Lessons learned from structures damaged by delayed ettringite formation and the French prevention strategy

Bruno Godart, Université Paris-Est, IFSTTAR, Paris, France Loïc Divet, Université Paris-Est, IFSTTAR, Paris, France

The Development of DEF

The expansive sulphate internal reaction due to Delayed Ettringite Formation (DEF) can damage concrete structures severely. The primary ettringite (a hydrous calcium trisulphoaluminate) is a normal reaction product formed from the reaction of C 3 A and C 4 AF with gypsum during the plastic stage of the hydration of Portland cement. However, when peak temperatures in concrete are over about 65°C, the sulphates may be incorporated in other cement phases. After concrete hardening, the very slow formation of higher volume secondary ettringite may occur as water is taken into the crystal structure which can lead to potentially disruptive expansion. DEF is defined as the formation of ettringite in a concrete after setting, and without any external sulphate supply, but with a water supply. DEF appears in concretes exposed to frequent humidity or contact to water, and subjected to a relatively high thermal treatment (> 65°C) or having reached equivalent temperatures for other reasons (massive cast-in-place concrete, concrete casting during summer, etc).

The first reported cases of DEF occurred in some precast concrete elements subjected to a heat treatment unsuited to the composition and the environment of the concrete. International examples of DEF include railway sleepers [START_REF] Tepponen | Damages in concrete railway sleepers in Finland[END_REF][START_REF] Heinz | Delayed ettringite formation in heat treated mortars and concretes[END_REF][START_REF] Vitouva | Concrete Sleepers in CSD tracks[END_REF][START_REF] Shayan | Microscopic feature of cracked and uncracked concrete railway sleepers[END_REF][START_REF] Oberholster | Cracked prestressed concrete railway sleepers : alkali-silica reaction or delayed ettringite formation[END_REF][START_REF] Mielenz | Investigation of prestressed concrete railway tie distress[END_REF][START_REF] Sahu | Delayed ettringite formation in Swedish concrete railroad ties[END_REF][START_REF] Silva | Diagnosis and prognosis of Portuguese concrete railway sleepers degradation -a combination of ASR and DEF[END_REF], and massive cast-in-place concrete components [START_REF] Collepardi | Damage by delayed ettringite formation[END_REF][START_REF] Hobbs | Cracking of concrete attributed to delayed ettringite formation[END_REF][START_REF] Thomas | Diagnosing delayed ettringite formation in concrete structures[END_REF][START_REF] Ingham | Delayed ettringite formation in concrete structures[END_REF]. DEF was first observed in France, in 1997 [START_REF] Divet | Existe-t-il un risque de développement d'une réaction sulfatique d'origine endogène dans les pièces en béton de grande masse ? Le cas du pont d'Ondes[END_REF], on bridges whose concrete had been cast on site. The bridge parts damaged by DEF were primarily massive structural elements (piers, crossbeams on piers or abutments, etc.) in contact with water or subjected to high moisture. DEF is distinct from the more traditional sulphate reactions where the sulphates attack the concrete from outside and create a progressive degradation from the surface towards the interior of the element. DEF affects the interior concrete without any ingress of external sulphates. It leads to a concrete swelling and the cracking of the structure.

Diagnosis of DEF

Laboratory studies were conducted using petrographic and mineralogical analysis [START_REF] Deloye | Utilisation du calcul automatique en analyse minéralogique quantitative[END_REF][START_REF] Afpc-Afrem | Durabilité des bétons -Méthodes recommandées pour la mesure des grandeurs associées à la durabilité[END_REF]. They included the following investigations: porosity, chemical analysis, X-ray diffraction, thermal analysis and scanning electron microscopy (on polished sections and broken surfaces). Several core samples were taken from the structures, with varying depths. The specimens were taken from both degraded zones and un-degraded zones to allow comparison between the two.

It is important to note that no alkali-silica reaction product was observed in the following investigated structures cases. The microscopic analyses showed that the studied concretes have been subjected to a major expansive internal sulphate reaction as ettringite formed after hardening. This check was conducted with the objective of avoiding any confusion with damages caused by the alkali-silica reaction. This is important as when both causes of deterioration are present, it is difficult to know which reaction is mainly responsible for the damage.

The internal sulphate reaction results in the presence of ettringite. This latter was present throughout the material and on all the examined surfaces. The ettringite that is generally observed in the concrete has a wide range of textures, but it is mainly poorly crystallized and massive. Furthermore this form is principally present at the cement paste/aggregate interfaces. Figure 1 Temperature is a fundamental parameter for the formation of delayed ettringite. For this reason it is necessary to evaluate the temperature within concrete during setting. Our starting hypothesis was that the temperature rise in a massive concrete element can be estimated on the basis of the temperature rise under adiabatic conditions. The temperature field T(X,t) was predicted at all points of coordinates X in the structure and at each time t. This prediction is based on a finite element calculation which requires the knowledge of the exothermic properties of the concrete, the geometry and the parameters that regulate exchanges between the environment and the concrete element. Modelling was conducted as a diagnostic aid to understand the damage observed in the structure after several years. Numerical simulation of the thermal effects was undertaken with the TEXO module which is part of the CESAR-LCPC finite element program of IFSTTAR.

Lessons learned from investigations of structures

Most of the following cases are presented in detail in the report of L. Divet [START_REF] Divet | Les réactions sulfatiques internes au béton : contribution à l'étude des mécanismes de la formation différée de l'ettringite[END_REF].

The Ondes bridge

This bridge, located near Toulouse, was completed and opened to traffic in 1955. It consists of four pre-tensioned beams which are joined together by a top slab and crossbeams (Fig. 2). It has a total length of 202 m and consists of five independent spans. The average height of the piers is 6.8 m. They are constructed on mass concrete which is set in the molassic substratum and they consist of two circular columns, 2 m in diameter, which are connected at the top by a cap beam (see Fig. 2). These latter are either hollow, at piers P1 and P3 supporting the moving bridge bearings, or solid, at piers P2 and P4 to which the fixed bridge bearings are fitted.

The damage only affects the solid cap beam of pier P2 (Fig. 3), the concrete of which was cast in August 1954. This is a solid parallelepiped reinforced concrete cap beam, measuring 1.5 m high, 8.2m long and 2.7 m wide. The first visible defects were observed during a detailed inspection conducted in 1982. The inspection report mentions the presence of cracking and the growth of vegetation at the ends of the pier cap beam. The previous inspection reports did not mention these defects. We can consequently assume that the damage first occurred in the early 1980s. The damage appeared as a network of widely-spaced multidirectional cracks. Cracking sometimes has a preferential orientation that depends on how the reinforcement is distributed. The crack widths vary from a few tenths of millimetres to a few millimetres according to the zones. The cracks are frequently emphasized by moisture and, occasionally, a whitish or greyish substance has been observed to exude from the cracks. The horizontal cracks appear to be more severe and are occasionally as much as 7 mm wide. Lastly, the damage is mainly present at the cap beam ends, on the facings that are directly exposed to bad weather and water dripping from the footpaths. No damage was found on the central part of the cap beam.

The concrete of the cap beams has a specified cement content of 350 kg/m 3 of Ordinary Portland Cement (CEM I type), with a water/cement ratio of 0.5 in mass. The aggregates are alluvial. The investigations conducted in the laboratory showed that the deteriorated concrete of pier P2 did not conform to the specifications. Analysis showed that the concrete contains a cement overdose of about 100 kg/m 3 (cement content of 430 kg/m 3 ). The mix is well composed of sand and gravels 5/15, but it incorporates gravels 25/50 that were not specified initially (see table 1).

The heating cycle of the concrete during its setting was then estimated by a numerical simulation using the TEXO module. For this, information was required on: meteorological data during the setting period of the concrete, geometry of the structural element, type and duration of use of the formwork, construction phases, exothermic properties of the concrete.

A three-dimensional thermal modelling was performed using this data (fig. 4). Figure 5 shows a plot of temperature versus time. The temperature increase was important for the cap beam of pier P2: the maximum temperature reached in the centre of the material was close to 80 °C, while for the cap beam of pier P4, the temperature did not exceed 70 °C. Moreover, the temperature in the concrete of pier P2 remained over 70°C during about 70 hours. In consequence the deterioration due to the DEF is limited to that in the cap beam of pier P2. This is the consequence of the initial heating of the concrete from the combination of several causes: use of an exothermic OPC for a massive concrete part, pouring of the concrete during summer time without any particular precautions and an overdose of the cement relative to the initial specified concrete composition.

As part of the remedial programme on the Ondes bridge, the waterproofing membrane was repaired and the expansion joints of the roadway and sidewalk were reformed in June 1995. On pier P2, the extremities of the cap beam were treated by an injection of the cracks and then protected by a surface coating. In July 1996, a discontinuous horizontal crack of width 0.2 mm was observed. Water inflows are also observed on the extremities, despite the repair made in 1995. The evolution of the phenomenon is confirmed in 1998 with the occurrence of new cracks and by the results of the monitoring of the global deformations that has been installed. This attests of the difficulty to implement durable repair solutions for this type of pathology, and justifies the interest in enforcing preventive measures during construction to avoid DEF.

The Bourgogne bridge

The Bourgogne bridge is a cable-stayed bridge built during the years 1989-1992 over the Saône River in Chalon/Saône (Fig. 6). It has a total length of 351 m and comprises 8 spans. The main span is 151.80 m long, and the pylons are 46 m high. The deck is made of a prestressed concrete slab. The lower part of each pylon is composed of a footing, a base and two legs. The dimensions (in metres) with respect to length, width and height are respectively for the footing: 15.20 x 6 x 2, for the base: 11.90 x 3 x 6.5, and for the legs: 2.20 x 3 x 6.

The footing is always immersed, the base is partially immersed in water but may be sometimes fully immersed during floods, and the legs are always above the water level of the river. The immersed parts of the base of the pylons P3 and P4 are affected by a multidirectional cracking like in the aerial parts (fig. 7). The width of the cracks reaches several millimetres on P3, and is lower than one millimetre on P4. A high cracking density is observed on the faces of the base, with a maximum density in the middle of the faces. The most important crack is observed on the top face of the base with a width of 1 cm. Some cracks also show an offset of the crack lips which is typical of a differential expansion inside the concrete; the most important offset is 2 to 3 mm on the top face of the P3 base. The lateral faces of the legs of pylons P3 and P4 present some cracks that are oriented horizontal, with widths varying between 0.2 and 0.4 mm. For the footing, the base and the legs, the cement content is 400 kg/m3, the water/cement ratio is 0.45 and the cement used is a CPA 55 R (CEMI 52.5 N type).

A finite element calculation of the temperature reached in the base and in the legs was conducted with the module TEXO, and with the help of adiabatic test results obtained on a reconstituted concrete. The results show that the temperature in the footing, the base and the two legs exceeds the 65 °C threshold, with respectively 79 °C for the footing and the base, 71°C for the lower part of the legs and 61 °C for the upper part of the legs. They show that the temperature of the base remains over 70 °C during 6 days, while those of the footing remains over 70 °C during 3 days. The laboratory investigations (particularly with the Scanning Electron Microscope) confirmed that the concrete of the basement was affected by DEF and that the concrete of the footing and legs was unaffected.

The Lodève Bridge This bridge was constructed during the years 1980-1981. It is a statically determinate prestressed beam structure with a three span deck which is 120 m long and 13 m wide. The deck is supported by two abutments and two piers constructed on footings. The damage is confined to the cap beams of the two piers. Each pier (Fig. 8) consists of a pillar whose crosssection fits within a rectangle measuring 3.1 x 7.5 m, with a height of 14.5 m in the case of the northern pier and 9.3 m in the case of the southern pier. Each pillar is topped by a cap beam which has a length of 14 m, a width of 3.5 m, and whose height varies between 1.1 m and 2.0 m. The extremities of the cap beams are covered with architectural facings consisting of six prefabricated reinforced concrete units. The two cap beams were built in August and September 1980.

The cap beam on the North pier has closely-spaced vertical cracks on both sides with a maximum crack width varying between 0.2 mm and 1.2 mm (fig. 9). The extremities of the cap beam are covered by the architectural facings which prevent their observation. Some cracks contain streaks of calcite, which indicates a circulation of water inside the concrete. Several horizontal cracks are also present and appear to be located at construction joints. The results of the 1997 crack survey for the cap beam of the north pier shows a considerable increase in cracking as compared with the 1989 inspection. The cap beam of the South pier suffers from the same type of damage, but seems less severely affected. The maximum crack width is only 0.3 mm, except on the east extremity where widths attain 1.7 mm. Furthermore, the network of cracks is denser and efflorescence can be observed in this zone. Lastly, the detailed inspection conducted in 1997 showed a lack of drainage from the supports (abutments and piers). Therefore, some parts of the supports are abnormally exposed to a humid environment.

The damage rating of a concrete structure may be determined with the help of a cracking index which corresponds to the summation of cracks intercepted on axes of variable lengths and normalized to a length of one metre (LCPC 1997). One of the cap beam of the Lodève bridge has been monitored with the help of cracking indexes and measurements of overall movement (LCPC 2009) since 2000. The cracking index measured in the central part of the cap beam is 0.55 mm/m and is stabilised. According to the classification (LCPC 1997), this value corresponds to a low cracking. For the extremity of the cap beam, the cracking index reached 1.64 in 2002 and 2.0 in 2007; it indicated an evolution of the phenomenon and the cracking is rated moderate to high. This was confirmed by the overall movement monitoring which gives a global deformation rate for the cap beam extremity of 0.2 mm/m/year.

The concrete of the piles and cap beams was made with an OPC cement (CEM I type) having a rapid hardening; the cement content was 400 kg/m 3 and the W/C ratio was 0.47. The aggregates are silico-calcareous.

A finite element calculation of the temperature reached in the cap beam and the pillar (module TEXO) was carried out to explain the difference of behaviour between the damaged cap beam and the non damaged pillar. Laboratory investigations (particularly with the Scanning Electron Microscope) confirmed that the concrete of the pillar was unaffected by DEF. In the cap beam, the calculated temperature increase was close to 80 °C in the centre and the temperature remained over 70 °C for nearly 5 days. In the pillar, which is less massive and has a particular geometry which favours the heat losses because of a larger surface area, the maximum calculated temperature reached 70 °C in a relatively small zone in the centre. Most of the concrete of the pillar reached a temperature below 65 °C. This was the main reason for the absence of DEF in the pillar but it is also less exposed to water ingress and humidity..

The Bellevue Bridge

This bridge was constructed in Nantes in 1988-1989 and opened in 1990 (Fig. 10). It is a continuous box girder bridge with 6 spans and a total length of 385 m, and has 7 supports (2 abutments and 5 piers). The 5 reinforced concrete piers are massive and are 6.3 m long by 3.0m wide. The damage is mostly confined to pier P6 on which multidirectional cracking typical of that caused by an internal swelling reaction is apparent (Fig. 11). In addition the map cracking is mainly localised within the tidal zone, where the average tidal range is between 3 and 4 metres. The four sides of the pier are affected and cracking seems to be progressing in the dry zone. The concrete of this pier was cast in August and September 1989. Pier P4 also exhibits map cracking, but this is less severe. Lastly, piers P3 and P5 have vertical cracks with widths reaching 0.8 mm. However, this cracking was observed immediately after construction and was attributed to thermal shrinkage of the concrete.

The cement content is 380 kg/m 3 , the W/C ratio is 0.54, and the aggregates are predominately from siliceous and silicate rocks. The cement is a Portland cement (CEM II/A type) composed of about 10 % of calcareous fillers (see table 1). The evolution of the temperature inside the concrete has not been estimated by a finite element calculation. The maximum temperature was estimated by the simplified method of calculation presented in the annex 4 of the LCPC guide (LCPC 2007). With the hypothesis that the temperature of the fresh concrete at the moment of pouring was equal to 20 °C, the maximum temperature at the centre of the pier was estimated to have reached 75 °C.

The Saint-Maurice de Beynost bridge This bridge, built in 1992, has a deck made of reinforced concrete beams and is composed of three independent spans made continuous on piers by the slab over the beams. The spans are skewed and have different length (14.92 m -17.44 m -9.47 m). The deck is resting on two abutments and two intermediary piers (fig. 12). Each abutment is composed of 4 rectangular columns topped by a cap beam. Each pier is composed of 5 columns of 5.4 m height and also topped by a cap beam.

The damage has occurred only on the extremity of the cap beam of the pier P2 which is exposed to the elements (fig. 13). A map cracking was observed during the first detailed inspection of the bridge in 1992. This cracking is evolving since the width of the cracks was below 0.3 mm in 1992 and reached up to 2 mm in the detailed inspection of 1997. From the documentation of the bridge, it was possible to find the dates of casting of the concrete in the different parts of the bridge. The cap beam of pier P2 was built in August 1982. The cement is an OPC (CEM I type), the W/C ratio is 0.49, and the concrete composition is close to the specified value of 350kg/m 3 given in table 1.

The calculation of the temperature increase with time in the bridge parts (pier and cap beam) was conducted with the TEXO. It was based on the meteorological data for the construction period and data obtained from adiabatic tests carried on the concrete composition reconstituted in laboratory. No damage has occurred in the piers, and the calculation shows that the maximum temperature reached at the centre of the piers was 61 °C. For the cap beam of the pier P2, laboratory investigations identified characteristic features of DEF, and the maximum temperature reached at the centre of this cap beam was 69 °C. The calculation shows also that the cooling of the cap beam was slow and that its concrete temperature remains greater than 60 °C for 44 hours.

The Cheviré Bridge

The Cheviré bridge was opened to traffic in May 1991 and is classified as an exceptional bridge (Fig. 14). It comprises two access viaducts composed of continuous box girders made of prestressed concrete having respectively a length of 603 m for the north viaduct and 798 m for the south. The central span has a length of 242 m and includes an independent steel box girder 164 m long, supported by cantilever prestressed concrete box girder. The total length of the bridge is 1562 m. The North viaduct is supporting by one abutment and nine piers, and the South one is supporting by one abutment and twelve piers. The hollow piers have a variable height (from 5.2 to 43.5 m) and comprise concrete bases at the bottom. The dimensions of these bases are: 3.15 x 3.90 x 4.00 m.

In 1998, map cracking was observed on the facings of some bases of the South access viaduct (Fig. 15). DEF was identified by laboratory investigations. The rate of deterioration of the bases is variable: some bases presented only some vertical cracks with a width lower than 0.3mm while others exhibited more or less developed map cracking (up to a width of 0.6 mm). The phenomenon is globally slow and the rate of opening of the cracks was 0.1 mm/m/year, from 2003 to 2006. The searches in the records showed that the damaged bases were cast during the summer 1988. The cracked faces are exposed to rain and capillary rises. The concrete was made with a Portland cement incorporating calcareous fillers (cement II/A type), a marine sand and gneiss aggregates.

The strategy for prevention

The French recommendations for the prevention of damage due to DEF (LCPC 2007) were established from data available in the scientific literature and from the detailed studies of damaged structures, like the six cases presented above. From this analysis, a strategy was developed based on categorising the structure according to the level of risk of occurrence of cracking that can be accepted, and related to the environmental conditions to which the structure is exposed during its service life. This strategy (Godart 2012) defines four levels of prevention with precautions that have to be applied related to risk and environmental categories. The precautions are mainly based on the limitation of the maximum temperature reached within the centre of the concrete pour during hardening of the concrete, and on the choice of an appropriate composition of the concrete.

The structures (or parts of them) are classified in 3 categories that are representative of the level of risk of cracking with respect to DEF that are acceptable for a given structure (or a part of it). The choice of the structure category is a function of the nature of the structure, its purpose, the consequences of the cracking in relation with the desired safety level, and its future maintenance: Category I refers to the structures for which the consequences of the occurrence of cracking are low or acceptable (for example: temporary structures), Category II gathers the structures for which the consequences of the occurrence of cracking would be detrimental (like resisting elements of most buildings and civil engineering structures), and Category III corresponds to structures for which the consequences of the occurrence of cracking would be unacceptable (like nuclear containment, dams or tunnels). Three exposure classes take into account the fact that water or a high ambient humidity are factors necessary for the development of DEF: the class XH1 concerns a dry or moderate humidity, the class XH2 is related with alternation of humidity and drying or with a high humidity, and the class XH3 considers structures in constant with water (permanent immersion, water stagnation on the surface, tidal zone,...).

Then four levels of prevention are determined according to the risk category of the structure and the exposure class XH to which the considered part of the structure is subjected. The choice of these levels of prevention indicated by the letters As, Bs, Cs and Ds, is the responsibility of the structure owner in setting the design specification. Table 3 is helpful when making this selection.

The principle of prevention relies primarily on the limitation of the heating of the concrete characterised by the maximum temperature, Tmax, likely to be reached within the structure and, also, by the duration of the period for which a high temperature is maintained. The limiting values of Tmax were defined by a group of expert on the basis of a literature review, a comparative study of factors encountered in damaged and non damaged bridges (Table 2), several French research thesis, and some foreign recommendations such as [START_REF] Bre | Delayed Ettringite formation: in-situ concrete[END_REF]. The precautions corresponding to the four levels of prevention are the following:

• Level As: Tmax < 85 °C. However in the case of a heat treatment applied on a precast element, it is authorized to go beyond the temperature Tmax = 85 °C up to 90 °C, provided that the length of the period during which the temperature exceeds 85°C is limited to 4 hours.

• Level Bs: Tmax < 75 °C. However if the maximum temperature reached in the concrete cannot remain lower than 75°C, then it must remain lower than 85 °C and at least one supplementary condition must be respected.

• Level Cs: Tmax < 70 °C. However if the maximum temperature reached in the concrete cannot remain lower than 70°C, then it must remain lower than 80 °C and at least one supplementary condition must be respected.

• Level Ds: Tmax < 65 °C. But if Tmax cannot remain lower than 65 °C, then it must remain lower than 75°C with the use of mineral additions and the validation of the concrete composition by an independent laboratory expert in DEF.

The supplementary condition should be selected from the following: o Equivalent active alkalis of the concrete < 3 kg/m 3 o Use of a cement conforming to the standard NF P 15-319 (ES) for sulphate resisting cements, and some additional conditions (see LCPC 2007) o Use of cements non conforming to the standard NF P 15-319 of the type CEM II/B-V or CEM II/B-S or CEM II/B-Q or CEM II/B (S-V) or CEM III/A or CEM V, with SO 3 of cement < 3 % and C 3 A of the clinker < 8 % o Use of fly ashes, slags, calcinated natural pozzolans or metakaolin in combination with a CEM I, with additions content > 20 %, and with SO 3 of cement < 3 % and C 3 A of the clinker < 8 % o Application of a performance test on the concrete mix according to (LCPC 2007) o For precast elements, the concrete mix and heating cycle identical or similar to that having at least 5 references of use in similar conditions without any problem.

  is a back-scattered electron (BSE) image showing ettringite-filled gap around aggregate particles.

Figure 1 :

 1 Figure 1: BSE image of a polished sample showing ettringite-filled gap around aggregate particles.

Figure 2 :

 2 Figure 2: Overall view of the Ondes Figure 3: View of the end of the cap beam bridge in 2005, after the treatment applied in 1995

Figure 4 :

 4 Figure 4: 3D modeling of the cap beam of pier P2 with CESAR-LCPC.

Figure 5 :

 5 Figure 5: Computed temperatures in the cap beam of pier P2

Figure 6 :

 6 Figure 6: Overall view of the Bourgogne Figure 7: Cracking of the aerial part bridge of the pylon base

Figure 8 :

 8 Figure 8: Overall view of a Figure 9: Network of cracks on the cap beam Lodève Bridge pier of the North pier

Figure 10 :

 10 Figure 10: Overall view of The Bellevue Figure 11: Pier 6 of the Bellevue Bridge Bridge with map cracking in the tidal zone.

Figure 12 :

 12 Figure 12: Partial view of the Saint Maurice de Beynost bridge

Figure 14 :

 14 Figure 14: General view of the Cheviré Bridge. (Courtesy G. Forquet)

Table 1 :

 1 Specified concrete composition of the different bridges (in kg/m 3 ).

	Material	Cement	Water	Filler	Sand	Gravel	Gravel	Gravel
	Ondes	350	175		750	1100
					(0/5)	(5/15)
	Bourgogne	400	180		730	376	700
					(0/4)	(4/12)	(12/20)
	Lodeve	400	188	140*	555	395	310
				(0/3)	(0/4)	(4/10)	(10/20)
	Bellevue	380	205		800	290	130	640
						(6/10)	(10/14)	(14/20)
	Beynost	350	170		750	250	930
					(0/5)	(5/10)	(10/20)
	Cheviré	385	185	75	730	285	130	630
						(6/10)	(10/14)	(14/20)
	* calcareous sand

Table 2 :

 2 Comparative study of factors encountered in the investigated bridges.

			Ondes	Bourgogne Lodève	Bellevue Beynost Cheviré
	Date of		1955	1990	1980	1988	1982	1988/89
	construction
	Casting of		August	August/	August/	August/	August	July/
	concrete			September	September	September		August
	Structural		Cap beam	Base of	Cap beam	Pier	Cap	Base of
	part			pylon			beam	pier
	Delay in		27 yrs	6 yrs	9 yrs	10 yrs	10 yrs	8 yrs
	occurrence
	of damage
	Environ-	Waterproofing	Immersed	Lack of	Immersed	Exposure	Rains and
	ment		Problem	& variable	drainage	& variable	to rains	capillarity
				immersion		immersion
	T max (°C)		80	79	80	80	69	75
	W/C		0.50	0.45	0.47	0.54	0.49	0.48
	ratio
	Nature of		CEM I	CEM I	CEM I	CEM II/A	CEM I CEM II/A
	cement
	Cement con-	430	400	400	380	350	385
	tent (kg/m 3 )
	SO 3 cement		2,5	2,8	2,6	2,5	3,4	2,5
	(%)
	C3A cement		11,2	8,2	9,8	7,0	10,4	7,0
	(%)

Table 3 :

 3 Levels of prevention defined by structure risk category with exposure

	Risk category of structure	Moisture/water exposure class of the structural part
		XH1	XH2	XH3
	I	As	As	As
	II	As	Bs	Cs
	III	As	Cs	Ds

Acknowledgment:

The authors thank Jonathan G.M. Wood for his comprehensive review of the English usage and terminology, and his valuable technical comments.


N Forcadel

A Ghorbel
email: ghorbel@cermics.enpc.fr

S Walha

Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model

Keywords: AMS Classification: 35B27, 35F20, 45K05, 47G20, 49L25, 35B10 Frenkel-Kontorova models, Traveling waves, Viscosity solutions, Maximum principle, Hull function

In this paper, we study the existence and uniqueness of traveling wave solution for the accelerated Frenkel-Kontorova model. This model consists in a system of ODE that describes the motion particles in interaction. The most important applications we have in mind is the motion of crystal defects called dislocations. For this model, we prove the existence of traveling wave solutions under very weak assumptions. The uniqueness of the velocity is also studied as well as the uniqueness of the profile which used different types of strong maximum principle. As far as we know, this is the first result concerning traveling waves for accelerated, spatially discrete system.

Introduction

In the present paper, we study the accelerated Frenkel-Kontorova model (F-K) which describes a chain of particles interacting by an harmonic potential. Besides its original aim of modeling crystal dislocations, the (F-K) model has many applications in physics such as the description of magnetic domain walls, atoms adsorbed on a crystalline surface or superionic conductors (see for instance the book of Braun and Kivshar [START_REF] Braun | The Frenkel-Kontorova model, Concepts, Methods and Applications[END_REF] for an introduction to this model). The goal of this work is to prove the existence and the uniqueness of traveling wave as well as the uniqueness of the velocity. This work is a generalization of the one of Al Haj et al. [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF] in which the authors study the fully overdamped case.

The study of traveling waves in reaction-diffusion equations has been introduced in pioneering works of Fisher [START_REF] Fisher | The advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky and Piskunov [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]. Existence of traveling waves solutions has been for instance obtained in [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics[END_REF][START_REF] Berestycki | Traveling wave solutions to combustion models and their singular limits[END_REF][START_REF] Fife | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF][START_REF] Kanel´ | Certain problems on equations in the theory of burning[END_REF]. More generally, there is a huge literature about existence, uniqueness and stability of traveling waves with various non linearities with applications in particular in biology and combustion and we refer for instance to the references cited in [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Chen | Traveling waves in discrete periodic media for bistable dynamics[END_REF]. There are also several works on discrete or nonlocal versions of reaction-diffusion equations (see for instance [START_REF] Bates | Traveling waves of bistable dynamics on a lattice[END_REF][START_REF] Bates | Traveling waves in a convolution model for phase transitions[END_REF][START_REF] Carpio | Wave solutions for a discrete reaction-diffusion equation[END_REF][START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF][START_REF] Chow | Traveling waves in lattice dynamical systems[END_REF][START_REF] De Masi | Travelling fronts in non-local evolution equations[END_REF][START_REF] Elmer | A variant of Newton's method for the computation of traveling waves of bistable differential-difference equations[END_REF][START_REF] Guo | Traveling wave solution for a lattice dynamical system with convolution type nonlinearity[END_REF][START_REF] Hankerson | Wavefronts for a cooperative tridiagonal system of differential equations[END_REF][START_REF] Mallet-Paret | The Fredholm alternative for functional differential equation of mixed type[END_REF][START_REF] Mallet-Paret | The global structure of traveling waves in spatially discrete dynamical systems[END_REF][START_REF] Wu | Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations[END_REF][START_REF] Zinner | Existence of traveling wavefront solutions for the discrete Nagumo equation[END_REF][START_REF] Zinner | Traveling wavefronts for the discrete Fisher's equation[END_REF] and the references cited therein) and on damped hyperbolic equation (see [START_REF] Dunbar | On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks[END_REF][START_REF] Gallay | Scaling variables and stability of hyperbolic fronts[END_REF][START_REF] Gallay | Scaling variables and asymptotic expansions in damped wave equations[END_REF][START_REF] Gallay | Stability of travelling waves for a damped hyperbolic equation[END_REF][START_REF] Hadeler | Hyperbolic travelling fronts[END_REF]) but, as far as we know, there is no result concerning hyperbolic discrete in space equations.

The Frenkel-Kontorova model

The classical Frenkel-Kontorova (F-K) model describes the dynamics of crystal defects. If u i (t) is the position of the particle i ∈ Z, then the classical (F-K) models is given by the following dynamics

m 0 d 2 u i dt 2 + du i dt = u i+1 + u i-1 -2u i -sin(2π(u i -L)) -sin(2πL)
where d 2 u i dt 2 denotes the acceleration of the ith particle, du i dt is its velocity, m 0 denotes the mass of the particles, -sin(2πL) is a constant driving force which will cause the movement of the chain of atoms and -sin(2π(u i -L)) describes the force created by a periodic potential whose period is assumed to be 1. We set

f L (v) = -sin(2π(v -L)) -sin(2πL)
and for all i ∈ Z Ξ i (t) = u i (t) + 2m 0 du i dt (t).

We replace it in equation (1.2) in order to obtain the following monotone system: for i ∈ Z and t ∈ (0, +∞),

(1.1)

     du i dt = 1 2m 0 (Ξ i -u i ) dΞ i dt = 2(u i+1 + u i-1 -2u i ) + f L (u i ) + 1 2m 0 (u i -Ξ i ).
We look for particular traveling wave solution of (1.1), which have the form

(1.2) u i (t) = φ 1 (i + c t) Ξ i (t) = φ 2 (i + c t).
If we replace (1.2) in (1.1), then the profile (φ 1 , φ 2 ) should satisfy

(1.3) cφ ′ 1 (z) = α 0 (φ 2 (z) -φ 1 (z)) cφ ′
2 (z) = 2(φ 1 (z + 1) + φ 1 (z -1) -2φ 1 (z)) + 2f L (φ 1 (z)) + α 0 (φ 1 (z) -φ 2 (z))

with z = i + c t and α 0 = 1 2 m 0 . We then have the following result.

Theorem 1.1 (Existence and uniqueness of traveling wave solution for Frenkel-Kontorova model).

There exists a constant α * (which will be made precised later on assumption (A)) such that for all α 0 ≥ α * , there exist a unique real c and two functions φ 1 : R → R and φ 2 : R → R that satisfy

(1.4)            cφ ′ 1 (z) = α 0 (φ 2 (z) -φ 1 (z)) cφ ′ 2 (z) = 2(φ 1 (z + 1) + φ 1 (z -1) -2φ 1 (z)) + 2f L (φ 1 (z)) + α 0 (φ 1 (z) -φ 2 (z)) φ 1 , φ 2 are non-decreasing over R φ 1 (-∞) = 0, φ 1 (+∞) = 1 φ 2 (-∞) = 0, φ 2 (+∞) = 1
in the classical sense if c = 0 and almost everywhere if c = 0. Moreover, if c = 0, then the two profiles are unique up to translation.

Main results for the general case

We now consider a generalization of system (1.3). Given a function F : [0, 1] N +1 → R, we consider the system c φ ′ 1 (z) = α 0 (φ 2 (z) -φ 1 (z)) c φ ′ 2 (z) = 2 F ((φ 1 (z + r i )) i=0,...,N ) + α 0 (φ 1 (z) -φ 2 (z)). In order to provide our results, we introduce some assumptions on F .

Assumption (A)

• Regularity of F : F is globally Lipschitz continuous over [0, 1];

• Monotonicity of F : F (X 0 , ..., X N ) is non-decreasing in X i for i = 0, and (1.5) 2 ∂F ∂X 0 + α 0 > 0.

We set F (v, ...., v) = f (v).

Assumption (B)

• Instability: f (0) = 0 = f (1) and there exists b ∈ [0, 1] such that

f (b) = 0, f |(0,b) < 0, f |(b,1) > 0 and f ′ (b) > 0.
• Smoothness: F is C 1 in a neighborhood of {b} N +1 .

We give the first main result concerning the existence of traveling wave.

Theorem 1.2 (Existence of a traveling wave). Under assumptions (A) and (B), there exist a real c and two functions φ 1 : R → R and φ 2 : R → R that solves

(1.6)            c φ ′ 1 (z) = α 0 (φ 2 (z) -φ 1 (z)) c φ ′ 2 (z) = 2 F ((φ 1 (z + r i )) i=0,...,N ) + α 0 (φ 1 (z) -φ 2 (z)) φ 1 , φ 2 are non-decreasing over R φ 1 (-∞) = 0, φ 1 (+∞) = 1 φ 2 (-∞) = 0, φ 2 (+∞) = 1
in the classical sense if c = 0 and almost everywhere if c = 0.

In order to prove the uniqueness of the traveling waves, we require some additional assumptions: Assumption (C): Inverse monotonicity close to {0} N +1 and E = {1} N +1 : There exists β 0 > 0 such that for a > 0, we have F (X + (a, ..., a)) < F (X) for all X, X + (a, ..., a) ∈ [0, β 0 ] N +1 F (X + (a, ..., a)) < F (X) for all X, X + (a, ..., a) ∈ [1 -β 0 , 1] N +1 Assumption (D+) i) All the r i have the same sign: We assume that r i ≤ 0, for all i ∈ {0, ..., N }.

ii) Strict monotonicity: F is increasing in X i + with r i + > 0.

Assumption (D-)

i) All the r i have the same sign: We assume that r i ≥ 0, for all i ∈ {0, ..., N }.

ii) Strict monotonicity: F is increasing in X i -with r i -< 0.

We give the second main results concerning the uniqueness of the velocity and of the profile. Theorem 1.3 (Uniqueness of the velocity and of the profile). We assume (A) and let (c, (φ 1 , φ 2 )), with φ 1 , φ 2 : R → [0, 1], be a solution of

(1.7)        c φ ′ 1 (z) = α 0 (φ 2 (z) -φ 1 (z)) c φ ′ 2 (z) = 2 F ((φ 1 (z + r i )) i=0,...,N ) + α 0 (φ 1 (z) -φ 2 (z)) φ 1 (-∞) = 0, φ 1 (+∞) = 1 φ 2 (-∞) = 0, φ 2 (+∞) = 1.
Then, we have the following properties.

) i) or ii) if c > 0 (resp. (D-) i) or ii) if c < 0), (φ 1 , φ 2
) is unique (up to translation) and φ 1 and φ 2 are increasing.

Remark 1.4. We note that F (X 0 , X 1 , X 2 ) = X 1 + X 2 -2X 0 -sin(2π(X 0 -L)) -sin(2πL)
satisfies assumptions (A), (B), (C) , (D+)ii) and (D-)ii). Then Theorem 1.1 is a direct application of Theorems 1.2 and 1.3.

For this paper, we define

(1.8) r * = max i=0...N |r i |
and we assume that r * > 0 (otherwise, the system reduce to a single ODE).

Organization of the paper

In Section 2, we give the definition of viscosity solution and of Hull function. We also recall some basic results about monotone functions. Section 3 is devoted to the proof of existence of traveling waves, namely Theorem 1.2. In Section 4, we study the question of uniqueness of the velocity by proving a comparison principal on the half line. Finally, in Section 5, we prove the uniqueness of the profile using different types of strong maximum principles.

Preliminary results

This section is divided into four subsections. The first one is devoted to the extension of the function F onto R N +1 . In the second subsection, we give the definition of viscosity solution while the notion of hull functions is recalled in the third one. Finally, we present some results about monotone functions in the last subsection.

Extension of F

To construct the traveling waves, we will use the hull functions constructed in [START_REF] Forcadel | Homogenization of accelerated Frenkel-Kontorova models with n types of particles[END_REF]. To do that, as in [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF], we will need to extend the function F by F which is defined over R N +1 and satisfied the following assumption:

Assumption ( Ã): a) Regularity: F is Lipschitz continuous over R N +1 . b) Periodicity: F (X 0 + 1, ..., X N + 1) = F (X 0 , ..., X N ) for every X = (X 0 , ..., X N ) ∈ R N +1 . c) Monotonicity: F (X 0 , ..., X N ) is non-decreasing in V i for i = 0 and (2.1) 2 ∂ F ∂X 0 + α 0 > 0.
Then, we have the following extension of the function F .

Lemma 2.1. Given a function F defined over Q = [0, 1] N +1 satisfying (A) and F (1, ..., 1) = F (0, ..., 0), there exists an extension F defined over R N +1 such that F|Q = F and F satisfies ( Ã).

Proof.

     (φ 1 , φ 2 ) is non-decreasing over R φ 1 (-∞) = 0, φ 1 (+∞) = 1 φ 2 (-∞) = 0, φ 2 (+∞) = 1.
Theorem 1.2 is then a direct application of the following result.

Proposition 2.3 (Existence of traveling waves). We assume that F satisfies ( Ã) and (B). Then there exist a real c and two functions φ 1 , φ 2 solutions of (2.2)

           c φ ′ 1 (z) = α 0 (φ 2 (z) -φ 1 (z)) c φ ′ 2 (z) = 2 F ((φ 1 (z + r i )) i=0,...,N ) + α 0 (φ 1 (z) -φ 2 (z)) φ 1 , φ 2 are non-decreasing over R φ 1 (-∞) = 0, φ 1 (+∞) = 1 φ 2 (-∞) = 0, φ 2 (+∞) = 1
in the classical sense if c = 0 and almost everywhere if c = 0.

Proof of Theorem 1.2. The proof of Theorem 1.2 is a direct consequence of Remark 2.2 and Proposition 2.3.

For simplicity of presentation, we call F as F in the rest of this section and in Section 3.

Viscosity solution

In this subsection, we give the definition of viscosity solution. We first recall the definition of the upper and the lower semi-continuous envelopes u * and u * :

u * (y) = lim sup x→y u(x) u * (y) = lim inf x→y u(x).
Definition 2.4 (Viscosity solution). Let c ∈ R and F be defined over R N +1 . Let u 1 : R → R and u 2 : R → R be two locally bounded and upper semi-continuous functions.

(u 1 , u 2 ) is called a sub-solution on an open set Ω ⊂ R of (2.3) c u ′ 1 (z) = α 0 (u 2 (z) -u 1 (z)) c u ′ 2 (z) = 2 F ((u 1 (z + r i )) i=0,...,N ) + α 0 (u 1 (z) -u 2 (z)) if for any test function ψ ∈ C 1 (Ω) such that (u 1 -ψ) (resp (u 2 -ψ)) reaches a local maximum at a point z ∈ Ω then we have c ψ ′ (z) ≤ α 0 (u 2 (z) -u 1 (z)) resp. c ψ ′ (z) ≤ 2 F ((u 1 (z + r i )) i=0,...,N ) + α 0 (u 1 (z) -u 2 (z)) .
Let u 1 : R → R and u 2 : R → R be two locally bounded and lower semi-continuous functions.

(u 1 , u 2 ) is called a super-solution of (2.3) on Ω if for any test function ψ ∈ C 1 (Ω) such that (u 1 -ψ) (resp (u 2 -ψ)) reaches a local minimum at a point z ∈ Ω then we have

c ψ ′ (z) ≥ α 0 (u 2 (z) -u 1 (z)) resp. c ψ ′ (z) ≥ 2 F ((u 1 (z + r i )) i=0,...,N ) + α 0 (u 1 (z) -u 2 (z)) .
Finally, a locally bounded functions (u 1 , u 2 ) is called a viscosity solution of (2.3) if ((u 1 ) * , (u 2 ) * ) is a sub-solution and ((u 1 ) * , (u 2 ) * ) is a super-solution.

Hull fonction

We present the notion of hull function for (1.3). This result has been proved in [START_REF] Forcadel | Homogenization of accelerated Frenkel-Kontorova models with n types of particles[END_REF]Theorem 1.10].

Proposition 2.5 (Existence of hull functions). Let F be a given function satisfying ( Ã) and let p > 0. Then there exists a unique λ p ∈ R such that there exists two locally bounded functions h p : R → R and g p : R → R satisfying (in the viscosity sense):

(2.4)                λ p h ′ p (x) = α 0 (g p (x) -h p (x)) λ p g ′ p (x) = 2F ((h p (x + p r i )) i=0...n ) + α 0 (h p (x) -g p (x)) h p (x + 1) = h p (x) + 1 g p (x + 1) = g p (x) + 1 h ′ p (x) ≥ 0 g ′ p (x) ≥ 0.
We then define (2.5)

φ 1 p (x) = h p (p x) φ 2 p (x) = g p (p x)
and c p = λ p p .

We now give some properties of the function (φ p 1 , φ p 2 ).

Lemma 2.6 (Properties of (φ p 1 , φ p 2 )). We assume that F satisfies ( Ã). Then the function (φ p 1 , φ p 2 ) defined in (2.5) satisfies in the viscosity sense

(2.6)                        c p (φ 1 p ) ′ = α 0 (φ 2 p -φ 1 p ) c p (φ 2 p ) ′ = 2F ((φ 1 p (x + p r i ) i=0,...,N )) + α 0 (φ 1 p -φ 2 p ) φ 1 p x + 1 p = φ 1 p (x) + 1, φ 2 p x + 1 p = φ 2 p (x) + 1 (φ 1 p ) ′ (x) ≥ 0, (φ 2 p ) ′ (x) ≥ 0.
Moreover, if c p = 0 then there exists M > 0 independent on p such that

(2.7) |(φ p i ) ′ | ≤ M |c p | for 0 < p < 1 r * and i = 1, 2.
Proof of Lemma 2.6. Equation (2.6) is obtained by the change of variables (2.5) in (2.4). We now prove (2.7). We fix p > 0 such that

1 p ≥ r * .
We first remark that the function ψ defined by ψ(x) = α 0 (φ 2 p (x) -φ 1 p (x)) is bounded (because ψ is continuous and periodic) then there exists M 1 > 0 such that

|ψ(x)| ≤ M 1 .
This implies that

|(φ p 1 ) ′ | ≤ M 1 |c p | .
On the other side, since φ 1 p is non-decreasing, we have

           φ 1 p (x + r i ) -φ 1 p (x) ≤ φ 1 p x + 1 p -φ 1 p (x) = 1 if r i ≥ 0 φ 1 p (x + r i ) -φ 1 p (x) ≤ φ 1 p x - 1 p -φ 1 p (x) = 1 if r i ≤ 0.
Moreover, using that F ∈ Lip(R N +1 ), we get

F ((φ 1 p (x + r i )) i=0,...,N ) -F ((φ 1 p (x)) i=0,...,N ) ≤ L 1 . . . 1 =: L 1 .
On the other hand, f is bounded (because f is Lipschitz continuous and periodic). Therefore

F ((φ 1 p (x + r i ) i=0,...,N )) ≤ L 1 + |f | L ∞ (R) and 2 F ((φ 1 p (x)) i=0,...,N ) + α 0 (φ 2 p (x) -φ 1 p (x)) ≤ 2(L 1 + |f | L ∞ (R) ) + M 1 =: M 2 . This implies that |(φ p 2 ) ′ | ≤ M 2 |c p | .
Taking M = max(M 1 , M 2 ), we get the desired result.

Useful results for monotone functions

In this subsection, we recall some results about monotone function that will be used later for the proof of Proposition 2.3. We state Helly's Lemma and the equivalence between viscosity and almost everywhere solution.

First we recall Helly's Lemma which gives the convergence of subsequence in the almost everywhere sense.

Lemma Finally, after the use of Lemma 2.7 we often need to apply the following lemma (which proof is very similar to the one of [START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF]Lemma 2.11]) in order to get a solution in the viscosity sense.

Lemma 2.8 (Equivalence between viscosity and a.e. solutions). We assume that F satisfies ( Ã).

Then φ 1 and φ 2 are viscosity solutions of

(2.8) 0 = α 0 (φ 2 (x) -φ 1 (x)), 0 = 2 F ((φ 1 (x + r i )) i=0....N ) + α 0 (φ 1 (x) -φ 2 (x))
if and only if φ 1 and φ 2 are solutions in the almost everywhere sense of the same equation.

Construction of a traveling wave

This section is divided into two subsections. In the first one, we control the velocity of propagation and give some properties on the plateau of the profiles. The second subsection is devoted to the proof of Proposition 2.3.

Preliminary results

We begin to show that the velocity c p is uniformly bounded in p. Proof. We consider the functions φ 1 p and φ 2 p given by (2.5) and satisfying (2.6). Let c p be the associated velocity given by (2.5). We assume by contradiction that when

p → p 0 ∈ [0, 1 r * ] lim p→p0 c p = +∞ (the case c p → -∞ being similar). Let φ1 p = φ 1 p (c p x) and φ2 p = φ 2 p (c p x) solution of (3.1)    ( φ1 p ) ′ = α 0 ( φ2 p -φ1 p ) ( φ2 p ) ′ = 2 F (( φ1 p (x + r i c p )) i=0,...,N ) + α 0 ( φ1 p -φ2 p ).
According to (2.7), we have ( φ1 for ε small enough. Using Ascoli theorem and diagonal extraction argument, we have, up to extract a subsequence, that φ1 p → φ1 φ2 p → φ2 .

p ) ′ = c p (φ 1 p ) ′ ≤ M ( φ2 p ) ′ = c p (φ 2 p ) ′ ≤ M for M independent of p. Since (3.
Moreover, by stability of viscosity solutions, φ1 and φ2 satisfy

( φ1 ) ′ (x) = α 0 ( φ2 (x) -φ1 (x)) ( φ2 ) ′ (x) = 2 F (( φ1 (x)) i=0,...,N ) + α 0 ( φ1 (x) -φ2 (x)) and φ1 (0) = b -ε. Since ( φ1 p ) ′ ≥ 0, ( φ2 p ) ′ ≥ 0, we have (φ 1 ) ′ ≥ 0, (φ 2 ) ′ ≥ 0. This implies that α 0 ( φ2 (x) -φ1 (x)) ≥ 0 2 f ( φ1 (x)) + α 0 ( φ1 (x) -φ2 (x)) ≥ 0. Therefore 2 f ( φ1 (x)) ≥ 0. In particular, 2 f ( φ1 (0)) = 2 f (b -ε) ≥ 0, which is a contradiction since f (b -ε) < 0 (see assumption (B)).
We continue with some properties on the plateau of the profiles. The following lemma shows that if one of the profile have a large enough plateau then the other profile has the same plateau.

Lemma 3.2 (Properties on the plateau of the profiles). Let (φ 1 , φ 2 ) be solution of

(3.2)      cφ ′ 1 (x) = α 0 (φ 2 (x) -φ 1 (x)) cφ ′ 2 (x) = 2 F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0, φ ′ 2 ≥ 0.
We assume that there exists a constant C, a point x 0 ∈ R and a > r * such that

φ 1 (x) = C ∀x ∈ (x 0 -a, x 0 + a) or φ 2 (x) = C ∀x ∈ (x 0 -a, x 0 + a). Then φ 1 (x) = φ 2 (x) = C ∀x ∈ (x 0 -a, x 0 + a). Proof. If φ 1 (x) = C ∀x ∈ (x 0 -a, x 0 + a), then φ ′ 1 (x) = 0 ∀x ∈ (x 0 -a, x 0 + a)
and the first equation of (3.2) implies the result.

Let us then assume that

φ 2 (x) = C ∀x ∈ (x 0 -a, x 0 + a).
We set

ψ 1 (x) = (φ 1 ) * (x + a) -(φ 1 ) * (x -a) and ψ 2 (x) = (φ 2 ) * (x + a) -(φ 2 ) * (x -a). Then (ψ 1 , ψ 2 ) is solution of (3.3)    cψ ′ 1 (x) ≥ α 0 (ψ 2 (x) -ψ 1 (x)) cψ ′ 2 (x) ≥ 2 [F (((φ 1 ) * (x + a + r i )) i=0,...,N ) -F (((φ 1 ) * (x -a + r i )) i=0,...,N )] +α 0 (ψ 1 (x) -ψ 2 (x)).
Since φ 1 and φ 2 are non-decreasing, we have ψ 1 ≥ 0 and ψ 2 ≥ 0. Moreover, ψ 2 (x 0 ) = 0. Hence, x 0 is a point of minimum of ψ 2 and the second equation of (3.3) implies that

0 ≥2 [F (((φ 1 ) * (x 0 + a + r i )) i=0,...,N ) -F (((φ 1 ) * (x 0 -a + r i )) i=0,...,N )] + α 0 ψ 1 (x 0 ) ≥2 [F (((φ 1 ) * (x 0 -a) + ψ 1 (x 0 ), (φ 1 ) * (x 0 -a + r i )) i=1,...,N ) -F (((φ 1 ) * (x 0 -a + r i )) i=0,...,N )] +α 0 ψ 1 (x 0 )
where we have used the monotony of F for the second inequality. We set

G(x) = 2F (((φ 1 ) * (x 0 -a) + x, (φ 1 ) * (x 0 -a + r i )) i=1,...,N ) + α 0 x.
Then, by assumption (1.5), G is strictly increasing. Using that

0 ≥ G(ψ 1 (x 0 )) -G(0),
we deduce that ψ 1 (x 0 ) = 0 (recall that ψ 1 ≥ 0). This implies that φ 1 is constant over (x 0 -a, x 0 +a) and by the first equation of (3.2), this constant is also equal to C.

In the proof of Proposition 2.3, we will need to pas to the limit for (φ 1 p , φ 2 p ). This is the goal of the following lemma. Lemma 3.3 (Passing to the limit for (φ 1 p , φ 2 p )). For every n ∈ N, let (c n , φ n 1 , φ n 2 ) be a solution of

(3.4)      c n (φ n 1 ) ′ (x) = α 0 (φ n 2 (x) -φ n 1 (x)) c(φ n 2 ) ′ (x) = 2 F ((φ n 1 (x + r i )) i=0,...,N ) + α 0 (φ n 1 (x) -φ n 2 (x)) (φ n 1 ) ′ ≥ 0, (φ n 2 ) ′ ≥ 0 satisfying        φ n 1 (x + 1) ≤ φ n 1 (x) + 1 φ n 2 (x + 1) ≤ φ n 2 (x) + 1 |c n | ≤ M 0 |c n (φ n 1 ) ′ | ≤ M 1 , |c n (φ n 2 ) ′ | ≤ M 2
where M 0 , M 1 and M 2 are positive constant. We also assume that there exists M 3 > 0 and x * ∈ R such that |φ n 1 (x * )| ≤ M 3 . Then there exists (c, φ 1 , φ 2 ) such that, up to extract a subsequence, c n → c, φ n 1 → φ 1 and φ n 2 → φ 2 a.e. and (c, φ 1 , φ 2 ) is a viscosity solution of

(3.5)      cφ ′ 1 (x) = α 0 (φ 2 (x) -φ 1 (x)) cφ ′ 2 (x) = 2 F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0, φ ′ 2 ≥ 0
Proof. Up to translate φ 1 , we assume that x * = 0. Since |c n | ≤ M 0 , up to extract a subsequence, we can assume that c n → c as n → +∞.

We study two cases for c.

|(φ n 1 ) ′ | ≤ 2 M 1 |c| and |(φ n 2 ) ′ | ≤ 2 M 2 |c| .
Using Ascoli's Theorem and the diagonal extraction argument, we can assume, up to a subsequence, that (φ n 1 ) n and (φ n 2 ) n converge locally uniformly on R respectively to φ 1 and φ 2 . By stability, φ 1 and φ 2 satisfy in the viscosity sense

(3.6)    cφ ′ 1 (x) = α 0 (φ 2 (x) -φ 1 (x)) cφ ′ 2 (x) = 2 F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0, φ ′ 2 ≥ 0.
Case 2: c = 0. We have

φ n 1 (1 + x) ≤ φ n 1 (x) + 1. This implies, using the fact that φ n 1 (0) ≤ M 3 , that (3.7) φ n 1 (x) ≤ ⌈x⌉ + M 3 for x ≥ 0 φ n 1 (x) ≥ -⌈|x|⌉ -M 3 for x ≤ 0
Using also the fact that 0

≤ |c n (φ n 1 ) ′ | ≤ M 1 , we get (3.8)    φ n 2 (x) ≤ ⌈x⌉ + M 1 α 0 + M 3 for x ≥ 0 φ n 2 (x) ≥ -⌈|x|⌉ -M 3 for x ≤ 0
Using Helly's Lemma (Lemma 2.7), up to extract a subsequence, we have φ n 1 → φ 1 a.e. and φ n 2 → φ 2 a.e.. This implies that

c n b2 b1 (φ n 1 ) ′ (x)dx = α 0 b2 b1 (φ n 2 (x) -φ n 1 (x))dx c n b2 b1 (φ n 2 ) ′ (x)dx = 2 b2 b1 F ((φ n 1 (x + r i )) i=0,...,N )dx + α 0 b2 b1 (φ n 1 (x) -φ n 2 (x))dx. for every b 1 ≤ b 2 . That is c n (φ n 1 (b 2 ) -φ n 1 (b 1 )) = α 0 b2 b1 (φ n 2 (x) -φ n 1 (x))dx c n (φ n 2 (b 2 ) -φ n 2 (b 1 ) = 2 b2 b1 F ((φ n 1 (x + r i )) i=0,...,N )dx + α 0 b2 b1 (φ n 1 (x) -φ n 2 (x))dx.
We have

     α 0 (φ n 2 (x) -φ n 1 (x)) → α 0 (φ 2 (x) -φ 1 (x)) a.e. 2 F ((φ n 1 (x + r i )) i=0,...,N ) + α 0 (φ n 1 (x) -φ n 2 (x)) → 2 F ((φ 1 (x + r i )) i=0,...,N )+ α 0 (φ 1 (x) -φ 2 (x)) a.e.
and (because of (3.7) and F is Lipschitz continuous and f is bounded)

|α 0 (φ n 2 (x) -φ n 1 (x))| ≤ M 1 |F ((φ n 1 (x + r i )) i=0,...,N )| ≤ C(1 + |x|), i.e. |F ((φ n 1 (x + r i )) i=0,...,N ) + α 0 (φ n 1 (x) -φ n 2 (x))| ≤ C(1 + |x|) + M 1 .
Thus, using Lebesgue's dominated convergence Theorem, we pass to the limit as n → +∞ and we get

0 = α 0 b2 b1 (φ 2 (x) -φ 1 (x))dx 0 = 2 b2 b1 F ((φ 1 (x + r i )) i=0,...,N )dx + α 0 b2 b1 (φ 1 (x) -φ 2 (x))dx. which implies that 0 = α 0 (φ 2 (x) -φ 1 (x)) a.e. 0 = 2F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) a.e. Since (φ n 1 ) ′ ≥ 0, (φ n 2 ) ′ ≥ 0, we get φ ′ 1 ≥ 0, φ ′ 2 ≥ 0 and so Lemma 2.8 implies (3.9)        0 = α 0 (φ 2 (x) -φ 1 (x)) 0 = 2F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0 φ ′ 2 ≥ 0
in the viscosity sense.

We finish this subsection with the following proposition which help us to identify the value of the plateau of the profiles. Proposition 3.4 (The value of the plateau of the profile are close to the zero of f ). We assume that F satisfies ( Ã) and let a > r * . For every ε > 0, there exists δ = δ(ε) such that for all function

(c, φ 1 , φ 2 ) solution of                    cφ ′ 1 (x) = α 0 (φ 2 (x) -φ 1 (x)) cφ ′ 2 (x) = 2 F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0, φ ′ 2 ≥ 0 φ 1 (x + 1) ≤ φ 1 (x) + 1, φ 2 (x + 1) ≤ φ 2 (x) + 1 |c| ≤ M 0 |cφ ′ 1 | ≤ M 1 , |cφ ′ 2 | ≤ M 2
and for all x 0 ∈ R satisfying

(φ 1 ) * (x 0 + a) -(φ 1 ) * (x 0 -a) ≤ δ(ε) or (φ 2 ) * (x 0 + a) -(φ 2 ) * (x 0 -a) ≤ δ(ε), we have dist(α 1 , {0, b} + Z) < ε for all α 1 ∈ [(φ 1 ) * (x 0 ), (φ 1 ) * (x 0 )] and dist(α 2 , {0, b} + Z) < ε for all α 2 ∈ [(φ 2 ) * (x 0 ), (φ 2 ) * (x 0 )].
Proof. The proof is decomposed into three steps.

Step 1: Construction of a sequence. We assume by contradiction that there exists ε > 0 such that for all δ n → 0 and (c n , φ n 1 , φ n 2 ) solution of (3.10)

               c n (φ n 1 ) ′ (x) = α 0 (φ n 2 (x) -φ n 1 (x)) c n (φ n 2 ) ′ (x) = 2 F ((φ n 1 (x + r i )) i=0,...,N ) + α 0 (φ n 1 (x) -φ n 2 (x)) (φ n 1 ) ′ ≥ 0, (φ n 2 ) ′ ≥ 0 φ n 1 (x + 1) ≤ φ n 1 (x) + 1, φ n 2 (x + 1) ≤ φ n 2 (x) + 1 |c n | ≤ M 0 |c n (φ n 1 ) ′ | ≤ M 1 , |c n (φ n 2 ) ′ | ≤ M 2
and there exists (

x n ) n ⊂ R satisfying (3.11) (φ n 1 ) * (x n + a) -(φ n 1 ) * (x n -a) ≤ δ n → 0 or (φ n 2 ) * (x n + a) -(φ n 2 ) * (x n -a) ≤ δ n → 0 and α n 1 ∈ [(φ n 1 ) * (x n ), (φ n 1 ) * (x n )] such that (3.12) dist (α n 1 , {0, b} + Z) ≥ ε > 0 or α n 2 ∈ [(φ n 2 ) * (x n ), (φ n 2 ) * (x n )] such that (3.13) dist (α n 2 , {0, b} + Z) ≥ ε > 0.
Up to translate the profile, we assume that (3.14)

x n ≡ 0 φ n 1 (0) ∈ [0, 1) for all n.
Step 2: Passing to limit n → +∞. Using Lemma 3.3, we deduce that there exists (c, φ 1 , φ 2 ) such that, up to extract a subsequence, c n → c, φ n 1 → φ 1 and φ n 2 → φ 2 a.e. and (c,

φ 1 , φ 2 ) is solution of (3.15)      cφ ′ 1 (x) = α 0 (φ 2 (x) -φ 1 (x)) cφ ′ 2 (x) = 2 F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0, φ ′ 2 ≥ 0
Step 3: Getting a contradiction. We pass to the limit in (3.11) with x n = 0. This implies that

(3.16) (φ 1 ) * (a) ≤ (φ 1 ) * (-a) or (φ 2 ) * (a) ≤ (φ 2 ) * (-a).
Since φ 1 and φ 2 are non-decreasing, we have

φ 1 = K 1 on [-a, a] or φ 2 = K 1 on [-a, a]. Using Lemma 3.2 we then get that φ 2 = φ 1 = K 1 on [-a, a].
Using (3.15), we deduce that for x = 0

0 = F ((φ 1 (x + r i )) i=0,...,N ) = F ((K 1 ) i=0,...,N ) = f (K 1 ). Hence K 1 ∈ {0, b} + Z. Although, since α n 1 ∈ [(φ n 1 ) * (0), (φ n 1 ) * (0)] then α n 1 → α 1 ∈ {K 1 }. But, passing to the limit in (3.12), yields dist (α 1 , {0, b} + Z) ≥ ε > 0,
which is a contradiction. Similarly, if we pass to the limit in (3.13), we then get

dist (α 2 , {0, b} + Z) ≥ ε > 0,
which is also a contradiction.

Proof of Proposition 2.3

We are now able to give the proof of Proposition 2.3.

Proof of Proposition 2.3.

Step 0: Introduction. Let p > 0 and (φ 1 p , φ 2 p ) (given by (2.5)) be two non-decreasing functions solution of (3.17)

c p (φ 1 p ) ′ (x) = α 0 (φ 2 p (x) -φ 1 p (x)) c p (φ 2 p ) ′ (x) = 2 F ((φ 1 p (x + r i )) i=0,...,N ) + α 0 (φ 1 p (x) -φ 2 p (x)). with (3.18) φ 1 p x + 1 p = 1 + φ 1 p (x) and φ 2 p x + 1 p = 1 + φ 2 p (x).
Up to translate φ 1 p , we assume that

(3.19) (φ 1 p ) * (0) ≤ b (φ 1 p ) * (0) ≥ b.
Our goal is to pass to the limit as p tends to zero.

Step 1: Passing to the limit p → 0. We want to apply Lemma 3.3. The only thing we have to show is that φ 1 p (0) is bounded. From (3.18) and (3.19), we deduce that

b -1 ≤ φ 1 p - 1 2p ≤ (φ 1 p ) * (0) ≤ b ≤ (φ 1 p ) * (0) ≤ φ 1 p 1 2p ≤ b + 1. Thus b -1 ≤ φ 1 p (0) ≤ b + 1.
Using Lemma 3.3, we then deduce that there exists (c, φ 1 , φ 2 ) such that, up to extract a subsequence, c p → c, φ 1 p → φ 1 and φ 2 p → φ 2 a.e. and (c,

φ 1 , φ 2 ) is solution of (3.20)      cφ ′ 1 (x) = α 0 (φ 2 (x) -φ 1 (x)) cφ ′ 2 (x) = 2 F ((φ 1 (x + r i )) i=0,...,N ) + α 0 (φ 1 (x) -φ 2 (x)) φ ′ 1 ≥ 0, φ ′ 2 ≥ 0.
We also note that (φ 1 ) * (0) ≤ b and (φ 1 ) * (0) ≥ b.

Step 2 : Properties of the limit (φ 1 , φ 2 ).

Step 2.1: The oscillation of (φ 1 , φ 2 ) is bounded. Let R > 0. For every p such that R ≤ 1 2p we have

φ 1 p (R) -φ 1 p (-R) ≤ φ 1 p 1 2p -φ 1 p -1 2p = 1.
Passing to the limit as p → 0, we get

φ 1 (R) -φ 1 (-R) ≤ 1.
Sending R → +∞, we deduce that

φ 1 (+∞) -φ 1 (-∞) ≤ 1.
We get in the same way that φ 2 (+∞) -φ 2 (-∞) ≤ 1.

Step 2.2: φ 1 (±∞) ∈ Z ∪ (b ± Z) and φ 2 (±∞) = φ 1 (±∞). We define

φ 1 n (x) = φ 1 (x -n), and φ 2 n (x) = φ 2 (x -n).
Since (3.20) is invariant by translation, we get that (φ 1 n , φ 2 n ) is still solution of (3.20). Moreover, since (φ 1 , φ 2 ) is non-decreasing and bounded, (φ 1 n , φ 2 n ) is also non-decreasing and bounded. Thus (φ 1 n , φ 2 n ) converges as n → +∞ and we denote by (φ 1 (-∞), φ 2 (-∞)) its limit. By stability of viscosity solution, we then get that

0 = α 0 (φ 2 (-∞) -φ 1 (-∞)) 0 = 2 F (((φ 1 (-∞)) i=0,...,N ) + α 0 (φ 1 (-∞) -φ 2 (-∞)).
The first equation implies that φ 1 (-∞) = φ 2 (-∞) while the second implies that f (φ 1 (-∞)) = 0 and so φ 1 (-∞) ∈ Z ∪ (b + Z). In the same way, we get φ 1 (+∞) ∈ Z ∪ (b + Z) and φ 1 (+∞) = φ 2 (+∞).

Step 3: φ 1 (±∞) / ∈ {b} + Z. Since

φ 1 (+∞) -φ 1 (-∞) ≤ 1 and (φ 1 ) * (0) ≤ b (φ 1 ) * (0) ≥ b, we obtain that φ 1 (-∞) ∈ {b -1, 0, b} and φ 1 (+∞) ∈ {b, 1, b + 1}. If φ 1 (+∞) = b + 1 then φ 1 (-∞) = b and if φ 1 (-∞) = b -1 then φ 1 (+∞) = b.
Thus, it is sufficient to exclude the cases φ 1 (±∞) = b. At the end, this will prove that that φ 1 (+∞) = 1 and φ 1 (-∞) = 0 (and so by step 2.2, φ 2 (+∞) = 1 and φ 2 (-∞) = 0). By contradiction, we assume that φ 1 (+∞) = b.

(the case φ 1 (-∞) = b being similar). Let x 0 = 2 r * , where r * = max i=0,...,N

|r i |. Since b = φ 1 (+∞) ≥ (φ 1 ) * (0) ≥ b then φ 1 (x) = b for all x > 0. Hence φ 1 (x 0 ) = φ 1 (x 0 ± a) = b for r * < a < 2 r * .
Step 3.1: Introduce z p and y p . For any ε > 0 small enough (ε < min(b, 1 -b)/2), let z p and y p ∈ R such that

(3.21) (φ 1 p ) * (z p ) ≤ b + ε (φ 1 p ) * (z p ) ≥ b + ε and (φ 1 p ) * (y p ) ≤ b -ε (φ 1 p ) * (y p ) ≥ b -ε. Let ψ 1 p (x) = (φ 1 p ) * (x + a) -(φ 1 p ) * (x -a) ψ 2 p (x) = (φ 2 p ) * (x + a) -(φ 2 p ) * (x -a). Note that (ψ 1 p , ψ 2
p ) is lower semi-continuous and solution of

(3.22)    c p (ψ 1 p ) ′ (x) ≥ α 0 (ψ 2 p (x) -ψ 1 p (x)) c p (ψ 2 p ) ′ (x) ≥ 2 (F (((φ 1 p ) * (x + a + r i )) i=0,...,N ) -F (((φ 1 p ) * (x -a + r i )) i=0,...,N )) +α 0 (ψ 1 p (x) -ψ 2 p (x)).
Moreover, we have

b + ε ∈ [(φ 1 p ) * (z p ), (φ 1 p ) * (z p )] such that dist (b + ε, {0, b} + Z) ≥ ε.
Then there exists δ(ε) (given by Proposition 3.4) independent of p such that (for a > r * )

(3.23) ψ 1 (z p ) ≥ δ(ε) > 0 and ψ 2 (z p ) ≥ δ(ε) > 0.
Similarly, we get that

(3.24) ψ 1 (y p ) ≥ δ(ε) > 0 and ψ 2 (y p ) ≥ δ(ε) > 0.
Using the uniform convergence of φ 1 p to φ 1 (see the second part of Lemma 2.7 if c = 0), we also get that

φ 1 p (x 0 ) → b and ψ 1 p (x 0 ) = (φ 1 p ) * (x 0 + a) -(φ 1 p ) * (x 0 -a) → 0 as p → 0.
Step 3.2: Equation satisfied by (ψ 1 , ψ 2 ) at its point of minimum. Since z p → +∞ as p → +∞ y p ≤ 0, we have x 0 ∈ [y p , z p ] for p small enough. We define

m 1 p = min x∈[yp,zp] ψ 1 p (x) = ψ 1 p (x 1 p ) ≥ 0 with x 1 p ∈ [y p , z p ] m 2 p = min x∈[yp,zp] ψ 2 p (x) = ψ 2 p (x 2 p ) ≥ 0 with x 2 p ∈ [y p , z p ].
Note that

(3.25) m 1 p = ψ 1 p (x 1 p ) ≤ ψ 1 p (x 0 ) → 0 as p → 0.
Since, by (3.23) and (3.24),

ψ 1 p (y p ) ≥ δ(ε) > 0, ψ 1 p (z p ) ≥ δ(ε) > 0,
we have x 1 p ∈ (y p , z p ) and from (3.22), we get that

0 = c p (ψ 1 p ) ′ (x 1 p ) ≥ α 0 (ψ 2 p (x 1 p ) -ψ 1 p (x 1 p )).
This gives that

m 1 p = ψ 1 p (x 1 p ) ≥ ψ 2 p (x 1 p ) ≥ m 2 p ≥ 0. Thus m 2
p → 0 as p → 0. Using that ψ 2 p (y p ) ≥ δ(ε) > 0 and ψ 2 p (z p ) ≥ δ(ε) > 0, we then get (3.26)

x 2 p ∈ (y p , z p ).

Then by (3.22), we have

(3.27)    0 = c p (ψ 1 p ) ′ (x 1 p ) ≥ α 0 (ψ 2 p (x 1 p ) -ψ 1 p (x 1 p )) 0 = c p (ψ 2 p ) ′ (x 2 p ) ≥ 2( F (((φ 1 p ) * (x 2 p + a + r i )) i=0,...,N ) -F (((φ 1 p ) * (x 2 p -a + r i )) i=0,...,N )) +α 0 (ψ 1 p (x 2 p ) -ψ 2 p (x 2 p )).
Using that

α 0 (ψ 1 p (x 2 p ) -ψ 2 p (x 2 p )) ≥ α 0 (ψ 1 p (x 1 p ) -ψ 2 p (x 1 p )) ≥ 0, we deduce that (3.28) 0 ≥ F (((φ 1 p ) * (x 2 p + a + r i )) i=0,...,N ) -F (((φ 1 p ) * (x 2 p -a + r i )) i=0,...,N ).
Step 3.3:

ψ 1 p (x 2 p + r i ) ≥ ψ 1 p (x 1 p ) = m 1 p for all i. Using (3.26) we get (3.29) b -ε ≤ (φ 1 p ) * (y p ) ≤ φ 1 p (x 2 p ) ≤ (φ 1 p ) * ((z p ) ≤ b + ε.
Using Lemma 3.3, we deduce that there exists φ 1 0 and φ 2 0 such that

φ 1 p (x 2 p + •) → φ 1 0 and φ 2 p (x 2 p + •) → φ 2 0
a.e. on R and (φ 1 0 , φ 2 0 ) is solution of

(3.30) c (φ 1 0 ) ′ (z) = α 0 (φ 2 0 (z) -φ 1 0 (z)) c (φ 2 0 ) ′ (z) = 2 F ((φ 1 0 (z + r i )) i=0,...,N ) + α 0 (φ 1 0 (z) -φ 2 0 (z)).
Using that

(3.31) 0 ≤ (φ 2 p ) * (x 2 p + a) -(φ 2 p ) * (x 2 p -a) = ψ 2 p (x 2 p ) ≤ ψ 2 p (x 1 p ) ≤ ψ 1 p (x 1 p ) = m 1 p → 0 as p → 0
we deduce that φ 2 0 = K 1 on (-a, a). Using Lemma 3.2, we deduce that

φ 1 0 = K 1 on (-a, a) with K 1 ∈ (b -ε, b + ε) (by (3.29)).
The second equation of (3.30) implies that f (K 1 ) = 0, which gives K 1 = b. We then deduce (using the uniform Lipschitz continuity or Helly's Lemma in the case c = 0), that sup

(x 2 p -a+δ,x 2 p +a-δ) φ 1 p (x) -b → 0 for all δ > 0, which implies (3.32) (φ 1 p ) * (x 2 p + a -δ), (φ 1 p ) * (x 2 p -a + δ) → b as p → 0
Using (3.21), we deduce, taking δ small enough (0 < δ ≤ a -r * ), that y p ≤ x 2 p + r i ≤ z p for all i which gives that (3.33)

ψ 1 p (x 2 p + r i ) ≥ ψ 1 p (x 1 p ) = m 1 p .
Step 3.4: Getting a contradiction. In this step, we assume that m 1 p > 0 and we want to get a contradiction. Set

k i = (φ 1 p ) * (x 2 p + r i + a) if r i ≤ 0 (φ 1 p ) * (x 2 p + r i -a) if r i > 0 and note that k i ∈ [(φ 1 p ) * (x 2 p -a), (φ 1 p ) * (x 2 p + a)] which implies that k i → b as p → 0 (by (3.32)).
Hence from (3.28),(3.33) and using the monotonicity of F , we get 0 ≥ F ((a i ) i=0,...,N ) -F ((c i ) i=0,...,N ) where

a i = k i if r i ≤ 0 k i + m 1 p if r i > 0 and c i = k i -m 1 p if r i ≤ 0 k i if r i > 0.
Therefore from the fact that k i → b and m 1 p → 0, we deduce that

a i → b and c i → b as p → 0. Since F is C 1 near {b} n+1 and c i + t (a i -c i ) = c i + t m 1 p , we have 0 ≥ 1 0 dt N i=0 (a i -c i ) ∂F ∂X i (c j + t(a j -c j ) j=0,...,N ) = 1 0 dt N i=0 m 1 p ∂F ∂X i ((c j + t m 1 p ) j=0,...,N ) .
Using that m 1 p > 0, we get 0

≥ 1 0 dt N i=0 ∂F ∂X i ((c j + t m 1 p ) j=0,...,N ) = f ′ (b) + 1 0 dt N i=0 ∂F ∂X i ((c j + t m 1 p ) j=0,...,N ) - N i=0 ∂F ∂X i (b, ..., b) . But F is C 1 near {b} N +1 and c i + t m 1 p → b for all i, thus 1 0 dt N i=0 ∂F ∂X i ((c j + t m 1 p ) j=0,...,N ) - N i=0 ∂F ∂X i (b, ..., b) → 0 as p → 0. This implies that 0 ≥ f ′ (b) > 0
which is a contradiction with assumption (B).

Step 5: m 1 p > 0. We split this step into two cases:

Case 1: F is strongly increasing in some direction. We assume that F satisfies

(3.34) ∂F ∂X i1 ≥ δ 0 > 0.
We assume by contradiction that m 1 p = 0. Thus

ψ 1 p (x 1 p ) = (φ 1 p ) * (x 1 p + a) -(φ 1 p ) * (x 1 p -a) = 0.
Since φ 1 p is non-decreasing, we get

φ 1 p| (x 1 p -a,x 1 p +a) = φ 1 p (x 1 p ) = b.
The first equation of (3.17) implies that

φ 2 p| (x 1 p -a,x 1 p +a) = b.
Let d 1 ≥ x 1 p + a be the first real number such that

φ 1 p (d 1 + η 1 ) > b for every η 1 > 0.
We choose 0 < η 1 < r i1 and set

x 1 = d 1 + η 1 -r i1 .
From the definition of d 1 , we deduce that

φ 1 p = φ 2 p = b on a neighborhood of x 1 .
hence (φ 2 p ) ′ (x 1 ) = 0. Moreover, we have

φ 1 p (x 1 + r i ) ≥ b for all i = i 1 φ 1 p (x 1 + r i1 ) = φ 1 p (d 1 + η 1 ) > b for i = i 1 .
Thus, the second equation of (3.17) implies that

0 = c (φ 2 p ) ′ (x 1 ) =2F ((φ 1 p (x 1 + r i )) i=0,....,N ) ≥2F (b, ..., φ 1 p (x 1 + r i1 ), ...., b) ≥f (b) + δ 0 (φ 1 p (d 1 + η 1 ) -b) =δ 0 (φ 1 p (d 1 + η 1 ) -b) > 0.
This is a contradiction.

Case 2: Create the monotonicity. In fact, we can always assume (3.34) for a modification F p of F , where

F p (X 0 , X 1 , ..., X N ) = F (X 0 , X 1 , ..., X N ) + p(X i1 -X 0 ).
Then the whole construction works for F replaced by F p with the additional monotonicity property (3.34) with δ 0 = p. Once we pass to the limit p → 0, we still get the same contradiction as in Step 3.4 and we recuperate the construction of traveling wave (φ 1 , φ 2 ) of (1.6) for the function F.

Uniqueness of the velocity c

In the first subsection, we prove a comparison principle on (-∞, r * ]) and then another one on [-r * , +∞)). These two comparison principles will be used in the second subsection to prove the uniqueness of the velocity.

Comparison principle on the half-line

   there exists β 0 > 0 such that if : Y = (Y 0 , ...., Y N ), Y + (a...., a) ∈ [0, β N +1 0 ] then F (Y + (a, ..., a)) < F (Y ) if a > 0. Let (u 1 , u 2 ) and (v 1 , v 2 ) : (-∞, r * ] 2 → [0, 1] 2 be
respectively a sub and a super-solution of

(4.2) c u ′ 1 (z) = α 0 (u 2 (z) -u 1 (z)) on (-∞, 0) c u ′ 2 (z) = 2 F ((u 1 (z + r i )) i=0,...,N ) + α 0 (u 1 (z) -u 2 (z)) on (-∞, 0)
We suppose that

u 1 ≤ β 0 on (-∞, 2r * ] u 2 ≤ β 0 on (-∞, 2r * ] and u 1 ≤ v 1 on [0, r * ] u 2 ≤ v 2 on [0, r * ].
Then Let (u 1 , u 2 ) and (v 1 , v 2 ) : [-r * , +∞) 2 → [0, 1] 2 be respectively a sub and a super-solution of (4.2) on (0, +∞). We assume that

u 1 ≤ v 1 on (-∞, r * ] u 2 ≤ v 2 on (-∞, r * ].
v 1 ≥ 1 -β 0 on [-2r * , +∞) v 2 ≥ 1 -β 0 on [-2r * , +∞) and u 1 ≤ v 1 on [-r * , 0] u 2 ≤ v 2 on [-r * , 0]. Then u 1 ≤ v 1 on [-r * , +∞] u 2 ≤ v 2 on [-r * , +∞]. Lemma 4.3 (Transformation of a solution of (4.2)). Let (u 1 , u 2 ), (v 1 , v 2 ) : (-∞, r * ] 2 → [0, 1]
2 be respectively a sub and a super-solution of (4.2).Then

û1 (x) = 1 -u 1 (-x) û2 (x) = 1 -u 2 (-x) and v1 (x) = 1 -v 1 (-x) v2 (x) = 1 -v 2 (-x)
are respectively a super and a sub-solution of (4.2) on (0 + ∞) with F, c, r i (for all i ∈ {0, ..., N }) replaced by F , ĉ and r, given by

(4.4)    F (X 0 , ...., X N ) = -F (1 -X 0 , ...., 1 -X N ) ĉ = -c ri = -r i with F : [0, 1] N +1 → R
f (v) = -f (1 -v).
Proof. Let (u 1 , u 2 ) : (-∞, r * ] 2 → [0, 1] 2 be a sub-solution of (4.2) and set

û1 (x) = 1 -u 1 (-x), û2 (x) = 1 -u 2 (-x).
We have

c û′ 1 (x) = c u ′ 1 (-x) ≤ α 0 (u 2 (-x) -u 1 (-x)) c û′ 2 (x) = c u ′ 2 (-x) ≤ -2 F ((u 1 (-x + r i )) i=0,...,N ) + α 0 (u 1 (-x) -u 2 (-x)) thus ĉ û′ 1 (x) ≥ α 0 (û 2 (x) -û1 (x)) ĉ û′ 2 (x) ≥ 2 F ((1 -û1 (x -r i )) i=0,...,N ) + α 0 (û 2 (x) -û1 (x)
). Hence (û 1 , û2 ) is a super-solution of (4.2) on (0, +∞). Similarly, we prove that (v 1 , v2 ) is a subsolution of the same equation on (0, +∞).

The proof of Corollary 4.2 is now very easy.

Proof of Corollary 4.2. Let (u 1 , u 2 ), (v 1 , v 2 ) : [-r * , +∞) 2 → [0, 1] 2 be a sub and a super-solution of (4.2) on (0, +∞) such that v 1 ≥ 1 -β 0 and v 2 ≥ 1 -β 0 on [-2r * , +∞). We set

û1 (x) = 1 -u 1 (-x), û2 (x) = 1 -u 2 (-x), v1 (x) = 1 -v 1 (-x) and v2 (x) = 1 -v 2 (-x).
Thus v1 , v2 ≤ β 0 on (-∞, 2r * ] and by Lemma 4.3 (û 1 , û2 ) and (v 1 , v2 ) are respectively a super and a sub-solution of (4.2). Since F satisfies (4.3), we deduce that F satisfies (4.1). By Theorem 4.1, we then get that v1 ≤ û1 on (-∞, r * ] v2 ≤ û2 on (-∞, r * ].

This implies that

u 1 ≤ v 1 on [-r * , +∞) u 2 ≤ v 2 on [-r * , +∞).
We now go back to the proof of Theorem 4.1

Proof of Theorem 4.1. Let (u 1 , u 2 ) and (v 1 , v 2 ) : (-∞, r * ] 2 → [0, 1] 2 be respectively a sub and a super-solution of (4.2) such that

u 1 ≤ β 0 on (-∞, 2r * ] u 2 ≤ β 0 on (-∞, 2r * ] and u 1 ≤ v 1 on [0, r * ] u 2 ≤ v 2 on [0, r * ]
Step 0:

Introduction. Let v1 = min(v 1 , β 0 ) v2 = min(v 2 , β 0 )
According to (4.1) we have F (β 0 , ..., β 0 ) ≤ 0.

Therefore the constant β 0 is a super-solution of (4.2) and then (v 1 , v2 ) is a super-solution of (4.2) on (-∞, 0) with u 1 ≤ v1 , u 2 ≤ v2 on [0, r * ]. Moreover, since v1 ≤ v 1 and v2 ≤ v 2 , it is sufficient to prove the comparison principle (Theorem 4.1) between (u 1 , u 2 ) and (v 1 , v2 ) with u 1 , v1 , u 2 , v2 ∈ [0, β 0 ]. For simplicity, we note (v 1 , v2 ) as (v 1 , v 2 ).

Step 1: Doubling the variables. We assume by contradiction that

M = sup x∈(-∞,r * ] max(u 1 (x) -v 1 (x), u 2 (x) -v 2 (x)) > 0
Let 0 < ε, α < 1 and we define

ψ 1 (x, y) = u 1 (x) -v 1 (y) - |x -y| 2 2ε + αx ψ 2 (x, y) = u 2 (x) -v 2 (y) - |x -y| 2 2ε + αx and M ε,α = sup x,y∈(-∞,r * ] max(ψ 1 (x, y), ψ 2 (x, y))
Since the function ψ 1 and ψ 2 are upper semi-continuous and satisfy ψ 1 (x, y), ψ 2 (x, y) → -∞ as |(x, y)| → +∞, we deduce that ψ 1 and ψ 2 reach their maximum respectively at (x 1 ε , y 1 ε ) and

(x 2 ε , y 2 ε ) ∈ (-∞, r * ] 2 . We also denote by (x ε , y ε , i ε ) ∈ (-∞, r * ] 2 × {1, 2} such that M ε,α = ψ iε (x ε , y ε ).
Moreover, for α small enough, we get that

M ε,α ≥ M 2 > 0.
Using also the fact that

u 1 (x 1 ε ) -v 1 (x 1 ε ) ≤ β 0 and u 2 (x 2 ε ) -v 2 (x 2 ε ) ≤ β 0 , we get then (4.5) |x ε -y ε | 2 2ε -αx ε ≤ β 0 .
Step 2: for all α and ε small enough, we have x ε , y ε ∈ (-∞, 0). By contradiction, assume that there exists α small enough and ε → 0 such that x ε ∈ [0, r * ] or y ε ∈ [0, r * ] and i ε = i.

We suppose that x ε ∈ [0, r * ] (the case y ε ∈ [0, r * ] being similar). Using (4.5), we deduce that ȳε ∈ [-2(β 0 + r * )ε, r * ]. Then x ε and y ε converge to x 0 ∈ [0, r * ] as ε → 0. We deduce that

0 < M 2 ≤ lim sup ε→0 (u i (x ε ) -v i (y ε )) ≤ u i (x 0 ) -v i (y 0 ) ≤ 0
which is a contradiction.

Step 3: Viscosity inequalities. Using that x → ψ 1 (x, y 1 ε ) reaches a maximum at point x 1 ε , we get that

c x 1 ε -y 1 ε ε + α ≤ α 0 (u 2 (x 1 ε ) -u 1 (x 1 ε )).
In the same way, we have

c x 1 ε -y 1 ε ε ≥ α 0 (v 2 (y 1 ε ) -v 1 (y 1 ε )).
Subtracting the two inequalities, we deduce that

(4.6) cα ≤ α 0 (u 2 (x 1 ε ) -v 2 (y 1 ε )) -(u 1 (x 1 ε ) -v 1 (y 1 ε ))
In the same way (using ψ 2 ) we get that

cα ≤2 (F ((u 1 (x 2 ε + r i )) i=0,...,N ) -F ((v 1 (y 2 ε + r i )) i=0,...,N ) (4.7) + α 0 ((u 1 (x 2 ε ) -v 1 (y 2 ε )) -(u 2 (x 2 ε ) -v 2 (y 2 ε ))
Step 4: Passing to the limit ε, α → 0. We set

u ε,k j,i = u j (x k ε + r i ), and v ε,k j,i = v j (x k ε + r i ).
The proof is split into two cases:

Case 1: ∃ ε, α → 0 such that i ε = 1.
In that case, equation (4.6) implies that

ψ 2 (x 2 ε , y 2 ε ) ≥ ψ 2 (x 1 ε , y 1 ε ) ≥ ψ 1 (x 1 ε , y 1 ε ) + cα α 0 ≥ M 2 + cα α 0 ≥ M 4
for α small enough. Hence, using classical arguments, we deduce that (4.8)

x 2 ε -y 2 ε 2 2ε , αx 2 ε → 0 as ε, α → 0.
Moreover, using that

ψ 1 (x 2 ε + r i , y 2 ε + r i ) ≤ ψ 1 (x 1 ε , y 1 ε ), we have u ε,2 1,i ≤ v ε,2 1,i + m ε + δ ε i where m ε = u 1 (x 1 ε ) -v 1 (y 1 ε ), and δ ε i = |x 2 ε -y 2 ε | 2 2ε -α(x 2 ε + r i ) (note that αx 1 ε ≤ 0). Since u ε,k j,i , v ε,k j,i ∈ [0, β 0 ] and M 1 2 ≤ m ε ≤ β 0 , we deduce that as ε, α → 0          u ε,k j,i → u k 1,i v ε,k j,i → v k 1,i m ε → m 0 = u 1 1,0 -v 1 1,0 δ ε i → 0 with u k j,i , v k j,i ∈ [0, β 0 ], 0 < M 1 2 ≤ m 0 ≤ β 0 and u 2 1,i ≤ v 2 1,i + m 0 . Passing to the limit in (4.7) implies that 0 ≤ 2 (F ((u 2 1,i ) i=0,...,N ) -F ((v 2 1,i ) i=0,...,N )) + α 0 ((u 2 1,0 -v 2 1,0 ) -(u 2 2,0 -v 2 2,0 )). We define m = m 0 -(u 2 1,0 -v 2 1,0 ) ≥ 0. Thus, using the monotony (1.5), we get 0 ≤ 2 (F (u 2 1,0 + m, (u 2 1,i ) i=1,...,N ) -F ((v 2 1,i ) i=0,...,N )) + α 0 ((u 1 1,0 -v 1 1,0 ) -(u 2 2,0 -v 2 2,0
)). Passing to the limit in (4.6) and using the fact that u

1 2,0 -v 1 2,0 ≤ u 2 2,0 -v 2 2,0 , we get that α 0 ((u 1 1,0 -v 1 1,0 ) -(u 2 2,0 -v 2 2,0 )) ≤ 0. Thus, up to redefine u 2 1,0 by u 2 1,0 + m ∈ [0, β 0 ], we get (4.9) 0 ≤ F (u 2 1,0 , (u 2 1,i ) i=1,...,N ) -F ((v 2 1,i ) i=0,...,N ). with u k j,i , v k j,i ∈ [0, β 0 ], 0 < M 1 2 ≤ m 0 ≤ β 0 , u 2 1,i ≤ v 2 1,i + m 0 and u 2 1,0 = v 2 1,0 + m 0 . Case 2: ∃ ε, α → 0 such that i ε = 2. Using that ψ 1 (x 2 ε + r i , y 2 ε + r i ) ≤ ψ 2 (x 2 ε , y 2 ε ), we get that u ε,2 1,i ≤ v ε,2 1,i + m ε + δ ε i where m ε = u 2 (x 2 ε ) -v 2 (y 2 ε ), and
δ ε i = -αr i . Since u ε,k j,i , v ε,k j,i ∈ [0, β 0 ] and M 1 2 ≤ m ε ≤ β 0 , we deduce that as ε, α → 0          u ε,k j,i → u k 1,i v ε,k j,i → v k 1,i m ε → m 0 = u 2 2,0 -v 2 2,0 δ ε i → 0 with u k j,i , v k j,i ∈ [0, β 0 ], 0 < M 1 2 ≤ m 0 ≤ β 0 and u 2 1,i ≤ v 2 1,i + m 0 . Passing to the limit in (4.7) implies that 0 ≤ 2 (F ((u 2 1,i ) i=0,...,N ) -F ((v 2 1,i ) i=0,...,N )) + α 0 ((u 2 1,0 -v 2 1,0 ) -(u 2 2,0 -v 2 2,0 )). We define m = m 0 -(u 2 1,0 -v 2 1,0 ) ≥ 0. Thus, using the monotony (1.5), we get 0 ≤2 (F (u 2 1,0 + m, (u 2 1,i ) i=1,...,N ) -F ((v 2 1,i ) i=0,...,N )) + α 0 ((u 2 1,0 -v 2 1,0 ) -(u 2 2,0 -v 2 2,0 ) + m) =2 (F (u 2 1,0 + m, (u 2 1,i ) i=1,...,N ) -F ((v 2 1,i ) i=0,...,N )) Thus, up to redefine u 2 1,0 by u 2 1,0 + m ∈ [0, β 0 ], we get (4.10) 0 ≤ F ((u 2 1,i ) i=0,...,N ) -F ((v 2 1,i ) i=0,...,N ). with u k j,i , v k j,i ∈ [0, β 0 ], 0 < M 1 2 ≤ m 0 ≤ β 0 , u 2 1,i ≤ v 2 1,i + m 0 and u 2 1,0 = v 2 1,0 + m 0 .
Step 5: Getting a contradiction. We claim that for all i, there exists l i , l ′ i ≥ 0 such that (4.11)

u 2 1,i + l i = v 2 1,i -l ′ i + m 0 , and u 2 1,i := u 2 1,i + l i ≤ β 0 v 2 1,i := v 2 1,i -l ′ i ≥ 0.
Recall that for all i ∈ {0, ..., N }, we have

     u 2 1,i , v 2 1,i ∈ [0, β 0 ] u 2 1,i ≤ v 2 1,i + m 0 u 2 1,0 -v 2 1,0 = m 0 ≤ β 0 .
If for some i, u 2 1,i = v 2 1,i +m 0 , then it suffices to take

l i = l ′ i = 0. Assume then that u 2 1,i < v 2 1,i +m 0 . Case 1: u 2 1,i , v 2 1,i ∈ (v 2 1,0 , u 2 1,0 ). Set l i = u 2 1,0 -u 2 1,i and l ′ i = v 2 1,i -v 2 1,0 . Then u 2 1,i = u 2 1,i + l i = u 2 1,0 ≤ β 0 v 2 1,i = v 2 1,i -l ′ i = v 2 1,0 ≥ 0, and u 2 1,i = v 2 1,i + m 0 . Case 2: u 2 1,i > u 2 1,0 and v 2 1,i > v 2 1,0 . Since u 2 1,i -v 2 1,0 > m 0 , then there exists l ′ i < v 2 1,i -v 2 1,0 such that u 2 1,i = v 2 1,i -l ′ i + m 0 and v 2 1,i = v 2 1,i -l ′ i > v 2
1,0 ≥ 0. Thus, it is sufficient to take l i = 0.

Case 3: u 2 1,i < u 2 1,0 and v 2 1,i < v 2 1,0 . This case can be treated as Case 2 by taking l ′ i = 0 and

l i < u 2 1,0 -u 2 1,i .
Finally, going back to (4.9) or (4.10), since F is non-decreasing, we deduce that

0 ≤ F ((u 2 1,i ) i=0,...,N ) -F ((v 2 1,i ) i=0,...,N ) ≤ F ((u 2 1,i ) i=0,...,N ) -F ((v 2 1,i ) i=0,...,N ) = F ((u 2 1,i ) i=0,...,N ) -F ((u 2 1,i -m 0 ) i=0,...,N ) < 0.
Last inequality takes place since F verifies (4.1) for u 2 1,i , u 2 1,i -m 0 ∈ [0, β 0 ] and m 0 > 0. Therefore, we get a contradiction.

Uniqueness of the velocity

In this subsection, we use Theorem 4.1 and Corollary 4.2 in order to prove the uniqueness of the velocity c. We set δ = min(β 0 , 1 4 ) where β 0 is given in assumption (C) and up to translate (φ 11 , φ 12 ) and (φ 21 , φ 22 ), we assume that

φ 11 (x) ≥ 1 -δ ∀ x ≥ -2r * φ 12 (x) ≥ 1 -δ ∀ x ≥ -2r * and φ 21 (x) ≤ δ ∀ x ≤ 2r * φ 22 (x) ≤ δ ∀ x ≤ 2r * .
This implies that

φ 21 (x) ≤ φ 11 (x) over [-r * , r * ] φ 22 (x) ≤ φ 12 (x) over [-r * , r * ].
Moreover, since c 1 < c 2 , we have

c 1 (φ 21 ) ′ (x) ≤ c 2 (φ 21 ) ′ (x) = α 0 (φ 22 (x) -φ 21 (x)) c 1 (φ 22 ) ′ (x) ≤ c 2 (φ 22 ) ′ (x) = 2 F ((φ 21 (x -r i ) i=0,...,N ) + α 0 (φ 21 (x) -φ 22 (x)).
Thus (c We set

         u 1 (t, x) = φ 11 (x + c 1 t) u 2 (t, x) = φ 12 (x + c 1 t) u 3 (t, x) = φ 21 (x + c 2 t) u 4 (t, x) = φ 22 (x + c 2 t).
Then for i = 1, j = 2 and i = 3, j = 4, we have (4.12)

∂ t u i (t, x) = α 0 (u j (t, x) -u i (t, x)) ∂ t u j (t, x) = 2 F ((u i (t, x + ri)) i=0,...,N ) + α 0 (u i (t, x) -u j (t, x)).
Moreover, at time t = 0, (4.13)

u 1 (0, x) = φ 11 (x) ≥ φ 21 (x) = u 3 (0, x) over R u 2 (0, x) = φ 12 (x) ≥ φ 22 (x) = u 4 (0, x) over R.
Then, applying the comparison principle for equation (4.12), we get

u 1 ≥ u 3 ∀ t ≥ 0 ∀ x ∈ R u 2 ≥ u 4 ∀ t ≥ 0 ∀ x ∈ R.
Taking x = y -c 1 t, yields φ 11 (y) ≥ φ 21 (y + (c 2 -c 1 )t) ∀t ≥ 0, ∀y ∈ R φ 12 (y) ≥ φ 22 (y + (c 2 -c 1 )t) ∀t ≥ 0, ∀y ∈ R.

Using that c 1 < c 2 and passing to the limit t → +∞, we get

φ 11 (y) ≥ φ 21 (+∞) = 1 ∀y ∈ R φ 12 (y) ≥ φ 22 (+∞) = 1 ∀y ∈ R.
But φ 11 (-∞) = 0 and φ 12 (-∞) = 0, hence a contradiction. Therefore c 1 ≥ c 2 . Similarly, we prove that c 2 ≥ c 1 . Thus c 1 = c 2 .

Uniqueness of the profile

This section is devoted to the proof of the uniqueness of the profiles (under assumption (D±)) using tow different types of strong maximum principle. Proof. We do the proof in the case where c > 0. By contradiction, assume that there exists x 0 < 0 such that φ 21 (x 0 ) > φ 11 (x 0 ) or φ 22 (x 0 ) > φ 12 (x 0 ).

Different types of strong maximum principle

Let w 1 (x) = φ 21 (x) -φ 11 (x) and w 2 (x) = φ 22 (x) -φ 12 (x). A simple computation gives that (5.1)

c w ′ 1 (x) ≥ α 0 (w 2 (x) -w 1 (x)) c w ′ 2 (x) ≥ 2(F ((φ 21 (x + r i )) i=0,...,N ) -F ((φ 11 (x + r i )) i=0,...,N ) + α 0 (w 1 (x) -w 2 (x)) .
Using that F is non-decreasing w.r.t. X i for all i = 0, we get

   c w ′ 1 (x) ≥ α 0 (w 2 (x) -w 1 (x)) , c w ′ 2 (x) ≥ 2 (F (φ 11 (x) + w 1 (x), (φ 11 (x + r i )) i=1,...,N ) -F (φ 11 (x), (φ 1 (x + r i )) i=1,...,N )) +α 0 (w 1 (x) -w 2 (x))
Let w(x) = w 1 (x) + w 2 (x). Then c w ′ (x) ≥ 2 (F (φ 11 (x) + w 1 (x), (φ 11 (x + r i )) i=1,...,N ) -F (φ 11 (x), (φ 11 (x + r i )) i=1,...,N ))

Since F is globally Lipschitz continuous (we denote by L its Lipschitz constant), we have

(5.2) w ′ (x) ≥ - 2 L c w 1 (x) ≥ - 2 L c w(x).
We note that y(x) := w(x 0 ) exp -2 L (x -x 0 ) c satisfied (5.2) for all x 0 ∈ R. Using the comparison principle, we deduce that

(5.3) w(x) ≥ w(x 0 ) exp - 2 L (x -x 0 ) c for all x ≥ x 0 .
Since w 1 (x 0 ) > 0 or w 2 (x 0 ) > 0, we have

w(x 0 ) > 0.
This implies that w(x) > 0 for all x ≥ x 0 .

In particular, for x = 0, we get Proof. a) We assume for simplicity of notation that i 0 = 1. As in the proof of Lemma 5.1, we define w 1 (x) = φ 21 (x) -φ 11 (x) and w 2 (x) = φ 22 (x) -φ 12 (x) which satisfy

(5.4) c w ′ 1 (x) ≥ α 0 (w 2 (x) -w 1 (x)) c w ′ 2 (x) ≥ 2(F ((φ 21 (x + r i )) i=0,...,N ) -F ((φ 11 (x + r i )) i=0,...,N )) + α 0 (w 1 (x) -w 2 (x)).
Using that w 1 (0) = 0, w 2 (0) = 0 and w 1 , w 2 ≥ 0 on R (hence 0 is a point of minimum of w 1 and w 2 ), we deduce that

0 ≥ α 0 (w 2 (0) -w 1 (0)) 0 ≥ 2(F ((φ 21 (r i )) i=0,...,N ) -F ((φ 11 (r i )) i=0,...,N )) + α 0 (w 1 (0) -w 2 (0)). Thus 0 ≥ 2 (F ((φ 21 (r i )) i=0,...,N ) -F ((φ 11 (r i )) i=0,...,N )) .
Using the fact that φ 21 (0) = φ 11 (0) and that F is monotone w.r.t. X i for all i = 0, we get

F ((φ 21 (r i )) i=0,...,N ) = F ((φ 11 (r i )) i=0,...,N ).
Since F is increasing w.r.t. X 1 , we deduce that φ 21 (r 1 ) = φ 11 (r 1 ), i.e. w 1 (r 1 ) = 0. Hence r 1 is a point of minimum of w 1 . The first equation of (5.4) then implies 0 ≥ w 2 (r 1 ) -w 1 (r 1 ) = w 2 (r 1 ).

Since w 2 ≥ 0, we deduce that w 2 (r 1 ) = 0, i.e.

φ 22 (r 1 ) = φ 12 (r 1 ).

Repeating the above argument replacing 0 by r 1 , we get that φ 21 (k r 1 ) = φ 11 (k r 1 ) and φ 22 (k r 1 ) = φ 12 (k r 1 ) for all k ∈ N.

b) We assume that c > 0 and that F satisfies (D+) ii) (the other case where c < 0 being similar). By contradiction, we suppose that there exists x ∈ R, such that φ 21 (x) > φ 11 (x) or φ 22 (x) > φ 12 (x).

Let k ∈ N big enough such that k r i+ > x. Using Lemma 5.1, and the fact that

φ 11 (k r i+ ) = φ 21 (k r i+ ) φ 12 (k r i+ ) = φ 22 (k r i+ ).
We get that

φ 21 (x) = φ 11 (x) φ 22 (x) = φ 12 (x).
which is a contradiction. Proof. We assume that c > 0 (the case c < 0 being similar). We define the functions w 1 (x) = φ 11 (x) -φ 21 (x) and w 2 (x) = φ 12 (x) -φ 22 (x) which satisfy

(5.5) c w ′ 1 (x) = α 0 (w 2 (x) -w 1 (x)) c w ′ 2 (x) = 2(F ((φ 11 (x + r i )) i=0,...,N ) -F ((φ 21 (x + r i )) i=0,...,N )) + α 0 (w 1 (x) -w 2 (x)).
By the first equation, we then deduce that

w ′ 2 = w ′ 1 + c α 0 w ′′ 1 .
The second equation then implies that

c 2 α 0 w ′′ 1 + 2cw ′ 1 = 2(F ((φ 11 (x + r i )) i=0,...,N ) -F ((φ 21 (x + r i )) i=0,...,N )).
Since φ 11 ≤ φ 21 on [-r * , 0] and r i ≤ 0 for all i = 0, then for all x ∈ [0, min i =0

(-r i )], we have

φ 11 (x + r i ) ≤ φ 21 (x + r i ) for i = 0. This implies that c 2 α 0 w ′′ 1 + 2cw ′ 1 ≤2(F (φ 21 (x) + w 1 (x), (φ 21 (x + r i )) i=1,...,N ) -F (φ 21 (x), (φ 21 (x + r i )) i=1,...,N )) ≤2 L |w 1 (x)|
where L is the Lipschitz constant of F . Moreover, w 1 (0) = 0, w ′ 1 (0) = α0 c (w 2 (0) -w 1 (0)) = 0 and y = 0 is a solution of c 2 α0 y ′′ + 2cy ′ = 2Ly, then using the comparison principle, we deduce that

w 1 ≤ 0 for all x ∈ [0, min i =0 (-r i )] i.e. φ 11 ≤ φ 21 for all x ∈ [0, min i =0 (-r i )].
Using the second equation of (5.5) and the fact that

φ 11 (x + r i ) ≤ φ 21 (x + r i ) for i = 0 for all x ∈ [0, min i =0 (-r i )], we deduce that c w ′ 2 (x) ≤2(F (φ 21 (x) + w 1 (x), (φ 21 (x + r i )) i=1,...,N ) -F (φ 21 (x), (φ 21 (x + r i )) i=1,...,N )) + α 0 (w 1 (x) -w 2 (x)) =G(w 1 (x)) -G(0) -α 0 w 2 (x)
where G(t) = 2F (φ 21 (x) + t, (φ 21 (x + r i )) i=1,...,N ) + α 0 t. Using that G is non-decreasing (see (1.5)) and w 1 (x) ≤ 0, we deduce that c w ′ 2 (x) ≤ -α 0 w 2 (x). Using again that w 2 (0) = 0 and y = 0 is a solution of w ′ (x) = -α0 c w 2 (x), we deduce by the comparison principle that w 2 ≤ 0 for all x ∈ [0, min

i =0 (-r i )].
We repeat the above argument several times, each on the new extended interval. We deduce that φ 11 ≤ φ 21 for all x ≥ -r * φ 12 ≤ φ 22 for all x ≥ -r * .

We use Lemma 5.1 and Lemma 5.3 in order to prove the Strong Maximum principle under (D±) i). Proof. The idea of the proof is to translate (φ 21 , φ 22 ) and then to compare it with (φ 11 , φ 12 ).

         φ 21 ≥ φ 11 on R φ 22 ≥ φ 12 on R φ 21 (0) = φ 11 (0) φ 22 (0) = φ 12 (0). Then φ 11 (x) = φ 21 (x) for all x in R φ 12 (x) = φ 22 (x) for all x in R.
Step 1: Family of solutions above (φ 11 , φ 12 ). For a ∈ R, we define φ a 21 (x) = φ 21 (x + a) φ a 22 (x) = φ 22 (x + a).

For some a > 0 large enough, we have Since F is non-decreasing in X i for i = 0, we have F ((φ a * 21 (x + r i )) i=0,...,N ) ≥F (φ a * 21 (x), (φ 11 (x + r i )) i=1,...,N ) ≥F (φ 11 (x) + k 1 (x), (φ 11 (x + r i )) i=1,...,N ) Thus (since x 0 is a point of minimum of k 2 and k 2 (x 0 ) = 0) (5.7) 0 ≥ 2(F (φ 11 (x 0 ) + k 1 (x 0 ), (φ 11 (x 0 + r i )) i=1,...,N ) -F ((φ 11 (x 0 + r i )) i=0,...,N )) + α 0 k 1 (x 0 )

We define G : R → R by Since k 1 (x 0 ) = 0 and k 1 (x) ≥ 0, we deduce that x 0 is a point of minimum of k 1 and so by the first equation of (5.6), we get 0 ≥ α 0 k 2 (x 0 ) > 0 wich is a contradiction. Proof. Assume that c > 0 (the proof when c < 0 being similar) and let (φ 1 , φ 2 ) be a solution of (1.7).

Step 1: (φ 1 , φ 2 ) are non-decreasing. The goal is to show that φ 1 (x + a) ≥ φ 1 (x) and φ 2 (x + a) ≥ φ 2 (x) for all a ≥ 0. As in the proof of Lemma 5.5, we deduce that for a ≥ 0 large enough and for all a ≥ a, we have Thus using the comparison principle (Theorem 4.1 and Corollary 4.2), we deduce that for all a ≥ a, we have φ a 1 (x) ≥ φ 1 (x) and φ a 1 (x) ≥ φ 1 (x) on R. Let a * 1 = inf{a ∈ R, φ ā 21 ≥ φ 11 on R for all ā ≥ a} a * 2 = inf{a ∈ R, φ ā 22 ≥ φ 12 on R for all ā ≥ a}. As in the proof of Lemma 5.5, we can prove that a * 1 = a * 2 = a * . We want to prove that a * = 0. By definition of a * , there exists some x 0 such that (5.9)

           φ a * 1 ≥ φ 1 on R φ a * 2 ≥ φ 2 on R φ a * 1 (x 0 ) = φ 1 (x 0 ) φ a * 2 (x 0 ) = φ 2 (x 0 ).
Then, using the Strong Maximum Principle Lemma 5.2 or Lemma 5.4 (note that, since c = 0, φ 1 , φ 2 ∈ C 1 and so the first equation of (1.7) gives that φ 1 ∈ C 2 , then we can apply Lemma 5.4), we get that φ a * 1 = φ 1 , i.e., φ 1 is periodic of period a * . But φ 1 (-∞) = 0 and φ 1 (+∞) = 1, thus a * = 0.

Step 2: (φ 1 , φ 2 ) are increasing. Let a > 0, we want to show that φ 1 (x + a) > φ 1 (x) and φ 2 (x + a) > φ 2 (x). From Step 1, we have φ 1 (x + a) ≥ φ 1 (x) and φ 2 (x + a) ≥ φ 2 (x). Assume that there exists x 0 such that φ 1 (x 0 + a) = φ 1 (x 0 ) or φ 2 (x 0 + a) = φ 2 (x 0 ). As in the proof of Lemma 5.5, we can prove that φ 1 (x 0 + a) = φ 1 (x 0 ) and φ 2 (x 0 + a) = φ 2 (x 0 ). Using the Strong Maximum Principle (Lemma 5.2 or Lemma 5.4), we get that a = 0, which is a contradiction. Thus φ 1 (x + a) > φ 1 (x) and φ 2 (x + a) > φ 2 (x) on R for any a > 0.

Proof of Theorem 1.3. The proof of the uniqueness of the velocity is a direct consequence of Proposition 4.4. For the uniqueness of the profiles, it suffices to use Lemma 5.5 and the Strong Maximum Principle (Lemma 5.2 or Lemma 5.4). Note that, since c = 0, φ 1 , φ 2 ∈ C 1 and so the first equation of (1.7) gives that φ 1 ∈ C 2 , then we can apply Lemma 5.4. Finally the strict monotonicity of φ 1 and φ 2 follows from Lemma 5.6.

( a )

 a Uniqueness of the velocity: Under the additional assumption (C), the velocity c is unique. (b) Uniqueness of (φ 1 , φ 2 ): If c = 0, then under the additional assumption (C) and (D+

Lemma 3 . 1 (

 31 Velocity c p is bounded). Under the assumption ( Ã) and (B), let c p be the velocity given by (2.5). Then there exists C > 0 such that |c p | ≤ C for 0 < p < 1 r * with r * = max i=0,...,N |r i | .

  1) is invariant by space translation, we assume that φ1 p (0) = b -ε.

Case 1 :

 1 c = 0. For n large enough, we have |c n | ≥ |c| 2 = 0. Hence for n large enough, we have

Theorem 4 . 1 (

 41 Comparison principle on (-∞, r * ]). Let F : [0, 1] N +1 → R satisfying (A) and assume that (4.1)

Corollary 4 . 2 (

 42 Comparison principle on [ -r * , +∞)). Let F : [0, 1] N +1 → R satisfying (A) β 0 > 0 such that if : Y = (Y 0 , ...., Y N ), Y + (a, ...., a) ∈ [1 -β 0 , 1] N +1 then F (Y + (a, ..., a)) < F (Y ) if a > 0.

  satisfying (A), (B) and (C), where b and f are replaced by b = 1 -b

Proposition 4 . 4 (

 44 Uniqueness of the velocity). Under assumptions (A), we consider the function F defined on [0, 1] N +1 . Let (c 1 , (φ 11 , φ 12 )) and (c 2 , (φ 21 , φ 22 )) be two solutions of (1.7), with φ 11 , φ 12 , φ 21 , φ 22 : R → [0, 1]. If F satisfies in addition (C), then c 1 = c 2 . Proof. Assume that (c 1 , (φ 11 , φ 12 )) and (c 2 , (φ 21 , φ 22 )) are solutions of (1.7) and assume by contradiction that c 1 < c 2 . We have φ 11 (-∞) = 0, φ 11 (+∞) = 1 φ 12 (-∞) = 0, φ 12 (+∞) = 1. and φ 21 (-∞) = 0, φ 21 (+∞) = 1 φ 22 (-∞) = 0, φ 22 (+∞) = 1.

Lemma 5 . 1 (φ 21 ≥

 5121 Half Strong Maximum Principle). Let F : [0, 1] N +1 → R satisfying assumption (A) and let (φ 11 , φ 12 ) and (φ 21 , φ 22 ) be respectively a viscosity sub and super-solution of (1.7), with φ 11 , φ 12 , φ 21 , φ 22 : R → [0, 1]. We assume that φ 11 on R φ 22 ≥ φ 12 on R φ 21 (0) = φ 11 (0) φ 22 (0) = φ 12 (0). If c > 0 (resp. c < 0), then φ 11 = φ 21 for all x ≤ 0 (resp. x ≥ 0) φ 12 = φ 22 for all x ≤ 0 (resp. x ≥ 0).

w 1 (

 1 0) > 0 or w 2 (0) > 0, i.e. φ 21 (0) > φ 11 (0) or φ 22 (0) > φ 12 (0), which is a contradiction. We now use Lemma 5.1 in order to get a Strong Maximum Principle under assumption (D±) ii).

Lemma 5 . 2 (φ 21 ≥

 5221 Strong Maximum Principle under (D±) ii)). Let F : [0, 1] N +1 → R satisfying (A). Let (φ 11 , φ 12 ) and (φ 21 , φ 22 ), with φ 11 , φ 12 , φ 21 , φ 22 : R → [0, 1], be respectively a viscosity sub and super-solution of (1.7) such that φ 11 on R φ 22 ≥ φ 12 on R φ 21 (0) = φ 11 (0) φ 22 (0) = φ 12 (0). a) If F is increasing w.r.t. X i0 for a certain i 0 = 0 then φ 21 (k r i0 ) = φ 11 (k r i0 ) for all k ∈ N φ 22 (k r i0 ) = φ 12 (k r i0 ) for all k ∈ N. b) If we suppose moreover that F satisfies (D+) ii) if c > 0 or (D-) ii) if c < 0 then φ 21 (x) = φ 11 (x)for all x ∈ R φ 22 (x) = φ 12 (x) for all x ∈ R.

Lemma 5 . 3 (

 53 Comparison Principle under (D±) i)). We assume that c > 0 (resp. c < 0) and let F satisfying (A) and (D+) i) (resp. (D-) i)). Let (φ 11 , φ 12 ) and (φ 21 , φ 22 ) be two solutions of (1.7), with φ 11 , φ 12 , φ 21 , φ 22 : R → [0, 1]. We assume that φ 11 , φ 21 ∈ C 2 and φ 12 , φ 22 ∈ C 1 and that φ 11 (0) = φ 21 (0) and φ 12 (0) = φ 22 (0). Suppose moreover that φ 21 (x) ≥ φ 11 (x) on [-r * , 0] (resp. on [0, r * ]) φ 22 (x) ≥ φ 12 (x) on [-r * , 0] (resp. on [0, r * ]) then φ 21 (x) ≥ φ 11 (x) for all x ≥ -r * (resp. x ≤ r * ) φ 22 (x) ≥ φ 12 (x) for all x ≥ -r * (resp. x ≤ r * ).

Lemma 5 . 4 (

 54 Strong Maximum Principle under (D±) i)). We assume that c > 0 (resp. c < 0) and F satisfies (A) and (D+) i) (resp. (D-) i)). Let (φ 11 , φ 12 ) and (φ 21 , φ 22 ) be respectively a viscosity sub and a super-solution of (1.7), with φ 11 , φ 12 , φ 21 , φ 22 : R → [0, 1]. We assume that φ 11 , φ 21 ∈ C 2 and φ 12 , φ 22 ∈ C 1 and that

Proof. 21 ≥

 21 Let c > 0 . Using Lemma 5.1, we deduce that φ 11 = φ 21 for all x ≤ 0 φ 21 = φ 22 for all x ≤ 0. By Lemma 5.3, we then deduce that φ 11 ≥ φ 21 for all x ≥ -r * φ 12 ≥ φ 22 for all x ≥ -r * . which gives the result. Lemma 5.5 (Ordering two solutions of (2.3) up to translation). We assume that c = 0 and let F : [0, 1] N +1 → R satisfiyng (A) and (C). Let (φ 11 , φ 12 ) and (φ 21 , φ 22 ) be respectively a viscosity sub and super-solution of (1.7), with φ 11 , φ 12 , φ 21 , φ 22 : R → [0, 1]. There exists a shift a * ∈ R and some x 0 ∈ [-r * , r * ] such that  φ 11 on R φ a * 22 ≥ φ 21 on R φ a * 21 (x 0 ) = φ 11 (x 0 ) φ a * 22 (x 0 ) = φ 12 (x 0 ), where φ a * 21 (x) = φ 21 (x + a * ) φ a * 22 (x) = φ 22 (x + a * ).

φ ā 21 ≥

 21 φ 11 on [-2r * , 2r * ] for all ā > a φ ā 22 ≥ φ 12 on [-2r * , 2r * ] for all ā > a. Using the comparison principle (Theorem 4.1 and Corollary 4.2 ), we deduce that for all ā ≥ a, we have φ ā 21 ≥ φ 11 on R φ ā 22 ≥ φ 12 on R.Step 2: There exists a * such that φ a * 21 and φ 11 touch atx 0 ∈ [-r * , r * ] , φ a * 22 and φ 12 touch at x 0 ∈ [-r * , r * ]. Let a * 1 = inf{a ∈ R, φ ā 21 ≥ φ 11 on R for all ā ≥ a} a * 2 = inf{a ∈ R, φ ā 22 ≥ φ 12 on R for all ā ≥ a}.We set a * = max(a * 1 , a * 2 ). We definek 1 (x) = φ a * 21 (x) -φ 11 (x) and k 2 (x) = φ a * 22 (x) -φ 12 (x) which satisfy (5.6) c k ′ 1 (x) ≥ α 0 (k 2 (x) -k 1 (x)) c k ′ 2 (x) ≥ 2(F ((φ a * 21 (x + r i )) i=0,...,N ) -F ((φ 11 (x + r i )) i=0,...,N )) + α 0 (k 1 (x) -k 2 (x)).We now prove that a * 2 = a * 1 . By contradiction, assume that a * 2 = a * 1 If a * 2 21 (x) > φ 11 (x) φ a * 22 (x) ≥ φ 12 (x) φ a * 22 (x 0 ) = φ 12 (x 0 ).

= 2 F

 2 (φ 11 (x 0 ) + y, (φ 11 (x 0 + r i )) i=1...N ) + α 0 y.Then 0 ≥ (G(k 1 (x 0 ) -G(0)).But k 1 (x 0 ) > 0 and, by assumption (1.5), G ′ (y) > 0, so we get a contradiction.If a * 1 > a * 2 , then      φ a * 22 (x) > φ 12 (x) φ a * 21 (x) ≥ φ 11 (x) φ a * 21 (x 0 ) = φ 11 (x 0 ).

Lemma 5 . 6 (

 56 21 (x) ≥ φ 11 (x) φ a * 22 (x) ≥ φ 12 (x) φ a * 21 (x 0 ) = φ 11 (x 0 ) φ a * 22 (x 0 ) = φ 12 (x 0 ). Monotonicity of the profiles). Assume that c > 0 (resp. c < 0) and let F : [0, 1] N +1 → R satisfying (A), (C) and (D+) i) or ii). Let φ 1 , φ 2 : R → [0, 1] be a solution of (1.7). Then φ 1 and φ 2 are increasing on R.

φ a 1

 1 (x) := φ 1 (x + a) ≥ φ 1 (x) and φ a 2 (x) := φ 2 (x + a) ≥ φ 2 (x) on [-2r * , 2r * ].

  The construction is made in[START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models[END_REF] Lemma 2.1]. The only thing to verify is that F satisfy (2.1) if F satisfy (1.5), but this is trivial by looking to the way the function F is constructed. We remark that, if (φ 1 , φ 2 ) is a traveling wave for equation(1.6) with F replaced by F , then (φ 1 , φ 2 ) is also a traveling wave of the same equation. This is a direct consequence of Lemma 2.1 and the fact that

	Remark 2.2.

  2.7 (Helly's Lemma). Let (g n ) n∈N be a sequence of non-decreasing functions on [a, b] verifying |g n | ≤ C. Then there exists a subsequence (g nj ) j∈N such that

	Proof. The first part of this lemma is the classical Helly's Lemma and a proof can be found in [2,
	Section 3.3, p. 70] while the second part is proved in [1, Lemma 2.10].
	g nj → g a.e. on [a, b]
	where g is non-decreasing on [a, b] and |g| ≤ C.
	Moreover, if (g n ) n∈N is a sequence of non-decreasing functions on a bounded interval I and if
	g n → g a.e. on I
	with g constant on I, then for every closed subset interval I ′ ⊂ I,
	g n → g uniformly on I ′ .

  1 , (φ 21 , φ 22 )) is a sub-solution of (1.7). Using Corollary 4.2, we deduce that φ 21 ≤ φ 11 over [-r * , +∞) φ 22 ≤ φ 12 over [-r * , +∞). ] φ 22 ≤ φ 12 over (-∞, r * ]. Therefore φ 21 ≤ φ 11 over R φ 22 ≤ φ 12 over R.

	Similarly, using Theorem 4.1, we get that
	φ 21 ≤ φ 11 over (-∞, r

*

ACKNOWLEDGMENTS

The first author was partially supported by ANR AMAM (ANR 10-JCJC 0106), ANR IDEE (ANR-2010-0112-01) and ANR HJNet (ANR-12-BS01-0008-01).