Pierre-Emmanuel Arduin

Elsa Negre

Camille Rosenthal-Sabroux

Knowledge and Decision for Smart Cities Initiatives -Cases of Paris and Nice

Keywords: Smart City, Knowledge, Decision Support System, Digital Knowledge Ecosystem I

INTRODUCTION

Since the early 90s, the development of Internet and communication technology has facilitated the generation of initiatives to create opportunities for communication and information sharing by local authorities. In our everyday life, we are more and more invaded by data and information. These flows of data and information have given rise to a huge mass of data to treat [START_REF] Negre | Smart Cities: A Salad Bowl of Citizens, ICT, and Environment[END_REF]. The world is becoming increasingly a digital world and people are affected by such changes. Today's world faces two important growths: urbanization and information technologies which mean that digital infrastructure infers an information environment that is "as imperceptible to us as water is to a fish" [START_REF] Mcluhan | Counterblast 1954[END_REF].

As pointed out by [START_REF] Lima | Visual Complexity: Mapping Patterns of Information[END_REF], "The complexity of connectedness of modern times requires new tools of analysis and exploration, but above all it demands a new way of thinking. It demands a pluralistic understanding of the world that is able to envision the widen structural plan and at the same time examine the intricate mesh of connections among its smallest elements. It ultimately calls for a holistic systems approach; it calls for network thinking." [17, p.45-46] There exists a kind of parallelism between technology and humans. On one hand, people are hyper-connected by using more and more technology (and information and knowledge flow between different stakeholders, with different uses and backgrounds [START_REF] Kennedy | Transforming Big Data into Knowledge: Experimental Techniques in Dynamic Visualization[END_REF]), and on the other hand (digital) systems are more and more user-centered [START_REF] Viitanen | Smart cities and green growth: outsourcing democratic and environmental resilience to the global technology sector[END_REF]. Thus, within cities, systems have to adapt to hyper-connected citizens, in a very particular environment, the one of cities in constant evolution where systems and humans are nested. Thus becoming "smart" is a challenge increasingly important for many cities or communities [START_REF] Kennedy | Transforming Big Data into Knowledge: Experimental Techniques in Dynamic Visualization[END_REF].

The definition of a smart city is indispensable to identify its perimeter and to understand which initiatives can be considered smart and which cannot. The definition and the comprehensive smart city framework are the necessary basis on which to build the smart city goals system. As the definition of a smart city changes depending on the practitioners [START_REF] Alawadhi | Aspirations and realizations: the smart city of Seattle[END_REF], in this paper we consider that cities are smart: "when investments in human and social capital and traditional and modern communication infrastructure fuel sustainable economic growth and a high quality of life, with a wise management of natural resources, through participatory governance" [START_REF] Caragliu | Smart cities in Europe[END_REF].

From our point of view, a smart city can be seen as a Digital Knowledge Ecosystem: (i) a knowledge ecosystem because there is a dynamic evolution of knowledge interactions between stakeholders (citizens, public administrations, etc.) to improve decision-making and innovation through improved evolutionary networks of collaboration [START_REF] Shrivastava | Knowledge Ecology: Knowledge Ecosystems for Business Education and Training[END_REF], and (ii) a digital ecosystem because it is "an open, loosely coupled, domain clustered, demand-driven, self-organizing, and agent based environment which each species is proactive and responsive for its own benefit and profit" [START_REF] Dong | Digital ecosystem ontology[END_REF].

The paper is organized as follows: in the next section, we explain our vision of knowledge and decision. Then we detail some existing approaches concerning smart cities initiatives. We present a comparison of two French (smart) cities: Paris and Nice in Section IV. We conclude in Section V.

II. KNOWLEDGE AND DECISION

This section introduces our vision of knowledge and the underlying knowledge ecosystems; and the ways of regarding decision making processes when considering smart cities.

A. Knowledge

As the authors of this article, we have got tacit knowledge, i.e. an individual cognitive construction that we have structured in information during a process of sense-giving. As the reader of this paper, you have interpreted this information perceiving forms and colors; you have absorbed words, data, during a process of sense-reading, possibly creating new tacit knowledge for you.

Drawing from the concepts of "sense-giving" and "sensereading" introduced by [START_REF] Polanyi | Sense-Giving and Sense-Reading[END_REF], [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF] states that: when datum is sense-given through interpretative framework, it becomes information, and when information is sense-read through interpretative framework, it becomes knowledge" ( [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF], p. 88). The concept of "interpretative framework" [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF] can be seen as a "mental model": "Mental models are personal, internal representations of external reality that people use to interact with the world around them. They are constructed by individuals based on their unique life experiences, perceptions and understandings of world. Mental models are used to reason and make decisions and can be the basis of individual behaviors." [START_REF] Jones | Mental models: an interdisciplinary synthesis of theory and methods[END_REF].

Consequently, knowledge is not an object processed independently of the person who is acting. Formalized and codified knowledge which is independent from individual is no more than information, which may lead to different interpretations, notably due to the existence of meaning variance [START_REF] Arduin | On the Measurement of Cooperative Compatibility to Predict Meaning Variance[END_REF].

Regarding smart cities and their underlying knowledge ecosystems, we consider these three fundamental postulates: (i) Knowledge is not an objectit results of the interpretation by someone of information [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF]; (ii) Knowledge is linked to the actionit is necessary to realize processes, which in turn create knowledge [START_REF] Grundstein | Distinguishing Knowledge from Information: A Prerequisite for Elaborating KM Initiative Strategy[END_REF], and (iii) City's knowledge includes two main categories of knowledgetacit and made-explicit knowledge [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF].

Nevertheless, when knowledge has been made explicit, is stable, well defined, and recognized by a given specific and homogenous population, it can be considered as an object, and managed as information.

B. Decision

Even if it has been extended since it first description, the classic decision making process of [START_REF] Simon | The new science of management decision[END_REF] stays the same: intelligence, design and choice. When intelligence and design are carried out by more than one individual, the choice is harder to consensually achieve. In a city, smart city initiatives have to involve citizens in their construction processes. That is the reason why participatory governance is so much important in a smart city: choices are shared choices.

Thereby where decision making could be seen as a constructed model, collaborative decision making, for us, may be regarded as the use of socio-technical media in order to improve the performance and the acceptability of decision making. This is particularly crucial in the case of smart city initiatives. The location of knowledge in the realization of collaborative decision making has to be regarded insofar as, according to [START_REF] Gzara-Yesilbas | Towards a knowledge repository for collaborative design process: focus on conflict management[END_REF] (p. 348), knowledge sharing leads to improve collaborative decision making: a "static view" has to be structured and constitutes the "collaborative knowledge". Knowledge having an important role in individual decision making, we consider that for collaborative decision making, knowledge has to be shared [START_REF] Arduin | From knowledge sharing to collaborative decision making[END_REF].

[24] introduces the "analyst" as the person who makes explicit the problem for the decision maker. The analyst is living the processes, he/she interferes in them, he/she is a "participant observer" [START_REF] Jordan | Ethnographic workplace studies and computer supported cooperative work[END_REF]. Indeed, [START_REF] Stockdale | An interpretive approach to evaluating information systems: A content, context, process framework[END_REF] notices that neglecting social activity leads to "meaningless conclusions". This is particularly true in the case of smart city initiatives, where citizens are stakeholders. [START_REF] Giffinger | Smart cities: Ranking of European mediumsized cities[END_REF] proposes a ranking of 70 European medium-sized cities by using six characteristics: smart economy (competitiveness), smart people (human and social capital), smart governance (participation), smart mobility (transport and ICT), smart environment (natural resources), and smart living (quality of life); where a smart city is "a city well performing in a forward-looking way in these six characteristics, built on the 'smart' combination of endowments and activities of selfdecisive, independent and aware citizens" (p. 11). Each characteristic is defined by a number of factors (33 in total) and each factor is described by a number of indicators. The indicators that "describe the factors of a smart city are derived from public and freely available data" [START_REF] Giffinger | Smart cities: Ranking of European mediumsized cities[END_REF]. This classification was followed by some others.

III. SMART CITY MODELLING APPROACHES

[19] suggests three conceptual dimensions of a smart city: technology (use of ICT to transform life and work within a city), people (human infrastructure, human capital and education), and community (support of government and policy). For [START_REF] Pardo | Conceptualizing smart city dimensions of technology, people and institutions[END_REF] "a city is smart when investments in human/social capital and IT infrastructure fuel sustainable growth and enhance a quality of life, through participatory governance" (p. 286).

According to [START_REF] Dameri | Searching for Smart City definition: a comprehensive proposal[END_REF], with the EU vision, the concept of smart city is based on four basic elements that composed the city: Land (territorial dimension: not limited to the administrative boundaries of the city), Infrastructures (Buildings, streets, traffic and public transports), People (students, workers, neighbors, friends, tourists, etc.), and Government (local, central, or European level).

Finally, [START_REF] Chourabi | Understanding smart cities: An integrative framework[END_REF] proposes a framework for analyzing smart city projects with eight dimensions that affect the design, implementation, and use of smart cities initiatives: (1) Management and organization: Organizational and managerial factors such as project size, leadership or change management;

(2) Technology: Technological challenges such as lack of IT skills; (3) Governance: Factors related to the implementation of processes with constituents who exchange information according to rules and standards in order to achieve goals and objectives; (4) Policy context: Political and institutional components that represent various political elements and external pressures; (5) People and communities: Factors related to the individuals and communities, which are part of the socalled smart city, such as the digital divide or the level of education; (6) Economy: Factors around economic variables such as competitiveness, innovation, entrepreneurship, productivity or flexibility; (7) Built infrastructure: Availability and quality of the ICT infrastructure and (8) Natural environment: Factors related to sustainability and better management of natural resources.

From our point of view, the latter is the most comprehensive and integrative framework for analyzing smart city projects, i.e., it has a sufficient number of analysis dimensions to be implemented on the field while being open enough to manage a lack of field data.

IV. CASES OF PARIS AND NICE

According to the model of [START_REF] Chourabi | Understanding smart cities: An integrative framework[END_REF], in this section, we establish a picture of Smart City initiatives in Paris and Nice [START_REF] Abid | Etude d'un référentiel d'indicateurs caractérisant une smart city (Study of a repository of indicators characterizing a smart city)[END_REF].

A. Paris

Paris is the capital and most-populous city of France. The Paris region had a GDP of €612 billion in 2012, accounting for 30.1% of the GDP of France, and ranking it as one of the wealthiest five regions in Europe; it is the banking and financial center of France [31].

1) Management and organization:

The city of Paris has put in place: (i) a mission "Smart and Sustainable City" whose job is to identify transversal topics such as Smart City and work with the various departments concerned, (ii) a strategic plan "Paris: Smart and sustainable city, prospects for 2020" (since June 2015) [START_REF]Paris intelligente et durable perspectives 2020 et au-delà[END_REF], as a platform, scalable and participative taking into account the views of citizens, (iii) creation of a Chief Data officer (during 2015) to achieve the best of available data, and (iv) an effective resilience (participation in the group of the 100 most resilient cities, creating a Chief Resilient Officer, etc.).

2) Technology: From a technological point of view, there exist: (i) some applications for dematerializing public services (the Web site www.paris.fr, for example, helps citizens doing procedures, finding facilities, aids, jobs, parks, etc.), (ii) the extension ".paris" is available for lovers of Paris and for economic operators of the city, (iii) Public numeric spaces with Internet access to better integrate technologies in citizen every-day life, (iv) Open Data portal since 2011 (opendata.paris.fr).

3) Governance: The smart city initiative within Paris is mainly directed in a vision of citizen participation and coproduction. The concept of Open City is central to this initiative, the human is at the core of the device through continuous interaction on various platforms: (i) "Paris Petitions" to launch a petition and to sign on subjects within the town hall (ii) "Madam the Mayor, I have an idea" is a platform where citizens propose ideas that will be discussed for possible implementation (traffic problems, urban development, etc.), (iii) "In my street" to report any anomalies, dirt, clutter, etc. (iv) many websites to propose projects concerning the Paris of tomorrow.

4) Policy Context:

The urban vision of Paris as a digital and smart city has its roots in the project "Grand Paris", set out in 2007 by the former President of the Republic Nicolas Sarkozy. But it was only in 2012, following the election of the new President Francois Hollande that a concrete vision is presented through the project "Grand Paris Numérique", which aims to be an innovative and progressive project that would fast-tracks Paris to modernity and technology. Finally, in July 2014 (election of the new Mayor), a Mission "Smart and Sustainable City" is formally established in the town hall. The creation of this mission organizes the Parisian speech around the "smart city" in a specific public action. A major communication is organized around the project "smart and sustainable city" to keep all stakeholders informed within the city (citizens, businesses, public administrations, etc.) and also to take advantage of the discussions triggered around the subject.

5) People and communities:

Citizens are the core of the project "smart and sustainable city" of Paris. They are now sought on several topics of discussion to propose, vote and decide on major projects in the city. Paris has chosen to integrate citizens in the processes of decision and action in a co-production logic that leverages the collective intelligence and creativity. The smart city initiative has the major objective of improving the quality of life of citizens in all areas of daily life. The city of Paris, in conjunction with its economic players, has adopted a policy of communication and information exchange on all aspects of the practical life of its inhabitants, visitors and/or companies (real-time traffic, optimized paths, neighborhood maps, air quality, etc.).

6) Economy:

The smart city initiative aims to make Paris, one of the start-up capitals through several mechanisms put in place to encourage innovation and creation thanks to aids (i) to finance ("Paris Finance Plus", "Paris Initiative Amorçage", "Scientipôle Initiative"), (ii) the installation of foreign entrepreneurs ("Paris French Tech Ticket"), (iii) the installation in an incubator ("Paris Landing Pack"), (iv) awards for innovative projects, (v) the installation.

7) Built infrastructures include: (i) free WIFI and 4G, (ii) "Autolib", rental of self-service electric vehicles, (iii) "Velib", rental of self-service bicycles, (iv) housing construction: 70 000 units per year, (v) "Grand Paris Express" to add 4 subway lines and expand 2 existing lines by 2030, (vi) a "green" bus network in the next 10 years.

8) Natural environment:

Paris is experimenting and implementing several innovative practices in energy consumption and resource conservation as part of its commitment to reduce its emissions of greenhouse gas by 75% to 2050 (i) electric vehicles and bicycles, (ii) use of renewable energy (eco-district, smart grid etc.), (iii) circular economy (redistribution of unsold, composting, etc.).

B. Nice

Nice is the fifth most populous city in France, is the capital of the Alpes Maritimes French department and is located on the south east coast of France on the Mediterranean Sea. The Nice metropolitan area had a GDP amounting to $47.7 billion, and $34,480 per capita, slightly lower than the French average [30].

1) Management and organization:

Since 2008, Nice is engaged in a smart city strategy. Its strategy is essentially based on technology and on encouraging innovation to monitor and manage all elements of the city.

2) Technology: From a technological point of view, there exist: (i) "Spot Mairie" for dematerializing public services since 2013 (Self-service collaborative cubicle that aims to make life easier for citizens by bringing public services closer to citizen life places), (ii) "Cityzi" for contactless payments with smartphones (Near Field Communication technology), (iii) "NICE Park" for smart parking since 2011 (find a parking place from a smartphone and know the optimized path to get there taking into account the traffic, public transport schedules, weather data, …), (iv) "Intelligence operation center" for real-time collecting and analyzing data concerning citizen mobility, air and water quality, noise, energy, …, (v) Public numeric spaces with Internet access to better integrate technologies in citizen every-day life, (vi) Open Data portal since 2014 (www.opendata.nicecotedazur.org).

3) Governance: Nice wants to be a pioneer for smart city initiatives and builds on innovation and information and communication technology (ICT). Stakeholders are: (i) the metropolitan area of Nice and the various municipalities that constitute it, (ii) teaching and research actors, and (iii) many (local and national) industrial and technological partners.

4) Policy Context:

Since 2008 (the beginning of the mandate of the current mayor and president of the metropolitan area of Nice), Nice is engaged in a smart city strategy and has established several partnerships with global ICT players to be more "connected" and more open to new technologies. This strategy aims to provide an advanced technological environment to attract investments and contribute to economic growth.

5) People and Communities:

Citizens and visitors benefit from major urban and technological undertaken projects ("Spot Mairie", "Cityzi", etc.) to facilitate their daily life and ensure the good quality of their environment. In addition, solutions that will facilitate the monitoring of elderly or sick persons are being tested (sensors for monitoring patients at home for example).

6) Economy:

Several devices are in place to encourage entrepreneurship, installation of businesses and innovation: (i) the site "Invest in Cote d'Azur" (www.investincotedazur.com), (ii) the first European science park: "Sophia-Antipolis", competitiveness Pole bringing together businesses, start-ups and scientific research centers, (iii) the urban project "Eco Valley", (iv) lending facilities for the creation of businesses and (v) support structures for business financing.

7) Built infrastructures include: (i) free WIFI and 4G, (ii) "Autobleu", rental of electric vehicles, (iii) "Vélobleu", rental of self-service bicycles, (iv) urban monitoring with 2 000 captors (between 2014 and 2016) to collect then analyze environmental data, (v) "Boulevard connecté" (since 2013) where a specific boulevard is equipped with 200 captors to collect the filling level of paper and glass containers in order to adjust the tours of garbage collection trucks, the brightness level to adjust the public lighting to real lighting needs, the real-time traffic, etc., (vi) a new tram line connecting the port to the airport. 8) Natural environment: From natural environment, there are: (i) optimizing the collection of waste (level sensors in glass or paper containers to prevent unnecessary tours of trucks, trucks equipped with GPS to optimize tours and better distribute the work, etc.), (ii) provision of electric vehicles and bicycles (self-service), (iii) "REFLEXE" (since 2011), smart grid for production-storage-consumption of energy, (iv) "Nice Grid" for a smart solar quarter (since 2011).

C. Comparison

In this subsection, we compare Paris and Nice (Table I synthetizes some geographic and demographic information of these two French cities).

Paris and Nice are making considerable efforts, through various initiatives, to become more innovative and smarter. The geographic, demographic and economic differs depending on the city and this shapes the smart city initiative set up (see Table II).

Paris considers people as being at the core of the device and positions it as a beneficiary but essentially as an important actor in all phases of the initiative from the consultation to decision making. Nice, meanwhile, plans to turn into agglomeration pioneer for innovation and advanced technology. Smart city projects of Paris and Nice have the same goal which is to ensure the requirements of comfort and well-being of citizens. Nevertheless, the two cities diverge as to the philosophy they have adopted for this purpose. In both cases, municipalities offer several aids and devices that encourage innovation and investments. The technology is used to facilitate the lives of citizens and to collect the maximum of data that will be used to rethink the management methods and the city.

However, the involvement of citizens of Nice in the decision making processes concerning smart city initiatives is very small compared to Paris. However, the latter does not encourage and does not invest as much as Nice in the implementation of technological tools. This difference may be partly due to the specificities of each of the two populations.

Indeed, Nice is characterized by an aging population, with few strategic jobs, disadvantaged by the low level of qualification of its young, not very attractive demographically [START_REF]Métropole Nice Côte d'Azur -Une croissance démographique à retrouver[END_REF]. These characteristics are a handicap for the city hampering citizens' involvement in the consultation and proposal phases around the city development topics. Paris meanwhile can benefit from the skills, ideas and actions of its citizens as the city has the largest number of students, companies and managers in France. As Digital Knowledge Ecosystems, smart cities improve their decision making processes by taking into account stakeholders' knowledge.

The divergence between Nice and Paris also lies on the vision of politicians in power on technology and its uses. Paris cautiously envisaged an extensive use of sensors and cameras to monitor the city and its inhabitants; unlike Nice which is related in the media to George Orwell's Big Brother watching citizens by blaming its possible detriment to liberties and privacy.

In both cases, the recovery and the use of data collected in the urban space is not specified. The questions of ethic and direct impacts of the data on urban policies remain open. Both experiences have much to gain by observing the methods and achievements from each other to question their approaches and adopt the relevant ideas.

V. CONCLUSION

Living together within a smart city may be natural for people, but induces investments and governance obligations for municipalities. Cities are becoming smart, relying on digital and social networks.

In this paper, we explained our vision of knowledge and decision. Then we detailed some existing approaches concerning smart cities initiatives. We finally presented a comparison of two French (smart) cities: Paris and Nice.

Within cities, systems have to adapt to hyper-connected citizens, in a very particular environment, the one of cities in constant evolution where systems and humans are nested. The approach used in this paper to study the cases of Paris and Nice lead to formal and quantitative conclusions (see Table II). Nevertheless, we have to warn researchers and municipalities on the fact that such approach, as well as those presented in Section III, neglect one of the major questions in smart cities: the recovery and the use of data collected in the urban space.

TABLE I

 I

		.	PARIS AND NICE
		Paris		Nice
	Region	Ile-de-France		Alpes Maritimes
	Mayor	Anne Hidalgo (2014-2020)	Christian Estrosi (2014-2020)
	Area	105,4 km² (40,7 sq. mi)		71.92 km 2 (27.77 sq. mi)
	Population	2 241 346 (18,2% of France)	343,629
	Urban	10 516 110		1,005,230
	Density	21 000 / km² (55 000/sq. mi)	4,800 / km 2 (12,000/sq. mi)
	Rank	1 st in France		5 th in France
	Website	www.paris.fr		http://www.nice.fr
				http://www.nicecotedazur.org

TABLE II

 II

		.	COMPARISON OF PARIS AND NICE INITIATIVES FOR
			BECOMING SMART CITIES
			Paris		Nice
	Management	The citizen is the core of	No direct application of
	and		smart city device		citizens as actors or
	organization					partners of the initiative
	Technology		Use of technologies	Opening	in	several
			primarily	for	the	technological experiments
			dematerialization	of	in all fields
			services
	Governance		Citizens	through	Actors of teaching and
			suggestions, plans, ideas	research, businesses and
			contribute and shape the	ICT contribute
			Smart City initiative
	Policy context	Since 2012			Since 2008
	People	and	People are the central	The main objective of the
	Communities	pivot of the project:	initiative is the human. The
			solicited and involved	technology is at the service
			upstream	and	of the welfare of citizens.
			beneficiaries
			downstream.
	Economy		Both cities have implemented several features to attract
			investments and encourage the holders of ideas and
			start-ups to settle.
	Built		Poorly	developed	Most	developed
	infrastructure	technological			technological
			infrastructures			infrastructures
	Natural		Energy consumption and	Environmentalist
	environment		resource conservation


Philippe Grillot

Simona Mancini

Michèle Grillot
email: michele.grillot@univ-orleans.fr-philippe.grillot@univ-orleans.fr-simona.mancini@univ-orleans.fr

Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects

Keywords: degenerate parabolic system, existence of solutions, stability, pattern formation, aggregation Mathematical Subject Classification: MSC 92C15 MSC 82B21 MSC 35J70

Introduction

The study and modeling of the adhesion process between cells is an active and complex research subject both in biology and in mathematics, and has implications, for example, with the cancer growth and the formation of neuronal connections in the embryon (see for example [START_REF] Mège | Regulation of cell-cell junctions by the cytoskeleton[END_REF], [START_REF] Lambert | Nucleation and growth of cadherin adhesions[END_REF], [START_REF] Wu | Transforming binding affinities from three dimensions to two with application to cadherin clustering[END_REF], [START_REF] Anguige | A one-dimensional model for cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion[END_REF], [START_REF] Colin | Modeling of the migration of endothelial cells on bioactive micro-patterned polymers[END_REF] and [START_REF]Cell mechanics: From single scale-based models to multi-scale modeling[END_REF]). The main protein involved in the growth of adhesion junctions between cells is called cadherin. Cadherins diffuse on the cell membrane and may link to other cadherins on the same membrane (cis-contact) as well as bind to other cadherins on another cell membrane (trans-contact). These localized contacts can then evolve and give rise to larger adhesion domain, called adhesion junctions. These adhesion junctions are linked to actin filaments and help to their polymerization and growth, which in turns leads to the cell migration. In the experimental framework we consider, the link with the actin filaments is inhibited, and therefore the adhesion junctions and the binding cadherins are fixed in space. More precisely, experimentally, a cell is left to expand on a microscope slide covered by a uniform substrate of fixed cadherin targets which mimic the membrane of a second cell. Thus, when a diffusing cadherin link to a target it stops diffusing and fixes on the position of the given target. It appears that, even without the influence of the actin filaments, contacts between cadherins are easier created next to already existent ones, this local effect creates aggregates of linked cadherins enforcing the cell-cell adhesion junctions. On the other hand, if the strength of the contact is not large enough, the initial link may break and the cadherin diffuses again.

From now on, we shall consider cadherins as particles and we describe their spatio-temporal dynamics on the substrate by a mathematical model at the mesoscopic scale. The unknown is the couple of density functions (u, v) representing the particle distributions of the diffusing particles, u = u(x, t), and of the fixed ones, v = v(x, t), where x ∈ Ω ⊂ R 2 is the bi-dimensional space position and t ≥ 0 is time, Ω is the bounded domain where the evolution takes place. We describe the evolution of the population of the diffusing particles, u, by a reaction-diffusion equation, and the evolution of the population of the fixed particles, v, by an non-linear ordinary differential equation, since the diffusion coefficient for v is null. Hence we are dealing with a degenerate system of reaction-diffusion equations. The reaction term represents the change of state of a particle, form free to fixed and vice-versa. It is thus composed of two terms, a gain one describing the growth of the fixed particle population v, and a loss one giving the amount of broken links. The gain term is proportional to the number of free targets (i.e. those targets which have no particle linked to), to the density of diffusing particles and to the aggregation probability function describing the probability of a diffusing particle to link to a target. The loss term is proportional to the density of fixed particles and to a unbinding probability function. The mathematical analysis of this problem is done in a general framework, but in the sequel we prove that the simple definition of the aggregation function as an affine function of v, and of the unbinding function as constant, is not in agreement with the experimental observations, since there is no pattern formations, so that no isolated adhesion junctions can be obtained.

In this paper, we are mainly interested by the theoretical study of the solutions (u, v). The main results we prove are the existence and uniqueness of solutions to the degenerate reactiondiffusion system and their positivity and boundedness. We also discuss the existence of solutions to the stationary problem and their stability in other to prove the existence of pattern formations. All these results are interesting in order to show that the proposed model makes sense in the biological framework and that it is, or not, consistent with the experimental results. For instance, we show that the simple affine aggregation function and the constant unbinding probability are not appropriate to recover the results shown by experiments. In fact, for this simple choice, the stationary solution turns out to be homogeneous in space and stable with respect to small perturbations, so that no pattern formation is possible, which would correspond in our model to no formation of isolated particle aggregates. A non-linear reaction term based on a more detailed description of the aggregation and unbinding phenomena, together with the comparison with biological experimental results, is given in [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF]. Although a large number of parabolic, degenerate systems or not, were already studied (see for exemple [START_REF] Amann | Highly degenerate quasilinear parabolic systems[END_REF][START_REF] Amann | Global existence for a class of highly degenerate parabolic systems[END_REF][START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF][START_REF] Rodriguez-Bernal | Attractors for partly dissipative reaction diffusion systems in R n[END_REF][START_REF] Quittner | Blow-up, Global Existence and Steady States[END_REF] and as well as the references therein), their results do not directly apply to our model because of the particular form of our reaction term. Note, in particular, that our reaction term behaves like a polynomial which doesn't satisfy the bound of [START_REF] Amann | Global existence for a class of highly degenerate parabolic systems[END_REF].

The paper is organized as follows. In section 2 we detail the mathematical model describing the particles dynamics and we announce the main results. Section 3 is devoted to the proofs. In section 4 we study the stability of the stationary solution to the problem. Finally, in section 5, we consider the particular case of an affine aggregation function and constant unbinding probability, and show that the stationary solution in this case is stable. This result is also illustrated by some numerical simulation.

The model and main result

We consider a bounded regular bi-dimensional space Ω ⊂ R 2 . Then, the unknown of our problem is the couple of density functions (u, v) = (u(t, x), v(t, x)) of both the diffusing and fixed particles. For instance, u = u(t, x) (respectively v = v(t, x)) represents the probability to find a diffusing (respectively fixed) particle at time t ≥ 0 in the position x ∈ Ω. Note that x is a two dimensional vector point. We assume the targets density on the substrate to be constant in time and space, and we denote it by ρ so that, up to a normalisation, 0 < ρ ≤ 1. From a biological point of view, it make sens to assume that there can't be more fixed particles than targets. Moreover, since u and v are density distributions, they must be non-negative. Thus it must be, for all time t ≥ 0 and all x ∈ Ω : 0 ≤ u(x) and 0 ≤ v(x) ≤ ρ.

(1)

Note that v(t, x) = ρ holds if and only if we have the critical situation in which fixed particles never unlink from the target. Recalling that, free particles diffuse and become fixed when linking to a target, and that fixed ones don't diffuse and become free when unbinding from a target, the respective density functions u and v must then satisfy the following degenerate system of reaction-diffusion equations :

∂ t u -σ∆u = -Q(u, v) ∂ t v = Q(u, v) (2)
in (0, +∞) × Ω, where σ > 0 is the diffusion coefficient.

In [START_REF] Amann | Global existence for a class of highly degenerate parabolic systems[END_REF], the function Q(u, v) is the reaction term which describes how particles pass from a state to the other one: diffusing to fixed and vice-versa. As explained previously, see section 1, the reaction term Q is composed by the difference of a gain term Q + , counting the diffusing particles that became fixed, and of a loss term Q -, counting the fixed particles that became free. The gain term Q + is thus proportional to the density of free targets ρ -v, to the density of free particles u and to an aggregation probability function F = F (v). Analogously, the loss term Q -is assumed to be proportional to the density of fixed particles v, and to the unbinding probability G = G(v). Hence Q is given by:

Q(u, v) = Q + (u, v) -Q -(u, v) = u(ρ -v)F (v) -vG(v) (3)
Since (1) should hold, let us consider v ∈ [0, ρ], and fix the assumption we need on F and G as follows:

a) ∀ v ∈ [0, ρ], 0 < F (v), G(v) ≤ 1
c) ∀ v ≤ 0, F (v) = F (0) , G(v) = G(0) d) ∀ v ≥ ρ, F (v) = F (ρ) , G(v) = G(ρ)
Note that, in the biological framework, the aggregation function F must be increasing in v, but this condition is not required for the existence positivity and boundedness proofs. In Section 5 we'll consider a particular case in which F is a strictly increasing affine function for v ∈ [0, ρ] and G is constant (see ( 30) and ( 31)). With these choices F and G satisfy assumptions a) and b), but also relation [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF], leading to a unique homogeneous stable solution at equilibrium.

Since Ω is bounded, the first equation in [START_REF] Amann | Global existence for a class of highly degenerate parabolic systems[END_REF] needs to be endowed by some boundary conditions. We choose to describe them by Neumann boundary conditions:

∂u ∂ν (t, x) = 0, (4)
biologically implying a zero flux on the boundary ∂Ω. Further, we complete system (2) by the initial condition, for x ∈ Ω, (u 0 , v 0 ) = (u(0, x), v(0, x)) representing the initial distribution of both diffusing and fixed particles. We impose u 0 and v 0 continuous in Ω and such that (1) holds ∀ x ∈ Ω, that is:

0 ≤ u 0 (x) and 0 ≤ v 0 (x) ≤ ρ. (5)
Since u 0 is a continuous function on a bounded domain Ω, we define its maximum on Ω as:

µ = max x∈Ω u 0 (x). (6)
Up to a multiplicative factor, we can assume the total initial density of particles to be normalized to 1, that is :

Ω u 0 (x) + v 0 (x) dx = 1 (7)
Since particles aren't loss or created, but just change their status from free to fixed and viceversa, (7) must hold at any time t ≥ 0, that is the following conservation property must hold:

Ω u(t, x) + v(t, x) dx = 1, ∀ t ≥ 0. ( 8
)
Proposition 1 The solution of system (2) completed by the boundary conditions (4) and with an initial data satisfying [START_REF] Colin | Modeling of the migration of endothelial cells on bioactive micro-patterned polymers[END_REF], satisfy the conservation property [START_REF] Ladyzenskaja | ceva : Linear and quasi-linear equations of parabolic type[END_REF].

Proof: Summing the equations in (2), integrating over Ω, and applying the Green formula, we get :

∂ ∂t Ω (u + v)dx -σ ∂Ω ∂u ∂ν dσ = 0,
and thanks to the boundary conditions ( 4) we obtain:

∂ ∂t Ω (u + v)dx = 0,
concluding the proof.

Following [START_REF] Bedjaoui | Critical blowup exponents for a system of reaction-diffusion equations with absorption[END_REF], by a classical solution (u, v) of ( 2), we mean (u, v) 2), where T ∈ (0, ∞] denotes its maximal existence time. Before enter in the details of the analysis of the solutions to system (2), we state here the main results, which will be proved in the next sections. We start by a result important from a biological point of view, since it ensures that the solutions to (2) remains non-negative and bounded.

∈ (C([0, T ) × Ω)) 2 ∩ (C 1,2 ((0, T ) × Ω) × C 1,0 ((0, T ) × Ω)) solution of (
Theorem 1 Assume that the initial data (u 0 , v 0 ) satisfies (5), then a classical solution (u, v) to (2) verifies ∀ t > 0, ∀ x ∈ Ω: 0 ≤ u(t, x) ≤ µ + ρt , 0 ≤ v(t, x) ≤ ρ. (9)
We express now our main result , which is also important since it ensures the existence of solutions to system (2).

Theorem 2 Let T > 0, let F and G satisfy conditions a) to d) and consider an initial data with both u 0 = u 0 (x) and v 0 = v 0 (x) in C 0 (Ω) and satisfying (5) for all x ∈ Ω. Then there exists two non-negative functions u ∈ W 1,p ((0, T ) × Ω), p > 1 and v ∈ L ∞ ((0, T ) × Ω), solutions to (2)-( 4) endowed by the initial conditions:

u(0, .) = u 0 and v(0, .) = v 0 (10)
Note that we don't have the unicity in Theorem 2. If we add some monotonicity conditions on F and G, then we have the following:

Theorem 3 Let T > 0, let F and G satisfy conditions a) to d) and be such that v → -u(ρ - v)F (v) + vG(v) is nondecreasing.
Let us consider an initial data (u 0 , v 0 ) such that both u 0 and v 0 are in C 0 (Ω) and satisfying (5) for all x ∈ Ω. Then there exists a unique solution (u, v) of (2)-( 4) and [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF],

with v(t, x) = 0 ∀(t, x) ∈ (0, T ) × ∂Ω and such that u ∈ C([0, T ) × Ω) ∩ C 1,2 ((0, T ) × Ω) and v ∈ C([0, T ) × Ω) ∩ C 1,0 ((0, T ) × Ω) are non-negative functions.
Moreover, note that both Theorem 1 and Theorem 3 imply that the solutions exist on (0, ∞) × Ω.

Once proved that the problem is well posed, we will consider (see section 4) the existence and stability of solutions at equilibrium, that is of solution of the stationary problem associated to (2)-( 4). We will prove that there exist at least one stationary solution which is homogeneous in space and satisfying [START_REF] Amann | Highly degenerate quasilinear parabolic systems[END_REF]. Moreover, under the condition:

ρF (v)G(v) + v(ρ -v)(F (v)G (v) -F (v)G(v)) > 0, (11)
we obtain uniqueness of the stationary homogeneous solution, which we denote by (U, V ), and also its stability.

Finally, defining F (v) by ( 30), and G(v) by ( 31), we will show that , the stationary solution (U, V ) is stable with respect to small perturbations, and thus we can't have formation of aggregate of particles. This can be overcome choosing a non-linear aggregation function and also a non-linear unbinding rate, as it is done in [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF].

Existence, uniqueness and boundedness

We consider first the positivity and boundedness of the solution (u, v) to ( 2), assuming that it exists. We want thus to prove Theorem 1.

Proof of Theorem 1

We shall detail the proof in three steps. First we prove the upper bound for v. Second we prove the bound for u. Third we prove the positivity of both u and v.

Proof:

Step I. We first prove by contradiction the upper bound for v, v ≤ ρ. Let us define the set

A = {t > 0 : ∃ x ∈ Ω such that v(t, x) > ρ},
and assume that there exists a time t 0 > 0 and a point x 0 ∈ Ω such that v(t 0 , x 0 ) > ρ. Hence, t 0 ∈ A and A is bounded from below by 0, so that it exists t 0 = inf A. Note that, by means of continuity arguments, for this time t 0 , it exists a point x 0 ∈ Ω such that v(t 0 , x 0 ) = ρ. Now, considering the equation for v in (2) and computing its value at the point (t 0 , x 0 ), we get, because of assumption a),

∂ t v(t 0 , x 0 ) = -G(ρ)ρ < 0.
Therefore, the function v(t, x 0 ) is decreasing in time in a neighborhood of t 0 . Hence, it exists η > 0 such that for all t ∈ [t 0 -η, t 0 ] we have v(t, x 0 ) > v(t 0 , x 0 ) = ρ. Thus t 0 -η ∈ A and t 0 -η < t 0 , contradicting the fact that t 0 = inf A. Hence, t 0 doesn't exists and A = ∅.

Step II. Having that for all time t ≥ 0 and all x ∈ Ω, v ≤ ρ, we now consider the upper bound for u, u ≤ µ + ρt.

We prove that the function (u -µ -ρt) + = max(0, u -µ -ρt) is equal to zero for all time t ≥ 0 and all x ∈ Ω, where µ is defined in [START_REF] Chueh | Positively invariant regions for systems of nonlinear diffusion equations[END_REF].

Note that if (u -µ -ρt) + = 0, then u ≥ µ + ρt, so that u > 0. Using the equation for u in (2), and since, from a), G(v) ≤ 1 and, from Step I, v ≤ ρ, we have :

∂ t (u -µ -ρt)(u -µ -ρt) + -σ∆(u -µ -ρt)(u -µ -ρt) + = (∂ t u -σ∆u)(u -µ -ρt) + -ρ(u -µ -ρt) + = -u(ρ -v)F (v)(u -µ -ρt) + + (vG(v) -ρ)(u -µ -ρt) + ≤ 0.
Hence, integrating over Ω :

1 2 d dt Ω ((u -µ -ρt) + ) 2 dx + σ Ω |∇(u -µ -ρt) + | 2 dx ≤ 0,
so that, for all t ≥ 0, it must be

(u -µ -ρt) + 2 2 ≤ (u(0, x) -µ) + 2 2 = 0.
Concluding this second step of the proof.

Step III. Let (u, v) be a solution of (2), under hypothesis (5) we prove that v ≥ 0 and u ≥ 0 for all times t ≤ T . Let T ∈ (0, T ). Multiplying equations of ( 2) respectively by u -= min(u, 0) and v -= min(v, 0), integrating on Ω and using Green's formula, we obtain:

1 2 d dt Ω [(u -) 2 + (v -) 2 ]dx + σ Ω |∇(u -)| 2 dx = - Ω (u -) 2 (ρ -v)F (v)dx + Ω v + G(v)u -dx + Ω v -G(v)u -dx + Ω u + v -(ρ -v)F (v)dx + Ω u -v -(ρ -v)F (v)dx - Ω (v -) 2 G(v)dx
where we have used that u = u + + u -and v = v + + v -. Because of a), c) and d), because of

Step I, and since u + v -≤ 0 and u -v + ≤ 0, we get:

1 2 d dt Ω [(u -) 2 + (v -) 2 ]dx ≤ Ω (ρ -v)F (v)[-(u -) 2 + u -v -]dx + Ω G(v)[-(v -) 2 + u -v -]dx. But u -v -≤ [(u -) 2 + ((v -) 2 ]
, implies:

d dt Ω [(u -) 2 + (v -) 2 ]dx ≤ Ω (ρ -v)F (v)(v -) 2 dx + Ω G(v)(u -) 2 dx. (12)
Recalling that v ∈ C([0, T ) × Ω), we denote by m = m(v, T ) the minimum of v in [0, T ] × Ω, and using the upper bound in a), we deduce from ( 12) that:

d dt Ω [(u -) 2 + (v -) 2 ]dx ≤ max(ρ -m, 1) Ω [(u -) 2 + (v -) 2 ]dx.
Since u 0 ≥ 0 and v 0 ≥ 0, the Gronwall's lemma allows us to conclude that

Ω [(u -) 2 + (v -) 2 ] ≤ 0,
so that nor u nor v can be negative on [0, T ] × Ω .

Proof of Theorem 2 and Theorem 3

We prove here the existence of solution to problem ( 2)-( 4) and [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF], by constructing two Cauchy sequences (u n ) n≥0 and (v n ) n≥0 converging to u and v solution to ( 2)-( 4) and ( 10).

Let T > 0 and let u 0 and v 0 be two continuous function on Ω and satisfying [START_REF]Cell mechanics: From single scale-based models to multi-scale modeling[END_REF]. For all (t, x) ∈ [0, T ] × Ω, let us define two sequences (u n ) n≥0 and (v n ) n≥0 by:

                 ∂ t u n+1 -σ∆u n+1 = -Q(u n+1 , v n ) in (0, T ) × Ω ∂ t v n+1 = Q(u n , v n+1 ) in (0, T ) × Ω ∂u n+1 ∂ν = 0 on (0, T ) × ∂Ω u n+1 (0, .) = u 0 in Ω v n+1 (0, .) = v 0 in Ω, (13)
with u 0 (t, x) = u 0 (x) and v 0 (t, x) = v 0 (x).

Note that those functions well exist and are continuous, in fact the first equation is a linear heat equation in u with a continuous coefficient (ρ -v n )F (v n ) and the second equation is a Riccati equation in time.

We'll prove that (u n ) n≥0 and (v n ) n≥0 are two Cauchy sequences, thus converging to two functions u and v, which are solution to (2)-( 4) and [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF]. We first need some technical results.

Proposition 2 Under the assumptions of the theorem both functions u n and v n defined by ( 14) and (13) satisfy the bound of Theorem 1, see [START_REF] Lambert | Nucleation and growth of cadherin adhesions[END_REF], that is:

0 ≤ u n (t, x) ≤ µ + ρt and 0 ≤ v n (t, x) ≤ ρ
for all (t, x) ∈ (0, T ) × Ω.

Proof:

The proof of this proposition is by induction using the same arguments as in the proof of Theorem 1. We omit it.

The goal is to prove that both sequences (u n ) and (v n ) are Cauchy sequences. For simplicity of notations, for n ∈ N * and t ∈ (0, T ), let us define by U n (t) and V n (t) the squares of the

L 2 -norms u n+1 -u n 2 L 2 (Ω) and v n+1 -v n 2 L 2 (Ω) , so that: U n (t) = Ω (u n+1 -u n ) 2 dx , V n (t) = Ω (v n+1 -v n ) 2 dx. (15)
We first prove some technical results.

Proposition 3 Under the assumptions of Theorem 2, there exists a constant k > 1 such that for all n ∈ N * and t ∈ (0, T ) :

U n (t) ≤ 3k U n (t) + k V n-1 (t) (16)
and

V n (t) ≤ 3k V n (t) + k V n-1 (t). ( 17
)
Note that ( 16) and ( 17) are completely analogous.

Proof :

The function Q(r, s) defined by ( 3) is a Lipschitz function in [0, γ] × [0, ρ], where

γ = µ + ρT.
Therefore there exists a constant k = k(γ, ρ) > 1 such that for all ((r, s), (r , s

)) ∈ ([0, γ]×[0, ρ]) 2 : |Q(r, s) -Q(r , s )| ≤ k(|r -r | + |s -s |). ( 18
)
Since both functions u n and u n+1 satisfy (13), we have:

∂ t (u n+1 -u n ) -σ∆(u n+1 -u n ) = -Q(u n+1 , v n ) + Q(u n , v n-1 ) (19)
in (0, T ) × Ω. Multiplying ( 19) by (u n+1 -u n ) and integrating on Ω, we obtain for all t ∈ (0, T ):

1 2 d dt Ω (u n+1 -u n ) 2 dx + σ Ω |∇(u n+1 -u n )| 2 dx (20) = Ω (Q(u n , v n-1 ) -Q(u n+1 , v n )) (u n+1 -u n ) dx.
Because of ( 18) and ( 15) we deduce from (20) :

1 2 U n (t)≤ k Ω (|u n+1 -u n | + |v n -v n-1 |) |u n+1 -u n |dx ≤ k Ω |u n+1 -u n | 2 dx + k 2 Ω |v n -v n-1 | 2 + |u n+1 -u n | 2 dx ≤ 3k 2 U n (t) + k 2 V n-1 (t)
which implies [START_REF] Rodriguez-Bernal | Attractors for partly dissipative reaction diffusion systems in R n[END_REF]. The proof of ( 17) is completely similar because the second term of the first member of (20) doesn't play any role.

The differential inequalities of Proposition 3 allow us to obtain over-bounds inequalities of U n and V n : Proposition 4 Under the assumptions of Theorem 2, we have for all n ∈ N * and t ∈ (0, T ) :

U n (t) ≤ kL ke 3kT n t n n! (21)
and similarly :

V n (t) ≤ kL ke 3kT n t n n! (22)
where k ≥ 1 is the Lipschitz constant of Q in (18) and L is defined by :

L = 4[max(γ, ρ)] 2 |Ω|,
where we denote by |Ω| the measure of Ω that is, |Ω| = Ω dx.

Proof:

We prove it by induction.

For n = 1, integrating ( 16) on (0, T ) for t ∈ (0, T ), we have:

U 1 (t) -U 1 (0) ≤ 3k t 0 U 1 (s)dx + k t 0 V 0 (s)ds.
Recalling ( 13) and [START_REF] Quittner | Blow-up, Global Existence and Steady States[END_REF], it is easily seen that U 1 (0) = 0. Note that by the same arguments we have U n (0) = 0 for all n ∈ N * . On the other hand, Proposition 2 implies:

V 0 (s) = Ω (v 1 (s, x) -v 0 (s, x)) 2 dx ≤ Ω (|v 1 (s, x)| + |v 0 (s, x)|) 2 dx ≤ 4ρ 2 |Ω|.
Thus, for all t ∈ (0, T ) :

U 1 (t) ≤ 4ρ 2 |Ω|kt + 3k t 0 U 1 (s)ds.
Gronwall's lemma leads us to the following inequality, for all t ∈ (0, T ) :

U 1 (t) ≤ 4ρ 2 |Ω|kt + 3k t 0 4ρ 2 |Ω|kse 3k(t-s) ds.
Integrating by parts we obtain for all t ∈ (0, T ) :

U 1 (t) ≤ 4kρ 2 |Ω|
t 0 e 3k(t-s) ds.

Since k ≥ 1 and since s ∈ [0, t] ⊂ [0, T ], we have t -s ≤ T and:

U 1 (t) ≤ 4kρ 2 |Ω|e 3kT t ≤ Lk 2 e 3kT t.
In the same way we have for all t ∈ (0, T ) :

V 1 (t) ≤ Lk 2 e 3kT t.
We prove now the induction step. Assume that ( 21) and ( 22) hold for a fixed n ≥ 1, that is, for all t ∈ (0, T ):

         U n (t) ≤ Lk n+1 e 3kT n t n n!
V n (t) ≤ Lk n+1 e 3kT n t n n! From Proposition 3, since ( 16) and ( 22) hold, we have for all t ∈ (0, T ) :

U n+1 (t) ≤ 3k U n+1 (t) + Lk n+2 e 3kT n t n n!
Therefore, integrating on (0, t) and recalling that U n+1 (0) = 0, we have:

U n+1 (t) ≤ 3k t 0 U n+1 (s)ds + Lk n+2 e 3kT n t n+1 (n + 1)! .
Gronwall's lemma implies for all t ∈ (0, T ) :

U n+1 (t) ≤ Lk n+2 e 3kT n t n+1 (n + 1)! + 3kLk n+2 e 3kT n t 0 s n+1 (n + 1)! e 3k(t-s) ds.

Integrating by parts we obtain for all t ∈ (0, T ):

U n+1 (t) ≤ Lk n+2 e 3kT n t 0 s n n! e 3k(t-s) ds,
and since t -s ≤ t ≤ T , we deduce for all t ∈ (0, T ):

U n+1 (t) ≤ Lk n+2 e 3kT (n+1) t n+1 (n + 1)! .
In the same way we obtain

V n+1 (t) ≤ Lk n+2 e 3kT (n+1) t n+1 (n + 1)! ,
which ends the proof.

We now have all the ingredients to prove that (u n ) n≥0 and (v n ) n≥0 are Cauchy sequences.

data (u 0 , v 0 ) satisfying [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF]. The first step is to determine the solutions to the stationary problem associated to ( 2)-( 4), which reads, for x ∈ Ω:

       ∆u = 0 in Ω ∂u ∂ν = 0 on ∂Ω Q(u, v) = 0 in Ω. (23)
These solutions are interesting since they give the solution at equilibrium of the considered problem. We have the following:

Proposition 5 Let |Ω| = 1
, under assumptions a) to d), there exists at least one couple (U, V ) solution of problem (23) satisfying the normalization condition (8) and bounds [START_REF] Amann | Highly degenerate quasilinear parabolic systems[END_REF]. Such couple (U, V ) satisfies, for V ∈]0, ρ[:

U = V G(V ) (ρ -V )F (V ) (24)
Moreover, the unicity of the couple (U, V ) holds if the additional condition [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF] is satisfied for all v ∈ (0, ρ).

Proof :

Multiplying the first equation by u, integrating on Ω and applying the Green formula, we obtain that the solution u of the first equation in ( 23) must be homogeneous in Ω, that is u(x) = U ∈ R for all x ∈ Ω. Replacing u = U in the last equation of (23), we then obtain that, if it exists, the solution v of Q(U, v) = 0 must be constant too, that is v(x) = V ∈ R for all x ∈ Ω. From the relation Q(U, V ) = 0 we easily deduce (24). We still have to prove the existence of such a solution satisfying (1) and [START_REF] Ladyzenskaja | ceva : Linear and quasi-linear equations of parabolic type[END_REF]. From the normalization condition [START_REF] Ladyzenskaja | ceva : Linear and quasi-linear equations of parabolic type[END_REF], and since |Ω| = 1, we get that U = 1 -V . So that V is implicitly defined by Q(1 -V, V ) = 0 which leads to the relation :

(1 -V )(ρ -V )F (V ) -V G(V ) = 0. ( 25
)
We then seek for a value V ∈]0, ρ[ satisfying (25). Let us define the following real function h(v) for v ∈]0, ρ[:

h(v) = vG(v) (ρ -v)F (v)
.

Then (25) reads:

h(v) + v -1 = 0.
Considering the limit values of the above functions at the boundaries v = 0 and v = ρ, we have :

h(0) + 0 -1 = -1 < 0 and lim v→ρ -h(v) + v -1 > 0.
So by means of continuity arguments we can conclude that there exists at least one value V ∈]0, ρ[, such that (25) is satisfied. Moreover, since U can be defined by (24), and thanks to assumption a) on F and G, it easily follows that U > 0. Thus both U and V satisfy (1).

To have uniqueness for V , we need h(v) to be strictly monotone. This is true if, for all v ∈]0, ρ[:

h (v) = ρF (v)G(v) + v(ρ -v) (F (v)G (v) -F (v)G(v)) (ρ -v) 2 F (v) 2 > 0.
Thus, if [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF] holds, h(v) is strictly increasing in (0, ρ), and so is the function h(v) + v -1, concluding the proof.

It is well known that for some reaction-diffusion systems, some of the stationary solutions, or equilibrium points, for some ranges of the parameters, may not be stable, and that other non-homogeneous equilibrium structures may be found. This is known in morphogenesis as the Turing instabilities and happens in particular when the ratio of the diffusion coefficients is very small, see for exemple [START_REF] Murray | Mathematical Biology I. An Introduction[END_REF][START_REF] Murray | Mathematical Biology II. Spatial Models and Biomedical Applications[END_REF]. In our problem, one of the diffusion coefficient is zero, and we are then interested to study the stability of the homogeneous stationary solution (U, V ), to see under which conditions patterns formation may occur. We have the following result: Proposition 6 Under hypothesis a) to d), and if the homogeneous stationary solution (U, V ) satisfies [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF], then (U, V ) is a stable equilibrium solution of system (2)-( 4).

Proof :

In order to study the stability of the equilibrium couple (U, V ), we must consider the linearized problem associated to ( 2)-( 4) around the equilibrium (U, V ). We need to evaluate the derivates of the reaction term Q(u, v) with respect to u and v in (U, V ). Denoting these terms respectively by :

A = ∂ u Q(U, V ) and B = ∂ v Q(U, V ),
and recalling that U may be expressed as in (24), we get:

A = (ρ -V )F (V ), B = -V (ρ -V ) (F (V )G (V ) -F (V )G(V )) -ρF (V )G(V ) (ρ -V )F (V ) .
Since hypothesis a) to d) and [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF] hold, then for V ∈]0, ρ[, A > 0 and B < 0 .

The linearized problem then reads :

∂ t ũ = σ ∆ũ -Aũ -Bṽ ∂ t ṽ = Aũ + Bṽ (26)
where ũ and ṽ are approximation of u and v such that u = U + ũ and v = V + ṽ. To solve this problem, under the Neumann boundary condition, we look to solution (ũ, ṽ) defined under the form : ũ = e λt û(k), ṽ = e λt v(k),

where û(k) and v(k) are the Fourier transforms of ũ and ṽ (i.e. k > 0 is the Fourier wavenumber), and λ are the eigenvalues which determines the temporal growth. It can easily be seen from ( 27) that, if λ < 0, then both ũ and ṽ goes to zero when t goes to infinity, so that the solution (u, v) converges back to the stationary homogeneous solution (U, V ), which is thus stable: a small perturbation of (U, V ) doesn't affect its stability. We thus want to prove that the eigenvalues λ are negative. Replacing (27) in (26) and simplifying e λt , we get, for each k, the following linear system :

(λ + A + σk 2 )û + Bv = 0 -Aû + (λ -B)v = 0. ( 28
)
Since we look for nontrivial solutions (û, v) to (28), the determinant of the associated matrix must be zero, so that the values for λ are determined has solutions of the characteristic polynomial associated to (28). That is, for each k, λ must solve :

λ 2 + λ(σk 2 + A -B) -Bσk 2 = 0. That is, λ = 1 2 -(σk 2 + A -B) ± (σk 2 + A -B) 2 + 4Bσk 2 (29)
Since B < 0 and

(σk 2 + A -B) 2 + 4Bσk 2 = A 2 + 2Aσk 2 -2AB + (B + σk 2 ) 2 > 0,
there always exist two real solutions to (29). Finally, since (σk 2 + A -B) > 0, in order to have λ < 0, we just have to check that the solution with the plus sign is negative. This holds if,

(σk 2 + A -B) 2 + 4Bσk 2 < (σk 2 + A -B),
which is true for all wavenumber k > 0, since under hypothesis c), B < 0, and conclude the proof.

A simple application

Getting back to the biology problem we inspired from, in this section we assume F (v) to be an increasing and bounded affine function with respect to v, modeling the fact that the probability of a particle to be linked is larger when there already are fixed particles in its neighborhood :

F (v) =        a/(a + bρ) if v < 0 a + bv a + bρ if 0 ≤ v ≤ ρ 1 if v > ρ (30)
where a > 0 represents the probability of a particle to naturally link to a target (i.e. if no other particle is fixed nearby) and b > 0 is a parameter which control the influence of other fixed particles in the neighborhood. Concerning G(v), we choose it to be homogeneous with respect to v :

G(v) = ε , ∀ v ∈ R, (31)
where ε > 0 is the unbinding rate.

Proposition 7 System (2)-( 4), endowed by a continuous initial data (u 0 , v 0 ) satisfying (1) and (8), and with F (v) and G(v) respectively defined by (30) and (31) has a unique homogeneous stable solution at equilibrium given by the couple (U, V ), with U ≥ 0 defined by (24) and V ∈ ]0, ρ[.

Proof :

It is clear that both F and G satisfy hypothesis a) to d). Hence the existence of the solution and relation (1) are ensured by Theorem 2 and Theorem 1. Moreover, the existence of the stationary homogeneous solution is proved in Proposition 5.

Concerning the stability of the stationary solution, from Proposition 6 we just have to prove [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF]. Since G (v) = 0, dividing [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems[END_REF] by G(V ) > 0, we obtain the stability of (U, V ) if the condition F (V ) > 0 holds. And this is the case, since F (v) = b > 0 for all v ∈]0, ρ[. Concluding the proof.

To conclude, we perform some numerical simulations defining the aggregation function F (v) as in (30) and the unbinding function as in (31). In Figure 1 we show the convergence of the distribution function v to the homogeneous function v(x) = V when the initial data is defined as a small perturbation of the stationary solution V : The evolution of the solution (u, v) towards the stable equilibrium state for a C 1 (Ω) initial data (u 0 , v 0 ) is shown in Figure 2. We choose here a sinusoidal initial data defined by : u 0 (x) = 1 -v 0 (x), v 0 (x) = 0.4 + (0.2 sin(3πx) cos(3πy)).

v 0 = V ± ṽ,
Both particle distributions evolves to a homogeneous stationary state, which density value can be computed numerically solving h(v) + v -1 = 0 and is given, for our choice of parameters, by V = 0.3107435 and U = 0.6892565. This result shows that this simple local model of aggregation can't produce clusters as they are shown by experiments. Thus this choice is not suitable for the description of the biological problem we consider. Non-linear and local aggregation and unbinding rates leading to patterns formations are considered in [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF] and shows very good agreements with experimental data.

b)F

  and G are locally Lipschitz Assumption a) reflects the fact that F and G are probabilities. Assumption b) is the minimal regularity assumption needed in the sequel. When needed we'll continuously prolong the definition of F and G to the whole real values v ∈ R just defining:

  with |ṽ| ≈ 10 -3 . With our choice of parameters the stationary value is V = 0.43086.

Figure 1 :

 1 Figure 1: The initial data (left) and the equilibrium solution (right) for the function v. The parameters used for the computation are : ρ = 1, a = 0.25, b = 0.5 and ε = 0.35, σ = 1 and T = 200.

Figure 2 :

 2 Figure 2: The initial data (blue) and the equilibrium solution (red) for the functions u (left) and v (right). The parameters used for the computation are : ρ = 0.7, a = 0.25, b = 0.5 and ε = 0.35, σ = 1 and T = 10.

Finally, we

  may compute the convergence rate towards the stable solution.

2 .

 2 Figure 3 shows the time convergence of the max(v), min(v) and of the mean value v m (t) towards V = v 1 , . The computation has been stopped when the maximum between the absolute errors |v 0 -max(v)|, |v 0 -min(v)| and |v 0 -v m | is smaller than 10 -3 . A linear regression study of the three black curves shows that the convergences are exponential with a standard deviation of the order of 10 -

Figure 3 :

 3 Figure 3: Convergence in time (black line) of the maximum (left), the minimum (middle) of v(t, •), and of mean value v m (t) (right) towards the value v 1 = 0.3107435 (blue line) for the parameters ρ = 0.7, a = 0.25, b = 0.5, ε = 0.35 and σ = 1.

Now that the problem is proved to be well-posed, we consider its behavior at equilibrium, and deal, in this section, with the stability of the solution (u, v) to (2)-(4) completed by an initial

Acknowledgments

This work was supported by the PEPS-MBI project "Modélisation d'Adhésion des Cadhérines" founded by the French research institutes CNRS, INRIA and INSERM, and by the Kibord project (ANR-13-BS01-0004) funded by the French Ministry of Research.

Proof of Theorem 2 :

Let t ∈ (0, T ). We have for all (n, m) ∈ (N * ) 2 , n > m :

Because of (21) we deduce:

, with C > 0, is a convergent serie, we can conclude that (u n ) n≥0 is a Cauchy sequence in L 2 ((0, T ) × Ω). Similarly we prove that (v n ) n≥0 is a Cauchy sequences in L 2 ((0, T ) × Ω). Therefore there exists two functions u and v in L 2 ((0, T ) × Ω) such that:

There also exists two subsequences, still denoted (u n ) and (v n ), respectively converging almost everywhere to u and v in (0, T ) × Ω. Finally, Proposition 2 implies:

for almost all (t, x) ∈ (0, T ) × Ω, so that both functions u and v are in L ∞ ((0, T ) × Ω) and this is the same for Q(u, v). Thus from classical parabolic equations arguments, see [START_REF] Ladyzenskaja | ceva : Linear and quasi-linear equations of parabolic type[END_REF], we have that u ∈ W 1,p ((0, T ) × Ω), v ∈ L ∞ ((0, T ) × Ω) and they are solutions of ( 2)-( 4) and [START_REF] Mancini | A phenomenological model of cell-cell adhesion mediated by cadherins[END_REF].

Note that system (2) doesn't satisfy classical comparison principle, and that the above theorems and proofs no more hold if ρ is not a constant. In order to get uniqueness of the solution (u, v) we need some more assumption on the aggregation function F (v) and/or on the unbinding function G(v). For instance, if F (v) is nonincreasing and G(v) nondecrasing, then the hypothesis of Theorem 3 are satisfied. This will be the case for the particular choice (30) and (31).

Proof of Theorem 3 :

The conditions on F and G allow us to use the appendix of [START_REF] Bedjaoui | Critical blowup exponents for a system of reaction-diffusion equations with absorption[END_REF] and then we have a maximum principle for the problem which implies the uniqueness. The existence holds from theorem 2 and from classical parabolic equations arguments, see [START_REF] Ladyzenskaja | ceva : Linear and quasi-linear equations of parabolic type[END_REF]. Note that because of the maximum principle the existence can also be proved using monotonous sequences.