Abdelkarim Tadibaght
email: tadibaghtabde@gmail.com

Abdelmounim El M'rini

Lionel Siame

Olivier Bellier

Olivier Bellier Orcid

Tsunami impact assessment for low-lying cities along the Northern Atlantic Coast of Morocco using MIRONE software

Keywords: 1755 Lisbon tsunami, Numerical modeling, MIRONE, Northern Atlantic coast, Morocco

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Coastal areas are exposed to quite a lot of natural hazards, from waves triggered by earthquake or extreme weather events to erosion including landslides. This situation is likely to increase in the context of global warming and sea-level rise combined to socio-economic growths, and seriously questions the adaptation of heavily urbanized areas located at low elevation. Within this context, there is a dire need to better characterize coastal hazards and associated risks located along the margin of the diffuse tectonic boundary accommodating the convergence between the Nubia and Iberia plates (Fig. 1). The Atlantic coast of Morocco is potentially threatened by tsunami waves trigged by earthquakes generated within the Gulf of Cadiz. Indeed, to the southwest of Cape Saint Vincent, oceanic earthquakes may be tsunamigenic, and generate waves that can reach the coasts of Portugal, Spain and Morocco, like during the so-called Lisbon event on November 1 st , 1755 (MW~8.5) [START_REF] Solares | Isoseismal map of the 1755 Lisbon earthquake obtained from Spanish data[END_REF][START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF][START_REF] Gutscher | What caused the great Lisbon earthquake?[END_REF][START_REF] Gutscher | The Gibraltar Arc seismogenic zone (part 2): constraints on a shallow east dipping fault plane source for the 1755 Lisbon earthquake provided by tsunami modeling and seismic intensity[END_REF]. The Atlantic coast of Morocco also gathers about 10.3 millions of inhabitants [START_REF] Laouina | Note de recherche--Le littoral dans le développement du Maroc et de sa politique atlantique[END_REF], which corresponds to 33% of the total country population according to last general population and housing census data (H.C.P, 2014). This region has always been an important attraction hub for population coming from the interior regions [START_REF] El Mrini | Evolution morphodynamique et impact des aménagements sur le littoral tétouanais entre Ras Mazari et Fnideq (Maroc nord occidental)[END_REF]. This region also hosts some of the largest cities with a heavy weight in the Moroccan economy: Tangier, Larache, Kenitra, Rabat, Mohammedia, Casablanca and El-Jadida. In fact, accounting for all the socio-economic assets, this coveted coastline is a structuring pole of the Moroccan economy (D.A.T, 2017), which therefore experiences significant forms of pressure. It is the site of a strong land speculation and anarchic urbanization, as a consequence of important industrialization, logistics and tourist development projects, in addition to fishery and agricultural activities [START_REF] Laouina | Note de recherche--Le littoral dans le développement du Maroc et de sa politique atlantique[END_REF]. On an area of approximately 25,000 km², corresponding to the tape ~10 km from the Atlantic shores, the average density is roughly 407 inhabitants / km², almost ten times the national value (Fig. 1) [START_REF] Laouina | Note de recherche--Le littoral dans le développement du Maroc et de sa politique atlantique[END_REF]. Since the early 2000s, Morocco has implemented a strategy for the development of tourism, through the Azur plans "vision 2010" and "vision 2020". These programs being essentially based on the development of coastal tourism, several resorts and hotel structures have thus been constructed along the coastlines concerned by this study. So, the combination of the Moroccan coast's exposition to the tectonic setting in the Gulf of Cadiz with density of population and economic assets should thus be an important motivation for adequately managing this region for protection against natural coastal hazards. In this work, our aim is to identify high-risk cities for tsunamis events using open-source models where quantitative assessments should also be carried out (e.g., [START_REF] Tadibaght | Quantitative risk assessment in El-Jadida (Northern Atlantic Coast of Morocco) for a tsunami scenario equivalent to that of the 1755 Lisbon earthquake[END_REF]. In this study, tsunami characteristics such as inundation boundaries, flow depths, inundation distance, and inundation areas were simulated (for some of the largest socio-economical areas) along the Atlantic coast of Morocco, allowing for a regional reappraisal of the tsunami risk assessment. Considering several potential earthquake sources located in the Gulf of Cadiz, and using MIRONE code [START_REF] Luis | Mirone: A multi-purpose tool for exploring grid data[END_REF], this characterization is achieved through a scenariobased approach like that described in [START_REF] Omira | Design of a sealevel tsunami detection network for the Gulf of Cadiz[END_REF], which considers an initial sea-surface perturbation created by an instantaneous seabed displacement (e.g., [START_REF] Mansinha | The displacement fields of inclined faults[END_REF]. It was executed on a series of bathymetric and topographic grid layers (300 m, 60 m, 15 m, 10 m and 2 m resolutions). The results are displayed as maps of Maximum Flow (Depth and Distance), and Maximum Wave Elevation from epicenter to the coast for the more exposed cities along the coast and discussed regarding historical data and previously published models.

Extreme coastal events and geodynamic setting of the Gulf of Cadiz region

Tsunamis generally comprise a series of superficial oceanic waves generated for example by a volcanic eruption, an earthquake, or a submarine landslide, and spread in the ocean at high speed together with a relatively small amplitude. Approaching coastal areas, the wave speed decreases while the amplitude increases in response to the spatial compression due to the shallowing bathymetry. When exceeding tens of meters, such coastal waves can drown large areas of low-lying regions like during the 2004 Sumatra [START_REF] Chadha | The tsunami of the great Sumatra earthquake of M 9.0 on 26 December 2004--Impact on the east coast of India[END_REF] and the 2011 Tohoku-oki [START_REF] Goto | The 2011 Tohoku-oki tsunami-Three years on[END_REF] events. Most of large tsunamis are generally associated with high magnitude (Mw≥8.5) earthquakes along subduction zones (e.g., [START_REF] Jing | A revised evaluation of tsunami hazards along the Chinese coast in view of the Tohoku-Oki Earthquake[END_REF].

Offshore of the Atlantic coasts of Morocco, the Southwestern Iberian margin region is characterized by submarine faults that are considered capable to release energy similar to that of a subduction zone (Gutscher et al., 2006). Around the Gulf of Cadiz, the countries that are exposed to potential tsunami events are Portugal, Spain, and Morocco (Fig. 1). The earthquake that caused the Lisbon catastrophe in 1755 also triggered a transoceanic tsunami (Roger et al., 2010a(Roger et al., , 2010b[START_REF] Roger | The transoceanic 1755 Lisbon tsunami in Martinique[END_REF] that was observed in most of countries of the Northern Atlantic Ocean, with major repercussions along the coastlines of Portugal, southwest Spain and Northern Morocco [START_REF] Solares | Isoseismal map of the 1755 Lisbon earthquake obtained from Spanish data[END_REF][START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF][START_REF] Baptista | Constrains on the source of the 1755 Lisbon tsunami inferred from numerical modelling of historical data on the source of the 1755 Lisbon tsunami[END_REF][START_REF] Gutscher | The Gibraltar Arc seismogenic zone (part 2): constraints on a shallow east dipping fault plane source for the 1755 Lisbon earthquake provided by tsunami modeling and seismic intensity[END_REF]. In the archives, historians described this tsunami and its consequences on European and Moroccan coasts, suggesting that its main source was a strong earthquake in the southwestern Iberian margin region. Nonetheless, the understanding of this tsunami is still debated and imposes a confrontation of the historical sources in terms of interpretation, since some parameters such as the actual location of the seismic epicenter, the height of waves or, the flooded zones are not well known (e.g., [START_REF] Baptista | Constrains on the source of the 1755 Lisbon tsunami inferred from numerical modelling of historical data on the source of the 1755 Lisbon tsunami[END_REF][START_REF] Omira | Design of a sealevel tsunami detection network for the Gulf of Cadiz[END_REF]. However, some studies suggested that the seismogenic source of the 1755 and 1969 (Mw=7.8) events could be the same [START_REF] Levret | The effects of the November 1, 1755 Lisbon earthquake in Morocco[END_REF].

Mentioning a great impact of the 1755 tsunami along the Moroccan shores, Soyris (1755) described tsunami wave height of 24 m and run-ups reaching about 15 m [START_REF] Blanc | The tsunami in Cadiz on 1 November 1755: A critical analysis of reports by Antonio de Ulloa and by Louis Godin[END_REF]. [START_REF] Barkan | Far field tsunami simulations of the 1755 Lisbon earthquake: Implications for tsunami hazard to the US East Coast and the Caribbean[END_REF] reported that the runup of this tsunami has reached 5 to 15 m along the Moroccan and Portuguese coasts. Therefore, the 1969 event is also worth mentioning in terms of the need to evacuate the shore-dwelling population as the Casablanca tide gauge recorded the first wave of tsunami which peaking at about 0.8 m [START_REF] Gjevik | Modeling tsunamis from earthquake sources near Gorringe Bank southwest of Portugal[END_REF]. There are other tsunami sources like the November 1941 tsunami event which had its epicenter along the Gloria fault [START_REF] Baptista | Tsunamis along the Azores Gibraltar plate boundary[END_REF] and that was recorded in several locations along the Moroccan coast with a maximum wave's amplitude reach up to 0.25 m in Casablanca and 0.45 m in Essaouira [START_REF] Kaabouben | On the moroccan tsunami catalogue[END_REF]. However, tsunami simulations for the 1941 event, including some Moroccan coastal areas, done a maximum wave's amplitude reach up to 0.09 m and 0.12 m respectively [START_REF] Baptista | New study on the 1941 Gloria Fault earthquake and tsunami[END_REF]. More recently, some seismic events occurred with lower magnitude in 2003 (Mw= 5.3), 2007 (Mw = 5.9) and 2009 (Mw=5.5), without triggering tsunamis [START_REF] Pro | The earthquakes of 29 July 2003, 12 February 2007, and 17 December 2009 in the region of Cape Saint Vincent (SW Iberia) and their relation with the 1755 Lisbon earthquake[END_REF]. However, [START_REF] Omira | The November 1st, 1755 tsunami in Morocco: Can numerical modeling clarify the uncertainties of historical reports[END_REF], using tsunami simulations, showed that the descriptions included in the historical archives may overestimate the tsunami characteristics. In addition, geomorphic and sedimentological markers of coastal extreme events have been described along the coasts of Portugal (Algarve: [START_REF] Andrade | Tsunami generated forms in the Algarve barrier islands (South Portugal)[END_REF]); (Lisbon: [START_REF] Scheffers | Tsunami relics on the coastal landscape west of Lisbon, Portugal[END_REF]; (Cascais-Caboda Roca: [START_REF] Oliveira | Transport of large boulders quarried from shore platforms of the Portuguese west coast[END_REF]), Spain (Cadiz: Dabrio et al., 1998;[START_REF] Luque | Tsunami deposits as paleoseismic indicators: examples from the Spanish coast[END_REF]; (Cabo of Trafalgar: Whelan and Kelletat, 2005), and Morocco (Tangier: El [START_REF] El Talibi | Geological evidence of highenergy marine flooding events on the Tangier coastal plain, Morocco. Euro-Mediterranean[END_REF], Rabat: Medina et al., 2011;Mhammdi et al., 2008, Loukkos zone: Mhammdi and[START_REF] Mhammdi | Recent tsunami deposits along the Moroccan Atlantic Coast[END_REF]. However, the origin of the triggering waves is still largely debated between tsunami or strong storm events [START_REF] Scheffers | Paleotsunami evidences from boulder deposits on Aruba, Curaçao and Bonaire[END_REF][START_REF] Hall | Patterns and rates of erosion produced by high energy wave processes on hard rock headlands: The Grind of the Navir, Shetland, Scotland[END_REF][START_REF] Morton | Coarse-clast ridge complexes of the Caribbean: a preliminary basis for distinguishing tsunami and storm-wave origins[END_REF][START_REF] Goff | The tsunami geomorphology of coastal dunes[END_REF]Goff et al., , 2012b;;Switzer and Burston, 2010;[START_REF] Hoffmann | Block and boulder accumulations along the coastline between Fins and Sur (Sultanate of Oman): tsunamigenic remains?[END_REF]. It should be mentioned that there are other studies on coastal geomorphology and sedimentology of tidal ranges and different energy systems (e.g., [START_REF] Dill | The Totumo mud volcano and its near-shore marine sedimentological setting (North Colombia) -From sedimentary volcanism to epithermal mineralization[END_REF][START_REF] Dill | Coastal landforms of "Meso-Afro-American" and "Neo-American" landscapes in the periglacial South Atlantic Ocean: With special reference to the clast orientation, morphology, and granulometry of continental and marine sediments[END_REF], which it can used with others like sedimentological, chemical and mineralogical approaches to find out the origin of recent or historiques events deposits and landforms (e.g., [START_REF] Dill | Trends and composition -A sedimentological-chemical-mineralogical approach to constrain the origin of Quaternary deposits and landforms -From the review to the manual[END_REF]. This leads us to distinguish each phenomenon from another (i.e., storms or tsunamis), as was done exactly in a research paper by Ruiz Muñoz et al., 2021, where a set of the easternmost mineralogical and geochemical evidence of the 1755 Lisbon tsunami event were found in the Western Mediterranean.

The studied area comprises large Moroccan cities between Tangier and El-Jadida along the Northern Atlantic coast of Morocco, which are potentially exposed to tsunami impact generated by earthquakes occurring in the Southwestern Iberian margin region (e.g., [START_REF] Blanc | Earthquakes and tsunami in November 1755 in Morocco: a different reading of contemporaneous documentary sources[END_REF][START_REF] Blanc | The tsunami in Cadiz on 1 November 1755: A critical analysis of reports by Antonio de Ulloa and by Louis Godin[END_REF]. In this region, a complex of faults is located at the end of the eastern Azores-Gibraltar plate boundary that separates the Nubian and Iberia continental plates. Along the Azores-Gibraltar plate boundary, the deformation pattern changes from ENE-WSW extension, close to the Azores triple junction to right lateral strike slip along the Gloria fault and compressional deformation from the Gorringe Bank through the Alboran Sea (e.g., [START_REF] Hayward | A seismic reflection and GLORIA study of compressional deformation in the Gorringe Bank region, eastern North Atlantic[END_REF] (Fig. 2). This area is tectonically active and produced tsunamis generated by a number of earthquakes (e.g., [START_REF] Baptista | Constrains on the source of the 1755 Lisbon tsunami inferred from numerical modelling of historical data on the source of the 1755 Lisbon tsunami[END_REF], like 1755 November 1 st [START_REF] Gutscher | The Gibraltar Arc seismogenic zone (part 2): constraints on a shallow east dipping fault plane source for the 1755 Lisbon earthquake provided by tsunami modeling and seismic intensity[END_REF], 1941 November 25 th (Mw 8) [START_REF] Buforn | Seismicity and focal mechanisms in south Spain[END_REF][START_REF] Buforn | Seismic sources on the Iberia-African plate boundary and their tectonic implications[END_REF], 1969 February 28 th (Mw 7.3), 1975 May 26 th (Mw7.9), and 2007 February12 th (Mw 6.3) [START_REF] Baptista | Maximum entropy analysis of Portuguese tsunami data; the tsunamis of 28.02. 1969 and 26.05[END_REF][START_REF] Ph | Numerical simulation of the 1969 tsunami along the Portuguese coast. Preliminary results[END_REF]. In this region, the main active structures are the Marquês do Pombal Fault [START_REF] Zitellini | The tectonic source of the 1755 Lisbon earthquake and tsunami[END_REF], Gorringe Bank Fault [START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF], Portimão Bank Fault [START_REF] Zitellini | The quest for the Africa--Eurasia plate boundary west of the Strait of Gibraltar[END_REF][START_REF] Lima | Impact of a 1755-like tsunami in Huelva, Spain[END_REF], Gulf of Cádiz Fault [START_REF] Gutscher | Evidence for active subduction beneath Gibraltar[END_REF][START_REF] Gutscher | The Gibraltar Arc seismogenic zone (part 2): constraints on a shallow east dipping fault plane source for the 1755 Lisbon earthquake provided by tsunami modeling and seismic intensity[END_REF]Horseshoe Fault (Gracia et al., 2003). Even if the source of the Lisbon event is still debated, [START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF] suggested that its origin zone may well be a structure located between the Coral Patch Ridge and the Gorringe Bank (Fig. 2). Recent studies based on seismic reflection and multibeam echosounder data indicated that three major faults can be possibly the source of 1755 event: the N60°E-striking thrust Gorringe Bank Fault [START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF][START_REF] Grandin | Simulations of strong ground motion in SW Iberia for the 1969 February 28 (M s= 8.0) and the 1755 November 1 (M~ 8.5)[END_REF], the NNE, trending Marquês do Pombal thrust fault [START_REF] Zitellini | Source of 1755 Lisbon earthquake and tsunami investigated[END_REF]Gracia et al., 2003), and the western Gulf of Cádiz Fault, possibly as part of an Nubian plate subduction beneath Gibraltar [START_REF] Gutscher | Evidence for active subduction beneath Gibraltar[END_REF]2006;[START_REF] Thiebot | The Gibraltar Arc seismogenic zone (part 1): constraints on a shallow east dipping fault plane source for the 1755 Lisbon earthquake provided by seismic data, gravity and thermal modeling[END_REF].

Methodology

To evaluate tsunami hazard parameters along the studied coastline, we applied a methodology illustrated by the simplified logical tree shown by Figure . 3. This approach aims to reproduce historic events such as the 1755 1 st November tsunami that affected the Gulf of Cadiz. To do so, the tsunami hydrodynamic simulation includes three main steps: wave generation, and propagation as well as inland inundation.

Figure 3: Logical approach to evaluate tsunami impact in study area.

Tsunami scenarios

To define the seismic scenarios capable to trigger tsunamis in the Southwestern Iberian region, we compiled potential tsunamigenic faults from the literature [START_REF] Blanc | Earthquakes and tsunami in November 1755 in Morocco: a different reading of contemporaneous documentary sources[END_REF] (Table 1 and Fig. 2) and used the focal mechanisms of some selected events from the International Seismological Center (ISC, 2020) catalogue for the period 1970-2007 (Mw >3).

We selected four earthquake scenarios that correspond to four faults in the Gulf of Cadiz (dashed red lines in Fig. 2), using indicators of seismotectonic activity such as fault traces in submarine Quaternary deposits and seismo-stratigraphic signatures as well as instrumental seismicity [START_REF] Hayward | A seismic reflection and GLORIA study of compressional deformation in the Gorringe Bank region, eastern North Atlantic[END_REF][START_REF] Zitellini | Source of 1755 Lisbon earthquake and tsunami investigated[END_REF][START_REF] Gràcia | Crustal architecture and tectonic evolution of the Gulf of Cadiz (SW Iberian margin) at the convergence of the Eurasian and African plates[END_REF][START_REF] Omira | Design of a sealevel tsunami detection network for the Gulf of Cadiz[END_REF][START_REF] Cunha | Optical dating of clastic deposits generated by an extreme marine coastal flood: The 1755 tsunami deposits in the Algarve (Portugal)[END_REF][START_REF] Bartolome | Evidence for active strike-slip faulting along the Eurasia-Africa convergence zone: Implications for seismic hazard in the southwest Iberian margin[END_REF][START_REF] Martínez-Loriente | Morphostructure, tectono-sedimentary evolution and seismic potential of the Horseshoe Fault, SW Iberian Margin[END_REF].

In these scenarios, we focused on assessment of tsunamis impact in the Gulf of Cadiz region and some cities of Northern Atlantic Morocco using typical faults parameters (Table 1): Gorringe Bank, Gulf of Cadiz, Marquês do Pombal, and Horseshoe faults. To model associated tsunami, the geometry of the faults was simplified to a rectangle and the slip amplitude was fixed accordingly [START_REF] Scholz | Scaling laws for large earthquakes; consequences for physical models[END_REF]. Therefore, the seismic moment definition of [START_REF] Aki | Generation and Propagation of G Waves from the Niigata Earthquake of June 16, 1964[END_REF] (Eq.1) and the Mo-Mw relation defined by Hanks and Kanamori, 1979 (Eq. 2) included in MIRONE software were used to compute the earthquake scenario's magnitude (Table .1).

Mo= µ × A × D

(1)

Log10 (Mo) = 1.5× Mw +9.05 (2)
where Mo is the seismic moment (Nm); A is the rupture area (km 2 ); µ is the crustal rigidity (Nm -2 ), D the fault slip (m) and Mw is the moment magnitude.

Table 1: Fault parameters of the tsunamigenic scenarios used in this study. μ: the shear modulus, L: the fault length in km, W: the fault width in km, and Mw: the seismic magnitude (modified from [START_REF] Omira | Design of a sealevel tsunami detection network for the Gulf of Cadiz[END_REF].

Scenarios

Simulation code

Numerical simulation is an effective method to estimate the areas that will be flooded, as well as the maximum wave height and the arrival travel times of a tsunami. We used MIRONE program that allows realizing numerical simulation of the hydrodynamic process with the TINTOL code (http://fctgmt.ualg.pt/mirone/downloads/windows.html, accessed December 2019) [START_REF] Luis | Mirone: A multi-purpose tool for exploring grid data[END_REF]. To compute the generation of a tsunami caused by an offshore seismic event, we used a linear elastic dislocation model [START_REF] Mansinha | The displacement fields of inclined faults[END_REF]. Because this option only computes the vertical component of the deformation produced by an earthquake, it is less prone to uncertainties in the parameters and simply consists in calculating the initial vertical movement of the sea surface floor associated with a seismic rupture. As for propagation and inundation, linear (Eq.3) and non-linear (Eq.4) differential models included in MIRONE software are used, respectively [START_REF] Luis | Mirone: A multi-purpose tool for exploring grid data[END_REF]. where ɳ is the free-surface displacement, H = ɳ + h is the total water depth and h is the still-water depth. P and Q are the horizontal components of the volume flux in the x and y-axes, respectively, and τϰ and τy are the bottom friction terms in the x-and y-axes, respectively.

Finally, to propagate this initial impulsion through the oceanic water body, from the source zone to the coast, we used a series of bathymetric/topographic grids layers were downloaded from the GEBCO database (General Bathymetric Chart of the Oceans, https://www.gebco.net/data_and_products/gridded_bathymetry_data/, accessed December 2019). The time step used for tsunami simulations was 1.5 s and the computations were carried out for 3000 s. This dataset was nested in a grid system with different resolutions for consecutive calculations of tsunami event generation, propagation in an open boundary condition and inland inundation. The main grid layer has a 300 m-resolution and is used for tsunami propagation in the Gulf of Cadiz. The second grid layers, improved by transplanting the main grid layer of 300 m using 'transplant 2nd grid tool' [START_REF] Luis | Mirone: A multi-purpose tool for exploring grid data[END_REF], have a high resolution (60 m, 15 m, 10 m and 2 m) and are used for the inland inundation of selected cities exposed to tsunami along the Atlantic coast of Morocco (El-Jadida, Casablanca and Mohammedia).

Results

For each earthquake scenario, tsunami simulation was produced to derive the following parameters: spatial and temporal distribution of Maximum Water Elevation (MWE), Maximum Inundation Distance (MID), and Maximum Flow Depths (MFD). The MWE shows the large-scale tsunami energy direction toward the coasts around the Gulf of Cadiz, whereas the MFD, and MID were computed to evaluate the local impact of simulated tsunamis in selected areas along the Northern Atlantic coast of Morocco.

Maximum Water Elevation and tsunami arrival travel time estimate

For the four earthquake scenarios, the results of MWE calculations are presented in Figures 4,5, 6 and 7 respectively, with the results of arrival travel time, from the epicenter source to the coast that locates the highest tsunami waves. In our simulations, all earthquake scenarios steered tsunami energy towards the Gulf of Cadiz region including the south-western coasts of the Iberian Peninsula and the Northern Atlantic coasts of Morocco.

However, there are differences in the main direction of maximum energy as well as in the time required to reach the coastal regions.

The scenario involving Cadiz Wedge in front of the Gulf of Cadiz Fault (Mw = 8.6: Table . 1) has been considered as a possible source of the 1755 event (e.g., [START_REF] Gutscher | What caused the great Lisbon earthquake?[END_REF]. In figure 4-a, the simulation indicates that the wave front generated by this scenario is directed towards the coasts of all cities around the Gulf of Cadiz, with a greater MWE in south-western Iberian Peninsula than along the Northern Atlantic coasts of Morocco. In this scenario, the tsunami impact would significantly affect coastline between Larache and El-Jadida. In the offshore area, the tsunami waves have equal amplitudes. Nevertheless, change in bathymetry affects the wave speed, which can abruptly produce a MWE reaching up to 6 m in some locations (Fig. 4 The Marquês do Pombal Fault is located in the active tectonic region between the Gorringe Bank and the Coral Patch Ridge zone (Fig. 2) [START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF][START_REF] Zitellini | Source of 1755 Lisbon earthquake and tsunami investigated[END_REF]. With a Mw of 8.1 (Table 1), the tsunami may cause severe destructions to coastal cities of Morocco, Portugal, and Spain. In figure 5-a, the simulation displays that the wave front generated by this scenario would be directed towards the coasts of Larache, Casablanca, Rabat-Sale, El-Jadida, Cadiz, Faro and Lisbon. MWE computations for Marquês do Pombal Fault along of the coasts exposed to this scenario may reach up to 24 m in some areas (Fig. 5-a). Examination of these results

indicates that the tsunami impact in some locations is low (i.e., Northern of Kenitra) because of the local bathymetry. Moreover, the estimated magnitude of 8.1 (Table . 1 ,Lisbon;2,Faro;3,Huelva;4,Cadiz;5,Tangier;6,Assilah;7,Larache;8,Kenitra;9,10,Mohammedia;11,Casablanca;12, The Gorringe Bank Fault may be the source for the 1755 earthquake because of its associated active uplift and shortening (e.g., [START_REF] Johnston | Seismic moment assessment of earthquakes in stable continental regions. III. New Madrid 1811--1812[END_REF]. In figure 6 ,Lisbon;2,Faro;3,Huelva;4,Cadiz;5,Tangier;6,Assilah;7,Larache;8,Kenitra;9,10,Mohammedia;11,Casablanca;12, In figure 7-a, we present the results of MWE and tsunami wave's directions for the Horseshoe Fault scenario.

First, this potential earthquake has an epicenter zone very close to the others and has almost the same orientation, except to that of Gulf of Cadiz Fault. Whereas the magnitude of this earthquake scenario is like that of Gorringe Bank Fault, it would create higher waves and larger tsunami energy over the coastal zones of Morocco. The

Impact and Mapping Inundation

The results of MID, MFD, and MIA computations and spatial distribution with arrival travel times on the exposed cities along the Northern Atlantic coasts of Morocco for the worst scenario associated to Horseshoe Fault are presented in figures 8 and 9, which display the results for the considered cities (El-Jadida, Casablanca, and Mohammedia). Figure 8-a shows the tsunami impact at El-Jadida. In this city, the MFD ranges from 5 to 6 m and the MID reaches 2.2 km and 2 km in the surrounding areas of the hypodrome and harbor, respectively (Fig. 8-b). We also present the tsunami impact generated by the Horseshoe Fault scenario at Casablanca, which produces a MFD ranging from 8 m to 10 m (Fig. 8-c) and a MID of about 2.8 km at the harbor (Fig. 8-d). This scenario produces height impact along the coast of Mohammedia city, with the MFD ranging from 8 m to 10 m (Fig. 8-e). The MID reaches 4.6 km at the location of the oil refinery "La Samir" and surrounding areas (city beach, and oil harbor) because of the flat topography and rivers (Dir and Mellah), where the inundation extends even more, flooding large land surfaces (Fig. 8-f).

Discussion

Comparison with other model results and historical data

The simulation maps show that maximum flow depth generated by the Horseshoe Fault scenario, which we assumed to be the scenario triggering the worst tsunami impact on considered coastal cities and surrounding areas. The harbors of these cities are completely flooded, the inundation depth reaching 6 m at El-Jadida and 10 m at both Casablanca and Mohammedia. Moreover, the maximum area flooded located at Mohammedia harbor (the main oil port of Morocco) and in "La Samir" refinery. This site constitutes an important issue as it comprises not only the refinery but also the largest thermal electric central in the country, which largely supplies the region of Casablanca.

The maximum flow depth decreases progressively from the coastline towards the inundation boundary because of the variation of topography and energy dissipation. Furthermore, simulation results are influenced by the fault strike [START_REF] Satake | Tsunami and its Hazard in the Indian and Pacific Oceans: Introduction[END_REF] and the basin bathymetry that constrains the tsunami energy distribution [START_REF] Omira | Evaluating tsunami impact on the Gulf of Cadiz coast (Northeast Atlantic)[END_REF]. The maximal tsunami amplitude is perpendicular to the fault strike [START_REF] Titov | Offshore forecasting of Alaska-Aleutian subduction zone tsunamis in Hawaii[END_REF], and the decrease in bathymetry amplifies the wave amplitude along the coasts. Therefore, the characteristic parameters (strike, dip and slip) of candidate tsunamigenic sources may affect the MWE and the rake or slip angle on the fault plane, which also plays an important role on the initial deformation. On the other hand, the results of three earthquake scenarios associated with the Horseshoe, Marquês do Pombal, and Gorringe Bank faults show some degrees of compatibility with both historical tsunami impact observations and description of geomorphic and sedimentary deposits in Spain [START_REF] Whelan | Boulder deposits on the southern Spanish Atlantic coast: possible evidence for the 1755 AD Lisbon tsunami[END_REF], Portugal [START_REF] Scheffers | Tsunami relics on the coastal landscape west of Lisbon, Portugal[END_REF] and along the Northern coast of Morocco [START_REF] Medina | The Rabat and Larache boulder fields; new examples of high-energy deposits related to storms and tsunami waves in north-western Morocco[END_REF][START_REF] Talibi | New sedimentary and geomorphic evidence of tsunami flooding related to an older events along the Tangier-Asilah coastal plain, Morocco[END_REF][START_REF] Dat | Élaboration de la stratégie nationale de gestion intégrée du littoral -Phase 2 : Diagnostic stratégique du littoral[END_REF]. If the Horseshoe Fault, with some differences in the maximum level of waves heights due to the limitation of MIRONE software (see section 5.2), appears to be the more impacting potential source, in agreement with previous models [START_REF] Omira | Design of a sealevel tsunami detection network for the Gulf of Cadiz[END_REF][START_REF] Omira | Tsunami vulnerability assessment of Casablanca-Morocco using numerical modelling and GIS tools[END_REF][START_REF] Lima | Impact of a 1755-like tsunami in Huelva, Spain[END_REF][START_REF] Mellas | Le risque tsunamique au Maroc: modélisation et évaluation au moyen d'un premier jeu d'indicateurs d'exposition du littoral atlantique[END_REF][START_REF] Benchekroun | Tsunami impact and vulnerability in the harbour area of Tangier, Morocco[END_REF][START_REF] El Moussaoui | Tsunami hazard and buildings vulnerability along the Northern Atlantic coast of Morocco--the 1755-like tsunami in Asilah test-site[END_REF], the simulation, the MWE in El-Jadida should be less than 10 m, which disagrees with historical records (e.g., [START_REF] Soyris | Competing mechanisms for boulder deposition on the southeast Australian coast[END_REF]. However, [START_REF] Blanc | Earthquakes and tsunami in November 1755 in Morocco: a different reading of contemporaneous documentary sources[END_REF] estimated that the MWE of the 1755 tsunami, reported along the shoreline of Morocco between Agadir and Tangier, was 2.5 m. Indeed, our results show that the worst energy of Horseshoe scenario is steered between Tangier and El-Jadida coasts. This is in agreement with the historical data that indicate that the tsunami impact was greater in Northern Atlantic coasts of Morocco than along the southern coasts [START_REF] Kaabouben | On the moroccan tsunami catalogue[END_REF]. However, these authors show that at El-Jadida city, the flooding limit reaches roughly 0.7 km, which is smaller than the 2 km inland calculated in this work. Nevertheless, according to the tsunami travel time calculated using the ray tracing technique by [START_REF] Omira | Design of a sealevel tsunami detection network for the Gulf of Cadiz[END_REF], as well as to evacuation model used in [START_REF] Tadibaght | Quantitative risk assessment in El-Jadida (Northern Atlantic Coast of Morocco) for a tsunami scenario equivalent to that of the 1755 Lisbon earthquake[END_REF], the Northern Atlantic population of Morocco can safely evacuate if the tsunami event were to occur in daytime and the evacuation were to start 1 hour after the triggering of tsunami waves or earlier.

Validity and limitations of tsunami simulation

Despite the improvement of the bathymetric/topographic data in the MIRONE software, the 300 m resolution (www.Gebco.net) is a limitation especially in hazard mapping at the local scale. Thus, the results obtained in this work may be influenced by protective structures in ports and by submarine canyons. The latter cannot be detected due to the low resolution of the input data, but may change the amplitude of wave breaking and the tsunami energy. However, according to [START_REF] Wynn | The Northwest African slope apron: a modern analogue for deep-water systems with complex seafloor topography[END_REF], maps of submarine canyons in Morocco show that they exist only in the south of El-Jadida. So, the effect of these canyons can be regarded as negligible.

We restricted the scenarios to four major faults (Gorringe Bank, Marquês do Pombal, Horseshoe, Gulf of Cádiz), considering that they are active and responsible for recent seismic events in the Gulf of Cadiz (e.g., [START_REF] Zitellini | Source of 1755 Lisbon earthquake and tsunami investigated[END_REF][START_REF] Silva | Micro-seismicity in the Gulf of Cadiz: Is there a link between micro-seismicity, high magnitude earthquakes and active faults?[END_REF]. The magnitude calculated for each scenario by MIRONE software can be influenced by the geometry of the source area and by the number of fault segments such as the complex seismic scenarios proposed by Fonseca, (2020), which also considers the accretionary prism as a potential source for seismic events in the Gulf of Cadiz. However, MIRONE software (Open-source tool) gives good results comparable with commercial software of more limited access (e.g., [START_REF] Omira | The November 1st, 1755 tsunami in Morocco: Can numerical modeling clarify the uncertainties of historical reports[END_REF]. Moreover, MIRONE is one of the most popular programs in the scientific community due to its high quality of cartography and graphing capabilities [START_REF] Luis | Mirone: A multi-purpose tool for exploring grid data[END_REF].

According to other results in the three scenarios Gorringe Bank, Marquês do Pombal and Horseshoe faults, the MWE higher than 10 meters and the distance inundation reach up to 1 km are overestimated and unrealistic values (e.g., [START_REF] Mellas | Le risque tsunamique au Maroc: modélisation et évaluation au moyen d'un premier jeu d'indicateurs d'exposition du littoral atlantique[END_REF][START_REF] Omira | The November 1st, 1755 tsunami in Morocco: Can numerical modeling clarify the uncertainties of historical reports[END_REF][START_REF] Omira | Probabilistic tsunami hazard in the Northeast Atlantic from nearand far-field tectonic sources[END_REF]. But, an event of such magnitude is still possible, since in Morocco there are cities like Mazagan (El-Jadida) that were affected by the 1755 event despite their geographical location exceeding 10 meters (e.g., [START_REF] Soyris | Competing mechanisms for boulder deposition on the southeast Australian coast[END_REF][START_REF] El Mrabet | The great earthquakes in the Maghreb region and their consequences on man and environment[END_REF].

Conclusion

In this paper, we simulated scenarios of tsunami flooding in some cities along the Northern Atlantic coasts of Morocco for the earthquake sources located in the offshore of the Gulf of Cadiz region. To do this, we used the parameters of scenario sources by the integration in MIRONE numerical code combining a series of bathymetric /topographic grid layers with an initial water elevation (sea-surface perturbation) generated using Mansinha's formulations and assuming an instantaneous seabed displacement. Using MIRONE software (Opensource tool), results indicate that the Horseshoe Fault is the worst scenario, which produced the largest tsunami toward the Northern Atlantic coasts of Morocco. Through our literature review and results, we can say that it is the most likely source of a potential tsunami. This scenario computed a MWE that may exceed 20 m in some area, the MFD and MID range respectively from 8 m to 10 m and 1.9 km to 4.6 km inland at "La Samir" in Mohammedia. Moreover, the computed MWE ≤ 10 m at EL-Jadida show a large disagreement with historical reports in some locations and agreement with numerical model results. Tsunami impact calculated at both Casablanca and Mohammedia is the same except the inland distance that is less at Casablanca. However, the economic and touristic areas are exposed to west-Iberian tsunamis in the Northern of Morocco. Therefore, economic and social impacts of a major tsunami along these coasts may have large consequences, either directly: power generating units and power and gas supply at Mohammedia; losses for industry, shipping and navigation, especially at Casablanca; loss of agricultural production by loss of crops and contamination of soils, especially at El-Jadida and surrounding areas, or indirectly: destruction of hospital and medical centre buildings and equipments, emergence of water vector-borne diseases following sea water inundation of freshwater bodies. On this basis and in absence of early warning system, risk education of coastal communities is one of the main preventive actions in the case of a future event.

Highlights

-Characteristics of potential tsunami sources around the Gulf of Cadiz.

-Simulation the impacts of tsunami on some cities along the Northern Atlantic coasts of Morocco.

-MIRONE software as reliable open-source tool for tsunami waves simulation.

Figure 1 :

 1 Figure 1: (a) Map of the study area showing the population density and the principal cities of Northern Morocco (based on Laouina, 2019), (b) tectonic boundary and regional overview of the Northeast Atlantic area.

Figure 2 :

 2 Figure2: Bathymetry map and tectonic interpretation of the Azores-Gibraltar area modified from[START_REF] Zitellini | The quest for the Africa--Eurasia plate boundary west of the Strait of Gibraltar[END_REF][START_REF] Bartolome | Evidence for active strike-slip faulting along the Eurasia-Africa convergence zone: Implications for seismic hazard in the southwest Iberian margin[END_REF]. Earthquake epicenter distribution (red dots) from ISC Catalogue and the 1 st November 1755 Lisbon model earthquakes used in this study (red dashed lines). The blue stars locate the epicentres of the most important events, and the black circles locate the sites where coastal mega-blocks have been convincingly described(1, 2, Cabo of Trafalgar; 3, Larache; 4, Val d'Or). Abr: GBF, Gorringe Bank Fault; MPF, Marquês do Pombal Fault; HF, Horseshoe Fault; GF, Gloria Fault; PB, Portimão Bank Fault; GCF, Gulf of Cádiz Fault.

  -a). The estimation of the arrival travel time for the tsunami waves is on the order of 65 min along the Northern coasts of Morocco (Fig. 4b).

Figure 4 :

 4 Figure 4: Results of the propagation for the Gulf of Cádiz Fault in south-western coasts of Iberian Peninsula and Northern of Morocco coasts, (a) wave height from the epicentre zone to the coast and (b) tsunami arrival travel times (TTT) at selected cities (1, Lisbon; 2, Faro; 3, Huelva; 4, Cadiz; 5, Tangier; 6, Assilah; 7, Larache; 8, Kenitra; 9, 10, Mohammedia; 11, Casablanca; 12,

  ) is less than for other faults considered in this study. Therefore, MWE results show sites where the tsunami energy is weaker (Kenitra and surrounding coasts) than in others. The arrival travel time estimation could be about 95 min along the Northern coasts of Morocco because of the long distance traveled (Fig.5-b).

Figure 5 :

 5 Figure 5: Results of the propagation for the Marquês do Pombal Fault in south-western coasts of Iberian Peninsula and Northern of Morocco coasts, (a) wave height from the epicentre zone to the coast and (b) tsunami arrival travel times (TTT) at selected cities (1, Lisbon; 2, Faro; 3, Huelva; 4, Cadiz; 5, Tangier; 6, Assilah; 7,

  -a, the simulation displays tsunami waves that are directed towards three different directions in the Gulf of Cadiz. Part of the waves created is directed towards the Portuguese coast, with a MWE ranging from 2 to 8 m at Faro and reach up to 20 m in the south of Lisbon (Fig.6-a). Another part is directed towards the Gibraltar Strait and is characterized by MWE lower than 6 m at Cadiz (Fig.6-a). The last part is directed towards the Moroccan coasts and is characterized by very high amplitudes up to 10 m close to Casablanca and surrounding areas (Fig.6-a). The estimated arrival travel time could be about 99 min along the studied coastal area (Fig.6-b).

Figure 6 :

 6 Figure 6: Results of the propagation for the Gorringe Bank Fault in south-western coasts of Iberian Peninsula and Northern of Morocco coasts, (a) wave height from the epicentre zone to the coast and (b) tsunami arrival

  principal directions of tsunami energy are to the South (El-Jadida), to South-Southeast (Mohammedia-Casablanca-Kenitra-Rabat), to the Southeast (Larache-Tangier-Assilah-Gibraltar), to the East (Cadiz), and to the Northeast (Huelva-Faro). Within this scenario, the MWE may reach up to 10 m, 23 m, 25 m, 10 m and 10-25 m according to these directions, respectively (Fig.7-a). The tsunami may significantly affect the coastlines of Casablanca, Mohammedia and Larache. During the simulation, the tsunami waves spread out and reached the Northern Atlantic coasts of Morocco in a shorter time (~79 min), unlike the other scenarios (Gorringe Bank and Marquês do Pombal faults) (Fig.7-b). The result of MWE computations reach about 23 m for Horseshoe Fault along most of the coasts exposed to this scenario, with a maximum energy towards Mohammedia and Larache coastal areas (Fig.7-a). Consequently, the tsunami wave near the Moroccan coastline, and the energy generated by the Horseshoe Fault could be very high. These results suggest that this scenario is the worst tsunami case for the Moroccan coasts, as well as in some other areas along the coastal areas of the south-western of Iberian Peninsula.

Figure 7 :

 7 Figure 7: Results of the propagation for the Horseshoe Fault in south-western coasts of Iberian Peninsula and Northern of Morocco coasts, (a) wave height from the epicentre zone to the coast and (b) tsunami arrival travel times (TTT) at selected cities (1, Faro; 2, Huelva; 3, Cadiz; 4, Tangier; 5, Assilah; 6, Larache; 7, Kenitra; 8, 0, Mohammedia; 10, Casablanca; 11,

Figure 8 :

 8 Figure 8: Flooding maps and computed inundation distances for the Horseshoe Fault scenario on the selected exposed cities along the Northern Atlantic coast of Morocco: (a) MFD and (b) MID at El-Jadida; (c) MFD and (d) MID at Casablanca; (e) MFD and (f) MID at Mohammedia.

Figure 9 :

 9 Figure 9: Results inundated area and maximum flow depths, for the Horseshoe Fault scenario along the harbors and principal locations in (a) Mohammedia, (b) Casablanca, (c) El-Jadida with high resolution inundation maps 2 m, 10 m and 15 m respectively and (d) maximum tsunami water elevation (MWE) along the coastline and tsunami arrival travel times (TTT) at selected locations (1-El-Jadida and 2/3-Casablanca & Mohammedia).

  analysis of tsunami propagation close to the shoreline indicate that waves have similar behavior for the three scenarios, most probably because of the proximity of the sources as well as their similar orientations. If Assilah and Larache are at equal distance (360 km) from the epicenter of Marquês do Pombal and Horseshoe faults, the difference in waves arrival travel time after earthquake that reach to 95 and 80 respectively (difference of 15 min), reflects the propagation velocity of the two wave trains consequently to bathymetry variations and differences in energies released towards the Moroccan coasts (see Fig. 5-b and Fig. 7-b). According to our

Acknowledgment

This work is part of the RiskMED project, Labex OT-Med (ANR-11-LABE-0061), supported by the Investissements d'Avenir, French Government project of the French National Research Agency (ANR) through the A*Midex project (ANR-1-IE-0001-02). The authors thank all reviewers for the constructive and valuable comments.

Declaration of competing Interest

The authors declare having no conflict of interest.


Michel Chipot
email: m.m.chipot@math.uzh.ch

Mingmin Zhang

On Some Model Problem for the Propagation of Interacting Species in a Special Environment

Keywords: AMS 2020 Subject Classification: 35J47, 35J57 Nonlinear elliptic systems, Coupled equations, Field-Road model, Truncated domain

The purpose of this note is to study the existence of a nontrivial solution for an elliptic system which comes from a newly introduced mathematical problem so called Field-Road model. Specifically, it consists of coupled equations set in domains of different dimensions together with some interaction of non classical type. We consider a truncated problem by imposing Dirichlet boundary conditions and an unbounded setting as well.

Introduction and notation

It has been observed in diffusion or propagation problems that the topography of the environment is playing a crucial rôle. We refer for instance to [START_REF] Berestycki | The influence of a line with fast diffusion on Fisher-KPP propagation[END_REF], [START_REF] Berestycki | Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion[END_REF] and the references there. In particular the roads are not only essential for human beings but also can be useful for other species. One can easily imagine that mosquitos could be interested in socialising with human beings along a road but many other situations seem to have occurred due to this element of civilisation. In this note we consider a model problem where the living space of our species consists in a field and one or several roads that we will assume to be unidimensional. We will also assume the roads to be straight but several extensions can be addressed very easily with our approach. On each domain -field or road-we will consider nonlinear diffusion equations which include for instance the Fisher-KPP types. We restrict ourselves to two species. The interaction with the two species is modelled by a particular flux condition (see [START_REF] Berestycki | The influence of a line with fast diffusion on Fisher-KPP propagation[END_REF], [START_REF] Berestycki | Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion[END_REF] and the set of equation below).

Let Ω be the open set of R 2 , defined for , L > 0 as Ω = (-, ) × (0, L).

We denote by Γ 0 the part of the boundary of Ω located on the x 1 -axis i.e. Γ 0 = (-, ) × {0} and by Γ 1 the rest of the boundary that is to say

Γ 1 = ∂Ω \Γ 0 .
When convenient we will identify Γ 0 to (-, ). In this setting Ω stands for a field and Γ 0 for a portion of a road. We would like to find a solution to the problem

           -D∆v = f (v) in Ω , v = 0 on Γ 1 , D ∂v ∂n = µu -νv on Γ 0 , -D u + µu = g(u) + νv in Γ 0 , u = 0 on ∂Γ 0 = {-, }.
(n denotes the outward unit normal to Ω ).

In the weak form we would like to find a couple (u, v) such that Note that this implies that for λ ≥ L f (respectively η ≥ L g ) the functions

     (u, v) ∈ H 1 0 (Γ 0 ) × V, Ω D∇v • ∇ϕ dx + Γ 0 νv(x 1 , 0)ϕ dx 1 = Ω f (v)ϕ dx + Γ 0 µuϕ dx 1 ∀ϕ ∈ V, Γ 0 D u ψ + µuψ dx 1 = Γ 0 νv(x 1 , 0)ψ dx 1 + Γ 0 g(u)ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ).
x → λx -f (x), ηx -g(x) (1.4)
are nondecreasing. In addition we will assume

f (0) = f (1) = 0, f > 0 on (0, 1), f ≤ 0 on (1, +∞).
(1.5)

g(0) = 0, g(m) ≤ 0. (1.6)
One should remark that under the assumptions above, (0, 0) is a solution to (1.1). We are interested in finding a nontrivial solution to the problem (1.1). For the notation and the usual properties on Sobolev spaces we refer to [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF], [START_REF] Chipot | Elliptic Equations: An Introductory Course[END_REF], [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF], [START_REF] Evans | Partial Differential Equations[END_REF], [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. We will also consider some extension in the case of a two-road problem which consists of three coupled equations with two interaction conditions on the upper-and lower-boundaries. Finally, we will address the case of an unbounded setting for the one-road problem.

Preliminary results

Lemma 2.1. Suppose that w is a measurable function on Γ 0 such that

0 ≤ w ≤ m.
Then under the assumptions above the problem

     v ∈ V, Ω D∇v • ∇ϕ dx + Γ 0 νv(x 1 , 0)ϕ(x 1 , 0) dx 1 = Ω f (v)ϕ dx + Γ 0 µwϕ dx 1 ∀ϕ ∈ V, (2.1)
possesses a minimal and a maximal solution with values in (0, µ ν m). Proof. Let us remark first that any nonnegative solution to (2.1) takes its values in (0, µ ν m).

Indeed if v is solution to (2.1) taking as test function ϕ = (v -k) + , k = µ ν m ≥ 1 one gets Ω D|∇(v -k) + | 2 dx = Ω D∇(v -k) • ∇(v -k) + dx = Ω D∇v • ∇(v -k) + dx = Ω f (v)(v -k) + dx + Γ 0 {µw -νv(x 1 , 0)}(v -k) + dx 1 ≤ Γ 0 {µw -νv}(v -k) + dx 1 ≤ 0,
since on the set where v ≥ k one has v ≥ µ ν m and {µw -νv} ≤ {µw -µm} ≤ 0. Next let us note that 0 is a subsolution to (2.1). Indeed this follows trivially from

Ω D∇0 • ∇ϕ dx + Γ 0 ν0ϕ dx 1 ≤ Ω f (0)ϕ dx + Γ 0 µwϕ dx 1 ∀ϕ ∈ V, ϕ ≥ 0.
On the other hand, for ϕ ∈ V , ϕ ≥ 0, one has also for k = µ ν m, since k ≥ 1

Ω D∇k • ∇ϕ dx + Γ 0 νkϕ dx 1 ≥ Ω f (k)ϕ dx + Γ 0 µwϕ dx 1
and the function constant equal to k = µ ν m is a supersolution to (2.1). For z ∈ L 2 (Ω ) we denote by y = S(z) the solution to

     y ∈ V, Ω D∇y • ∇ϕ dx + Ω λyϕ dx + Γ 0 νy(x 1 , 0)ϕ dx 1 = Ω f (z)ϕ dx + Ω λzϕ dx + Γ 0 µwϕ dx 1 ∀ϕ ∈ V, (2.2)
where we have chosen λ ≥ L f . Note that the existence of a unique solution y to the problem above is an immediate consequence of the Lax-Milgram theorem.

First we claim that the mapping S is continuous from L 2 (Ω ) into itself. Indeed setting y = S(z ) one has by subtraction of the equations satisfied by y and y

Ω D∇(y -y ) • ∇ϕ dx + Ω λ(y -y )ϕ dx + Γ 0 ν(y -y )ϕ dx 1 = Ω {f (z) -f (z )}ϕ dx + Ω λ(z -z )ϕ dx ∀ϕ ∈ V.
(2.3) Taking ϕ = y -y one derives easily

λ Ω |y -y | 2 dx ≤ Ω |f (z) -f (z )||y -y | dx + λ Ω |z -z ||y -y | dx ≤ (L f + λ) Ω |z -z ||y -y | dx.
By the Cauchy-Schwarz inequality we obtain then

|S(z) -S(z )| 2,Ω ≤ L f + λ λ |z -z | 2,Ω
which shows the continuity of the mapping S (| | 2,Ω denotes the usual L 2 (Ω )-norm).

We show now that the mapping S is monotone. Indeed suppose that z ≥ z and as above denote by y the function S(z ). Taking ϕ = -(y -y ) -in (2.3) we get (1.3), (1.4)). This shows that (y -y ) -= 0 and the monotonicity of the mapping S.

Ω D|∇(y -y ) -| 2 dx + Ω λ((y -y ) -) 2 dx + Γ 0 ν((y -y ) -) 2 dx 1 = - Ω [λ(z -z ) + {f (z) -f (z )}](y -y ) -dx ≤ 0, since -{f (z) -f (z )} ≤ L f |z -z | ≤ λ(z -z ) (see
We consider now the following sequences (Cf. [START_REF] Amann | Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[END_REF]) :

y 0 = 0, y 0 = µ ν m = k,
y n = S(y n-1 ), y n = S(y n-1 ), n ≥ 1.

(2.4)

One has

y 0 = 0 ≤ y 1 ≤ • • • ≤ y n ≤ y n ≤ • • • ≤ y 1 ≤ y 0 = µ ν m = k. (2.5)
Indeed since y 1 = S(y 0 ) = S(0) one has for ∀ϕ ∈ V , ϕ ≥ 0,

Ω D∇y 1 • ∇ϕ dx + Ω λy 1 ϕ dx + Γ 0 νy 1 (x 1 , 0)ϕ dx 1 = Ω f (y 0 )ϕ dx + Ω λy 0 ϕ dx + Γ 0 µwϕ dx 1 ≥ Ω D∇y 0 • ∇ϕ dx + Ω λy 0 ϕ dx + Γ 0 νy 0 (x 1 , 0)ϕ dx 1 ,
since y 0 is a subsolution to (2.1). Using this inequality with ϕ = (y 1 -y 0 ) -one derives easily

Ω D∇(y 1 -y 0 ) • ∇(y 1 -y 0 ) -dx + Ω λ(y 1 -y 0 )(y 1 -y 0 ) -dx + Γ 0 ν(y 1 -y 0 )(y 1 -y 0 ) -dx 1 ≥ 0.
Thus it follows that y 1 ≥ y 0 . With a similar proof one gets that y 1 ≤ y 0 . Applying S n-1 to these inequalilties leads to S n-1 (y 0 ) = y n-1 ≤ y n = S n-1 (y 1 ) , S n-1 (y 0 ) = y n-1 ≥ y n = S n-1 (y 1 ).

Furthermore from y 0 ≤ y 0 one derives by applying S n to both sides of the inequality

y n ≤ y n .
This completes the proof of (2.5). Then for some functions v, v in L 2 (Ω ) one has

y n → v, y n → v in L 2 (Ω ).
Clearly v and v are fixed point for S and thus (see (2.2)) solutions to (2.1). This completes the proof of the lemma.

We denote by λ 1 = λ 1 (Ω ) the first eigenvalue of the Dirichlet problem in Ω and by ϕ 1 the corresponding first eigenfunction positive and normalised. More precisely (λ 1 , ϕ 1 ) is such that

     -∆ϕ 1 = λ 1 ϕ 1 in Ω , ϕ 1 = 0 on ∂Ω , ϕ 1 (0, L 2 ) = 1. . (2.6)
We suppose that for s > 0 small enough one has

λ 1 ≤ f (s) Ds . (2.7)
Then one has :

Lemma 2.2. Under the assumptions of the preceding lemma and (2.7), for > 0 small enough, the maximal solution v to (2.1) satisfies

ϕ 1 ≤ v.
In particular v is bounded away from 0.

Proof. Due to (2.7) one has for > 0 small enough

Dλ 1 ϕ 1 ≤ f ( ϕ 1 ).
This allows us to show that ϕ 1 is a subsolution to (2.1). Indeed, for ϕ ∈ V , ϕ ≥ 0 it holds after integration by parts

Ω D∇( ϕ 1 ) • ∇ϕ dx + Γ 0 ν ϕ 1 (x 1 , 0)ϕ dx 1 = Ω D∇( ϕ 1 ) • ∇ϕ dx = Ω ∇ • (D∇( ϕ 1 )ϕ) -D∆( ϕ 1 )ϕ dx = ∂Ω D∂ n ( ϕ 1 )ϕdσ + Ω Dλ 1 ( ϕ 1 )ϕ dx ≤ Ω f ( ϕ 1 )ϕ dx + Γ 0 µwϕ dx 1 .
(n denotes the outward unit normal to Ω , note that ∂ n ( ϕ 1 ) ≤ 0). Thus ϕ 1 is a positive subsolution to (2.1).

Then one argues as in the preceding lemma introducing the sequence defined for small by :

y 0 = ϕ 1 ≤ µ ν m, y 0 = µ ν m, y n = S(y n-1 ), y n = S(y n-1 ), n ≥ 1.
One has with the same proof as above

y 0 = ϕ 1 ≤ y 1 ≤ • • • ≤ y n ≤ y n ≤ • • • ≤ y 1 ≤ y 0 = µ ν m.
The result follows from the fact that y n → v. This completes the proof of the lemma.

One has also :

Lemma 2.3. Suppose that f (s) s is decreasing on (0, +∞). (2.8)
If v 1 , v 2 are positive solutions to (2.1) corresponding to w 1 , w 2 respectively then

w 1 ≤ w 2 implies v 1 ≤ v 2 .
In particular (2.1) has a unique positive solution.

Proof. Denote by θ a smooth function such that

θ(t) = 0 ∀t ≤ 0, θ(t) = 1 ∀t ≥ 1, θ (t) ≥ 0. Set θ (t) = θ( t ). Clearly v 1 θ (v 1 -v 2 ), v 2 θ (v 1 -v 2 ) ∈ V.
From the equations satisfied by v 1 , v 2 one gets, setting

θ = θ (v 1 -v 2 ), Ω D∇v 1 • ∇(v 2 θ ) dx + Γ 0 νv 1 (x 1 , 0)(v 2 θ ) dx 1 = Ω f (v 1 )(v 2 θ ) dx + Γ 0 µw 1 (v 2 θ ) dx 1 , Ω D∇v 2 • ∇(v 1 θ ) dx + Γ 0 νv 2 (x 1 , 0)(v 1 θ ) dx 1 = Ω f (v 2 )(v 1 θ ) dx + Γ 0 µw 2 (v 1 θ ) dx 1 .
By subtraction we obtain

Ω D{∇v 2 • ∇(v 1 θ ) -∇v 1 • ∇(v 2 θ )} dx = Ω f (v 2 )(v 1 θ ) -f (v 1 )(v 2 θ ) dx + Γ 0 µ(w 2 v 1 -w 1 v 2 )θ (v 1 -v 2 ) dx 1 .
Clearly the last integral above is nonnegative so that one has

Ω D{∇v 2 • ∇(v 1 θ ) -∇v 1 • ∇(v 2 θ )} dx ≥ Ω f (v 2 )(v 1 θ ) -f (v 1 )(v 2 θ ) dx.
By a simple computation writing θ for θ (v 1 -v 2 ) one derives

Ω f (v 2 )(v 1 θ ) -f (v 1 )(v 2 θ ) dx ≤ Ω D{∇v 2 • ∇(v 1 θ ) -∇v 1 • ∇(v 2 θ )} dx = Ω D{∇v 2 • ∇(v 1 -v 2 )θ v 1 -∇v 1 • ∇(v 1 -v 2 )θ v 2 } dx = Ω D{v 1 ∇v 2 -v 2 ∇v 1 } • ∇(v 1 -v 2 )θ dx = Ω D{v 1 ∇v 2 -v 2 ∇v 2 + v 2 ∇v 2 -v 2 ∇v 1 } • ∇(v 1 -v 2 )θ dx = Ω D∇v 2 • ∇(v 1 -v 2 )(v 1 -v 2 )θ dx - Ω Dv 2 |∇(v 1 -v 2 )| 2 θ dx ≤ Ω D∇v 2 • ∇(v 1 -v 2 )(v 1 -v 2 )θ dx.
Let us set γ (t) = t 0 sθ (s)ds in such a way that the inequality above reads

Ω f (v 2 )(v 1 θ ) -f (v 1 )(v 2 θ ) dx ≤ Ω D∇v 2 • ∇γ (v 1 -v 2 ) dx.
From the equation satisfied by v 2 , since γ (v 1 -v 2 ) ∈ V and v 2 , γ are nonnegative one has

Ω D∇v 2 • ∇γ (v 1 -v 2 ) dx ≤ Ω f (v 2 )γ (v 1 -v 2 ) dx + Γ 0 µw 2 γ (v 1 -v 2 ) dx 1 .
Since for some constant C γ (t)

≤ 0 sθ ( s ) 1 ds ≤ C
the right hand side of the two inequalities above goes to 0 when → 0. Since when → 0 one has

θ (v 1 -v 2 ) → χ {v 1 >v 2 } the characteristic function of the set {v 1 > v 2 } = {x ∈ Ω | v 1 (x) > v 2 (x)} one gets {v 1 >v 2 } f (v 2 )v 1 -f (v 1 )v 2 dx ≤ 0.
But on the set of integration thanks to (2.8) one has f

(v 2 )v 1 -f (v 1 )v 2 > 0 hence the set of integration is necessarily of measure 0, i.e. v 1 ≤ v 2 .
This completes the proof of the lemma.

3 The main result Proof. As mentioned above it is of course clear that (0, 0) is solution to (1.1). Set

K = {v ∈ L 2 (Γ 0 ) | 0 ≤ v ≤ m}.
For u ∈ K, let v be the unique positive solution to (2.1) associated to w = u. For η ≥ L g let U = T (u) the solution to

U ∈ H 1 0 (Γ 0 ), Γ 0 D U ψ + µU ψ + ηU ψ dx 1 = Γ 0 νv(x 1 , 0)ψ + g(u)ψ + ηuψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ). (3.1)
The existence of U is a consequence of the Lax-Milgram theorem.

We claim that T is continuous on K ⊂ L 2 (Γ 0 ). Indeed suppose that u n → u in K. Denote by v n the solution to (2.1) associated to u n i.e. satisfying

Ω D∇v n • ∇ϕ dx + Γ 0 νv n (x 1 , 0)ϕ dx 1 = Ω f (v n )ϕ dx + Γ 0 µu n ϕ dx 1 ∀ϕ ∈ V. (3.2)
Since v n and u n are bounded, taking ϕ = v n in the equality above one gets easily

Ω D|∇v n | 2 dx + Γ 0 νv n (x 1 , 0) 2 dx 1 ≤ C,
where C is a constant independent of n. Thus, up to a subsequence, there exists v ∈ V such that

v n v in H 1 (Ω ), v n → v in L 2 (Ω ), v n (. , 0) → v(. , 0) in L 2 (Γ 0 ).
Passing to the limit in (3.2) it follows from Lemma 2.3 that v = v the solution to (2.1) corresponding to w = u. By uniqueness of the limit one has convergence of the whole sequence and in particular v n (. , 0) → v(. , 0) in L 2 (Γ 0 ).

Passing to the limit in (3.1) written for u = u n one derives

T (u n ) → T (u) in L 2 (Γ 0 ).
We can show also that T is monotone. Indeed, suppose that u 1 ≥ u 2 and set

U i = T (u i ), i = 1, 2. One has Γ 0 D U 1 ψ + µU 1 ψ + ηU 1 ψ dx 1 = Γ 0 νv 1 (x 1 , 0)ψ + g(u 1 )ψ + ηu 1 ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ), Γ 0 D U 2 ψ + µU 2 ψ + ηU 2 ψ dx 1 = Γ 0 νv 2 (x 1 , 0)ψ + g(u 2 )ψ + ηu 2 ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ).

By subtraction it comes

Γ 0 D (U 1 -U 2 ) ψ + µ(U 1 -U 2 )ψ + η(U 1 -U 2 )ψ dx 1 = Γ 0 ν(v 1 (x 1 , 0) -v 2 (x 1 , 0))ψ + (g(u 1 ) -g(u 2 ))ψ + η(u 1 -u 2 )ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ). Choosing ψ = -(U 1 -U 2 )
-and taking into account that, by Lemma 2.3, v 1 (x 1 , 0)-v 2 (x 1 , 0) ≥ 0 and that for η ≥ L g , (g(u 1 ) -g(u 2 )) + η(u 1 -u 2 ) ≥ 0 (Cf. (1.3), (1.4)), one gets

Γ 0 D |{(U 1 -U 2 ) -} | 2 + µ{(U 1 -U 2 ) -} 2 + η{(U 1 -U 2 ) -} 2 dx 1 ≤ 0.
Thus (U 1 -U 2 ) -= 0 and T (u 1 ) ≥ T (u 2 ).

Next we assert that T maps K into itself. Indeed, if U 0 = T (0) one has, with an obvious notation for v 0

Γ 0 D U 0 ψ + µU 0 ψ + ηU 0 ψ dx 1 = Γ 0 νv 0 (x 1 , 0)ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ). Taking ψ = -U - 0 one deduces easily since v 0 > 0 that U 0 = T (0) ≥ 0. Similarly if U m = T (m) one has, with an obvious notation for v m Γ 0 D U m ψ + µU m ψ + ηU m ψ dx 1 = Γ 0 νv m (x 1 , 0)ψ + g(m)ψ + ηmψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ). Thus choosing ψ = (U m -m) + it comes since g(m) ≤ 0, v m ≤ µ ν m Γ 0 D |{(U m -m) + } | 2 + (µ + η){(U m -m) + } 2 dx 1 = Γ 0 (νv m -µm)(U m -m) + dx 1 ≤ 0.
From which it follows that U m ≤ m. By the monotonicity of T it results that T maps the convex

K into itself. But clearly T (K) ⊂ C 1 2 (Γ 0 ) is relatively compact in L 2 (Γ 0 )
. Thus, by the Schauder fixed point theorem (see [START_REF] Evans | Partial Differential Equations[END_REF], [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], [START_REF] Chipot | Elliptic Equations: An Introductory Course[END_REF]), T has a fixed point in K which leads to a nontrivial solution to (1.1). This completes the proof of the theorem.

If it is clear at this point that the solution we constructed is non degenerate in v it is not clear that the same holds for u. In fact we have : Proof. Suppose that u ≡ 0. Due to the second equation of (1.1) one has v(x 1 , 0) = 0 and from the first equation of (1.1) we get

Ω D∇v • ∇ϕ dx = Ω f (v)ϕ dx ∀ϕ ∈ V. (3.3)
Consider then a small ball B = B x 0 centered at x 0 ∈ Γ 0 . Set ṽ = v in Ω ∩ B, 0 in the rest of the ball.

Let ϕ ∈ D(B). One has by (3.3),

B D∇ṽ • ∇ϕ dx = Ω ∩B D∇v • ∇ϕ dx = Ω ∩B f (v)ϕ dx = B f (ṽ)ϕ dx ∀ϕ ∈ D(B)
.

Thus -D∆ṽ = f (ṽ) in B.
It is clear that ṽ and thus f (ṽ) are bounded and one has f (ṽ) ∈ L ∞ (B) ⊂ L p (B) ∀p. From the usual regularity theory it follows that ṽ ∈ W 2,p (B) ⊂ C 1,α (B). Since f (ṽ) ≥ 0, f (ṽ) ≡ 0 it follows that ṽ > 0 in B (see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]). Hence a contradiction. This shows the impossibility for u to be identical to 0 and this completes the proof of the proposition.

Remark 1. One can easily show (see [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF], [START_REF] Chipot | Asymptotic Issues for Some Partial Differential Equations[END_REF]) that

λ 1 = λ 1 (Ω ) = π 2 2 + π L 2 .
Thus for a smooth function f it is clear that (2.7) is satisfied if

λ 1 = λ 1 (Ω ) < f (0) D ,
i.e. for and L large enough.

Note that (2.8) (see also (1.5)) is satisfied in the case of the Fisher equation i.e. for

f (v) = v(1 -v)
the Lipschitz character of f being used only on a finite interval.

For z ∈ L 2 (Ω ) we denote by y = S(z) the solution to

       y ∈ V, Ω D∇y • ∇ϕ dx + Ω λyϕ dx + Γ 0 νy(x 1 , 0)ϕ dx 1 + Γ 0 ν y(x 1 , L)ϕ dx 1 = Ω f (z)ϕ dx + Ω λzϕ dx + Γ 0 µ uϕ dx 1 + Γ 0 µ wϕ dx 1 ∀ϕ ∈ V,
where λ ≥ L f . The existence of a unique solution y to the problem above follows from the Lax-Milgram theorem. Then reproducing the arguments of Lemma 2.1 it is easy to show that S is continuous and monotone. Introducing the sequence defined in (2.4) one concludes as in the Lemma 2.1 to the existence of a minimal and maximal solution v and v.

Then one has :

Lemma 4.2. Under the assumptions of the preceding lemma and (2.7), for > 0 small enough, the maximal solution v to (4.3) satisfies

ϕ 1 ≤ v.
In particular v is bounded away from 0.

Proof. Due to (2.7) one has for > 0 small enough

Dλ 1 ϕ 1 ≤ f ( ϕ 1 ).
Then for ϕ ∈ V , ϕ ≥ 0 it holds after integration by parts

Ω D∇( ϕ 1 ) • ∇ϕ dx + Γ 0 ν ϕ 1 (x 1 , 0)ϕ dx 1 + Γ 0 ν ϕ 1 (x 1 , L)ϕ dx 1 = Ω D∇( ϕ 1 ) • ∇ϕ dx = Ω ∇ • (D∇( ϕ 1 )ϕ) -D∆( ϕ 1 )ϕ dx = ∂Ω D∂ n ( ϕ 1 )ϕdσ + Ω Dλ 1 ( ϕ 1 )ϕ dx ≤ Ω f ( ϕ 1 )ϕ dx + Γ 0 µũϕ dx 1 + Γ 0 µ wϕ dx 1 .
(n denotes the outward unit normal to Ω , note that ∂ n ( ϕ 1 ) ≤ 0). Thus ϕ 1 is a positive subsolution to (4.3) and one concludes as in the proof of Lemma 2.2.

Analogous to Lemma 2.3 one has :

Lemma 4.3. Suppose that f satisfies (2.8). If v 1 , v 2 are positive solutions to (4.3) corresponding to (u 1 , w 1 ) and (u 2 , w 2 ) respectively then

u 1 ≤ u 2 and w 1 ≤ w 2 implies v 1 ≤ v 2 .
In particular (4.3) has a unique positive solution.

Proof. Denote by θ a smooth function such that

θ(t) = 0 ∀t ≤ 0, θ(t) = 1 ∀t ≥ 1, θ (t) ≥ 0. Set θ (t) = θ( t ). Clearly v 1 θ (v 1 -v 2 ), v 2 θ (v 1 -v 2 ) ∈ V.
From the equations satisfied by v 1 , v 2 one gets, setting

θ = θ (v 1 -v 2 ), Ω D∇v 1 • ∇(v 2 θ ) dx + Γ 0 νv 1 (x 1 , 0)(v 2 θ ) dx 1 + Γ 0 ν v 1 (x 1 , L)(v 2 θ ) dx 1 = Ω f (v 1 )(v 2 θ ) dx + Γ 0 µu 1 (v 2 θ ) dx 1 + Γ 0 µ w 1 (v 2 θ ) dx 1 , Ω D∇v 2 • ∇(v 1 θ ) dx + Γ 0 νv 2 (x 1 , 0)(v 1 θ ) dx 1 + Γ 0 ν v 2 (x 1 , L)(v 1 θ ) dx 1 = Ω f (v 2 )(v 1 θ ) dx + Γ 0 µu 2 (v 1 θ ) dx 1 + Γ 0 µ w 2 (v 1 θ ) dx 1 .
By subtraction we obtain

Ω D{∇v 2 • ∇(v 1 θ ) -∇v 1 • ∇(v 2 θ )} dx = Ω f (v 2 )(v 1 θ ) -f (v 1 )(v 2 θ ) dx + Γ 0 µ(u 2 v 1 -u 1 v 2 )θ (v 1 -v 2 ) dx 1 + Γ 0 µ (w 2 v 1 -w 1 v 2 )θ (v 1 -v 2 ) dx 1 .
Clearly the last two integrals above are nonnegative so that one has

Ω D{∇v 2 • ∇(v 1 θ ) -∇v 1 • ∇(v 2 θ )} dx ≥ Ω f (v 2 )(v 1 θ ) -f (v 1 )(v 2 θ ) dx.
Then the rest of the proof is like in Lemma 2.3.

Then we can now show :

Theorem 4.1. Under the assumptions above the problem (4.1) admits a nontrivial solution.

Proof. It is of course clear that (0, 0, 0) is solution to (4.1). Set

K = {u ∈ L 2 (Γ 0 ) | 0 ≤ u ≤ m}, K = {w ∈ L 2 (Γ 0 ) | 0 ≤ w ≤ m }.
For (u, w) ∈ K × K , let v be the unique positive solution to (4.3) associated to ( u, w) = (u, w).

For η ≥ L g , ξ ≥ L h , let (U, W ) = T (u, w) be the solution to            (U, W ) ∈ H 1 0 (Γ 0 ) × H 1 0 (Γ 0 ), Γ 0 D U ψ + µU ψ + ηU ψ dx 1 = Γ 0 νv(x 1 , 0)ψ + g(u)ψ + ηuψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ), Γ 0 D W φ + µ W φ + ξW φ dx 1 = Γ 0 ν v(x 1 , L)φ + h(w)φ + ξwφ dx 1 ∀φ ∈ H 1 0 (Γ 0 ).
The existence of (U, W ) is a consequence of the Lax-Milgram theorem.

We show as in Theorem 3.

1 that T is continuous on K × K ⊂ L 2 (Γ 0 ) × L 2 (Γ 0 ). Indeed suppose that u n → u in K and w n → w in K . Denote by v n the solution to (4.3) associated to (u n , w n ) i.e. satisfying Ω D∇v n • ∇ϕ dx + Γ 0 νv n (x 1 , 0)ϕ dx 1 + Γ 0 ν v n (x 1 , L)ϕ dx 1 = Ω f (v n )ϕ dx + Γ 0 µu n ϕ dx 1 + Γ 0 µ w n ϕ dx 1 ∀ϕ ∈ V. (4.4)
Since v n , u n and w n are bounded, taking ϕ = v n in the equality above one gets easily

Ω D|∇v n | 2 dx + Γ 0 νv n (x 1 , 0) 2 dx 1 + Γ 0 ν v n (x 1 , L) 2 dx 1 ≤ C,
where C is a constant independent of n. Thus, up to a subsequence, there exists v ∈ V such that

v n v in H 1 (Ω ), v n → v in L 2 (Ω ), v n (. , 0) → v(. , 0) in L 2 (Γ 0 ), v n (. , L) → v(. , L) in L 2 (Γ 0 ).
Passing to the limit in (4.4) one derives as in Theorem 3.

1 that T (u n , w n ) → T (u, w) in L 2 (Γ 0 ) × L 2 (Γ 0 ).
We can show also that T is monotone. Indeed, suppose that (u 1 , w 1 ) ≥ (u 2 , w 2 ) in the sense that u 1 ≥ u 2 and w 1 ≥ w 2 and set (U i , W i ) = T (u i , w i ), i = 1, 2. First, for U i one has

Γ 0 D U 1 ψ + µU 1 ψ + ηU 1 ψ dx 1 = Γ 0 νv 1 (x 1 , 0)ψ + g(u 1 )ψ + ηu 1 ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ), Γ 0 D U 2 ψ + µU 2 ψ + ηU 2 ψ dx 1 = Γ 0 νv 2 (x 1 , 0)ψ + g(u 2 )ψ + ηu 2 ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ).

By subtraction it comes

Γ 0 D (U 1 -U 2 ) ψ + µ(U 1 -U 2 )ψ + η(U 1 -U 2 )ψ dx 1 = Γ 0 ν(v 1 (x 1 , 0) -v 2 (x 1 , 0))ψ + (g(u 1 ) -g(u 2 ))ψ + η(u 1 -u 2 )ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ). Choosing ψ = -(U 1 -U 2 ) -and taking into account that, by Lemma 4.3, v 1 (x 1 , 0)-v 2 (x 1 , 0) ≥ 0 and that for η ≥ L g , (g(u 1 ) -g(u 2 )) + η(u 1 -u 2 ) ≥ 0 (Cf. (1.3), (1.4)), one gets Γ 0 D |{(U 1 -U 2 ) -} | 2 + µ{(U 1 -U 2 ) -} 2 + η{(U 1 -U 2 ) -} 2 dx 1 ≤ 0. Thus (U 1 -U 2 ) -= 0 and U 1 ≥ U 2 . Similarly one shows that W 1 ≥ W 2 .
Next we assert that T maps K × K into itself. Indeed, if (U 0 , W 0 ) = T (0, 0) one has, with an obvious notation for v 0

Γ 0 D U 0 ψ + µU 0 ψ + ηU 0 ψ dx 1 = Γ 0 νv 0 (x 1 , 0)ψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ). Γ 0 D W 0 ψ + µW 0 φ + ξW 0 φ dx 1 = Γ 0 ν v 0 (x 1 , L)φ dx 1 ∀φ ∈ H 1 0 (Γ 0 ). Taking ψ = -U - 0 , φ = -W - 0 , one deduces easily since v 0 (x 1 , 0) ≥ 0 and v 0 (x 1 , L) ≥ 0 that U 0 ≥ 0 and W 0 ≥ 0. Similarly, if (U 1 , W 1 ) = T (m, m ) one has, with an obvious notation for v 1 Γ 0 D U 1 ψ + µU 1 ψ + ηU 1 ψ dx 1 = Γ 0 νv 1 (x 1 , 0)ψ + g(m)ψ + ηmψ dx 1 ∀ψ ∈ H 1 0 (Γ 0 ), Γ 0 D W 1 φ + µ W 1 φ + ξW 1 φ dx 1 = Γ 0 ν v 1 (x 1 , L)φ + h(m )φ + ξm φ dx 1 ∀φ ∈ H 1 0 (Γ 0 ).
Thus choosing ψ = (U 1 -m) + and φ = (W 1 -m ) + , due to (4.2) and

v 1 ≤ µ ν m = µ ν m it comes Γ 0 D |{(U 1 -m) + } | 2 + (µ + η){(U 1 -m) + } 2 ≤ Γ 0 (νv 1 (x 1 , 0) -µm)(U 1 -m) + dx 1 ≤ 0. Γ 0 D |{(W 1 -m ) + } | 2 + (µ + ξ){(W 1 -m ) + } 2 ≤ Γ 0 (ν v 1 (x 1 , L) -µ m )(W 1 -m ) + dx 1 ≤ 0.
From which it follows that U 1 ≤ m, W 1 ≤ m . By the monotonicity of T it results that T maps the convex

K × K into itself. But clearly T (K × K ) ⊂ C 1 2 (Γ 0 ) × C 1 2 (Γ 0 ) is relatively compact in L 2 (Γ 0 ) × L 2 (Γ 0 )
. Thus, by the Schauder fixed point theorem (see [START_REF] Evans | Partial Differential Equations[END_REF], [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], [START_REF] Chipot | Elliptic Equations: An Introductory Course[END_REF]), T has a fixed point in K × K which leads to a nontrivial solution (u, v, w) to (4.1). This completes the proof of the theorem.

Remark 2. One can show as in Proposition 3.1 that u, w are also non degenerate in the sense that u ≡ 0, w ≡ 0.

The case of an unbounded domain

The goal of this section is to show that when = +∞ it remains possible to define and find a nontrivial solution to problem (1.1). Let us introduce some notation. For convenience we will denote by V the space V defined in section 1. Similarly we will indicate the dependence in for Γ 0 i.e Γ 0 = Γ 0 = (-, ) × {0}.

When convenient we will set I = (-, ). In addition we set

Ω ∞ = R × (0, L), Γ ∞ 0 = R × {0}, Γ ∞ 1 = R × {L}, V ∞ = {v ∈ H 1 oc (Ω ∞ ) | v = 0 on Γ ∞ 1 }, where H 1 oc (Ω ∞ ) = {v | v ∈ H 1 (Ω 0 ) ∀ 0 > 0}.

Then we have

Theorem 5.1. Under the assumption above, in particular (1.2), (1.3), (1.5), (1.6), (2.7), (2.8) there exists (u, v) nontrivial solution to

             (u, v) ∈ H 1 0 (Γ ∞ 0 ) × V ∞ , Ω 0 D∇v • ∇ϕ dx + I 0 νv(x 1 , 0)ϕ dx 1 = Ω 0 f (v)ϕ dx + I 0 µuϕ dx 1 ∀ϕ ∈ V 0 , ∀ 0 , I 0 D u ψ + µuψ dx 1 = I 0 νv(x 1 , 0)ψ dx 1 + I 0 g(u)ψ dx 1 ∀ψ ∈ H 1 0 (I 0 ), ∀ 0 .
(5.1)

(We identify Γ ∞ 0 with R. Recall that I = (-, )).
Proof. Let (u , v ) be a solution to (1.1). We can find such a solution for every > 0 (Cf. Theorem 3.1). One notices that for ≥ one has

Ω ⊂ Ω , H 1 0 (Ω ) ⊂ H 1 0 (Ω ),
(we suppose the functions of H 1 0 (Ω ) extended by 0 outside Ω ). By definition of λ 1 = λ 1 (Ω ) one has

λ 1 (Ω ) = inf H 1 0 (Ω )\{0} Ω |∇v| 2 dx Ω v 2 dx
, and thus clearly

λ 1 (Ω ) ≤ λ 1 (Ω ) ∀ ≥ .
Let us assume for some 1 > 0 (Cf. (2.7))

λ 1 (Ω 1 ) ≤ f (s) Ds for s > 0 small enough. ( 5.2)
Then for any ≥ 1 one has for for s > 0 small enough

λ 1 (Ω ) ≤ f (s) Ds .
Moreover, since it is easy to see that ϕ 1 defined in (2.6) is given by

ϕ 1 = sin π 2 (x 1 + ) sin π L x 2 ,
one has 0 ≤ ϕ 1 ≤ 1 and if (5.2) holds one has

Dλ 1 (Ω ) ϕ 1 ≤ f ( ϕ 1 )
for > 0 small enough independently of ≥ 1 . We suppose from now on that this is fixed such that if (u , v ) is a solution to (1.1) constructed as in Theorem 3.1 one has

ϕ 1 ≤ v
and in particular for every ≥ 1

(sin π 4 ) 2 ≤ ϕ 1 ≤ v a.e. x ∈ (- 2 , 2 ) × ( L 4 , 3L
).

(5.3)

One should also notice that independently of one has

0 ≤ u ≤ m, ϕ 1 ≤ v ≤ µ ν m. (5.4)
We assume from now on ≥ 1 and for 0 ≤ -1 we define ρ by

ρ = ρ(x 1 ) =            1 on I 0 , x 1 + 0 + 1 on (-0 -1, -0 ), -x 1 + 0 + 1 on ( 0 , 0 + 1), 0 outside I 0 +1 ,
whose graph is depicted below. Clearly ρ 2 v = ρ 2 (x 1 )v ∈ V and from the first equation of (1.1) one gets

Ω D∇v • ∇(ρ 2 v ) dx + Γ 0 νρ 2 v 2 (x 1 , 0) dx 1 = Ω f (v )ρ 2 v dx + Γ 0 µu ρ 2 v (x 1 , 0) dx 1 . (5.5)
One should notice that in the integrals in Ω one integrates only on Ω 0 +1 and for the ones in Γ 0 on I 0 +1 . Then remark that

Ω ∇v • ∇(ρ 2 v ) dx = Ω |∇v | 2 ρ 2 + 2ρv ∇v • ∇ρ dx,
and

Ω |∇(ρv )| 2 dx = Ω |ρ∇v + v ∇ρ| 2 dx = Ω |∇v | 2 ρ 2 + 2ρv ∇v • ∇ρ + v 2 |∇ρ| 2 dx.
From this it follows that

Ω ∇v • ∇(ρ 2 v ) dx = Ω |∇(ρv )| 2 dx - Ω v 2 |∇ρ| 2 dx.
Thus, since the second integral of (5.5) is nonnegative, it comes

D Ω 0 +1 |∇(ρv )| 2 dx ≤ D Ω 0 +1 v 2 |∇ρ| 2 dx + Ω 0 +1 f (v )ρ 2 v dx + I 0 +1 µu ρ 2 v (x 1 , 0) dx 1 .
Using the definition of ρ and in particular the fact that ρ = 1 on Ω 0 we get easily by (5.4)

Ω 0 |∇v | 2 dx ≤ C (5.6)
where C is independent of . Taking now ψ = ρ 2 u in the second equation of (1.1) we get

I 0 +1 D u (ρ 2 u ) + µρ 2 u 2 dx 1 = I 0 +1 νv (x 1 , 0)ρ 2 u + g(u )ρ 2 u dx 1 .
Arguing as above we derive easily

I 0 +1 u (ρ 2 u ) dx 1 = I 0 +1 |(ρu ) | 2 dx 1 - I 0 +1 u 2 ρ 2 dx 1 .
This leads to

I 0 +1 D |(ρu ) | 2 + µρ 2 u 2 dx 1 ≤ I 0 +1
D u 2 ρ 2 dx 1 + νv (x 1 , 0)ρ 2 u + g(u )ρ 2 u dx 1 .

Integrating only on I 0 in the fisrt integral i.e. where ρ = 1 we obtain

I 0 (u ) 2 + u 2 dx 1 ≤ C (5.7)
where C is some other constant independent of . It results from (5.6), (5.7) that (u , v ) is bounded in H 1 (I 0 ) × V 0 independently of . Thus there exists a subsequence of (u , v ) that we will denote by (u n,0 , v n,0 ) such when n → ∞ u n,0 u 0 in H 1 (I 0 ), v n,0 v 0 in V 0 , u n,0 → u 0 in L 2 (I 0 ), v n,0 → v 0 in L 2 (Ω 0 )

v n,0 (., 0) → v 0 (., 0) in L 2 (I 0 ). (Note that a function of V 0 extended by 0 belongs to V ). Clearly -as a subsequence of (u , v )the sequence (u n,0 , v n,0 ) is bounded in H 1 (I 0 +1 ) × V 0 +1 independently of n and one can extract a subsequence that we still label by n and denote by (u n,1 , v n,1 ) such that

Considering the equations

u n,1 u 1 in H 1 (I 0 +1 ), v n,1 v 1 in V 0 +1 , u n,1 → u 1 in L 2 (I 0 +1 ), v n,1 → v 1 in L 2 (Ω 0 +1 )
v n,1 (., 0) → v 1 (., 0) in L 2 (I 0 +1 ).

Note that (u 1 , v 1 ) = (u 0 , v 0 ) on I 0 × Ω 0 . Clearly (u 1 , v 1 ) satisfies

Ω 0 +1 D∇v 1 • ∇ϕ dx + I 0 +1 νv 1 (x 1 , 0)ϕ dx 1 = Ω 0 +1 f (v 1 )ϕ dx + I 0 +1 µu 1 ϕ dx 1 ∀ϕ ∈ V 0 +1 , I 0 +1 D u 1 ψ + µu 1 ψ dx 1 = I 0 +1 νv 1 (x 1 , 0)ψ dx 1 + I 0 +1
g(u 1 )ψ dx 1 ∀ψ ∈ H 1 0 (I 0 +1 ).

By induction one constructs a sequence (u n,k , v n,k ) extracted from the preceding converging toward (u k , v k ) satisfying the equations above where we have replaced 0 + 1 by 0 + k. Then using the usual diagonal process it is clear that the sequence (u n,n , v n,n ) will converge toward a solution to (5.1) nontrivial thanks to (5.3). This completes the proof of the theorem.

Figure 1 :

 1 Figure 1: The domain Ω for one-road problem

(1. 1 )

 1 Here we assume that D, D , m, µ, ν are positive constants, are Lipschitz continuous functions i.e. such that for some positive constantsL f , L g it holds |f (x) -f (y)| ≤ L f |x -y|, |g(x) -g(y)| ≤ L g |x -y| ∀x, y ∈ R. (1.3)

Theorem 3 . 1 .

 31 Suppose that (1.2)-(1.6),(2.7),(2.8) hold, then the problem (1.1) admits a nontrivial solution.

Proposition 3 . 1 .

 31 Let (u, v) be the solution constructed in Theorem 3.1. One has u ≡ 0.

Figure 3 :

 3 Figure 3: The graph of the function ρ(x 1 )

Ω 0 D∇v 0 D u ψ + µu ψ dx 1 = I 0 νv (x 1 , 0

 00010 • ∇ϕ dx + I 0 νv (x 1 , 0)ϕ dx 1 = Ω 0 f (v )ϕ dx + I 0 µu ϕ dx 1 ∀ϕ ∈ V 0 , I 0)ψ dx 1 + I 0 g(u )ψ dx 1 ∀ψ ∈ H 1 0 (I 0 ),with (u , v ) replaced by (u n,0 , v n,0 ), one can pass to the limit in n and see that(u 0 , v 0 ) ∈ H 1 (I 0 ) × V 0 satisfies Ω D∇v 0 • ∇ϕ dx + I 0 νv 0 (x 1 , 0)ϕ dx 1 = Ω 0 f (v 0 )ϕ dx + I 0 µu 0 ϕ dx 1 ∀ϕ ∈ V 0 , I 0 D u 0 ψ + µu 0 ψ dx 1 = I 0 νv 0 (x 1 , 0)ψ dx 1 + I 0g(u 0 )ψ dx 1 ∀ψ ∈ H 1 0 (I 0 ).

Acknowledgement : Part of this work was performed when the first author was visiting USTC in Hefei. We thank the university and the school of mathematical sciences for their hospitality and their support.

Some extension

We would like to show that our results extend in the case of a so called two road problem. More precisely set Γ 0 = (-, ) × {L}, Γ 1 = ∂Ω \{Γ 0 ∪ Γ 0 }, We consider the problem of finding a (u, v, w) solution to

Here we assume that D, D , D , µ, ν, µ , ν are positive constants, f, g, h are Lipschitz continuous functions with Lipschitz constants L f , L g , L h respectively (Cf.

(1.3)), which implies that for λ ≥ L f , η ≥ L g and ξ ≥ L h the functions

are nondecreasing. We will suppose that f satisfies (1.5) and that

Since Γ 0 and Γ 0 are playing exactly identical roles there is no loss of generality in assuming for instance µ ν ≥ µ ν .

Then for

we will assume

One should notice the following properties

Then with small variants we can reproduce the results we had in the preceding sections. First we have Lemma 4.1. Suppose that u, w are measurable functions on Γ 0 and Γ 0 respectively such that

Then under the assumptions above the problem

possesses a minimal and a maximal solution with values in (0, µ ν m). Proof. Let us remark first that any nonnegative solution to (4.3) takes its values in (0, µ ν m).