Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations

In this article it is proved that the dynamical properties of a broad class of semilinear parabolic problems are sensitive to arbitrarily small but smooth perturbations of the nonlinear term, when the spatial dimension is either equal to one or two. This topological instability is shown to result from a local deformation of the global bifurcation diagram associated with the corresponding elliptic problems. Such a deformation is shown to systematically occur via the creation of either a multiple-point or a new fold-point on this diagram when an appropriate small perturbation is applied to the nonlinear term. More precisely, it is shown that for a broad class of nonlinear elliptic problems, one can always find an arbitrary small perturbation of the nonlinear term, that generates a local S on the bifurcation diagram whereas the latter is e.g. monotone when no perturbation is applied; substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem. The rigorous proof of the latter instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for the applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily $C^1$ nonlinear terms, that prevent in particular the use of linearization techniques, and for which the family of semigroups may exhibit non-dissipative properties.


Introduction
Bifurcations of local attractors arising in semilinear parabolic problems and related bifurcations in semilinear elliptic problems, have been thoroughly studied in the literature since the works of [Rab73,CR73,MM76,Hen81,Sat73,Sat80], and a large portion of the subsequent works has been devoted to the study of the qualitative changes occurring within a fixed family when a bifurcation parameter is varied; see e.g. [MW05,MW13,HI11,Kie12] and references therein. For semilinear elliptic problems, the study of qualitative changes of the global shape of the bifurcation diagram with respect to variations in other degrees of freedom such as the "shape" of the nonlinearity, has also attracted a certain attention, but such questions have been mainly addressed in the context of two-parameter families of elliptic problems [BCT88a, BCT88b, BRR80, BF82, DPF90, CHMP75, Du00, DL01, KK73, KL99, Lio82, MS80, She80, SW82]; see however e.g. [Dan88,Dan08,Hen05,NS94] for the related effects of the variation of the domain.
A rigorous framework is introduced below to give a precise sense, from a topological and functional analysis points of view, to the notion of robustness of the family of phase portraits associated with a given family of semilinear parabolic problems (that do not necessarily exhibit a global attractor) with respect to general perturbations of this family; see Definition 2.5 below. Within this framework, we are able to prove that the dynamical properties of a broad class of semilinear parabolic problems can turn out to be sensitive to arbitrarily small perturbations of the nonlinear term, when the spatial dimension d is equal to one or two. This is the content of Theorem 3.2 proved below, which constitutes the main result of this article. The proof of this theorem is articulated around a combination of techniques regarding the generation of discontinuities in the minimal branch borrowed from [CEP02], the growth property of the branch of minimal solutions as recalled in Proposition 3.1 below, and a general continuation result from the Leray-Schauder degree theory, formulated as Theorem A.1 below and proved in Appendix A for the sake of completeness. The latter provides conditions from the Leray-Schauder degree theory, under which the existence of an unbounded continuum of steady states for the corresponding family of semilinear elliptic problems, 1 can be ensured.
It is worth mentioning that the proof of Theorem 3.2 provides furthermore the mechanism at the origin of the aforementioned topological instability. The latter results from a local deformation of the λ-bifurcation diagram (associated with −∆u = λg(u), u| ∂Ω = 0) by the creation of a multiple-point or a new fold-point on it when an appropriate small perturbation is applied. This is accomplished under assumptions on g that prevents the use of linearization techniques.

A framework for the topological robustness of families of semilinear parabolic problems
In Section 2.1 that follows, the perturbed Gelfand problem serves as an illustration of perturbed bifurcation problems arising in families of semilinear elliptic equations regarding the dependence of the global bifurcation diagram to perturbations of the nonlinear term [KK73]. Such a dependence problem is of fundamental importance to understand, for instance, how the multiplicity of solutions of such equations varies as the nonlinearity is subject to small disturbances, or is modified due to model imperfections [BF82,GS79,KK73].
We will see in Section 2.3 how such problems can be naturally related to the study of topological robustness of the corresponding families of semilinear parabolic equations. The latter notion of topological robustness is related to the more standard notion of structural stability encountered for semilinear parabolic problems as recalled in Section 2.2. However as we will see, the former, based on the notion of topological equivalence between parameterized families of semigroups such as introduced in Definition 2.2 (see Section 2.3), adopts a more global point of view than the latter. It allows us, in particular, to deal with semilinear parabolic problems which are not necessarily dissipative 2 such as encountered in combustion theory [BE89,QS07] or in plasma physics [BB80,Tem75]; see [Fil05]. More specifically, the framework introduced in Section 2.3 below allows us to deal with semigroups for which some trajectories undergo finite-time blow up, or do not exhibit a compact absorbing set nor even a bounded absorbing set for all solutions 3 .
2.1. The perturbed Gelfand problem as a motivation. Given a smooth bounded domain Ω ⊂ R d , the perturbed Gelfand problem, consists of solving the following nonlinear eigenvalue problem (2.1) −∆u = λ exp(u/(1 + εu)), in Ω, u = 0, on ∂Ω, of unknown λ > 0, and u in some functional space. We refer to [BE89, Cha57, F-K69, Gel63, JL73, Tai95, Tai98, QS07] for more details regarding the physical contexts where such a problem arises. We first recall how general features regarding the structure of the solution set of (2.1) (parameterized by λ), can be easily derived by application of Theorem A.1, and the theory of semilinear elliptic equations [Caz06]; and we point out some open questions related to the exact shape of this solution set when the nonlinearity is varied by changing ε. Such open questions although not directly addressed in this article, motivate, in part, the results obtained in Theorem 3.2 proved below.
Let α ∈ (0, 1) and consider the Hölder spaces V = C 2,α (Ω) and E = C 0,α (Ω). It is well known (see e.g. [GT98,Chapter 6]) that given f ∈ E and λ ≥ 0, there exists a unique u ∈ V of the following Poisson problem, This allows us to define a solution map S : E → V given by S(f ) = u, where u ∈ V is the unique solution to (2.2). By composing S with the compact embedding i : , and consider the equation, The mapping G is a completely continuous perturbation of the identity and solutions of the equation G(λ, u) = 0 correspond to solutions of (2.1). For any neighborhood U ⊂ X of 0 E , the function u = 0 is the unique solution to (2.3) with λ = 0. Moreover, In the sense that they do not exhibit a bounded absorbing set [Tem97]. 3 The solutions u(t) that are well defined in some Hilbert space for all t > 0 but are not bounded in time, are sometimes referred as "grow-up" solutions; see [Ben10]. and therefore from Theorem A.1 (see Appendix A), there exists a global curve of nontrivial solutions which emanates from (0, 0 E ). Here deg(G(0, ·), U , 0 E ) stands for the classical Leray-Schauder degree of G(0, ·) with respect to U and 0 E ; see e.g. [Dei85,Nir01]. From the maximum principle these solutions are positive in Ω. Since u = 0 is the unique solution for λ = 0 (up to a multiplicative constant), the corresponding continuum of solutions is unbounded in (0, ∞) × E, from Theorem A.1. From [Lio82, Theorem 2.3], it is known that there exists a minimal positive solution of (2.1) for all λ > 0; cf. also Proposition 3.1 below. Furthermore, there exists λ ♯ such that for every λ ≥ λ ♯ , there exists only one positive solution of (2.1), u λ (cf. [CS84]), and that the branch λ → u λ is increasing; see [Ama76] and see again Proposition 3.1.
For λ small enough, i.e. for 0 < λ ≤ λ ♯ for some λ ♯ > 0, it can be proved that the same conclusions about the uniqueness of positive solutions, as well as about the monotony of the corresponding branch, hold. The problem is then to know what happens for λ ∈ (λ ♯ , λ ♯ ). Theorem A.1 may give some clues in that respect. For instance, since Theorem A.1 ensures that the branch of solutions is a continuum, in case where the existence of three solutions for some λ ′ ∈ (λ ♯ , λ ♯ ) is guaranteed, then such a continuum is necessarily S-like shaped, with multiple turning points (not necessarily reduced to two turning points).
The exact shape of this continuum, for general domains, is however a challenging problem. For instance it is known that for ε ≥ 1/4, the problem (2.1) has a unique positive solution for every λ > 0 whatever the spatial dimension d is, the branch of solutions being a monotone function of λ; see e.g. [BIS81,CS84]. However, if d = 2 and Ω is the unit open ball of R 2 , then it has been proved in [DL01] that there exists ε * > 0 such that for 0 < ε < ε * this continuum is exactly S-shaped (with exactly two turning points) when represented in a "(λ, · ∞ )-plane" classically used for representing bifurcation diagrams such as arising in elliptic problems for domains with radial symmetries [JL73]. The global bifurcation curve can become however more complicated than S-shaped, when Ω is the unit ball in higher dimension; see [Du00] for 3 ≤ d ≤ 9.
In the case d = 1, a lower bound of the critical value ε * > 0, for which for all 0 < ε < ε * the continuum is S-shaped, has been derived in [KL99]. It ensures in particular that ε * ≥ 1

4.35
when Ω = (−1, 1), which gives rather a sharp bound of ε * in that case, since ε * ≤ 1 4 from the general results of [BIS81,CS84]. Numerical methods with guaranteed accuracy to enclose a double turning point [Min04] strongly suggest that this theoretical bound can be further improved. Based on such numerical methods and the aforementioned theoretical results, it can be reasonably conjectured that in dimension d = 1 (Ω = (−1, 1)) for ε > 1/4 the λ-bifurcation diagram 4 does not present any turning point (monotone branch), whereas once ε < 1/4, an S-shaped bifurcation occurs. We observe here that a continuous change in the parameter ε can lead to a qualitative change of its λ-bifurcation diagram on its whole: from a monotone curve to an S-shaped curve as ε crosses 1/4 from above. It is thus reasonable to conjecture that ε * = 1/4, for d = 1.
From the numerical results of [Min04], it can be inferred that ε * ∈ (0.238, 0.2396] if d = 3 and Ω is the unit ball; emphasizing the dependence of such a critical value on the dimension of the physical space. Remark 2.1. It should be noticed that the result of [Du00] for two-dimensional balls, combined with a domain perturbation technique due to [Dan88], implies that, even in dimension 2, if Ω is the union of several balls touched slightly, then the number of positive solutions of (2.1) may be greater than three for some values of λ. This suggests that the λ-bifurcation diagram is not necessarily S-shaped, even in dimension 2.

2.2.
Structural stability for dissipative semilinear parabolic problems. The qualitative changes of the λ-bifurcation diagram recalled above for the perturbed Gelfand problem [Min04], when ε is varying, is reminiscent with the so-called cusp bifurcation observed in two-parameter families of autonomous ordinary differential equations (ODEs) [Kuz04]. Indeed, if we consider the paradigmatic example,ẋ = β 1 + β 2 x − x 3 , where x ∈ R, β = (β 1 , β 2 ) ∈ R 2 , we can easily exhibit two bifurcation curves in the (β 1 , β 2 )-plane. These bifurcation curves are given by 2 , β 2 > 0}, on which saddle-node bifurcation occur (i.e. collision and disappearance of two equilibria [Kuz04]). This two curves divide the parameter plane into two regions: inside the "dead-end" formed by γ + and γ − , there are three steady states, two stable and one unstable; and outside this corner, there is a single steady state, which is stable. A crossing of the cusp point β = (0, 0) from outside the "dead-end," leads to an unfolding of singularities [Arn81, Arn83, CT97, GS85] which consists more exactly to an unfolding of three steady states from a single stable equilibrium; see also [Kuz04].
The qualitative changes described at the end of the previous section may be therefore interpreted in that terms; see also [MN07, Fig. 1]. Singularity theory is a natural framework to study the effects on the bifurcation diagram of small perturbations or imperfections to a given (static) model [GS79,GS85]. In that spirit, geometrical connections between a double turning point and a cusp point have been discussed for certain nonlinear elliptic problems in [BCT88a, BF82, MS80, SW82], but a general treatment of the effects of arbitrary perturbations on bifurcation diagrams arising for such problems has not been fully achieved, especially when the perturbations are not necessarily smooth; see however [Dan08,Hen05] for related problems.
In that respect, it is tempting to describe the aforementioned qualitative changes in terms of structural instability such as defined in classical dynamical systems theory [AM87, Arn83,Sma67]. Nevertheless, as we will see in Section 2.3, such topological ideas have to be recast into a formalism which takes into account the functional settings for which the parabolic and corresponding elliptic problems are considered; see Definitions 2.1, 2.2 and 2.5 below. This formalism will be particularly suitable for problems such as arising in combustion theory for which the associated semigroups are not necessarily dissipative. To better appreciate this distinction with the standard theory, we recall briefly below the notion of structural stability such as encountered for dissipative infinite-dimensional systems.
The concept of structural stability has been originally introduced for finite-dimensional C 1vector fields in [AP37]. A system is said to be structurally stable if roughly speaking a small, sufficiently smooth 5 perturbation of this system preserves its dynamics up to a homeomorphism, i.e. up to a bijective continuous change of variables (with continuous inverse) that transforms the phase portrait of this system into that of the perturbed system; see e.g. [AM87, Arn83, KH97, New11, Sma67]. The existence of such an homeomorphism is related to the study of conjugacy or topological equivalence problems; see e.g. [AM87, Arn83, CGVR06, CR13, KH97, Sma67] and references theirein.
Structural stability has also been investigated for various types of infinite dimensional dynamical systems, mainly dissipative. As a rule for such dynamical systems, one investigates structural stability of the semiflow restricted to a compact invariant set, usually the global attractor, rather than the flow in the original state space [HMO02, Definition 1.0.1]; an exception can be found in e.g. [Lu] where the author considered the semiflow in a neighborhood of the global attractor. In the context of reaction-diffusion problems, the problem of structural stability concerns, which is assumed to generate a semigroup {S(t)} t≥0 which possesses a global attractor A g in some Banach space X [BP97, HMO02,Lu]. It may be formulated as the existence problem of an homeomorphism H : A g → A g for arbitrarily small perturbations g of g in some topology T of C 1 (R, R), that aims to satisfy the following properties where { S(t)} t∈≥0 denotes the semigroup generated by u t − ∆u = g(u), u| ∂Ω = 0. The topology T may be chosen as the compact-open topology or the finer topology of Whitney; see [Hir76] for general definitions of these topologies, and [BP97] for questions regarding the genericity of structurally stable reaction-diffusion problems of type (3.4), making use of the Whitney topology. Note that in (2.5b) 6 , the restriction of the dynamics to the global attractor, allows us to consider backward trajectories onto the attractor giving rise to flows onto the attractor; see e.g. [Rob01].
A necessary condition in order that a parabolic equation, generating a semigroup, possesses a global attractor in a Banach X, is to satisfy a dissipation property, i.e. to verify the existence of an absorbing ball in X for this semigroup (see e.g. [MWZ02, Theorem 3.8]), which in particular prevents any blow-up in finite or infinite time 7 (at least in X); see [Hal88,Rob01,SY02,Tem97] for classical conditions on g ensuring the existence of such dissipative semigroups in cases where X is a Hilbert space. In case of structural stability, it is worthwhile to note that the set of In specific applications, families of semigroups S = {S λ } λ∈Λ depending upon a parameter λ in some metric space Λ, and possessing a global attractor A λ for each λ ∈ Λ, may arise. In such a context, the notion of A-stability has been introduced [HMO02, Defnition 1.0.2] to study in particular the loss of structural stability within the family when the control parameter λ is varied, i.e. to study the occurrence of S λc at some critical value λ c such that for any neighborhood U of λ c , there exists S µ ∈ S which is non equivalent in the sense of (2.5) to S λc for some µ ∈ U . The loss of A-stability refers therefore to a notion of bifurcation of global attractors 8 . However, the underlying assumption implying that the related semigroups generated by a family of semilinear parabolic problems are dissipative may be viewed as too restrictive, since in many situations where such families arise (such as in combustion theory) blow-up in finite or infinite time may occur for certain trajectories; see [BE89, Ben10, CH98, Fil05, QS07].
2.3. Topological robustness and topological instability of families of semilinear parabolic problems. To deal with the problem of topological equivalence between families of semigroups which are not necessarily dissipative, we start by introducing several intermediate concepts that we illustrate on some examples borrowed from the literature. Let us first consider a family F f := {f λ } λ∈Λ of functions I → R, where Λ is some metric space, and I is some unbounded interval of R, along with the associated family of semilinear parabolic problems, in Ω, u = 0, on ∂Ω, 6 Note that (2.5b) may be substituted by the more general condition requiring that for all t ∈ R, and for all φ ∈ Ag, H(S(t)φ) = S(γ(t, φ))H(φ), with γ : R × Ag → R an increasing and continuous function of the first variable. Although this condition is often encountered in the literature, its use is not particularly required when one deals with semilinear parabolic problems as in the present article; see Remark 2.4 below. 7 Note that some authors, e.g. [Hal88], have referred to dynamical systems with this property as having bounded dissipation. 8 See [MW05] for the notion of bifurcation of local attractors.
where Ω is an open bounded subset of R d , with additional regularity assumptions on its boundary and f λ when needed. In general, such problems may generate a family of semigroups acting on a phase space X which does not necessarily agree with the functional space Y on which the solutions of are considered. Such a feature leads us to naturally introduce the following Definition 2.1. Let Λ be some metric space. Let X and Y be two Banach spaces and Ω an open bounded subset of R d , such that −∆u = f λ (u), u| ∂Ω = 0 makes sense in Y . A family F f := {f λ } λ∈Λ * will be said to be (X; Y )-admissible relatively to Λ * ⊂ Λ and Ω, if there exists a subset Λ * ⊂ Λ, such that for all λ ∈ Λ * the following properties are satisfied: Remark 2.2. When the domain Ω is fixed, we will simply say that a family functions is admissible without referring to Ω. We will make also often the abuse of language that consists of saying that given an admissible family of functions, the corresponding family of elliptic problems is also admissible.
We first provide an example of a family of superlinear elliptic problems which is not (C 1 (Ω); H 1 0 (Ω))admissible, but weakly (C 1 (Ω); H 1 0 (Ω))-admissible. Example 2.1. It may happen that E f λ = E f λ for some λ ∈ Λ * . The Gelfand problem [Gel63,Fuj69], , is an illustrative example of such a distinction that may arise between the set of equilibrium points and the set of steady states, depending on the functional setting adopted. It is indeed well known that for Y = H 1 0 (B 1 (0)) there exists λ * > 0 such that for λ > λ * there is no solution to (2.7), even in a very weak sense [BCMR96], whereas for λ ∈ [0, λ * ] there exists at least a solution (in Y ) so that E f = ∅; see [BV97] and Proposition 3.1 below. Let us take Λ * to be [0, λ * ]. Let us choose X to be the subspace of all radial functions in a fractional power space D(A β p ) associated with A p , where A p stands for the Laplace operator considered as an unbounded operator on L p (B 1 (0)) (under Dirichlet conditons), with domain SY02]. For p > d and 1 > β > (d + p)/(2p) we know that X is compactly embedded in C 1 (B 1 (0)); see e.g. [SY02,Lemma 37.7]. Then for all λ ∈ [0, λ * ] it can be proved that (P f λ ) is well posed in X (with f λ (x) = λ exp(x)) for such a choice of p and β, and by choosing where u λ (t; u 0 ) denotes the solution of (P f λ ) emanating from u 0 at t = 0.
Nevertheless this family is weakly (C 1 (B 1 (0)); H 1 0 (B 1 (0)))-admissible relatively to [0, λ * ]. This results from the fact that the singular steady state u λ ♯ can be approximated by a sequence of equilibria in X for the relevant topology [BV97,JL73].
Definition 2.2. Let Λ be a metric space and I be an unbounded interval of R. Let N (I, R) be some set of functions from I to R. Consider two families {f λ } λ∈Λ * and { f λ } λ∈ Λ * of N (I, R), which are (X; Y )-admissible relatively to Λ * and Λ * respectively, and for each λ ∈ Λ * and λ ∈ Λ * denote by {S λ (t)} t≥0 and { S λ (t)} t≥0 , the semigroups acting on D(f λ ) and D( f λ ) respectively. We denote by S f and by S f , the respective family of such semigroups.
Then S f and S f are called topologically equivalent if there exists an homeomorphism such that H(λ, u) = (p(λ), H λ (u)) where p and H λ satisfy the following two conditions: (i) p is an homeomorphism from Λ * to Λ * , (ii) for all λ ∈ Λ * , H λ is an homeomorphism from D(f λ ) to D( f p(λ) ), such that, In case of such equivalence, the families of problems {(P f λ )} λ∈Λ * and {(P f λ )} λ∈ Λ * will be also referred as topologically equivalent.
Remark 2.4. Note that the relation of topological equivalence given by (2.13) may be relaxed as follows, is an increasing and continuous function of the first variable. This second approach is an extension of the concept of topological orbital equivalence, classically encountered in finite-dimensional dynamical systems theory, which allows, in particular, for systems presenting periodic orbits of different periods, to be equivalent; avoiding by this way the so-called problem of modulii; see [KH97].
Remark 2.5. The terminologies of Definition 2.3 and Definition 2.4 will be also adopted for one-parameter families of steady states in Y as introduced in Definition 2.1 (ii).
Based on these definitions, simple criteria of non topological equivalence between two families of semigroups are then given by the following proposition whose obvious proof is left to the reader.
Proposition 2.2. Assume Λ is some unbounded interval of R. Let S f and S f be two families of semigroups as defined in Definition 2.2. Let E f and E f be the corresponding families of equilibria. Then S f and S f are not topologically equivalent if one of the following conditions are fulfilled.
(i) E f is constituted by a single unbounded continuum in Λ × X, and E f is the union of at least two disjoint unbounded continua in Λ × X. (ii) E f and E f are each constituted by a single continuum, and the set of fold-points of E f and E f are not in one-to-one correspondence. (iii) E f and E f are each constituted by a single continuum, and there exists an integer n ≥ 3 such that the set of multiple-points with n branches of E f and E f are not in one-to-one correspondence.
We can now introduce the following notion of topological robustness and related notion of topological instability (to small perturbations) of families of semigroups generated by one-parameter families of semilinear parabolic problems.
Definition 2.5. Let Λ be a metric space and I be an unbounded interval of R. Let N (I, R) be a set of functions from the interval I to R. Consider a family F f = {f λ } λ∈Λ * of N (I, R) which is (X; Y )-admissible relatively to Λ * ⊂ Λ (cf. Defintion 2.1), and for each λ ∈ Λ * we denote by {S λ (t)} t≥0 the semigroup acting on D(f λ ) ; and we denote by S f the corresponding family of semigroups. Consider T to be a topology on N (I, R).
We will say that F f is ((X; Y )-admissible) T-stable with respect to perturbations in the Ttopology, if for each λ ∈ Λ * , there exists in this topology a neighborhood U * λ of f λ such that for any neighborhood U λ ⊂ U * λ , (i) ∃ Λ * ⊂ Λ such that, In case where F f is T-stable, we will say furthermore that S f is T-topologically robust in X, with respect to perturbations in the T-topology, if (ii) for any family F f = { f λ } λ∈ Λ * such that for all λ ∈ Λ * , f λ ∈ U λ ; S f and S f are topologically equivalent in the sense of Definition 2.2.
Given a T-stable family F f , if for any U λ neighborhood of f λ such as provided by (i), there exists f λ ∈ U λ such that S f and S f are not topologically equivalent, then S f will be called topologically unstable with respect to small perturbations in T.

Topologically unstable families of semilinear parabolic problems: Main result
Similar conjectures such as recalled in Section 2.1 regarding the qualitative change of a"cusp type" for the λ-bifurcation diagram have been pointed out in other semilinear elliptic problems (see e.g. [BCT88b,ZWS07]), but general conditions on the nonlinear term under which a given family of semilinear parabolic problems is topologically unstable with respect to small perturbations 9 remain to be clarified; see however [Dan08].
It is the purpose of Theorem 3.2 below to identify such conditions. As already mentioned its proof relies on a combination of Theorem A.1 proved in Appendix A of this article, the growth property of the branch of minimal solutions as recalled in Proposition 3.1 below, and methods of generation of a discontinuity in the minimal branch borrowed from the proof of [CEP02, Theorem 1.2]. Theorem 3.2 allows us to conclude to the existence of a broad class of topologically unstable families of semilinear parabolic problems, not necessarily related to a specific type of bifurcation, and for situations where a global attractor is not guaranteed to exist. Figure 1 below depicts some typical λ-bifurcation diagrams of the corresponding families of semilinear elliptic problems concerned with Theorem 3.2.
It is worth mentioning that the proof of Theorem 3.2 provides furthermore the mechanism at the origin of the aforementioned topological instability. The latter results from a local deformation of the λ-bifurcation diagram (associated with −∆u = λg(u), u| ∂Ω = 0) by the creation of a multiple-point or a new fold-point on it when an appropriate small perturbation is applied. This is accomplished under assumptions on g that prevents the use of linearization techniques; see Remark 3.2 below.
To prepare the proof of Theorem 3.2, we first recall classical results about the solution set of, (i) For every λ ∈ [0, λ * ), there exists a unique minimal solution u λ ≥ 0, u λ ∈ H 1 0 (Ω)∩L ∞ (Ω) of (3.1). The solution u λ is minimal in the sense that any supersolution v ≥ 0 of (3.1) Remark 3.1. [Caz06, Theorem 3.4.1] is in fact proved for functions g which are C 1 . From this proof, it is not difficult to see however that the conclusions of this proposition still hold by assuming f to be locally Lipschitz instead of C 1 .
We are now in position to prove our main theorem.
is well defined for some α ∈ (0, 1) and is constituted by a continuum without multiple-points on it.
Assume furthermore that the set of fold-points of E g given by satisfies one of the following conditions (iii) F = ∅ and E g is constituted only by its minimal branch.
Assume finally that the family of functions F g := {λg} λ∈[0,λ * ) is (X; C 2,α (Ω))-admissible relatively to [0, λ * ) for some Banach space X, and T-stable in the sense of Definition 2.5, with T denoting the C 0 -compact-open topology on C(R + , R + * ). Let S g be the corresponding family of semigroups {S λ (t)} λ∈[0,λ * ) associated with Then S g is topologically unstable with respect to small perturbations 10 in T. Furthermore, such a perturbation g can be chosen such that g − g is with compact support, and g is C 1 , and increasing such that E g contains a new fold-point or a new multiple-point compared with E g , for either λ ∈ (0, λ m ), or λ ∈ (0, λ ♯ ), or λ ∈ (0, λ * ), depending on whether case (i), case (ii), or case (iii), is concerned.
Case (i). Let F = {(λ, u λ ) : (λ, u λ ) is a fold-point of E g } and assume first that F = ∅ and satisfies the condition (i) as stated in the theorem. Let us choose ε > 0 and λ ′ such that, 0 < λ ′ − 2ε ≤ λ m := min{λ : (λ, u λ ) ∈ F} and M + ε < u λm ∞ . For such λ ′ and ε, and the corresponding perturbation g of g described above; by applying Theorem A.1 and by adopting a similar reasoning as used in Section 2.1 for the perturbed Gelfand problem, it can be proved that there exists an unbounded continuum in Λ * × V of nontrivial solutions of, which emanates from (0, 0 V ), with here V = C 2,α (Ω). Let λ ∈ [λ ′ , λ ′ + ε] be the parameter value at which the discontinuity of λ → u λ occurs. Let C be the unbounded continuum of E g which contains (0, 0 V ). By construction of g and assumption on E g , we deduce that then necessarily there exists a branch of solutions of (3.5) that branches {(λ, u λ )} λ ′ ≤λ< λ at some point (λ, u λ ) for λ ∈ [λ ′ , λ), leading to the existence of a multiple-point of E g which turns out to be a signature of topological instability of S g from the assumption on E g and Proposition 2.2-(iii).
Denote by A + λ the point ( λ, lim λ↓ λ u λ ) which exists from same arguments of compactness. Sim- Since u + λ = lim λ↓ λ u λ , λ < λ m by construction, and the map λ → u λ is increasing from = ∅ for all γ > 0 is impossible. 11 such that g − g is with compact support and g − g ∞ ≤ ε. 12 In [CEP02] the authors have proved the existence of such a discontinuity in the L ∞ (Ω)-norm for solutions considered in C 2 (Ω) which is therefore valid for solutions considered in C 2,α (Ω).
By construction E g ∩ Γ + λm = E g ∩ Γ + λm , and therefore the set of fold-points in Γ + λm of E g and E g are identical. We have just proved the existence of a fold-point of E g in (0, λ m ) × X which no longer exists (in an homeomorphic sense) for E g by definition of λ m . From Proposition 2.2-(i), we conclude that S g and S g are thus not topologically equivalent.
Case (ii). The proof follows the same lines than above by working with (0, λ ♯ ) instead of (0, λ m ), and localizing the the perturbation on C ∩ Γ − λ ♯ . Case (iii). If F = ∅, λ ′ may be chosen arbitrary in (0, λ * ), and we can proceed as preceding to create a fold-point of E g whereas E g does not possess any fold-point (F = ∅).
In all the cases, we are thus able to exhibit g such that g − g ∞ ≤ ε and S g and S g are not topologically equivalent, for any choice of ε > 0. We have thus proved that S g is topologically unstable in the sense of Definition 2.5. The proof is complete.
Remark 3.2. If we assume g to be C 1 instead of continuous and locally Liptchitz, it can be shown that necessarily ( λ, u λ ) obtained in the proof above, is degenerate in the sense that λ 1 (−∆ − λg ′ ( u λ )I) = 0, and the linearized equation has a nontrivial solution. Then under further assumptions on g and appropriate a priori bounds, the existence of a fold-point at ( λ, u λ ) can be ensured using e.g. [CR75, Theorem 1.1]; see also [CR73,OS99].
The regularity assumption on g of Theorem 3.2 prevents the use of such linearization techniques. Theorem A.1 serves here as a substitutive ingredient to cope with the lack of regularity caused by our assumptions on g. At the same time, it is unclear how to weaken further these assumptions, since the proof of Theorem 3.2 provided above has made a substantial use of the growth property of the minimal branch such as recalled in Proposition 3.1 above; see also Remark 3.1.
The possibility of creation of a discontinuity in the minimal branch by arbitrarily small perturbations of the nonlinearity, has played a crucial role in the proof of Theorem 3.2. This is made possible when the spatial dimension is equal to one or two, due to the following observation regarding a specific Poisson equation used in the creation of a discontinuity in the minimal branch such as proposed in [CEP02].
Given r > 0, we denote by B r the ball of R d of radius r, centered at the origin. For 0 < ρ < R, the solution Ψ ρ of the following Poisson equation satisfies for ρ < R/2, where the behavior of K(ρ) as ρ → 0 is of the form This can be proved by simply writing down the analytic expression of the solution to (3.6); see [CEP02,Lemma 3.1]. When d ≥ 3, K(ρ) converges to a constant (depending on d) as ρ → 0. This removal of the singularity at 0 for K, is responsible of the "sufficiently large" requirement regarding the perturbation of the nonlinearity, in order to achieve a discontinuity in the minimal branch by the techniques of [CEP02] in dimension d ≥ 3. Whether this point is purely technical or more substantial, is still an open problem.
Appendix A. Unbounded continuum of solutions to parametrized fixed point problems, in Banach spaces We communicate in this appendix on a general result concerning the existence of an unbounded continuum of fixed points associated with one-parameter families of completely continuous perturbations of the identity map in a Banach space. This theorem is rooted in the seminal work of [LS34] that initiated what is known today as the Leray-Schauder continuation theorem. Extensions of such a continuation result can be found in [FMP86,MP84] for the multi-parameter case. Theorem A.1 below, summarizes such a result in the one-parameter case whose proof is given here for the sake of completeness. Under a nonzero condition on the Leray-Schauder degree to hold at some parameter value, Theorem A.1 ensures in particular the existence of an unbounded continuum of solutions to nonlinear problems for which the nonlinearity is not necessarily Fréchet differentiable.
Results similar to Theorem A.1 below, regarding the existence of an unbounded continuum of solutions to nonlinear eigenvalue problems, have been obtained in the literature, see e.g. In what follows the λ-section of a nonempty subset A of R + × E, will be defined as:  Then there exists a continuum C + ⊆ S + (i.e. a closed and connected subset of S + ) such that the following properties hold: In order to prove this theorem, we need an extension of the standard homotopy property of the Leray-Schauder degree [Dei85,Nir01] to homotopy cylinders that exhibit variable λ-sections. This is the purpose of the following Lemma.
Proof. We may assume, without loss of generality, that O = ∅ and that λ 1 = inf{λ : O λ = ∅} and λ 2 = sup{λ : O λ = ∅}. Consider ε > 0 and the following superset of O in R × E, , with some arbitrary fixed λ * ∈ [λ 1 , λ 2 ]. Then H is a completely continuous perturbation of the identity 13 in R × E. We denote by E the set R × E in what follows.
Since H(λ, u) = 0 E if and only if λ = λ * and u = G(λ, u), and since λ * ∈ [λ 1 , λ 2 ] and G(λ, u) = u on ∂O by assumptions, we can conclude that, H t (λ, u) = 0 ⇔ λ = λ * and u = G(λ, u) , and from our assumptions, we conclude again that H t (λ, u) = 0 E for all (λ, u) ∈ ∂O ε and all t ∈ [0, 1]. By applying now the classical homotopy invariance principle [Dei85,Nir01] to the family Then 0 E does not belong to H(∂O ε ∪ K) since the cancelation of H is possible only on the λ * -cross section, while K does not intercept this section by construction and 0 E ∈ H(∂O ε ) from (A.5). By applying now the excision property of the Leray-Schauder degree [Dei85,Nir01] with such a K, we obtain, The interest of (A.8) relies on the fact that the degree is by this way expressed on a cartesian product which allows us to apply the cartesian product formula (see Lemma B.1), which gives in our case By applying now (A.9), (A.8) and (A.7) and by recalling that deg(H, O ε , 0 E ) is independent of λ * , we have thus proved that for arbitrary λ * ∈ [λ 1 , λ 2 ], deg(I − G(λ * , ·), O λ * , 0 E ) is also independent of λ * . The proof is complete.
Remark A.1. The introduction of O ε in the proof given above is needed to work within an open bounded subset of a Banach space, here R × E, and thus to work within the framework of the Leray-Schauder degree 14 . The Dugundgi theorem is used to appropriately extend F on O ε in order to apply the Leray-Schauder degree techniques.
The last ingredient to prove Theorem A.1, is the following separation lemma from point set topology (Lemma A.2 below). A separation of a topological space X is a pair of nonempty open subsets U and V , such that U ∩ V = ∅ and U ∪ V = X. A space is connected if it does not admit a separation. Two subsets A and B are connected in X if the exists a connected set Y ⊂ X, such that A ∩ Y = ∅ and B ∩ Y = ∅. Two nonempty subsets A and B of X are separated if there exists a separation U, V of X such that A ⊆ U and B ⊆ V. There exists a relationship between these concepts in the case where X compact, this is summarized in the following separation lemma.
Lemma A.2. (Separation lemma) If X is compact and A and B are not separated, then A and B are connected in X.
The proof of this lemma may be found in [Dei85,Lemma 29.1]; see also [Kur68]. As a result if two subsets of a compact set are not connected, they are separated. We are now in position to prove Theorem A.1.
Proof of Theorem A.1.
Let U be as defined in Theorem A.1. Consider case (b) first. We want to apply Lemma A. since (λ * , u * ) ∈ C + and C + is the maximal connected subset of S + . We may therefore apply where Λ R denotes the projection of S + 2R onto [λ 0 , ∞). Now deg(I − G(λ 0 , ·), O λ 0 , 0) = 0 by (c 1 ) and the assumptions of Theorem A.1. We obtain therefore a contradiction from (c 4 ) when (A.10) is applied for λ = λ * .
The case C + = S + 2R , may be treated along the same lines and is left to the reader. The proof is complete. This appendix contains auxiliaries lemmas used in the previous Appendix. We first start with the cartesian product formula for the Leray-Schauder degree.
Lemma B.2. (Dugundgi) Let E and X be Banach spaces and let f : C → K a continuous mapping, where C is a closed subset of E, and K is a convex subset of X. Then there exists a continuous mapping f : E → K such that f (u) = f (u) for all u ∈ C.
Proof. (Sketch) For each u ∈ E\C, let r u = 1 3 dist(u, C), and B u := {v ∈ E : v − u < r u }. Then diam(B u ) ≤ dist(B u , C), and {B u } u∈E\C is a open cover of E\C which admits a local refinement {O λ } λ∈Λ : i.e. λ∈Λ O λ ⊃ E\C, for each λ ∈ Λ there exists B u such that B u ⊃ O λ , and every u ∈ E\C has a neighborhood U such that U intersects at most finitely many elements of {O λ } λ∈Λ (locally finite family).
By construction, the above sum over Λ contains only finitely many terms and thus γ is continuous. Now define f by, Then it can be shown that f is continuous.
Remark B.3. Conclusion of Lemma B.2 always hold with K = hull(f (C)), the convex hull of f (C).