Emilie Caldeira
email: emilie.caldeira@u-clermont1.fr.

Does the system of allocation of intergovernmental transfers in Senegal eliminate politically

Keywords: JEL Classi…cation: H20, H77, O12 Intergovernmental transfers, political economy, decentralization, Senegal

motivated targeting? 2011

Introduction

While …scal decentralization has been adopted by a large part of the developing world, there is a broad consensus in the literature that the bene…ts expected from decentralization will not materialize if the system of intergovernmental …scal transfers does not rely on an equitable and e¢ cient horizontal allocation mechanism. Although a huge literature on the determinants of allocation of central grants in developed countries exists, this kind of study is still limited for developing countries, especially Subsaharan countries. This paper attempts to …ll the gap in the empirical literature by raising the issue of how such transfers are allocated across local governments in a Subsaharan country, Senegal.

The traditional theoretical view on central transfers is that they should be guided by equity and e¢ ciency considerations, a welfare maximizing government wanting to reallocate resources between richer and poorer jurisdictions and to correct for externalities (Buchanan, 1950, Oates, 1972, Gramlich, 1977). Actually, a number of empirical studies have pointed out that political factors are more relevant in explaining the allocation of grants. Beginning with Wright (1974), economic historians examined the question of how transfers were allocated amongst American states during the New Deal in the 1930s and found that political variables explained this allocation considerably better than economic considerations.

As Banful (2010) notices, empirical studies have concerned an array of countries like Albania (Case, 2001), Argentina (Porto and Sanguinetti, 2001), Australia (Bungey, Grossman, and Kenyon, 1991, Worthington and Dollery, 1998), Canada (Albouy, 2010), Ghana (Banful, 2010, Miguel and Zaidi, 2003), India (Khemani, 2007, Cole, 2009, Arulampalam et al., 2009), Israel (Alperovich, 1984), Japan (Meyer and Naka, 1999), Portugal (Pereira, 1996, Veiga and Pinho, 2007), Russian Federation (Treisman, 1996), Sweden (Dahlberg and Johansson, 2002), Tanzania (Boex, 2003) and the United States (Anderson and Tollison, 1991, Wallis, 1998). An international comparison of these works shows that, besides local expenditure needs and …scal capacity, other factors including electoral concerns and political in ‡uence play important roles in the horizontal allocation of grants. Central governments appear to be opportunistic, using transfers to maximize their chances of re-election or partisan, allocating grants to further interests of their political support groups (Cox, 1986, Grossman, 1994, Case, 2001, Banful, 2010).

A common view is that basing the allocation of …scal transfers on a formula limits the discretionary power of politicians in distributing central grants so that this strategy has been widely adopted in the developing world (Banful, 2010). In this paper, we test the e¤ectiveness of formulas in eliminating discretion by analyzing how transfers are allocated across local governments in Senegal where a formula allocation mechanism is employed. More precisely, we intend to tackle the three following issues: (1) Does the Senegalese allocation formula allow a distribution of …scal transfers that conforms to the dictates of normative theory, in particular, to the equity principle? (2) Is such an allocation system actually su¢ cient to eliminate politically motivated targeting of grants? (3) If not, what kind of political factors explain the allocation of …scal resources? This paper adds to the existing empirical literature studying horizontal allocation of transfers and we believe its contribution to be twofold. First, to our knowledge, this is one of the …rst papers to exploit an original micro-level public …nance panel data from a Subsaharan country to test political economy theories of …scal transfers' allocation. It allows us to see to what extent results obtained for developed countries can be observed for a developing country.1 Besides, Senegal is a particularly interesting case since a received wisdom says that transfers allocation is determined by political a¢ nity between the central and local governments, as it is the case in many African countries (Banful, 2010), and some mayors deplore a discriminatory and opaque distribution of grants.2 Second, we employ a consistent econometric method which generates credible empirical results. Indeed, we use the …xed e¤ects vector decomposition (FEVD) estimator developed by Plümper and Troeger (2007) to avoid failure to control for heterogeneity of local governments and ine¢ ciency in estimating the e¤ect of variables that have little within variance, common issues in this kind of study. Moreover, to test whether equity concerns are dominant in the allocation of transfers, we compute an innovative poverty index at local level using the Demographic and Health Survey (DHS) and its geographic data.

After having brie ‡y reviewed the literature on normative and political economy determi-nants of …scal transfers, we rigorously investigate the empirical importance of such determinants in the distribution of central grants across local governments in Senegal by estimating a panel data for 67 communes from 1997 to 2009. (1) Estimation results tend to show that equity concerns do not impact the allocation of intergovernmental grants in Senegal, leading to the conclusion that the resources distribution system does not follow the dictates of the normative theory. (2) We also …nd evidences of politically motivated targeting of transfers despite the formula-based system. (3) In particular, our results highlight three kinds of political motivation. First, our analysis suggests that transfers allocation follows a tactical redistribution, which consists in targeting swing communes to achieve electoral success. Second, local governments which are better represented in parliament seem to receive larger grants, con…rming one of the most consistent empirical results in this literature.

Third, ethnic fractionalization seems to be positively correlated with per capita transfers which might indicate that the central government uses …scal resources as a way to pacify potentially troubled areas.

Section 2 o¤ers a brief review of the literature on the determinants of intergovernmental transfers. Section 3 presents the institutional background of transfers in Senegal, the empirical model, strategy and …ndings. Section 4 concludes with some policy implications.

2 Normative and political economy determinants of intergovernmental …scal transfers: A literature review Three stands of the literature consider factors that may in ‡uence the distribution of central transfers across local governments (Boex and Martinez-Vazquez, 2005). First, public …nance literature provides normative dictates on how intergovernmental transfers should be allocated. Second, voter-choice models in public choice literature explain how electoral concerns could a¤ect the central government's …scal choices in distributing …scal resources to local jurisdictions. Third, political economy arguments contribute to understand the allocation of grants by considering non-electoral arguments.

First, there is a consensus in the local public …nance literature that a system of …scal transfers should be designed to achieve equity and e¢ ciency in the allocation of resources (Buchanan, 1950, Oates, 1972, Gramlich, 1977). The central government is assumed to be a 'benevolent planner', which maximizes social welfare. The rationale behind the equity principle is the need for a reduction in horizontal …scal imbalances existing between local jurisdictions. Thereby, transfers should compensate the unequal access to local public goods and services generated by the uneven distribution of resources across the country. The pursuit of this objective leads to expect a pro-poor allocation of grants. However, most empirical studies …nd that wealthier local governments receive greater transfers, indicating that political considerations outweigh those of equity (see Kraemer, 1997, Alm and Boex, 2002, Wallis, 1998, Meyer and Naka, 1999). The economic e¢ ciency objective seeks to correct for externalities and compensate spillover e¤ects among jurisdictions in the provision of some local public services. This incentive objective would result in the central government providing greater grants in response to higher expenditure needs. 3 In empirical studies, local expenditure needs and costs generally have a positive impact on the level of transfers received by a local government. Exceptions include Nigeria (Alm and Boex, 2002) and Mexico (Kraemer, 1997) where expenditure needs have no e¤ect and a negative impact on grants received, respectively. 4 Second, the literature on targeted redistribution distinguishes between patronage, which consists in rewarding political supporters, and tactical redistribution, which aims at achieving electoral success. Greater political support for the central government in a particular jurisdiction can be rewarded by greater transfers. For instance, Miguel and Zaidi (2003) …nd evidence from Ghana of core supporters'targeting, districts from where the ruling party won all the parliamentary seats. 5 This patronage can also be tactical, however. Indeed, Cox (1986) argues that the optimal strategy for risk-averse opportunistic candidates is to favor partisan jurisdictions to maintain existing political supports. Similarly, Bungey, Grossman, and Kenyon (1991) and Leyden (1992) show that party closeness between central and local politicians increases returns in term of central government's support, and therefore the level of transfers. Case (2001) interprets empirical …ndings of greater grants in Albania to juris-dictions where the President received more votes in the past election as evidence of targeting of …ef districts, considered as more 'pivotal'. If the central government can reward its supporters or target it to maintain political support, it can also favor its opponents (Treisman, 1996) or 'swing'jurisdictions, where the distance of vote shares between the largest parties is small. Electoral results in these jurisdictions are assumed to be determined by how much they receive in resource transfers from the center. Arulampalam, Dasgupta, Dhillon, and Dutta (2009) construct a model where the federal government allocates transfers to states that are aligned with the incumbent party but also swing. Using Indian panel data, they validate their theoretical model. 6 In the context of an African country, Banful (2010) also …nds evidence that per capita grants are higher in Ghanaian districts where vote margins in the previous presidential election were lower. Following the predictions of the well known model of opportunistic political budgetary cycles provided by Rogo¤ and Sibert (1988), the central government is also expected to transfer more resources in election years to increase its likelihood of re-election. According to Worthington and Dollery (1998), grants in local election years would be more productive due to a heightened awareness of policies but, in the case of central elections, the returns from purchasing political capital in this manner would be o¤set by direct political bene…ts of central direct expenditure so that transfers should decrease.

Third, beyond electoral considerations, political decision-making processes are likely to be captured by powerful interest groups. Assuming that local o¢ cials try to extract as much resources as possible from the center, those with higher bargaining power may receive larger grants. The fact that local jurisdictions with higher political representation bene…t from greater transfers is one of the most robust empirical …ndings (Wright, 1974, Porto and Sanguinetti, 2001, Khemani, 2007). 7 Smaller jurisdictions are also expected to receive greater per capita transfers, which may be caused by scale economies or by their potentially higher lobbying power. In particular, this bias may be explained by an urge to secure 6 Cole (2009) also …nds that state governments in India supply greater subsidized agriculture loans to electoral districts where the ruling party had a narrow margin of victory (or loss) and Johansson (2003) provides theoretical and empirical evidence that swing Swedish municipalities receive larger grants than other groups.

7 For instance, Atlas (1995) shows that the allocation of per capita federal net spending in the United States from 1972 to 1990 was a¤ected by states'per capita congressional representation and highlights that the institutional basis of political representation a¤ects spending allocations across states. broad political support. 8 In the context of a developing country, central governments may also use economic means to deal with social con ‡icts.9 A common argument in favor of decentralization is that local governments are enabled to allocate public spending in line with the preferences of heterogeneous local communities. However, Tranchant (2010) shows that, while this hypothesis may be relevant for local majorities, it is not the case for local minorities, who are not in a position of strength. In this context, …scal decentralization can increase local con ‡icts, marginalized ethnic minorities clashing against powerful local majorities. Hence, the central government may use transfers as an instrument to pacify potentially troubled areas like ethnically fractionalized jurisdictions (Treisman, 1996). 10

3 The determinants of intergovernmental …scal transfers in Senegal: Empirical evidence

In this section, we intend to determine whether the Senegalese allocation system conforms to the dictates of the normative theory and if this system eliminates politically motivated targeting of transfers. We …rst investigate the institutional background of intergovernmental transfers in Senegal, then we specify our econometric model and strategy. Lastly, we present our principal …ndings resulting from the estimation of a panel data for 67 communes from 1997 to 2009. which are subdivided into 67 communes, 43 communes d'arrondisements which are further divided into 320 communautés rurales. 13 Local governments are endowed with legal personality and bene…t from the administration principle according to which local jurisdictions manage themselves freely by elected councils. The commune has to ensure the best living conditions for the whole population. 14 The competencies of Senegalese communes range from the maintenance of communal properties or the management of local public works to environmental protection with, for instance, the adoption of measures limiting pollution, and include the management of local employments and assistance to places of worship. Table 1 presents Senegalese communes' revenue sources and their evolution. Local own-revenue represents around 85% of local resources and are divided into tax and non-tax own-revenue.

Intergovernmental transfers in Senegal: Institutional background

The …rst one is mainly constituted by taxes on property, patent, advertisement, water, electricity and waste removal while the second one is related to the public domains'occupation (markets, car parks, tow pound...) and to some administrative services. Central transfers represent around 10% of the total local resources and have become more important in ab-1 1 Since 2007, the president is elected for a …ve-year term renewable once. 1 2 Local o¢ cials are elected for a six-year mandate but local elections, initially planned for May 2008, were reported in March 2009 due to the modi…cation of the regional administrative zoning.

1 3 There is also an administrative level between regions and communes: the departments (départements) but they are managed by a representative of the central government. Our empirical work will help to determine if these criteria are su¢ cient to allow an e¢ cient and equitable distribution of resources and to forbid the incentives of politicians to divert resources for personal gain.

The basic empirical model

To test whether the allocation of transfers is in ‡uenced by economic considerations and the presence and the nature of any politically motivated targeting of resource, we use the general empirical framework followed in the literature (Boex and Martinez-Vazquez, 2005).

Per capita amounts of transfers received by a local government are regressed upon sets of equity/e¢ ciency and public choice variables that may impact center's …scal decision.

To determine whether the Senegalese system conforms to the dictates of the normative theory we consider …scal incapacity and expenditure needs. We …rst examine the impact of a local government's …scal incapacity on the size of the transfers it receives by using a DHS poverty index (see the following section 3.3.1). If equity concerns are at play in the allocation of transfers, we should …nd a positive coe¢ cient associated with …scal incapacity. Second, we include local expenditure needs and costs in our econometric model (e¢ ciency principle).

The variables that are generally used to measure local expenditure needs are demographic variables such as the size of the school-aged population, the economically dependent population, or urbanization (Meyer and Naka, 1999, Wallis, 1998). 16 The population density is also commonly used to measure the per capita cost of providing local public goods. 17 We retain two variables; urbanization rate and population density. While we unambiguously expect a negative coe¢ cient associated with population density, it is not possible to know a priori which of rural or urban sets of needs dominate. 18 We then consider the in ‡uence of electoral concerns in the allocation of grants. We introduce a qualitative dummy variable that indicates if the central and local governments are of the same political party. This variable allows us to test the existence of patronage and Cox (1986) model, according to which the optimal strategy of political candidates is to favor their supporters. We also include a variable that measures the di¤erence in vote shares expressed in absolute values between the central government party and its main opponent, in the last local election in each commune (see Case, 2001 and Dahlberg and Johansson, 2002). 19 With this variable, we test the prediction that swing communes are targeted by the incumbent party. Moreover, two dummy variables for elections, central and local, are included in order to account for time-periods when ability to purchase political capital may vary from the norm (Veiga and Pinho, 2007). Since elections are held at the beginning of the year, we consider the year before elections. Following the prediction of Worthington and Dollery (1998), local and national election years should exhibit a positive and negative coe¢ cient, respectively.

Lastly, we introduce the number of deputies represented by departments, a local government with larger political representation per capita being expected to extract larger per capita transfer. We also consider the relative population size of the commune to test the existence of a bias in favor of smaller jurisdictions. In the particular context of a developing country, central decisions may also concern ethnic fractionalization, transfers being a potential instrument to avoid local con ‡icts. 20 Finally, we de…ne the following empirical model:

T ransf ers it = N ormative it + 'Electoral it + P olitical it + " it ; (1)
Indicators used for each category are summarized in Table 2.

Econometric framework

Before introducing our econometric strategy, we present our composite measure of local …scal incapacity, based on a DHS poverty index.

An indicator of local incapacity

Following studies that rely on composite measures of local incapacity, we propose to estimate a poverty index using the 1997 and 2005 DHS with their geographic data. 21 Due to the abundance of household survey data on asset ownership and the considerable biases measurement error associated with reported income or consumption, a substantial body of literature has developed an asset-based measure of wealth. Filmer and Pritchett (2001) concluded that the DHS wealth index actually performed better than the traditional consumption or expenditure index in explaining di¤erences in economic status. Hence, in the footsteps of Filmer and Pritchett (1999) and Sahn and Stifel (2003), we compute a composite poverty indicator from the DHS surveys. 22 However, since the DHS wealth index has been criticized as being too urban in its construction, we propose, as suggested by Rutstein (2008), to produce a single national-level composite index from wealth indexes that have been separately constructed for the urban and rural areas. Table 3 summarizes the assets included in the Principal Components Analysis (PCA) 23 for each index and their coe¢ cients for the DHS 2005. 24 Conversion adjustments are made to map urban and rural indexes onto the national index. 25 Then, we divide the national index into quintiles of the national household population and determine the percentage of the poor in each department. Table 4 gives the 'pro…le'of a household that belongs to the …rst quintile in 2005 and Figures 1 gives the percentage of poor households by region.

Econometric issues and strategy

Our econometric model is quite similar to those considered in the literature and su¤ers from several defects. We then present our econometric strategy.

2 2 The general methodology used to calculate the wealth index is given in Filmer and Pritchett (2001). The speci…c approach used in the DHS is described in Rutstein and Johnson (2004).

2 3 It is a technique for extracting from a set of variables those few orthogonal linear combinations of the variables that capture the common information most successfully (for a detailed explanation, see Filmer and Pritchett, 2001).

2 4 Based on descriptive statistics, we thought that possession of most livestock would be exclusively rural and some other items such as computer or internet access would be exclusively urban. The construction of the national index uses the set of indicator variables that the rural and urban areas have in common and is restricted to those that correlate with wealth in the same direction 2 5 The level and distribution adjustment values are found by regressing the value of each household's area-speci…c index scores onto its national index score. For instance, with the DHS 2005, we have: U rban : W In = 0:24 + 0:41W Iu

(2)

Rural : W In = 1:09 + 0:37W Ir (3)
where W In; W Iu; and W Ir are the national, urban-speci…c, and rural-speci…c wealth index scores, respectively. We also use a quadratic form of the regressions since it improves the …t a little.

Econometric issues First, we correct for all time-invariant community characteristics, observed or unobserved, and all 'year e¤ects'. 26 Local governments di¤er in ways that are captured only imperfectly by our economic and political variables and these persistent differences may produce signi…cant di¤erences in transfers. So, we include a commune-speci…c e¤ect, i . Then, omitted variables that vary over time but are constant between communes can in ‡uence the amount of transfers available and received and, at the same time, the …scal capacities of communes. By introducing time dummies (T t ) we correct this potential endogeneity bias due to omitted variables. We should limit the arti…cial positive correlation between the …scal capacity and transfers, both increasing because of their common correlation with a third variable, for instance, the national economic conjuncture. 27 Second, we consider endogeneity bias due to reverse causality. This turns out to be important for a number of variables, especially for …scal incapacity. Indeed, local spending, in part …nanced by central transfers, may reduce local poverty. Hence, if jurisdictions with higher transfers tend to have higher …scal capacity, then, estimates will show that jurisdictions with higher …scal capacity receive larger transfers. However, in our case, the …scal incapacity index represents a more permanent status than does either income or consumption so that transfers at time t probably cannot a¤ect the level of wealth at time t. If doubts remain, we propose to use the lagged value of …scal incapacity indicator to test the robustness of our estimations. Concerning political variables, if we believe that politicians use public resources to buy support, we must also believe that transfers have an e¤ect on electoral results. However, since we use values of electoral outcomes that are determined before the start of a …scal year, transfers are unable to a¤ect past results so that the coe¢ cient of political a¢ liation variable should not be biased. We could also consider the possibility of a 'vote with feet', where people are encouraged to migrate when they perceive situations to be better in another jurisdiction, such as a higher level of transfers. However, we think that this potential bias is limited since Tiebout's model rests on assumptions of perfect mobility and information, which are seldom found in developing countries (Bardhan, 2002). Finally, since we regress central transfers on explanatory variables of which some are observed on a more aggregate level, we introduce department cluster (Moulton, 1990).

Econometric strategy A panel data approach allows us to control for the potentially large number of unmeasured explanatory variables by estimating a '…xed-e¤ects'(FE) model.

However, the FE model does not allow the estimation of time-invariant variables and results from its ine¢ ciency in estimating the e¤ect of variables that have little within variance (Baltagi, 2001, Wooldridge, 2002, Hsiao, 2003). 28 In order to assess coe¢ cients of timeconstant or rarely changing variables, and to control for commune speci…c e¤ects, we propose to use the FEVD estimator developed by Plümper and Troeger (2007). 29 This estimator, based on a three-step procedure, allows a decomposition of the unit …xed e¤ect ( i ) into two parts; a part explained by the time-invariant variables and an unexplained part, b h i . 30

Regression (1) becomes:

T ransf ers it = N ormative it + 'Electoral it + P olitical it + T t + b h i + " it ; (4)
Based on Monte Carlo simulations, Plümper and Troeger (2007) …nd that the vector decomposition model performs far better than pooled OLS, Random-E¤ect (RE), and the Hausman-Taylor procedure. 31 To capture the potential persistency in central transfers, we also consider the dynamic version of our model by introducing the lagged dependent variable. This yields the inconsistency of …xed-e¤ect estimators (see Nickell, 1981) so that we use the GMM-System estimator (Blundell and Bond, 1998).

2 8 In our case, for instance, ethnic variables, which are time-invariant variables, are dropped in the …xed e¤ect model so that it is impossible to determinate whether these variables a¤ect the allocation of …scal transfers.

2 9 Rarely changing variables are de…ned as having a low within variance. Our variables of political and ethnic factors, …scal capacity, urbanization rate and population are rarely changing variables with a little within variance.

3 0 A formal explanation of this estimator is given in Plümper and Troeger (2007).

3 1 An alternative answer to assess coe¢ cients of time-constant variables and to control for commune-speci…c e¤ects is to use a RE model. However, this estimator implies orthogonality between explanatory variables and the error term, a hypothesis that does not seems to be relevant in our case. The Hausman test actually con…rms that we should use FE estimators.

Data and estimation results

Data for this study come from a variety of sources. Data on …scal transfers are drawn Committee. We constructed a panel data for 67 communes from 1997 to 2009. 32 Table 5 gives estimation results. First, we test the static model with the FEVD estimator.

To test the robustness of our estimations, we introduce progressively a trend (2) and a department cluster (3), we use alternative indicators (4), 33 the lagged value of the …scal incapacity indicator (5) and year dummies instead of the trend variable (6). Then, we estimate the dynamic model with one-step robust GMM-System estimator (7). 34 Does the central government assist jurisdictions with poor tax bases and greater socioeconomic needs as it should according to normative guidance? Equity concerns seem not to be at play in determining the allocation of …scal transfers. Indeed, we …nd a negative and signi…cant coe¢ cient associated with our indicator of …scal incapacity indicating that poorer local governments receive smaller intergovernmental transfers. This result is consistent with most empirical studies (Kraemer, 1997, Alm and Boex, 2002, Wallis, 1998, Meyer and Naka, 1999). Estimation results support the hypothesis that expenditure needs have an e¤ect on transfer allocation. Indeed, the urbanization rate is negatively correlated with …scal transfers suggesting that transfers alleviate special rural problems. However, population density, used to measure the per capita cost of providing local public goods, seems to be positively correlated with …scal transfers, probably capturing urban needs since a higher 3 2 Note that, when we introduce the number of representatives in parliament, the panel data include data for 67 communes from 1998 to 2009 since we only have results of legislative elections since 1998. When we consider political a¢ liation variables, the panel data include data from 2002 to 2009 since we have local election results for 2002 and 2009.

3 3 We consider P Sit, the score of the president in o¢ ce at the previous local election instead of P Ait and ethnic polarization, EPit; instead of EFit (see Table 3).

3 4 We assume the weak exogeneity of the lagged dependant variable and the strict exogeneity of other explanatory variables. The lagged levels of variables are instruments in regressions in level as well as in regressions in di¤erence. Following Roodman (2009) we collapse instruments and limit their number to avoid non optimal weight matrix, biased standard errors, and incorrect overidenti…cation tests. With the Hansen test and AR(2) test, we conclude that orthogonality conditions are respected. degree of urbanization is generally associated with higher population density.

Is the allocation of …scal transfers also guided by political logic despite the formula-based system? Our results actually suggest that normative guidance is not the only motivation that determines the distribution of grants across communes. First, as regards electoral concerns, while supporter communes do not appear to receive more transfers, greater resources seem to be provided to local governments that are more swing. Indeed, coe¢ cient associated with our proxy for swing communes is always negative and signi…cant at 1% level. Contrary to the …ndings of Case (2001) and Miguel and Zaidi (2003), the center does not seem to target more resources to its supporters. Transfers are not targeted to areas in which political support is concentrated to maximize return in terms of votes, as predicted by Cox (1986).

Our result is closer to Cole (2009), who …nds that politicians, who care about winning election, target swing jurisdictions. It also reinforces the …ndings of Banful (2010) suggesting that swing districts can be targeted in an African context. We cannot clearly con…rm the predictions of Worthington and Dollery (1998). Indeed, the year before national election is negatively but not always signi…cantly correlated with grants and coe¢ cient associated with the year before local elections is positive but rarely signi…cant. Second, other political and ethnic considerations play important roles in the distribution of per capita transfers across communes. This last point is con…rmed by the positive association between transfers and political economy determinants such as representatives in parliament, population and ethnic fractionalization. Indeed, greater representation per voter seems actually to result in greater per capita transfers, which is consistent with empirical …ndings in the literature (Wright, 1974, Porto and Sanguinetti, 2001, Khemani, 2007). Population is generally negatively and signi…cantly correlated with …scal transfers, probably denoting a disproportionate lobbying power of smaller communes. Beyond the e¤ect of political considerations, ethnic fractionalization is positively correlated with per capita …scal transfers. It indicates that the central government may use transfers as a paci…cation instrument as Treisman (1996) suggested.

Our results are robust to the introduction of a trend variable, a cluster department and year dummies. Moreover, the impact of …scal incapacity remains negative when we correct for potential simultaneity bias and our conclusions do not change when we use alternative indicators of political a¢ liation and ethnic fractionalization. Finally, our …ndings are similar when we consider the dynamic model even if the lagged dependant variable appears to be not signi…cant, which is not surprising in a context of developing country where central transfers are instable.

Conclusion

This paper exploits an original public …nance panel data allowing us to test political economy theories of …scal transfers for a developing country, Senegal. The estimation of a panel data for 67 communes from 1997 to 2009 suggests that the allocation system in Senegal does not conform to the dictates of normative theory. In particular, equity concerns do not appear to a¤ect the allocation of intergovernmental …scal transfers. On the contrary, wealthier local governments seem to receive greater intergovernmental transfers. Moreover, our results tend to show that the allocation of transfers follows a tactical redistribution by targeting swing communes and to con…rm that local governments with more political power receive larger transfers. The Senegalese case also emphasizes the importance of ethnic considerations and the fact that the central government may use transfers as a tactical instrument to pacify fractionalized areas. Our …ndings are consistent with those observed in other countries. Indeed, intergovernmental transfers are generally allocated in a needs-equalizing way but are counterequalizing when it comes to …scal capacity, and political economy factors are consistently a driving force in determining the distribution of intergovernmental …scal transfers.

Our work adds to empirical evidence from around the world that has shown several instances in which politicians in central government make …scal decisions by optimizing their electoral objectives and being in ‡uenced by political factors beyond economic considerations.

It shows that results found for developed countries can actually be observed for a developing country. This study also highlights that an allocation system based on a formula can be insu¢ cient to eliminate politically motivated allocation of transfers. Eliminating discretion seems to require more than a formula. Delegating responsibility for the distribution of resources across local governments to an independent agency could help to mitigate such distortions (Khemani, 2007). 18,732 14,662 36,736 21,189 26,000 27,885 35,419 37,982 43,601 47,443 44,629,219 49,422,824 Local non-tax own-revenue

Normative factors

Fiscal incapacity

We evaluate the revenue incapacity, RI dt , using the DHS Wealth Index (see section 3.3.1) in department d on year t.

Expenditure needs

U dt , the urbanization rate in department d on year t, de…ned as the percentage of population living in urban area, D it , the population density in commune i on year t.

Electoral concerns

Other political factors

Political representation N it , the number of representatives in parliament by department d on year t.

Population size P it , the population size of the jurisdiction i on year t.

Ethnical factors

EF

it , the ethnic fractionalization in communes i on year t de…ned as the probability that two individuals randomly drawn from a jurisdiction are from di¤erent ethnic group. We also consider alternative indicator by using ethnic polarization,

EP

it .

(3) (4)

(5) (6)

  Senegal has shown a remarkable political stability since its independence in 1960, which was strengthened by peaceful presidential transitions. Abdou Diouf served …ve terms as President. He was defeated in February 2000 by opposition leader Abdoulaye Wade, by direct popular vote in the majority (two-round) system. The regime of Abdoulaye Wade follows four decades of Socialist Party rule. He was re-elected in February 2007, at the end of the seven-year term. 11 Local governments are directly elected by local population from a list in one round. Local elections occurred, the same day for all communes, in March 2002 and 2009. 12 The fact that the voice of the opposition was loudly heard in the latest local elections, including the defeat of Wade's own son, Karim, in Dakar, could be a sign that Senegal's democracy remains relatively strong. Our dataset covers all national and local elections which occurred in the period 1997-2009. Decentralization has been implemented since the beginning of the independence in 1960 to move government closer to citizens. However, 1990 marked a turning point in the process with the abolition of the tutelary power of the center within the communes'executive. The last step is constituted by the adoption in 1996 of the new laws of decentralization: law 96-06 carrying the Local Government Code, law 96-07 de…ning transferred competencies, and law 96-09 …xing the territorial administration. Senegal is divided into eleven regions (régions)

  1 4 See law 96-06 with the Local Governments Code. solute terms. One of the main features of …scal decentralization in Senegal is the increasing level of revenue at local level. However, local resources remain insu¢ cient to provide local basic public services 15 and important inequalities appear between communes: the resources of the ten poorest communes represent 1 % of the resources of the …ve richest ones. There is also an important variability across jurisdictions, which stays relatively constant over time.In 2009, communes such as Ranerou, Oussouye and Foundiougne received more than four times the national average, while other jurisdictions like Pikine or Bargny received a transfer per capita ten times smaller than that amount.State subsidies should mitigate the lack of resources and reduce horizontal …scal imbalances. The design and implementation of transfers deserves serious concern, in particular, in developing countries (seeBird and Smart, 2002, for a survey of central transfers systems adopted in a number of developing countries). Senegal employs a formula-based resources allocation mechanism. Finances law …xes a minimum amount of transfers as an annual percentage of central tax revenue. This then amount depends on a percentage of the Value Added Tax collected for the bene…t of central government. Transfers are distributed between local authorities according to criteria annually …xed by decree after consulting the National Council of Development of Local Authorities. This council, in charge of the follow-up of the decentralization process, is constituted by the Senegalese President, the members of the government and representatives of deconcentrated services of the central government and of local governments. The allocation's criteria are twofold. First, there is a compensation criterion: around 80% of the total transfer is distributed according to the cost of local public spending induced by the transfer of competences and responsibilities from the central government to the local authorities. In practice, the central government only considers spending made by the local government the previous year. Second, the rest of the amount is divided into two parts; a …rst part (70 %) is distributed in equal shares between jurisdictions and a second part (30 %) is distributed towards the demographic importance of each jurisdiction.

  from the Municipal Development Partnership, and local characteristics like population, area, urbanization rate and ethnic composition come from the General Population and Housing Census and from the National Institute of Statistics and Demography of the Ministry of the Economy and Finance of Senegal. The results of legislative elections come from the National Assembly and the results of local elections come from the Independent National Electoral

  g lo b a l lo c a l re ve nu g lo b a l lo c a l re ve nu g lo b a l lo c a l re ve nu g lo b a l lo c a l re ve nu g lo b a l lo c a l re ve nu

  dummy variable which takes the value one if commune i has the same political a¢ liation as the president in o¢ ce.We consider alternatelyP Sit , the score of the president in o¢ ce at the previous local election.Tactical distributionDVit measures the di¤erence in vote shares expressed in absolute values between the central government party and its main opponent, in the last local election in each commune. variables, which take the value one the year before of national and local election, respectively.

Table 1 :

 1 Local revenue of Senegalese communes

Table 2 :

 2 Determinants of the horizontal allocation of per capita local government transfers

		i on year t.
	Indicator used	T it , per capita amount of transfers received by a commune
	Variables	Central transfers

Table 4 :

 4 Pro…le of a household who belongs to the …rst quintile in 2005

	Mean	0.01	0.001	0.002	0.04	20.39	49.92	25.24	0.07	0.22	4.15	6.92	0.84	0.41	18.10	9.27	0.41	1.31	0.50	0.28	0.55	2.61
	Mean	Plough 0.09 Motorcycle /scooter	Horse 0.07 Car /truck	Cow 0.11 Commercial van/truck	Donkey 0.13 Boat	Hand cart 0.04 Source of drinking water	Refrigerator 0.01 Time to get to drinking water	Telephone 0.01 Type of toilet facility	Antenne 5 tv 0.001 Has electricity	Canal tv 0.001 Share toilet with others	Washing machine 0.001 Means of sewage disposal	Video 0.003 Means of disposing of water	Air-conditioner 0.002 Frequency of water outages	Computer 0.003 Duration of water outages	Internet access 0 Main material of the ‡oor	Sheep 0.41 Type of cooking fuel	Poultry 0.59 Have bednet for sleeping	Bicycle 0.15 Place for hand washing	Radio 0.71 Items present : water tap	Television 0.03 Items present : soap	Cell phone 0.06 Items present : basin	Cooker 0.11 Number of rooms for sleeping

Table 5 :

 5 Estimation results -Fixed e¤ects vector decomposition and GMM-System

	(1)
	Dep. var.: Per cap. transfers

The …ndings of[START_REF] Miguel | Do politicians reward their supporters? Regression discontinuity evidence from Ghana[END_REF] concerning Ghana already suggest that in African democracies ruling parties are able to reward their supporters and use the advantages of incumbency to win subsequent elections.

See, for instance, Le ministre Aliou Sow brocardé par Alioune Sarr, le Pcr de Notto Diobasse, Le Peuple, October 18th 2010 (http://lepeuple-sn.com) or Fonds de dotation : Aliou Sow fait la part belle à sa collectivité selon le PCR de Ndindy, Rewmi, November 2nd 2010 (http://www.rewmi.com/).

Moreover, Albouy (2010) shows that providing higher grant levels to jurisdictions that pay higher central taxes is a mechanism for reducing ine¢ cient migration.

We also have to note that transfers pursue a vertical …scal balance objective, that is, ensure that the revenues and expenditures of each level of government are approximately equal[START_REF] Weingast | Second generation …scal federalism: The implications of …scal incentives[END_REF].

[START_REF] Pereira | A politico-economic approach to intergovernmental lump-sum grants[END_REF] also …nds that intergovernmental grants in Portugal were designed to reward central governments'political support.

Empirical work on lump-sum grants in Portugal[START_REF] Pereira | A politico-economic approach to intergovernmental lump-sum grants[END_REF] supports the politico-economic hypothesis and rejects the hypothesis that economies of scale are the main explanatory cause for the observed regressivity of per capita transfers. Indeed, he shows that per capita grants decrease with the population size of communities even after controlling for the e¤ect of economies of scale which might be captured by the density variable.

 9 In particular, Senegal has to deal with a violent separatist movement in the southern region of the Casamance.1 0 Note that[START_REF] Montalvo | Ethnic polarization, potential con ‡ict and civil war[END_REF] suggested that ethnic polarization measures are more appropriate than fractionalization indices to capture social con ‡ict.

5 Tax own-revenue of all communes represents on average only 6 % of the central tax revenue

6 Generally, instead of using physical infrastructure measures such as hospital beds or the number of schools, it is preferable to measure the number of citizens with a certain need, such as infant mortality or school-aged population, which are free from incentive problems.1 7 Presumably, the lower the population density in a jurisdiction, the higher the unit cost of delivering social services, since the provision of public services increases with a more dispersed population. It may also be more costly to deliver government services across a larger land area.1 8 Indeed, rural areas may su¤er from inadequate transport or electri…cation while urban areas have special needs associated with congestion, pollution or urban blight[START_REF] Treisman | The politics of intergovernmental transfers in post-soviet Russia[END_REF].

9 Two main concepts are used in the literature. Swing voters are de…ned as those with weak party preferences, while swing jurisdictions are those where the distance of vote shares between the central government party and its main opponent is minimal. We focus on the second one.

2 0 We also consider ethnic polarization[START_REF] Montalvo | Ethnic polarization, potential con ‡ict and civil war[END_REF]. 2 1 It represents around 8000 representative households for each survey.

6 Fiscal transfers have an important characteristic to lead a relevant statistical analysis: their amount varies signi…cantly over time, allowing the use of panel econometric methods.2 7 When we introduce dummy variables for election years we cannot introduce time dummies, so we will add Tt, a trend variable which accounts for the common trend in local governments'transfers.

Acknowledgements I am grateful to the Municipal Development Agency of Senegal for their valuable help in collecting data, especially to Mamadou Ndiaye. I also thank the Municipal Development Partnership, for their warm welcome, helpful comments and discussions. Thanks to G. Rota-Graziosi and M. Foucault for excellent advises and encouragements, to the participants of 16th Annual Conference of African Econometric Society (July 2011, Kenya) and to anonymous referees for their fruitful comments and suggestions. Any remaining errors are mine.


Matthieu Alfaro

Danielle Hilhorst

Interface dynamics of the porous medium equation with a bistable reaction term

Keywords: degenerate diffusion, singular perturbation, sharp interface limit, population dynamics. 2

We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.

Introduction

In this paper we consider the rescaled porous medium equation with a bistable reaction term

(P ε )            u t = ε∆(u m ) + 1 ε f (u) in Ω × (0, ∞) ∂(u m ) ∂ν = 0 on ∂Ω × (0, ∞) u(x, 0) = u 0 (x)
in Ω , 1 and study the sharp interface limit as ε → 0. Here Ω is a smooth bounded domain in R N (N ≥ 2), ν is the Euclidian unit normal vector exterior to ∂Ω and m > 1.

We assume that f is smooth, has exactly three zeros 0 < a < 1 such that

f ′ (0) < 0 , f ′ (a) > 0 , f ′ (1) < 0 , (1.1)
and that

1 0 mu m-1 f (u) du > 0 . (1.2)
The above assumption implies that the speed of the underlying degenerate travelling wave is positive (see subsection 3.1), so that the region enclosed by the limit interface is expanding (see below). This explains why the requirement (1.2) is convenient for the study of invasion processes.

As far as the initial data is concerned, we assume that 0 ≤ u 0 ≤ M (for some M > a) is a C 2 (Ω) function with compact support Supp u 0 := Cl{x ∈ Ω : u 0 (x) > 0} ⊂⊂ Ω .

Furthermore we define the initial interface Γ 0 by Γ 0 := {x ∈ Ω : u 0 (x) = a} , and suppose that Γ 0 is a smooth hypersurface without boundary, such that, n being the Euclidian unit normal vector exterior to Γ 0 , Γ 0 ⊂⊂ Ω and ∇u 0 (x) = 0 if x ∈ Γ 0 , (1.3)

u 0 > a in Ω (1) 0 , u 0 < a in Ω (0) 0 , (1.4)
where Ω

(1) 0 denotes the region enclosed by Γ 0 and Ω (0) 0 the region enclosed between ∂Ω and Γ 0 .

Problem (P ε ) possesses a unique weak solution u ε as it is explained in Section 2. As ε → 0, by formally neglecting the diffusion term, we see that, in the very early stage, the value of u ε quickly becomes close to either 1 or 0 in most part of Ω, creating a steep interface (transition layers) between the regions {u ε ≈ 1} and {u ε ≈ 0}. Once such an interface develops, the diffusion term is large near the interface and comes to balance with the reaction term. As a result, the interface ceases rapid development and starts to propagate in a slower time scale. Therefore the limit solution ũ(x, t) will be a step function taking the value 1 on one side of the moving interface, and 0 on the other side.

We shall prove that this sharp interface limit, which we denote by Γ t , obeys the law of motion

(P 0 ) V n = c * on Γ t Γ t t=0 = Γ 0 ,
where V n is the normal velocity of Γ t in the exterior direction, and c * the positive speed of the underlying travelling wave (see subsection 3.1). Problem (P 0 ) possesses a unique smooth solution on [0, T max ) for some T max > 0.

We denote this solution by Γ = ∪ 0≤t<T max (Γ t × {t}). From now on, we fix 0 < T < T max and work on [0, T ]. We set

Q T := Ω × (0, T ) ,
and, for each t ∈ [0, T ], we denote by Ω

t the region enclosed by the hypersurface Γ t , and by Ω (0) t the region enclosed between ∂Ω and Γ t . We define a step function ũ(x, t) by

ũ(x, t) := 1 in Ω (1) t 0 in Ω (0) t for t ∈ [0, T ] , (1.5)
which represents the formal asymptotic limit of u ε as ε → 0.

Our main result, Theorem 1.1, describes both the emergence and the propagation of the layers. First, it gives the profile of the solution after a very short initial period: the solution u ε quickly becomes close to 1 or 0, except in a small neighborhood of the initial interface Γ 0 , creating a steep transition layer around Γ 0 (generation of interface). The time needed to develop such a transition layer, which we will denote by t ε , is O(ε| ln ε|). Then the theorem states that the solution u ε remains close to the step function ũ on the time interval [t ε , T ] (motion of interface).

Theorem 1.1 (Generation and motion of interface). Assume m ≥ 2. Define µ as the derivative of f (u) at the unstable equilibrium u = a, that is

µ = f ′ (a) .
Let η ∈ (0, min(a, 1a)) be arbitrary. Fix α 0 > 0 arbitrarily small. Then there exist positive constants ε 0 and C such that, for all ε ∈ (0, ε 0 ) and for all (x, t) such that t ε ≤ t ≤ T , where

t ε := µ -1 ε| ln ε| , (1.6)
we have

u ε (x, t) ∈            [1 -2ε, 1 + η] if x ∈ Ω (1) t \ N Cε| ln ε| (Γ t ) [0, 1 + η] if x ∈ Ω [0, η] if x ∈ Ω (0) t \ N α 0 (Γ t ) , (1.7)
where N r (Γ t ) := {x ∈ Ω : dist(x, Γ t ) < r} denotes the r-tubular neighborhood of Γ t .

Remark 1.2 (About the thickness of the interface). Since the construction of super-solutions is much more involved than that of sub-solutions, the statement (1.7) is more accurate in Ω

(1) t than in Ω (0)

t . More precisely, on the one hand, (1.7) shows that the convergence to 1 is uniform "inside the interface" except in O(ε| ln ε|) tubular neighborhoods of the sharp interface limit; on the other hand, (1.7) only shows that the convergence to 0 is uniform "outside the interface" except in O(1) tubular neighborhoods of the sharp interface limit.

Remark 1.3 (About the assumption m ≥ 2). Note that the sub-and supersolutions we shall construct to study the motion of interface allow m > 1. Nevertheless, since we consider not well-prepared initial data, we need to quote a generation of interface result from [START_REF] Alfaro | Generation of interface for an Allen-Cahn equation with nonlinear diffusion[END_REF] which is valid only for m ≥ 2 (if 1 < m < 2 the partial differential equation is not only degenerate but also singular). When initial data have a "suitable shape", the restriction m ≥ 2 can be removed.

As a direct consequence of Theorem 1.1, we have the following convergence result.

Corollary 1.4 (Convergence). Assume m ≥ 2. As ε → 0, the solution u ε converges to ũ in ∪ 0<t≤T (Ω (i) t × {t}), where i = 0, 1.
For the relevance of nonlinear diffusion in population dynamics models, we refer the reader to Gurney and Nisbet [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF], Gurtin and Mac Camy [START_REF] Gurtin | On the diffusion of biological populations[END_REF]: density dependent diffusion is efficient to study the dynamics of a population which regulates its size below the carrying capacity set by the supply of nutrients. Since density dependent equations degenerate at points where u = 0, a loss of regularity of solutions occurs and their support propagates at finite speed.

Let us mention some earlier works on problems involving nonlinear diffusion that are related to ours. Feireisl [START_REF] Feireisl | Front propagation for degenerate parabolic equations[END_REF] has studied the singular limit of (P ε ) in the whole space R N , which allows to reduce the issue to the radially symmetric case. Hilhorst, Kersner, Logak and Mimura [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF] have investigated the singular limit of the equation posed in a bounded domain of R N , with f (u) of the Fisher-KPP type. Note that the authors in [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF] assume the convexity of Ω [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF] 0 which allows them to construct a single super-solution for both the generation and the motion of interface. Here, we dot not make such a geometric assumption.

The organization of this work is as follows. In Section 2, we briefly recall known results concerning the well-posedness of Problem (P ε ). Section 3 is the body of the paper: we construct sub-and super-solutions to study the motion of the transition layers. Finally, we prove Theorem 1.1 in Section 4.

Comparison principle, well-posedness

Since the diffusion term degenerates when u = 0 a loss of regularity of solutions occurs. We define below a notion of weak solution for Problem (P ε ), which is very similar to the one proposed by Aronson, Crandall and Peletier [START_REF] Aronson | Stabilization of solutions of a degenerate nonlinear diffusion problem[END_REF] for the one dimensional problem with homogeneous Dirichlet boundary conditions. Concerning the initial data, we suppose here that u 0 ∈ L ∞ (Ω) and u 0 ≥ 0 a.e. Note that in this subsection, and only in this subsection, we assume, without loss of generality, that ε = 1, which yields the Problem (P )

         u t = ∆(u m ) + f (u) in Ω × (0, ∞) ∂(u m ) ∂ν = 0 on ∂Ω × (0, ∞) u(x, 0) = u 0 (x) in Ω . Definition 2.1. A function u : [0, ∞) → L 1 (Ω) is a solution of Problem (P ) if, for all T > 0, (i) u ∈ C [0, ∞); L 1 (Ω) ∩ L ∞ (Q T ) ; (ii) for all ϕ ∈ C 2 (Q T ) such that ϕ ≥ 0 and ∂ϕ ∂ν = 0 on ∂Ω, it holds that Ω u(T )ϕ(T ) - Q T (uϕ t + u m ∆ϕ) = Ω u 0 ϕ(0) + Q T f (u)ϕ .
(2.1)

A sub-solution (respectively a super-solution) of Problem (P ) is a function satisfying (i) and (ii) with equality replaced by ≤ (respectively ≥).

Theorem 2.2 (Existence and comparison principle). The following properties hold.

(i) Let u -and u + be a sub-solution and a super-solution of Problem (P ε ) with initial data u - 0 and u + 0 respectively. If

u - 0 ≤ u + 0 a.e. then u -≤ u + in Q T ;
(ii) Problem (P ) has a unique solution u on [0, ∞) and

0 ≤ u ≤ max(1, u 0 L ∞ (Ω) ) in Q T ; (2.2) (iii) u ∈ C(Q T ).
The proof of the theorem above can be performed in the same lines as in [3, Theorem 5] (see also [START_REF] Kalašnikov | The nature of the propagation of perturbations in problems of nonlinear heat conduction with absorption[END_REF] and [START_REF] Bertsch | Positivity versus localization in degenerate diffusion equations[END_REF] for related results). The continuity of u ε follows from [START_REF] Dibenedetto | Continuity of weak solutions to a general porous medium equation[END_REF].

The following result turns out to be an essential tool when constructing smooth sub-and super-solutions of Problem (P ε ).

Lemma 2.3. Let u be a continuous nonnegative function in Ω × [0, T ]. Define Ω ⋆ t = {x ∈ Ω : u(x, t) > 0} and Γ ⋆ t = ∂Ω ⋆ t for all t ∈ [0, T ]. Suppose the family Γ := ∪ 0<t≤T Γ ⋆
t × {t} is sufficiently smooth and let ν ⋆ t be the outward normal vector on Γ ⋆ t . Suppose moreover that

(i) ∇(u m ) is continuous in Ω × [0, T ] (ii) L[u] := u t -∆(u m ) -f (u) = 0 in {(x, t) ∈ Ω × [0, T ] : u(x, t) > 0} (iii) ∂(u m ) ∂ν ⋆ t = 0 on ∂Ω ⋆ t , for all t ∈ [0, T ] .
Then u is a solution of Problem (P ). Similarly a sub-solution (respectively a super-solution) of Problem (P ) is a function satisfying (i) and (ii)-(iii) with equality replaced by ≤ (respectively ≥).

The proof of this result can be found in [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF].

3 Motion of the transition layers

Materials

Underlying travelling waves. Hosono [START_REF] Hosono | Traveling wave solutions for some density dependent diffusion equations[END_REF] has investigated travelling wave solutions for the degenerate one dimensional equation

u t = (u m ) xx + f (u) .
He proved that there exists a unique travelling wave (c * , U ), that the sign of the velocity c * is that of 1 0 u m-1 f (u) du, and that the profiles vary with the sign of the velocity. More precisely, for c * < 0, the front is smooth and U ∈ C ∞ (R), whereas, for c * > 0, we only have

(U m-1 ) ′ ∈ L ∞ (R), but (U m-1 ) ′ / ∈ C(R).
These different behaviors of the travelling waves are in contrast with the density independent diffusion models, where fronts are smooth whatever their velocities are (see [START_REF] Fife | The approach of solutions of nonlinear diffusion equations to traveling front solutions[END_REF]).

In the present paper, the assumption (1.2) implies that c * > 0. More precisely the following holds (see [START_REF] Hosono | Traveling wave solutions for some density dependent diffusion equations[END_REF] for details). The travelling wave (c * , U ) is the solution of the auxiliary problem

                       (U m ) ′′ + c * U ′ + f (U ) = 0 on (-∞, ω) U (-∞) = 1 U (0) = a U ′ < 0 on (-∞, ω) (U m ) ′ (ω) = 0 U ≡ 0 on [ω, ∞) , (3.1)
for some ω > 0. As z → -∞, terms are exponentially decaying:

max 1 -U (z), |U ′ (z)|, |U ′′ (z)| ≤ Ce -λ|z| for z ≤ 0 , (3.2)
for some positive constants C and λ. As z ր ω, we have

lim zրω U m-1 ′ (z) = - m -1 m c * and lim zրω (U m-1 ) ′′ (z) = - (m -1) 2 m 2 f ′ (0) , (3.3
) and U ′ (ω) ∈ [-∞, 0). Moreover, for a positive constant which we denote again by C, there holds

|(U m ) ′′ (z)| ≤ C|U ′ (z)| = -CU ′ (z) for all z ∈ (-∞, ω) .
(3.4)

The cut-off signed distance function. Another classical ingredient in similar situations (see [START_REF] Serrin | The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables[END_REF] or [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]) is a cut-off signed distance function d which we now define. Let d(•, t) be the signed distance function to Γ t , namely

d(x, t) := -dist(x, Γ t ) for x ∈ Ω (1) t dist(x, Γ t ) for x ∈ Ω (0) t , (3.5)
where dist(x, Γ t ) is the distance from x to the hypersurface Γ t . We remark that d(

ζ(s) =      s if |s| ≤ d 0 -2d 0 if s ≤ -2d 0 2d 0 if s ≥ 2d 0 .
We then define the cut-off signed distance function d by

d(x, t) := ζ d(x, t) . (3.7) Note that if |d(x, t)| < d 0 then |∇d(x, t)| = 1 , (3.8)
that d is constant (= 2d 0 ) in a neighborhood of ∂Ω, and that the equation of motion (P 0 ) yields

if |d(x, t)| < d 0 then d t (x, t) + c * = 0 . (3.9)
Moreover, there exists a constant C > 0 such that

|∇d(x, t)| + |∆d(x, t)| ≤ C for all (x, t) ∈ Q T .
(3.10)

Construction of sub-solutions

Equipped with the travelling wave (c * , U ) and the signed distance function d, we are looking for sub-solutions in the form

u - ε (x, t) := (1 -ε)U d(x, t) + ε| ln ε|p e t ε = (1 -ε)U (z - ε (x, t)) , (3.11)
where

z - ε (x, t) := d(x, t) + ε| ln ε|p e t ε .
(3.12)

Lemma 3.1 (Sub-solutions). Let p > 0 be arbitrary. Then, for ε > 0 small enough, u - ε is a sub-solution for Problem (P ε ).

Proof. In this proof (and only in this proof) we set u - ε = u and z

- ε = z. Note that Ω ⋆ t = {x ∈ Ω : d(x, t) < -ε| ln ε|p e t + εω} , (3.13)
where Ω ⋆ t is defined as in Lemma 2.3. It follows that u ≡ 0 near the boundary ∂Ω so that the Neumann boundary condition (iii) in Lemma 2.3 is fulfilled. Since (U m ) ′ (ω) = 0 we see that ∇(u m ) is continuous in Ω×[0, T ]. Therefore, by virtue of Lemma 2.3, it is enough to prove that

εL ε [u] := εu t -ε 2 ∆(u m ) -f (u) ≤ 0 in {(x, t) : d(x, t) < -ε| ln ε|pe t + εω} = {(x, t) : z(x, t) < ω}.
By using straightforward computations we get

εu t = (1 -ε) d t + ε| ln ε|p e t U ′ (z) ε 2 ∆(u m ) = (1 -ε) m |∇d| 2 (U m ) ′′ (z) + (1 -ε) m ε∆d(U m ) ′ (z) ,
where z = z(x, t). Then using the ordinary differential equation (U m ) ′′ + c * U ′ + f (U ) = 0, we see that

εL ε [u] = E 1 + • • • + E 4 ,
with

E 1 := (1 -ε) d t + c * -(1 -ε) m-1 ε∆d(mU m-1 )(z) + ε| ln ε|p e t U ′ (z) E 2 := (1 -ε) m (1 -|∇d| 2 )(U m ) ′′ (z) E 3 := ((1 -ε) -(1 -ε) m ) (U m ) ′′ (z) E 4 := -f ((1 -ε)U (z)) + (1 -ε)f (U (z)) .
In the following we shall denote by C some positive constants which do not depend on ε > 0 small enough (and may change from place to place).

We start with some observations on the term E 4 . Note that

f ((1 -ε)u) -(1 -ε)f (u) = -εuf ′ (θ) + εf (u) , (3.14)
for some θ ∈ ((1ε)u, u). Hence

|E 4 | ≤ Cε .
Moreover, since f (1) = 0 and f ′ (1) < 0, it follows from (3.14) that, for u sufficiently close to 1,

f ((1 -ε)u) -(1 -ε)f (u) ≥ βεu , (3.15)
for some β > 0. Hence since U (-∞) = 1, by choosing γ ≫ 1 we see that

E 4 ≤ -βεU (z) ≤ - 1 2
βε for all z ≤ -γ .

(3.16)

In the following we distinguish three cases, namely (3.17), (3.19) and (3.20).

Assume that

-ε| ln ε|pe t -εγ ≤ d(x, t) < -ε| ln ε|pe t + εω , (3.17)
which in turn implies that -γ ≤ z < ω. Since U ′ < 0 on (-∞, ω) and U ′ (ω) ∈ [-∞, 0), it holds that U ′ (z) ≤ -α, for some α > 0. If ε > 0 is small enough (3.8) shows that E 2 = 0; from (3.4) we deduce that |E 3 | ≤ -CεU ′ (z); moreover we have |E 4 | ≤ Cε. In view of (3.9), E 1 reduces to

E 1 = (1 -ε) -(1 -ε) m-1 ε∆d(mU m-1 )(z) + ε| ln ε|p e t U ′ (z) . (3.18) Since | -(1 -ε) m-1 ε∆d(mU m-1 )(z)| ≤ Cε, inequality E 1 ≤ 1 2 p ε| ln ε|U ′ (z) holds.
Collecting theses estimates we have

L ε [u] ≤ ( 1 2 p ε| ln ε| -Cε)U ′ (z) + Cε ≤ -1 4 p αε| ln ε| + Cε ≤ 0 , if ε > 0 is sufficiently small. Assume that -d 0 ≤ d(x, t) < -ε| ln ε|pe t -εγ , (3.19)
which in turn implies that z < -γ so that (3.16) implies E 4 ≤ 0 . Here again (3.8) shows that E 2 = 0, from (3.4) we deduce that |E 3 | ≤ -CεU ′ (z), and E 1 reduces to (3.18). Hence we collect

L ε [u] ≤ U ′ (z) [-(1 -ε)εC + (1 -ε)p ε| ln ε| -Cε] ≤ 0 , for ε > 0 small enough. Assume that -2d 0 ≤ d(x, t) < -d 0 , (3.20)
which in turn implies that, for ε > 0 small enough, z ≤ -d 0 2ε . In this range (3.8) and (3.9) no longer apply but the exponential decay (3.2) shows that

|E 1 | + |E 2 | + |E 3 | ≤ Ce -λ d 0 2ε . Last E 4 ≤ -1 2 βε (see (3.16)) shows that, for ε > 0 small enough, L ε [u] ≤ 0.
The lemma is proved.

Construction of super-solutions

The construction of super-solutions is more involved: since we want them to be positive it is no longer possible to use the natural travelling wave (c * , U ) which is compactly supported. Therefore we shall first consider slightly larger speeds c > c * which provide faster travelling wave solutions which tend to +∞ in -∞; then a small modification will provide us positive and more regular functions which are "nearly" travelling wave solutions. Before making this argument more precise, let us note that, as it will clearly appear below, the possibility of the above strategy follows from [START_REF] Hosono | Traveling wave solutions for some density dependent diffusion equations[END_REF].

Let η ∈ (0, min(a, 1a)) be arbitrary. Let α 0 > 0 be fixed. Let us recall that we have fixed 0 < T < T max , where T max denotes the time when the first singularities occur in (P 0 ). Therefore we can select ρ > 0 small enough so that the following holds. First the smooth solution (Γ c t ) of the free boundary problem

(P 0 c ) V n = c := c * + ρ on Γ c t Γ c t t=0 = Γ 0 ,
exists at least on [0, T ]. Secondly, if we denote by d c (x, t) the cut-off signed distance function associated with Γ c := ∪ 0≤t≤T (Γ c t ×{t}) then, for all (x, t)

∈ Q T , d(x, t) ≥ α 0 =⇒ d c (x, t) ≥ α 0 2 . (3.21)
Since c > c * , as explained in [12, Remark 3.1], there exists a faster travelling wave (c, V ) which satisfies the same requirements as (c * , U ) in the auxiliary problem (3.1), except that V (-∞) = +∞ rather than U (-∞) = 1. In particular, V is still compactly supported from one side.

Next, for all n ≥ 1, following the construction which comes before Proposition 4.1 in [START_REF] Hosono | Traveling wave solutions for some density dependent diffusion equations[END_REF] (it consists in slightly modifying the above travelling wave (c, V )), we can consider (c, U n ) such that (i) U n satisfies the ordinary differential equation

(U n m ) ′′ + cU n ′ + f (U n ) = 0 on some (-∞, Z n ) , where U n ′ < 0 holds (ii) U n is constant equal to some (δ n ) 1 m-1 > 0 on [Z n , ∞) (iii) (U n ) m-1 ∈ C 1 (R)
together with U n (0) = a and U n (-∞) = +∞. Moreover δ n → 0 as n → ∞, so that we can fix n 0 ≫ 1 such that (δ n 0 )

1 m-1 ≤ η.
As a conclusion, if we denote U n 0 , δ n 0 and Z n 0 by W , δ and Z we are now equipped with (c,

W ) such that W m-1 ∈ C 1 (R) and                    (W m ) ′′ + cW ′ + f (W ) = 0 on (-∞, Z) W (-∞) = +∞ W (0) = a W ′ < 0 on (-∞, Z) W ≡ δ 1 m-1 ≤ η on [Z, ∞) . (3.22)
We are now looking for super-solutions in the form

u + ε (x, t) := W d c (x, t) -ε| ln ε|Ke t ε . (3.23)
In the sequel we set

z + ε (x, t) := d c (x, t) -ε| ln ε|Ke t ε . (3.24)
Remark 3.2 (The sub-domain Σ). We shall consider below a sub-domain Σ whose slice at time t, namely σ t := {x : (x, t) ∈ Σ}, is the region enclosed between ∂Ω and (more or less) Γ c t . We shall prove that u + ε is a super-solution in Σ. Thanks to (4.4) this will be sufficient for our purpose (see Section 4).

Denote by -θ the point where W (-θ) = 1 + η. For each 0 ≤ t ≤ T , define the open set

σ t := {x ∈ Ω : d c (x, t) > ε| ln ε|Ke t -εθ} = {x : z + ε (x, t) > -θ} ,
and the sub-domain Σ := ∪ 0<t<T (σ t × {t}) .

Note that the lateral boundary of Σ is made of ∂ out Σ := ∂Ω × (0, T ) and ∂ in Σ := ∪ 0<t<T (s t × {t}) where s t denotes the smooth hypersurface

s t := {x ∈ Ω : d c (x, t) = ε| ln ε|Ke t -εθ} = {x : z + ε (x, t) = -θ} .
Lemma 3.3 (Super-solutions in Σ). Let η ∈ (0, min(a, 1a)) be arbitrary and let α 0 > 0 be fixed. Then, for all K > 0, all ε > 0 small enough, u + ε is such that

(i) L ε [u + ε ] := (u + ε ) t -ε∆((u + ε ) m ) - 1 ε f (u + ε ) ≥ 0 in Σ (ii) ∂((u + ε ) m ) ∂ν = 0 on ∂ out Σ = ∂Ω × (0, T ) (iii) u + ε ≡ 1 + η on ∂ in Σ = ∪ 0<t<T (s t × {t}) .
Proof. In this proof (and only in this proof) we put u + ε = u and z + ε = z. Recall that d c is constant near the boundary ∂Ω so that the Neumann boundary condition (ii) is fulfilled. Moreover the Dirichlet boundary condition (iii) is clear from the definition of s t and the fact that W (-θ) = 1 + η.

Therefore it remains to prove that εL ε

[u] = εu t -ε 2 ∆(u m ) -f (u) ≥ 0 in Σ = {(x, t) : z(x, t) > -θ}. If z(x, t) ≥ Z then L ε [u] = L ε [δ 1 m-1 ] ≥ 0.
We now assume that z(x, t) ∈ (-θ, Z), i.e. ε| ln ε|Ke tεθ < d c (x, t) < ε| ln ε|Ke t + εZ .

(3.25)

Straightforward computations combined with the ordinary differential equation (W m ) ′′ + cW ′ + f (W ) = 0 yield

εL ε [u] = (d c t + c)W ′ -ε| ln ε|Ke t W ′ -ε∆d c (W m ) ′ + (1 -|∇d c | 2 )(W m ) ′′ .
If ε > 0 is small enough, then (3.25) combined with (3.8) and (3.9) -with d c and c playing the roles of d and c * -shows that the above equality reduces to εL ε [u] = -εW ′ | ln ε|Ke t + ∆d c (mW m-1 ) .

Since W ′ ≤ 0 we have εL ε [u] ≥ 0, for ε > 0 small enough. The lemma is proved.

4 Proof of Theorem 1.1

A generation of interface property

We first state a result on the generation of interface.

Lemma 4.1 (Generation of interface). Assume m ≥ 2. Let η > 0 be arbitrary small. Then, for all x ∈ Ω, we have, for ε > 0 small enough,

0 ≤ u ε (x, t ε ) ≤ 1 + η , (4.1)
and there exists M 0 > 0 such that, for ε > 0 small enough,

u 0 (x) ≥ a + M 0 ε| ln ε| =⇒ u ε (x, t ε ) ≥ 1 -ε (4.2) u 0 (x) ≤ a -M 0 ε| ln ε| =⇒ u ε (x, t ε ) ≤ ε , (4.3)
where t ε = µ -1 ε| ln ε|.

Proof. We only give an outline since the arguments can be found in [START_REF] Alfaro | The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system[END_REF], [START_REF] Alfaro | Generation of interface for an Allen-Cahn equation with nonlinear diffusion[END_REF]. Denote by Y (τ ; ξ) the solution of the bistable ordinary differential equation Y τ = f (Y ) on (0, ∞) supplemented with the initial condition Y (0; ξ) = ξ. Modulo a change of the time variable, we can use the sub-and supersolutions constructed in [START_REF] Alfaro | Generation of interface for an Allen-Cahn equation with nonlinear diffusion[END_REF] to deduce that, for some C * > 0, for ε > 0 small enough,

Y t ε ε ; u 0 (x) -ε 2 C * (e µt ε /ε -1) + ≤ u ε (x, t ε ) ≤ Y t ε ε ; u 0 (x) + ε 2 C * (e µt ε /ε -1) + .
Next a straightforward modification of [1, Lemma 3.9] -which specifies the instability of the equilibrium Y ≡ a and the stability of the equilibria Y ≡ 0, Y ≡ 1-shows that for ε > 0 small enough, for all ξ ∈ (-1, 2), we have

Y (µ -1 | ln ε|; ξ) ≤ 1 + η ,
and that

ξ ≥ a + ε| ln ε| =⇒ Y (µ -1 | ln ε|; ξ) ≥ 1 -ε ξ ≤ a -ε| ln ε| =⇒ Y (µ -1 | ln ε|; ξ) ≤ ε .
The combination of the above arguments completes the proof of the lemma.

Proof of the main theorem

We are now in the position to prove Theorem 1.1. To that purpose we show that solutions are in between the propagation of interface sub-and super-solutions at time t ε .

Proof. Assume m ≥ 2. Let η ∈ (0, min(a, 1a)) be arbitrary. First note that the comparison principle directly implies that inequality (4.1) persists for later times, i.e.

u ε (x, t) ∈ [0, 1 + η] for all x ∈ Ω and all t ε ≤ t ≤ T . (4.4)
Since ∇u 0 = 0 everywhere on Γ 0 = {x ∈ Ω : u 0 (x) = a} and since Γ 0 is a compact hypersurface, we can find a positive constant M 1 such that

if d(x, 0) ≤ -M 1 ε| ln ε| then u 0 (x) ≥ a + M 0 ε| ln ε| if d(x, 0) ≥ M 1 ε| ln ε| then u 0 (x) ≤ a -M 0 ε| ln ε| , (4.5)
with M 0 the constant which appears in Lemma 4.1.

We first investigate the behavior "inside the interface". In view of (3.13), we can choose p > 0 large enough so that, for ε > 0 small enough, the subsolution u - ε defined in (3.11) is such that the set {x : u - ε (x, 0) > 0} is included in {x : d(x, 0) ≤ -M 1 ε| ln ε|}. Therefore, from the correspondence (4.5), the estimate (4.2) and the fact that u - ε (x, 0) ≤ 1ε, we deduce that, for all x ∈ Ω,

u - ε (x, 0) ≤ u ε (x, t ε ) . It follows from the comparison principle that u - ε (x, t -t ε ) = (1 -ε)U d(x, t -t ε ) + ε| ln ε|p e t-t ε ε ≤ u ε (x,
and prove below that u ε (x, t) ≥ 1 -2ε, for t ε ≤ t ≤ T . Note that, for ε > 0 small enough, d(x, tt ε ) = d(x, t) + c * t ε , (4.9) which, combined with (4.6) and (4.7), implies

u ε (x, t) ≥ (1 -ε)U -2 λ | ln ε| ≥ (1 -ε)(1 -Cε 2 ) ≥ 1 -2ε ,
where we have used (3.2).

Last we investigate the behavior "outside the interface". Fix α 0 > 0 arbitrarily small. For such α 0 > 0, we follow the strategy of subsection 3.3 to construct super-solutions in Σ. More precisely define K = 2M 1 , where M 1 > 0 is the constant that appears in (4.5); then construct u + ε (x, t) as in (3.23). In view of Lemma 3.3 (ii), (iii), the super-solution u + ε (x, t) and the solution u ε (x, t + t ε ) satisfy a Neumann boundary condition on ∂ out Σ andtaking advantage of (4.4)-are ordered on ∂ in Σ. Last we claim that (note that d c (x, 0) = d(x, 0) since Γ c 0 = Γ 0 )

u ε (x, t ε ) ≤ W d(x, 0) -2M 1 ε| ln ε| ε = u + ε (x, 0) , (4.10)
for all x ∈ σ 0 = {x : d(x, 0) > 2M 1 ε| ln ε| -εθ}. Indeed, (4.5) and (4.3) show that u ε (x, t ε ) ≤ ε and the conclusion (4.10) follows from the fact that δ

α 0 3 -δ 1 m- 1 1 m- 1 .

 31111 2M 1 e T ε| ln ε| ε = ≤ η , since W (+∞) = δ Conclusion follows from (4.11). Theorem 1.1 is proved.

  t) ,(4.6) for all (x, t) ∈ Ω × [t ε , T ]. We choose C ≫ max(c * µ -1 , p e T , λ) so that -Cε| ln ε| + c * µ -1 ε| ln ε| + ε| ln ε|p e t-t ε Cε| ln ε| (Γ t ), i.e.

	ε	≤ -	C 2	| ln ε| ≤ -	2 λ	| ln ε| , (4.7)
						(1) t	\
	N

where λ > 0 is the constant appearing in (3.2). We take x ∈ Ω d(x, t) ≤ -Cε| ln ε| ,

  ≤ W . Hence the comparison principle yieldsu ε (x, t) ≤ W d c (x, tt ε ) -ε| ln ε|2M 1 e t-t ε ε = u + ε (x, tt ε ) , (4.11) for all (x, t) ∈ Σ with t ε ≤ t ≤ T . We take x ∈ Ω \ N α 0 (Γ t ), i.e. d(x, t) ≥ α 0 ,(4.12)and prove below that u ε (x, t) ≤ η, for t ε ≤ t ≤ T . From (3.21) we deduce that (x, t) ∈ Σ so that (4.11) applies. Note also that d c (x, tt ε ) = d c (x, t) + ct ε . Therefore we infer from (3.21) that, for ε > 0 small enough, d c (x, tt ε ) ≥ α 0 3 which in turn implies u +

	1
	m-1 (0)

t ε (x, tt ε ) ≤ W

The first author is supported by the French Agence Nationale de la Recherche within the project IDEE (ANR-2010-0112-01).

AMS Subject Classifications:

35K65, 35B25, 35R35, 92D25.