ϵ} for π_{ϵ,f_{ϵ}}.

Before stating our next convergence result let us note that there is a positive real constant C_{1} such that for all ϵ > 0, r > 0 and u ∈ L^{2}(Ω)

(3)  |e^{−A_{ϵ}r}u|_{ϵ} ≤ (C_{1}r^{−1/2} + 1)|u|_{L^{2}}.

Moreover, for all u ∈ L_{s}^{2}(Ω)

(4)  |e^{−A_{0}r}u|_{ϵ} ≤ (C_{1}r^{−1/2} + 1)|u|_{L^{2}}.

We will need these estimates in the proofs to follow.

We can now state our second convergence theorem:

Theorem 2.11. Let (ϵ_{n})_{n∈ℕ} and (t_{n})_{n∈ℕ} be arbitrary sequences of positive numbers convergent to zero. Moreover, let u_{0} ∈ H_{s}^{1}(Ω) and (u_{n})_{n∈ℕ} be a sequence in H^{1}(Ω) such that

|u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Then

|u_{n}π_{ϵ_{n}}t_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Proof. We may assume that t_{n} ≤ 1 for all n ∈ ℕ. For every t ∈ ]0, 1] we have

u_{n}π_{ϵ_{n}}t − u_{0} = e^{−tA_{ϵn}} u_{n} + ∫_{0}^{t} e^{−(t−s)A_{ϵn}} (f_{ϵ_{n}}(u_{n}π_{ϵ_{n}}s) − f_{ϵ_{n}}(u_{0})) ds
+ ∫_{0}^{t} e^{−(t−s)A_{ϵn}} f_{ϵ_{n}}(u_{0}) ds − u_{0}.

Therefore the estimate (3) implies that

|u_{n}π_{ϵ_{n}}t − u_{0}|_{ϵ_{n}} ≤ |e^{−tA_{ϵn}} u_{n}|_{ϵ_{n}} + ∫_{0}^{t} C_{1}L((t − s)^{−1/2} + 1)|u_{n}π_{ϵ_{n}}s − u_{0}|_{ϵ_{n}} ds
+ ∫_{0}^{t} C_{1}((t − s)^{−1/2} + 1)|f_{ϵ_{n}}(u_{0})|_{L^{2}} + |u_{0}|_{ϵ_{n}}.

By hypothesis (A2)2 we have that the sequence f_{ϵ_{n}}(u_{0}), n ∈ ℕ, is bounded in L^{2}(Ω). Since the sequence (|u_{n}|_{ϵ_{n}})_{n∈ℕ} is bounded there is a positive real constant C_{2} such that for all n ∈ ℕ and every t ∈ ]0, 1]

|u_{n}π_{ϵ_{n}}t − u_{0}|_{ϵ_{n}} ≤ C_{2} + C_{2} ∫_{0}^{t}(t − s)^{−1/2}|u_{n}π_{ϵ_{n}}s − u_{0}|_{ϵ_{n}} ds.

Thus an application of Henry’s Inequality ([24], Lemma 7.1.1) implies that

|u_{n}π_{ϵ_{n}}t − u_{0}|_{ϵ_{n}} ≤ C_{2} + C_{2} ∫_{0}^{t} ρ(t − s) ds,

where

ρ(x) := ∑_{n=1}^{∞} \frac{(C_{2}Γ(β))^{n}}{Γ(nβ)} x^{nβ−1}

with β := 1/2. The function ρ: ]0, ∞[ → ]0, ∞[ is well defined and continuous on ]0, ∞[ and it satisfies the estimate

ρ(x) ≤ C_{2}x^{−1/2} + C_{3} for x ∈ ]0, 1],

where C_{3} is a constant. It follows that there is a constant M such that for all n ∈ ℕ and every t ∈ ]0, 1]

|u_{n}π_{ϵ_{n}}t − u_{0}|_{ϵ_{n}} ≤ M.

It follows that for all n ∈ ℕ and every s with 0 < s ≤ 1

|f_{ϵ_{n}}(u_{n}π_{ϵ_{n}}s)|_{L^{2}} ≤ |f_{ϵ_{n}}(u_{n}π_{ϵ_{n}}s) − f_{ϵ_{n}}(u_{0})|_{L^{2}} + |f_{ϵ_{n}}(u_{0})|_{L^{2}} ≤ LM + |f_{ϵ_{n}}(u_{0})|_{L^{2}} ≤ C_{4}

for some constant C_{4}. Since

u_{n}π_{ϵ_{n}}t_{n} − u_{0} = e^{−t_{n}A_{ϵn}} u_{n} − u_{0} + ∫_{0}^{t_{n}} e^{−(t_{n}−s)A_{ϵn}} f_{ϵ_{n}}(u_{n}π_{ϵ_{n}}s) ds

we conclude that

|u_{n}π_{ϵ_{n}}t_{n} − u_{0}|_{ϵ_{n}} ≤ |e^{−t_{n}A_{ϵn}} u_{n} − u_{0}|_{ϵ_{n}} + C_{5} ∫_{0}^{t_{n}}(t_{n} − s)^{−1/2} ds

for some constant C_{5}. Now an application of Theorem 2.9 completes the proof of the theorem. □

Remark 2.12. Let (ϵ_{n})_{n∈ℕ} and (t_{n})_{n∈ℕ} be arbitrary sequences of positive numbers convergent to zero. Let u_{0} ∈ L_{s}^{2}(Ω) and (u_{n})_{n∈ℕ} be a sequence in L^{2}(Ω) converging to u_{0} in L^{2}(Ω). Then

|u_{n}π_{ϵ_{n}}t_{n} − u_{0}|_{L^{2}(Ω)} → 0 as n → ∞.

The proof of this assertion is analogous to (and easier than) the proof of Theorem 2.11.

We can also state the following generalization of Theorem 5.1 in [33]: Theorem 2.13. Let (ϵ_{n})_{n∈ℕ} be an arbitrary sequence of positive numbers convergent to zero. Moreover let t ∈ ]0, ∞[ and (t_{n})_{n∈ℕ} be a sequence in ]0, ∞[ converging to t. Finally, let u_{0} ∈ H_{s}^{1}(Ω) and (u_{n})_{n∈ℕ} be a sequence in H^{1}(Ω) such that

|u_{n} − u_{0}|_{L^{2}} → 0 as n → ∞.

Then

|u_{n}π_{ϵ_{n}}t_{n} − u_{0}π_{0}t|_{ϵ_{n}} → 0 as n → ∞.

Proof. We modify the initial part of proof of Theorem 5.1 in [33] and therefore use the notation of that proof: write | · |_{n} := | · |_{ϵ_{n}}, A_{n} := A_{ϵ_{n}}, A := A_{0}, π_{n} := π_{ϵ_{n}}, π := π_{0} and u := u_{0}. Let b ∈ [0, ∞[ be arbitrary with t_{n} ∈ ]0, b], n ∈ ℕ, and t ∈ ]0, b]. For every t ∈ [0, b] we have, by the variation-of-constants formula,

u_{n}π_{n}t − uπt = e^{−A_{n}t}u_{n} − e^{−At}u+

∫_{0}^{t} e^{−A_{n}(t−s)}(f_{ϵ_{n}}(u_{n}π_{n}s) − f_{ϵ_{n}}(uπs)) ds + ∫_{0}^{t}(e^{−A_{n}(t−s)}f_{ϵ_{n}}(uπs) − e^{−A(t−s)}f_{0}(uπs)) ds.

Define the function g_{n}: [0, b] × [0, b] → ℝ as follows: If 0 < s < t then set

g_{n}(t, s) = |e^{−A_{n}(t−s)}f_{ϵ_{n}}(uπs) − e^{−A(t−s)}f_{0}(uπs)|_{n}

and set g_{n}(t, s) = 0 otherwise. The function g_{n} restricted to the set of (s, t) with 0 < s < t is continuous. Thus g_{n} is measurable on [0, b] × [0, b]. By Fubini’s theorem the function

c_{n}(t) := ∫_{0}^{b} g_{n}(t, s) ds = ∫_{0}^{t} g_{n}(t, s) ds

is a.e. defined and measurable on [0, b]. Set

a_{n}(t) := |e^{−A_{n}t}u_{n} − e^{−At}u|_{n} + c_{n}(t) for t ∈ ]0, b]

and a_{n}(0) := 0. It follows that a_{n} is measurable on [0, b]. Using the estimates (3) and (4) we obtain

|g_{n}(t, s)| ≤ 2C_{2}(C_{1}(t − s)^{−1/2} + 1) for 0 < s < t

where

C_{2} := max{ sup_{s∈[0,b]} sup_{n∈ℕ}|f_{ϵ_{n}}(uπs)|_{L^{2}}, sup_{s∈[0,b]} |f_{0}(uπs)|_{L^{2}}}.

Note that, by hypothesis (A2)

|f_{ϵ_{n}}(uπs)|_{L^{2}} ≤ |f_{ϵ_{n}}(uπs) − f_{ϵ_{n}}(u)|_{L^{2}} + |f_{ϵ_{n}}(u)|_{L^{2}} ≤ L|uπs − u|_{n} + |f_{ϵ_{n}}(u)|_{L^{2}} ≤ M

for some constant M < ∞, independent of n ∈ ℕ and s ∈ [0, b]. Similarly, we may assume that

|f_{0}(uπs)|_{L^{2}} ≤ M, s ∈ [0, b].

This shows that C_{2} < ∞.

The remaining part of the proof is now almost identical to the corresponding part of the proof of Theorem 5.1 in [33] and so we omit it here. □

We thus obtain the following corollary: Corollary 2.14. Let (ϵ_{n})_{n∈ℕ} be an arbitrary sequence of positive numbers convergent to zero. Moreover let t ∈ [0, ∞[ and (t_{n})_{n∈ℕ} be a sequence in [0, ∞[ converging to t. Finally, let u_{0} ∈ H_{s}^{1}(Ω) and (u_{n})_{n∈ℕ} be a sequence in H^{1}(Ω) such that

|u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Then

|u_{n}π_{ϵ_{n}}t_{n} − u_{0}π_{0}t|_{ϵ_{n}} → 0 as n → ∞.

Proof. We may assume, of course, that all numbers t_{n} are positive. If t = 0, then the assertion follows from Theorem 2.11. If t > 0, the assertion follows from Theorem 2.13. □

We also need the following Proposition 2.15. Let (ϵ_{n})_{n∈ℕ} be an arbitrary sequence of positive numbers convergent to zero. Moreover, let (t_{n})_{n∈ℕ} be a sequence in [0, ∞[ converging to ∞. For every n ∈ ℕ let σ_{n}: [−t_{n}, 0] → H^{1}(Ω) be a solution of π_{ϵ_{n}}. Assume that there is a C ∈ [0, ∞[ such that

|σ_{n}(t)|_{ϵ_{n}} ≤ C for all n ∈ ℕ and t ∈ [−t_{n}, 0].

Then there exists a subsequence of (σ_{n})_{n∈ℕ}, denoted by the same symbol (σ_{n})_{n∈ℕ}, and there is a full left solution σ: ]−∞, 0] → H_{s}^{1}(Ω) of π_{0}, such that for every t ∈ ]−∞, 0],

|σ_{n}(t) − σ(t)|_{ϵ_{n}} → 0 as n → ∞.

Remark 2.16. Note that for every t ∈ ]−∞, 0], there is some n_{0} with −t_{n} ≤ t for all n ≥ n_{0}, so the sequence (σ_{n}(t)) is well defined for n ≥ n_{0}.

Proof. The proof of this result is completely analogous to that of Corollary 5.2 in [33]. We therefore omit the details. □

Note that, for every fixed ϵ > 0, the space H^{1}(Ω) is a Hilbert space under the scalar product

(u, v)_{ϵ} := a_{ϵ}(u, v) + ⟨u, v⟩, u, v ∈ H^{1}(Ω).

Since H_{s}^{1}(Ω) is a closed subspace of H^{1}(Ω), there is a map

Q_{ϵ}: H^{1}(Ω) → H^{1}(Ω)

which is the orthogonal projector onto H_{s}^{1}(Ω) with respect to the scalar product (·, ·)_{ϵ}. This projector will play a crucial role in the sequel. Note that, for every u ∈ H^{1}(Ω),

|u|_{ϵ}^{2} = |Q_{ϵ}u|_{ϵ}^{2} + |(I − Q_{ϵ})u|_{ϵ}^{2},

where, as usual, I is the identity map on H^{1}(Ω).

Remark 2.17. If Ω is a product domain, i.e. if Ω = U × V with U ⊂ ℝ^{M} and V ⊂ ℝ^{N}, and if κ denotes the measure of V then, as it is easily proved, the projector Q_{ϵ} has the explicit form

(Q_{ϵ}u)(x) = \frac{1}{κ} ∫_{V} u(x, y) dy, for u ∈ H^{1}(Ω) and x ∈ U

i.e. Q_{ϵ} is the mean-value operator with respect to the variable y ∈ V. No such explicit form of Q_{ϵ} is known for general domains Ω. Definition 2.18. Given a subset V of H_{s}^{1}(Ω), η > 0 and ϵ > 0, define the ‘inflated’ subsets ]V [_{ϵ,η} and [V]_{ϵ,η} of H^{1}(Ω) as follows:

]V [_{ϵ,η} := { u ∈ H^{1}(Ω) | Q_{ϵ}u ∈ V and |(I − Q_{ϵ})u|_{ϵ} < η },
[V]_{ϵ,η} := { u ∈ H^{1}(Ω) | Q_{ϵ}u ∈ V and |(I − Q_{ϵ})u|_{ϵ} ≤ η }.

Lemma 2.19. Let V be a subset of H_{s}^{1}(Ω). Then for every ϵ > 0 and η > 0

cl_{ϵ}]V [_{ϵ,η} = [cl V]_{ϵ,η} .

Proof. Since [cl V]_{ϵ,η} is closed in H^{1}(Ω), we obtain that cl_{ϵ}]V [_{ϵ,η} ⊂ [cl V]_{ϵ,η}. Conversely, let u ∈ [cl V]_{ϵ,η}. This implies that Q_{ϵ}u ∈ cl V and |(I − Q_{ϵ})u|_{ϵ} ≤ η. Therefore there exists a sequence (w_{n})_{n∈ℕ}, w_{n} ∈ V, such that

|w_{n} − Q_{ϵ}u|_{H^{1}}→ 0 as n → ∞.

Define

z_{n} := (I − Q_{ϵ})\frac{n}{n + 1}u and v_{n} := w_{n} + z_{n}.

It is clear that v_{n} ∈ ]V [_{ϵ,η} and |v_{n} − u|_{ϵ} → 0 for n → ∞. The lemma is proved. □

The following result is an immediate consequence of Proposition 2.15 and the proof of Corollary 5.2 in [33]: Lemma 2.20. Let S be a closed bounded set in H_{s}^{1}(Ω), η > 0 and (ϵ_{n})_{n∈ℕ} be a sequence of positive numbers converging to zero. For every n ∈ ℕ let σ_{n}: J_{n} → [S]_{ϵ_{n},η} be a solution of π_{ϵ_{n}}. If J_{n} ≡ ℝ for every n ∈ ℕ (resp. if J_{n} = [−t_{n}, 0], n ∈ ℕ, where t_{n} → ∞), then there is a subsequence of (σ_{n})_{n∈ℕ}, again denoted by (σ_{n})_{n∈ℕ} and there is a solution σ: ℝ → S (resp. σ: ]−∞, 0] → S) of π_{0} such that for every t ∈ ℝ (resp. every t ∈ ]−∞, 0])

|σ_{n}(t) − σ(t)|_{ϵ_{n}} → 0 as n → ∞

so

|Q_{ϵ_{n}}σ_{n}(t) − σ(t)|_{H^{1}} → 0 and |(I − Q_{ϵ_{n}})σ_{n}(t)|_{ϵ_{n}} → 0 as n → ∞.
□

3. A Conley index continuation theorem

In this section, we will state our basic Conley index continuation theorem for thin domain problems. The proof of this theorem will be given in the Appendix.

We will first review some basic definitions and results concerning the Conley index theory for semiflows defined on a metric space. The reader is referred to [37] or [39] for the proofs of several of the results stated below and for further details on the subject.

Given a topological space Z and an arbitrary set Y, we define the quotient space Z/Y as follows: fix an arbitrary p ∉ Z. Define the set Z/Y as

Z/Y := (Z \ Y) ∪ {p}

and the map q: Z → Z/Y as

q(z) = \begin{cases}z & if z ∈ Z \ Y, \\ p & otherwise. \end{cases}

We commonly write [z] instead of q(z). We also write [Y] instead of p.

Call a subset V of Z/Y open in Z/Y if and only if q^{−1}(V) is open in Z. This defines a (quotient) topology on Z/Y. Note that if Y ∩ Z ≠ ∅, then q is a (surjective) identification map. If Y ∩ Z = ∅, then Z/Y = Z ∪ {p} and V ⊂ Z/Y is open in Z/Y if and only if V ∩ Z is open in Z.

For the rest of this section, unless specified otherwise, let X be a metric space and let π be a local semiflow on X.

Suppose that Y is a subset of X. We define the following subsets of X:

A_{π}^{+}(Y) :={ u ∈ X | uπ [0, ω_{u}[ ⊂ Y },
A_{π}^{−}(Y) :={ u ∈ X | there exists a solution σ: ]−∞, 0] → X through u with
σ(]−∞, 0]) ⊂ Y },
A_{π}(Y) :=A_{π}^{+}(Y) ∩ A_{π}^{−}(Y).

A subset Y of X is called invariant (respectively positively invariant, respectively negatively invariant) relative to π if Y = A_{π}(Y) (respectively Y = A_{π}^{+}(Y), respectively Y = A_{π}^{−}(Y)).

Let N be a closed subset of X such that K := A_{π}(N) is closed and K ⊂ Int N. Then N is called an isolating neighborhood of K relative to π and K is called an isolated invariant set relative to π.

Let B ⊂ X be a closed set and u ∈ ∂B. The point u is called a strict egress (respectively strict ingress, respectively bounce-off) of B, if for every solution σ: [−δ_{1}, δ_{2}] → X through u, with δ_{1} ≥ 0 and δ_{2} > 0, the following properties hold:

(1) There exists an ϵ_{2} ∈ ]0, δ_{2}[ such that σ(t) ∉ B (respectively σ(t) ∈ Int B, respectively σ(t) ∉ B), for t ∈ ]0, ϵ_{2}].

(2) If δ_{1} > 0, then there exists an ϵ_{1} ∈ ]0, δ_{1}[ such that σ(t) ∈ Int B (respectively σ(t) ∉ B, respectively σ(t) ∉ B), for t ∈ [−ϵ_{1}, 0[.

The set of all strict egress points (respectively strict ingress, respectively bounce-off) of the closed set B will be denoted by B^{e} (respectively B^{i}, respectively B^{b}). A closed set B ⊂ X is called an isolating block, if ∂B = B^{e} ∪ B^{i} ∪ B^{b} and B^{−} := B^{e} ∪ B^{b} is closed.

Let N be a closed subset of X. Then N is called strongly π-admissible if the following properties are satisfied:

(1) Whenever uπt ∈ N for all t ∈ [0, ω_{u}[ then ω_{u} = ∞.

(2) Whenever (u_{n})_{n∈ℕ} is a sequence in X and (t_{n})_{n∈ℕ} is a sequence in [0, ∞[ such that t_{n} → ∞ as n → ∞ and u_{n}π [0, t_{n}] ⊂ N for all n ∈ ℕ, then the sequence (u_{n}πt_{n})_{n∈ℕ} of endpoints has a convergent subsequence.

Theorem 3.1. Let K be an isolated π-invariant set and N be a strongly π-admissible isolating neighborhood of K. Then there exists an open set V such that B := cl V is an isolating block such that K ⊂ V ⊂ B ⊂ N and ∂V = ∂B.

Proof. By Theorem 5.1 in Chapter 1 of [39] there exists an isolating block B′ such that K ⊂ Int B′ ⊂ B′ ⊂ N. Set V := Int B′ and B := cl V. It is easily verified that B := cl V is an isolating block such that K ⊂ V ⊂ B ⊂ N and ∂V = ∂B. □

Let N, Y be subsets of X such that Y ⊂ N. The set Y is called N-positively invariant relative to π, if whenever u ∈ X, t ≥ 0 are such that uπ [0, t] ⊂ N and u ∈ Y, then uπ [0, t] ⊂ Y. Definition 3.2. Let N be a closed set in X and N_{1} and N_{2} be closed subsets of N. The pair (N_{1}, N_{2}) is called a pseudo-index pair in N if:

(1) N_{1} and N_{2} are N-positively invariant;

(2) whenever u ∈ N_{1} and uπt_{0} ∉ N for some t_{0} ∈ [0, ∞[, then there exists a t′ ∈ [0, t_{0}] such that uπ [0, t′] ⊂ N and uπt′ ∈ N_{2}.

A pseudo index pair (N_{1}, N_{2}) in N is called an index pair in N if A_{π}(N) is closed and A_{π}(N) ⊂ Int(N_{1} \ N_{2}).

An isolating block B determines a special index pair in B, namely (B, B^{−}).

Let K be an isolated invariant set, N be a strongly π-admissible isolating neighborhood of K and (N_{1}, N_{2}) be an index pair in N. Then the homotopy type of the pointed space (N_{1}/N_{2}, [N_{2}]) depends only on the semiflow π and the isolated invariant set K (see Theorem 10.1 in [39]). The Conley index, h(π, K), of the isolated invariant set K with respect to π is defined to be the homotopy type of (N_{1}/N_{2}, [N_{2}]). Remark 3.3. If N is a strongly π-admissible isolating neighborhood relative to π, we will sometimes write h(π, N) to denote h(π, A_{π}(N)). This will not lead to confusion.

The following result is obvious: Proposition 3.4. Let π and π′ be local semiflows on the metric space X. Let Y be a subset of X on which π and π′ coincide. Suppose N is closed in X and N ⊂ Y. Then A_{π}(N) = A_{π′}(N). Moreover, N is a strongly admissible isolating neighborhood relative to π if and only if N is a strongly admissible isolating neighborhood relative to π′ and in this case

h(π, N) = h(π′, N).

We can now state the main result of this section: Theorem 3.5. Let ϵ_{0} > 0 and (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} be a family of maps satisfying hypothesis (A1). Let N be a bounded isolating neighborhood for π_{0,f_{0}}. Then for every η > 0 there exists an ϵ^{c} = ϵ^{c}(η) > 0 such that for every ϵ ∈ ]0, ϵ^{c}] the set [N]_{ϵ,η} is a strongly admissible isolating neighborhood relative to π_{ϵ,f_{ϵ}} and

(5)  h(π_{ϵ,f_{ϵ}}, [N]_{ϵ,η}) = h(π_{0,f_{0}}, N).

Theorem 3.5 can also be reworded in terms of the semiflows π̃_{ϵ,f̃_{ϵ}} generated by the original reaction-diffusion equations on the squeezed domains Ω_{ϵ}: Theorem 3.6. Let ϵ_{0} > 0 and (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} be a family of maps satisfying hypothesis (A1). For ϵ > 0 define f̃_{ϵ}: H^{1}(Ω_{ϵ}) → L^{2}(Ω_{ϵ}) by f̃_{ϵ} := Φ_{ϵ}^{−1} ∘f_{ϵ} ∘Φ_{ϵ}. Let N be a bounded isolating neighborhood for π_{0,f_{0}}. Then for every η > 0 there exists an ϵ^{c} = ϵ^{c}(η) > 0 such that for every ϵ ∈ ]0, ϵ^{c}] the set Φ_{ϵ}^{−1}([N]_{ϵ,η}) is a strongly admissible isolating neighborhood relative to π̃_{ϵ,f̃_{ϵ}} and

h(π̃_{ϵ,f̃_{ϵ}}, Φ_{ϵ}^{−1}([N]_{ϵ,η})) = h(π_{0,f_{0}}, N).

Proof. The isomorphism Φ_{ϵ} conjugates the local semiflow π̃_{ϵ,f̃_{ϵ}} to the local semiflow π_{ϵ,f_{ϵ}}. Thus whenever S is a strongly admissible isolating neighborhood with respect to π_{ϵ,f_{ϵ}}, then Φ_{ϵ}^{−1}(S) is a strongly admissible isolating neighborhood with respect to π̃_{ϵ,f̃_{ϵ}} and

h(π_{ϵ,f_{ϵ}}, S) = h(π̃_{ϵ,f̃_{ϵ}}, Φ_{ϵ}^{−1}(S)).

Now Theorem 3.5 completes the proof. □

Theorem 3.5 yields the following corollary, which was stated, somewhat less precisely, in the Introduction: Corollary 3.7. For every η > 0 the family (K_{ϵ,η})_{ϵ∈[0,ϵ^{c}(η)]} of invariant sets, where K_{ϵ,η} := A_{π_{ϵ}}([N]_{ϵ,η}), ϵ ∈ ]0, ϵ^{c}(η)], and K_{0,η} = K_{0} := A_{π_{0}}(N), is upper semicontinuous at ϵ = 0 with respect to the family | · |_{ϵ} of norms. In other words,

lim_{ϵ→0^{+}} sup_{u∈K_{ϵ,η}} inf_{v∈K_{0}} |u − v|_{ϵ} = 0.

Proof. If the corollary is not true, then there are numbers η and β > 0, a sequence ϵ_{n} → 0^{+} and a sequence u_{n} ∈ K_{ϵ_{n},η}, n ∈ ℕ, such that

inf_{v∈K_{0}} |u_{n} − v|_{ϵ} > β.

For every n ∈ ℕ let σ_{n}: ℝ → [N]_{ϵ_{n},η} be a solution of π_{ϵ_{n}} with σ_{n}(0) = u_{n}. Using Lemma 2.20 we may assume that whenever t ∈ ℝ, then |σ_{n}(t) − σ(t)|_{ϵ_{n}} → 0 as n → ∞, where σ: ℝ → N is a solution of π_{0}. In particular, |u_{n} − v|_{ϵ_{n}} → 0 as n → ∞ where v := σ(0) ∈ K_{0}. This is a contradiction, which proves the corollary. □

Theorem 3.5 will be proved in the Appendix. In the next section we discuss a few applications of this result. More applications will be given in a forthcoming publication.

4. Applications

4.1. Connection lifting. Until further notice assume that ϵ_{0} > 0 and that (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} is a family of maps satisfying hypothesis (A1). Write π_{ϵ} := π_{ϵ,f_{ϵ}} for ϵ ∈ [0, ϵ_{0}]. Using Theorem 3.5 we will show in this subsection that, under certain hypotheses, heteroclinic orbits of the limit semiflow π_{0} can be ‘lifted’ to the semiflows π_{ϵ} and thus, by conjugacy, to the corresponding semiflows π̃_{ϵ} on the squeezed domains Ω_{ϵ}, for ϵ > 0 small. We first treat an abstract problem and then apply the results obtained to an equation considered by Fiedler and Rocha.

Proposition 4.1. Suppose that the local semiflow π_{0} is gradient-like with respect to a Liapunov function V_{0} : H_{s}^{1}(Ω) → ℝ. Assume also that N, N′ and N′′ are bounded isolating neighborhoods with respect to π_{0} with N′ ∩ N′′ = ∅, N′ ∪ N′′ ⊂ N and there are points u′ and u′′ such that V_{0}(u′) > V_{0}(u′′), {u′} = A_{π_{0}}(N′) and {u′′} = A_{π_{0}}(N′′). Finally, assume that

(6)  h(π_{0}, N) ≠ h(π_{0}, N′) ∨ h(π_{0}, N′′)

and there are no equilibria of π_{0} lying in N \(N′∪N′′). Then there is a solution σ_{0} : ℝ → N of π_{0} with {u′} = α(σ_{0}) and {u′′} = ω(σ_{0}). Remark 4.2. Recall that, given a metric space X and an arbitrary map σ: ℝ → X, the α-limit set α(σ) (resp. the ω-limit set ω(σ)) of σ is defined as the set of all points x ∈ X for which there is a sequence (t_{n})_{n∈ℕ} in ℝ with t_{n} → −∞ (resp. t_{n} → ∞) as n → ∞ such that σ(t_{n}) → x as n → ∞.

Proof. From (6) we see that there is a solution σ_{0} : ℝ → N of π_{0} with σ_{0}(ℝ) ⊄ N′ ∪ N′′. Since π_{0} is gradient-like, it follows that α(σ_{0}) and ω(σ_{0}) are nonempty sets of equilibria of π_{0}. Our assumptions imply that the only equilibria of π_{0} lying in N are u′ and u′′. In particular, σ_{0} is a nonconstant solution, so V_{0} is strictly decreasing along σ_{0}. This concludes the proof. □

Using Theorem 3.5 we can, in some sense, ‘lift’ the connection σ_{0} to the semiflows π_{ϵ}, for ϵ > 0 small: Theorem 4.3. Assume the hypotheses of Proposition 4.1. In addition, suppose that for some ϵ_{0} > 0 and every ϵ ∈ ]0, ϵ_{0}] the local semiflow π_{ϵ} is gradient-like with respect to a Liapunov function V_{ϵ}: H^{1}(Ω) → ℝ. Assume that whenever ϵ_{n} → 0^{+}, (u_{n})_{n∈ℕ} is a sequence in H^{1}(Ω) and |u_{n} − u_{0}|_{ϵ_{n}} → as n → ∞, where u_{0} ∈ H_{s}^{1}(Ω), then V_{ϵ_{n}}(u_{n}) → V_{0}(u_{0}) as n → ∞. Then for every η > 0 there is an ϵ^{η} > 0 such that for every ϵ ∈ ]0, ϵ^{η}] there exists a solution σ_{ϵ}: ℝ → [N]_{ϵ,η} of π_{ϵ} such that α(σ_{ϵ}) ⊂ [N′]_{ϵ,η} and ω(σ_{ϵ}) ⊂ [N′′]_{ϵ,η}.

Proof. Choose numbers γ_{1} and γ_{2} such that

V_{0}(u′′) < γ_{1} < γ_{2} < V_{0}(u′).

Given η > 0, Theorem 3.5 implies that there is an ϵ^{η} > 0 such that for ϵ ∈ ]0, ϵ^{η}]

\begin{aligned}h(π_{ϵ}, [N]_{ϵ,η}) & = h(π_{0}, N) \\ h(π_{ϵ}, [N′]_{ϵ,η}) & = h(π_{0}, N′) \\ h(π_{ϵ}, [N′′]_{ϵ,η}) & = h(π_{0}, N′′) \end{aligned} ; (7)

and so there exists a solution σ_{ϵ}: ℝ → [N]_{ϵ,η} of π_{ϵ} such that σ_{ϵ}(0) ∈ [N \ (N′ ∪ N′′)]_{ϵ,η}.

An application of Proposition 2.15 shows that we may also assume that for ϵ ∈ ]0, ϵ^{η}] there are no equilibria of π_{ϵ} lying in [N \ (N′ ∪ N′′)]_{ϵ,η}. This implies that

α(σ_{ϵ}) ∪ ω(σ_{ϵ}) ⊂ [N′]_{ϵ,η} ∪ [N′′]_{ϵ,η}

and so

α(σ_{ϵ}) ∪ ω(σ_{ϵ}) ⊂ A_{π_{ϵ}}([N′]_{ϵ,η} ∪ [N′′]_{ϵ,η}) = A_{π_{ϵ}}([N′]_{ϵ,η}) ∪ A_{π_{ϵ}}([N′′]_{ϵ,η})

for ϵ ∈ ]0, ϵ^{η}]. The latter equality follows since N′ and N′′ are disjoint.

Using Proposition 2.15 again we may assume that

V_{ϵ}(u′′_{ϵ}) < γ_{1} < γ_{2} < V_{ϵ}(u′_{ϵ})

whenever ϵ ∈ ]0, ϵ^{η}], u′_{ϵ} ∈ A_{π_{ϵ}}([N′]_{ϵ,η}) and u′′_{ϵ} ∈ A_{π_{ϵ}}([N′′]_{ϵ,η}). By the same token, we may assume that for every ϵ ∈ ]0, ϵ^{η}] there is a t_{ϵ} such that γ_{1} < V_{ϵ}(σ_{ϵ}(t_{ϵ})) < γ_{2}. We also have

V_{ϵ}(u′′_{ϵ}) < V_{ϵ}(σ_{ϵ}(t_{ϵ})) < V_{ϵ}(u′_{ϵ})

whenever ϵ ∈ ]0, ϵ^{η}], u′_{ϵ} ∈ α(σ_{ϵ}) and u′′_{ϵ} ∈ ω(σ_{ϵ}). Thus whenever ϵ ∈ ]0, ϵ^{η}] and u′_{ϵ} ∈ α(σ_{ϵ}) then for every u′′_{ϵ} ∈ A_{π_{ϵ}}([N′′]_{ϵ,η}) we obtain that

V_{ϵ}(u′′_{ϵ}) < γ_{1} < V_{ϵ}(σ_{ϵ}(t_{ϵ})) < V_{ϵ}(u′_{ϵ})

so u′_{ϵ} ∉ A_{π_{ϵ}}([N′′]_{ϵ,η}). Hence α(σ_{ϵ}) ⊂ A_{π_{ϵ}}([N′]_{ϵ,η}). Similarly we prove that ω(σ_{ϵ}) ⊂ A_{π_{ϵ}}([N′′]_{ϵ,η}). □

Assuming hyperbolicity, we can refine the results of Theorem 4.3. To this end, we need the following result: Proposition 4.4. Assume that for every ϵ ∈ [0, ϵ_{0}] the map f_{ϵ} is Fréchet differentiable. Moreover, suppose that, whenever u_{0}, v ∈ H_{s}^{1}(Ω), ϵ_{n} → 0^{+} as n → ∞ and (u_{n})_{n∈ℕ} is a sequence in H^{1}(Ω) with |u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞, then

|Df_{ϵ_{n}}(u_{n})v − Df_{0}(u_{0})v|_{L^{2}} → 0 as n → ∞.

Let u_{0} be a hyperbolic equilibrium of π_{0} and N be a bounded isolating neighborhood of {u_{0}}. Then for every η > 0 there is an ϵ_{1} = ϵ_{1}(η), 0 < ϵ_{1} ≤ ϵ_{0}, such that for every ϵ ∈ ]0, ϵ_{1}] there is only a finite number of equilibria of π_{ϵ} in the set [N]_{ϵ,η} and all of them are hyperbolic.

Proof. In view of Proposition 2.8 it is not restrictive to assume that the family (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfies hypothesis (A2). Let η > 0 be arbitrary. We claim that for ϵ > 0 small, every equilibrium of π_{ϵ} in the set [N]_{ϵ,η} is hyperbolic. If the claim is not true, then we obtain a strictly decreasing sequence ϵ_{n} → 0^{+} and a sequence u_{n} ∈ [N]_{ϵ_{n},η}, n ∈ ℕ, such that u_{n} is a nonhyperbolic equilibrium of π_{ϵ_{n}}. It follows that for every n ∈ ℕ there is a v_{n} ∈ D(A_{ϵ_{n}}) with |v_{n}|_{ϵ_{n}} = 1 and

A_{ϵ_{n}}v_{n} = Df_{ϵ_{n}}(u_{n})v_{n}.

For ϵ ∈ [0, ϵ_{0}] define F_{ϵ} := Df_{ϵ}(u_{n}) if there is an n ∈ ℕ with ϵ = ϵ_{n} and set F_{ϵ} := Df_{ϵ}(u_{0}) otherwise. Since |u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞ it follows that the family (F_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfies hypothesis (A2). Since σ_{n}(t) ≡ v_{n} is a solution of π_{ϵ_{n},F_{ϵn}} for every n ∈ ℕ, we may use Proposition 2.15 and assume, taking a subsequence if necessary, that |v_{n} − v_{0}|_{ϵ_{n}} → 0 as n → ∞, where v_{0} is an equilibrium of π_{0,F_{0}}. It follows that |v_{0}|_{H^{1}} = 1 and

A_{0}v_{0} = Df_{0}(u_{0})v_{0}.

Thus u_{0} is not hyperbolic, a contradiction, proving our claim. Since for ϵ > 0 small, every equilibrium of π_{ϵ} in the set [N]_{ϵ,η} is hyperbolic and since the largest π_{ϵ}-invariant set in [N]_{ϵ,η} is compact, it follows that there can be only a finite number of equilibria in [N]_{ϵ,η}. This proves the proposition. □

Combining Theorem 4.3 and Proposition 4.4 we arrive at the following Corollary 4.5. Assume the hypotheses of Theorem 4.3. In addition, suppose that for every ϵ ∈ [0, ϵ_{0}] the map f_{ϵ} is Fréchet differentiable and that

|Df_{ϵ_{n}}(u_{n})v − Df_{0}(u_{0})v|_{L^{2}} → 0 as n → ∞

whenever u_{0}, v ∈ H_{s}^{1}(Ω), ϵ_{n} → 0^{+} as n → ∞ and (u_{n})_{n∈ℕ} is a sequence in H^{1}(Ω) with |u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞. Finally, assume that the equilibria u′ and u′′ are hyperbolic.

Under these assumptions, for every η > 0 there is an ϵ^{η} > 0 such that for every ϵ ∈ ]0, ϵ^{η}] there exists a solution σ_{ϵ}: ℝ → [N]_{ϵ,η} and hyperbolic equilibria u′_{ϵ} ∈ [N′]_{ϵ,η} and u′′_{ϵ} ∈ [N′′]_{ϵ,η} of π_{ϵ} such that {u′_{ϵ}} = α(σ_{ϵ}) and {u′′_{ϵ}} = ω(σ_{ϵ}). □

For the rest of this subsection suppose that M = 1 so Ω ⊂ ℝ×ℝ^{N} and let P: ℝ×ℝ^{N} → ℝ be the projection onto the first coordinate. Define J := P (Ω). Suppose also that Ω has connected vertical sections, i.e. that for every x ∈ J the x-section

Ω_{x} := { y ∈ ℝ^{N} | (x, y) ∈ Ω }

is connected.

For every x ∈ J let p(x) > 0 be the N-dimensional Lebesgue measure of the vertical section Ω_{x}. Define the space

H_{p} := { v | v: J → ℝ and p^{1/2}v ∈ L^{2}(J)}.

This is a Hilbert space under the scalar product (u, v)_{H_{p}} = ∫_{J} puv dx.

Furthermore, define the space

V_{p} := { v ∈ H_{p} | v′ ∈ L_{loc}^{1}(J) and p^{1/2}v′ ∈ L^{2}(J) }.

This is a Hilbert space under the scalar product (u, v)_{V_{p}} = ∫_{J} puv dx+∫_{J} pu′v′ dx. If p and 1/p are bounded on J, then H_{p} is isomorphic to L^{2}(J) and V_{p} is isomorphic to H^{1}(J). An obvious modification of the proof of Proposition 3.6 in [34] shows that a function u: Ω → ℝ lies in L_{s}^{2}(Ω) if and only if there is a function ũ: J → ℝ such that u(x, y) = ũ(x) a.e. in Ω and p^{1/2}ũ ∈ L^{2}(J). The assignment u ↦ ũ defines an isomorphism Φ: L_{s}^{2}(Ω) → H_{p}. Moreover, Φ(H_{s}^{1}(Ω)) = V_{p} and Φ|_{H_{s}^{1}(Ω)} is an isomorphism of H_{s}^{1}(Ω) onto V_{p}.

A modification of the proof of Theorem 6.6 in [33] yields a characterization of the operator A_{0}. In fact, u ∈ D(A_{0}) ⊂ H_{s}^{1}(Ω) and A_{0}u = w ∈ L_{s}^{2}(Ω) if and only if, defining ũ := Φ(u) ∈ V_{p} and w̃ := Φ(w) ∈ H_{p}, we have that

(1) The distributional derivative (pũ′)′ of pũ′ is an L_{loc}^{1}(J)-function with (pũ′)′ = −pw̃;

(2) (pũ′)(x) = 0 for x ∈ ∂J.

Assume that p(x) ≥ c > 0 on J and p ∈ C^{3}(cl J). It follows that u ∈ D(A_{0}) and A_{0}u = w if and only if the functions ũ := Φ(u) and w̃ := Φ(w) are such that u ∈ H^{2}(Ω), ũ′(x) = 0 for x ∈ ∂J and w̃ = −ũ′′ − p′ũ′/p a.e. in J.

Let f: ℝ × ℝ × ℝ^{N} × ℝ → ℝ, (ϵ, x, y, u) ↦ f(ϵ, x, y, u) be a function of class C^{2} for which there are constants β, γ and C ∈ [0, ∞[ such that for all (ϵ, x, y, u) ∈ ℝ×ℝ×ℝ^{N} ×ℝ the following estimates are satisfied:

(1) |∂_{ϵ}f(ϵ, x, y, u)| ≤ C(1 + |u|^{β}),

(2) |∇_{y}f(ϵ, x, y, u)| ≤ C(1 + |u|^{β}),

(3) |∂_{u}f(ϵ, x, y, u)| ≤ C(1 + |u|^{γ}),

(4) |∂_{ϵ}∂_{u}f(ϵ, x, y, u)| ≤ C(1 + |u|^{β}),

(5) |∇_{y}∂_{u}f(ϵ, x, y, u)| ≤ C(1 + |u|^{β}),

(6) |∂_{u}∂_{u}f(ϵ, x, y, u)| ≤ C(1 + |u|^{γ}).

If n := M +1 > 2, we also assume that β ≤ 2^{∗}/2 and γ ≤ (2^{∗}/2)−1, where 2^{∗} := 2n/(n−2).

Given ϵ ≥ 0 and a function u: Ω → ℝ define the function f̂_{ϵ}(u): Ω → ℝ by

f̂_{ϵ}(u)(x, y) := f(ϵ, x, ϵy, u(x, y)) for (x, y) ∈ Ω.

Lemma 4.6. The following conditions are satisfied:

(1) If u ∈ H^{1}(Ω), then f̂_{ϵ}(u) ∈ L^{2}(Ω).

(2) The family (f̂_{ϵ})_{ϵ∈[0,1]} of maps satisfies hypothesis (A1).

(3) For every ϵ ∈ [0, 1] the map f̂_{ϵ} is Fréchet differentiable and

|Df̂_{ϵ_{n}}(u_{n})v − Df̂_{0}(u_{0})v|_{L^{2}} → 0 as n → ∞

whenever u_{0}, v ∈ H_{s}^{1}(Ω), ϵ_{n} → 0^{+} as n → ∞ and (u_{n})_{n∈ℕ} is a sequence in H^{1}(Ω) with |u_{n} − u_{0}|_{ϵ_{n}} → 0 as n → ∞.

Proof. Assertions 1 and 2 follows from Proposition 2.6. The proof of that proposition, using the estimates on ∂_{ϵ}∂_{u}f, ∇_{y}∂_{u}f and ∂_{u}∂_{u}f yields assertion 3. □

Define the function F: ℝ × ℝ × ℝ^{N} × ℝ → ℝ by

F (ϵ, x, y, s) := ∫_{0}^{s} f(ϵ, x, y, r) dr for (ϵ, x, y, s) ∈ ℝ × ℝ × ℝ^{N} × ℝ.

An easy calculation shows that for every ϵ > 0 the semiflow π_{ϵ} := π_{ϵ,f̂_{ϵ}} is gradient-like with respect to the Liapunov function V_{ϵ}: H^{1}(Ω) → ℝ defined by

V_{ϵ}(u) := (1/2)a_{ϵ}(u, u) − ∫_{Ω} F (ϵ, x, ϵy, u(x, y)) dx dy, for u ∈ H^{1}(Ω).

Moreover, the semiflow π_{0} := π_{0,f̂_{0}} is gradient-like with respect to the Liapunov function V_{0} : H_{s}^{1}(Ω) → ℝ defined by

V_{0}(u) := (1/2)a_{0}(u, u) − ∫_{Ω} F (0, x, 0, u(x, y)) dx dy, for u ∈ H_{s}^{1}(Ω).

It is easily seen that whenever ϵ_{n} → 0^{+}, (u_{n})_{n∈ℕ} is a sequence in H^{1}(Ω) and |u_{n} −u_{0}|_{ϵ_{n}} → as n → ∞, where u_{0} ∈ H_{s}^{1}(Ω), then V_{ϵ_{n}}(u_{n}) → V_{0}(u_{0}) as n → ∞.

Furthermore, it is clear that the isomorphism Φ|_{H_{s}^{1}(Ω)} conjugates the semiflow π_{0} to the semiflow π̃_{0} generated by the following scalar semilinear parabolic equation with Neumann boundary conditions:

(8)  \begin{matrix}ũ_{t} = ũ_{xx} + p′ũ_{x}/p + f(0, x, 0, ũ), & t > 0, x ∈ J, \\ ũ_{x} = 0, & t > 0, x ∈ ∂J. \end{matrix}

Therefore Proposition 3.2 in Chapter 2 of [39] implies that a set N is a strongly admissible isolating neighborhood with respect to π_{0} if and only if Ñ := Φ(N) is a strongly admissible isolating neighborhood with respect to π̃_{0}, and, in this case, Aπ̃_{0}(Ñ) = Φ(A_{π_{0}}(N)) and h(π_{0}, A_{π_{0}}(N)) = h(π̃_{0}, Aπ̃_{0}(Ñ)).

Notice that the equation (8) satisfies the hypotheses of [14].

Therefore, we obtain the following result: Theorem 4.7. Let p and f satisfy the above hypotheses. Assume ũ′ and ũ′′ are hyperbolic equilibria for the equation (8) such that h(π̃_{0}, {ũ′}) = Σ^{i(ũ′)} and h(π̃_{0}, {ũ′′}) = Σ^{i(ũ′′)} with i(ũ′) = i(ũ′′) + 1. Moreover assume that the connections from ũ′ to ũ′′ are not blocked in the sense of [14].

Under these assumptions there exist bounded isolating neighborhoods N, N′ and N′′ with respect to π_{0} with N′ ∩ N′′ = ∅, N′ ∪ N′′ ⊂ N such that {Φ^{−1}(ũ′)} = A_{π_{0}}(N′) and {Φ^{−1}(ũ′′)} = A_{π_{0}}(N′′). Furthermore for every η > 0 there is an ϵ^{η} > 0 such that for every ϵ ∈ ]0, ϵ^{η}] there exists a solution σ_{ϵ}: ℝ → [N]_{ϵ,η} and hyperbolic equilibria u′_{ϵ} ∈ [N′]_{ϵ,η} and u′′_{ϵ} ∈ [N′′]_{ϵ,η} of π_{ϵ} such that {u′_{ϵ}} = α(σ_{ϵ}) and {u′′_{ϵ}} = ω(σ_{ϵ}).

Proof. Lemma 1.7 in [14] and the remarks preceding the statement of this theorem imply that the hypotheses of Proposition 4.1 hold. Note that the conditions of Theorem 4.3 are also satisfied. Therefore the result follows from Lemma 4.6 and Corollary 4.5. □

4.2. Monotonicity. Now recall that, given ϵ ≥ 0, we denote by (λ_{ϵ,j})_{j∈ℕ} the repeated sequence of eigenvalues of the operator A_{ϵ}.

As it was proved in [33], for every j ∈ ℕ the family (λ_{ϵ,j})_{ϵ>0} is monotone decreasing and converges to λ_{0,j}. Note that, λ_{ϵ,1} ≡ 0. On the other hand, if j ≥ 2, then, as we will prove now, the family (λ_{ϵ,j})_{ϵ>0} is strictly monotone decreasing in many cases. We begin with the following result, which gives a necessary and sufficient condition for strict monotonicity of a given family (λ_{ϵ,j})_{ϵ>0} of eigenvalues: Theorem 4.8. (1) Suppose that there is a j ∈ ℕ and there are numbers 0 < ϵ_{2} < ϵ_{1},

such that λ_{ϵ_{2},j} = λ_{ϵ_{1},j}. Then there is a w ∈ H_{s}^{1}(Ω), w ≠ 0, such that

(9)  ⟨∇w, ∇v⟩ = λ⟨w, v⟩ for all v ∈ H^{1}(Ω),

where λ = λ_{ϵ_{2},j}.

(2) Conversely, if there exist a λ ∈ ℝ and a w ∈ H_{s}^{1}(Ω), w ≠ 0 such that (9) is satisfied, then there is a unique r ∈ ℕ with λ = λ_{0,r} < λ_{0,r+1}. Furthermore, there is an ϵ_{0} > 0 with the property that

λ_{ϵ,r} ≡ λ for all ϵ ∈ [0, ϵ_{0}].

Proof. Assume the hypothesis of the first part of the theorem. Let E be the (j-dimensional) subspace of H^{1}(Ω) spanned by the vectors w_{ϵ_{2},k}, k = 1, …, j. Then, by the min-max principle,

λ_{ϵ_{1},j} ≤ max_{u∈E\\{0\}} \frac{a_{ϵ_{1}}(u, u)}{⟨u, u⟩}.

Hence there is a w ∈ E, ⟨w, w⟩ = 1, such that

λ_{ϵ_{1},j} ≤ a_{ϵ_{1}}(w, w) = a_{ϵ_{2}}(w, w) − (\frac{1}{ϵ_{2}^{2}} − \frac{1}{ϵ_{1}^{2}} ) ⟨∇_{y}w, ∇_{y}w⟩.

Now

a_{ϵ_{2}}(w, w) = ∑_{k=1}^{j} λ_{ϵ_{2},k}|⟨w, w_{ϵ_{2},k}⟩|^{2}.

We also have

λ_{ϵ_{2},j} = λ_{ϵ_{2},j}⟨w, w⟩ = λ_{ϵ_{2},j} ∑_{k=1}^{j} |⟨w, w_{ϵ_{2},k}⟩|^{2}.

Since λ_{ϵ_{2},j} = λ_{ϵ_{1},j} we obtain

λ_{ϵ_{2},j} ∑_{k=1}^{j} |⟨w, w_{ϵ_{2},k}⟩|^{2} ≤ ∑_{k=1}^{j} λ_{ϵ_{2},k}|⟨w, w_{ϵ_{2},k}⟩|^{2} − (\frac{1}{ϵ_{2}^{2}} − \frac{1}{ϵ_{1}^{2}} ) ⟨∇_{y}w, ∇_{y}w⟩,

i.e.

∑_{k=1}^{j}(λ_{ϵ_{2},j} − λ_{ϵ_{2},k})|⟨w, w_{ϵ_{2},k}⟩|^{2} ≤ − (\frac{1}{ϵ_{2}^{2}} − \frac{1}{ϵ_{1}^{2}} ) ⟨∇_{y}w, ∇_{y}w⟩.

Since the left-hand side of this inequality is nonnegative while the right-hand side is nonpositive, both sides must be equal to zero. This implies, on the one hand, that ⟨∇_{y}w, ∇_{y}w⟩ = 0 so w ∈ H_{s}^{1}(Ω) and, on the other hand, that ⟨w, w_{ϵ_{2},k}⟩ = 0, whenever λ_{ϵ_{2},j} − λ_{ϵ_{2},k} > 0. Thus w lies in the eigenspace of the eigenvalue λ (with respect to the pair (a_{ϵ_{2}}, b)). In other words, for all v ∈ H^{1}(Ω),

⟨∇w, ∇v⟩ = ⟨∇_{x}w, ∇_{x}v⟩ = ⟨∇_{x}w, ∇_{x}v⟩ + \frac{1}{ϵ_{2}^{2}} ⟨∇_{y}w, ∇_{y}v⟩ = λ⟨w, v⟩.

This proves the first part of the theorem. Now assume the hypothesis of the second part of the theorem. Since w ∈ H_{s}^{1}(Ω) it follows that for every ϵ > 0

a_{ϵ}(w, v) = ⟨∇w, ∇v⟩ = λ⟨w, v⟩ for all v ∈ H^{1}(Ω).

Thus for every ϵ ≥ 0 the number λ is an eigenvalue of the operator A_{ϵ}. In particular, let r ∈ ℕ be the largest number with λ = λ_{0,r}. Then λ_{0,r} < λ_{0,r+1} so there exists an ϵ_{0} > 0 such that λ = λ_{0,r} < λ_{ϵ,r+1} for ϵ ∈ ]0, ϵ_{0}]. This implies that, for every ϵ ∈ ]0, ϵ_{0}], we have that λ_{ϵ,r} = λ, since otherwise

λ_{ϵ,r} < λ < λ_{ϵ,r+1}

so λ is not an eigenvalue of A_{ϵ}, a contradiction. The theorem is proved. □

The following result implies that strict monotonicity of all the families (λ_{ϵ,j})_{ϵ>0}, j ≥ 2, is an open dense (and so generic) property with respect to C^{1}-perturbations of the boundary of Ω: Theorem 4.9. Suppose that there is a nonempty open set J in ℝ^{M} and there are disjoint open sets U and U′ in ℝ^{M} × ℝ^{N} such that Ω ∩ (J × ℝ^{N}) = U ∪ U′, and for every x ∈ J, the vertical section

U_{x} := { y | (x, y) ∈ U }

is nonempty and connected. For x ∈ J let p(x) > 0 be the N-dimensional Lebesgue measure of the vertical section U_{x}. Assume that p ∈ C^{1}(J) and ∇p ≢ 0. Then for every j ∈ ℕ with j ≥ 2 the family (λ_{ϵ,j})_{ϵ>0} is strictly monotone decreasing.

Proof. If the theorem is not true, then, by Theorem 4.8, there is a λ ∈ ℝ, λ ≠ 0 and a w ∈ H_{s}^{1}(Ω), w ≠ 0, such that

⟨∇w, ∇v⟩ = λ⟨w, v⟩ for all v ∈ H^{1}(Ω).

By the usual regularity theory it follows that w is real analytic on Ω and Δw(x, y) = −λw(x, y) for all (x, y) ∈ Ω. Since w ∈ H_{s}^{1}(Ω) it follows that ∇_{y}w(x, y) ≡ 0 for all (x, y) ∈ Ω. Since U_{x} is connected for all x ∈ J it follows that there is a function w̃: J → ℝ with w̃(x) ≡ w(x, y) for all (x, y) ∈ U. This immediately implies that w̃ is real analytic on J and

(10)  Δ_{x}w̃(x) = −λw̃(x) for x ∈ J.

Now given ṽ ∈ C_{0}^{∞}(J) define the function v: Ω → ℝ by

v(x, y) = \begin{cases}ṽ(x) & if (x, y) ∈ U, \\ 0 & otherwise. \end{cases}

Using our assumptions it is easy to prove that v ∈ C^{∞}(Ω) with ∇_{y}v ≡ 0 and

∇_{x}v(x, y) = \begin{cases}∇ṽ(x) & if (x, y) ∈ U, \\ 0 & otherwise. \end{cases}

Thus v and ∇v are bounded on Ω and so v ∈ H^{1}(Ω).

We therefore obtain

⟨∇w, ∇v⟩ = ∫_{U} ∇w · ∇v dx dy = ∫_{J} p(x)∇w̃(x) · ∇ṽ(x) dx

= − ∫_{J} p(x)ṽ(x)Δw̃(x) dx − ∫_{J} ṽ(x)∇p · ∇w̃ dx
= λ ∫_{J} p(x)ṽ(x)w̃(x) dx − ∫_{J} ṽ(x)∇p · ∇w̃ dx.

On the other hand,

⟨∇w, ∇v⟩ = λ⟨w, v⟩ = λ ∫_{U} wv dx dy = λ ∫_{J} p(x)ṽ(x)w̃(x) dx.

This shows that

∫_{J} ṽ(x)∇p · ∇w̃ dx ≡ 0 for all ṽ ∈ C_{0}^{∞}(J)

so

∇p · ∇w̃ ≡ 0 on J.

Since ∇p ≢ 0 it follows that for some nonempty open connected subset J′ of J,

∇w̃(x) ≡ 0 on J′.

Hence w̃ is constant on J′. Since λ ≠ 0, formula (10) implies that w̃(x) ≡ 0 on J′ so

w(x, y) ≡ 0 for (x, y) ∈ Ω′ := U ∩ (J′ × ℝ^{N}).

Since J′ is nonempty and U_{x} is nonempty for every x ∈ J, it follows that Ω′ is nonempty. Since w is analytic, it follows that w ≡ 0, which is a contradiction. The theorem is proved. □

4.3. Bifurcation at resonance. For the rest of this subsection, suppose that M = N = 1 so Ω ⊂ ℝ × ℝ and let r ∈ ℕ, r ≥ 2, be such that λ_{0,r} < λ_{0,r+1}. Consider the following hypothesis:

(HR) f: ℝ → ℝ is a function of class C^{2} such that f(0) = 0, f′(0) = λ_{0,r} and there are constants p and C ∈ ]0, ∞[ such that

|f′′(s)| ≤ C(|s|^{p} + 1)

for all s ∈ ℝ.

The following result is well-known, cf. Theorem 5.3 and Proposition 5.5 in [33]: Lemma 4.10. Let f satisfy hypothesis (HR). Then

(1) f ∘ v ∈ L^{2}(Ω) whenever v ∈ H^{1}(Ω).

(2) The Nemitski operator f̂: H^{1}(Ω) → L^{2}(Ω) given by

f̂(v) = f ∘ v

is well-defined, Lipschitzian on bounded subsets of H^{1}(Ω) and it maps bounded subsets of H^{1}(Ω) into bounded subsets of L^{2}(Ω). Moreover, f̂(H_{s}^{1}(Ω)) ⊂ L_{s}^{2}(Ω).

(3) f̂ is Fréchet differentiable and

Df̂(v)(u) = f′(v) · u,

for all v and for all u in H^{1}(Ω).

(4) Df̂ is continuous and Lipschitzian on bounded subsets of H^{1}(Ω).

□

For the rest of this subsection assume hypothesis (HR) and suppose that

(11)  H_{s}^{1}(Ω) ⊂ L^{∞}(Ω) with continuous inclusion.

This latter property is satisfied, e.g., if Ω is a nicely decomposed domain in the sense defined in [33].

We write π_{ϵ} := π_{ϵ,f̂} for ϵ ≥ 0. Since f̂(0) = 0, the set K_{0} = {0} is an invariant set for π_{ϵ}, for ϵ ≥ 0. We will prove in this subsection that, under appropriate conditions, there is a family u_{ϵ}, of nontrivial equilibria of π_{ϵ}, ϵ > 0 small, bifurcating from the trivial equilibrium 0 of π_{0}. The precise statement is given in Theorem 4.15. Our main tool will be Theorem 3.5 and the Conley index product formula introduced in [38] or Chapter 2 of [39]. To set up the framework, define

L := A_{0} − λ_{0,r}I.

Recall that (w_{0,j})_{j∈ℕ} is a complete orthonormal system on L_{s}^{2}(Ω).

Define the following linear subspaces of L_{s}^{2}(Ω):

X_{1} := span{ w_{0,j} | λ_{0,j} = λ_{0,r} },
X_{2} := span{ w_{0,j} | λ_{0,j} < λ_{0,r} },
X_{3} := cl_{L^{2}} span{ w_{0,j} | λ_{0,j} > λ_{0,r} }.

Notice that m := dim X_{1} < ∞, m ≥ 1 and dim X_{2} = r − m. Furthermore, X_{1}, X_{2} and X_{3} are mutually orthogonal and L-invariant subspaces of L_{s}^{2}(Ω). Moreover

(12)  L_{s}^{2}(Ω) = X_{1} ⊕ X_{2} ⊕ X_{3}.

For each i = 1, 2, 3 the restriction of the operator L to X_{i} will be denoted by L_{i}. Since σ(L) = {λ − λ_{0,r} | λ ∈ σ(A_{0}) }, we have

λ ∈ σ(L_{1}) implies that λ = 0,
λ ∈ σ(L_{2}) implies that λ < 0,
λ ∈ σ(L_{3}) implies that λ > 0.

In what follows we will write e_{j} := w_{0,r−m+j}, j = 1, …, m, so

X_{1} = span{e_{1}, …, e_{m}}.

We will also write g := f̂ − λ_{0,r}I. Notice that we are in the setting of Theorem 2.1 in Chapter 2 of [39]. Therefore for each i = 1, 2, 3 there exists a neighborhood V_{i} of zero in X_{i}^{1/2} := H_{s}^{1}(Ω) ∩ X_{i} such that an m-dimensional local center manifold close to zero can be described by a mapping

ξ: V_{1} → V_{2} ⊕ V_{3}.

Furthermore, the coordinate representation π_{ξ} of the semiflow π_{0} on the center manifold is the flow generated by the ordinary differential equation

(13)  u̇ + L_{1}u = E_{1}g(u + ξ(u)),

where u ∈ V_{1} and E_{1} denotes the projection of L_{s}^{2} onto X_{1} induced by direct sum described in (12). We have the following result: Proposition 4.11. Assume hypotheses (HR) and (11). Moreover, suppose that

(14)  f(s) = λ_{0,r}s + as^{k} + β(s),

where k ≥ 2 is a natural number, a ∈ ℝ and β(s) = O(s^{k+1}) as s → 0. Then the following statements hold:

(1) If a < 0 and k is odd, then the set {0} is an isolated invariant set relative to π_{ξ}

and

h(π_{ξ}, {0}) = Σ^{0}.

(2) If k is even, m = 1 and a⟨w_{0,r}^{k}, w_{0,r}⟩ ≠ 0, then the set {0} is an isolated invariant

set relative to π_{ξ} and

h(π_{ξ}, {0}) = ¯0.

Proof. Notice that all norms are equivalent on the finite-dimensional space X_{1}. Write I_{m}^{k} := { 1, …, m }^{k} and let α = (α_{1}, …, α_{k}) denote a generic element of I_{m}^{k}. Recall that

X_{1} = span{ e_{1}, e_{2}, …, e_{m} }.

Define the map ϕ: X_{1} → X_{2}^{1/2} ⊕ X_{3}^{1/2} by

ϕ(u) := ∑_{α∈I_{m}^{k}}⟨u, e_{α_{1}}⟩⟨u, e_{α_{2}}⟩ … ⟨u, e_{α_{k}}⟩v(α_{1}, …, α_{k}),

with v(α_{1}, …, α_{k}) ∈ D(A_{0}) and orthogonal to e_{i} for all i = 1, …, m to be determined later. Obviously ϕ is of class C^{∞}. By (11) there are a constant C and a δ_{0} > 0 such that |ϕ(u)(x, y)| ≤ C|u|_{L^{2}}^{k} for all u ∈ X_{1} and all (x, y) ∈ Ω. Moreover, |β(u(x, y)+ϕ(u)(x, y))| ≤ M|u|_{L^{2}}^{k+1} for all u ∈ X_{1} with |u|_{L^{2}} ≤ δ_{0} and all (x, y) ∈ Ω. It follows that

g(u + ϕ(u)) = au^{k} + β_{1}(u),

with |β_{1}(u)|_{L^{2}} = O(|u|_{L^{2}}^{k+1}) as |u|_{L^{2}} → 0. Consequently,

E_{1}g(u + ϕ(u)) = a⟨u^{k}, e_{1}⟩e_{1} + ⋯ + a⟨u^{k}, e_{m}⟩e_{m} + β_{2}(u),

with β_{2}(u) = O(|u|_{L^{2}}^{k+1}) as |u|_{L^{2}} → 0. Define the map Δ: X_{1} → L^{2}(Ω) by

(15) Δ(u) := Dϕ(u) [L_{1}u − E_{1}g(u + ϕ(u))] − Lϕ(u) + g(u + ϕ(u)) − E_{1}g(u + ϕ(u)).

Notice that L_{1}u = 0. It is a simple task to prove that |Dϕ(u) [−E_{1}g(u + ϕ(u))] |_{L^{2}} = O(|u|_{L^{2}}^{k+1}) as |u|_{L^{2}} → 0. Therefore

Δ(u) = ∑_{α∈I_{m}^{k}}⟨u, e_{α_{1}}⟩⟨u, e_{α_{2}}⟩ … ⟨u, e_{α_{k}}⟩h(α_{1}, …, α_{k}) + β_{3}(u),

where

h(α_{1}, …, α_{k}) := −Lv(α_{1}, …, α_{k}) + ae_{α_{1}} … e_{α_{k}} − a ∑_{i=1}^{m}⟨e_{α_{1}} … e_{α_{k}}, e_{i}⟩e_{i}

and |β_{3}(u)|_{L^{2}} = O(|u|_{L^{2}}^{k+1}) as |u|_{L^{2}} → 0. Now choose v(α_{1}, …, α_{k}) ∈ D(A_{0}), orthogonal to e_{i}, for i = 1, …, m in such a way that h(α_{1}, …, α_{k}) = 0. Since L = A_{0} − λ_{0,r}, the vector v(α_{1}, …, α_{k}) with the above property exists and is unique. With this choice, we have

|Δ(u)|_{L^{2}} = O(|u|_{L^{2}}^{k+1}).

Therefore, we are in the conditions of Theorem 2.3 in Chapter 2 of [39]. In other words,

|ξ(u) − ϕ(u)|_{H_{s}^{1}} = O(|u|_{L^{2}}^{k+1}).

Reasoning as above, we see that the reduced equation (13) on the center manifold becomes:

(16)  u̇ = E_{1}g(u + ξ(u)) = E_{1}g(u + ϕ(u)) + β_{4}(u) = aE_{1}(u^{k}) + β_{5}(u),

with |β_{4}(u)|_{L^{2}} = O(|u|_{L^{2}}^{k+1}) and |β_{5}(u)|_{L^{2}} = O(|u|_{L^{2}}^{k+1}) as |u|_{L^{2}} → 0.

Suppose first that a < 0 and k is odd. Consider the positive definite functional V : X_{1} → ℝ given by V (u) = ⟨u, u⟩/2. Let t ↦ uπ_{ξ}t be a solution of π_{ξ}. Then

\frac{d}{dt}V (uπ_{ξ}t) |_{t=0} = a⟨E_{1}(u^{k}), u⟩ + ⟨β_{5}(u), u⟩ = a⟨u^{k}, u⟩ + β_{6}(u),

where β_{6}(u) = O(|u|_{L^{2}}^{k+2}) as |u|_{L^{2}} → 0. There is a constant C such that

|u|_{L^{2}} ≤ C|u|_{L^{k+1}} for all u ∈ X_{1}.

Therefore there exist positive constants M and δ such that whenever |u|_{L^{2}} ≤ δ then

|β_{6}(u)| ≤ M|u|_{L^{2}}|u|_{L^{2}}^{k+1} ≤ MC^{k+1}|u|_{L^{2}}⟨u^{k}, u⟩.

Choosing δ such that MC^{k+1}δ = −a/2 we see that

\frac{d}{dt}V (uπ_{ξ}t) |_{t=0} ≤ (a/2)⟨u^{k}, u⟩ < 0

if |u|_{L^{2}} ≤ δ and u ≠ 0.

It follows that the closed ball

B := { u ∈ X_{1} | ⟨u, u⟩ ≤ δ }

is an isolating block for π_{ξ} with B^{−} = ∅ and the largest invariant set in B is {0}. Thus h(π_{ξ}, {0}) = h(π_{ξ}, B) = Σ^{0}. This completes the proof of the first part of the proposition.

Now assume that k is even, m = 1 and a⟨w_{0,r}^{k}, w_{0,r}⟩ ≠ 0. Note that e_{m} = w_{0,r}. Writing u = ye_{m} we see that (16) is equivalent to the one-dimensional equation

ẏ = a⟨e_{m}^{k}, e_{m}⟩y^{k} + γ(y),

where γ(y) = O(|y|^{k+1}) as y → 0. Since k is even and a⟨e_{m}^{k}, e_{m}⟩ ≠ 0 , it is clear that {0} is an isolated invariant set of the latter equation with Conley index ¯0. This proves the second part of the proposition. □

Applying the index-product formula of [38] (or [39]) we thus arrive at the following Proposition 4.12. Under each of the alternative hypotheses of Proposition 4.11 the set K_{0} = {0} is an isolated invariant set for π_{0} and

h(π_{0}, K_{0}) = \begin{cases}Σ^{r−m} & if k is odd and a < 0, \\ ¯0 & k is even, m = 1 and a⟨w_{0,r}^{k}, w_{0,r}⟩ ≠ 0. □ \end{cases}

The case for which m = 1 and a⟨w_{0,r}^{k}, w_{0,r}⟩ = 0 can be treated as in Section 2.4 of Chapter 2 in [39]. The index depends on the higher order terms in the expansion of the map f. We only treat a special case: Proposition 4.13. Assume hypotheses (HR) and (11). Moreover, suppose that m = 1, a⟨w_{0,r}^{k}, w_{0,r}⟩ = 0 and

(17)  f(s) = λ_{0,r}s + as^{k} + bs^{2k−1} + β(s),

where k is an even positive integer, a and b ∈ ℝ are arbitrary and β(s) = O(s^{2k}) as s → 0. Then for all b < 0 with |b| large enough the set {0} is an isolated invariant set relative to π_{ξ} and

h(π_{ξ}, {0}) = Σ^{0}.

Consequently, the set {0} is an isolated invariant set relative to π_{0} and

h(π_{0}, {0}) = Σ^{r−1}.

Proof. Note again that e_{m} = w_{0,r}. Let ϕ: X_{1} → X_{2}^{1/2} ⊕ X_{3}^{1/2} be a map defined by

ϕ(u) = ⟨u, e_{m}⟩^{k}v + ⟨u, e_{m}⟩^{2k−1}w,

where v, w ∈ D(A_{0}), v and w orthogonal to e_{m}, Lv = ae_{m}^{k} + a⟨e_{m}^{k}, e_{m}⟩e_{m} = ae_{m}^{k} and Lw = ake_{m}^{k−1}v − ak⟨e_{m}^{k}, v⟩e_{m} + be_{m}^{2k−1} − b⟨e_{m}^{2k−1}, e_{m}⟩e_{m}.

Proceeding exactly as in Section 2.4 of Chapter 2 in [39], we see that the reduced equation on the center manifold is

u̇ = E_{1}g(u + ξ(u)) = ⟨u, e_{m}⟩^{2k−1} (ak⟨e_{m}^{k}, v⟩ + b⟨e_{m}^{2k−1}, e_{m}⟩) e_{m} + γ(u),

where |γ(u)|_{L^{2}} = O(|u|_{L^{2}}^{2k}) as |u|_{L^{2}} → 0. An equivalent ordinary differential equation on ℝ reads as follows:

ẏ = (ak⟨e_{m}^{k}, v⟩ + b⟨e_{m}^{2k−1}, e_{m}⟩) y^{2k−1} + γ_{1}(y),

where |γ_{1}(y)| = O(|y|^{2k}) as |y| → 0. Since ⟨e_{m}^{2k−1}, e_{m}⟩ > 0, it follows that ak⟨e_{m}^{k}, v⟩ + b⟨e_{m}^{2k−1}, e_{m}⟩ < 0 for b < 0 and |b| large enough. It clearly follows that in this case the set K_{ξ} := {0} is an isolated invariant set relative to π_{ξ} and its index is Σ^{0}. An application of the index product formula concludes the proof. □

Proposition 4.14. For every ϵ > 0 let π_{ϵ} be as above. Suppose that the family (λ_{ϵ,r})_{ϵ>0} is strictly monotone decreasing. Then there exists an ϵ_{1} > 0 such that for all ϵ ∈ ]0, ϵ_{1}], the set {0} is an isolated invariant set relative to π_{ϵ} and

h(π_{ϵ}, {0}) = Σ^{r}.

Proof. Since λ_{0,r} < λ_{0,r+1}, our assumption implies that

λ_{ϵ,r} < λ_{0,r} < λ_{ϵ,r+1}

for all ϵ > 0 small enough. This obviously implies the assertion. □

We can now state the main result of this subsection: Theorem 4.15. Assume any of the alternative hypotheses of Proposition 4.11 or else assume the hypotheses of Proposition 4.13. Moreover, suppose that the family (λ_{ϵ,r})_{ϵ>0} is strictly monotone decreasing. Suppose that N_{0} is a bounded isolating neighborhood of {0} for π_{0}. Then for every η > 0 there exists an 0 < ϵ_{0} = ϵ_{0}(η), such that for every ϵ ∈ ]0, ϵ_{0}],

K_{ϵ,η} := A_{π_{ϵ}}([N_{0}]_{ϵ,η})

contains a nontrivial equilibrium u_{ϵ} of π_{ϵ} and a nonconstant full solution σ_{ϵ} of π_{ϵ} whose ωlimit set or α-limit set is equal to {0}. In other words, 0 is connected to a set of nontrivial equilibria of π_{ϵ}.

Proof. Fix η > 0. By Proposition 4.14, there exists an ϵ_{1} > 0 such that for all ϵ ∈ ]0, ϵ_{1}], the set {0} is an isolated invariant set relative to π_{ϵ} and

h(π_{ϵ}, {0}) = Σ^{r}.

Since Σ^{r} ≠ Σ^{r−m} and Σ^{r} ≠ ¯0 Theorem 3.5 and Proposition 4.12 (or else Proposition 4.13) implies that

h(π_{ϵ}, K_{ϵ,η}) ≠ Σ^{r}

so K_{ϵ,η} ≠ {0} for all ϵ > 0 small enough. Since π_{ϵ} is gradient-like, we obtain the existence of a nontrivial equilibrium u_{ϵ}. The existence of the connection σ_{ϵ} follows from the irreducibility of K_{ϵ,η} (see Lemma 11.4 and Theorem 11.6 in [39]). □

4.4. C-shaped domains. In this subsection we will define a class of very simple twodimensional domains, which we call C-shaped, and show that all of the alternative assumptions of Theorem 4.15 are satisfied for an appropriate choice of the domain Ω belonging to that class. Definition 4.16. Let a_{i} and b_{i}, i = 1, 2, 3, be real numbers with a_{1} < 0, a_{2} > 0, a_{3} > 0 and 0 < b_{1} < b_{2} < b_{3}. The C-shaped domain with parameters a_{i} and b_{i}, i = 1, 2, 3 is the following set

Ω := (]a_{1}, a_{2}[ ×]0, b_{1}[) ∪ (]a_{1}, 0[ ×]0, b_{3}[) ∪ (]a_{1}, a_{3}[ ×]b_{2}, b_{3}[).

(See Figure 1.) Setting Ω_{1} := ]a_{1}, 0[ ×]0, b_{3}[, Ω_{2} := ]0, a_{2}[ ×]0, b_{1}[, Ω_{3} := ]0, a_{3}[ ×]b_{2}, b_{3}[ and Z := {0} × ℝ we obtain a nice decomposition of Ω in the sense of [33]. In this case J_{1} = ]a_{1}, 0[, J_{2} = ]0, a_{2}[ and J_{3} = ]0, a_{3}[, while p_{1}(x) ≡ b_{3}, p_{2}(x) ≡ b_{1} and p_{3}(x) ≡ b_{3} − b_{2}.

[figure: a_{2}; Ω_{1}; Ω_{2}; Ω_{3}; a_{3}; a_{1}; b_{1}; b_{2}; b_{3}; Figure 1.]

For the rest of this section let Ω be a C-shaped domain with parameters a_{i} and b_{i}, i = 1, 2, 3. Theorem 6.6 in [33] provides conditions for a pair (λ, u) to be an eigenvalueeigenvector pair of the operator A_{0} on Ω. In fact, (λ, u) is an eigenvalue-eigenvector pair of A_{0} if and only if u ≢ 0 and there exist absolutely continuous functions u_{l}: J_{l} → ℝ, l = 1, 2, 3 and a null set S in ℝ × ℝ such that whenever l = 1, 2, 3, then u(x, y) = u_{l}(x) for (x, y) ∈ Ω_{l} \ S, u_{l}, u′_{l} ∈ L^{2}(J_{l}) and the following properties hold:

(1) u′′_{l} = −λu_{l}, l = 1, 2, 3, in the sense of distributions;

(2) The limits: u_{l}(0) := lim_{x→0} u_{l}(x), l = 1, 2, 3, exist and u_{1}(0) = u_{2}(0) = u_{3}(0);

(3) The limits: u′_{l}(a_{l}) := lim_{x→a_{l}} u′_{l}(x) and u′_{l}(0) := lim_{x→0} u′_{l}(x), l = 1, 2, 3, exist and u′_{1}(a_{1}) = u′_{2}(a_{2}) = u′_{3}(a_{3}) = 0 while b_{3}u′_{1}(0) = b_{1}u′_{2}(0) + (b_{3} − b_{2})u′_{3}(0).

Notice that condition 1 is equivalent to

u_{l}(x) = α_{l} cos \sqrt{λ}x + β_{l} sin \sqrt{λ}x, l = 1, 2, 3, x ∈ J_{l},

where α_{l} and β_{l}, l = 1, 2, 3, are arbitrary real numbers. By a simple calculation we thus obtain the following

Proposition 4.17. A pair (λ, u) is an eigenvalue-eigenvector pair of A_{0} if and only if u ≢ 0 and there exist functions u_{l}: J_{l} → ℝ, l = 1, 2, 3, real numbers α and β_{l}, l = 1, 2, 3, and a null set S in ℝ × ℝ such that whenever l = 1, 2, 3 then u(x, y) = u_{l}(x) for all (x, y) ∈ Ω_{l} \ S and conditions

(18)  u_{l}(x) = α cos \sqrt{λ}x + β_{l} sin \sqrt{λ}x, l = 1, 2, 3, x ∈ J_{l},
(19)  b_{3}\sqrt{λ}β_{1} = b_{1}\sqrt{λ}β_{2} + (b_{3} − b_{2})\sqrt{λ}β_{3}

and

(20)  −α\sqrt{λ} sin a_{l}\sqrt{λ} + β_{l}\sqrt{λ} cos a_{l}\sqrt{λ} = 0, l = 1, 2, 3

hold.

Remark 4.18. Proposition 4.17 immediately implies that λ = 0 is a simple eigenvalue of A_{0}. Now let λ > 0 be an arbitrary eigenvalue of A_{0}. If sin a_{l}\sqrt{λ} = 0 for l = 1, 2, 3, then β_{l} = 0 for l = 1, 2, 3 by condition (20), so by condition (18) λ is a simple eigenvalue. If sin a_{i}\sqrt{λ} ≠ 0 for some i, then α = β_{i} cot a_{i}\sqrt{λ} and so conditions (18) and (19) imply that the eigenspace of λ is at most two-dimensional. Thus there are no eigenvalues of A_{0} of multiplicity higher than two.

The following result is a strict monotonicity criterion for C-shaped domains:

Proposition 4.19. Let λ > 0 be an eigenvalue of A_{0} and let r ∈ ℕ, r ≥ 2, be such that λ = λ_{0,r} < λ_{0,r+1}. Then the following properties are equivalent:

(1) sin(a_{l}\sqrt{λ}) = 0 for l = 1, 2, 3;

(2) There is an ϵ_{0} > 0 such that λ_{ϵ,r} ≡ λ for ϵ ∈ [0, ϵ_{0}].

Proof. Suppose property 1 holds. Set u(x, y) ≡ u(x) := cos \sqrt{λ}x, for (x, y) ∈ Ω. Then u ≢ 0 and so Proposition 4.17 (with α = 1 and β_{l} ≡ 0) implies that the pair (λ, u) is an eigenvalue-eigenvector pair of the operator A_{0}. In particular we have that u ∈ H_{s}^{1}(Ω).

The divergence theorem implies that for every v ∈ H^{1}(Ω)

⟨∇u, ∇v⟩ = ∑_{l=1}^{3} ∫_{Ω_{l}} u′(x)∂_{x}v(x, y) dx dy

= ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′(x)v(x, y)ρ_{l}(x, y)dσ(x, y) − ∑_{l=1}^{3} ∫_{Ω_{l}} u′′(x)v(x, y) dx dy

= ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′(x)v(x, y)ρ_{l}(x, y)dσ(x, y) + λ ∑_{l=1}^{3} ∫_{Ω_{l}} u(x)v(x, y) dx dy

= ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′(x)v(x, y)ρ_{l}(x, y)dσ(x, y) + λ⟨u, v⟩,

where σ is the 1-dimensional Hausdorff measure on ℝ^{2} and ρ_{l}(x, y) is the x-component of the outer normal vector to ∂Ω_{l} at (x, y) ∈ ∂Ω_{l}. Now ρ_{l}(x, y) = 0 on the horizontal part of ∂Ω_{l}, while, by the definition of u, u′(x) = 0 for (x, y) lying on the vertical part of ∂Ω_{l}. Thus

∑_{l=1}^{3} ∫_{∂Ω_{l}} u′(x)v(x, y)ρ_{l}(x, y)dσ(x, y) = 0

and so

⟨∇u, ∇v⟩ = λ⟨u, v⟩ for all v ∈ H^{1}(Ω).

Now Theorem 4.8 implies property 2.

Assume, conversely, that property 2 holds. Then Theorem 4.8 implies that there is a u ∈ H_{s}^{1}(Ω), u ≢ 0, such that

(21)  ⟨∇u, ∇v⟩ = λ⟨u, v⟩ for all v ∈ H^{1}(Ω).

In particular, (λ, u) is an eigenvalue-eigenvector pair of A_{0}. By Proposition 4.17 there exist functions u_{l}: J_{l} → ℝ, real numbers α and β_{l}, l = 1, 2, 3, and a null set S in ℝ×ℝ such that whenever l = 1, 2, 3 then u(x, y) = u_{l}(x) for all (x, y) ∈ Ω_{l} \ S and conditions (18), (19) and (20) hold. Formula (21) and the divergence theorem imply that for every v ∈ H^{1}(Ω)

λ⟨u, v⟩ = ⟨∇u, ∇v⟩ = ∑_{l=1}^{3} ∫_{Ω_{l}} u′_{l}(x)∂_{x}v(x, y) dx dy

= ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′_{l}(x)v(x, y)ρ_{l}(x, y)dσ(x, y) − ∑_{l=1}^{3} ∫_{Ω_{l}} u′′_{l}(x)v(x, y) dx dy

= ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′_{l}(x)v(x, y)ρ_{l}(x, y)dσ(x, y) + λ ∑_{l=1}^{3} ∫_{Ω_{l}} u_{l}(x)v(x, y) dx dy

= ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′_{l}(x)v(x, y)ρ_{l}(x, y)dσ(x, y) + λ⟨u, v⟩.

This implies that

(22)  ∑_{l=1}^{3} ∫_{∂Ω_{l}} u′_{l}(x)v(x, y)ρ_{l}(x, y)dσ(x, y) = 0 for all v ∈ H^{1}(Ω).

Letting v in (22) be the restriction to Ω of an arbitrary nonnegative test function on ℝ^{2} whose support is included in the horizontal strip ℝ × ]b_{1}, b_{2}[ and which is nontrivial on \{0\} × ]b_{1}, b_{2}[ we see that u′_{1}(0) = 0, i.e. β_{1}\sqrt{λ} = 0, so β_{1} = 0. Similarly, we prove that β_{2} = β_{3} = 0. Hence α ≠ 0 and so condition (20) implies that sin(a_{l}\sqrt{λ}) = 0 for l = 1, 2,

3. The proposition is proved. □

The following result shows that, by choosing the parameters a_{i} and b_{i}, i = 1, 2, 3, appropriately, we can satisfy all the alternative hypotheses of Theorem 4.15: Proposition 4.20. Suppose that b_{l} = l for l = 1, 2, 3 and let λ be an arbitrary real number.

(1) If −a_{1} = a_{2} = a_{3} =: a and cos a\sqrt{λ} = 0, then λ is a double eigenvalue of A_{0}, whose eigenspace is formed by all functions u for which there are constants β_{2}, β_{3} ∈ ℝ such that: 1.1. u(x, y) = (1/3)(β_{2} + β_{3}) sin \sqrt{λ}x a.e. on Ω_{1}; 1.2. u(x, y) = β_{l} sin \sqrt{λ}x a.e. on Ω_{l}, l = 2, 3.

(2) If −a_{1} = a_{2} =: a, cos a\sqrt{λ} = 0 and cos a_{3}\sqrt{λ} ≠ 0, then λ is a simple eigenvalue of A_{0}, whose eigenspace is formed by all functions u for which there is a constant β_{2} ∈ ℝ such that: 2.1. u(x, y) = (1/3)β_{2} sin \sqrt{λ}x a.e. on Ω_{1}; 2.2. u(x, y) = β_{2} sin \sqrt{λ}x a.e. on Ω_{2}; 2.3. u(x, y) = 0 a.e. on Ω_{3}; If e is an arbitrary eigenfunction of λ and k ∈ ℕ is even, then

(23)  ⟨e^{k}, e⟩ = ∫_{Ω} e^{k+1}(x, y) dx dy ≠ 0.

(3) If a_{2} = a_{3} =: a, cos a\sqrt{λ} = 0 and cos a_{1}\sqrt{λ} ≠ 0, then λ is a simple eigenvalue of A_{0}, whose eigenspace is formed by all functions u for which there is a constant β_{2} ∈ ℝ such that: 3.1. u(x, y) = 0 a.e. on Ω_{1}; 3.2. u(x, y) = β_{2} sin \sqrt{λ}x a.e. on Ω_{2}; 3.3. u(x, y) = −β_{2} sin \sqrt{λ}x a.e. on Ω_{3}; If e is an arbitrary eigenfunction of λ and k ∈ ℕ is even, then

(24)  ⟨e^{k}, e⟩ = ∫_{Ω} e^{k+1}(x, y) dx dy = 0.

In all the above cases, if r is such that λ = λ_{0,r} < λ_{0,r+1} then the family (λ_{ϵ,r})_{ϵ>0} is strictly monotone decreasing, so, in particular, λ_{ϵ,r} < λ for all ϵ > 0.

Proof. Consider the first case. If u is any function satisfying properties 1.1 and 1.2 then conditions (18), (19) and (20) of Proposition 4.17 are satisfied with α = 0 and β_{1} = (1/3)(β_{2} +β_{3}), so, by that proposition, λ is an eigenvalue of A_{0} and u lies in its eigenspace.

Now consider the second case. Suppose first that λ is an eigenvalue of A_{0} and let u be a corresponding eigenvector. Then u is determined by constants α and β_{l}, l = 1, 2, 3, so that conditions (18), (19) and (20) of Proposition 4.17 hold. Since cos a\sqrt{λ} = 0, we conclude that α = 0 and since cos a_{3}\sqrt{λ} ≠ 0 we also conclude that β_{3} = 0. Thus u must necessarily satisfy properties 2.1, 2.2 and 2.3, so λ is a simple eigenvalue. On the other hand, letting u ≢ 0 be an arbitrary function satisfying these properties, we conclude from Proposition 4.17 that (λ, u) is an eigenvalue-eigenvector pair of A_{0}. Thus λ is, indeed, a simple eigenvalue of A_{0}. Formula (23) is obtained by a simple integration.

The third case is proved in the same way.

The last statement of the proposition is a consequence of Proposition 4.19. The proposition is proved. □

5. Appendix

In this section we prove Theorem 3.5. In view of Proposition 2.8 it is not restrictive to assume that the family (f_{ϵ})_{ϵ∈[0,ϵ_{0}]} satisfies hypothesis (A2). Write π_{ϵ} := π_{ϵ,f_{ϵ}} for all ϵ ∈ [0, ϵ_{0}].

We need a number of preliminary results. Proposition 5.1. For all sufficiently small ϵ > 0 the set [N]_{ϵ,η} is a strongly π_{ϵ}-admissible isolating neighborhood for π_{ϵ}.

Proof. Theorem 4.4 in [39] implies that [N]_{ϵ,η} is a strongly π_{ϵ}-admissible set for every η > 0 and ϵ > 0. Suppose that there are an η > 0 and a sequence (ϵ_{n})_{n∈ℕ} of positive numbers converging to zero such that for each n ∈ ℕ the set [N]_{ϵ_{n},η} is not an isolating neighborhood for π_{ϵ_{n}}. Then for each n ∈ ℕ there exists a solution σ_{n}: ℝ → H^{1}(Ω) of π_{ϵ_{n}} such that

σ_{n}(ℝ) ⊂ [N]_{ϵ_{n},η} and σ_{n}(0) ∈ ∂ [N]_{ϵ_{n},η}.

By Lemma 2.20 there exists a subsequence of (σ_{n})_{n∈ℕ} denoted again by (σ_{n})_{n∈ℕ} such that for every t ∈ ℝ

(25)  |Q_{ϵ_{n}}σ_{n}(t) − σ(t)|_{H^{1}} → 0 and |(I − Q_{ϵ_{n}})σ_{n}(t)|_{ϵ_{n}} → 0

as n → ∞, where σ: ℝ → N is a solution of π_{0}. Setting t = 0 in (25), we conclude that |(I − Q_{ϵ_{n}})σ_{n}(0)|_{ϵ_{n}} < η and so Q_{ϵ_{n}}σ_{n}(0) ∈ ∂N for all n large enough. This implies that σ(0) ∈ ∂N, a contradiction to the fact that N is an isolating neighborhood for π_{0}. The proposition is proved. □

Assume now that K_{0} := A_{π_{0}}(N) = ∅. We claim that K_{ϵ,η} := A_{π_{ϵ}}([N]_{ϵ,η}) = ∅ for every η > 0 and all sufficiently small ϵ > 0. Otherwise, we find an η > 0 and a sequence (ϵ_{n})_{n∈ℕ}, of positive numbers converging to zero such that for each n ∈ ℕ,

K_{ϵ_{n},η} ≠ ∅.

Let σ_{n}: ℝ → [N]_{ϵ_{n},η} be a full solution of π_{ϵ_{n}}. Lemma 2.20 implies that there exists a solution σ: ℝ → N of π_{0}. Since K_{0} is the empty set, we obtain a contradiction. This proves our claim.

Thus Theorem 3.5 holds if K_{0} = ∅. Therefore let us assume from now on that K_{0} ≠ ∅. We shall prove Theorem 3.5 by appropriately modifying the proof of the usual Conley index continuation theorem as given in [37] or [39].

Assume that K_{0} ≠ ∅ is an isolated invariant set of π_{0}. By Theorem 3.1, we may choose an open set U_{0} in H_{s}^{1}(Ω) such that N_{0} := cl U_{0} is an isolating block for π_{0} and such that K ⊂ U_{0} ⊂ N_{0} ⊂ N and ∂U_{0} = ∂N_{0}.

Let α: [0, ∞[ → [1, 2[ be a monotone increasing C^{∞}-diffeomorphism. Let s^{+} : N_{0} → ℝ ∪ { ∞ } be given by

s^{+}(u) := sup{ t ≥ 0 | uπ_{0} [0, t] ⊂ N_{0} },

t^{+} : U_{0} → ℝ ∪ { ∞ } by

t^{+}(u) := sup{ t ≥ 0 | uπ_{0}[0, t] ⊂ U_{0} } > 0

and F: H_{s}^{1}(Ω) → [0, 1] defined by

F (u) := min{ 1, d(u, A_{π_{0}}^{−}(N_{0})) },

where d(u, A_{π_{0}}^{−}(N_{0})) is the distance (in H_{s}^{1}(Ω)) of the point u from the set A_{π_{0}}^{−}(N_{0}). Finally, we define g^{−}: N_{0} → ℝ as follows:

g^{−}(u) := sup{ α(t)F (uπ_{0}t) | t ∈ [0, s^{+}(u)] , if s^{+}(u) < ∞ and t ∈ [0, ∞[ , if s^{+}(u) = ∞ }.

Whenever (u_{n})_{n∈ℕ} is a sequence in N_{0} with g^{−}(u_{n}) → 0 as n → ∞, then, by admissibility, there is a subsequence of (u_{n})_{n∈ℕ} converging to an element of A_{π_{0}}^{−}(N_{0}). Given a > 0, b > 0 define

V (a, b) := { u ∈ U_{0} | g^{−}(u) < a, t^{+}(u) > b }.

Note that V (a, b) is open in H_{s}^{1}(Ω), K_{0} ⊂ V (a, b) and admissibility implies that we can choose a_{0} > 0 and b_{0} > 0 such that cl V (a_{0}, b_{0}) ⊂ U_{0}.

Given a > 0, b > 0, ϵ > 0 and α > 0 define

V_{ϵ,α}(a, b) := ]V (a, b)[_{ϵ,α} .

Lemma 5.2. Fix positive numbers a, b, α, δ, η and M such that a ≤ a_{0} and b ≥ b_{0}. Then

K(ϵ, η, a, b) := A_{π_{ϵ}}([cl V (a, b)]_{ϵ,η}) ⊂ V_{ϵ,α}(δ, M)

for all sufficiently small ϵ > 0.

Proof. If the lemma is not true then there are positive numbers a, b, α, δ, η and M such that a ≤ a_{0} and b ≥ b_{0} and a sequence (ϵ_{n})_{n∈ℕ} of positive numbers converging to zero such that for all n ∈ ℕ

K(ϵ_{n}, η, a, b) ⊄ V_{ϵ_{n},α}(δ, M).

For each n ∈ ℕ, there is a solution σ_{n}: ℝ → [cl V (a, b)]_{ϵ_{n},η} of π_{ϵ_{n}} with σ_{n}(0) ∉ V_{ϵ_{n},α}(δ, M). By Lemma 2.20 there exists a subsequence of (σ_{n})_{n∈ℕ} which we will denote again by (σ_{n})_{n∈ℕ} and there exists a solution σ: ℝ → cl V (a, b) ⊂ N_{0} of π_{0} such that for every t ∈ ℝ,

(26)  |Q_{ϵ_{n}}(σ_{n}(t)) − σ(t)|_{H^{1}} → 0 and |(I − Q_{ϵ_{n}})(σ_{n}(t))|_{ϵ_{n}} → 0

as n → ∞. Thus σ(ℝ) ⊂ K_{0} and |(I − Q_{ϵ_{n}})(σ_{n}(0))|_{ϵ_{n}} < α for all n large enough. Since V (δ, M) is open in H_{s}^{1}(Ω) it follows that Q_{ϵ_{n}}(σ_{n}(0)) ∈ V (δ, M) for all n large enough. In other words, σ_{n}(0) ∈ V_{ϵ_{n},α}(δ, M) for all n large enough, a contradiction proving the lemma. □

Let ϵ > 0 and ν > 0 be arbitrary. Given u ∈ ]U_{0}[_{ϵ,ν}, define

t_{ϵ,ν}^{+}(u) := sup{ t ≥ 0 | uπ_{ϵ}[0, t] ⊂ ]U_{0}[_{ϵ,ν} } > 0.

Lemma 5.3. Fix ν > 0. Let ϵ_{n} > 0, u_{n} ∈ ]U_{0}[_{ϵ_{n},ν}, n ∈ ℕ, and u ∈ U_{0} be such that ϵ_{n} → 0 and

|u_{n} − u|_{ϵ_{n}} → 0 as n → ∞.

Then

t_{ϵ_{n},ν}^{+}(u_{n}) → t^{+}(u) as n → ∞.

Proof. Let C ∈ ]0, ∞[ be such that t^{+}(u) < C. Since uπ_{0}t^{+}(u) ∈ ∂U_{0} = ∂N_{0} and N_{0} is an isolating block for π_{0}, there exists an s ∈ ℝ with C > s > t^{+}(u) such that uπ_{0}s ∉ N_{0}. By Corollary 2.14 we have |u_{n}π_{ϵ_{n}}s − uπ_{0}s|_{ϵ_{n}} → 0 and so |Q_{ϵ_{n}}(u_{n}π_{ϵ_{n}}s) − uπ_{0}s|_{H^{1}} → 0 as n → ∞. This implies that Q_{ϵ_{n}}(u_{n}π_{ϵ_{n}}s) ∉ N_{0}, so u_{n}π_{ϵ_{n}}s ∉ [N_{0}]_{ϵ_{n},ν} and therefore t_{ϵ_{n},ν}^{+}(u_{n}) < s < C for n large enough.

Now let t^{+}(u) > C > 0. Then uπ_{0} [0, C] ⊂ U_{0}. We claim that u_{n}π_{ϵ_{n}} [0, C] ⊂ ]U_{0}[_{ϵ_{n},ν} for all n large enough. Suppose that this is not true. Then we may assume that for every n ∈ ℕ

u_{n}π_{ϵ_{n}} [0, C] ⊄ ]U_{0}[_{ϵ_{n},ν}

and so there exists a t_{n} ∈ [0, C] such that u_{n}π_{ϵ_{n}}t_{n} ∉ ]U_{0}[_{ϵ_{n},ν}. Again we can assume that there exists a t ∈ [0, C] such that t_{n} → t as n → ∞. Now Corollary 2.14 implies that |u_{n}π_{ϵ_{n}}t_{n} − uπ_{0}t|_{ϵ_{n}} → 0 as n → ∞. Thus |Q_{ϵ_{n}}(u_{n}π_{ϵ_{n}}t_{n}) − uπ_{0}t|_{H^{1}} → 0 and |(I − Q_{ϵ_{n}})(u_{n}π_{ϵ_{n}}t_{n})|_{ϵ_{n}} → 0 as n → ∞. It follows that Q_{ϵ_{n}}(u_{n}π_{ϵ_{n}}t_{n}) ∈ U_{0} and |(I − Q_{ϵ_{n}})(u_{n}π_{ϵ_{n}}t_{n})|_{ϵ_{n}} < ν for all n large enough.

This shows that u_{n}π_{ϵ_{n}}t_{n} ∈ ]U_{0}[_{ϵ_{n},ν} for all large n, contradicting the choice of the sequence (t_{n})_{n∈ℕ}. This proves our claim, which, in turn, implies that t_{ϵ_{n},ν}^{+}(u_{n}) > C for all sufficiently large n. The lemma is proved. □

Definition 5.4. Fix positive real numbers M, M′, β, ν, η, ρ, a and b. For all positive numbers ϵ, α and δ define the following subsets of H^{1}(Ω):

N_{1}(α, δ, ϵ) := [cl V (a, b)]_{ϵ,η} ∩ cl_{ϵ}{ v | ∃ u ∈ V_{ϵ,α}(δ, M) and t ≥ 0 such that
uπ_{ϵ}[0, t] ⊂ ]U_{0}[_{ϵ,β} and uπ_{ϵ}t = v },
N_{2}(α, δ, ϵ) := N_{1}(α, δ, ϵ) ∩ { u ∈ ]U_{0}[_{ϵ,ν} | t_{ϵ,ν}^{+}(u) ≤ M′ },
Ê(ϵ) := [{ u ∈ U_{0} | t^{+}(u) ≤ 5M } ∩ cl V (a, b)]_{ϵ,η} ,
Ĉ(ϵ, α, δ) := { u ∈ ]U_{0}[_{ϵ,ν} | t_{ϵ,ν}^{+}(u) ≤ 4M } ∩ N_{1}(α, δ, ϵ),
E(ϵ, α, δ) := [{ u ∈ U_{0} | t^{+}(u) ≤ 3M } ∩ cl V (δ, M)]_{ϵ,α} ,
C(ϵ, ρ, α, δ) := { u ∈ ]U_{0}[_{ϵ,α} | t_{ϵ,α}^{+}(u) ≤ 2M } ∩ cl_{ϵ} V_{ϵ,ρ}(δ, M).

Remark 5.5. If α ≤ β, α ≤ η, δ ≤ a and M ≥ b then clearly V_{ϵ,α}(δ, M) ⊂ N_{1}(α, δ, ϵ). Moreover, if a ≤ a_{0}, b ≥ b_{0} and η < ν then [cl V (a, b)]_{ϵ,η} ⊂ ]U_{0}[_{ϵ,ν} and whenever v lies in K(ϵ, η, a, b), the largest invariant set in [cl V (a, b)]_{ϵ,η}, then, by the definition of t_{ϵ,ν}^{+}, we have t_{ϵ,ν}^{+}(v) = ∞. In particular, K(ϵ, η, a, b) ∩ N_{2}(α, δ, ϵ) = ∅.

Proposition 5.6. Assume that M > b, M′ > b, a ≤ a_{0}, b ≥ b_{0} and η < ν < β.

(1) For all sufficiently small positive α, δ and ϵ: (a) (N_{1}(α, δ, ϵ), N_{2}(α, δ, ϵ)) is a pseudo-index pair in [cl V (a, b)]_{ϵ,η}; (b) the following inclusions are satisfied:

E(ϵ, α, δ) ⊂ Ĉ(ϵ, α, δ) ⊂ Ê(ϵ)

(2) For every α > 0 and all sufficiently small positive ρ, δ and ϵ

C(ϵ, ρ, α, δ) ⊂ E(ϵ, α, δ).

Proof. Proceeding exactly as in the proof of Lemma 12.5 in [39] we see that the sets N_{1}(α, δ, ϵ) and N_{2}(α, δ, ϵ) are closed in H^{1}(Ω) and [cl V (a, b)]_{ϵ,η}-positively invariant relative to π_{ϵ}.

Suppose that part 1 of the proposition does not hold. Then for some choice of the constants M, M′, β, ν, η, a and b as above there are sequences (δ_{n})_{n∈ℕ}, (α_{n})_{n∈ℕ} and (ϵ_{n})_{n∈ℕ} of positive numbers converging to zero and a sequence (u_{n})_{n∈ℕ} such that

(27)  u_{n} ∈ N_{1}(α_{n}, δ_{n}, ϵ_{n}) ∩ ∂_{ϵ_{n}} [cl V (a, b)]_{ϵ_{n},η} \ N_{2}(α_{n}, δ_{n}, ϵ_{n}) for all n ∈ ℕ

or

(28)  u_{n} ∈ Ĉ(ϵ_{n}, α_{n}, δ_{n}) \ Ê(ϵ_{n}) for all n ∈ ℕ

or

(29)  u_{n} ∈ E(ϵ_{n}, α_{n}, δ_{n}) \ Ĉ(ϵ_{n}, α_{n}, δ_{n}) for all n ∈ ℕ.

In the first two cases u_{n} ∈ N_{1}(α_{n}, δ_{n}, ϵ_{n}), so there exist a v_{n} ∈ V_{ϵ_{n},α_{n}}(δ_{n}, M) and a t_{n} ≥ 0 such that

v_{n}π_{ϵ_{n}} [0, t_{n}] ⊂ ]U_{0}[_{ϵ_{n},β}

and ũ_{n} := v_{n}π_{ϵ_{n}}t_{n} is such that |u_{n} − ũ_{n}|_{ϵ_{n}} < 2^{−n}. Since g^{−}(Q_{ϵ_{n}}v_{n}) < δ_{n} → 0, we may assume, taking subsequences if necessary, that |Q_{ϵ_{n}}v_{n} − v|_{H^{1}} → 0 for some v ∈ A_{π_{0}}^{−}(N_{0}). Since |(I − Q_{ϵ_{n}})v_{n}|_{ϵ_{n}} → 0, it follows that |v_{n} − v|_{ϵ_{n}} → 0.

We claim that there is a subsequence of (v_{n}π_{ϵ_{n}}t_{n})_{n∈ℕ}, denoted again by (v_{n}π_{ϵ_{n}}t_{n})_{n∈ℕ}, and there is a ṽ ∈ A_{π_{0}}^{−}(N_{0}) such that

|v_{n}π_{ϵ_{n}}t_{n} − ṽ|_{ϵ_{n}} → 0 as n → ∞.

First assume that the sequence (t_{n})_{n∈ℕ} is bounded. By taking subsequences, if necessary, we may assume that there exists a t ∈ [0, ∞[ such that t_{n} → t as n → ∞. It follows from Corollary 2.14 that

|v_{n}π_{ϵ_{n}}t_{n} − vπ_{0}t|_{ϵ_{n}} → 0 as n → ∞.

Since A_{π_{0}}^{−}(N_{0}) is an N_{0}-positively invariant set relative to π_{0}, it follows that

ṽ := vπ_{0}t ∈ A_{π_{0}}^{−}(N_{0}).

If (t_{n})_{n∈ℕ} is unbounded then we may assume that t_{n} → ∞ as n → ∞. Let σ_{n}: [−t_{n}, 0] → H^{1}(Ω) be a solution through v_{n}π_{ϵ_{n}}t_{n} relative to π_{ϵ_{n}} given by

σ_{n}(s) = v_{n}π_{ϵ_{n}}(t_{n} + s), for every s ∈ [−t_{n}, 0].

By Lemma 2.20 there exist a subsequence of (σ_{n})_{n∈ℕ}, denoted again by (σ_{n})_{n∈ℕ}, and a solution σ: ]−∞, 0] → H_{s}^{1}(Ω) of π_{0} lying in N_{0} such that for each s ∈ ]−∞, 0],

(30)  |σ_{n}(s) − σ(s)|_{ϵ_{n}} → 0 as n → ∞.

Let ṽ := σ(0) ∈ A_{π_{0}}^{−}(N_{0}). Letting s = 0 in (30), we obtain

|v_{n}π_{ϵ_{n}}t_{n} − ṽ|_{ϵ_{n}} → 0 as n → ∞.

This completes the proof of our claim.

It follows that |ũ_{n} − ṽ|_{ϵ_{n}} → 0 and so |u_{n} − ṽ|_{ϵ_{n}} → 0 as n → ∞. Thus

(31)  |(I − Q_{ϵ_{n}})u_{n}|_{ϵ_{n}} < η and |(I − Q_{ϵ_{n}})u_{n}|_{ϵ_{n}} < ν for all n large enough.

Moreover, since u_{n} ∈ [cl V (a, b)]_{ϵ_{n},η} we also have Q_{ϵ_{n}}u_{n} ∈ cl V (a, b) ⊂ U_{0}. Thus u_{n} ∈ ]U_{0}[_{ϵ_{n},ν} for all n large enough and so Lemma 5.3 and the continuity of of t^{+} imply that t_{ϵ_{n},ν}^{+}(u_{n}) → t^{+}(ṽ) and t^{+}(Q_{ϵ_{n}}u_{n}) → t^{+}(ṽ) as n → ∞.

Suppose that (27) holds. Then relations (27) and (31) imply that Q_{ϵ_{n}}u_{n} ∈ ∂ cl V (a, b) ⊂ U_{0}, for all n large enough and so ṽ ∈ ∂ cl V (a, b) ⊂ U_{0}. Since t_{ϵ_{n},ν}^{+}(u_{n}) > M′ for all n we also conclude that t^{+}(ṽ) ≥ M′ > b. Since g^{−}(ṽ) = 0 < a, we see that ṽ ∈ V (a, b). However V (a, b) ∩ ∂ cl V (a, b) = ∅. This contradiction proves part 1(a).

Suppose now that (28) holds. Then t_{ϵ_{n},ν}^{+}(u_{n}) ≤ 4M for all n and so t^{+}(ṽ) ≤ 4M < 5M. We thus conclude that t^{+}(Q_{ϵ_{n}}u_{n}) < 5M for all n large enough. All this clearly implies that u_{n} ∈ Ê(ϵ_{n}) for all n large enough, a contradiction proving the second inclusion in 1(b).

Finally assume that (29) holds. Therefore for all n ∈ ℕ

(32)  Q_{ϵ_{n}}u_{n} ∈ { u ∈ U_{0} | t^{+}(u) ≤ 3M } ∩ cl V (δ_{n}, M) and |(I − Q_{ϵ_{n}})u_{n}|_{ϵ_{n}} ≤ α_{n}.

Hence there exists a v_{n} ∈ V (δ_{n}, M) such that |Q_{ϵ_{n}}u_{n} − v_{n}|_{H^{1}} < 2^{−n}. Since g^{−}(v_{n}) → 0, we may again assume that there is a v ∈ A_{π_{0}}^{−}(N_{0}) such that |v_{n} − v|_{H^{1}} → 0 as n → ∞. Therefore

|Q_{ϵ_{n}}u_{n} − v|_{H^{1}} → 0 as n → ∞.

Together with (32) this shows that |u_{n} − v|_{ϵ_{n}} → 0 as n → ∞. For all n large enough we have g^{−}(v_{n}) < δ_{n} < a and t^{+}(v_{n}) > M > b so v_{n} ∈ V (a, b). This implies that v ∈ cl V (a, b) ⊂ U_{0}. Since t^{+}(Q_{ϵ_{n}}u_{n}) ≤ 3M, the continuity of t^{+} implies that t^{+}(v) ≤ 3M. Recall that u_{n} ∈ ]U_{0}[_{ϵ_{n},ν} for all n large enough. Lemma 5.3 now shows that t_{ϵ_{n},ν}^{+}(u_{n}) → t^{+}(v) as n → ∞. Hence we can assume that t_{ϵ_{n},ν}^{+}(u_{n}) ≤ 4M for all n large enough.

Since N_{1}(α_{n}, δ_{n}, ϵ_{n}) is closed, we obtain from Remark 5.5 and Lemma 2.19 that, for all n large enough,

u_{n} ∈ [cl V (δ_{n}, M)]_{ϵ_{n},α_{n}} = cl_{ϵ_{n}} ]V (δ_{n}, M)[_{ϵ_{n},α_{n}} = cl_{ϵ_{n}} V_{ϵ_{n},α_{n}}(δ_{n}, M) ⊂ N_{1}(α_{n}, δ_{n}, ϵ_{n}).

This implies that u_{n} ∈ Ĉ(ϵ_{n}, α_{n}, δ_{n}). This contradiction completes the proof of part 1 of the proposition.

If the second part of the proposition is not true then there exist numbers M > 0, α > 0 and sequences of positive numbers (δ_{n})_{n∈ℕ}, (ρ_{n})_{n∈ℕ} and (ϵ_{n})_{n∈ℕ}, converging to zero and a sequence (u_{n})_{n∈ℕ} such that

u_{n} ∈ C(ϵ_{n}, ρ_{n}, α, δ_{n}) and u_{n} ∉ E(ϵ_{n}, α, δ_{n}).

Therefore for every n ∈ ℕ we have u_{n} ∈ ]U_{0}[_{ϵ_{n},α}, t_{ϵ_{n},α}^{+}(u_{n}) ≤ 2M and there exists a v_{n} ∈ V_{ϵ_{n},ρ_{n}}(δ_{n}, M) such that |u_{n} − v_{n}|_{ϵ_{n}} < 2^{−n}. Therefore Q_{ϵ_{n}}v_{n} ∈ V (δ_{n}, M). Hence for every n ∈ ℕ,

g^{−}(Q_{ϵ_{n}}v_{n}) < δ_{n} and t^{+}(Q_{ϵ_{n}}v_{n}) > M

and so there exists a subsequence of (v_{n})_{n∈ℕ}, denoted again by (v_{n})_{n∈ℕ}, and a v ∈ A_{π_{0}}^{−}(N_{0}) such that |Q_{ϵ_{n}}v_{n} − v|_{H^{1}} → 0 as n → ∞.

Since |(I − Q_{ϵ_{n}})v_{n}|_{ϵ_{n}} < ρ_{n}, we conclude that |v_{n} − v|_{ϵ_{n}} → 0 and so |u_{n} − v|_{ϵ_{n}} → 0 as n → ∞. For all n large enough we have Q_{ϵ_{n}}v_{n} ∈ V (a, b). Therefore v ∈ cl V (a, b) ⊂ U_{0}. Recall that u_{n} ∈ ]U_{0}[_{ϵ_{n},α}. Hence Lemma 5.3 implies that t_{ϵ_{n},α}^{+}(u_{n}) → t^{+}(v) as n → ∞. Therefore t^{+}(v) ≤ 2M. This inequality and the continuity of t^{+} imply that t^{+}(Q_{ϵ_{n}}u_{n}) ≤ 3M for all n large enough. Since u_{n} ∈ cl_{ϵ_{n}} V_{ϵ_{n},ρ_{n}}(δ_{n}, M) = [cl V (δ_{n}, M)]_{ϵ_{n},ρ_{n}} it follows that for all n large enough,

Q_{ϵ_{n}}u_{n} ∈ cl V (δ_{n}, M) and |(I − Q_{ϵ_{n}})u_{n}|_{ϵ_{n}} < α.

Therefore we have proven that u_{n} ∈ E(ϵ_{n}, α, δ_{n}) for all n large enough, a contradiction. The proposition is proved. □

Lemma 5.7. Let N_{1} and N_{2} be closed subsets in H_{s}^{1}(Ω) such that N_{2} ⊂ N_{1}. Then for every ϵ > 0 and η > 0 the pointed spaces N_{1}/N_{2} and [N_{1}]_{ϵ,η} / [N_{2}]_{ϵ,η} have the same homotopy type.

Proof. Let T_{ϵ}: [N_{1}]_{ϵ,η} / [N_{2}]_{ϵ,η} → N_{1}/N_{2} be the map induced by the projection Q_{ϵ}. Obviously T_{ϵ} ∘ ι = I_{N_{1}/N_{2}}, where ι: N_{1}/N_{2} → [N_{1}]_{ϵ,η} / [N_{2}]_{ϵ,η} is the inclusion induced map. Let H: [N_{1}]_{ϵ,η} / [N_{2}]_{ϵ,η} × [0, 1] → [N_{1}]_{ϵ,η} / [N_{2}]_{ϵ,η} be the map induced by the function

K: [N_{1}]_{ϵ,η} × [0, 1] → [N_{1}]_{ϵ,η} , (v, t) ↦ Q_{ϵ}v + (I − Q_{ϵ})(1 − t)v.

It is clear that H is a homotopy between I_{[N_{1}]_{ϵ,η}/[N_{2}]_{ϵ,η}} and ι ∘ T_{ϵ}. This concludes the proof. □

We can now complete the proof of Theorem 3.5: Choose positive numbers β, ν, η and M with M > b_{0} and η < ν < β. Now choose positive numbers α′ < η, δ′ < a_{0} and ϵ′ such that for all positive α ≤ α′, δ ≤ δ′ and ϵ ≤ ϵ′ all assertions of part 1 of Proposition 5.6 hold, where we fix a := a_{0}, b := b_{0} and M′ := 4M. Now choose positive numbers ρ′ < α′, δ′′ ≤ δ′ and ϵ′′ ≤ ϵ′ such that for all positive ρ ≤ ρ′, δ ≤ δ′′ and ϵ ≤ ϵ′′ part 2 of Proposition 5.6 holds, where we fix α := α′.

Notice also that 2M > b_{0}, ρ′ < α′ and α′ < β (α′ < η < ν < β). Therefore we are able to apply Proposition 5.6 with M and M′ replaced by 2M, η := ρ′, ν := α′, a := δ′′, b := M. Thus we obtain positive numbers α′′′ ≤ ρ′, δ′′′ < δ′′ and ϵ′′′ ≤ ϵ′′ such that for all positive α ≤ α′′′, δ ≤ δ′′′ and ϵ ≤ ϵ′′′ the pair (Ñ_{1}(α, δ, ϵ), Ñ_{2}(α, δ, ϵ)) is a pseudo-index pair in [cl V (δ′′, M)]_{ϵ,ρ′}.

Here,

Ñ_{1}(α, δ, ϵ) = [cl V (δ′′, M)]_{ϵ,ρ′} ∩ cl_{ϵ}{ v | ∃ u ∈ V_{ϵ,α}(δ, 2M) and t ≥ 0 such
that uπ_{ϵ}[0, t] ⊂ ]U_{0}[_{ϵ,β} and uπ_{ϵ}t = v }

and

Ñ_{2}(α, δ, ϵ) = Ñ_{1}(α, δ, ϵ) ∩ { u ∈ ]U_{0}[_{ϵ,α′} | t_{ϵ,α′}^{+}(u) ≤ 2M }.

Fix α := α′′′, δ := δ′′′ and write ρ := ρ′. We now conclude that for ϵ ∈ ]0, ϵ′′′[

A_{1} := Ñ_{1}(α, δ, ϵ) ⊂ [cl V (δ′′, M)]_{ϵ,ρ} ⊂ A_{2} := [cl V (δ′′, M)]_{ϵ,α′}
⊂ A_{3} := N_{1}(α′, δ′′, ϵ) ⊂ A_{4} := [cl V (a_{0}, b_{0})]_{ϵ,η} .

B_{1} := Ñ_{2}(α, δ, ϵ) ⊂ [cl V (δ′′, M)]_{ϵ,ρ} ∩ { u ∈ ]U_{0}[_{ϵ,α′} | t_{ϵ,α′}^{+}(u) ≤ 2M }
= C(ϵ, ρ, α′, δ′′) ⊂ B_{2} := E(ϵ, α′, δ′′)
⊂ B_{3} := Ĉ(ϵ, α′, δ′′) = N_{2}(α′, δ′′, ϵ)
⊂ B_{4} := Ê(ϵ).

Notice that for each i = 1, 2, 3, 4, B_{i} ⊂ A_{i}. Therefore we can consider the pointed spaces A_{i}/B_{i}. The two sequences of inclusions described above induce continuous maps

Γ_{1} : A_{1}/B_{1} → A_{2}/B_{2}, Γ_{2} : A_{2}/B_{2} → A_{3}/B_{3}, and Γ_{3} : A_{3}/B_{3} → A_{4}/B_{4}.

Using Lemma 5.2 let us now choose a positive number ϵ^{c} ≤ ϵ′′′ such that

K(ϵ, η, a_{0}, b_{0}) ⊂ V_{ϵ,α′}(δ′′, M) ∩ V_{ϵ,α}(δ, 2M)

for ϵ ∈ ]0, ϵ^{c}]. Then Remark 5.5 implies that (A_{1}, B_{1}) and (A_{3}, B_{3}) are index pairs for K(ϵ, η, a_{0}, b_{0}) and it follows from Theorem 9.4 in [39] that Γ_{2} ∘ Γ_{1} is an isomorphism in the homotopy category of the pointed spaces. The proof of Theorem 2.3 in [39] implies that

(cl V (δ′′, M), cl V (δ′′, M) ∩ { u ∈ U_{0} | t^{+}(u) ≤ 3M })

and

(cl V (a_{0}, b_{0}), cl V (a_{0}, b_{0}) ∩ { u ∈ U_{0} | t^{+}(u) ≤ 5M })

are index pairs for K_{0}. Now, by Lemma 5.7, we conclude that there exist maps

Γ_{4} : cl V (δ′′, M)/(cl V (δ′′, M) ∩ { u ∈ U_{0} | t^{+}(u) ≤ 3M }) → A_{2}/B_{2}

and

Γ_{5} : cl V (a_{0}, b_{0})/(cl V (a_{0}, b_{0}) ∩ { u ∈ U_{0} | t^{+}(u) ≤ 5M }) → A_{4}/B_{4}

which are isomorphism in the homotopy category of pointed spaces. Moreover, Theorem 9.4 in [39] allows us to conclude that Γ_{5}^{−1} ∘ Γ_{3} ∘ Γ_{2} ∘ Γ_{4} is also an isomorphism in the same category. Consequently Γ_{3} ∘ Γ_{2} is an isomorphism. Now Lemma 12.4 in [39] implies that Γ_{1}, Γ_{2} and Γ_{3} are all isomorphisms in the homotopy category of pointed spaces. We conclude that (5) also holds for the case K_{0} ≠ ∅.

The proof of Theorem 3.5 is complete. □

Acknowledgment

A part of this work was completed while the first author was visiting the Institute of Mathematics of the Jagiellonian University. We thank Professor Roman Srzednicki for his hospitality and many delightful discussions.

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Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13.560-970 São Carlos SP, Brazil

E-mail address: mdccarbi@icmc.sc.usp.br

Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 18055 Rostock, Germany

E-mail address: krzysztof.rybakowski@mathematik.uni-rostock.de