Katarzyna Świrk
email: katarzynaświrk@agh.edu.pl

Maria Elena Gálvez

Monika Motak

Teresa Grzybek

Magnus Rønning

Patrick Da Costa

Syngas production from dry methane reforming over yttrium-promoted nickel-KIT-6 catalysts

Keywords: KIT-6, nickel, yttrium, dry methane reforming

Introduction

Nowadays, fossil fuels are used as major energy sources. However, coal, natural gas and oil combustion results in greenhouse gas emissions, which, once released into the atmosphere, may cause important hazards to earth climate and human health. Carbon dioxide, one of the main components of exhaust gases, can be however utilized as raw material through different chemical processes, such as dry reforming of methane (DRM) [1]. In DRM, CO2 reacts with CH4, yielding synthesis gas (mixture of H2 and CO) as final product [1][START_REF] Wang | Synthesis, characterization and catalytic performance of MgO-coated Ni/SBA-15 catalysts for methane dry reforming to syngas and hydrogen[END_REF][START_REF] Zhang | Nickel catalysts supported on MgO with different specific surface area for carbon dioxide reforming of methane[END_REF]. The lack of selective and stable catalysts for DRM, together with the high endothermicity of this reaction, are the main obstacles for the scale-up and commercialization of this chemical CO2 utilization process. Further catalyst development is therefore needed.

Nickel-based catalysts have been extensively studied in the last years [START_REF] Kawi | Progress in Synthesis of Highly Active and Stable Nickel-Based Catalysts for Carbon Dioxide Reforming of Methane[END_REF][START_REF] Hu | The characterization of a highly effective NiO/MgO solid solution catalyst in the CO2 reforming of CH4[END_REF][START_REF] Min | Carbon dioxide reforming of methane on Ni-MgO-Al2O3 catalysts prepared by sol-gel method: Effects of Mg/Al ratios[END_REF][START_REF] Fan | Utilization of greenhouse gases through dry reforming: Screening of nickel-based bimetallic catalysts and kinetic studies[END_REF]. Nickel is more readily available, active and relatively inexpensive compared to noble metals, i.e. Pt, Pd, Ru, Rh and Ir [START_REF] Fan | Utilization of greenhouse gases through dry reforming: Screening of nickel-based bimetallic catalysts and kinetic studies[END_REF].

Active phase sintering and coking due to the simultaneous occurrence of carbon forming reactions are the main two drawbacks of nickel catalysts [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF][START_REF] Zhang | Development of stable bimetallic catalysts for carbon dioxide reforming of methane[END_REF]. However, the choice of a proper support may inhibit these processes, through improved textural properties and enhanced dispersion of the active phase, leading to controlled selectivity and higher stability.

A wide variety of metal oxides, such as Al2O3, MgO, CeO2, ZrO2 or SiO2, have been used as supports in the preparation of Ni-catalysts for DRM. Alumina shows excellent textural properties.

Moreover, in Ni/Al2O3 catalyst a spinel phase (NiAl2O4) can be sometimes formed, which can considerably hinder carbon formation. However, as reported by Becerra et al. [START_REF] Becerra | Stable Ni/Al2O3 catalysts for methane dry reforming Effects of pretreatment[END_REF], the formation of this NiAl2O3 phase may lead to low catalytic activity, especially at relatively low temperatures. MgO is a support with high Lewis basicity. Therefore, Ni/MgO catalysts are particularly able to adsorb CO2 molecules on their surface. Nevertheless, the activation of side reactions, such as reverse watergas shift is very likely on this kind of materials [START_REF] Zanganeh | Dry reforming of methane to synthesis gas on NiO-MgO nanocrystalline solid solution catalysts[END_REF][START_REF] Zanganeh | Preparation of nanocrystalline NiO-MgO solid solution powders as catalyst for methane reforming with carbon dioxide: Effect of preparation conditions[END_REF][START_REF] Jafarbegloo | One-pot synthesis of NiO-MgO nanocatalysts for CO2 reforming of methane: The influence of active metal content on catalytic performance[END_REF]. Additionally, Ni/MgO-Al2O3 catalysts show good catalytic behavior. Modification with 20 wt.% of nickel was positive in terms of catalytic activity, although enhanced formation of carbon deposit was reported [START_REF] Alipour | Effect of Ni loadings on the activity and coke formation of MgO-modified Ni/Al2O3 nanocatalyst in dry reforming of methane[END_REF]. The possible reason of the latter may be the inferior Ni dispersion, while modifying the support with high content of this metal. Furthermore, MgAl2O4 spinel may be also considered as a support. Alvar and Rezaei [START_REF] Alvar | Mesoporous nanocrystalline MgAl2O4 spinel and its applications as support for Ni catalyst in dry reforming[END_REF] examined this mesoporous material, subsequently modified with nickel, in dry reforming of methane.

The catalyst showed high specific surface area and nanocrystalline structure, which benefited in catalytic activity, as well as in stability tests of DRM (no deactivation after 50 h). CeO2 is known to have high oxygen storage capacity. Its use as support in the preparation of Ni-catalysts for DRM can lead to improved stability through the participation of labile oxygen in the oxidation of the carbon formed during DRM [START_REF] Odedairo | Metal-support interface of a novel Ni-CeO2 catalyst for dry reforming of methane[END_REF]. Asami et al. [START_REF] Asami | CO2 reforming of CH4 over ceria-supported metal catalysts[END_REF] reported almost negligible deactivation of a Ni/CeO2 catalyst during DRM at 850ºC. At lower reaction temperatures, i.e. 700ºC, a decrease in stability was however registered. Moreover, cerium may be used as promoter in DRM process. As studied by Dębek et al. [START_REF] Dębek | Low temperature dry methane reforming over Ce, Zr and CeZr promoted Ni-Mg-Al hydrotalcite-derived catalysts[END_REF] the addition of this metal to Ni-based Mg,Al-mixed oxides derived from hydrotalcites resulted in better activity compared to non-promoted material. This contributed to enhanced reducibility by introduction of new oxygen pairs. The authors also tested Zr-promoted hydrotalcite catalysts [START_REF] Dębek | Low temperature dry methane reforming over Ce, Zr and CeZr promoted Ni-Mg-Al hydrotalcite-derived catalysts[END_REF]. The addition of zirconium significantly improved stability in the DRM test at 550°C, since it inhibited the reaction of direct methane decomposition. Moreover, modification with both, Ce and Zr, resulted in better stability and, in the same time, decreased catalytic activity. This could be ascribed to higher amount of basic sites with strong strength [START_REF] Dębek | Influence of Ce/Zr molar ratio on catalytic performance of hydrotalcite-derived catalysts at low temperature CO2 methane reforming[END_REF]. Zirconium oxides are also used as carriers for DRM due to their appropriate basicity and weak acid properties. As studied by Rezaei et al. [START_REF] Rezaei | Effects of K2O Promoter on the Activity and Stability of Nickel Catalysts Supported on Mesoporous Nanocrystalline Zirconia in CH4 Reforming with CO2[END_REF] zirconia supports promoted with K2O showed improved catalytic activity and stability, which was due to enhanced basicity in the materials. Moreover, Tunisian natural clays also may find their application as supports in CO2 reforming reaction. Liu et al. [START_REF] Liu | Ceria and zirconia modified natural clay based nickel catalysts for dry reforming of methane[END_REF] examined the influence of ceria and zirconia promotion (added separately) on Cu-pillared and Fe-pillared clays.

The modification with metal oxides increased the number of medium and weak basic sites, together with Niº crystalline size, which resulted in increased catalytic activity during DRM test, and a presence of carbon forming reactions. Silica mesoporous materials are also considered attractive carriers due to their high specific surface area, mesopore-enriched porosity and good thermal stability [START_REF] Huang | Catalytic performances of Ni/mesoporous SiO2 catalysts for dry reforming of methane to hydrogen[END_REF]. Huang et al. [START_REF] Huang | Catalytic performances of Ni/mesoporous SiO2 catalysts for dry reforming of methane to hydrogen[END_REF] studied SiO2 supports loaded with Ni. The use of SiO2 allowed to obtain very high dispersion of the Ni-phase and controlled crystallite sizes, leading to improved catalytic performance. Lovell et al. [START_REF] Lovell | Ni-SiO2 Catalysts for the Carbon Dioxide Reforming of Methane: Varying Support Properties by Flame Spray Pyrolysis[END_REF] studied SiO2 supports, which were prepared through flame spray pyrolysis and subsequently loaded with nickel. The best catalytic performance in DRM was registered for the catalysts having the highest surface area and showing the smallest Ni crystallite size of the series. Recently, considerable attention has also been paid to the ordered mesoporous silica SBA-15.

Gálvez et al. [START_REF] Gálvez | Enhanced catalytic stability through non-conventional synthesis of Ni/SBA-15 for methane dry reforming at low temperatures[END_REF] studied Ni/SBA-15 catalysts, which were synthesized by three different ways (impregnation, co-precipitation and co-precipitation with ascorbic acid). The pre-reduction of the Niphase using ascorbic acid resulted in a very accurate control of Ni-crystallite size, leading to Niparticles placed inside the mesopores of the SBA-15 structure, resulting in enhanced activity, selectivity and stability.

Promotion with different metals can also positively influence catalytic behavior of DRM catalyst. Samarium-promoted Ni-SBA-15 catalysts were tested in DRM by Taherian et al. [START_REF] Taherian | Promotional effect of samarium on the activity and stability of Ni-SBA-15 catalysts in dry reforming of methane[END_REF]. The promotion with Sm resulted in enhanced stability and catalytic activity due to the increase in dispersion of NiO and its better interaction with SBA-15. Erdogan et al. [START_REF] Erdogan | SBA-15 supported mesoporous Ni and Co catalysts with high coke resistance for dry reforming of methane[END_REF] examined the effect of nickel and cobalt addition into SBA-15 support. The formation of Ni-Co alloy hindered agglomeration of nickel particles, resulting in suppressed carbon deposit formation. Kaydouh et al. [START_REF] Kaydouh | Highly active and stable Ni/SBA-15 catalysts prepared by a "two solvents" method for dry reforming of methane[END_REF] studied SBA-15 loaded with Ni and Ce, which were highly active and stable in DRM. However, the promoting effect of Ce was not clear, since no significant differences in terms of Ni-reducibility and catalytic activity were observed for the Ce-promoted and the non-promoted catalysts. Albarazi et al. [START_REF] Albarazi | Synthesis strategies of ceria-zirconia doped Ni/SBA-15 catalysts for methane dry reforming[END_REF][START_REF] Albarazi | Hydrogen and syngas production by methane dry reforming on SBA-15 supported nickel catalysts: On the effect of promotion by Ce0.75Zr0.25O2 mixed oxide[END_REF] examined the DRM activity of Ni/SBA-15 catalysts modified with Ce 0.75 Zr 0.25 O 2 . The presence of Ce and Zr promoters resulted in higher catalytic activity, i.e. higher CH 4 conversions, and improved stability in the low temperature range (600-630°C). B. Li et al. [START_REF] Li | Methane reforming with CO2 using nickel catalysts supported on yttria-doped SBA-15 mesoporous materials via sol-gel process[END_REF] prepared Ni,Y-modified SBA-15. The authors reported that yttrium caused a decrease in Niº size, linked to enhanced reducibility of the Niphase in the calcined catalyst. This was related with an increased number of oxygen vacancies. J.F.

Li et al. [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF] also reported a positive effect of Y-promotion of Ni/SBA-15 catalysts. The authors prepared different catalysts using 6, 3 and 9 wt.% yttrium loading, for a fixed amount of nickel (9 wt.%). The 9%Y-Ni/SBA-15 catalyst evidenced the lowest carbon formation and the highest catalytic activity of the series.

Due to its high specific surface area and very particular pore size distribution, KIT-6 ordered mesoporous silica with Ia3d symmetric structure has been recently considered as support in the preparation of catalysts for various processes [START_REF] Gopinath | Efficient mesoporous SO4 2-/Zr-KIT-6 solid acid catalyst for green diesel production from esterification of oleic acid[END_REF][START_REF] Kumaresan | Mesoporous Ti-KIT-6 molecular sieves: Their catalytic activity in the epoxidation of cyclohexene[END_REF][START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF][START_REF] Pirez | Tunable KIT-6 Mesoporous Sulfonic Acid Catalysts for Fatty Acid Esteri fi cation[END_REF][START_REF] Subhan | Enhanced desulfurization characteristics of Cu-KIT-6 for thiophene[END_REF][START_REF] Aslam | Facile fabrication of Ni-based KIT-6 for adsorptive desulfurization[END_REF][START_REF] Varkolu | Ni nanoparticles supported on mesoporous silica (2D, 3D) architectures: highly efficient catalysts for the hydrocyclization of biomass-derived levulinic acid[END_REF][START_REF] Guo | Catalytic properties and stability of cubic mesoporous LaxNiyOz/KIT-6 catalysts for CO2 reforming of CH4[END_REF][START_REF] Grams | Mesoporous silicas as supports for Ni catalyst used in cellulose conversion to hydrogen rich gas[END_REF]. For example, Ni-KIT-6 were examined in pyrolysis of cellulose [START_REF] Grams | Mesoporous silicas as supports for Ni catalyst used in cellulose conversion to hydrogen rich gas[END_REF], Ni-La/KIT-6 catalysts were tested in CO2 reforming [START_REF] Guo | Catalytic properties and stability of cubic mesoporous LaxNiyOz/KIT-6 catalysts for CO2 reforming of CH4[END_REF]; Ni-V/KIT-6, Ni-Ce/KIT-6, Ni-La/KIT-6 and Ni-Mn/KIT-6 in CO methanation [START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF]; Cu/KIT-6 for thiophene removal [START_REF] Subhan | Enhanced desulfurization characteristics of Cu-KIT-6 for thiophene[END_REF]; SO4 2-/Zr-KIT-6 for esterification of oleic acid [START_REF] Gopinath | Efficient mesoporous SO4 2-/Zr-KIT-6 solid acid catalyst for green diesel production from esterification of oleic acid[END_REF]; and Ti/KIT-6 for epoxidation of cyclohexene [START_REF] Kumaresan | Mesoporous Ti-KIT-6 molecular sieves: Their catalytic activity in the epoxidation of cyclohexene[END_REF]. However, to the best of our knowledge there are no studies reporting the use of yttrium as promoter on Ni-containing KIT-6 supported catalysts for dry reforming applications.

Thus, the aim of the present study was to prepare Ni,Y-modified KIT-6 catalysts and evaluate their activity and selectivity in dry reforming of methane. Catalysts with different Y loadings were thus prepared, characterized and tested in DRM at moderate temperature, i.e. 600-750°C. The fresh, calcined, reduced and spent catalysts were characterized by means of X-ray fluorescence (XRF), Fourier transformed infrared spectroscopy (FT-IR), thermal gravimetric analysis/ differential scanning calorimetry-mass spectrometry (TGA/DSC-MS), N2 adsorption-desorption analysis, transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HRTEM), X-ray diffraction (XRD), and hydrogen temperature-programmed reduction (TPR-H2).

Experimental

2.1.Catalyst preparation

The mesoporous KIT-6 material was synthesized as described elsewhere [START_REF] Pirez | Tunable KIT-6 Mesoporous Sulfonic Acid Catalysts for Fatty Acid Esteri fi cation[END_REF]. In short, 7.47 g of aqueous solution of HCl (37%) was dissolved in 144 cm 3 of distilled water. Then 4 g of P123 triblock polymer were added and the solution was stirred for 4h. Afterwards, the temperature of solution was increased to 35ºC and butanol (4g) was added dropwise. The solution was stirred for another 1h.

Finally, TEOS (Tetraethyl orthosilicate, 8.6 g) was introduced while stirring at 35ºC. After 24h, the mixture was transposed into a Teflon bottle for hydrothermal synthesis. The synthesis was carried out under static conditions at 90ºC for 24h. The obtained product was filtered and dried at 100ºC overnight. In order to remove the template, calcination was performed at 600ºC for 6h.

Incipient wetness impregnation was used to prepare the KIT-6 supported Ni-Y catalysts. A mixture of aqueous solutions of Ni(NO3)2•6H2O and Y(NO3)3•6H2O was used as precursors in order to achieve the fixed amount of nickel (12 wt.%), and different content of yttrium in each sample (4 wt.%, 8 wt.% or 12 wt.%, respectively). After the impregnation, the samples were dried overnight and then calcined in air at 550ºC for 5 h in order to remove the nitrates and to form NiO on the catalyst as described elsewhere [START_REF] Dębek | Low temperature dry methane reforming over Ce, Zr and CeZr promoted Ni-Mg-Al hydrotalcite-derived catalysts[END_REF].

Catalysts characterization

XRF (Supermini200) was performed to determine the elemental composition of the catalysts. The measurements were carried out under vacuum at 36.5ºC. A proportional counter detector (PC) was used under P-10 gas (mixture of 10% CH4/Ar) with flow of 24.7 cm 3 /min. The samples were diluted in boric acid with a ratio of 1:200, and then pelletized under pressure of 10 bar. Just before tests the samples were covered with 6µm polypropylene film. FT-IR spectra were obtained in a Fouriertransform infrared spectrometer Thermo Nicolet 380, in the range of 4000 to 400 cm -1 . The samples were pelletized together with KBr at the ratio of 1:100. TGA/DSC-MS experiments were performed in a Netzsch STA 449C Jupiter TGA/DSC and NetzschAëlos QMS 403 MS. For this purpose, 8 mg of fresh sample was used to measure weight loss, DSC signal and current as a function of temperature (from room temperature to 800ºC). The tests were performed in air using a total flow of 100 ml/min. The products analyzed by mass spectrometry were m/e= 44 (CO2 or N2O) and 18 (H2O). TEM analyses were performed in JOEL JEM-100XCII microscope. HRTEM analysis was carried out in a JEM-2100Plus Transmission Electron Microscope. N2 adsorption-desorption isotherms were acquired with the help of a Micromeritics TriStar II device at -195°C. Prior to analysis, the samples were outgassed at 300ºC for 3 h [START_REF] Subhan | Enhanced desulfurization characteristics of Cu-KIT-6 for thiophene[END_REF]. Specific surface area and pore diameters were respectively calculated using the Brunauer-Emmett-Teller (BET) and Barrett-Joyner-Halenda (BJH) method. XRD patterns were recorded by a PANalytical-Empyrean diffractometer equipped with a Cu Kα (λ=0.15406 nm) radiation source. The measurements were carried out in the 2Θ range of 3 to 90º.

Temperature-programmed reduction (TPR-H2) measurements were performed in a BEL Japan BELCAT-M apparatus, equipped with a TCD detector. Typically, 60 mg of each catalyst was first degassed in helium at 100ºC for 2 h, and then reduced in the mixture of 5 % H2/Ar at temperature range from 100ºC to 750ºC (heating rate of 7.5ºC/min).

2.3.DRM catalytic activity

Dry reforming of methane tests were carried out at atmospheric pressure in a fixed bed quartz reactor with a thermocouple type K in contact with the catalytic bed and protected by a quartz shield.

Before reaction, all the catalysts were reduced in situ at 750ºC for 1h in a mixture of 5% H2/Ar (heat rate 10ºC/min) in order to have all the nickel particles reduced (Ni°) [START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF]. The catalytic activity was evaluated in the temperature range of 750-600ºC with a temperature step every 50ºC, and 0.5 h step at each temperature to obtain steady state. In order to check initial stability of the catalysts, additional DRM tests were carried out for 5 hours at 700ºC. In both types of experiments, the total flow was 100 cm 3 /min and the composition of the inlet gas mixture was CH4/CO2/Ar=1/1/8 (GHSV=20,000 h -1 ).

The outlet gases were analyzed by gas chromatograph (490 Varian Micro-GC).

The respective conversions of CH4, CO2, and H2/CO molar ratio where calculated as follows:

(1)

(2)

, 100 [%] , , , 4 4 4 4
× - = initial CH final CH initial CH conversion n n n CH , 100 [%] , , , 2 2 2 2 × - = initial CO final CO initial CO conversion n n n CO (3)
Where nCH4,initial and nCO2,initial are the inlet numbers of moles of methane and carbon dioxide, respectively. And nCH4,final, nCO2,final, nH2,final, nCO,final are outlet number of moles of methane, carbon dioxide, hydrogen and carbon oxide, respectively.

Results and discussion

Catalyst characterization

Textural and structural properties

The composition of the different KIT-6 supported catalysts is given in Table 1. Silicon is the main component with a content of ca. 90% in every sample. Nickel loading resulted Ni contents close to the nominal value, i.e. 12 wt.% vis-à-vis 7-10 wt.% measured in the catalysts after calcination.

Only 50% of the nominal Y loading was found in the calcined catalysts. Final yttrium amounts are 2 wt.% to 7 wt.% in comparison to nominal loads ranging from 4 wt.% to 12 wt.%, respectively (Table 1).

The FT-IR spectra acquired for the different catalysts are presented in Fig. 1. The peaks observed at 464, 960, 807 and 1083 cm -1 correspond to the bending vibration of Si-O-Si, the stretching vibrations of Si-OH, symmetric and asymmetric stretching vibrations of Si-O-Si, respectively. The mentioned vibrations are assigned to KIT-6, and related to the formation of condensed framework of silica. FT-IR analysis thus confirms the successful synthesis of KIT-6 material. In addition, the bands at 1640 and 3435 cm -1 are due to the presence of absorbed water molecules, namely the stretching vibration of O-H [START_REF] Guo | Catalytic properties and stability of cubic mesoporous LaxNiyOz/KIT-6 catalysts for CO2 reforming of CH4[END_REF]. The double peak at 2858 and 2978 cm -1 can be ascribed to the P123 template [START_REF] Prabhu | Synthesis and characterization of aluminium incorporated mesoporous KIT-6: Efficient catalyst for acylation of phenol[END_REF][START_REF] Dinari | Preparation and characterization of nanocomposite materials based on polyamide-6 and modified ordered mesoporous silica KIT-6[END_REF]. This peak disappeared for Ni,Y-modified samples, as a consequence of the calcination of KIT-6 at 600ºC. The peak at 1380 cm -1 can be ascribed either to the remaining presence of template molecules or to or the N-O vibration of the nitrates used in the impregnation [START_REF] Aslam | Facile fabrication of Ni-based KIT-6 for adsorptive desulfurization[END_REF]. Since this peak looks more intense for the Ni-Y modified catalysts, it can be therefore linked to the presence of TEM images of the calcined support are presented in Fig. 3. The characteristic la3d structure of KIT-6 can be clearly observed, further confirming the successful synthesis of this ordered mesoporous material [START_REF] Gopinath | Efficient mesoporous SO4 2-/Zr-KIT-6 solid acid catalyst for green diesel production from esterification of oleic acid[END_REF][START_REF] Kumaresan | Mesoporous Ti-KIT-6 molecular sieves: Their catalytic activity in the epoxidation of cyclohexene[END_REF][START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF][START_REF] Pirez | Tunable KIT-6 Mesoporous Sulfonic Acid Catalysts for Fatty Acid Esteri fi cation[END_REF][START_REF] Subhan | Enhanced desulfurization characteristics of Cu-KIT-6 for thiophene[END_REF][START_REF] Aslam | Facile fabrication of Ni-based KIT-6 for adsorptive desulfurization[END_REF][START_REF] Varkolu | Ni nanoparticles supported on mesoporous silica (2D, 3D) architectures: highly efficient catalysts for the hydrocyclization of biomass-derived levulinic acid[END_REF][START_REF] Guo | Catalytic properties and stability of cubic mesoporous LaxNiyOz/KIT-6 catalysts for CO2 reforming of CH4[END_REF]. Highly ordered mesopores create a bi-continuous network of channels.

This structure is further confirmed at the sight of the obtained N2-adsorption isotherms, as discussed later on. The pore size, measured by using ImageJ software, varies from 3.8 to 6.3 nm.

Fig. 4A shows N2 adsorption-desorption isotherms for the modified KIT-6 supports. All catalysts revealed type IV isotherms with an H1 hysteresis loop, which is typical of such ordered mesostructured materials [START_REF] Pirez | Tunable KIT-6 Mesoporous Sulfonic Acid Catalysts for Fatty Acid Esteri fi cation[END_REF][START_REF] Guo | Catalytic properties and stability of cubic mesoporous LaxNiyOz/KIT-6 catalysts for CO2 reforming of CH4[END_REF]. The maintained shape of the isotherms shows that the structure of the catalysts does not change after upon nickel and yttrium loading. The pore distribution was bimodal in all cases (Fig. 4B), revealing two types of pores (i) narrow pores (of ca. 3.5 nm) and (ii) wide pores (with mean diameter between 4.5 to 4.8 nm). The textural properties are summarized in Table 1. The highest SBET was obtained for KIT-6 support (456 m 2 /g), and after further modification with Ni and Y the specific surface area and pore volumes decreased, as a consequence of pore blockage occurring to a certain extent upon metal loading.

XRD patterns for KIT-6 support of freshly prepared samples and after calcination at 600ºC for 6h are presented in Fig. 5. The wide reflection at ca. 2ϴ = 23º can be attributed to SiO2 [START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF]. Fig. 6A shows the Y-promoted materials after calcination at 550ºC. In all patterns the characteristic XRD reflection of amorphous silica (2ϴ = 23º) and the reflections typical of NiO (2ϴ = 37.3, 43.3, 63, 75.4, 79.4º) are observed. Additionally, a small shoulder at ca. 2ϴ = 29º was registered. The presence of this shoulder can be linked to the existence of a segregated Y2O3 phase [START_REF] Li | Methane reforming with CO2 using nickel catalysts supported on yttria-doped SBA-15 mesoporous materials via sol-gel process[END_REF]. No shift towards higher Bragg angles has been detected in the XRD diffractograms, which may point to the absence of Ni-Y mixed phase.

Reducibility of KIT-6 based catalysts

Fig. 7 shows TPR-H2 results for the synthesized KIT-6 materials. For the (Ni)/KIT-6 catalysts, three wide poorly resolved reduction peaks were observed at 373, 433 and 650°C. The first one is associated with reduction of NiO species weakly interacting with KIT-6 [START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF][START_REF] Grams | Mesoporous silicas as supports for Ni catalyst used in cellulose conversion to hydrogen rich gas[END_REF], the second refers to reduction of bulk NiO [START_REF] Cao | Highly dispersed Ni nanoparticles on 3D-mesoporous KIT-6 for CO methanation: Effect of promoter species on catalytic performance[END_REF][START_REF] Varkolu | Ni nanoparticles supported on mesoporous silica (2D, 3D) architectures: highly efficient catalysts for the hydrocyclization of biomass-derived levulinic acid[END_REF], while the third may be attributed to NiO located in the pores of the support [START_REF] Varkolu | Ni nanoparticles supported on mesoporous silica (2D, 3D) architectures: highly efficient catalysts for the hydrocyclization of biomass-derived levulinic acid[END_REF]. As a consequence of yttrium addition, a shift towards higher reduction temperatures and the change in peaks shape can be clearly observed. Narrow peaks appear at around 418°C, their intensity increasing with increasing Y-loading. Table 2 presents H2 consumption obtained for the reduction of bulk NiO (I type) and reduction of NiO located inside pores (type II) as calculated from Fig. 7. The reducibility of the bulk NiO increases with the addition of yttrium, whereas the opposite was observed for NiO located inside the pores. The values were decreasing with the yttrium loading, which may suggest that sintering is occurring with less of Niº inside the pores. Moreover, the ratio between the H2 consumption for NiO inside pores to the total consumption is decreasing for the samples with yttrium. The yttrium promotion affects reducibility of nickel species, and all the tested catalysts could be reduced at 750ºC.

Only minor textural changes were found for the catalysts after TPR-H2 tests, as proven by the N2 adsorption-desorption experiments (Table 1). The shape of the isotherms and quantity adsorbed remained stable. The specific surface area was slightly lower for these samples, and there were no changes of pore volume and pore diameters. 2. An increase in nickel crystallite size after reduction is generally observed, which becomes more evident for the Y-modified catalysts. The Niº crystallite size varies from 4 nm for non-promoted catalyst (Ni)/KIT-6 to 13 nm for the highest Y-loading catalyst (Ni-12Y)/KIT-6.

TEM images for the reduced catalysts are shown in Fig. 8. Compared to the micrographs of the unmodified KIT-6 support (Fig. 3), it can be concluded that the ordered silica mesoporous structures were well preserved and that metallic Ni particles were present, which is in accordance with the results obtained by XRD (Fig. 6B). Two populations of Niº particles can be observed, one inside the channel and another outside the mesoporous structure. The average particle size inside the pores, measured by ImageJ software, was 14 nm and 9 nm for (Ni)/KIT-6 and (Ni-8Y)/KIT-6, respectively.

For the Y-modified sample nickel particles are better dispersed in the channels of the KIT-6 support than for in (Ni)/KIT-6 catalyst, where the particles are present only in some spots of KIT-6 support. The positive role of yttrium has already been reported in the literature. J.F. Li et al. [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF] observed that 9 wt.% yttrium promotion of a SBA-15 catalyst led to the highest conversion among the tested catalysts with yttrium loading varying from 3 wt.% and 9 wt.%. This observation was explained by the positive influence of yttrium on Ni dispersion. B. Li et al. [START_REF] Li | Methane reforming with CO2 using nickel catalysts supported on yttria-doped SBA-15 mesoporous materials via sol-gel process[END_REF] also reported that surface oxygen vacancies are initiated from yttrium incorporation into SBA-15, which activated the sites and endorsed CH4 and CO2 conversion. The optimal ratio for Y/Si was found to be 0.04. Y/Si molar ratio of 0.08 resulted in the destruction of the mesoporous structure of the support, hindering the catalytic activity. Similar conclusions can be derived from Fig. 9. At 750°C the maximum CH4 and CO2 conversions are 66.1 and 72.3%, respectively. The H2/CO molar ratio is presented in Fig. 9C. The obtained values are lower than unity, which suggests the occurrence of side reactions. More carbon oxide (CO) was formed than hydrogen (H2) and the highest values were observed for (Ni-8Y)/KIT-6

(Y/Si=0.05) which is in agreement with the literature [START_REF] Li | Methane reforming with CO2 using nickel catalysts supported on yttria-doped SBA-15 mesoporous materials via sol-gel process[END_REF]. Moreover, higher CO2 conversions with respect to CH4 were also reported and ascribed to the presence of parallel reactions, such as reverse water-gas shift [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF]. [START_REF] Dębek | Methane dry reforming over hydrotalcite-derived Ni-Mg-Al mixed oxides: the influence of Ni content on catalytic activity, selectivity and stability[END_REF] observed clear correlation between nickel particle size and formed carbon. The relatively small Ni particles hinder the carbon forming reaction, i.e. direct methane decomposition (CH4 = C + 2H2). This reaction is favorable at relatively low temperature, at 550ºC [START_REF] Dębek | Methane dry reforming over hydrotalcite-derived Ni-Mg-Al mixed oxides: the influence of Ni content on catalytic activity, selectivity and stability[END_REF]. Thus, the values obtained from the carbon balance (at 700ºC) may not directly relate to the presence of methane decomposition in Y-promoted sample, which will be further discussed. suggests that the synthesized support is resistant to DRM conditions. On the other hand, graphitic filament carbon was formed in all the catalysts, and it arises from the methane decomposition reaction, and possibly from CO disproponation (2CO = CO2 + C) [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF]. The former is strongly dependent on the size of Ni particles, and it is favored at low-temperature DRM. The nickel particles observed in the micrographs are predominantly located in the channels of KIT-6, and additionally on the external surface of the carrier. KIT-6 promoted with yttrium clearly shows better dispersion of nickel in the channels, although additionally larger nickel particles were observed on the external surface of the support. The latter is responsible for the formation of surface carbon [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF], and as presented in TEM images, greater amount was observed in spent (Ni-8Y)/KIT-6 catalyst. Table 2 summarizes the average particle size inside the pores for (Ni)/KIT-6 and (Ni-8Y)/KIT-6 spent catalysts, i.e. 11 nm and 8 nm, respecively. The average size of larger particles, located on external surface, was 26 nm and 51 nm for (Ni)/KIT-6 and (Ni-8Y)/KIT-6, respectively. Thus, the yttrium promotion improves dispersion of small Ni° particles located inside pores of the support. However, in case of Ni° particles on the external surface, the particles are bigger for Y-promoted catalysts compared to the un-promoted one. The latter may be the reason of the higher amount of carbon formed on (Ni-8Y)/KIT-6 during the reaction, as confirmed by TEM images (Fig. 10) and in agreement with the literature [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF]. HRTEM is presented in Fig. 11 for (Ni-8Y)/KIT-6 sample. Metallic nickel, plane (111) with d-spacing of 1.98Å is observed. No yttrium was detected in this image, which is in good agreement with XRD results (separate phase of Ni°). Also, well distributed particles of Ni° are shown in Fig. 11B. The particles may be located inside the channels, or on the surface of the KIT-6 support.

Fig. 11C shows a Y2O3 with an intereticular distance dhkl equal to 3.06 Å corresponding to a (222) plane. No presence of nickel was registered in the micrograph, which shows no interaction between these two metals in the catalyst. The Y2O3 was observed before in SBA-15 modified samples in the studies of B. Li et al. [START_REF] Li | Methane reforming with CO2 using nickel catalysts supported on yttria-doped SBA-15 mesoporous materials via sol-gel process[END_REF] and J. F. Li et al. [START_REF] Li | Y2O3-promoted NiO/SBA-15 catalysts highly active for CO2/CH4 reforming[END_REF]. However, the obtained value may also arise from (021) plane of monolitic Y2Si2O7 (dhkl=3.05 Å).

Fig. 12 presents XRD patterns of the samples after DRM. In comparison to the diffractograms shown in Fig. 6B, reflections at 2ϴ ca. 27º appeared. They may be assigned to graphitic carbon. The average Ni° crystallite sizes calculated from Scherrer equation (Table 2) are 5 nm, 9 nm, 18 nm, and 23 nm for (Ni)/KIT-6, (Ni-4Y)/KIT-6, (Ni-8Y)/KIT-6, and (Ni-12Y)/KIT-6, respectively. These values are higher compared to the crystal size calculated for reduced samples. A possible reason may be the formation of nickel agglomerates on the outer surface, due to increasing loading of yttrium. This is consistent with the TEM results. Bigger Niº crystal size increases activity towards direct methane decomposition. However, the values obtained for the Y promoted sample are not in agreement with the calculated carbon balance from the initial stability test at 700ºC, indicating that the other dominant reaction at this specific temperature is occurring rather than carbon formation.

Conclusions

KIT-6 ordered mesoporous silica with Ia3d symmetric structure was synthesized and modified with nickel and yttrium. (Ni-8Y)/KIT-6, corresponding to Y/Si=0.05, presented the highest activity in dry methane reforming at temperatures from 600 to 750ºC, compared to (Ni)/KIT-6, (Ni-4Y)/KIT-6 and (Ni-12Y)/KIT-6. The prepared catalysts showed characteristic absorption spectra for silica material, larger Ni crystallite size, and higher reducibility of bulk NiO for Y-promoted materials than (Ni)/KIT-6. TEM analysis of reduced and spent catalysts revealed the stable structure of the support.

For promoted catalysts, TEM images showed incorporation of metallic nickel into the channels of the support, as well as presence of larger Ni particles on the external surface. For the former, better dispersion was observed in the sample promoted with nickel and yttrium as compared to the sample only modified with nickel. For the latter, nickel agglomerates are present on the outer surface, and they contribute to a higher extent of filamentous carbon formation on Y-modified catalysts compared to catalysts without addition of Y. HRTEM analysis showed no presence of Ni,Y alloy formation in spent catalysts. Additionally, separate phases of Niº, Y2O3, and possibly Y2Si2O7 were observed.

  Fig.2shows TGA/DSC-MS curves acquired for the different materials prepared. The KIT-6

Fig. 6B

  Fig. 6B presents XRD patterns for the reduced materials. Apart from the SiO2 reflection, and the

Fig. 9A

  Fig. 9A to C present results of catalytic tests over KIT-6 modified catalysts. The increase of

3. 3 .

 3 Fig. 10 shows TEM images of (A) KIT-6 support, (B) (Ni)/KIT-6, and (C) (Ni-8Y)/KIT-6 after

Fig. 2 .

 2 Fig. 2. TGA/DSC-MS curves for KIT-6 freshly synthesized catalysts: KIT-6 (A), (Ni)/KIT-6 (B),

Fig. 3 .

 3 Fig. 3. TEM images of KIT-6 calcined support.

Fig. 4 .

 4 Fig. 4. Isotherms of N2 adsorption-desorption (A) and distribution of pore diameters (B) for KIT-6

Fig. 5 .Fig. 6 .

 56 Fig. 5. XRD patterns of freshly synthesized KIT-6 and after calcination at 600ºC for 6h.

Fig. 7 .

 7 Fig. 7. TPR-H2 of reduced KIT-6 modified samples.

Fig. 8 .

 8 Fig. 8. TEM images of (Ni)/KIT-6 (A) and (Ni-8Y)/KIT-6 reduced catalysts.

Fig 9 .

 9 Fig 9. Dry methane reforming catalytic tests of KIT-6 modified catalysts (CH4/CO2/Ar=1/1/8,

Fig. 10 .

 10 Fig. 10. TEM micrographs of catalysts after dry methane reforming KIT-6 (A), (Ni)/KIT-6 (B), (Ni-8Y)/KIT-6 (C).

Fig. 11 .

 11 Fig. 11. HRTEM micrographs of (Ni-8Y)/KIT-6 catalyst after dry methane reforming.

Fig. 12 .

 12 Fig. 12. XRD patterns of spent catalysts KIT-6, (Ni)/KIT-6, (Ni-4Y)/KIT-6, (Ni-8Y)/KIT-6,

Table 2

 2 Nickel crystal size (from XRD and TEM analysis) and reducibility (TPR-H2).

	Niº crystal size from XRD analysis	Average Niº crystal size TEM analysis		Reducibility
	Catalyst		Reduced catalysts [nm]	Spent catalysts [nm]		H2 consumption [µmol H2/g]
	Reduced	Spent
	catalysts [nm]	catalysts [nm]
			Inside	External	Inside	Extrenal	I type	II type	I type/
			pores	surface	pores	surface	bulk NiO	NiO inside pores	(I type + II type)
	KIT-6
	np	np	nd	nd	nd	nd	np	np	np
	(Ni)/KIT-6
	4	5	14	66	11	26	0.030	0.016	0.35
	(Ni-4Y)/KIT-6
	6	9	nd	nd	nd	nd	0.031	0.015	0.32
	(Ni-8Y)/KIT-6
	7	18	9	83	8	51	0.034	0.008	0.19
	(Ni-12Y)/KIT-6
	13	23	nd	nd	nd	nd	0.052	0.011	0.17

Acknowledgements

K. Świrk would like to acknowledge the French Embassy in Poland and InnoEnergy PhD School for the financial support during her co-tutelle doctoral studies at UPMC. K. Świrk would also like to express her gratitude to the KinCat Catalysis Group at NTNU and the Research Council of Norway for the possibility to conduct this research during an Erasmus+ internship. Thanks are due to Sandra Casale (UPMC) for TEM, and Patricia Beaunier (UPMC) for HRTEM expertise. Part of the work was supported by AGH grant 11.11.210.373.

pore volumes derived from BJH desorption method,

3)

average pore size distribution obtained from BJH desorption method.


Matthieu Alfaro

Arnaud Ducrot

Grégory Faye

Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations

Keywords: extinction, propagation, threshold phenomena, sharp threshold phenomena. AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic equations), 35B40 (Asymptotic behavior of solutions)

Introduction

In this work we consider the solution u = u(t, x) of the reaction-diffusion equation

u t = ∆u + f (u), t > 0, x ∈ R N , (1.1)
supplemented with some radially symmetric compactly supported initial data. Typically, f is a so-called ignition or bistable nonlinearity. It is well established [START_REF] Flores | The stable manifold of the standing wave of the Nagumo equation[END_REF], [START_REF] Zlatǒs | Sharp transition between extinction and propagation of reaction[END_REF] that, roughly speaking, "small" initial data lead to extinction, whereas "large" initial data lead to propagation, which is referred as a threshold phenomenon. It was more recently proved [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF], [START_REF] Poláčik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF], [START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations[END_REF][START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF] that, in many situations, a sharp threshold phenomenon does occur. Here, we say that there is a sharp threshold behavior when, for any strictly increasing family of initial data exhibiting extinction for sufficiently small values of the parameter and propagation for sufficiently large values of the parameter, there is exactly one member of the family for which neither extinction nor propagation occurs. For example, for smooth unbalanced bistable nonlinearity, in one space dimension, it is well known that at the threshold value, the corresponding solution of (1.1) converges to the unique ground state centered at the origin [START_REF] Flores | The stable manifold of the standing wave of the Nagumo equation[END_REF], [START_REF] Zlatǒs | Sharp transition between extinction and propagation of reaction[END_REF], [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF], [START_REF] Poláčik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF], [START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations[END_REF][START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF].

By a ground state of this equation we mean a positive stationary solution that decays to zero at infinity, that is a solution of 0 = ∆u + f (u), x ∈ R N , u > 0, lim |x|→+∞ u(x) = 0.

As far as we know, no quantitative estimate of this (sharp) threshold phenomenon exists in the literature. The goal of the present work is to fill this gap by providing such estimates in some asymptotic regimes.

Initiation of propagating fronts or pulses has a long history with multiple applications in both physical and natural sciences, e.g. flame combustion, epidemic outbreaks, ecological or bacterial invasion. We can in particular mention applications in neuroscience where one is typically interested to quantify the depolarization effect of local stimulation of an electrode on a nerve fiber [START_REF] Terman | A free boundary problem arising from a bistable reaction-diffusion equation[END_REF], [START_REF] Mckean | Stabilization of the standing wave in a simple caricature of the nerve equation[END_REF], [START_REF] Neu | Initiation of propagation in a one-dimensional excitable medium[END_REF] or to characterize the critical stimulus amplitude and timing needed to generate waves in visual cortex [START_REF] Faye | Threshold of front propagation in neural fields: An interface dynamics approach[END_REF]. Several approaches have been developed to derive criteria for initiation of propagation in bistable or excitable medium in one space dimension. For McKean-type nonlinearity f (u) = -u + H(u -a), a ∈ (0, 1/2), and H the Heaviside step function, an associated free boundary problem was studied by looking at the evolution of m(t) = sup {x > 0 | u(t, x) = a}. By characterizing the asymptotic limiting behavior of m(t), it was possible to obtain sharp threshold of propagation [START_REF] Terman | A free boundary problem arising from a bistable reaction-diffusion equation[END_REF], [START_REF] Mckean | Stabilization of the standing wave in a simple caricature of the nerve equation[END_REF] for some classes of initial conditions which cross the threshold u = a only twice, and we refer to [START_REF] Moll | Calculation of the threshold surface for nerve equations[END_REF] for a numerical treatment of the problem. More precisely, there is a trichotomy. If m(t) is well defined for all time and lim sup t→+∞ m(t) = +∞, then there is propagation. If there exists T > 0 such that lim inf t→T m(t) = 0, then there is extinction. Finally, if m(t) remains uniformly bounded for all times, then the corresponding solution u(t, x) converges to the unique ground state to (1.1). Another method to derive quantitative conditions for initiation of propagation in the one-dimensional case was obtained by rewriting the reaction-diffusion equation (1.1) as a gradient-flow of the energy

E(u) = 1 2 |∇u| 2 + F (u) dx, (1.2)
where F (u) = -u 0 f (v)dv, and projecting this gradient flow onto an approximate solution space [START_REF] Neu | Initiation of propagation in a one-dimensional excitable medium[END_REF]. A common choice for the approximation space consists of the amplitude and width of Gaussian profiles. This projects the infinite dimensional dynamical systems (1.1) into a two-dimensional space and the criterion for initiating propagation takes the form of separatrices given by the stable manifold of the ground state which is a saddle node in this two-dimensional projected space. The idea that the ground state's stable manifold forms a threshold surface separating initial conditions leading to propagation or extinction was used [START_REF] Idris | Analytical Approach to Initiation of Propagating Fronts[END_REF] to derive estimates of the sharp threshold by approximating the stable manifold by its tangent linear space. Let us finally remark that the free boundary approach was recently applied to neural field equation with Heaviside step nonlinearity where it is possible to obtain explicit formula for the sharp threshold [START_REF] Faye | Threshold of front propagation in neural fields: An interface dynamics approach[END_REF] in this nonlocal setting.

Previous works on (sharp) threshold phenomena for (1.1) rely on different tools such as the so-called zero number argument [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF], the combination of parabolic Liouville theorems and results on exponential separation and principal Floquet bundle [START_REF] Poláčik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF], or energy methods [START_REF] Muratov | Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations[END_REF][START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF], [START_REF] Alfaro | Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue[END_REF]. Let us mention that the work [START_REF] Alfaro | Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue[END_REF], primarly concerned with a population dynamics model [START_REF] Kanarek | Allee effects, adaptive evolution, and invasion success[END_REF] for evolutionary rescue, shows in particular that the problem

u t = ∆u + u(u -θ(t))(1 -u)
where θ(t) 0 as t → +∞, enjoys the so-called hair trigger effect: for any nonnegative and nontrivial initial data, the solution tends to 1 as t → +∞, locally uniformly in x ∈ R N . Since quantitative estimates for the bistable sharp threshold were missing, the main assumption was +∞ 0 θ(s)ds < +∞, which may be "not optimal". Our approach is rather more direct, in the spirit of [START_REF] Zlatǒs | Sharp transition between extinction and propagation of reaction[END_REF], and consists in constructing new and elaborate sub-and supersolutions that enable to capture the underlying time-space scaling of the problem. The sharp quantitative estimates we obtain may lead to refine existing results, such as those in [START_REF] Alfaro | Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue[END_REF] for instance.

We state below our assumptions and main results.

Assumption 1.1 (Nonlinearity f ). The function f : R → R is locally Lipschitz continuous.
There is a threshold θ ∈ (0, 1) such that

f (u) = 0 for all u ∈ (-∞, 0] ∪ {θ} ∪ [1, ∞), (1.3)
and

f (u) > 0, ∀u ∈ (θ, 1), and
     f (u) < 0, ∀u ∈ (0, θ), (BISTABLE CASE) or f (u) = 0, ∀u ∈ (0, θ), (IGNITION CASE).
(1.4)

In the bistable case, we further require

1 0 f (s)ds > 0. (1.5)
Moreover, the following structure conditions hold.

(i) There are r + > 0 and δ + ∈ (θ, 1) such that

f (u) ≤ r + (u -θ), ∀u ∈ [θ, δ + ].
(ii) There are r -> 0 and δ -∈ (θ, 1) such that

f (u) ≥ r -(u -θ) , ∀u ∈ (-∞, δ -].
Notice that the usual cubic bistable nonlinearity

f (u) = ru(u -θ)(1 -u), ∀u ∈ [0, 1],
where r > 0, satisfies the above set of assumptions as soon as θ < 1 2 . Also, an ignition linearity satisfies the above items (i) and (ii) as soon as

0 < lim inf u→θ + f (u) u -θ ≤ lim sup u→θ + f (u) u -θ < +∞.
Now, for ε ∈ (0, 1 -θ) and L > 0, we consider the family of initial data φ ε L given by φ

ε L (x) = (θ + ε)1 B L (x), x ∈ R N , (1.6)
wherein 1 A denotes the characteristic function of the set A, and B L the open ball of center 0 ∈ R N , and radius L. We denote by u ε L the solution of (1.1) starting from the initial data φ ε L . Then this family of solutions enjoys the so-called threshold property.

Proposition 1.2 (Threshold property). Let Assumption 1.1 hold. Let ε ∈ (0, 1 -θ) be given. Then there exist L ε > 0 and L ε > 0 such that

lim t→+∞ u ε L (t, x) = 0 uniformly in R N if 0 < L < L ε , 1 locally uniformly in R N if L > L ε .
In a one dimensional framework, such threshold results were initiated by the seminal works of Kanel' [START_REF] Kanel | Stabilization of the solutions of the equations of combustion theory with finite initial functions[END_REF], Aronson and Weinberger [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation[END_REF], Fife and McLeod [START_REF] Fife | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF]. We also refer to Du and Matano [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF] for more recent developments.

In arbitrary dimension and in our context, the existence of L ε for the ignition case can, for instance, by found in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] or [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF]. Note that the study of the ignition case is sufficient to conclude to the existence of L ε for the bistable case due to the comparison principle. Notice also that the existence (as well as some estimates) of such small radii leading to extinction will also be proved in this work for monostable nonlinearities (see Theorem 6.2), and thus for the ignition case due to comparison arguments.

In arbitrary dimension, the proof of the existence of L ε can, for instance, be found in [START_REF] Poláčik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF]Lemma 3.5] in the case where f is bistable, of the class C 1 , and satisfies f (0) < 0. In our context, since f is Lipschitz continuous on [0, 1], we can use a small C 1 -modification from below of the nonlinearity f and construct a suitable subsolution converging to 1 when L is large enough. Hence the existence of L ε follows from the comparison principle for the bistable case, and thus for the ignition case due to comparison arguments. Now, for each ε ∈ (0, 1 -θ), we consider the quantities 0 < L ext ε ≤ L prop ε given by

L ext ε := sup L > 0 : lim t→∞ u ε L (t, •) = 0 uniformly in R N , L prop ε := inf L > 0 : lim t→∞ u ε L (t, •) = 1 in C 0 loc (R N ) .
In this work we derive sharp estimates for the above quantities in the asymptotic ε → 0. We roughly prove that

L ext ε and L prop ε ≈ ln 1 ε as ε 1.
Our precise result reads as follows.

Theorem 1.3 (Quantitative estimates of the threshold). Let Assumption 1.1 hold. Then there are two constants 0 < C -< C + such that

C -≤ lim inf ε→0 + L ext ε ln 1 ε ≤ lim sup ε→0 + L prop ε ln 1 ε ≤ C + .
Moreover, the constants C ± can be chosen as

C -= 1 √ r + , C + = 2 √ r - ,
where r ± are as in Assumption 1.1, items (i) and (ii).

When L ext ε = L prop ε := L ε for all small enough ε > 0 , we say that the threshold is sharp. Theorem 1.3 immediately yields the following corollary.

Corollary 1.4 (Sharp threshold). Let Assumption 1.1 hold. If the threshold is sharp then

1 √ r + ≤ lim inf ε→0 + L ε ln 1 ε ≤ lim sup ε→0 + L ε ln 1 ε ≤ 2 √ r - .
Let us recall that the sharp threshold phenomena was first analysed by Zlatǒs [START_REF] Zlatǒs | Sharp transition between extinction and propagation of reaction[END_REF] in the one dimensional setting.

In the bistable case, the threshold is known to be sharp when we further assume that f is of the class C 1 and f (0) < 0, see [START_REF] Poláčik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF], which is the case of the usual cubic nonlinearity (1.8). We also refer to [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF]Theorem 4] for other conditions insuring that the threshold is sharp in the bistable case.

In the ignition case, when N = 1, the threshold is known to be sharp when f is nondecreasing in some right neighborhood of θ, see [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF]Theorem 1.4], which is the case of the nonlinearity (1.7). When N ≥ 2, we refer to [15, Theorem 5, Theorem 6] for conditions ensuring that the threshold is sharp in the ignition case.

We now provide a corollary about the existence and the behavior as ε → 0 for the sharp threshold for the two following prototype nonlinearities

Ignition: f (u) = r(u -θ)(1 -u)1 (θ,1) (u), (1.7)
and Bistable:

f (u) = ru(u -θ)(1 -u)1 (0,1) (u), (1.8)
we shall use for numerical validations in Section 5. Here r > 0 and θ ∈ (0, 1) are given parameters with θ < 1/2 in the bistable case (1.8). In these two prototype situations, our main results above rewrite as follows.

Corollary 1.5 (Sharp threshold for prototype nonlinearities (1.7) and (1.8)). Let ε ∈ (0, 1-θ) be given. We consider problem (1.1) with the nonlinearities (1.7) and (1.8). Then the family of initial data φ ε L defined in (1.6) exhibits a sharp threshold L ε that satisfies

C -≤ lim inf ε→0 L ε ln 1 ε ≤ lim sup ε→0 L ε ln 1 ε ≤ C + ,
where the constants C ± read as

C -= 1 r(1 -θ) , C + = 2 r(1 -θ) for (1.7), while C -= 1 rθ(1 -θ) , C + = 2 rθ(1 -θ) for (1.8).
The organization of the paper is as follows. In Section 2 and Section 3 we inquire on extinction and non extinction phenomena in some related toy models. Next, in Section 4, we build on these preliminary results to prove Theorem 1.3 and Corollary 1.5, thus providing a quantitative estimate of the threshold phenomena. Section 5 is devoted to numerical explorations that not only validate Theorem 1.3 and Corollary 1.5 but also enable to make some conjectures on the best constants C ± . Last, in Section 6, we present some results on the threshold phenomena in the case of a degenerate monostable nonlinearity "not enjoying the so-called hair trigger effect", the typical example being f (u) = ru p (1 -u), where p > p F := 1 + 2 N . These results are not sharp with respect to ε 1 as in Theorem 1.3 above. It however reflects the main order for the length of the sharp threshold for small ε and we believe our construction of adequate sub and supersolutions is instructive.

Extinction in a toy model

In this section, we consider the piecewise linear function

g(u) = (u -θ) + .
Herein θ > 0 is a given and fixed parameter and the subscript + is used to denote the positive part of a real number. Let N ≥ 1 be a given integer. Consider the solution w = w(t, x) of the semi-linear problem

w t = ∆w + g(w), t > 0, x ∈ R N , (2.1)
together with the initial data

w(0, x) = (θ + ε)1 B L (x), x ∈ R N , (2.2)
wherein ε > 0, L > 0, and B L ⊂ R N denotes the ball of radius L centred at the origin.

Proposition 2.1 (Extinction). Let δ > θ be given. Consider the time

T ε := ln δ -θ ε . (2.3)
Then, for any 0 < γ < 1, there exists ε 0 > 0 small enough such that, for each ε ∈ (0, ε 0 ) and for each 0

< L < γ ln 1 ε , the solution w = w(t, x) of (2.1)-(2.2) satisfies sup x∈R N w (T ε , x) ≤ θ, (2.4)
and is thus going to extinction at large times.

Proof. Consider the solution v = v(t, x) of the heat equation

v t = ∆v, t > 0, x ∈ R N ,
starting from w(0, •). Denote by Γ = Γ(t, x) the heat kernel on R N , given by

Γ(t, x) = 1 (4πt) N 2 exp - |x| 2 4t , t > 0, x ∈ R N ,
where

| • | is used to denote the Euclidean norm in R N . Then v(t, •) = (θ + ε)Γ(t, •) * 1 B L , so that for all t > 0 V (t) := v(t, •) L ∞ (R N ) = θ + ε (4πt) N 2 |x|<L e -|x| 2 4t dx = θ + ε (4π) N 2 |x|< L √ t e -|x| 2 4 dx.
We now construct a supersolution to (2.1) in the form W (t, x) := v(t, x)ϕ(t), with ϕ(0) = 1 and ϕ(t) > 0. From W t -∆W = vϕ and the expression of g this requires

ϕ (t) ≥ ϕ(t) - θ v(t, x) + , ∀(t, x) ∈ (0, +∞) × R N ,
and thus

ϕ (t) ≥ ϕ(t) - θ V (t) +
, ∀t ∈ (0, +∞).

We now choose ϕ as the solution of the Cauchy problem

ϕ (t) = ϕ(t) - θ V (t) , ϕ(0) = 1, that is ϕ(t) = e t 1 - t 0 e -s θ V (s) ds .
Observe that V (0)ϕ(0) > θ and denote by T > 0 the first time where V (T )ϕ(T ) = θ (obviously we let T = +∞ if such a time does not exist). Then ϕ(t) -θ

V (t) + = ϕ(t) -θ V (t)
for all t ∈ [0, T ), and thus W (t, x) = v(t, x)ϕ(t) is a supersolution to (2.1) on the time interval (0, T ). In particular, if T < +∞, it follows from the comparison principle that w(T, •) ≤ W (T, •) ≤ θ, and we are done provided T ≤ T ε , a condition we aim at reaching below. Now, observe that the condition T < +∞ rewrites as there exists t > 0 satisfying the equation

F L (t) := θ 1 - e -t A L (t) + ε - t 0 e -s θ A L (s) ds = 0,
wherein we have set

A L (t) := 1 (4π) N 2 |x|< L √ t e -|x| 2 4 dx = C N L 2 √ t 0 r N -1 e -r 2 dr, for some constant C N > 0. Since F L (0) = ε and F L (t) = θe -t A L (t) A 2 L (t) < 0, the condition T ≤ T ε is equivalent to require F L (T ε ) < 0, that also reads as 1 + ε θ < Tε 0 e -s A L (s) ds + e -Tε A L (T ε ) . (2.5)
Note that the right-hand side of the above expression is decreasing with respect to L. We now select

L ε = γ T ε ln 1 ε , (2.6)
for some constant γ > 0 to be chosen later, and aim at proving that (2.5), with L = L ε , is satisfied for ε > 0 small enough. To do so, set

G ε := Tε 0 e -s A Lε (s) ds.
Integrating by parts yields

G ε = Tε 0 e -s A Lε (s) ds = 1 - e -Tε A Lε (T ε ) + C N 2 N +1 L N ε Tε 0 e -s A 2 Lε (s) 1 s N +2 2 e -L 2 ε 4s ds.
Next the change of variable z = s/L 2 ε yields

G ε = 1 - e -Tε A Lε (T ε ) + C N 2 N +1 1 γ 2 ln 1 ε 0 e -L 2 ε z A 2 1 (z) 1 z N +2 2 e -1 4z dz. Hence (2.5) rewrites 2 N +1 C N θ ε < H ε := 1 γ 2 ln 1 ε 0 e -L 2 ε z-1 4z A 2 1 (z) 1 z N +2 2
dz.

(2.7)

Observe that the function z ∈ (0, +∞) → L 2 ε z + 1 4z is decreasing then increasing and reaches its minimal value L ε at z = 1 2Lε . Next one has, as ε → 0,

2L ε = 2γ ln(δ -θ) ln 1 ε + ln 2 1 ε = 2γ ln 1 ε 1 + O ln -1 1 ε .
We now restrict to 0 < γ < 2 and select k > 0 small enough so that γ < 2 1+k . It follows from the above that, for 0 < ε 1,

1 2L ε , 1 + k 2L ε ⊂ 0, 1 γ 2 ln 1 ε . Since L 2 ε z + 1 4z ≤ L ε 2 (2 + k), ∀z ∈ 1 2L ε , 1 + k 2L ε , it follows that H ε ≥ 1+k 2Lε 1 2Lε e -Lε 2 (2+k) z N +2 2 dz ≥ e -Lε 2 (2+k) 1+k 2Lε N +2 2 k 2L ε = C N,k e -Lε 2 (2+k) L N 2 ε ∼ C N,γ,k e -γ(1+ k 2 ) ln 1 ε ln 1 ε N 2
, as ε → 0. In the above estimate, C N,k and C N,γ,k denote positive constants independent of ε > 0 small enough, depending on (N, k) and (N, γ, k) respectively. As a result, by restricting further to 0 < γ < 1 and k > 0 small enough so that γ(1 + k 2 ) < 1, we get that condition (2.7) is satisfied for 0 < ε 1. This completes the proof of the result.

Non extinction in a toy model

In this section, we fix θ ∈ (0, 1) and we consider the solution w = w(t, x) of the problem

w t = ∆w + w -θ, t > 0, x ∈ R N , (3.1)
together with the initial data

w(0, x) = (θ + ε)1 B L (x), x ∈ R N . (3.2)
Proposition 3.1 (Non extinction). Let θ < α < α < 1 be given. Consider the time

T ε := ln α -θ ε . (3.3)
Let γ > 2 be given. Then for any 0 < k < 1 -2 γ , there exists ε 0 > 0 small enough such that, for each ε ∈ (0, ε 0 ) and for each L > γ ln 1 ε , the solution w = w(t, x) of (3.1)-(3.2) satisfies min |x|≤kLε w(T ε , x) ≥ α .

Proof. Notice that the function w = w(t, x) is explicitly given by

w(t, x) = θ + e t (v(t, x) -θ), (3.4)
where v = v(t, x) is the solution of the heat equation

v t = ∆v, t > 0, x ∈ R N ,
with the initial datum v(0, x) = w(0, x). Hence

w(T ε , x) = θ + e Tε ε - θ + ε (4πT ε ) N/2 |y|≥L e -|x-y| 2 4Tε dy = θ + (α -θ) 1 - θ + ε ε(4πT ε ) N/
-y| ≥ (1 -k)|y|, w(T ε , x) ≥ θ + (α -θ) 1 - θ + ε ε(4πT ε ) N/2 |y|≥L e -(1-k) 2 |y| 2 4Tε dy ≥ θ + (α -θ) 1 - θ + ε ε(1 -k) N π N/2 |z|≥ (1-k)L √ 4Tε e -|z| 2 dz . (3.5)
We now select

L ε = γ T ε ln 1 ε , (3.6)
for some constant γ > 2. In view of (3.5), it is enough to conclude the proof to reach

θ + (α -θ) 1 - θ + ε ε(1 -k) N π N/2 |z|≥ (1-k)Lε √ 4Tε e -|z| 2 dz ≥ α , for 0 < ε 1, that is I ε := |z|≥ (1-k) 2 γ ln 1 ε e -|z| 2 dz ≤ ε α -α α -θ (1 -k) N π N/2 θ + ε . (3.7)
On the other hand, by denoting by A N > 0 some constant depending on N , one has for

X > 0, |z|≥X e -|z| 2 dz = A N ∞ X r N -1 e -r 2 dr ∼ A N 2 X N -2 e -X 2 as X → ∞,
so that, as ε → 0,

I ε ∼ A N 2 (1 -k) 2 γ ln 1 ε N -2 exp - (1 -k) 2 4 γ 2 ln 1 ε .
Since (1-k)γ 2 > 1, the above implies I ε = o(ε) as ε → 0, which validates (3.7), and thus concludes the proof.

Quantitative estimates of the (sharp) threshold phenomena

In this section, we complete the proof of Theorem 1.3 and Corollary 1.5.

Proof of Theorem 1.3. Let Assumption 1.1 hold. Let us select θ < δ < min(δ -, δ + ), where δ -, δ + are provided by Assumption 1.1 (i), (ii) respectively.

By an immediate time-space rescaling argument we have that the threshold values

L ext ε = L ext ε (f ) and L prop ε = L prop ε
(f ) under investigation are given by

L ext ε = 1 √ r + L ext ε (f /r + ), L prop ε = 1 √ r - L prop ε (f /r -),
where L ext/prop ε (f /r) are the threshold values associated with the nonlinearity f /r. Let us first enquire on L ext ε (f /r + ). Let us denote by w ε L = w ε L (t, x) the solution to

w t = ∆w + 1 r + f (w), starting from φ ε L (x) = (θ + ε)1 B L (x)
. By combining Assumption 1.1 (i) and comparison with the ordinary differential equation (use the supersolution t → θ + εe t ), we immediately have the upper bound

w(t, x) ≤ δ, ∀(t, x) ∈ [0, T ε ] × R N ,
where the time T ε was defined in (2.3). This in turn implies that w is a subsolution to (2.1) on (0, T ε ) × R N . As a result, the extinction result Proposition 2.1 applies: for any 0 < γ < 1, we have, for 0 < ε 1, L ext ε (f /r + ) ≥ γ ln 1 ε and thus

L ext ε = L ext ε (f ) ≥ γ √ r + ln 1 ε .
This rewrites as

1 √ r + ≤ lim inf ε→0 L ext ε ln 1 ε .
Let us now enquire on L prop ε (f /r -). Let us denote by w ε L = w ε L (t, x) the solution to

w t = ∆w + 1 r -f (w), starting from φ ε L (x) = (θ + ε)1 B L (x). Set g(w) = 1 r -f (w) and note that due to Assumption 1.1 (ii), one has g(w) ≥ w -θ, ∀w ∈ (-∞, δ].
Let η ∈ (0, δ-θ) be small enough and define a Locally Lipschitz continuous function g : R → R such that

g(w) = g(w) for w ∈] -∞, θ] ∩ [δ, ∞), w -θ for w ∈ [θ, θ + η],
g(w) ≤ g(w) for w ∈ [θ + η, δ] and 1 0 g(s)ds > 0. Denote w = w(t, x) the solution of the problem w t = ∆ w + g( w), w(0, x) = w(0, x), so that w(t, x) ≤ w(t, x). Fix η ∈ (0, η) and consider the time

T ε = ln η ε . Since g(w) ≥ w -θ, ∀w ∈ (-∞, θ + η],
and for all ε ≤ η one has w(t, x) ≤ θ + η on [0, T ε ] × R N , using Proposition 3.1, one obtains that, for any given γ > 2 and k > 1 -2 γ , there exists ε 0 > 0 such that, for all ε ∈ (0, ε 0 ) and

L > L ε := γ ln 1 ε , one has w(T ε , x) ≥ (θ + η )1 B kLε .
Define by L prop η the propagating threshold associated to the equation with the nonlinearity g.

Hence, for all ε small enough such that kL ε > L prop η , one has w(t, x) → 1 as t → +∞ locally uniformly in space and therefore w(t, x) → 1 as t → +∞ locally uniformly in space. As a consequence we obtain

L prop ε (f /r -) ≤ L ε for all ε < < 1.
Hence we get lim sup

ε→0 L prop ε (f /r -) ln 1 ε ≤ γ, ∀γ > 2.
This rewrites as

lim sup ε→0 L prop ε ln 1 ε ≤ 2 √ r - ,
and concludes the proof of Theorem 1.3.

Proof of Corollary 1.5. In view of Corollary 1.4, it is sufficient to prove that the threshold is sharp. In the bistable case, the threshold is known to be sharp when we further assume that f is of the class C 1 and f (0) < 0, see [START_REF] Poláčik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF], which is the case of the usual cubic nonlinearity (1.8). On the other hand, the ignition case (1.7) can be handled using the results of [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF]. Indeed fix ε ∈ (0, 1 -θ) and fix L > L prop ε . We consider the family of solutions u ε L = u ε L (t, x) of (1.1) starting from (1.6). Recall that it is non decreasing with respect to L. Then, since u ε L (t, •) → 1 as t → +∞, locally uniformly in R N , there exists t > 0 such that the energy, see (1.2), satisfies E u ε L (t, •) < 0. Finally, since u ε L (t, •) → 0 as L → 0, the existence of the sharp threshold L ε ε directly follows from the application of the results of Muratov and Zhong [START_REF] Muratov | Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations[END_REF] for ignition nonlinearities and the family of initial data {u ε L (t, •)} L>0 . Notice that this argument also applies in the bistable case and in some monostable situations (see Corollary 6.5).

Numerical explorations

We explored sharp threshold of propagation numerically for the one dimensional case, that is N = 1 throughout this section. We first report on numerical explorations of the validity range of our predictions from Corollary 1.4 for nonlinearities satisfying Assumption 1.1 and more precisely Corollary 1.5 for (1.7) and (1.8). We also show some evidence for the existence of a universal constant C(θ) such that lim

ε→0 + L ε ln 1 ε = C(θ),
for both ignition and bistable nonlinearities. Finally, we investigate other dependencies.

Numerical validation of Corollary 1.5

We used second order finite differences in space and a splitting method in time where the diffusion part of the equation is evaluated with a Crank-Nicolson scheme while the nonlinear term is either evaluated exactly (for the ignition case) or with a Runge-Kuttta method of order two (for the bistable nonlinearity). Time step and space discretization were set respectively to δt = 0.02 and δx = 0.02. We used direct numerical simulations to evaluate the critical threshold L ε using the following strategy. For fixed nonlinearity f and ε > 0, we varied the size L > 0 of the support of the initial condition:

φ ε L = (θ + ε)1 [-L,L] ,
and ran our numerical scheme from t = 0 to some final time t = T . For each L > 0, we evaluated the maximum of the numerically computed solution at the last time step. We then discriminated the precise value of L for which there is a transition from extinction (maximum being close to zero) to propagation (maximum being close to one). As we expect the sharp threshold L ε to be of the order ln 1 ε as ε → 0 + and the transition to occur for times of the order ln 1 ε , we took a spatial domain of computation [-100, 100] and a final time of computation T = 100. This allowed us to cover a range for ε from 10 -3 to 10 -1 .

Ignition nonlinearity. We present our results for the ignition nonlinearity function

f (u) = (u -θ)(1 -u), u ∈ [0, 1],
for several values of θ ∈ (0, 1). Numerically computed values of L ε are plotted in Figure 1 (left) as a function of ln 1 ε together with a corresponding linear fit for several representative values of θ. Measured slopes are presented in Table 1 for a larger range of θ ranging from 0.1 to 0.6. Our numerical results seem to indicate that lim

ε→0 + L ε ln 1 ε = 1 √ 1 -θ ,
as it can be inferred from the last line in Table 1. Bistable nonlinearity. We present our results for the bistable nonlinearity function

θ = 0.1 θ = 0.2 θ = 0.3 θ = 0.4 θ = 0.5 θ = 0.6 s m 1.
f (u) = u(u -θ)(1 -u), u ∈ [0, 1],
for several values of θ ∈ (0, 1/2). Numerically computed values of L ε are plotted in Figure 1 (right) as a function of ln 1 ε together with a corresponding linear fit for several representative ε for several values of θ for the ignition nonlinearity (left) and bistable nonlinearity (right). We also superimposed in plane line a linear fit whose slopes s m are reported in Table 1 for the ignition nonlinearity (resp. in Table 2 for the bistable nonlinearity). values of θ. Measured slopes are presented in Table 2 for a larger range of θ ranging from 0.1 to 0.4. As predicted by our main Theorem 1.3, we observe that θ(1 -θ)s m ∈ [START_REF] Alfaro | Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue[END_REF][START_REF] Aronson | Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation[END_REF]. Contrary to the ignition case, it seems that θ(1 -θ)s m does depend on θ and, moreover, in a nonlinear fashion. Our measured values did not allow us to conjecture an elaborated guess for this dependence. We report in Figure 2

Other dependencies -Bistable nonlinearity

Here, we report on some other dependencies of the critical threshold L in the case of the bistable nonlinearity

f (u) = u(u -θ)(1 -u).
We have considered the following family of compactly supported initial conditions:

φ L = 1 [-L,L] , L > 0,
such that the amplitude of the initial condition is fixed to the stable steady state 1. For each θ ∈ (0, 1/2), we numerically computed the corresponding critical threshold L by varying the size L > 0 of the support. We report in Figure 3 our numerical findings. We recover that 2 for the bistable nonlinearity (1.8).

in the limit θ → 0 + , the critical threshold L → 0. Indeed, as θ → 0 + , the nonlinearity approaches f (u) = u 2 (1 -u) for which the hair trigger effect is known [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. Our numerics suggest that the dependence of L near θ = 0 is linear, and we get lim

θ→0 + L θ ≈ 4.85.
On the other hand, when θ → 1 2 -, we obtain L → +∞. In the limit, we have 1 0 f (u)du = 0, and there does not exist any ground state (homoclinic solution to the steady state 0) to equation (1.1). Instead, there is a one-parameter family of stationary monotone interfaces between u = 0 and u = 1. Our numerics suggest that the dependence of L near θ = 1 2 is logarithmic, and we get lim

θ→ 1 2 - L ln 1 1-2θ ≈ 0.99.

Degenerate monostable case

In this section we consider u = u(t, x) the solution of the Cauchy problem

u t = ∆u + f (u), t > 0, x ∈ R N , u(0, x) = ε1 B L (x), x ∈ N , (6.1)
where ε ∈ (0, 1] and L > 0 are given parameter, and where f is a monostable degenerate nonlinearity, that is assumed to satisfy the following set of assumptions.

u→0 + f (u) u p ≤ lim sup u→0 + f (u) u p < +∞, (6.2)
for some p > 1.

When 1 < p ≤ p F , where p F := 1 + 2 N is the so called Fujita exponent [START_REF] Fujita | On the blowing up of solutions of the Cauchy problem for u t = ∆u + u 1+α[END_REF], it is well known [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] that the reaction-diffusion equation u t = ∆u + f (u) enjoys the hair trigger effect: for any nonnegative and nontrivial initial data, the solution tends to 1 as t → +∞, locally uniformly in x ∈ R N . On the other hand, when p > p F , it is known [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] that some "small enough" initial data imply extinction, whereas some "large enough" initial data imply propagation. Our goal is here to provide some quantitative estimates on this threshold phenomenon.

The first result of this section provides an lower estimate for the radius of extinction.

Theorem 6.2 (Extinction). Let Assumption 6.1 hold with p > 1 + 2 N . Then there is a constant C -= C -(f, N ) > 0 such that for all ε ∈ (0, 1], as soon as

L < C - ε p-1 2 , (6.3)
the solution to (6.1) satisfies

u(t, •) L ∞ (R N ) = O t -N 2 ,
as t → +∞. (6.4)

Proof. Throughout this proof we fix ε ∈ (0, 1] and we consider the solution v = v(t, x) of the heat equation

v t = ∆v, t > 0, x ∈ R N , starting from v(0, •) = u(0, •) = ε1 B L , so that V (t) := v(t, •) L ∞ (R N ) = ε (4π) N 2 |x|< L √ t e -|x| 2 4 dx =: εA L (t).
Following the proof of Proposition 2.1, we seek a supersolution to (1.1) for t > 0 and x ∈ R N in the form U (t, x) := v(t, x)ϕ(t), with ϕ(0) = 1 and ϕ(t) > 0. Since there is K > 0 such that f (u) ≤ Ku p for all u ∈ R, it is enough to have

ϕ (t) ϕ p (t) = Kε p-1 A p-1 L (t),
which is solved as

ϕ(t) = 1 1 -(p -1)Kε p-1 t 0 A p-1 L (s)ds 1 p-1 .
Since we need ϕ to be global, the condition we obtain reads as

1 > (p -1)Kε p-1 +∞ 0 A p-1 L (s)ds.
Notice that, for a given L > 0, there exists some positive constant

C N,L such that A L (t) ∼ C N,L t N 2
as t → +∞ so that the above improper integral does converge since p > 1 + 2 N . The above condition rewrites as

1 > (p -1)Kε p-1 1 (4π) (p-1)N 2 +∞ 0 |x|< L √ s e -|x| 2 4 dx p-1 ds, or, letting s = L 2 t, 1 > (p -1)Kε p-1 1 (4π) (p-1)N 2 L 2 +∞ 0 |x|< 1 √ t e -|x| 2 4 dx p-1 dt =: CL 2 ε p-1 ,
for some constant C = C(p, K, N ) > 0. Hence, when this condition is fulfilled, we are equipped with a global supersolution U (t, x) = v(t, x)ϕ(t) with ϕ bounded which ensures that (6.4) holds. This concludes the proof.

Our next result is concerned with a lower estimate for the radius leading to propagation. Contrary to the previous result that provides an estimate valid for any ε ∈ (0, 1], here our lower estimate is valid for 0 < ε 1 and it reads as follows.

Theorem 6.3 (Propagation). Let Assumption 6.1 hold. Then there exist C + = C + (f, N ) > 0 and ε 0 > 0 small enough such that, for all ε ∈ (0, ε 0 ), there is L ε > 0 satisfying

L ε ∼ C + ε p-1 2 ln 1 ε 1 2
as ε → 0, (6.5) such that, for all L > L ε , the solution to (6.1) satisfies lim t→+∞ u(t, x) = 1 locally uniformly in R N . (6.6)

Proof. From Assumption 6.1, there are δ > 0 and r > 0 such that f (u) ≥ ru p for all 0 ≤ u ≤ δ.

Up to a time-space rescaling argument we can assume, without loss of generality, that r = 1. As a direct consequence of Proposition 1.2 (for the ignition case), Problem (6.1) enjoys the threshold phenomena that ensures that for any α ∈ (0, 1) there exists a radius L α > 0 such that for all L ≥ L α the function w L = w L (t, x) defined as the resolution of (6.1) with the initial datum w(0, x) = α1 B L (x) satisfies w L (t, x) → 1 as t → +∞ locally uniformly for x ∈ R N . Now to prove Theorem 6.3, let us consider Y (τ, ξ), the solution of the ODE Cauchy problem, where ξ ∈ R serves as a parameter,

∂ τ Y (τ, ξ) = (Y ) p + , Y (0, ξ) = ξ, namely Y (τ, ξ) =      ξ if τ > 0 and ξ ≤ 0, 1 -(p-1)τ + 1 ξ p-1 1 p-1 if τ > 0, ξ > 0 and τ < 1 (p-1)ξ p-1 .
For 0 < ε < δ we set

T ε := 1 (p -1)ε p-1 - 1 (p -1)δ p-1 . (6.7)
And observe that one has 0 < Y (τ, ξ) < δ, ∀τ ∈ (0, T ε ), ∀ξ ∈ (0, ε).

Again we consider the solution

v = v(t, x) of the heat equation starting from v(0, •) = u(•, 0) = ε1 B L , and define u -(t, x) := Y (t, v(t, x)), 0 ≤ t ≤ T ε , x ∈ R N . For 0 < t ≤ T ε , x ∈ R N , we have u - t -∆u --f (u -) = Y τ + v t Y ξ -∆vY ξ -|∇v| 2 Y ξξ -f (Y ) ≤ Y τ + v t Y ξ -∆vY ξ -|∇v| 2 Y ξξ -Y p = -|∇v| 2 Y ξξ ≤ 0,
since one can easily check that Y ξξ (τ, ξ) ≥ 0 for all τ > 0, ξ > 0 with τ < 1 (p-1)ξ p-1 . Hence, for each ε ∈ (0, δ), the comparison principle applies and yields (6.9) Now fix 0 < δ < δ and k ∈ (0, 1). We look for a condition ensuring that, for all x ∈ B kL , we have Y (T ε , v(T ε , x)) ≥ δ , that reads as

u(T ε , x) ≥ Y (T ε , v(T ε , x)), ∀x ∈ R N . (6.8) Next, we have v(T ε , x) = ε - ε (4πT ε ) N/2 |y|≥L e -|x-
-(p -1)T ε + ξ 1-p ε 1 1-p ≥ δ .
Recalling the definition of T ε in (6.7) this rewrites as (δ) p-1 > 0, the above inequality becomes

ξ ε ≥ ε 1 + ε p-1 1 (δ ) p-1 - 1 (δ) p-1
ξ ε ≥ ε 1 + ε p-1 C 1 1-p = ε - ε p C p -1 + O ε 2p-1 .
In view of (6.9) it is sufficient to have, for 0 < ε 1,

ε (1 -k) N π N/2 |z|≥ (1-k)L √ 4Tε e -|z| 2 dz ≤ ε p C p -1 + O ε 2p-1 . (6.10)
We now select

L ε = γ T ε ln 1 ε , (6.11)
for some constant γ > 0. As in the proof of Proposition 3.1, if I ε denotes the integral in the left hand side of (6.10), we have, as ε → 0,

I ε ∼ A N 2 (1 -k) 2 γ ln 1 ε N -1 exp - (1 -k) 2 4 γ 2 ln 1 ε ,
for some constant A N > 0 independent of ε small enough. Using this we see that (6.10) holds true, for 0 < ε 1, as soon as γ > 0 is large enough so that (1 -k) 2 4 γ 2 > p -1. (6.12)

We have thus proved that, for any k ∈ (0, 1) and γ > 0 such that (6.12) holds, we have that, for all 0 < ε 1, min |x|≤kLε u(T ε , x) ≥ α .

As a consequence, choosing ε small enough so that kL ε ≥ L α , the comparison principle ensures that (6.6) holds true, as soon as ε > 0 is sufficiently small. This completes the proof of the theorem.

As in Section 1, introducing for each ε ∈ (0, 1] the radii L ext ε and L prop ε , from Theorem 6.2 and Theorem 6.3, we immediately infer the following. Note that in the one dimensional case, namely N = 1, it is known that the threshold is sharp, that is L ext ε = L prop ε := L ε for any ε ∈ (0, 1]. We refer to Du and Matano [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF] for the proof of this sharpness. We also refer to [15, Theorem 7, Theorem 8, Theorem 9] for further results and conditions insuring that the threshold is sharp in the multi-dimensional degenerate monostable case.

As a special case, when f (u) = ru p (1 -u)1 [0,1] (u) with p > 1 + 2 N , the threshold is sharp (see the proof of Corollary 1.5), and the above corollary provides rather refined estimates of this quantity in the asymptotic ε → 0, that can be reformulated as follows. L ε < +∞.

Figure 1 :

 1 Figure 1: Numerically computed values of L ε as a function of ln 1ε for several values of θ for the ignition nonlinearity (left) and bistable nonlinearity (right). We also superimposed in plane line a linear fit whose slopes s m are reported in Table1for the ignition nonlinearity (resp. in Table2for the bistable nonlinearity).

  the numerically computed values of θ(1 -θ)s m for several values of θ ∈ (0, 1/2). θ = 0.1 θ = 0.15 θ = 0.2 θ = 0.25 θ = 0.3 θ = 0.35 θ = 0.4

Figure 2 :

 2 Figure 2: Numerically computed values of θ(1 -θ)s m for several values of θ ∈ (0, 1/2) as reported in Table2for the bistable nonlinearity (1.8).

Assumption 6 . 1 (Figure 3 :

 613 Figure 3: Numerically computed values of L as a function of θ for the bistable nonlinearity and amplitude of the initial condition fixed to the stable steady state 1. We also superimposed in plane lines a linear fit near θ = 0 (blue line) and a logarithmic fit near θ = 1/2 (red line). The measured slope s 0 m of the linear fit near θ = 0 is s 0 m = 4.85. The measured slope s 1/2 m of logarithmic fit of L as a function of ln 1 1-2θ is s 1/2 m = 0.99.

  y| 2 4Tε dy.As a result, for any k ∈ (0, 1), for all |x| ≤ kL one has, since |y| ≥ L ensures that |x| ≤ kL ≤ k|y| and |x -y|≥ (1 -k)|y|, v(T ε , x) ≥ ε -ε (4πT ε ) N/2 |y|≥L e -(1-k) 2 |y| 2 4Tε dy ≥ ε -ε (1 -k) N π N/2 |z|≥ (1-k)L √ 4Tε e -|z| 2 dz := ξ ε .

Corollary 6 . 4 (

 64 Threshold radii for degenerate monostable case). Let Assumption 6.1 hold with p > 1 + 2 N . Then the threshold radii L ext ε

Corollary 6 . 5 (

 65 Sharp threshold for prototype degnerate monostable nonlinearity). Let ε ∈ (0, 1] be given. Consider the prototype nonlinearityf (u) = ru p (1 -u)1 (0,1) (u), p > 1 + 2 N ,where r > 0. Then the problem (6.1) exhibits a sharp threshold L ε that satisfies0 < lim inf ε→0 ε

  2 |y|≥L e -|x-y| 2 4Tε dy , from the definition of T ε in (3.3). As result, for all |x| ≤ kL one has, since |y| ≥ L ensures that |x| ≤ kL ≤ k|y| and |x

Table 1 :

 1 Ignition nonlinearity. Measured slopes s m of the linear fit of L ε as a function of ln1 ε together with their rescaled values √ 1 -θs m . See Figure1(left). Note that √ 1 -θs m 1 appears to be independent of θ.

	√ 1 -θs m	11 1.05	1.20 1.07	1.24 1.04	1.32 1.02	1.43 1.01	1.57 0.99

Table 2 :

 2 Bistable nonlinearity. Measured slopes s m of the linear fit of L ε as a function of ln1

	s m	5.12	4.43	4.04	3.78	3.62	3.52	3.45
	θ(1 -θ)s m	1.53	1.58	1.61	1.63	1.66	1.67	1.69

ε together with their rescaled values θ(1 -θ)s m . See Figures 1 (right) and 2. Note that θ(1 -θ)s m ∈ [1, 2] as predicted by our main Theorem 1.3.

Acknowledgements. M. Alfaro is supported by the ANR i-site muse, project michel 170544IA (n • ANR idex-0006).