Henri Berestycki

Juliette Bouhours

Guillemette Chapuisat

Front blocking and propagation in cylinders with varying cross section

Keywords: 2010 Mathematics Subject Classification: 35B08, 35B30, 35B40, 35C07, 35K57, 92B05, 92C20 Reaction-diffusion equations, travelling waves, invasion fronts, bistable equation, blocking, propagation, stationary solutions

In this paper we consider a bistable reaction-diffusion equation in unbounded domains and we investigate the existence of propagation phenomena, possibly partial, in some direction or, on the contrary, of blocking phenomena. We start by proving the well-posedness of the problem. Then we prove that when the domain has a decreasing cross section with respect to the direction of propagation there is complete propagation. Further, we prove that the wave can be blocked as it comes to an abrupt geometry change. Finally we discuss various general geometrical properties that ensure either partial or complete invasion by 1. In particular, we show that in a domain that is "starshaped" with respect to an axis, there is complete invasion by 1.

1 Introduction

The problem

In this paper we consider the following parabolic problem

∂ t u(t, x) -∆u(t, x) = f (u(t, x)), for t ∈ R, x ∈ Ω, ∂ ν u(t, x) = 0, for t ∈ R, x ∈ ∂Ω, (1.1)
where Ω is an unbounded domain such that

Ω = (x 1 , x ′ ), x 1 ∈ R, x ′ ∈ ω(x 1 ) ⊂ R N -1 . (1.2)
Throughout the paper, we assume ω not to depend on x 1 for x 1 < 0 and f is a bistable nonlinearity. We call such domains "cylinder-like". We give more precisions on the general assumptions in the next section but one can see in Figure 1 various examples of the type of domains Ω we have in mind in dimension 2. The aim of this paper is to describe invasions coming from -∞. That is, waves propagating from x 1 = -∞ in the direction of increasing x 1 . These waves have implications for the initial value problem for initial data with, say, compact support. We can also envision domains with two different axes, with the direction of propagation changing as t → -∞ and as t → +∞. Think for instance of "bent cylinders". But for simplicity, in this paper we only present the results in the framework of a single axis. The main results in this paper are the following.

• We first study the existence of transition fronts in this setting, i.e the existence and uniqueness of an entire solution u of the parabolic problem (1.1), such that u(t, x) -ϕ(x 1 -ct) → 0 as t → -∞ uniformly in Ω, (1.3) where (ϕ, c) is the unique bistable travelling front solution of:

ϕ ′′ + cϕ ′ + f (ϕ) = 0 in R, ϕ(-∞) = 1, ϕ(+∞) = 0, ϕ(0) = θ.
(1.4)

• We establish the blocking of the solution if there exists a ∈ R such that the measure of Ω ∩ {a < x 1 < a + 1} is small,

• We prove that there is axial partial propagation when Ω contains a straight cylinder in the x 1 -direction, whose cross section is a ball of sufficiently large radius.

• We give conditions on the geometry of Ω under which there is complete invasion that is, u(t, x) → 1 as t → ∞ at every point x:

• In the case of general domains with decreasing cross section,

• Under the condition of partial convexity and domains with increasing cross section,

• For domains "star-shaped" with respect to an axis.

Let us specify some definitions on propagation and blocking properties. Let u ∞ (x) := lim t→+∞ u(t, x) for all x ∈ Ω (see Theorem 1.4 in the next section).

Definition 1.1 (Blocking and partial propagation). -Let u be the solution of problem (1.1) satisfying (1.3). We say that u is blocked when u ∞ (x 1 , x ′ ) → 0 as x 1 → +∞, (x 1 , x ′ ) ∈ Ω.

Otherwise we say that there is partial propagation in Ω.

Definition 1.2 (Axial partial propagation). -Let u be the solution of problem (1.1) satisfying (1.3), we say that there is axial partial propagation if inf

R×B ′ R u ∞ > 0,
for some R > 0 and where B ′ R is the ball of radius R centred at 0 in R N -1 .

Definition 1.3 (Complete invasion

). -Let u be the solution of problem (1.1) satisfying (1.3), we say that there is complete invasion

if u ∞ ≡ 1 in Ω.
This problem is of interest in different domains of application. In population dynamics, to study the evolution of a population going through an isthmus or for fish populations going through a straight for example. In medical sciences to model the evolution of a depolarisation wave in the brain or to study ventricular fibrillations. These motivations from biology and medicine are detailed in section 2, where we also discuss the interpretations of our results in these contexts.

Previous mathematical results

In this paper, we are interested in the existence and properties of invasion fronts in the direction x 1 (see [START_REF] Berestycki | Generalized travelling waves for reactiondiffusion equations[END_REF] for definition) for a bistable reactiondiffusion equation in unbounded domains. This problem was partly studied in [START_REF] Chapuisat | Existence and nonexistence of traveling wave solutions for a bistable reaction-diffusion equation in an infinite cylinder whose diameter is suddenly increased[END_REF] in special cases. There the authors proved that in two or three dimensions, if Ω is the succession of two rectangles (or parallelepipeds) with different widths r and R, then one can find conditions on r and R such that the solution is blocked when it goes from the small rectangle to the large rectangle. The authors used a specific symmetrization method that does not work for other domains. In [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF], the authors proved the existence of generalised transition fronts in exterior domains, which pass an obstacle. They analysed the interactions between travelling wave solutions and different kind of obstacles. In particular, they gave geometrical conditions on the shapes of the obstacles under which there is complete invasion by the most stable state. They also constructed an obstacle that blocks the bistable travelling wave in some area of the domain. In this case, there is only partial invasion.

In this paper, we investigate the question of propagation/blocking properties for rather general "cylinder-like" geometries. Roughly speaking, by this we mean to say that there is an axis, say the x 1 -axis, in the direction of which the domain is unbounded (both as x 1 → -∞ and x 1 → +∞). Furthermore, for simplicity, we assume that the domain is a genuine cylinder in the part {x 1 < 0}. We first construct a generalised transition front using the techniques of [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF]. Then the main purpose of this paper is to analyse the geometrical conditions, under which, the wave is blocked or propagates and, in the latter case, whether the invasion is partial or complete.

Related questions have been studied previously in the literature. In [START_REF] Grindrod | One-way blocks in cardiac tissue: a mechanism for propagation failure in purkinje fibres[END_REF] Grindrod and Lewis approximate the normal speed of the interface of a front using the Eikonal approximation and numerical simulations to point out that the presence of a symmetric narrowing and widening will decrease the average speed and that an abrupt widening could lead this speed to 0. More recently in [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF][START_REF] Lou | Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed[END_REF], Lou, Matano and Nakamura study the effect of undulating boundary on the normal speed of the interface of a front analysing a curvature-driven motion of plane curves in dimension 2. They give necessary and sufficient conditions on the undulations of the domain for the existence of generalised travelling waves in this context.

Notations and assumptions

Here are some notations that we will use throughout this paper. First R -= (-∞, 0) and R + = (0, +∞). The space dimension is N ≥ 2 and we write the space variable as

x = (x 1 , x ′ ) ∈ R N with x 1 ∈ R and x ′ ∈ R N -1 .
We will use balls both in R N as in R N -1 . Thus to distinguish the two cases, we use the notation

B R (x 0 ) := x ∈ R N , |x -x 0 | < R
for the ball of radius R and centred at x 0 in R N whereas

B ′ R (x 0 ) := x ′ ∈ R N -1 , |x ′ -x 0 | < R
denotes the ball of radius R and centred at x 0 in R N -1 . If x 0 = 0, we will simply write B R and B ′ R . The outward unit normal at x ∈ ∂Ω will be denoted

ν(x) = (ν 1 (x), ν ′ (x)) ∈ R × R N -1 .
Now let us state precisely the conditions on the bistable nonlinearity f . First we assume that f ∈ C 1,1 ([0, 1]) and there exists θ ∈ (0, 1) such that

f (0) = f (θ) = f (1) = 0, f ′ (0) < 0, f ′ (1) < 0, (1.5a)
f (s) < 0 for all s ∈ (0, θ), f (s) > 0 for all s ∈ (θ, 1).

(1.5b)

Moreover we assume that 1 is "more stable" than 0 in the sense that:

1 0 f (s)ds > 0. (1.6)
We let β ∈ (θ, 1) denotes the real number:

β = inf x ∈ [0, 1], x 0 f (s)ds > 0 . (1.7)
As mentioned before, (ϕ, c) will always denote the one-dimension travelling front with ϕ(0) = θ, i.e. (ϕ, c) is solution of (1.4).

Concerning the domain Ω ⊂ R N , we have in mind to consider cylinders around the x 1 -axis with varying cross section but the various results are valid for much more general domains and our only general assumptions are that Ω is a uniformly C 2,α domain of R N and

(1.8)

Ω ∩ x ∈ R N , x 1 < 0 = R -× ω where ω is an open subset of R N -1 . (1.9)
Condition (1.9) means that the portion of Ω lying in a half space (here {x 1 < 0}) is a straight cylinder. Thus, the change in the geometry of the domain takes place in the half space R + × R N -1 .

Main results

We first prove the well-posedness of the problem with the following theorem.

Theorem 1.4. Let Ω ⊂ R N and f satisfy conditions (1.8)-(1.9) and (1.5)-(1.6), let (ϕ, c) be the unique solution of (1.4), with c > 0. Then there exists a unique entire solution of

∂ t u(t, x) -∆u(t, x) = f (u(t, x)), for t ∈ R, x ∈ Ω, ∂ ν u(t, x) = 0, for t ∈ R, x ∈ ∂Ω, such that u(t, x) -ϕ(x 1 -ct) → 0 as t → -∞ uniformly in Ω.
Moreover, u satisfies u t (t, x) > 0, 0 < u(t, x) < 1 for all (t, x) ∈ R × Ω and lim

t→+∞ u(t, .) = u ∞ in C 2 loc (Ω)
where u ∞ is stationary solution, that is, verifies

∆u ∞ + f (u ∞ ) = 0 x ∈ Ω, ∂ ν u ∞ = 0 x ∈ ∂Ω. (1.10)
This theorem is essentially due to [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF], who considered an exterior domain. But their proof readily extends to our case. We sketch it in section 3. Remark 1.5. This result also applies to bent and right-ended cylinders as those illustrated in section 3.4

Most of this paper is devoted to analyse the large time behaviour of u(t, x), that is understanding when it is blocked, when there is partial propagation or axial partial propagation or when there is complete invasion. We start with the case of domains with decreasing cross section for which we can prove that there is complete invasion. Theorem 1.6 (Complete propagation into decreasing domains). Assume that for all x ∈ ∂Ω, ν 1 (x) ≥ 0, where ν 1 (x) is the first component of the outward unit normal at x. Under the same assumptions on Ω, f and (ϕ, c) as in Theorem 1.4, the solution u of (1.1) satisfying (1.3) propagates to 1 in Ω in the sense of Definition 1.3. That is

u(t, •) → 1 as t → +∞ in Ω.
Moreover, if we assume that

Ω ∩ x ∈ R N , x 1 > l = (l, +∞) × ω r ,
for some l > 0, ω r ⊂ R N -1 , then c is the asymptotic speed of propagation, i.e

∀ ĉ > c, lim t→+∞ sup x 1 >ĉt u(t, x) = 0, ∀ ĉ < c, lim t→+∞ inf x 1 <ĉt u(t, x) = 1.
This theorem means that there exists a generalised transition front representing an invasion of 0 by 1 for problem (1.1). Remark 1.7. If we assume that Ω is a cylinder-like domain with axis on the x 1 axis (i.e w(•) defined in (1.2) is open, connected, smooth and bounded), the condition on ν 1 (x) is fulfilled as soon as the cross section is non-increasing with respect to x 1 , but it applies to more general domains (see the upper middle figure of Figure 1 for an example in dimension 2).

Next we study the existence of blocking phenomena and prove the following Theorem 1.8 (Blocking by a narrow passage). Under the same assumptions on Ω, f and (ϕ, c) as in Theorem 1.4 and let a, b be fixed with -∞ < a < b < +∞. There exists ε > 0 small enough depending on

Ω ∩ {x ∈ R N , b < x 1 < b + 1} (1.11) such that if |Ω ∩ {x ∈ R N , x 1 ∈ (a, b)}| < ε, (1.12)
then the unique solution u of (1.1) satisfying (1.3) is blocked in the right part of the domain in the sense of Definition 1.1. That is

u(t, •) → u ∞ in Ω as t → +∞ with u ∞ (x) → 0 as x 1 → +∞.
A particular case of this theorem is the example of abruptly opening cylinders in the direction of propagation x 1 . More precisely, for a fixed right hand half of the domain Ω + := Ω ∩ {x 1 > 0}, there exists ε > 0 small enough such that if Ω ∩ {x 1 < 0} ⊂ R × B ε , the advancing wave is blocked. From the proof, essentially, we will see that it is blocked at the abrupt passage from a narrow cylinder into a wide domain. On the other hand, Theorem 1.13 establishes that for a large left cylinder, a generalised travelling front can never be blocked. Thus rescaling the space variable to fix a given measure to the left cylinder and enlarge the right side of the domain does not necessarily work. Figure 2 displays examples of 2D domains where this blocking phenomenon arises. Remark 1.9. Notice that rescaling f into λf for some λ positive, then ε becomes ε √ λ , which means that if we increase the amplitude of f then, the threshold ε decreases and blocking will take place in thinner channels. Furthermore, if we add a diffusion coefficient D > 0 in front of the second order term (scaled to 1 in this paper) then ε becomes √ Dε, which means that if D increases then the solution is blocked for larger diameters.

The blocking phenomenon as well as these scalings can be interpreted in the following manner. As the species moves through a narrow passage, it all of a sudden emerges into a widen open space. As it emerges, diffusion disperses the species, possibly to lower values of density where the reaction becomes adverse. Therefore, too narrow a passage does not allow for the species to reconstitute a strong enough basis to invade the right hand half of the domain and the invasion wave is stuck at this passage. Clearly, a large diffusion accentuates this problem, while a stronger f , by allowing large densities to transit through the channel has the opposite effect. Now considering domains that are not narrow, we show that this does not happen if the cross section is large enough. We prove that when Ω contains a straight cylinder in the x 1 -direction of radius large enough, there always is propagation.

Theorem 1.10 (Axial partial propagation). Under the same assumptions as in Theorem 1.4, there exists R 0 > 0 such that if R × B ′ R ⊂ Ω for an R > R 0 , the unique solution u of (1.1) satisfying (1.3), propagates along the x 1 -axis in the sense of Definition 1.2. That is

u(t, •) → u ∞ in Ω as t → +∞ with inf R×B ′ R u ∞ > 0.
We give more details about how this constant is determined in section 6.

Remark 1.11. Notice that in this theorem the invasion may be partial. Indeed depending on the shape of Ω, there could be subdomains where the solution u ∞ is close to 0 (see [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF][START_REF] Bouhours | Robustness of a liouville type theorem in exterior domain[END_REF] and Figure 9 in section 6 of this paper for examples of domains where it happens).

In the following theorems, we focus on the geometry Ω such that the invasion is complete, i.e. u ∞ ≡ 1. The next theorems are proved using a sliding method, which is inspired from [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF]. Actually, to prove the next one in section 7 we introduce a new variant of this method by sliding cylinders. Theorem 1.12 (Complete propagation in "axially star-shaped" domains). We assume that the conditions of Theorem 1.10 are fulfilled. Moreover we assume that at each point on the boundary x = (x 1 , x ′ ) ∈ ∂Ω, the outward unit normal ν makes a non-negative angle with the direction x ′ . More precisely, writing ν = (ν 1 , ν ′ ) (with ν ′ ∈ R N -1 ), we assume:

ν ′ • x ′ ≥ 0 for all points x = (x 1 , x ′ ) ∈ ∂Ω. (1.13)
Then, the invasion is complete, namely, u ∞ ≡ 1 in Ω.

We call a domain that satisfies the property (1.13) a star-shaped domain with respect to the x 1 -axis. One can look at Figure 1 The previous Theorem covers a large range of geometries for Ω, however the following Theorem completes this result for another type of geometry. It relies on a different proof.

Theorem 1.13 (Complete propagation into increasing domains). Under the same assumptions as in Theorem 1.4, there exists

R 1 > 0 such that if R × B ′ R ⊂ Ω, for R > R 1 , Ω r L := {(x 1 , x ′ ) ∈ Ω, x 1 > L} is convex, {x ∈ Ω, x 1 < L + R} ⊂ R × B C , ∀ x = (x 1 , x ′ ) ∈ ∂Ω x 1 < L + R ⇒ ν 1 (x) ≤ 0
for some L, C > 0, then the invasion is complete in the sense of Definition 1.3, namely u ∞ ≡ 1 in Ω. Here ν 1 (x) is the first component of the outward unit normal at x.

In the proof of this theorem we use a solution of the stationary elliptic equation in a ball with Dirichlet conditions. We then slide this ball in (R × B ′ R ) ∪ Ω r L to get a lower estimate on the solution u ∞ in this domain and then conclude with the maximum principle and Hopf Lemma. This proof could be adapted to somewhat more general domains as soon as it satisfies a sliding ball assumption, i.e every point in the domain could be touch, from within the domain, by a ball of radius R -δ for some δ > 0 such that for

x ∈ ∂Ω ∩ B R , ν(x) • x ≥ 0.
We thus point out two different types of behaviour of the solution in the case of widening cylinders. If the diameter of the cylinder before the change in geometry is small and the change is abrupt, we prove that there is blocking, whereas if the cylinder is wide enough before the change of geometry there is propagation even if the domain abruptly increases. One question that remains open is the existence of a sharp threshold, when considering, say a monotone 1-parameter family of domains. The question is thus to know whether the solution is blocked below a critical threshold and propagates above it.

In this paper we consider mainly generalised transition waves. But of course our results and methods bear important consequences for associated Cauchy problems.

Corollary 1.14 (The associated Cauchy problem). Let u be the solution of the following Cauchy problem

     ∂ t u(t, x) -∆u(t, x) = f (u(t, x)), for t ∈ (0, +∞), x ∈ Ω, ∂ ν u(t, x) = 0, for t ∈ (0, +∞, ) x ∈ ∂Ω, u(0, x) = u 0 (x), for x ∈ Ω.

Then

• Under the same assumptions as in Theorem 1.8 on Ω, f and (ϕ, c), the propagation of the solution is blocked for any u 0 , of which the support is contained in the left part of the domain , i.e supp{u 0 } ⊂ {x ∈ Ω, x 1 < a},

• Under the same assumptions as in either Theorem 1.6, Theorem 1.10, Theorem 1.12 or Theorem 1.13 on Ω, f and (ϕ, c), the propagation results would still hold if the domain {x ∈ Ω, u 0 (x) > θ + δ} is large enough, for some δ > 0.

This article is organised as follows. In section 2 we detail the motivations from biology and medicine behind this problem. We indicate the interpretations of our findings in these contexts. We prove the existence of transition fronts (Theorem 1.4) in section 3. We devote section 4 to the proof of Theorem 1.6 on the propagation of fronts in decreasing domains. Section 5 is devoted to the proof of Theorem 1.8, that is the blocking of the front if Ω contains a narrow passage. Then we derive a sufficient condition under which there is (possibly partial) propagation in section 6. There, we prove Theorem 1.10. Next we turn to further complete propagation results. We prove Theorem 1.12 in section 7 and Theorem 1.13 in section 8.

Motivations from medicine and biology and interpretations of the results

Equation (1.1) is classical in physics, for the representation of phase transitions, and in biology. A particularly important motivation for studying the effect of geometry on propagation of waves in equation (1.1) comes from the modelling of Cortical Spreading Depressions (CSDs) in the brain. These are transient and large depolarisations of the membrane of neurons that propagate slowly (with a speed of the order of 3mm/min) in the brain. CSDs are due to abnormal ionic exchanges between the intra-and extra-cellular space of the neuronal body that slowly diffuse in the brain. There are two stable states, the normal polarised rest state and the totally depolarised state, with a threshold on the ionic disturbances for passage from one stable state to the other. After a depolarisation lasting 3 or 4 minutes, several mechanisms take place that repolarise the neurons. Mathematical studies of CSDs focus generally on the depolarisation phase. Having in view this phase, CSDs are modelled by a bistable reaction-diffusion equation like (1.1), where 0 ≤ u ≤ 1 represents the degree of depolarisation. In this context, u ≡ 0 stands for the normal polarised state and u ≡ 1 represents the completely depolarised state. Thus, a depolarisation is represented by an invasion of the state u ≡ 1. The complete depolarisation mechanism only takes place in the grey matter -the part of the brain that contains the neuronal bodies -while simple ionic disturbances diffuse everywhere and are absorbed in the white matter -the part of the brain where no neuron body and only axons are to be found. In this context, the domain Ω in equation (1.1) can be thought of as representing a portion of grey matter of the brain. The Neumann boundary conditions here is a classical simplification resulting from considering that the grey matter is isolated.

The question of existence of and mechanisms to break CSDs in human brain is of great importance for the understanding of strokes or migraines with aura among other pathologies. CSDs were first observed by Leão in 1944 [START_REF] Leao | Pial circulation and spreading depression of activity in the cerebral cortex[END_REF]. When neurons are depolarised, they cannot propagate nerve impulses and this generates several symptoms. For example, CSDs are suspected of being responsible for the aura during migraines with aura [START_REF] Lauritzen | Pathophysiology of the migraine aura. the spreading depression theory[END_REF][START_REF] Lauritzen | Cortical spreading depression in migraine[END_REF][START_REF] Hadjikhani | Mechanisms of migraine aura revealed by functional mri in human visual cortex[END_REF]. The aura is a set of hallucinations, mainly visual and sensory, that appears during some migraine attacks. In ischemic strokes in rodents, it has been proved that each CSD increases the neurological damage by approximately 30% [START_REF] Mies | Correlation between peri-infarct dc shifts and ischeamic neuronal damage in rat[END_REF] and therapies aiming at blocking the appearance of CSDs have shown very promising results in rodents [START_REF] Keyser | Clinical trials with neuroprotective drugs in acute ischaemic stroke: are we doing the right thing?[END_REF][START_REF] Nedergaard | Gap junctions are required for the propagation of spreading depression[END_REF]. Unfortunately they turned out to be inefficient in humans.

Moreover there are obvious difficulties in observing CSDs in the human brain. These difficulties are related to the complex geometry of the human cortex. With EEG, each electrode records brain electrical activity from a cortical area of several square centimeters whereas the depolarized area during CSDs is much smaller (around few square milimeters). Alternative medical imaging methods record accompagnying symptoms of CSDs (cerebral blood flow increase or decrease) but not the depolarization itself. CSDs can be recorded only in acutely injured human brain undergoing neurosurgery, using cortical electrode strips similar to the technology applied in epilepsy surgery. But in such a pathological situation, the brain reacts quite differently from a healthy brain.

For many years, it was believed that CSDs were an artifact produced in animal experiments and without significance for human neurological conditions. Their occurence in human cortex was thus a matter of debate [START_REF] Aitken | Spreading depression in human hippocampal tissue in vitro[END_REF][START_REF] Gorji | Spreading depression in human neocortical slices[END_REF][START_REF] Mayevsky | Cortical spreading depression recorded from the human brain using a multiparmetric monitoring system[END_REF][START_REF] Mclachlan | Spreading depression of leao in rodent and human cortex[END_REF][START_REF] Šramka | Functional ablation by spreading depression: possible use in human stereptactic surgery[END_REF][START_REF] Strong | Spreading and synchronous depressions of cortical activity in acutely injured human brain[END_REF][START_REF] Back | Failure to demonstrate peri-infarct depolarizations by repetitive mr diffusion imaging in acute human stroke[END_REF]. However in the recent years, a medical consensus considering that CSDs occur in human brain has gained ground (see [START_REF] Lauritzen | Clinical relevance of cortical spreading depression in neurological disorders: migraine, malignant stroke, subarachnoid and intracranial hemorrhage, and traumatic brain injury[END_REF] and references therein). Consequently the question of how to block these CSDs became important. For migraines with aura, CSDs have been observed to become extinct when propagating into a sulcus [START_REF] Bowyer | Analysis of meg signals of spreading cortical depression with propagation constrained to a rectangular cortical strip. ii. gyrencephalic swine model[END_REF], that is in the bottom of a convolution. Generally the grey matter is thicker in a sulcus than in the rest of the cortex. For a CSD during a stroke, understanding how far the CSD will propagate in a specific patient could help in anticipating the dammage caused by the stroke. It could also help in the selection of patients for future clinical trial.

Mathematical modelling may help in understanding these phenomena since experiments on rodents cannot be conclusive for humans due to the differences of the brain morphologies. The grey matter of the human brain composes a thin layer at the periphery of the brain with its thickness subject to abrupt and large variations. The rest of the human brain is composed of white matter. On the opposite, the rodent brain is almost round and entirely composed of grey matter (see Figure 3). The variations of the grey matter thickness in the human brain could explain the extinction of CSDs. Indeed, as we establish it here, very abrupt variations of the domain may block propagation Chapuisat and Grenier [START_REF] Chapuisat | Existence and nonexistence of traveling wave solutions for a bistable reaction-diffusion equation in an infinite cylinder whose diameter is suddenly increased[END_REF] first stated the hypothesis that the morphol- ogy of the human brain may prevent the propagation of CSDs over large distances, unifying several observations by biologists. Several effects may be combined. First, variations of the grey matter thickness may prevent propagation of CSDs. This point is studied numerically in [START_REF] Dronne | Examples of the influence of the geometry on the propagation of progressive waves[END_REF] and proved in a special case in [START_REF] Chapuisat | Existence and nonexistence of traveling wave solutions for a bistable reaction-diffusion equation in an infinite cylinder whose diameter is suddenly increased[END_REF]. Furthermore, the absorbing effect of the white matter may stop the propagation of CSDs in areas of the brain where the grey matter is too thin. This phenomenon is studied in [START_REF] Chapuisat | Existence and nonexistence of curved front solution of a biological equation[END_REF] and more recently in [START_REF] Berestycki | Traveling fronts guided by the environment for reaction-diffusion equations[END_REF]. Finally the convolution of the human brain may also enhance the absorption effect of the white matter and thus the blocking of the CSDs. This point is studied numerically in [START_REF] Pocci | Numerical study of the stopping of aura during migraine[END_REF]. Here we prove in a general setting that abrupt transitions from thin to large regions will block waves.

These results can be interpreted in a biological framework. If we assume that the mechanisms triggering CSDs are the same in humans and in rodents, CSDs could be initiated in the human brain but only in areas where the grey matter is large enough and these CSDs will not propagate over large distances since they will be blocked by a sudden enlargement of the grey matter, or if the grey matter becomes too thin as compared to the white matter, or if they go through a sharp convolution of the brain. On the opposite, in rodent brains that are like balls of grey matter, CSDs could be triggered anywhere. This may explain why it has been difficult to observe CSDs in human brains during a stroke.

In the case of migraines with aura, the symptoms of aura correspond to a dysfunction of certain neurons. When translating the symptoms described by patients on a map of the cerebral areas, it is observed that the dysfunction of neurons propagate at a speed of 3mm/min which makes biologists and physicians think that the aura is due to CSDs. It has been shown that the aura always stops at the bottom of a very deep sulkus. This phenomenon has been studied numerically in [START_REF] Grenier | A numerical study of the blocking of migraine by rolando sulcus[END_REF] and it strengthens the hypothesis of the role of the morphology of the human brain in the propagation of CSDs. However a complete theoretical study of the influence of the geometry of the grey matter on the propagation of CSDs has never been carried out up to now.

Another motivation for the study of the effect of geomery on propagation in equation (1.1) comes from the study of ventricular fibrillations where blocking phenomena in domains with varying thickness is also of great importance. Ventricular fibrillation is a state of electrical anarchy in part of the heart that leads to rapid chaotic contractions of the heart. It is fatal unless a normal rhythm can be restored by defibrillation. One of the medical hypotheses is that ventricular fibrillations are triggered by the generation of a circular excitation wave that are thus trapped in a region of the heart. These waves are not present in a healthy heart and an important medical problem is to understand how these waves can be triggered. Ashman and Hull [START_REF] Ashman | Essentials of electrocardiography[END_REF] suggested that a myocardial infarct could create a circular area of cardiac tissue where excitation waves can propagate only in one way. Once an excitation wave would enter this area, it would be trapped and the propagation of the wave would trigger ventricular fibrillations.

Hence an important question is to understand how excitation waves can propagate only in one direction in cardiac fibres. Cranefield [START_REF] Cranefield | The Conduction of the Cardiac Impulse: The Slow Response and Cardiac Arrhythmias[END_REF] asserted that asymmetry of conduction in a normal cardiac fibre occurs due to variations of the diameter of the fibre. This asymmetry is strengthened by depression of excitability.

Here too, mathematical models of excitation waves in the cardiac tissue may shed light on these phenomena. Cardiac cells as brain cells are excitable and the propagation of the depolarisation initialising the excitation wave can be modelled by a bistable reaction-diffusion equation as in (1.1). One of the important problems is thus to understand if the geometry of a fibre (or rather a fibre bundle) may trigger an asymmetry in the propagation of the travelling front and if geometrical properties of the domain may even block propagation of travelling waves in one direction but not in the other. This is precisely what we establish here. Indeed, our findings show that in the presence of an abrupt transition, waves can be blocked in one direction (the direction in which the domain is widening) whereas they propagate in the opposite one (the direction in which it is narrowing). Related earlier works include numerical studies [START_REF] Joyner | Mechanisms of unidirectional blocks in cardiac tissues[END_REF][START_REF] Miller | A simple model of delay, block and one way conduction in Purkinje fibers[END_REF] that supported this hypothesis. Grindrod and Lewis [START_REF] Grindrod | One-way blocks in cardiac tissue: a mechanism for propagation failure in purkinje fibres[END_REF] studied the propagation of the excitation in a Purkinje fibre bundle using an eikonal equation obtained as an asymptotic limit of a model of propagation. They used biological values for all parameters and computed the approximated speed of the front. In the context of this asymptotic limit, they proved that both an abrupt increase in the diameter of the bundles as well as an heterogeneous depression of cardiac tissues may prevent propagation of fronts in one direction preserving the propagation in the other direction.

Lastly equation (1.1) is also relevant to describe spatially structured population dynamics. Bistable equations, of the type we study here, model populations that are subject to an Allee effect. One can think of the invasion by a population of fishes, through a straight into wide ocean. Invasions of species (plants, trees, animals) subject to Allee effect, going through an isthmus and then into a large area also lead to the kind of geometric effects we study here.

In [START_REF] Roques | A population facing climate change: joint influences of allee effects and environmental boundary geometry[END_REF], the authors studied numerically the evolution of a population in such a complex domain as it faces climate change. In this framework, the nonlinearity in equation (1.1) is space dependent: f = f (x, u) with, say f (x, u) = -γu when |u| ≥ A (where γ and A are positive constants). Furthermore, to model the effect of climate change one considers f = f (x -ct, u) where c is the exogeneously given climate change velocity. The domain involves a transition region with an opening angle α so that when α is small, the transition is gradual whereas when α is close to π/2 the transition is abrupt. It is shown numerically in [START_REF] Roques | A population facing climate change: joint influences of allee effects and environmental boundary geometry[END_REF] that there exists a critical angle α 0 ∈ (0, π/2) such that when α > α 0 there is extinction of the population, whereas if α < α 0 , the species survives. This property is analogous to what we prove here. Using our methods here we can derive the results when α is either cloe to 0 or to π/2. Nevertheless, the complete description of [START_REF] Roques | A population facing climate change: joint influences of allee effects and environmental boundary geometry[END_REF] with a specific critical angle is still an open problem.

3 Existence and uniqueness of the entire solution

The proof of Theorem 1.4 follows directly from the work of the first author with François Hamel and Hiroshi Matano on waves passing an obstacle [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF].

We will give here the main points needed to adaptat the arguments but for a detailed proof we refer to [7, section 2.3-2.4-3]. The idea is to construct sub-and supersolutions of problem (1.1) for negative time t < T < 0 for some T < 0 and to use these sub-and supersolutions to construct the entire solution. For the uniqueness result one uses the monotonicity in time and a comparison principle.

Construction of the entire solution

The construction of the entire solution follows exactly the same steps as in [7, section 2.3]. In our setting, the super-and subsolutions are defined by

w + (t, x) = 2ϕ(-ct -ξ(t)), if x 1 > 0, ϕ(x 1 -ct -ξ(t)) + ϕ(-x 1 -ct -ξ(t)), if x 1 ≤ 0, (3.1)
and

w -(t, x) = 0, if x 1 > 0, ϕ(x 1 -ct + ξ(t)) -ϕ(-x 1 -ct + ξ(t)), if x 1 ≤ 0, (3.2)
where the function ξ is such that

ξ(t) = M e λ(ct+ξ(t)) , ξ(-∞) = 0, with λ = 1 2 c + c 2 -4f ′ (0)
and M a positive constant to be determined. Thus,

ξ(t) = 1 λ ln 1 1 -M c -1 e λct for all t ≤ 1 λc ln c M .
If we assume that ct + ξ(t) ≤ 0 which is true as soon as t ≤ 1 λc ln c c+M , it can be proved that there exist T < 0 and M > 0 large enough such that w + is a supersolution and w -is a subsolution of problem (1.1) for all t < T and x ∈ Ω. Now the idea is to construct a sequence of solutions u n of the Cauchy problem defined for -n ≤ t < +∞, such that u n converges toward an entire solution as n → +∞. Following [7, section 2.4], we define u n to be the solution of (1.1) for all t ≥ -n with the initial condition

u n (-n, x) = w -(-n, x).
Since w -≤ w + and using the comparison principle one proves that w -≤ u n ≤ w + and

u n (t, x) ≥ u n-1 (t, x) for all t ∈ [-n + 1, +∞), x ∈ Ω.
Using the monotonicity of the sequence and parabolic estimates as n → +∞, we see that u n converges to an entire solution u of (1.1) for all t ∈ R, x ∈ Ω, and

w -(t, x) ≤ u(t, x) ≤ w + (t, x) for all t ∈ (-∞, T ], x ∈ Ω.
Since ξ(-∞) = 0, we have thus proved that there exists an entire solution to (1.1) such that |u(t, x) -ϕ(x 1 -ct)| → 0 as t → -∞ uniformly in x ∈ Ω.

Time monotonicity

As in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF], we will make use of the property that u t > 0 for all t ∈ R and x ∈ Ω. This can be proved using the maximum principle and the fact that w - t > 0. Hence (u n ) t (t, x) > 0 for all t ∈ (-n, +∞), x ∈ Ω.

Letting n → +∞ and using the strong maximum principle, one concludes that u t (t, x) > 0 for all t ∈ R, x ∈ Ω.

Since 0 < u < 1, using monotonicity and usual parabolic estimates, we obtain that u(t, .) → u ∞ in C 2 loc (Ω) and u ∞ is a solution of (1.10). In the following, we will need more informations on u t so for 0 < η ≤ 1 2 , we introduce Ω η (t) = {x ∈ Ω : η ≤ u(t, x) ≤ 1 -η}.

Lemma 3.1. For all η ∈ (0, 1 2 ], there exists δ > 0 such that u t (t, x) ≥ δ for all t ∈ (-∞, T η ] and x ∈ Ω η (t).

(3.3)

Proof of Lemma 3.1. The proof of the lemma is the same as in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] section 3 except that the domain is changed. We will therefore give only the main ideas.

First using (1.3), there exist T η ∈ R and M η > 0 such that,

Ω η (t) ⊂ {x ∈ Ω, |x 1 -ct| < M η } ⊂ R -× ω for all -∞ < t ≤ T η . (3.4)
Now argue by contradiction and assume that there exist t k ∈ (-∞, T η ] and

x k ∈ Ω η (t) such that u t (t k , x k ) → 0 as k → +∞.
There are two possibilities:

• If (t k ) k is bounded, then x k is also bounded and up to extraction of a subsequence, we can assume that t k → t * and x k → x * so u t (t * , x * ) = 0. But u t > 0 on Ω so x * ∈ ∂Ω and using the parabolic Hopf lemma, ∂ ν u t = (∂ ν u) t < 0 and we obtain a contradiction with the boundary condition.

• If (t k ) k is unbounded, then x k is also unbounded which means here that

x 1,k tends to -∞. Defining u k (t, x) = u(t + t k , x 1 + x 1,k , x ′
) and using parabolic estimates, we get that u k → u * as k → +∞ in C 1,2 loc (R×R×ω). Then, we conclude as in the previous case.

General comparison principle and uniqueness of the solution

We now want to prove that the entire solution satisfying condition (1.3) is unique. In a sense, this result is an extension of the parabolic comparison principle to the case when the initial condition is given at t = -∞, which we now state in the following more general form.

Lemma 3.2 (General comparison principle). Assume u and v are super-and subsolutions of (1.1) with limiting condition (1.3), that is:

       ∂ t u -∆u ≥ f (u) in R × Ω, ∂ ν u ≥ 0 on R × ∂Ω, lim t→-∞ inf x∈Ω u(t, x) -ϕ(x -ct) ≥ 0. and        ∂ t v -∆v ≤ f (v) in R × Ω, ∂ ν v ≤ 0 on R × ∂Ω, lim t→-∞ sup x∈Ω v(t, x) -ϕ(x -ct) ≤ 0 (3.5)
Then v ≤ u for all t ∈ R and x ∈ Ω.

This comparison principle indeed contains the uniqueness of the entire solution satisfying (1.3).

Proof of Lemma 3.2. In the following proof, we will note Lw := w t -∆wf (w).

Choose η > 0 small enough so that

f ′ (s) ≤ -γ for s ∈ [-2η, 2η] ∪ [1 -2η, 1 + 2η] (3.6)
for some γ > 0. For any ε ∈ (0, η) we can find

t ε ∈ R such that v(t, x) -u(t, x) ≤ ε for -∞ < t ≤ t ε and x ∈ Ω.
We let T * = min(T η -σε, t ε ). For any t 0 ∈ (-∞, T * ], we define

z(t, x) = u(t + t 0 + σε(1 -e -γt ), x) + εe -γt
where the constant σ > 0 will be specified later. Then

v(t 0 , x) ≤ z(0, x) for x ∈ Ω. (3.7)
Let us prove that z is a supersolution in the time range t ∈ [0, T * -t 0 ] if σ is sufficiently large (independently of ε).

Lz ≥ σγεe -γt u t -γεe -γt + f (u) -f (u + εe -γt ) = εe -γt (σγu t -γ -f ′ (u + θεe -γt )),
for some θ = θ(t, x) with 0 < θ < 1. If x ∈ Ω η [t + t 0 + σε(1 -e -γt )], then using the preceding lemma 3.1, we see that

Lz ≥ εe -γt (σγδ -γ -max 0≤s≤1 f ′ (s)). Therefore, Lz > 0 if σ is chosen sufficiently large (independently of ε > 0). If x ∈ Ω η [t + t 0 + σε(1 -e -γt )], we have u(t + t 0 + σε(1 -e -γt ), x) + θεe -γt ∈ [0, 2η] ∪ [1 -η, 1 + η]. Consequently, f ′ (u + θεe -γt ) ≤ γ and Lz ≥ εe -γt (-γ + γ) = 0.
Applying the classical comparison principle for t ∈ [0, T * -t 0 ] and x ∈ Ω, we get v(t 0 + t, x) ≤ z(t, x) for t ∈ [0, T * -t 0 ], x ∈ Ω.

Rewriting t 0 + t as t for simplicity of notation, we see that

v(t, x) ≤ u(t + σε(1 -e -γ(t-t 0 ) ), x) + εe -γ(t-t 0 ) for t ∈ [t 0 , T * ], x ∈ Ω where t 0 is chosen arbitrarily in (-∞, T * ]. Letting t 0 → -∞, we derive v(t, x) ≤ u(t + σε, x)
for all t ∈ (-∞, T * ],

x ∈ Ω and we can once again apply a classical comparison principle and conclude that this inequality holds for all t ∈ R. Finally, letting ε → 0, we have proved v ≤ u.

More general domains

The proofs we have sketched here actually work for more general domains.

For instance we can consider bent cylindrical type domains in which we have existence and uniqueness of an entire solution of problem (1.1) satisfying (1.3). By slightly modifying the sub-and supersolutions w -and w + for x 1 < 0, we can extend these results to U-shaped cylindrical domains. Of course in these type of domains, condition (1.3) is satisfied for only one of the end of the cylinder near x 1 → -∞. We also have the same existence and uniqueness proof for domains that only have one infinite "end" near x 1 → -∞ and the remaining part is bounded. These different types of domains are illustrated in Figure 4.

Propagation in domains with decreasing cross section

In this section we consider propagation in the direction of a narrowing domain. We formulate this condition by requiring the first component of the outward normal to be non-negative, see figure 5 for examples in 2D. Here we prove Theorem 1.6. We start by proving that the unique solution u of (1.1) such that (1.3) is satisfied cannot be blocked and propagates to 1 at large times. The proof is based on the following lemma. Lemma 4.1. Under the same assumption as in Theorem 1.6, in particular that the domain Ω satisfies ν 1 (x) ≥ 0 for all x ∈ ∂Ω, let (ϕ, c) be the travelling wave solution defined by (1.4) and u be the unique solution of (1.1) such that (1.3) is satisfied. Then for all t ∈ R and x ∈ Ω

ϕ(x 1 -ct) ≤ u(t, x). Proof of Lemma (4.1). Define v(t, x) = ϕ(x 1 -ct) for all (t, x) ∈ R × Ω, v satisfies the following problem:      ∂ t v -∆v = f (v) in R × Ω, ∂ ν v ≤ 0 on R × ∂Ω, v(t, x) -ϕ(x 1 -ct) → 0 as t → -∞ uniformly in x ∈ Ω. (4.1) Indeed, for all t ∈ R, x ∈ ∂Ω, ∂ ν v(t, x) = ϕ ′ (x 1 -ct)ν 1 (x).
As assumed in Theorem 1.6, ν 1 (x) ≥ 0 for all x ∈ ∂Ω and since ϕ is decreasing, we see that ∂ ν v ≤ 0 in R × ∂Ω. Applying the general comparison principle of Lemma 3.2, we get v ≤ u.

Next, we make precise the behaviour of the solution u at large time, i.e when t → +∞. Since u is bounded between 0 and 1 and ϕ(x 1 -ct) → 1 when t → +∞ uniformly locally for x ∈ Ω, we have u(t, x) → 1 uniformly locally for x ∈ Ω.

To complete the proof of Theorem 1.6, we now furthermore assume that Ω ∩ {x ∈ R N , x 1 > l} = (l, +∞) × ω r for a given l > 0, ω r ⊂ R N -1 . We will prove that the 1D speed c is the asymptotic spreading speed i.e. for all ĉ > c lim t→+∞ sup

x 1 >ĉt u(t, x) = 0, (4.2) for all ĉ < c lim t→+∞ inf x 1 <ĉt u(t, x) = 1. (4.3)
The lower bound for the speed of propagation in (4.3) immediately results from u(t, x) ≥ ϕ(x 1 -ct) for all (t, x) ∈ R × Ω. We now turn to the proof of (4.2), which gives an upper bound for the speed of propagation. Let us fix ĉ > c. For all η ∈ (0, θ 2 ), introduce f η (see Figure 6) a smooth function in [η, 1+η] such that f (η) = f (θ) = f (1 + η) = 0, f < 0 on (η, θ), f > 0 on (θ, 1 + η) and

f η ≡ f in [2η, 1 -η], f η > f in [η, 2η) ∪ (1, 1 + η]. (4.4)
Consider the 1-D travelling front solution ϕ η of speed c η associated with f η , that is (ϕ η , c η ) satisfies the equation

ϕ ′′ η (z) + c η ϕ ′ η (z) + f η (ϕ η ) = 0 for all z ∈ R, ϕ η (-∞) = 1 + η, ϕ η (+∞) = η. (4.5)
It is well know that c η → c as η → 0. We denote η 0 > 0 the threshold such that c η < ĉ for all η ∈ (0, η 0 ). Let η ∈ (0, η 0 ) be fixed. Since |u(t, x) -ϕ(x 1ct)| → 0 as t → -∞ and lim +∞ ϕ = 0, there exist T < 0 and L > 0 such that for all t < T and x 1 > L, u(t, x) ≤ η.

Let us consider the following problem

         v t (t, x) -∆v(t, x) = f (v(t, x)), for t ∈ [T, +∞[, x ∈ [l, +∞[×ω r , ∂ ν v(t, x) = 0, for t ∈ [T, +∞[, x ∈ [l, +∞[×∂ω r , v(t, l, x ′ ) = u(t, l, x ′ ), for t ∈ [T, +∞[, x ′ ∈ ω r v(T, x) = u(T, x), for x ∈ [l, +∞[×ω r . (4.6)
It is constructed in such a way that u is a solution of it and v = ϕ η (x 1 -c η t-ξ) is a supersolution if ξ ∈ R is chosen large enough so that ϕ η (l -ξ) ≥ 1.

According to the comparison principle, v ≥ u. Thus

lim t→+∞ sup x 1 >ĉt u(t, x) ≤ lim t→+∞ sup x 1 >ĉt v(t, x) = η since ĉ > c η .
As η can be chosen arbitrarily small in (0, η 0 ), we infer from this that lim t→+∞ sup

x 1 >ĉt u(t, x) = 0 and (4.2) is proved. Remark 4.2.
The same kind of result can be obtained for smooth perturbations of straight cylinders. Precisely, using techniques from [START_REF] Bouhours | Robustness of a liouville type theorem in exterior domain[END_REF], one can easily prove that if Ω ε is a family of domains that converges in the C 2,α topology to a straight cylinder as ε → 0, then u ε the solution of (1.1) with condition (1.3) converges to 1 everywhere. Moreover if ν 1 (x) ≤ 0 for all x ∈ ∂Ω and Ω ε ∩ {x 1 > l} = (l, +∞) × ω ε,r , the solution u ε propagates with the 1-D speed c.

Blocking by a narrow passage

In this section we prove Theorem 1.8. There are earlier results on the existence of non constant stable steady states for bistable reaction-diffusion equations with Neumann boundary conditions when there is a narrow passage in a number of cases. Matano [START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF] first showed the existence of non-constant stable solutions in the case of a bounded domain having the shape of an hourglass. The first author of the present paper together with Hamel and Matano [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] established an analogous result for an exterior domain having a narrow passage to a confined region. The paper [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] exhibits a non constant stable steady state in an exterior domain with narrow passage where the area in which the solution is close to 0 is bounded whereas the area where the solution is close to 1 is unbounded.

Our proof is inspired from [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF]. Here, we construct a non constant stable steady state for domains where the two areas are unbounded (the one where the solution is close to 0 and the one where the solution is close to 1). Stability is obtained since the solution is constructed by minimization of an energy.

Actually, we derive a more precise result here. We will show that once the measure of Ω ∩ {b < x 1 < b + 1} is fixed, there exists ε > 0 small enough such that if |Ω ∩ {a < x 1 < b}| < ε then there exists a supersolution w of the elliptic problem on Ω ∩ {x 1 > a} such that w = 1 on Ω ∩ {x 1 = a} and w goes to 0 in Ω, as x 1 → +∞.

Thus, we consider the reduced problem in Ω ′ = {x ∈ Ω, x 1 > a}, (see Figure 7).

     ∆w + f (w) = 0 in Ω ′ , ∂ ν w = 0 on ∂Ω ′ \{x 1 = a}, w ≡ 1 on {x 1 = a}, (5.1)
Such a w blocks the propagation of the solution u of (1.1) satisfying (1.3). Indeed, by the comparison principle, u(t, x)

≤ w(x) in Ω ′ , whence u ∞ (x 1 , x ′ ) → 0 as x 1 → +∞.
To find a solution of (5.1) we proceed in two steps. We first restrict our analysis to a bounded domain 7) and prove in section 5.1 that there exists a function 0

Ω ′ R = Ω ′ ∩ {x 1 < R}, R ≥ b + 1 (see figure
< w R < 1 in Ω ′ R ε Ω 0 R b a R ΩR x1 Ω 0 Ω ΩT ΩT Ω 0 R b a R ΩR ε x1 Ω 0 Ω
, Ω ′ = {x ∈ Ω, x 1 > a}, Ω ′ R = {x ∈ Ω, a < x 1 < R}, Ω T = {x ∈ Ω, a < x 1 < b} and Ω R = {x ∈ Ω, b < x 1 < R} solution of:          ∆w + f (w) = 0 in Ω ′ R , ∂ ν w = 0 on Γ = ∂Ω ′ R \ {x 1 = a} ∪ {x 1 = R} , w ≡ 1 on {x 1 = a}, w ≡ 0 on {x 1 = R}. (5.2)
Then we prove in section 5.2 that w R → w ∞ as R → +∞ with w ∞ → 0 as x 1 → +∞ and conclude, using the general comparison principle, that the solution u of (1.1)-(1.3) is blocked by w ∞ in the right hand part of the domain.

Reduced problem

For D ⊂ R N , we introduce the energy functional defined for all w ∈ H 1 (D), by

J D (w) = D 1 2 |∇w| 2 + F (w)dx, (5.3)
where F (t) = 1 t f (s)ds. Note that using (1.6) and extending f linearly outside [0, 1], F (t) > 0 for all t ∈ R and that F has the shape illustrated in Figure 8.

01 s F (s) κ(s -1) 2

Let Γ

L = {x ∈ ∂Ω ′ R , x 1 = a} and Γ R = {x ∈ ∂Ω ′ R , x 1 = R}, we define H 1 1,0 = w ∈ H 1 (Ω ′ R ), w ≡ 1 on Γ L and w ≡ 0 on Γ R .
Finding a local minimizer of J Ω ′ R in an open subset of H 1 1,0 yields a stable solution of (5.2).

We start by proving that the constant function

w 0 ≡ 0 on Ω R = {x ∈ Ω, b < x 1 < R} is a local minimum of J Ω R in H 1 (Ω R ).
Proposition 5.1. There exist δ > 0 and α > 0 such that for all R ≥ b + 1, for all w ∈ H 1 (Ω R ) with w H 1 (Ω R ) ≤ δ one has

J Ω R (w) ≥ J Ω R (w 0 ) + α w 2 H 1 (Ω R ) . (5.4)
Proof of Proposition 5.1. We first observe that,

J Ω R (w) = Ω R 1 2 |∇w| 2 + F (0) + F ′ (0)w + F ′′ (0) 2 w 2 + η(w)w 2 , (5.5)
where η : R → R is a continuous function such that η(s) → 0 as s → 0. This can also be written as

J Ω R (w) = J Ω R (w 0 ) + Ω R 1 2 |∇w| 2 - f ′ (0) 2 w 2 + η(w)w 2 . (5.6)
To exploit (5.6), we require the following lemma:

Lemma 5.2. If η : R → R is the function defined in (5.5), we have,

Ω R η(z)z 2 ≤ µ(z) z 2 H 1 (Ω R ) , (5.7)
where µ(z) → 0 as z H 1 (Ω R ) → 0

Proof of Lemma 5.2. By extending f linearly at infinity, we may assume that F is quadratic at infinity, which implies that η is a bounded function from R to R, i.e there exists C ∈ R such that |η(s)| ≤ C for all s ∈ R. Let us fix p > 0 sufficiently small so that

H 1 (Ω R ) ⊂ L p+2 (Ω R ) independently of R ≥ b + 1.
Since η is bounded and η(s) → 0 as s → 0, for all γ > 0 there exists ξ > 0 such that

|η(t)| ≤ γ + ξt p . (5.8)
Hence,

Ω R η(z)z 2 ≤ γ z 2 H 1 (Ω R ) + ξ z p+2 L p+2 (Ω R ) . (5.9)
By using Sobolev embedding we infer that

Ω R η(z)z 2 ≤ (γ + ξ z p H 1 (Ω R ) ) z 2 H 1 (Ω R ) . (5.10)
Since γ can be chosen as small as we want, this inequality yields Lemma 5.2.

As f ′ (0) < 0, we can fix α = 1 2 min( 1 2 , -f ′ (0) 2 ) > 0. Thus, Ω R 1 2 |∇w| 2 - f ′ (0) 2 w 2 ≥ 2α w 2 H 1 (Ω R )
and from the previous lemma, there exists δ > 0 small enough such that for all w 2 H 1 (Ω R ) ≤ δ, one has η(w) < α. Substituting in (5.6) yields

J Ω R (w) ≥ J Ω R (w 0 ) + α w 2 H 1 (Ω R ) . (5.11)
and proposition 5.1 is proved. Now going back to the domain Ω ′ R , we extend the function w 0 by

w 0 (x) = w 0 (x 1 ) := |x 1 -b| b-a for x 1 ∈ [a, b), x ∈ Ω, 0 for x 1 ∈ [b, R], x ∈ Ω.
(5.12)

The function w 0 is in H 1 1,0 . The next step is to prove the following proposition:

Proposition 5.3. There exists δ > 0, such that for all w ∈ H 1 1,0 with w -

w 0 H 1 (Ω ′ R ) = δ one has J Ω ′ R (w) > J Ω ′ R (w 0 ). (5.13)
Proof of Proposition 5.3. In the following, all constants will be denoted generically by C. For all w ∈ H 1 1,0 ,

J Ω ′ R (w) -J Ω ′ R (w 0 ) = J Ω T (w) -J Ω T (w 0 ) (1)
+ J Ω R (w) -J Ω R (w 0 ) (2)
.

If w -w 0 H 1 (Ω ′ R ) = δ, one has that w -w 0 H 1 (Ω R ) = w H 1 (Ω R
) ≤ δ and one can use Proposition 5.1 to prove that the second term (2) above is bounded from below, i.e

J Ω R (w) -J Ω R (w 0 ) ≥ α w -w 0 2 H 1 (Ω R ) .
For term [START_REF] Aitken | Spreading depression in human hippocampal tissue in vitro[END_REF], as f (0) = 0, f ′ (0) < 0 and F (1) = 0, there exists κ > 0 such that

F (s) ≥ κ(s -1) 2 ,
for all s ∈ R (see Figure 8). This implies that

J D (w) ≥ ν w -1 2 H 1 (D) , (5.14)
for all D compact domain in Ω ′ R , w ∈ H 1 (D), where ν = min{κ, 1 2 }. Moreover

J Ω T (w 0 ) = Ω T 1 2 |∇w 0 | 2 + F (w 0 ) ≤ C|Ω T | and |Ω T | ≤ ε.
(5.15)

From (5.14) and (5.15), we obtain

J Ω T (w) -J Ω T (w 0 ) ≥ ν w -1 2 H 1 (Ω T ) -Cε. Then we also use ν w -1 2 H 1 (Ω T ) ≥ ν 2 w -w 0 2 H 1 (Ω T ) -ν w 0 -1 2 H 1 (Ω T ) ≥ ν 2 w -w 0 2 H 1 (Ω T ) -Cε. Therefore there exists γ > 0 such that J Ω ′ R (w) -J Ω ′ R (w 0 ) ≥ γ w -w 0 2 H 1 (Ω ′ R ) -Cε.
Taking ε > 0 small enough, we get

J Ω ′ R (w) -J Ω ′ R (w 0 ) > 0.
This completes the proof of Proposition 5.3 From Proposition 5.3, we get the existence of a local minimizers w R of the energy functional J Ω ′ R that belongs to H 1 0,1 such that w R -w 0 < δ. This yields a stable solution of (5.2). For δ > 0 small enough, we have 0 < w R < 1 using the maximum principle. Remark 5.4. Note that in this proof we could actually choose any fonction g(x ′ ) on Ω ∩ {x 1 = a} and make the same construction and prove that there exists a solution of

         ∆w + f (w) = 0, in Ω ′ R , ∂ ν w = 0 on Γ = ∂Ω ′ R \ {x 1 = a} ∪ {x 1 = R} , w ≡ g(x ′ ) on {x 1 = a}, w ≡ 0 on {x 1 = R}.
for ε small enough. The next step will be to study the behaviour as R → +∞. Let us recall that δ is fixed for all R ≥ b + 1.

Construction of a particular supersolution

In this section we construct a particular supersolution of (1.1). We start with the following Proposition: Proposition 5.5. Let w R be the minimizer of the energy functional J Ω ′ R defined in the previous section. When R → +∞, up to a subsequence, w R converges to a solution w ∞ of

     ∆w ∞ + f (w ∞ ) = 0 in Ω ′ , ∂ ν w ∞ = 0 on Γ = ∂Ω ′ \{x 1 = a}, w ∞ ≡ 1 on {x 1 = a}, (5.16)
such that w ∞ → 0 as x 1 → +∞.

Proof of Proposition 5.5: As 0 < w R < 1 and using Schauder estimates, there exists R n → +∞ as n → +∞ such that w Rn → w ∞ in C 2 loc as n → +∞ and w ∞ is a solution of (5.16). It remains to prove that the limit w ∞ satisfies w ∞ → 0 as x 1 → +∞. Using Fatou's Lemma:

w 0 -w ∞ 2 H 1 (Ω ′ ) ≤ lim inf n→+∞ (w 0 -w Rn )1 {-1<x 1 <Rn} 2 H 1 (Ω ′ ) ≤ δ 2
Then arguing by contradiction, we assume that there exist η > 0 and a sequence (x n = (x n,1 , x ′ n )) n∈N such that x n ∈ Ω for all n ∈ N, x n,1 → +∞ as n → +∞ and w ∞ (x n ) > η, for all n ∈ N. As

w ∞ ∈ C 2 loc , it implies that |∇w ∞ | ≤ C on every compact set K. For all x ∈ B(x n , η 2C ) ∩ Ω, |w ∞ (x) -w ∞ (x n )| ≤ max x∈B(xn, η 2C )∩Ω ∇w ∞ |x -x n | So w ∞ (x) ≥ η 2 , for all x ∈ B(x n , η 2C ) ∩ Ω, for all n ∈ N. This yields w ∞ -w 0 L 2 (K) ≥ η 2 |B η 2C ∩ Ω| × ( number of x n ∈ K).
As Ω is assumed to be uniformly C 1 , w ∞ -w 0 L 2 (K) ≥ δ for K large enough, which is impossible.

We now extend w ∞ by 1 outside Ω ′ , i.e

w∞ (x) = w ∞ (x) if x ∈ Ω ′ , 1 if x ∈ Ω\Ω ′ .
(5.17)

Then w∞ is a supersolution of the parabolic problem

v t -∆v = f (v) in Ω, ∂ ν v = 0 on ∂Ω, (5.18)
Moreover lim

t→-∞ inf x∈Ω w∞ -ϕ(x 1 -ct) ≥ 0. Indeed for x 1 ≤ a, w∞ -ϕ(x 1 -ct) = 1 -ϕ(x 1 -ct) ≥ 0 for all t ∈ R and for x 1 ≥ a, lim t→-∞ ϕ(x 1 -ct) = 0 uniformly in x 1 ≥ a.
Using the generalised maximum principle 3.2, we obtain u(t, x) ≤ w∞ (x) for all t ∈ R and x ∈ Ω. As u is increasing in t and bounded between 0 and w∞ (x), u(t, x) → u ∞ (x) as t → +∞ for all x ∈ Ω and passing to the limit in time, u ∞ (x) ≤ w∞ (x) for all x ∈ Ω. We have thus proved Theorem 1.8.

Partial propagation in domains containing a sufficiently large cylinder

In this section we investigate the propagation properties of u when Ω contains an infinite cylinder in the x 1 -direction with large enough radius. We will prove Theorem 1.10. We assume here that R × B ′ R ⊂ Ω where R > R 0 for some R 0 which we now define.

We consider the following semi-linear elliptic equation with Dirichlet condition in the ball B R of radius R in R N with N ≥ 2:

-∆z = f (z) in B R , z = 0 on ∂B R . (6.1)
For large R, positive solutions can be found by minimization of the energy functional:

J(z) := B R |∇z(x)| 2 2 - z(x) 0 f (s)ds dx in H 1 0 (B R ).
It is known [START_REF] Berestycki | Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans R N[END_REF] that there exists a threshold R 0 > 0 such that for R < R 0 , there is no positive solution of (6.1) while for all R ≥ R 0 , there exists at least one positive solution of (6.1). Moreover, for all R ≥ R 0 , there exists a maximum positive solution which we denote by z = z R in the sequel. Incidentally, from a result of Rabinowitz [START_REF] Rabinowitz | Pairs of positive solutions of nonlinear elliptic partial differential equations[END_REF], it can be inferred that for all R > R 0 , Problem (6.1) has at least two distinct solutions.

The last positivity property is obtained from u ∞ > 0 in Ω and by using that u ∞ satisfies the Neuman boundary condition, and passing to the limit in it, if need be. This contradicts the strong maximum principle. In order to conclude that the state u ≡ 1 invades the whole domain, one has to require further assumptions on the geometry of Ω . Indeed in domains as illustrated in Figure 9 there is propagation in the sense given above but the propagation could be blocked when entering some parts of the domain. One can find in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] or [START_REF] Bouhours | Robustness of a liouville type theorem in exterior domain[END_REF] arguments leading to partial propagation in exterior domains and examples of domains where the propagation is actually only partial. We now turn to this question of finding conditions under which the invasion by 1 is complete. Since u(t, •) → u ∞ locally uniformly in Ω as t → ∞, the limit u ∞ is a solution of the following stationary problem

As a conclusion, inf

     -∆u ∞ = f (u ∞ ) in Ω, ∂ ν u ∞ = 0 on ∂Ω, u ∞ (x) → 1 as x 1 → -∞, (6.4)
We are thus led to finding geometric assumptions on Ω that imply that u ∞ ≡ 1.

It is interesting to note that this is related to a classical result of Matano [START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF] and Casten and Holland [START_REF] Casten | Instability results for reaction diffusion equations with Neumann boundary conditions[END_REF] regarding the following non-linear elliptic Neumann boundary value problems in bounded domains D: We have shown in Theorem 1.8 that there exists a stable non constant solution of the previous stationary problem for domain containing a transition area of small enough measure. We have also shown in Theorem 1.6 that u ∞ ≡ 1 for propagation in the direction of a decreasing domain. In the next sections we derive two different types of geometrical conditions that garantee the complete invasion by 1.

-∆u = f (u) in D, ∂ ν u = 0 on ∂D. ( 6
7 Complete propagation for domains that are "star-shaped" with respect to an axis

The object of this section is to prove the result of complete invasion of Theorem 1.12, that is u ∞ ≡ 1 in Ω. We consider a domain Ω such that Ω ⊃ R×B ′ R , with R > R 0 , R 0 > 0 defined in section 6. Moreover we assume that at each point on the boundary x = (x 1 , x ′ ) ∈ ∂Ω, the outward unit normal ν makes a non-negative angle with the direction x ′ . To prove the result, we introduce here a new method which rests on sliding cylinders.

As in section 6, for all R > R 0 , we denote z = z R the maximal positive solution of

-∆z = f (z) in B R , z = 0 on ∂B R .
It satisfies 0 < z R < 1 in B R and is radially symmetric and decreasing with respect to |x|. We extend z R by 0 outside B R . We denote z R (x) = w R (|x|) where w R is defined on R + . In particular, we know that sup z R = z R (0) = w R (0) > θ.

By shifting the axis of the cylinder in orthogonal directions and rotating the cylinder, we will now construct from w R a new axisymmetric function that is a subsolution. Let h ≥ 0 a given real and e ∈ S N -2 a given unit vector defining a direction in R N -1 . We define, for all x ′ ∈ R N -1 :

ψ h,e (x ′ ) := w R (|x ′ -he|) so that ψ h,e (x ′ ) > 0 on B ′ R (he) := B ′ R + he, and
ϕ h (x ′ ) := max e∈S N -2 ψ h,e (x ′ ). (7.1) Note that, since w R (|x ′ |) is axisymmetric and decreasing, ϕ h (x ′ ) = w R (|x ′ - h x ′ |x ′ | |) for x ′ = 0.
Hence ϕ h is also axisymmetric but not decreasing away from the origin anymore. Moreover ϕ

h (x ′ ) > 0 if and only if h -R < |x ′ | < h + R.
Finally since ϕ h is a supremum of a family of subsolutions, it is a generalised subsolution, as defined in [START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF]. But we will only use classical subsolutions. We consider it to be defined on all of R N , so that it is independent of the variable x 1 .

By Lemma 6.2, we know that u ∞ (x) > w R (|x ′ |) ≡ ϕ 0 (x ′ ) for all x ∈ Ω. We claim that u ∞ (x) ≥ ϕ h (x ′ ) for all x ∈ Ω and all h ≥ 0. Hence u ∞ (x) ≥ ϕ |x ′ | (x ′ ) = w R (0) > θ and we will prove later on that this implies u ∞ ≡ 1.

So let us assume by contradiction that there exist h 0 > 0 and x 0 ∈ Ω such that u ∞ (x 0 ) < ϕ h (x ′ 0 ). We define

h ⋆ = sup{h ≥ 0 , ∀h ′ ∈ [0, h] ∀x ∈ Ω u ∞ (x) ≥ ϕ h ′ (x ′ )} (7.2)
Clearly, h ⋆ ≥ 0 is well defined and finite. By continuity, we know that u ∞ (x) ≥ ϕ h ⋆ (y) for all x ∈ Ω. By definition of h ⋆ , there exists a sequence h n ց h ⋆ and a sequence of point

x n ∈ Ω such that u ∞ (x n ) < ϕ hn (x ′ n ) where x n = (x 1,n , x ′ n ).
Without loss of generality, we can assume that the sequence is bounded. Indeed, if the sequence (x n ) n is not bounded, then we can shift the origin by (x n ) n so that now x 1,n = 0 but the domain and the function change with n. We can pass to the limit on these by a standard compactness argument. Therefore, we can assume that the sequence (x n ) n converges to some point p = (0, p ′ ) ∈ Ω. In the limit, we get u ∞ (p) ≤ ϕ h ⋆ (p ′ ) and since u ∞ ≥ ϕ h ⋆ everywhere, we get u ∞ (p ′ ) = ϕ h ⋆ (p ′ ). First, we know from Lemma 6.2, that there exists δ > 0 such that u ∞ (x 1 , 0) ≥ max ϕ + δ for all x 1 ∈ R, so that it is impossible that p ′ = 0. It is also impossible that p ∈ Ω. Indeed, there exists some e ∈ S N -2 such that u ∞ (p ′ ) = ϕ h ⋆ (p ′ ) = ψ h ⋆ ,e (p ′ ) > 0 and u ∞ ≥ ψ h ⋆ ,e everywhere. In view of the strong maximum principle (recall that u ∞ -ψ h ⋆ ,e satisfies some linear elliptic equation, since u ∞ and ψ h ⋆ ,e satisfy the same semi-linear equation), we infer that u ∞ ≡ ψ h ⋆ ,e . But this is impossible as ψ h ⋆ ,e vanishes somewhere.

It remains to consider the case that p ∈ ∂Ω. Since u ∞ > 0, we know that h ⋆ -R < |p ′ | < h ⋆ + R. Let us show that, necessarily, h ⋆ ≤ |p ′ | < h ⋆ + R. Indeed, suppose to the contrary that h ⋆ -R < |p ′ | < h ⋆ . Recall that p ′ = 0. In the region h ⋆ -R < r < h ⋆ , the function ϕ h is decreasing with h. Thus, for small enough h ⋆ -h > 0 we see that ϕ h (p ′ ) > ϕ h ⋆ (p ′ ) = u ∞ (p). But this is in contradiction with the definition of h ⋆ that requires u ∞ ≥ ϕ h for such an h. We have reached a contradiction and, therefore, we know that h ⋆ ≤ |p ′ | < h ⋆ + R.

We have now reached a situation where u ∞ ≥ ϕ h ⋆ everywhere, u

∞ (p) = ϕ h ⋆ (p ′ ) = w R (|p ′ -h ⋆ p ′ |p ′ | |) = w R (||p ′ | -h ⋆ |).
We compute the outward normal derivative using the fact that w R is decreasing to get

∂ ν ϕ h ⋆ (p ′ ) = -|w ′ R (|p ′ -h ⋆ p ′ |p ′ | |)| p ′ |p ′ | • ν ≤ 0.
The term p ′ |p ′ | comes from the derivative of |p ′ -h ⋆ p ′ |p ′ | | with respect to p ′ and the information that h ⋆ ≤ |p ′ | < h ⋆ + R. The last inequality follows from our assumption

ν • p ′ |p ′ | ≥ 0 on ∂Ω
The proof is thereby complete. Now we want to prove that u ∞ ≡ 1 in Ω. Assuming that u ∞ ≡ 1, there exists a sequence (x n ) n in Ω such that u ∞ (x n ) → inf If 0 ∈ ∂D, it contradicts Hopf's Lemma and if 0 ∈ D, then ∆u(0) ≤ 0, but f (u(0)) > 0 which is impossible.

Remarks 7.1.

• The assumption (1.13) is a star-shaped type property. It essentially says that any section in planes orthogonal to the x 1 -axis is star shaped with respect to the trace of the axis in this plane. Hence the name "star shaped with respect to an axis"

• We can also treat the case when "Ω ends to the right". Namely Ω is a straight cylinder as x 1 → -∞, and say is bounded to the right. This proposition proves the complete propagation of 1 in a large variety of domains Ω but examples of domains that are not covered by this theorem are presented in figure 11. Some domains, as the figure on the left of Figure 11, that do not satisfy starshape assumption with respect to an axis nevertheless satisfy the conditions of Theorem 1.13 and thus satisfy the complete propagation property as we will now prove in the next section.

Complete propagation in the direction of an increasing domain

In this section we prove Theorem 1.13. Thus we assume that Ω satisfies the following assumptions: R × B ′ R ⊂ Ω (8.1a)

where R > R 1 and R 1 is defined below. We further assume that there exists L > 0 such that ,

Ω r L := {(x 1 , x ′ ) ∈ Ω, x 1 > L} is convex, (8.1b)
and that there exists C > 0 such that {x ∈ Ω,

x 1 < L + R} ⊂ (-∞, L + R) × B C , (8.1c)
and ν 1 (x) ≤ 0 for all x ∈ ∂Ω with x 1 < L + R. (8.1d)

We will first prove that u ∞ is close to 1 in Ω r L+R := Ω ∩ (x 1 , x ′ ) ∈ R × R N -1 , x 1 > L + R using a sliding method. Then we will extend this estimate to all of Ω using a backward travelling front as subsolution and finally we will conclude that u ∞ ≡ 1.

As in section 6, we denote z = z R the maximal positive solution of

-∆z = f (z) in B R , z = 0 on ∂B R , (8.2)
We recall that it satisfies 0 < z R < 1 in B R , and is radially symmetric and decreasing with respect to |x|. We extend z R by 0 outside B R and denote z R (x) = w R (|x|) where w R is defined on R + . In particular, we know that sup z R = z R (0) = w R (0) > θ. We further recall from Lemma 6.1 that for all x = (x 1 , x ′ ) ∈ Ω, we have u ∞ (x) ≥ w R (|x ′ |). By a result of [START_REF] Berestycki | Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans R N[END_REF] we know that this maximal solution w R converges to 1 locally uniformly as R → +∞. Thus, there exist R 1 > R 0 , where R 0 > 0 is defined in section 6, and δ > 0 such that w R (0) -2δ > β for all R > R 1 , with β > θ defined by (1.7) in the Introduction. Let η > 0 be such that w R (|x|) > w R (0) -δ for all x ∈ B η . We will prove that u ∞ > w R (0) -δ in Ω r L+R using the following definition.

Definition 8.1 (Sliding ball assumption). We say that x ∈ Ω verifies the sliding ball assumption if there exists a ∈ B η (x), x 0 ∈ R and γ : [0, 1] → Ω continuous, such that γ(0) = (x 0 , 0, ..., 0), γ(1) = a,

and ∀ t ∈ [0, 1) ∀ b ∈ B R (γ(t)) ∩ ∂Ω (b -γ(t)) • ν ≥ 0 (H2)
where ν is the outward unit normal at b ∈ ∂Ω.

Assumption (H2) means that there exists a continuous path γ from a a point near x to (x 0 , 0, ..., 0) a point of the x 1 -axis, such that for all x t on this path B R (x t ) ∩ Ω is star shaped with respect to the center x t .

Figure 1 :

 1 Figure 1: Examples of "cylinder-like" domains Ω. Propagation is considered in the direction of increasing x 1 .

Figure 2 :

 2 Figure 2: Examples of domains in which the wave is blocked at the passage into the wide region, for small enough ε > 0.

  (upper left, lower left and middle) for examples of such domains in dimension 2. More examples and counterexamples of such domains are given at the end of section 7.

Figure 4 :

 4 Figure 4: Examples of 2D domains for which existence and uniqueness of an entire solution of Problem (1.1), satisfying (1.3), hold. Left: domain with one end ; center: bent cylindrical type domain ; right: U-shaped cylindrical domain.

Figure 5 :

 5 Figure 5: Examples of 2D domains concerned by theorem 1.6.

Figure 6 :

 6 Figure 6: Example of function f η that satisfies (4.4).

Figure 7 :

 7 Figure 7: Illustrations of the various domains used in the proof of Theorem 1.8 for two different Ω satisfying assumptions (1.12) and (1.11) in Theorem 1.8. Ω is the initial domain, Ω ′ = {x ∈ Ω, x 1 > a}, Ω ′ R = {x ∈ Ω, a < x 1 < R}, Ω T = {x ∈ Ω, a < x 1 < b} and Ω R = {x ∈ Ω, b < x 1 < R}

Figure 8 :

 8 Figure 8: The function F (s) = 1 s f (t)dt, where f is a bistable function for s ∈ [0, 1] and is extended linearly outside [0, 1].

  δ and the proof of Theorem 1.10 is thereby complete.Remark 6.3. Note that, Lemma 6.2 actually proves a somewhat stronger result than the one stated in Theorem 1.10

Figure 9 :

 9 Figure 9: Example of domain in dimension 2 where propagation could be blocked in the small area on the top of Ω.

  x∈Ω u ∞ (x) ∈ [w R (0), 1) as n → +∞. We recall that w R (0) > θ. Letting u n (x) := u ∞ (x n + x) and D n = {x ∈ R N , x + x n ∈ Ω}, we know that u n → u and D n → D as n → +∞ such that -∆u = f (u) in D, ∂ ν u = 0 on ∂D.

Figure 10 :Figure 11 :

 1011 Figure 10: Examples of domains that satisfy the sliding cylinder assumption in dimension 2.

Figure 12 :

 12 Figure 12: Domains Ω that satisfy (8.1)

Figure 3 :

 3 Morphology of the brain, cross sections. Left, human brain: grey matter is only a thin layer at the periphery. Right, rodent brain: it is almost entirely composed of grey matter. images are from the University of Wisconsin and Michigan State Comparative Mammalian Brain Collections, and from the National Museum of Health and Medicine, available at the following website http://www.brainmuseum.org/sections/index.html.Preparation of all these images and specimens have been funded by the National Science Foundation, as well as by the National Institutes of Health.

	These

  .5)These authors have shown that if D is a convex and bounded smooth domain, then the only stable solutions are constants. It is a natural question to ask whether there are extensions of this result to unbounded domains. What further conditions imposed on the solution and which geometrical conditions on the unbounded domain D garantee that the only stable solutions are constants? Our results on complete propagation are partial answers to this question for the solution u ∞ . More general formulations are an interesting open problem.

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 321186 -ReaDi -"Reaction-Diffusion Equations, Propagation and Modelling" held by Henri Berestycki. This work was also partially supported by the French National Research Agency (ANR), within the project NONLOCAL ANR-14-CE25-0013. Henri Berestycki was also partly supported by an NSF FRG grant DMS-1065979. Part of this work was carried out while he was visiting the Department of Mathematics at the University of Chicago. Juliette Bouhours was supported by a PhD fellowship "Bourse hors DIM" of the "Région Ile de France".

The maximum solution satisfies 0 < z R < 1 in B R and is radially symmetric and decreasing with respect to |x| [START_REF] Gidas | Symmetry of positive solutions of nonlinear elliptic equations in R n[END_REF]. We extend z R by 0 outside B R and denote z R (x) = w R (|x|) where w R is defined on R + . In particular, we know that sup z R = z R (0) = w R (0) > θ.

Using the function w R we will now derive the following estimate from below on u ∞ for all R ≥ R 0 : Lemma 6.1. For all x = (x 1 , x ′ ) ∈ Ω, we have

Proof. This is obtained by a simple sliding argument. We know that for fixed R, we have max w R < 1. Therefore, for large enough a > 0, we have

Both functions u a and z R are solutions of the same elliptic equation in B R and z R = 0 while u a > 0 on the boundary of B R . We also know that u a > z R for large enough a. We now start decreasing a. From the strong maximum principle we infer that the function z R has to stay below u a for all a ∈ R. Consequently,

which proves the claim.

Next, we prove that u ∞ is bounded from below by a positive number up to the boundary of R × B ′ R . This is stated in the following lemma. Lemma 6.2. There exists δ > 0 such that

Proof. Assume by contradiction that for all n ∈ N * there exists

Lemma 8.2. For all x ∈ Ω that satisfies the sliding ball assumption, we have

Proof. The proof is the same as the one of Lemma 6.1 with the novelty here that we slide the maximal solution w R along the path between the x 1 -axis and a and compare it to the solution u ∞ . So we consider x ∈ Ω that satisfies the sliding ball assumption and an associated path γ from (x 0 , 0) on the x 1 -axis to a ∈ B η (x) satisfying the properties in the definition. Let w x R be the maximal solution of the Dirichlet boundary value problem (8.2) in B R (x). We know from Lemma 6.1 that w

We will prove by contradiction that t

and once again we have

By the strong maximum principle, it is impossible that x ∈ B R (γ(t ⋆ )) ∩ Ω so x ∈ B R (γ(t ⋆ )) ∩ ∂Ω and from Hopf's Lemma, we have ∂ ν v(x) < 0. But

since w R is decreasing. But this contradicts the definition of the sliding ball assumption.

We thus get that u ∞ ≥ w a R and since x ∈ B η (a), we have u ∞ (x) ≥ w R (0) -δ by definition of η and δ.

We claim that if

Indeed let us choose x ∈ Ω with x 1 ≥ L + R. Thanks to the previous lemma, it is sufficient to prove that x satisfies the sliding ball assumption. Since Ω r L is convex, as assumed in (8.1), we define a = x, x 0 = x 1 and γ(t

L but this contradicts the definition of the outward unit normal, for ε small.

Now we will prove that u ∞ (x) ≥ w R (0) -2δ > θ for all x ∈ Ω and thus u ∞ ≡ 1 in Ω following the same proof than in the previous section. This ends the proof of Theorem 1.13.

Let us define

) as follows:

with 0 < f δ ≤ f in (θ, w R (0) -2δ). And consider (ϕ δ , c δ ) the travelling wave solution between 0 and w R (0) -2δ, invading 0 from the right, i.e (ϕ δ , c δ ) is solution of

we have u ∞ ≥ ε on Ω for ε sufficiently small and thus we can translate ϕ δ on the right such that, ∀x ∈ Ω, u ∞ (x) ≥ ϕ δ (x 1 ).

We can now apply a parabolic comparison principle to u ∞ and v(t, x) = ϕ

This yields u ∞ (x) ≥ ϕ(x 1 -c δ t) for all t ≥ 0 and x ∈ Ω with -M ≤ x 1 ≤ L + R and passing to the limit for t → ∞, we obtain u ∞ (x) ≥ w R (0) -2δ on {x ∈ Ω, -M ≤ x 1 ≤ L + R} and thus on all Ω. Finally we conclude that u ∞ ≡ 1 as in the previous section. Remark 8.4. This proof can be adapted for more general domains than the ones considered in Theorem 1.13. The clue is to prove that the sliding ball assumption can be verified for any point of the domain far to the right.


T X Nguyen

S Bouchareb

V Griseri

L Berquez

Post-Electronic Irradiation Measurements by PEA and FLIMM Methods on Dielectric Films

Dielectric materials are frequently used in satellite's structure and especially as thermal blanket which are protecting the electronic devices embarked. Their behavior in relation to an electronic irradiation due to space environment must be clearly defined in order to prevent electrostatic discharges. Such discharges can be very harmful for the equipment. To study these materials an irradiation chamber called 'Matspace' has been built in the laboratory. In this study, we will focus our work on the analysis of electron behavior post irradiation in PolyTetraFluoroEthylene films of various thicknesses as they are storing charges for long period of time and they are widely employed in space industry. Space charge distribution will be recorded by mean of two complementary techniques: the Pulsed Electro-Acoustic method and the Focused Laser Intensity Modulated Method.

I. INTRODUCTION

Dielectric materials that are used in space industry can be subject to electrostatic discharges once the quantity of injected charges surpasses a critical amount [START_REF] Griseri | Behavior of dielectric in a charging space environment and related anomalies in fight[END_REF]. In order to prevent such phenomena, studies on electron injection effect in polymeric materials need to be performed in laboratory. To do so, we decided to use several techniques of space charge detections to characterize some samples post-irradiation. Therefore measurements on PolyTetraFluoroEthylene (PTFE) performed by Pulse Electro-Acoustic (PEA) device and Focused Laser Intensity Modulated Method (FLIMM) will be presented in this paper. As it will be shown, PEA is more efficient when charges are located in the bulk whereas FLIMM is more accurate when charges are remaining close to the surface. The complementarities of both techniques will be also demonstrated. To characterize the electron beam several simple tools have been installed in the chamber and will be introduced in the next part.

II. EXPERIMENTALS SET-UPS A. The Irradiation Chamber: MATSPACE

The irradiation chamber called 'MATSPACE' is made of a vacuum chamber (Φ 500 mm, x 700 mm) and a thermionic electron gun that can produce electrons up to 100 kV (Fig. 1). The source delivers electrons in the vertical direction. The sample holder is located perpendicularly to the beam. In an area of about 10 cm diameter the irradiation can be considered to be homogeneous. The spreading of the beam is realized by the electromagnetic lens. Several samples can be irradiated simultaneously and then analyzed by various tools. The vacuum maintained in the chamber during irradiation is of about 10 -6 mbar.

B. Space Charge Measurement Set-ups

In this work two methods have been selected to study the space charge distribution post-irradiation in the air. We will show that they supply complementary information. As soon as the irradiation was over, the chamber is open and the two samples are analyzed simultaneously by : -The classic Pulsed Electro-Acoustic (PEA) device [START_REF] Maeno | Calibration of the pulsed electroacoustic method for measuring space charge density[END_REF][START_REF] Griseri | PEA method : evolution and development perspectives for space charge measurement[END_REF] which is composed of a pulse generator (50-400 V, 400 Hz) delivering pulses of 9 ns width, a detection system using a Poly-Vinylidene Fluoride (PVDF) transducer (9 nm thick, g 33 = 216.10 -3 V.m/N) to transform the acoustic wave in electric signal, two 30 dB amplifiers and a Digital Storage Oscilloscope (Lecroy) connected to a computer. The reference signal that is necessary for the mathematical treatment is always recorded on the sample previous irradiation. No samples treatments are required.

-The Focused Laser Intensity Modulated Method (FLIMM) cell [START_REF] Petre | FLIMM and FLAMM methods : localization of 3-D space charg at the micrometer scale[END_REF] that consists of irradiating the surface of a sample using an intensity modulated laser diode (45 mW, 658 nm). The laser beam induces a non-uniform thermal gradient in the bulk. Under the effect of the thermal wave a pyroelectric current due to the presence of space charge is created. A low noise preamplifier converts the current into voltage then the signal is extracted from the noise by a lock in amplifier. A mathematical deconvolution treatment is used to recover the space charge profile. Previous sample irradiation, two 16 mm diameter gold metalized electrodes of about 30 nm-thick are coated on each side. The pyroelectric current depends on the laser beam modulation frequency, the local temperature in the sample and the charge function due to space charge or polarization. The current I (A) is described in the following equation:

I(f)=j.2π.f.A/L ∫ r(z).T(z,f).dz (1) 0 L R(z)=p(z)-(α x -α ε ).ε.ε 0 .E(z) (2
) With f (Hz) the frequency, A (m 2 ) the thermal spot size, L (m) the thickness, T(Z, f) (K) the simulated temperature, r(z) (C.m -3 ) the total charge, p(z) (C.m -3 .K -1 ) the pyroelectric coefficient, E(z) (V.m -1 ) the internal electric field, α x α ε (K -1 ) the dependencies of the local expansions and the electric permittivity with the temperature in the bulk.

C. Electron flux control system

The beam flux is controlled during the irradiation by recording the current on a picoamperemeter which is connected to an electrode located in a faraday cage arrangement (Fig.

III. RESULTS AND DISCUSSION

A. Experimental Protocol Before sample irradiation, the electron beam energy and the flux deliver need to be settled. The energy is controlled by the applied voltage between the tungsten filament and the anode (Fig. 1). The flux is controlled by collecting the current as presented in the previous section. However we found convenient to use a phosphorescent screen that allows us to control the size and the homogeneity of the irradiated area especially when the energy of the beam is modified.

For each experiment two PTFE samples of various thicknesses have been selected. For the PEA studies samples of 200 µm have been selected whereas sample of 100 µm were required for the FLIMM experiments.

All the irradiations have been performed for 10 min under vacuum. The irradiations conditions are reported in table I. Just after the irradiation, samples are brought back to atmospheric pressure and the space charge distribution measured simultaneously by PEA and FLIMM.

B. Current Measurements

In each case, the current detected by the electrode is characterized by a steep slope at the beginning and at the end of the irradiation (Fig. 2.b). The current remain quite stable during the whole irradiation period. However it was quite difficult to keep the same flux intensity for all the tests. This could eventually be adjusted by modifying the size of the irradiated area acting on the electromagnetic lens.

C. Space Charge Measurements Results obtained by PEA

In all the cases, space charge profiles recorded by PEA show one negative peak due to the injected charges in the bulk and two positive peaks of induced charges at the interface sample/electrode. The amplitude of the peaks depends on the amount of injected charges that varies with the irradiation flux intensity. As expected, the position of the negative peak is directly linked with the electron beam energy.

In the range 0-100 keV we observe three types of charge profile. Between 15 and 50 keV the peak maximum is shifted from 10 to 21 µm (table I). The negative peak of injected charges is detected near the irradiated surface and nearly all the positive induced charges are detected on the same side (Fig. 3.a and 3.b). Due to the configuration it is difficult to separate the positive and negative charges that are located in the vicinity of the interface sample/electrode [START_REF] Griseri | Charge Build-Up and Transport in Electron Beam Irradiated Polymers in a New Irradiation Chamber[END_REF]. Actually, the flux of the 15 and 20 keV electron beam are higher than 100 pA/cm 2 whereas the flux of the 30 keV electron beam is of about 34 pA/cm 2 and despite this large discrepancy the negative charge profile remains in the same order of amplitude as seen in fig. 3.a. The same flux was used during the irradiation under 50 and 60 keV and the charge profile detected in the second case is higher in amplitude (Fig. 3.b and 3.c). We can see that the peak of charge is shifted toward the bulk and seems to be fully revealed.

When the energy is increased the peak due to the negative charges is detected deeper in the bulk and it becomes easier to distinguish the positive and negative charges that are therefore no so close. At first sight, it seems that during the irradiation charges are stored in the whole irradiated area Fig. 3.d. Actually a shoulder in the charge profile corresponding to the injected charges appears near the surface above 70 keV and two peaks are clearly detected in the bulk when 100 keV electrons are injected (Fig. 3.e). This means that the electron beam delivers two groups of electrons of different energy. The energy of the electrons that are stored in the peak located near the surface is estimated to be of about 60 keV. The maximum of this first peak is detected at about 31 µm from the surface.

The presence of two batches of electrons with various energies is not expected as the electron beam is supposed to be quasimono-energetic. It is suspected that there are some reflections of a part of the beam on the chamber wall that could explain the presence of lower energetic electrons. Results obtained by FLIMM Just after irradiation samples were positioned in the FLIMM cell and the pyroelectric currents recorded. The real and imaginary parts of these currents are collected then they are deconvoluted to obtain the charge density profiles.

In Fig. 4.a we can see the shape of the experimental currents and the calculated ones in the case of the PTFE irradiated under 30 keV. The charge profile deduced is shown in Fig. 4.b. We can see a negative charge distribution that is due to the injected electrons. The two peaks observed at about 6 and 12.5 µm seem to show that the particles introduced in the material belongs to two groups of electrons with different energies as already mentioned before. The peaks at the surface and in the bulk could be due to the deconvolution treatment and should not be considered. The signal is rather flat in the bulk above about 40 µm that is the limit of detection by FLIMM that is why the signal won't be presented above this distance in some figure. The space charge distribution in the case of the 40 and 50 keV electron beam is similar to the one detected after the 30 keV irradiation (Fig. 5). The peaks position is shifted to the bulk proportionally to the irradiation energy. The second peaks detected after the 40 and 50 keV irradiation have been detected respectively at about 15 and 20 µm. This result is in good agreement with those obtained by PEA (Table I).

The space charge distribution recorded on the 15 keV irradiated sample show a single peak that correspond to the first one located at about 7 µm (Fig. 6.a). The first peak is not detected; it might be too close to the surface. On the contrary after the 60 keV electron beam irradiation only the first peak is detected at about 9 µm (Fig. 6.b). According to the PEA results the second peak is expected to be located at about 29 µm which is out of the resolution limit of the FLIMM method. Above 60 keV the first and second peaks are too deep in the bulk to be recorded by FLIMM. That is why no results are presented in that case.

C. Comparison with theoretical data

Experimentally we have seen that the penetration depth of the electrons depends on the energy. It is directly dependent on the electron-material interaction phenomena that can be reached by various simulation methods. In this paper, we are using the results provided by ESTAR program available online [6]. The calculation is based on the Continuous Slowing Down Approximation (CSDA) that represents the path length that an electron would cross when slowing down from its original energy to a stop.

The maximum penetration depth estimated by ESTAR and PEA, the first and second peaks position obtained by PEA and FLIMM are given in table I and plotted in Fig. 7. It is observed that in all cases, the second peak position correspond to the ESTAR data. From PEA data it seems that electrons go further in the bulk than predicted with ESTAR. This could be due to the resolution but it is important to notice that ESTAR does not take into account the electric field due to the injected charges. This effect is probably quite important during the irradiation as the back side of the sample is grounded and the irradiated surface is left at a floating potential. It is therefore not surprising that the electrons driven by the field are detected a bit deeper. For PTFE (ρ=2.15g.cm -2 ) and data recorded by PEA and FLIMM.

IV. C ONCLUSIONS AND PERSPECTVES

In this paper we have presented space charge distribution profiles recorded by PEA and FLIMM on PTFE samples irradiated in vacuum under the same conditions. It has been shown that it is possible to get a good resolution by FLIMM when the charges are located near the surface (above 7 µm) but not further than 25 µm that was the case for the electron in the range [15-50 keV]. On the contrary, PEA profiles are better defined above 29 µm that correspond to the electron energy above 60 keV. These experiments show that both techniques are complementary and can be used to study the same type of materials with different thicknesses.

In the future we intend to install the both methods in-situ in order to get a better understanding on the charge transport in dielectrics used in spatial applications. ACKNOWLEDGMENT T.X. Nguyen thanks the USTH for the PhD grant provided.

FilamentFig. 1 .

 1 Fig. 1. Schematic representation of the thermionic electron gun and sample holder arrangement in the irradiation chamber 'MATSPACE'.

Fig. 2 .

 2 Fig. 2. a) Electron collecting electrode in a Faraday cage arrangement used to determine the flux of the electron beam. b) Current detected by the electron flux control system versus time during the irradiation under 90 kV.

Fig. 3 .

 3 Fig. 3. Charge density (ρ) and electric field (E) profiles detected by PEA post irradiation under a) 15-20 keV, b) 30-40-50 keV, c) 60 keV, d) 70-80 keV, e) 90-100 keV on PTFE 200 µm thick samples.

Fig. 4 .

 4 Fig. 4. Data obtain by FLIMM measurements on a PTFE 100 µm thick sample after an irradiation of 30 keV during 10 min. a) FLIMM currents, b) Charge density profile.

Fig. 5 .

 5 Fig. 5. Data obtain by FLIMM measurements on a PTFE 100 µm thick sample after an irradiation during 10 min under 40 and 50 keV.

Fig. 6 .

 6 Fig. 6. Data obtain by FLIMM measurements on a PTFE 100 µm thick sample after an irradiation during 10 min under a) 15 keV, b) under 60 keV.

Fig. 7 .

 7 Fig. 7. Penetration depth versus the electron energy predicted by ESTAR [6]For PTFE (ρ=2.15g.cm -2 ) and data recorded by PEA and FLIMM.