MD Aristoteles Giagounidis

Susanna Leto Di Priolo

Susanne Ille

Pierre Fenaux

A European survey on the detection and management of iron overload in transfusiondependent patients with MDS

Keywords: MIDIS, Myelodysplastic syndromes, Transfusions, Iron overload, Iron chelation, Survey

To better understand the detection and management of iron overload in transfusion-dependent patients with myelodysplastic syndromes (MDS), a 15-minute web-or paper-based survey was conducted among 338 European physicians from 27 countries. Respondents had a mean of 18 years' clinical experience. 46% and 27% of physicians noted that detecting and treating iron overload were either 'very important' or 'important', respectively. The main reason for not actively exploring iron overload was related to poor patient prognosis, while the main reasons for not initiating iron chelation therapy were poor patient prognosis and older patient age. 37% and 31% of physicians believed that treating iron overload in these patients was 'very important' or 'important', respectively. 90% of physicians prescribed iron chelation therapy, and 38% of transfusion-dependent patients received iron chelation therapy. The key reasons for not initiating iron chelation therapy related to poor patient prognosis (72%), patient age ≥85 years (50%) and comorbidities (34%). The views of these experienced MDS physicians reflect available international MDS treatment guidelines.

Introduction

Red blood cell (RBC) transfusions are a common therapy to treat symptomatic anemia that affects most patients with myelodysplastic syndromes (MDS) [START_REF] Hellström-Lindberg | Management of anemia associated with myelodysplastic syndrome[END_REF]. Several retrospective studies have shown that patients who become RBC transfusion dependent have a significantly shorter overall survival than those who are not dependent on transfusions [START_REF] Malcovati | Prognostic factors and life expectancy in myelodysplastic syndromes classified according to WHO criteria: a basis for clinical decision making[END_REF][START_REF] Sanz | Independent impact of iron overload and transfusion dependency on survival and leukemic evolution in patients with myelodysplastic syndrome[END_REF]. This decreased survival may in part be due to patients accumulating excess iron and/or to intrinsically more severe bone marrow disease than in non-dependent patients. International Prognostic Scoring System (IPSS) Low-and Intermediate-1 (Int-1)-risk patients, who may have prolonged survival, receive blood transfusions for many years and may be exposed to a greater risk of iron overload, leading to progressive dysfunction in organs such as the liver, heart and endocrine glands [START_REF] Malcovati | Prognostic factors and life expectancy in myelodysplastic syndromes classified according to WHO criteria: a basis for clinical decision making[END_REF][START_REF] Bennett | Consensus statement on iron overload in myelodysplastic syndromes[END_REF].

Guidelines from Spain, [START_REF] Arrizabalaga | Guía clínica de quelación del paciente con síndrome mielodisplásico [Clinical guide to chelation therapy for patients with myelodysplastic syndrome (Spanish Guidelines)[END_REF] The Italian Society of Hematology, [START_REF] Santini | Clinical management of myelodysplastic syndromes: update of SIE, SIES, GITMO practice guidelines[END_REF] The UK MDS Guidelines Group, [START_REF] Bowen | Guidelines for the diagnosis and therapy of adult myelodysplastic syndromes[END_REF] The National Comprehensive Cancer Network in the USA [START_REF]NCCN Clinical Practice Guidelines in Oncology v.2: Myelodysplastic Syndromes[END_REF] and The MDS Foundation [START_REF] Bennett | Consensus statement on iron overload in myelodysplastic syndromes[END_REF] all recommend the use of iron chelation therapy in patients with Low-or Int-1-risk MDS who have received more than 20 to 50 transfusions. In Europe, these recommendations, although not based on controlled clinical trials demonstrating superiority of iron chelation versus observation alone, have an impact on the perceptions and prescriptions of iron chelation therapy in patients with MDS. There is currently some controversy regarding the use of iron chelation therapy in these patients, particularly because they have a median age of 70 years, and many patients with a poor prognosis may not survive long enough to develop the clinical consequences of iron overload [START_REF] Deloughery | Iron: The fifth horseman of the apocalypse?[END_REF][START_REF] Tefferi | Iron chelation therapy in myelodysplastic syndrome -Cui bono?[END_REF][START_REF]Myelodysplastic syndromes: Information and education[END_REF]. There is some argument, however, that even patients with higher-risk MDS may also benefit from iron chelation therapy through improved outcome of allogeneic stem cell transplantation, possible decreases in infection, and, although controversial, delayed leukemic transformation in some cases [START_REF] Santini | Clinical management of myelodysplastic syndromes: update of SIE, SIES, GITMO practice guidelines[END_REF][START_REF] Pullarkat | Objectives of iron chelation therapy in myelodysplastic syndromes: more than meets the eye?[END_REF].

In order to understand what physicians treating transfusion-dependent patients with MDS in

Europe believe to be the reasons for their willingness or reservations towards detecting and managing iron overload, a survey of European physicians has been conducted. The MDS Iron-overload Detection Insight Survey (MIDIS) was conducted by the MDS Foundation and the European School of Haematology (ESH) in partnership with Novartis Oncology. The results of the survey are presented here.

Design and Methods

Inclusion/exclusion criteria

To participate in the study, respondents had to be physicians from a European country, either fully qualified, in training or completing their internship, and seeing at least one transfusiondependent patient with MDS in an average 6-month period.

MIDIS was initiated at the 2008 meeting of the American Society of Hematology and was also distributed at other international conferences via e-mails, letters and flyers, as well as links to the survey hosted on the MDS Foundation and ESH websites.

Study questionnaire

MIDIS was a quantitative survey that comprised up to 33 structured questions that could be completed either on paper or online in approximately 15 minutes. The questions were reviewed by a study steering committee, comprising the authors and external experts, prior to being included in the survey. The questions related to: profiling and demographics of the respondents (qualification status, primary specialty, work setting, number of transfusiondependent patients with MDS seen, involvement in MDS); detection of iron overload (frequency of serum ferritin testing, influence of serum ferritin tests on treatment decisions; importance of detecting iron overload, potential reasons for and reservations against detection); and treatment of iron overload with chelation therapy (importance of treatment, use of iron chelation therapy, markers for therapy initiation, number of patients who receive iron chelation therapy, potential reasons for and reservations against iron chelation therapy).

Questions relating to detection and treatment of iron overload were answered using a scale from 1 (indicating not important at all) to 7 (very important). Questions relating to barriers and triggers for detecting and treating iron overload were also answered using a scale from 1 (indicating not a barrier or trigger at all) to 7 (a strong barrier or trigger).

Analysis

Collection and analysis of the MIDIS results were undertaken by the HealthCare Division of GfK SE. The results for both respondents' background, and barriers and triggers toward detection and treatment of iron overload, were analyzed quantitatively.

Results

Respondent demographics

A total of 338 physicians from 27 European countries participated in the survey (Table 1).

Their profile shows that they were experienced clinicians specializing mostly in hematology (Table 2). A total of 88% were involved in diagnosing patients with MDS, 87% initiated treatment for MDS, and the same proportion was involved in the maintenance of treatment for MDS. The survey was completed by the respondents themselves.

Detection of iron overload

On a scale from 1 (not important at all) to 7 (very important), the majority of the respondents believed that detection of iron overload in transfusion-dependent patients with MDS was 'very important' (score 7; 46% of respondents) or 'important' (score 6; 27%; Fig. 1). The mean score for the subjective importance of detecting iron overload in transfusion-dependent patients with MDS was 6.1. On a similar 1-7 scale, most respondents believed that it was 'very likely' (score 7; 40%) or 'likely' (score 6; 25%) that iron overload would be detected at their institution (mean score = 5.8). Approximately half of the respondents (53%) said that they monitored serum ferritin levels in their transfusion-dependent patients every 3 months, with 26% responding that the frequency of serum ferritin monitoring depended on the rate of transfusions that the patient was receiving. Respondents said that in 46% of cases, the results of serum ferritin monitoring 'always' influenced their decisions on how they treated their patients, and 'sometimes' influenced this in 47% of cases.

Fig. 1 Subjective importance of detection of iron overload

The factors that positively influenced detection of iron overload included: patients having received >20 RBC units (relevant trigger in 76% of respondents); introduction of serum ferritin as a standard test (76%); and awareness of the potential risks of frequent transfusions (75%). On a scale from 1 (not an aid at all) to 7 (a strong aid), the mean scores for these factors were 6.1, 6.1 and 6.0, respectively.

The most important reasons for not detecting iron overload were poor patient prognosis, irregular serum ferritin monitoring, as well as lack of awareness about guidelines and risks of iron overload (Fig. 2). Other reasons included a low priority for iron-overload screening, and perceptions about the importance of iron overload in MDS.

Percentages in each row may not add up to 100% because of rounding

Respondents' opinions were polarized about whether or not regularly monitoring serum ferritin levels was a reason for not detecting iron overload. While 30% said that this 'strongly prevented' or 'prevented' them from detecting iron overload, a similar proportion of respondents (38%) believed that this 'did not', or 'did not at all', prevent them from detecting iron overload (Fig. 2).

Treatment of iron overload with iron chelation therapy

On a scale from 1 (not important at all) to 7 (very important), respondents said that they believed that treating iron overload in transfusion-dependent patients with MDS was 'very important' (score 7; 37% of the respondents) or 'important' (score 6; 31%) the mean score was 5.9 (Fig. 3). The survey showed that 90% of respondents prescribed iron chelation therapy to their transfusion-dependent patients with MDS. Nonetheless, not all of these patients received iron chelation therapy; out of a mean of 34 transfusion-dependent patients with MDS seen in each of the respondents' practices or institutions in an average 6-month period, 13 (38%) received iron chelation therapy.

Fig. 3 Subjective importance of treating iron overload

The factor that most led respondents to initiate iron chelation therapy was patient age of 55-64 years (relevant trigger in 77% of respondents). On a scale from 1 (does not lead the respondent to initiate iron chelation therapy at all) to 7 (strongly leads to initiation of iron chelation therapy), the mean score for this factor was 5.9. Other factors that most led respondents to initiate iron chelation therapy included: serum ferritin levels >1000 ng/mL (76%; mean score 6.0); candidacy for allogeneic stem cell transplantation (76%; mean score 6.0); need to prevent organ dysfunction (74%; mean score 6.0); ≥2 RBC units transfused per month (71%; mean score 5.9); lifetime transfusions of >20 RBC units (68%; mean score 5.8); and convenience of oral iron chelation therapy (67%; mean score 5.8).

The majority of respondents who prescribe iron chelation therapy (92%) felt that a patient's serum ferritin reaching a specific level was a marker at which to initiate iron chelation therapy. The mean serum ferritin level at which these respondents initiated iron chelation therapy was 1130 ng/mL. More than half of the respondents who prescribe iron chelation therapy (56%) said that reaching a certain number of blood units transfused was also a marker (mean of 21 units). Other markers for initiating iron chelation therapy included: patients having received a particular number of transfusions (mean of 18 transfusions; 41% of respondents who prescribe iron chelation therapy); and the rate of increase of a patient's serum ferritin level (33% of respondents who prescribe iron chelation therapy).

The strongest reasons for not initiating iron chelation therapy were mostly related to poor patient prognosis and patient age ≥85 years (Fig. 4). There were polarized opinions regarding the importance of some of the reasons for not initiating iron chelation therapy. For example, while 31% of respondents replied that a high-risk MDS classification was a strong barrier to initiating iron chelation therapy, 25% thought that this did not represent a significant barrier.

Similarly, while 28% said that expected non-compliance was a strong barrier to initiating iron chelation therapy, 23% thought this was a weak barrier. Some of the reasons for not initiating iron chelation therapy specifically related to the age of the patients being treated. Older patient age was less of a reason to initiate iron chelation therapy, and more of a reason not to, than younger patient age.

Percentages in each row may not add up to 100% because of rounding

Discussion

This survey was undertaken to better understand what physicians perceive to be the key reasons for investigating iron overload and managing iron overload when treating transfusion-dependent patients with MDS, and the reasons that they do not. The respondents who took part in MIDIS were from multiple European countries, and the majority were experienced physicians who appeared to have a keen interest in the MDS therapy area. Over a quarter had more than 25 years' experience working as physicians and the majority were actively involved in the diagnosis and treatment of patients with MDS. The survey has the known limitations of optional surveys; i.e., it was performed on invitation at conferences or through e-mail alerts and weblinks. Therefore, the survey may have been biased towards respondents with an interest in MDS, and who may have had a greater understanding of the management of iron overload than the general population of hematologists. Furthermore, MIDIS is not a representative sample of the hematologists from each country, and the number of respondents from each country does not accurately reflect the proportion of that country's hematologists in Europe.

Nonetheless, the results of the survey provide a valuable insight into the practices and beliefs of European physicians treating transfusion-dependent patients with MDS. MIDIS showed that approximately 70% of the respondents believed that detecting and treating iron overload were important in patients with MDS. A total of 76% of physicians felt that a serum ferritin level >1000 ng/mL was a relevant trigger to initiate iron chelation therapy. Over 50% of the respondents monitored serum ferritin levels quarterly as a surrogate marker for iron overload in their patients and, overall, 38% of transfusion-dependent patients with MDS were actively being treated for iron overload with iron chelation therapy. In Europe the prevalence of MDS patients with Low/Int-1-risk MDS (according to the IPSS) is reported to be approximately 70% [START_REF] Malcovati | Prognostic factors and life expectancy in myelodysplastic syndromes classified according to WHO criteria: a basis for clinical decision making[END_REF], and 39-50% of those patients are transfusion dependent [START_REF] Brechignac | Quality of life and economic impact of red blood cell (RBC) transfusions on patients with myelodysplastic syndromes (MDS)[END_REF]. Therefore, approximately one-third of patients will be transfusion dependent and may be candidates for iron chelation therapy, according to published guidelines. The 38% response by the survey respondents reflects their active iron chelation treatment strategy. This is confirmed by a recent study showing that 41% of eligible patients with lower-risk MDS received iron chelation therapy in clinical practice [14]. In all, the views expressed in the survey are in line with a number of international guidelines that recommend that serum ferritin levels should be assessed 3-4 times per year in transfusion-dependent patients with MDS, and that iron chelation therapy should be initiated when their serum ferritin level reaches 1000 ng/mL, depending on the transfusion rate [START_REF] Bennett | Consensus statement on iron overload in myelodysplastic syndromes[END_REF][15][16][17]. Interestingly, fewer than a quarter of the respondents considered concerns about potential reductions in patient quality of life, the cost of iron chelation therapy and possible side effects of therapy as strong reasons for not initiating treatment. Only 23%, 19% and 11% of respondents, respectively, reported these as barriers or strong barriers to initiating treatment. Furthermore, the issue of potentially reduced quality of life was polarized; the proportion of respondents who considered this to be a barrier was similar to the proportion who did not (23% and 26%, respectively).

Participating physicians were asked whether considering iron overload to be a lesser problem in patients with MDS was a barrier to detecting iron overload. While 17% thought this was a barrier or strong barrier, 30% were strongly opposed to that view. This reflects the need for evidence-based data regarding the role of iron chelation in MDS. Serum ferritin is a wellknown acute-phase reactant that might overestimate the iron load in the presence of other diseases, in particular infectious episodes [18]. More accurate measurements of iron overload include liver and heart magnetic resonance imaging; however, these are time consuming and not available at many centers. This may be one reason why one third of MIDIS respondents said that they believed that irregular monitoring of serum ferritin levels was a strong barrier to detecting iron overload, while 38% thought that this did not prevent detection.

The survey respondents were reluctant to administer iron chelation therapy to patients aged ≥85 years. It is difficult to assess how transfusion dependence affects the overall survival of this elderly patient population with Low-or Int-1-risk MDS. Nonetheless, it is interesting to note that the proportion of physicians opting against iron chelation therapy for this population was higher than that of physicians opting against treatment for patients with a life expectancy of less than one year (Figure 4). Instead, patient age of 55-64 years was a strong reason for initiating iron chelation therapy.

In

Fig. 2

 2 Fig.2The most important reservations about the detection of iron overload

Fig. 4

 4 Fig.[START_REF] Bennett | Consensus statement on iron overload in myelodysplastic syndromes[END_REF] The most important reservations about initiating iron chelation therapy. Abbreviations: RAEB refractory anemia with excess blasts; CMML chronic myelomonocytic leukemia; WHO World Health Organization

  Figure 1

Percentages in each row may not add up to 100% because of rounding Figure 4

 4 Figure 2

Table 1

 1 Countries represented in the survey

	Country	Responses Country	Responses
	Austria	8	Macedonia	6
	Belgium	22	Malta	1
	Bulgaria	4	Netherlands	2
	Croatia	4	Norway	2
	Czech Republic	19	Poland	11
	Denmark	5	Portugal	9
	Estonia	4	Romania	6
	Finland	1	Serbia	1
	France	43	Slovakia	19
	Germany	20	Spain	45
	Greece	16	Sweden	4
	Hungary	3	Switzerland	7
	Italy	39	UK	36
	Lithuania	1

Total number of responses 338 Table 2 Respondent characteristics Characteristic n = 338

	Mean age, years	44
	Male : female, n (%)	183 (54) : 155 (46)
	Primary specialty, n (%)
	Hematologist	312 (92)
	Oncologist	8 (2)
	General practitioner	4 (1)
	Transfusionist	2 (0.6)
	Cytogeneticist	1 (0.3)
	Other physician	11 (3)
	Qualification, n (%)
	Fully qualified	293 (87)
	In training or completing internship	45 (13)
	Work place, n (%)
	Teaching hospital	199 (59)
	General hospital	95 (28)
	Office-based, cancer center, private hospital or other	44 (13)
	Experience as a physician, mean number of years*	18
	<16 years' experience, n (%)	159 (47)
	≥16 years' experience, n (%)	177 (52)
	Mean number of patients with MDS seen per month, n	18
	As a proportion of total patients seen, %	12

Mean number of transfusion-dependent patients with MDS seen over a 6-month period, n 19 *Two (1%) respondents did not answer this question

  summary, this European survey has demonstrated that physicians believe that iron overload is a relevant clinical issue in transfusion-dependent patients with MDS that is worthy of investigation and treatment. Concerns over age and patient prognosis are key factors in the treatment decisions taken by physicians. An ongoing, randomized Phase III trial in patients with Low-and Int-1-risk MDS on iron overload in MDS will help to clarify the role of iron chelation therapy in this patient population. 14. Raptis A, Duh MS, Wang S-T, Dial E, Fanourgiakis I, Fortner B, Paley C, Mody-Patel N, Corral M, Scott J (2010) Treatment of transfusional iron overload in patients with myelodysplastic syndrome or severe anemia: data from multicenter clinical practices.

		Transfusion 50:190-199
		15. Suzuki T, Tomonaga M, Miyazaki Y, Nakao S, Ohyashiki K, Matsumura I, Kohgo Y,
		Niitsu Y, Kojima S, Ozawa K (2008) Japanese epidemiological survey with consensus
		statement on Japanese guidelines for treatment of iron overload in bone marrow failure
		syndromes. Int. J Hematol 88:30-35
		16. Wells RA, Leber B, Buckstein R, Lipton JH, Hasegawa W, Grewal K, Yee K, Olney HJ,
		Larratt L, Vickars L, Tinmouth A (2008) Iron overload in myelodysplastic syndromes:
		a Canadian consensus guideline. Leuk Res 32:1338-1353
		17. Gattermann N, Porter J, Lopes LF, Seymour J (2005) Consensus statement on iron
		overload in myelodysplastic syndromes. Hematol. Oncol. Clin. North Am 19(Suppl
		1):18-25
		18. Jabbour E, Garcia-Manero G, Taher A, Kantarjian HM (2009) Managing iron overload
		in patients with myelodysplastic syndromes with oral deferasirox therapy. Oncologist.
		14:489-496
	1-2:	3 4	5	6	7:
	ot important				Very important
	(at all)
	0%	2% 5%	20%	27%	46%

Table 1

 1 Countries represented in the survey

	Country	Responses Country	Responses
	Austria	8	Macedonia
	Belgium	22	Malta
	Bulgaria	4	Netherlands
	Croatia	4	Norway
	Czech Republic	19	Poland	11
	Denmark	5	Portugal
	Estonia	4	Romania
	Finland	1	Serbia
	France	43	Slovakia	19
	Germany	20	Spain	45
	Greece	16	Sweden
	Hungary	3	Switzerland
	Italy	39	UK	36
	Lithuania	1
	Total number of responses		338

Table Table 2

 Table Respondent characteristics

	Characteristic	n = 338
	Mean age, years	44
	Male : female, n (%)	183 (54) : 155 (46)
	Primary specialty, n (%)
	Hematologist	(92)
	Oncologist	8 (2)
	General practitioner	4 (1)
	Transfusionist	2 (0.6)
	Cytogeneticist	1 (0.3)
	Other physician	11 (3)
	Qualification, n (%)
	Fully qualified	(87)
	In training or completing internship	45 (13)
	Work place, n (%)
	Teaching hospital	(59)
	General hospital	95 (28)
	Office-based, cancer center, private hospital or other	44 (13)
	Experience as a physician, mean number of years*	18
	<16 years' experience, n (%)	(47)
	≥16 years' experience, n (%)	(52)
	Mean number of patients with MDS seen per month, n	18
	As a proportion of total patients seen, %	12
	Mean number of transfusion-dependent patients with MDS	19
	seen over a 6-month period, n
	*Two (1%) respondents did not answer this question

Table 2 *Conflict of interest Click here to download Conflict of interest: COI form confirmation letter.doc

 2

Acknowledgements

This study was sponsored by Novartis Farma SpA. Financial support for medical editorial assistance was provided by Novartis Farma SpA. We thank Kathy Heptinstall, Didi Jasmin,

The MDS Foundation and The European School of Haematology for their assistance and support with MIDIS and the manuscript. We thank Roy Mazucco for medical editorial assistance with this manuscript.

Authors' contribution

AG and PF drafted the manuscript. SI analysed the data. AG, DJ, SLdP, SI and PF contributed to the study design, data interpretation, and reviewed and provided their comments on this manuscript.

Conflict of Interest

AG reports receiving consultancy and speaker bureau fees, and is a member of the board of directors or advisory committee for Novartis Pharma AG, Amgen Inc, GlaxoSmithKline plc,


Sepideh Mirrahimi
email: mirrahimi@ann.jussieu.fr

Guy Barles
email: barles@lmpt.univ-tours.fr

Benoît Perthame
email: benoit.perthame@upmc.fr

‡ Panagiotis

E Souganidis
email: souganidis@math.uchicago.edu

A singular Hamilton-Jacobi equation modeling the tail problem

Keywords: Reaction-diffusion equations, Asymptotic analysis, Hamilton-Jacobi equation, Survival threshold, Population biology, quasi-variational inequality, Free boundary. AMS Class. No: 35B25, 35K57, 49L25, 92D15

We study the long-time/long-range behavior of reaction diffusion equations with negative square root-type reaction terms. In particular we investigate the exponential behavior of the solutions after space and time are scaled in a hyperbolic way by a small parameter. This leads to a new type of quasi-variational inequality for a Hamilton-Jacobi equation. The novelty is that the obstacle, which defines the open set where the solutions of the reaction diffusion equation do not vanish in the limit, depends on the solution itself. Counter-examples show a nontrivial lack of uniqueness for the variational inequality depending on the conditions imposed on the free boundary of this open set. Both Dirichlet and state constraints boundary conditions play a role. When the competition term does not change sign, we can identify the limit while, in general, we only obtain lower and upper bounds.

Although models of this type are rather old and extinction phenomena are as important as blow-up, our motivation comes from the so-called "tail problem" in population biology. One way to avoid meaningless exponential tails is to impose a singular mortality rate below a given survival threshold. Our study shows that the precise form of this singular mortality term is asymptotically irrelevant and that, in the survival zone, the population profile is impacted by the survival threshold (except in the very particular case when the competition term is nonpositive).

Introduction

We study the asymptotic behavior, as ε → 0, of the solutions to reaction-diffusion equations (with singular reaction term) of the form

n ε,t -ε∆n ε = 1 ε n ε R -1 ε (β ε n ε ) 1/2 in R d × (0, +∞), n ε = e u 0 ε /ε on R d × {0}, (1)
where u 0 ε : R d → R is a given function, R : R d → R represents a linear logistic growth/death rate and the survival threshold parameter β, which models a singular death term, is given, for some u m < 0, by

β ε = e um/ε . (2)
The positive parameter ε is introduced by a hyperbolic scaling (x, t) → (x/ε, t/ε) with the aim to describe the long time and long range behavior of the unscaled problem (corresponding to ε = 1). The limiting behavior of scaled reaction-diffusion equations with KPP-type reaction has been studied extensively in, among other places, the theory of front propagation ([4, 19, 12]) using the so called WKB-(exponential) change of the unknown.

The novelty of the problem we are considering here is the presence of the negative square root term. To the best of our knowledge, the first study of such nonlinearity goes back to [START_REF] Evans | Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities[END_REF] where it is proved that local extinction occurs, i.e., the solution can vanish in a domain and stay positive in another region. For this reason β is thought to represent a survival threshold. That a solution of a parabolic problem can vanish locally is a surprising effect and as singular as the blow-up phenomena for supercritical reactions terms ( [START_REF] Quittner | Superlinear parabolic problems. Blow-up, global existence and steady state[END_REF]). In population biology such behavior prevents the so-called "tail problem" where very small (and thus meaningless) populations can generate artifacts ( [START_REF] Gauduchon | Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations[END_REF]). Although the mathematical analysis of the limit of (1) turns out to be a full subject in itself, our primary motivation comes from qualitative questions in population dynamics.

Indeed [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF] is the simplest model for studying the effect of "cutting the tail" but many other problems are relevant in ecology. Along the same lines, in the context of front propagation, one may consider the modified Fisher-KPP equation

n ε,t -ε∆n ε = 1 ε n ε (1 -n ε ) - 1 ε (β ε n ε ) 1/2 in R d × (0, +∞),
and ask the question whether the square root term changes fundamentally the study in [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF] and [START_REF] Gauduchon | Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations[END_REF] of the propagation of the invading/combustion fronts. In the context of speciation, an elementary model in adaptive evolution is the non-local reaction-diffusion equation

n ε,t -ε∆n ε = 1 ε n ε R(x, I ε ) - 1 ε (β ε n ε ) 1/2 in R d × (0, +∞) with I ε (t) = ψ(x)n ε (x, t)dx,
where n ε is the population density of individuals with phenotypical trait x, R represents the net growth rate, ψ is the consumption rate of individuals and I(t) is the total consumption of the resource at time t. The survival threshold was introduced in [START_REF] Gauduchon | Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations[END_REF]. Finally ε may represent large time and small mutations as studied in [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF][START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF]. It is known that under some assumptions the density concentrates as an evolving Dirac mass for the fittest trait. In biological terms this means that one or several dominant traits survive while others become extinct. Phenomena such as the discontinuous jumps of the fittest trait, non smooth branching and fast dynamics compared to stochastic simulations, motivated [START_REF] Gauduchon | Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations[END_REF] to improve the model by including a survival threshold. Numerical results confirm that this modification gives dynamics comparable to stochastic models. It is interesting to investigate rigorously whether the dynamics of the Dirac concentration points are really changed by the survival threshold and to explain why its specific form (n

1/2 ε
versus n γ ε with 0 < γ < 1) seems irrelevant. A way to approach these questions for (1) is through the asymptotic analysis of n ε . Since, as in the classical case, i.e., the Fisher-KPP equation without the square root term (see [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF]), n ε decays exponentially, the limit is better described using the Hopf-Cole transformation

u ε = ε ln n ε , (3)
which, for u 0 ε = ε ln n 0 ε , leads to the "viscous" Hamilton-Jacobi initial value problem

   u ε,t -ε∆u ε -|Du ε | 2 = R -exp u m -u ε 2ε in R d × (0, +∞), u ε = u 0 ε in R d × {0}. (4)
Throughout the paper we assume that there exist C > 0 and

u 0 ∈ C 0,1 (R d ) such that R C 0,1 ≤ C and u 0 C 0,1 ≤ C, (5)
u 0 ε ∈ C(R d ), u 0 ε ≤ C and u 0 ε -→ ε→0 u 0 in C(R d ). (6)
In the limit ε → 0, it is easy to see, at least formally, that any local uniform limit of the family (u ε ) ε>0 will satisfy, in the sense of the Crandall-Lions viscosity solutions ( [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF]), the Hamilton-Jacobi free boundary problem

           u t = |Du| 2 + R in Ω ⊂ R d × (0, ∞), u = -∞ in Ω c ∩ (R d × (0, ∞)), u ≥ u m in Ω, u = u 0 in Ω ∩ (R d × {0}), (7)
with the space-time open set Ω defined by

Ω = Int (x, t) ∈ R d × (0, ∞) : lim ε→0 u ε (x, t) > -∞ .
Notice that ( 7) is an obstacle problem with an obstacle depending on the solution itself. As a matter of fact the open set Ω plays an important role and, hence, the problem may be better stated in terms of the pair (u, Ω). The difficulty is that [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF] has several viscosity solutions (see Appendix A for examples) depending on the sense the boundary conditions are achieved and the sign of R.

Next we discuss the two boundary conditions arising in [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF]. The first is the Dirichlet boundary condition in the third relation in [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF]. Its precise form is lim

(x,t)→(x 0 ,t 0 )∈∂Ω u(x, t) = u m . ( 8
)
The second is the state constraint boundary condition (see [START_REF] Soner | Optimal control with state-space constraint[END_REF]), which is natural in view of the second equality in [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF]. It states that u is a supersolution in Ω and a subsolution in Ω .

The basic questions we are considering in this paper are: • What boundary condition should be satisfied by the limits of the family (u ε ) ε>0 on ∂Ω? Dirichlet or state constraint? The latter appears to play a fundamental role. To the best of our knowledge, there are no results available for state constraint problems with time varying and non smooth domains. Most of the technicalities in the paper stem from this difficulty.

• Does the limit ε → 0 select a particular solution to [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF], i.e., is there a natural selection? Is the limit of the family (u ε ) ε>0 the maximal subsolution or minimal solution to (7)?

• Do the limits of the family (u ε ) ε>0 depend on the specific form of the survival threshold, i.e., can we replace (β ε n ε ) 1/2 by (β ε n ε ) γ with γ ∈ (0, 1) without affecting the outcome? An important ingredient of our analysis is the asymptotics, as ε → 0, of the solution

u 1 ε of    u 1 ε,t = ε∆u 1 ε + |Du 1 ε | 2 + R in R d × (0, +∞), u 1 ε = u 0 ε in R d × {0}, (10)
which is obtained, after the Hopf-Cole transformation

u 1 ε = ε ln n 1 ε , (11)
from the simplified reaction diffusion equation

     n 1 ε , t -ε∆n 1 ε = n 1 ε ε R in R d × (0, +∞), n 1 ε = exp(ε -1 u 0 ε ) in R d × {0}. (12)
In view of ( 5) and ( 6), it follows from [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF] that, as ε → 0, the sequence (u 1 ε ) ε converge locally uniformly to u 1 ∈ C(R d × (0, ∞)), which is the unique viscosity solution of the eikonal -type equation

     u 1 t = |Du 1 | 2 + R in R d × (0, +∞), u 1 = u 0 in R d × {0}. (13)
The maximum principle yields n ε ≤ n 1 ε , which in turn implies that u ε ≤ u 1 ε and, in the limit (this is made precise later), u ≤ u 1 . It also follows from (4), at least formally, that, as ε → 0,

u ε → -∞ in (R d × (0, ∞))\Ω 1 ,
where

Ω 1 = {(x, t) | u 1 (x, t) > u m }. ( 14
)
It turns out that the case of nonpositive rate R is particularly illuminating and the above questions can be answered completely and positively using u 1 (see Section 2). The problem is, however, considerably more complicated when R takes positive values. In this case we introduce an iterative procedure that builds sequences of sub and supersolutions (Section 3). This construction gives the complete limit of u ε when R is constant (Section 4). The limit is not the maximal subsolution of [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF] and the Dirichlet condition is not enough to select it. In Section 5, we consider strictly positive spatially dependent R and provide a complete answer in terms of the iterative procedure. The relative roles of the Dirichlet and state constraint boundary conditions appear clearly in this case. In Section 6 we summarize our results. In the three part Appendix we present some examples of nonuniqueness as well as the proofs of few technical facts used earlier.

We conclude the introduction with the definition and the notation of the half-relaxed limits that we will be using throughout the paper. To this end, if (w ε ) ε>0 is a family of bounded functions, the upper and lower limits, which are denoted by w and w respectively, are given by w(x) = lim sup ε→0,y→x w ε (y) and w(x) = lim inf ε→0,y→x

w ε (y). ( 15
)
2 Nonpositive growth rate

Here we assume

R ≤ 0 in R d , (16)
and show that the behavior of the family (u ε ) ε , in the limit ε → 0, can be described completely in terms of solution u 1 of (13), which carries all the necessary information. More precisely, we can state the Theorem 2.1. Assume ( 5), ( 6) and [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF]. As ε → 0, the family (u ε ) ε>0 converges, locally uniformly in Ω 1 and in

R d × (0, ∞) \Ω 1 , to u(x, t) = u 1 (x, t) for (x, t) ∈ Ω 1 , -∞ for (x, t) ∈ R d × (0, ∞) \Ω 1 , (17)
with u 1 and Ω 1 defined by ( 13) and ( 14) respectively. In particular, u(x, t) → u m as (x, t) → ∂Ω 1 .

Before we begin with the proof, we present and discuss below several remarks and observations which are important to explain the meaning of the results.

Firstly, by "uniform convergence" to -∞, we mean lim sup ε→0,y→x,s→t u ε (y, s) = -∞. Secondly, the u associated with the open set Ω 1 is the maximal solution to [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF]. Indeed any other solution u, with the corresponding open set Ω, satisfies u ≤ u 1 and thus Ω ⊂ Ω 1 and u ≤ u. It also satisfies the Dirichlet and state constraint boundary conditions. To verify the latter we notice, using the standard optimal control formula ([15, 13, 1]), that

u 1 (x, t) = sup (x(s),s)∈R d ×[0,∞) x(t)=x t 0 - | ẋ(s)| 2 4 + R(x(s)) ds + u 0 (x(0)) : x ∈ C 1 ([0, t]; R d ) .
If x(•) is an optimal trajectory, the dynamic programming principle implies that, for any 0 < τ < t,

u 1 (x, t) = t τ - | ˙ x(s)| 2 4 + R( x(s)) ds + u 1 ( x(τ ), τ ).
Since R is nonpositive, u 1 is decreasing along the optimal trajectory. It follows that, if u 1 (x, t) > u m , then, for all 0 ≤ τ < t, u 1 ( x(τ ), τ ) > u m .

Hence, for all (x, t)

∈ Ω 1 , u(x, t) = sup (x(s),s)∈Ω 1 x(t)=x t 0 - | ẋ(s)| 2 4 + R(x(s)) ds + u 0 (x(0)) : x ∈ C 1 ([0, t]; R d ) ,
and, therefore, u verifies the state constraint condition. Finally, the limit u does not depend on the details of the singular death term. In particular it is the same if we replace in [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF]

n ε exp(ε -1 u m ) 1/2 by n γ ε exp(ε -1 γu m ) with 0 < γ < 1. Hence, the value γ = 1/2 is irrelevant.
We continue with the Proof of Theorem 2.1. As already discussed in the introduction, we know that u ε ≤ u 1 ε but we cannot obtain directly the other inequality in the limit ε → 0. It is therefore necessary to introduce a pair of auxiliary functions v A ε and v A,1 ε which converge, as ε → 0, in C(R d × (0, ∞)) to max(u 1 , -A). Using this information for appropriate values of the parameter A, we then prove that, as ε → 0, u ε → u 1 locally uniformly in the open set

A = {(x, t) : u 1 (x, t) > u m }, (18)
and u ε → -∞ locally uniformly in the open set

B = {(x, t) : u 1 (x, t) < u m }. ( 19
)
To this end, for any A such that 0

< A < -u m , (20)
we consider the functions v A ε and v A,1 ε given by

n ε + exp( -A ε ) = exp( v A ε ε ) and n 1 ε + exp( -A ε ) = exp( v A,1 ε ε ). ( 21
)
We have:

Proposition 2.2. Assume ( 5), ( 6), ( 16) and (20). As ε → 0, the families

(v A,1 ε ) ε>0 and (v A ε ) ε>0 converge in C(R d × [0, ∞)) to the unique solution v A,1 = max(u 1 , -A) of      min v A,1 + A, v A,1 t -|Dv A,1 | 2 -R = 0 in R d × (0, ∞), v A,1 = max(u 0 , -A) on R d × {0}. (22)
We postpone the proof to the end of this section and next we prove the convergence of the family (u ε ) ε in the sets A and B. We begin with the former.

Fix (x 0 , t 0 ) ∈ A. By the definition of A we have u 1 (x 0 , t 0 ) > u m and, hence, we can choose A such that u 1 (x 0 , t 0 ) > -A > u m . Proposition 2.2 yields that, as ε → 0 and uniformly in any neighborhood of (x 0 , t 0 ),

v A ε → v A,1 = max(-A, u 1 ) = u 1 .
Using the latter, the choice of A and the fact that

u ε = v A ε + ε ln 1 -exp(ε -1 (-A -v A ε )) ,
we deduce that, as ε → 0, u ε → u 1 uniformly in any neighborhood of (x 0 , t 0 ). Next we consider the limiting behavior in the set B. To this end, observe that, using ( 3) and ( 11), we find u ε ≤ u 1 ε and, thus, passing to the limit in the viscosity sense, u ≤ u 1 and

u < u m in B.
Assume that, for some (x 0 , t 0 ) ∈ B, u(x 0 , t 0 ) > -∞. Since u is upper semicontinuous (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]), there exists a family (φ α ) α>0 of smooth functions such that u -φ α attains a strict local maximum at some (x α , t α ) and, as α → 0, (x α , t α ) → (x 0 , t 0 ), u(x α , t α ) ≥ u(x 0 , t 0 ) and u(x α , t α ) → u(x 0 , t 0 ). It follows that there exists points (x α,ε , t α,ε ) such that u ε -φ α attains a local maximum at (x α,ε , t α,ε ), (x α,ε , t α,ε ) → (x α , t α ) as ε → 0, and, in view of (4), at (x α,ε , t α,ε ),

φ α,t -ε∆φ α -|Dφ α | 2 -R ≤ -exp((2ε) -1 (u m -u ε )).
Letting ε → 0 we find that at (x α , t α )

φ α,t -|Dφ α | 2 -R ≤ lim sup ε→0 [-exp((2ε) -1 (u m -u ε (x α,ε , t α,ε )))].
The definition of u yields lim sup

ε→0 u ε (x α,ε , t α,ε ) ≤ u(x α , t α )
and, since, for α sufficiently small, u(x, t) < u m , we have

u(x α , t α ) < u m and lim sup ε→0 [-exp((2ε) -1 (u m -u ε (x α,ε , t α,ε )))] = -∞ and, finally, at (x α , t α ), φ α,t -|Dφ α | 2 -R ≤ -∞,
which is not possible because φ α is a smooth function.

The claim about the uniform convergence on compact subsets is an immediate consequence of the upper semicontinuity of u and the previous argument.

We conclude the section with the proof of Proposition 2.2. Since it is long, before entering in the details, we briefly describe the main steps. We begin by establishing independent of ε bounds on the family (v A ε ) ε . Then we show that the half-relaxed limits v α and v α are respectively sub and supersolutions of (22). We conclude by identifying the limit.

Proof of Proposition 2.2. By the definition of v A ε , we have v A ε > -A and, thus, the family (v A ε ) ε is bounded from below.

To prove an upper bound we first notice that, on R d × {0},

v A ε = u 0 ε + ε ln(1 + e -A-u 0 ε ε ) and v A ε = -A + ε ln(1 + e A+u 0 ε ε ), (23) hence, v A ε ≤ max(u 0 ε + ε ln(2), -A + ε ln(2)) on R d × {0}, and, finally, in view of (6), v A ε ≤ C A on R d × {0}, for C A > 0 such that max(-A, u 0 ε ) ≤ C A . Moreover, since R ≤ 0, we have v A ε,t -ε∆v A ε -|Dv A ε | 2 = n ε n ε + exp( -A ε ) R - (β ε n ε ) 1/2 n ε + exp( -A ε ) ≤ 0 in R d × (0, ∞). ( 24
)
It follows from the maximum principle that

v A ε ≤ C A + ε ln(2) in R d × (0, ∞).
Next we show that v A is a supersolution of (22). Since u m < -A and

(β ε n ε ) 1/2 n ε + exp( -A ε ) ≤ (β ε n ε ) 1/2 2n ε exp( -A ε ) 1/2 = 1 2 exp( u m + A 2ε ),
as ε → 0 and uniformly on R d × (0, ∞), we have

(β ε n ε ) 1/2 n ε + exp( -A ε ) → 0. ( 25
)
From ( 16), (24) and 0

≤ n ε n ε + exp( -A ε ) ≤ 1, we then deduce that, in R d × (0, ∞), v A ε,t -ε∆v A ε -|Dv A ε | 2 ≥ R -O(ε), (26)
while by the definition of v A ε we also have

v A ε + A ≥ 0. (27)
Combining ( 26) and ( 27) and using the basic stability properties of the viscosity solutions (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]) we find that the lower semicontinuous function v A is a viscosity supersolution of (22).

To prove that v A is a subsolution to (22), following classical arguments from the theory of viscosity solutions (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]), we fix a smooth φ and assume that v A -φ has a strict local maximum at (x 0 , t 0 ). It follows that there exists a family, which for notational simplicity we denote again by ε, of points (x ε , t ε ) ε>0 in R d × (0, ∞) such that v A ε -φ has a local maximum at (x ε , t ε ), and, as ε → 0, (x ε , t ε ) → (x 0 , t 0 ) and v A ε (x ε , t ε ) → v A (x 0 , t 0 ). We also know, still using (24) and (25), that v A ε solves

v A ε,t -ε∆v A ε -|Dv A ε | 2 = 1 -exp( -A -v A ε ε ) R -O(ε).
It then follows that, at (x ε , t ε ),

φ t -ε∆φ -|Dφ| 2 -1 -exp(ε -1 (-A -v A ε ))R ≤ O(ε). ( 28
) Recall that lim ε→0 v A ε (x ε , t ε ) = v A (x 0 , t 0 ) ≥ -A. Hence, if v A (x 0 , t 0 ) > -A, then lim ε→0 exp(ε -1 (-A -v A ε (x ε , t ε ))) = 0.
From this and (28) we deduce that, if v A (x 0 , t 0 ) > -A, then, at (x 0 , t 0 ),

φ t -|Dφ| 2 -R ≤ 0.
Next we show that v and v satisfy the appropriate initial conditions. Indeed, in view of ( 6) and (23), we know that, as ε → 0,

v A ε → max(-A, u 0 ) on R d × {0}.
It also follows from a classical argument in theory of viscosity solutions ( [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Barles | Wavefront propagation for reaction-diffusion systems of PDE[END_REF]) that, on

R d × {0}, v A -max(-A, u 0 ) ≤ 0 and v A -max(-A, u 0 ) ≥ 0.
and, hence, v A and v A satisfy respectively the discontinuous viscosity subsolution and supersolution initial condition corresponding to (22). We already know from the definition of v A and v A that v A ≤ v A , while from the comparison property for (22) in the class of semicontinuous viscosity solutions (see [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]) we conclude from the steps above that 1 is the unique continuous viscosity solution of ( 22) and, consequently, the families v A ε and v A,1 ε converge, as ε → 0 and locally uniformly, to v A,1 . Combining ( 3) and ( 21) we find

v A ≤ v A in R d × (0, ∞). Hence v A = v A = v A,
v A,1 ε = u 1 ε + ε ln(1 + exp -A-u 1 ε ε ) and v A,1 ε = -A + ε ln(1 + exp A+u 1 ε ε ).
Moreover, as we already explained it in the introduction (see ( 12)-( 13)), we know that, as ε → 0, u 1 ε → u 1 locally uniformly. Hence, always for A < -u m , we obtain that, as ε → 0,

v A,1 ε → max(u 1 , -A) locally uniformly in R d × [0, ∞).
It also follows that the family (v A ε ) ε>0 converges, as ε → 0, locally uniformly to v A,1 = max(u 1 , -A).

General rate

When R changes sign, the situation is much more complicated and (17) does not hold in general.

In this case we are able to provide only inequalities for the half-relaxed limits of the family (u ε ) ε>0 . These estimates are used later to characterize the limit when R is positive. Fix u 0 , δ > 0 and recall that u 1 is the solution of ( 13) with u 1 = u 0 on R d × {0}. We introduce next the family (

u δ i [u 0 ], C δ i [u 0 ], Ω δ i [u 0 ]
) i∈Z + which is defined iteratively. To this end, for i = 1, let

u δ 1 [u 0 ] = u 1 , C δ 1 [u 0 ] = R d × [0, ∞) and Ω δ 1 [u 0 ] = {(x, t) ∈ R d × [0, ∞) : u δ 1 [u 0 ](x, t) > u m -δ}, (29)
and, given

u δ i [u 0 ], C δ i [u 0 ] and Ω δ i [u 0 ], u δ i+1 [u 0 ] : R d × [0, ∞) → R ∪ {-∞} is defined by u δ i+1 [u 0 ](x, t) = sup t 0 - | ẋ(s)| 2 4 + R (x(s)) ds + u 0 (x(0)) : x ∈ C 1 ([0, t]; R d ), (x(s), s) ∈ Ω δ i [u 0 ] for all s ∈ [0, t], x(t) = x , (30)
with

C δ i+1 [u 0 ] = {(x, t) ∈ Ω δ i [u 0 ] : u δ i+1 [u 0 ](x, t) > -∞} (31)
and

Ω δ i+1 [u 0 ] = {(x, t) ∈ Ω δ i [u 0 ] : u δ i+1 [u 0 ](x, t) > u m -δ} ⊂ C δ i+1 [u 0 ]. ( 32
)
It follows that, in general,

C δ i+1 [u 0 ] ⊆ Ω δ i [u 0 ]
. The inclusion may be, however, strict, i.e., they may exist points (x, t) ∈ Ω δ i [u 0 ] which cannot be connected to R d × {0} by a C 1 trajectory staying, for all s ∈ [0, t] in Ω δ i [u 0 ]. (See Figure 1.)

Ω δ i [u 0 ] D (t, x) Figure 1: An example of the space-time set Ω δ i [u 0 ]. The point (x, t) ∈ Ω δ i [u 0 ] cannot be connected to R d × {0} by a C 1 trajectory (x(s), s) s∈[0,t] staying within Ω δ i [u 0 ].
More generally, for the points in the grey area, called D, there is no admissible trajectory. We have indeed

C δ i+1 [u 0 ] = Ω δ i [u 0 ] \ D.
Moreover ( 5), (29) and classical considerations from the optimal control theory ([15, 13, 1, 8]) yield that, for all i ∈ Z + , the sets

C δ i [u 0 ] and Ω δ i are open and u δ i [u 0 ] ∈ C(C δ i [u 0 ]
). Note that the state constraint boundary condition, i.e., the requirement that the trajectories stay inside the domain, is hidden in the control formula. We do not write it, however, explicitly, because, to the best of our knowledge, there is no general theory, as in [START_REF] Soner | Optimal control with state-space constraint[END_REF], for state constraint problem with time varying and nonsmooth domains. Note that in our context we have no regularity properties for these domains.

Given

C δ i+1 [u 0 ] as in (31), it turns out that u δ i+1 [u 0 ] is the minimal viscosity solution to    u δ i+1,t [u 0 ] = |Du δ i+1 [u 0 ]| 2 + R in C δ i+1 [u 0 ], u δ i+1 [u 0 ] = u 0 in C δ i+1 [u 0 ] ∩ (R d × {0}). (33)
Indeed using standard arguments from optimal control theory (see, for example, [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]), we may easily see that u δ i+1 [u 0 ] satisfies the dynamic programming principle. The latter, as usual, implies that u δ i+1 [u 0 ] is a viscosity solution of (33). The proof of the fact that u δ i+1 [u 0 ] is a minimal solution to (33) in Appendix B.

The family (u δ i [u 0 ]) i∈Z + ,δ>0 is nonincreasing in both i and δ. Therefore there exists

U δ [u 0 ] ≥ -∞, which is itself nonincreasing in δ, such that, as i → +∞, u δ i [u 0 ] ց U δ [u 0 ] in R d × [0, ∞).
Let U [u 0 ] be the limit, as δ → 0, of the family (U δ [u 0 ]) δ>0 and, for µ > 0, consider the nonincreasing (in δ) family of sets

Ω δ [u 0 ] = i∈Z + Ω δ i [u 0 ] and Ω[u 0 -µ] = δ>0 Ω δ [u 0 -µ]. (34)
We have:

Theorem 3.1. Let n ε be the solution to (1), u ε = ε ln(n ε ) and assume [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF]. Then, for any µ > 0,

u ≤ U [u 0 ] in R d × [0, ∞) and U [u 0 -µ] + µ ≤ u in Ω[u 0 -µ]. (35)
Before we present the proof we remark that, by definition,

u δ i [u 0 ] = -∞ in C δ i [u 0 ] c . Therefore U δ [u 0 ] = -∞ in (Ω δ [u 0 ]) c and, finally, U [u 0 ] = -∞ in (Ω[u 0 ]) c = i,δ C δ i [u 0 ] c = δ Ω δ [u 0 ] c , and, hence, u = -∞ in (Ω[u 0 ]) c . Moreover, since u δ i [•] ≥ u m -δ in Ω δ [•]
, by passing to the limit i → ∞ and δ → 0 we also obtain

U [•] ≥ u m in Ω[•].
An important question is whether, as

µ → 0, U [u 0 -µ] → U [u 0 ]
. This is, in general, not true. A counterexample can be found for u 0 = u m and R > 0. Then Ω

δ 1 [u 0 -µ] cannot touch R d × {0} and u δ i [u 0 -µ] ≡ -∞. Therefore U [u 0 -µ] ≡ -∞ for any µ > 0. On the other hand, u δ i [u 0 ] > u m and U [u 0 ] = u 1 .
We continue with the Proof of Theorem 3.1. First we show by induction that, for all δ > 0 and i

∈ Z + , u ≤ u δ i [u 0 ]. Since n 1
ε is a supersolution to (1), it follows from the comparison principle that n ε ≤ n 1 ε and, hence,

u ≤ u δ 1 [u 0 ] = u 1 . Next we assume that u ≤ u δ i [u 0 ]
, and, arguing by contradiction, we show, following an argument similar to that in Section 2, that u

≤ u δ i+1 [u 0 ] = -∞ in (Ω δ i [u 0 ]) c . To this end, suppose that, for some (x 0 , t 0 ) ∈ (Ω δ i [u 0 ]) c , u(x 0 , t 0 ) > -∞.
Since u is upper semicontinuous, there exists a family (φ α ) α>0 of smooth functions such that u -φ α attains a strict local maximum at (x α , t α ) and, as α → 0, (x α , t α ) → (x 0 , t 0 ), u(x α , t α ) ≥ u(x 0 , t 0 ), and, consequently, u(x α , t α ) → u(x 0 , t 0 ). It follows that there exist points (x α,ε , t α,ε ) ∈ (R d × (0, ∞)) where u ε -φ α attains a local maximum and, as ε → 0, (x α,ε , t α,ε ) → (x α , t α ).

Moreover, in view of (4), at (x α,ε , t α,ε ),

φ α,t -ε∆φ α -|Dφ α | 2 -R ≤ -exp((2ε) -1 (u m -u ε )).
Letting ε → 0 yields, at (x α , t α ),

φ α,t -|Dφ α | 2 -R ≤ lim sup ε→0 (-exp[(2ε) -1 (u m -u ε (x α,ε , t α,ε ))]).
Since, by the definition of u, we have lim sup ε→0 u ε (x α,ε , t α,ε ) ≤ u(x α , t α ), the induction hypothesis yields that, for α small enough, u(

x α , t α ) ≤ u δ i [u 0 ](x α , t α ) ≤ u m -δ/2. It follows that lim sup ε→0 (-exp[(2ε) -1 (u m -u ε (x α,ε , t α,ε ))]) = -∞,
and, hence, at (x α , t α ),

φ α,t -|Dφ α | 2 -R ≤ -∞, which, of course, is not possible because φ α is a smooth function. Hence we have u = -∞ in (Ω δ i ) c and, in particular, u = -∞ on ∂Ω δ i [u 0 ]. Next we show that u ≤ u δ i+1 [u 0 ] = -∞ in (C δ i+1 [u 0 ]) c . To this end, let (x, t) ∈ (C δ i+1 [u 0 ]) c \ (Ω δ i [u 0 ]) c .
Note that the existence of such a point means that (x, t) cannot be connected to

R d × {0} by a C 1 -trajectory staying in Ω δ i [u 0 ]. Hence (x, t) belongs to a connected component D of ω δ i [u 0 ] = {(y, s) ∈ Ω δ i [u 0 ] : s ≤ t},
As in (21), for A > 0, we define w A ε by n ε + exp -A ε = exp w A ε ε . Arguing as in the previous section, we deduce that, for all A > 0, w A = max(-A, u) and min(w

A + A, w A t -|Dw A | 2 -R) ≤ 0,
and, in view of (36), min(w

A + A, w A t -|Dw A | 2 -R) ≤ 0 in D, w A = -A in ∂ p D,
which admits -A + C 1 t as a supersolution for some C 1 > 0.

It follows from the comparison principle that, for all A > 0,

u ≤ w A ≤ -A + C 1 t in D. Letting A → ∞ yields u = -∞ in D and, consequently, u(x, t) = -∞. Observe that u = -∞ in (C δ i+1 [u 0 ]) c implies that u = -∞ on ∂C δ i+1 [u 0 ] ∩ (R d × [0, ∞)).
Finally we show that

u ≤ u δ i+1 [u 0 ] in C δ i+1 [u 0 ].
To this end, define z ε by n ε + exp

u δ i+1 [u 0 ] ε = exp zε ε and notice that z ε = u δ i+1 [u 0 ] + ε ln exp u ε -u δ i+1 [u 0 ] ε + 1 = u ε + ε ln exp u δ i+1 [u 0 ] -u ε ε + 1 .
It follows that z = max(u, u δ i+1 [u 0 ]). We claim that z is a subsolution of 4) and the stability of viscosity subsolutions, we find that u satisfies the viscosity subsolution criteria for (37) at (x, t). Finally, since the maximum of two subsolutions is always a subsolution, we obtain that z is a subsolution of (37).

z t -|Dz| 2 -R ≤ 0 in C δ i+1 [u 0 ]. ( 37
) Indeed u δ i+1 [u 0 ] is a subsolution to (37) in C δ i+1 [u 0 ] by definition. Moreover if, for some (x, t) ∈ C δ i+1 [u 0 ], u(x, t) = -∞, using (
We proceed by noticing that, since, in view of the above,

u = -∞ on ∂C δ i+1 ∩ (R d × (0, +∞)), it follows that z = u δ i+1 [u 0 ] on ∂C δ i+1 [u 0 ] and, hence, z ≤ u δ i+1 [u 0 ] on ∂C δ i+1 [u 0 ]
. Therefore, using again the comparison principle for (33), we obtain

z ≤ u δ i+1 [u 0 ] in C δ i+1 [u 0 ]
and we conclude that u ≤ u δ i+1 [u 0 ].

Finally, since for all δ > 0 and i ∈ Z + , we have u ≤ u δ i [u 0 ] it follows that, for all δ > 0, u ≤

lim i→∞ u δ i [u 0 ] = U δ [u 0 ]. After letting δ → 0 we obtain u ≤ lim δ→0 U δ [u 0 ] = U [u 0 ] in R d × (0, ∞),
which concludes the proof of the first part of the claim.

For the second part we need the following lemma which is essentially a result from [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF] that we adapt to our context (see also [START_REF] Barles | Discontinuous viscosity solutions of first order Hamilton-Jacobi Equations: A guided visit[END_REF]). Its proof is postponed to the end of this section. Lemma 3.2. For all i ∈ Z + the lower semicontinuous function

v δ i = max(u δ i [u 0 -µ] + 2δ, u), is a supersolution of    v δ i,t -|Dv δ i | 2 -R ≥ 0 in Ω δ i [u 0 -µ], v δ i = u 0 in {u 0 -µ > u m -δ} ∩ (R d × {0}). ( 38
) Since u δ i+1 [u 0 -µ] is a minimal solution of (33) in C δ i+1 [u 0 -µ] ⊂ Ω δ i [u 0 -µ] with u δ i+1 [u 0 -µ] = u 0 -µ on R d × {0} (see Appendix B), it follows that u δ i+1 [u 0 -µ] ≤ v δ i -µ in C δ i+1 [u 0 -µ],
and, hence,

u δ i+1 [u 0 -µ] + µ ≤ max(u δ i [u 0 -µ] + 2δ, u) in C δ i+1 [u 0 -µ]. Letting i → ∞ yields U δ [u 0 -µ] + µ ≤ max(U δ [u 0 -µ] + 2δ, u) in Ω δ [u 0 -µ].
Choosing µ > 2δ we also get

U δ [u 0 -µ] + 2δ < U δ [u 0 -µ] + µ in Ω δ [u 0 -µ],
and, therefore,

U δ [u 0 -µ] + µ ≤ u in Ω δ [u 0 -µ].
Finally letting δ → 0 we obtain

U [u 0 -µ] + µ = lim δ→0 U δ [u 0 -µ] + µ ≤ u in Ω[u 0 -µ].
We conclude with the Proof of Lemma 3.2. The key idea of the proof comes from [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF] and [START_REF] Barles | Discontinuous viscosity solutions of first order Hamilton-Jacobi Equations: A guided visit[END_REF] and relies on the property that, for concave Hamiltonians, the maximum of two supersolutions is supersolution. Here we reprove this fact in the context of semicontinuous supersolutions in a space-time domain. To this end, fix i ∈ Z + and (x,

t) ∈ C δ i [u 0 -µ]. Since C δ i [u 0 -µ] is an open set, there exists ρ > 0 such that B ρ (x, t) ∈ C δ i [u 0 -µ],
where B ρ (x, t) denotes the open ball of radius ρ centered at (x, t).

For α > 0, we define

u δ,α i (x, t) = inf (y,s)∈Bρ(x,t) {u δ i [u 0 -µ](y, s) + (2α) -1 (|x -y| 2 + |t -s| 2 )} and u δ,α,β i = u δ,α i * χ β ,
where χ β is a standard smoothing mollifier. Since u δ,α i is an inf-convolution of the continuous function u δ i (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]), it is locally Lipschitz continuous and semi-concave with semi-concavity constant 1/α.

It follows that u δ,α,β i is a smooth semi-concave function with semi-concavity constant 1/α and lim inf

(y,s)→(ȳ,s) α, β→0 u δ,α,β i (y, s) = u δ i [u 0 -µ](ȳ, s).
Finally, using Jensen's inequality and the concavity of the Hamiltonian, we obtain that, for some K > 0, u δ,α,β i is a smooth and, hence, a classical supersolution to

u δ,α,β i,t -ε∆u δ,α,β i -|Du δ,α,β i | 2 -R * χ β ≥ -Kα -ε/α in B ρ (x, t). (39)
To prove (38) we show that the smooth approximations v δ,α,β,ε i of v δ i in B ρ (x, t) given by

n ε + exp u δ,α,β i + 2δ ε = exp v δ,α, β,ε i ε (40)
are almost supersolutions to (38) for α, β and ε small. Notice that in (40) we use 2δ instead of δ.

Replacing n ε by exp

v δ,α,β,ε i ε -exp u δ,α,β i +2δ ε in (1) we get Rn ε -β ε √ n ε = v δ,α,β,ε i,t -ε∆v δ,α,β,ε i -|Dv δ,α,β,ε i | 2 exp(ε -1 v δ,α,β i ) -u δ,α,β i,t -ε∆u δ,α,β i -|Du δ,α,β i | 2 exp(ε -1 (u δ,α,β i + 2δ)),
and, in view of (40),

v δ,α,β,ε i,t -ε∆v δ,α,β,ε i -|Dv δ,α,β,ε i | 2 = (u δ,α,β i,t -ε∆u δ,α,β i -|Du δ,α,β i | 2 -R * χ β ) exp(ε -1 (u δ,α,β i + 2δ -v δ,α,β,ε i )) + (R * χ β -R) exp(ε -1 (u δ,α,β i + 2δ -v δ,α,β i )) + R -β ε n 1/2 ε exp(-ε -1 v δ,α,β,ε i ).
Using that, in view of (40

), exp(ε -1 (u δ,α,β i + 2δ -v δ,α,β,ε i
)) ≤ 1, and (39) we find

v δ,α,β,ε i,t -ε∆v δ,α,β,ε i -|Dv δ,α,β,ε i | 2 -R ≥ -Kα -ε/α + (R * χ β -R) exp(ε -1 (u δ,α,β i + 2δ -v δ,α,β i )) -β ε n 1/2 ε exp(ε -1 v δ,α,β i ). Define v δ,α,β i (ȳ, s) = lim inf ε→0 (y,s)→( ȳ,s) v δ,α,β,ε i (y, s).
Letting ε → 0 and using the stability of viscosity supersolutions we obtain

v δ,α,β i,t -|Dv δ,α,β i | 2 -R ≥ -Kα (41)
+ lim inf ε→0 (y,s)→( ȳ,s) [(R * χ β -R) exp(ε -1 (u δ,α,β i + 2δ -v δ,α,β i )) -β ε n 1/2 ε exp(ε -1 v δ,α,β i )].
Recalling that

u δ i + 2δ > u m + δ in Ω δ i [u 0 -µ],
we deduce that, as ε, α, β → 0, in B ρ (x, t),

β ε n 1/2 ε exp(-ε -1 v δ,ε i ) = β ε n 1/2 ε (n ε + exp(ε -1 u δ,ε i )) -1 ≤ (1/2)β ε exp(-(2ε) -1 u δ,α,β i ) → 0. (42)
Moreover, as ε, β → 0, we also have

R * χ β -R → 0, exp(ε -1 (u δ,α,β i + 2δ -v δ,α,β i )) < 1 and v δ i (η, s) = lim inf α,β→0 (y,s)→( ȳ,s) v δ,α,β i (y, s). (43)
Using ( 41), ( 42), (43), and the stability of viscosity supersolutions we find

v δ i,t -|Dv δ i | 2 -R ≥ 0 in B ρ (x, t).
Since all the above hold for all (x, t) ∈ Ω δ i [u 0 -µ], it follows that the lower semicontinuous function

v δ i is a supersolution to v δ i,t -|Dv δ i | 2 -R ≥ 0 in Ω δ i [u 0 -µ], v δ i = u 0 for {u 0 -µ > u m -δ} ∩ (R d × {0}).

Constant rate

Here we assume that the rate is a constant, i.e.,

R(x) = R in R d , (44)
and, in addition, setting O = {x ∈ R d : u 0 (x) > u m }, we have

O = {x ∈ R d : u 0 (x) ≥ u m }. (45)
We have:

Theorem 4.1. Assume (44) and (45). Then

lim ε→0 u ε (x, t) = U [u 0 ](x, t) locally uniformly in R d × [0, ∞) \ {(x, t) | U [u 0 ](x, t) = u m } , (46)
with

Ω[u 0 ] = {(x, t)| sup y∈O {- |x -y| 2 4t + Rt + u 0 (y)} ≥ u m }, (47)
and

U [u 0 ] =    sup y∈O -|x-y| 2 4t + Rt + u 0 (y) if (x, t) ∈ Ω[u 0 ], -∞ otherwise. (48)
We notice that, if R < 0, then one can obtain (46) from ( 17) and the dynamic programming principle. We also remark that, in particular, Theorem 4.1 shows that the limit of the family (u ε ) ε>0 is not, in general, given by [START_REF] Quittner | Superlinear parabolic problems. Blow-up, global existence and steady state[END_REF]. We refer to Appendix A for an explicit example.

(0, c) Proof of Theorem 4.1. When the rate R is constant, after one iteration of (30), ( 31) and (32) we find both the set Ω δ [u 0 ] and the function U δ [u 0 ], since for all i > 1, j > 2 and δ > 0,

(0, b) (0, a) (t, x)
Ω δ i [u 0 ] = C δ j [u 0 ] = Ω δ [u 0 ] and u δ i [u 0 ] = U δ [u 0 ]
. Indeed, every optimal trajectory in C δ 2 is a straight line connecting a point in Ω δ 2 to a point in

I δ = {x ∈ R d : u 0 (x) > u m -δ} and, hence, it is included in Ω δ
2 . This follows from the observation that

φ(x, t) = - |x -c| 2 4t + Rt + u 0 (c)
is concave in (x, t) and, therefore, all the optimal trajectories of the points in Ω δ 2 are included in Ω δ 2 . It follows that Ω δ 2 = C δ 3 , u δ 2 = u δ 3 and consequently Ω δ 2 = Ω δ 3 . By iteration we obtain, for all i > 2, Ω δ 2 = Ω δ i = Ω δ and u δ 2 = u δ i = U δ . Using (32) and (30) we see that, for all i ≥ 2,

Ω δ [u 0 ] = Ω δ i [u 0 ] = {(x, t) : sup y∈I δ {- |x -y| 2 4t + Rt + u 0 (y)} > u m -δ}, (49)
and

U δ [u 0 ] = u δ i [u 0 ](x, t) =    sup y∈I δ -|x-y| 2 4t + Rt + u 0 (y) if (x, t) ∈ Ω δ [u 0 ], -∞ otherwise. ( 50
)
It is easy to verify that (33) holds, since, for all i > 2 and δ > 0,

u δ i,t [u 0 ] -|Du δ i [u 0 ]| 2 -R = 0 in Ω δ i [u 0 ] = C δ i+1 [u 0 ].
Letting δ → 0 in (49) and (50) we obtain (47)-( 48). (See Figure 2.) We also have

Ω[u 0 -µ] = δ>0 Ω δ [u 0 -µ] = {(x, t) : sup y∈J µ {- |x -y| 2 4t + Rt + u 0 (y)} ≥ u m + µ}, with J µ = {(x, t) : u 0 ≥ u m + µ}.
It follows that

∪ µ>0 Ω[u 0 -µ] = {(x, t) : sup y∈O {- |x -y| 2 4t + Rt + u 0 (y)} > u m }, (51)
and

lim µ→0 + U [u 0 -µ] =      sup y∈O {-|x-y| 2 4t + Rt + u 0 (y)} for (x, t) ∈ ∪ µ>0 Ω[u 0 -µ],
-∞ otherwise.

(

) 52
We also notice that

sup y∈O {- |x -y| 2 4t + Rt + u 0 (y)} = sup y∈ Ō {- |x -y| 2 4t + Rt + u 0 (y)}. ( 53
)
Comparing ( 47), ( 48) with ( 51), (52) and using (53) we deduce that

lim µ→0 U [u 0 -µ](x, t) = U [u 0 ](x, t) for U [u 0 ](x, t) = u m ,
and, consequently,

lim ε→0 u ε (x, t) = U [u 0 ](x, t) locally uniformly in R d × [0, ∞) \ {(x, t) | U [u 0 ](x, t) = u m } .

Strictly positive rate

In this section we study the limiting behavior of the family (u ε ) ε>0 when

R ≥ a > 0 in R d , (54)
and show that, in general, the limit is not given by [START_REF] Quittner | Superlinear parabolic problems. Blow-up, global existence and steady state[END_REF]. For this we need to assume that, for sufficiently small µ > δ > 0, there exists ρ δ,µ > 0 such that lim µ→0 lim δ→0 ρ δ,µ = 0 and, if u 0 (y) > u m -δ, then sup

|y-z|≤ρ δ,µ u 0 (z) > u m -δ + µ. ( 55
)
Notice that it is important that ρ δ,µ is chosen independently of y. If u 0 ∈ C 1 , (55) implies u m is never a local maximum of u.

We have Theorem 5.1. Assume (54) and (55). Then

lim ε→0 u ε = U [u 0 ] locally uniformly in ∪ µ>0 Ω[u 0 -µ]. (56)
Recall that, in view of Theorem 3.1, we already know that lim

ε→0 u ε = -∞ in Ω[u 0 ] c . Proof of Theorem 5.1. For h > h = µ 2a + 1 2 µ 2 a 2 + ρ 2 δ,µ a , (x, t) ∈ R d × [0, ∞), i ≥ 1 and µ, δ > 0, we have u δ i [u 0 ](x, t) ≤ u δ i [u 0 -µ](x, t + h). (57)
We postpone the proof of this inequality to Appendix C and we continue with the ongoing one. Letting i → +∞ and δ, µ → 0 we find, for all h > 0 and t > 0,

U [u 0 ](•, •) ≤ lim µ→0 + U [u 0 -µ](•, • + h). (58)
Hence, for all (x, t)

∈ ∪ µ>0 Ω[u 0 -µ], U [u 0 ](x, t) ≤ lim µ→0 + U [u 0 -µ](x, t + h) ≤ u(x, t + h) ≤ u(x, t + h), U [u 0 ](x, t) ≤ lim inf h→0 + u(x, t + h) ≤ lim sup h→0 + u(x, t + h).
The definitions of u and u also imply that lim inf

h→0 + u(x, t + h) = u(x, t) and lim sup h→0 + u(x, t + h) = u(x, t).
Combining all the above we obtain

U [u 0 ] ≤ u ≤ u in ∪ µ>0 Ω[u 0 -µ].
This last inequality and (35

) yield u = u = U [u 0 ] in ∪ µ>0 Ω[u 0 -µ],
and, hence,

lim ε→0 u ε = U [u 0 ] in ∪ µ>0 Ω[u 0 -µ].

Conclusions

We showed that the local uniform limit, as ε → 0, for the parabolic problem (1) with finite time extinction is naturally analyzed using the Hopf-Cole change of variables (3). The formal limit is the variant (7) of the standard eikonal equation. The new feature is the resulting quasi-variational inequality with an obstacle that depends on the solution itself. The quasi-variational inequality admits many solutions (see Appendix A) and the difficulty is to select the correct additional information. This is easy when the rate R is negative, as shown in Section 2. Indeed, in this case it is enough to enforce the Dirichlet boundary condition on the boundary of the unknown open set Ω where the liminf of the family (u ε ) ε>0 is finite. This is due to the fact that, for concave Hamiltonians, the supremum of two supersolutions is still a supersolution.

When the rate R is positive we do not have easy supersolutions at hand, and the answer is more elaborate. It requires an induction argument which allows us to identify again the limit of the family (u ε ) ε>0 . The key ingredient is a free boundary problem defined through the level set of the solution. The boundary condition for the resulting equation involves state constraints which leads us to study the problem using the related control problem.

If the growth/death rate R changes sign, we can only bound from above and below the half-relaxed limits of the family (u ε ) ε>0 by sub and supersolutions ū and u respectively.

In terms of the biological motivation, our results qualitatively mean that the specific form of the survival threshold (a square root here) is irrelevant for the asymptotic problem. It also shows that the solution is deeply influenced by the survival threshold except when R is nonpositive. This confirms earlier numerical simulations in [START_REF] Gauduchon | Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations[END_REF].

We conjecture that these upper and lower solutions are in fact equal and the correct setting (implying uniqueness) is to find a pair (u, Ω) for which we can impose both Dirichlet and state constraints boundary conditions. Both establishing directly these boundary conditions for the half-limits of the family (u ε ) ε>0 as well as developing a theory of state constraints boundary conditions for time varying, non-smooth domains are challenging mathematical issues.

A Non-uniqueness

To explain the difficulty associated with [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF], we present here counter-examples for uniqueness and elaborate further conditions. Recall that the problem is to find pairs (u, Ω) such that u is a viscosity solutions to [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF].

A first source for non-uniqueness is the value of u on ∂Ω. Indeed assume that R and u 0 are such that there exists a unique viscosity solution u 1 of ( 13) or, more generally, with u 1 defined in ( 12) and [START_REF] Evans | Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities[END_REF]. For all η ≥ u m , we introduce the pair (w η , Ω η ) given by Ω η = {(x, t) : u 1 (x, t) ≥ η} and w η (x, t) =

   u 1 (x, t) if (x, t) ∈ Ω η , -∞ otherwise.
It can be easily verified that (w η , Ω η ) is a viscosity solution of [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF]. In order to avoid this artefact, one can add the Dirchlet boundary condition (8) which appeared throughout our constructions. However in the next example we see that this Dirichlet condition is not enough to obtain uniqueness. In fact a state constraint boundary condition is hidden behind the property u 1 = -∞ in the complement of Ω η and we do not take it into account here.

Let R(x) = 1 and u 0 (x) = -x 2 .

A simple computation shows that the solution u 1 to (13) is given by

u 1 (x, t) = t - x 2 1 + 4t .
Therefore the first truncation of u 1 , given by

u(x, t) =      t -x 2 1+4t for t -x 2 1+4t ≥ u m , -∞ otherwise,
with Ω = {(x, t) : u(x, t) > -∞}, is a viscosity solution of [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF]. As a matter of fact this is the maximal subsolution to (7), (8) but it does not satisfy the state constraint boundary condition. To see this choose u m = -0.04. The point [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF] is included in Ω since u(1, 2) = 0.2 > -0.04. The optimal trajectory associated to this point, giving the value u(1, 2) = 0.2, is the straight line connecting (0, 0.4) to [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]. But u 0 (0.4) = -0.16 < -0.04. So the point (0, 0.4) is not included in Ω. Therefore a part of the optimal trajectory of the point (1, 2) is not included in Ω. Hence u does not satisfy the state constraint condition.

Following the arguments in Section 4 we can find a viscosity solution to [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF] and [START_REF] Cardaliaguet | A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable[END_REF]. Indeed using (48) it is possible to compute explicitly the function

U [u 0 ] = lim δ→0 U δ [u 0 ] = lim δ→0 u δ 2 [u 0 ] to find ȗ(x, t) =                t -x 2 1+4t if -x 2 (1+4t) 2 ≥ u m , t -x 2 1+4t ≥ u m , t -(x- √ -um) 2 4t + u m if x > 0, -x 2 (1+4t) 2 ≤ u m , t ≥ (x- √ -um) 2 4t , t -(x+ √ -um) 2 4t + u m if x < 0, -x 2 (1+4t) 2 ≤ u m , t ≥ (x+ √ -um) 2 4t
, -∞ otherwise, with Ω = {(x, t) : ȗ(x, t) > -∞}.

From Theorem 4.1 we know that ȗ is indeed the pointwise limit of the family (u ε ) ε>0 outside the exceptional set {(x, t) : ȗ(x, t) = u m }.

However, in general u = ȗ. Consider, for instance, the value u m = -0.04. Then u(2, 1) = 0.2, ȗ(2, 1) = 0.15, u(2.21, 1) = 0.02, ȗ(2.21, 1) = -∞. and, consequently, Ω Ω.

On the other hand, according to Section 4, the state constraint boundary condition is satisfied for ȗ, which motivates our conjecture in Section 6.

B u δ i [u 0 ] is a minimal solution of (33) in

C δ i [u 0 ]
Here we prove that u δ i [u 0 ] is a minimal solution of (33) in C δ i [u 0 ] by considering a supersolution w ∈ C δ i [u 0 ] of (33) and showing that

u δ i [u 0 ] ≤ w in C δ i [u 0 ]. ( 59
)
To this end, we fix (x, t) ∈ C δ i [u 0 ] and assume that (γ(•), •) : [0, t] → Ω δ i-1 [u 0 ] is a C 1 -trajectory with (γ(t), t) = (x, t). Since C δ i [u 0 ] is the set of points that can be connected by a C 1 -trajectory in Ω δ i-1 [u 0 ] to some point in R d × {0}, it follows that γ is included in C δ i [u 0 ]. For the supersolution w, we define, for s ∈ [0, t], the (clearly) lower semicontinuous function ϕ(s) = w(γ(s), s) and we observe that ϕ is a viscosity supersolution of

ϕ ′ ≥ - | γ| 2 4 + R(γ) in (0, t). (60)
We postpone the proof of this claim to the end of the present paragraph and we proceed noticing that the function

  which does not touch R d × {0}. (See Figure 1.) Therefore ∂ p D ⊂ ∂Ω δ i [u 0 ], where ∂ p D = {(y, s) ∈ ∂D : s < t} is the parabolic boundary of D. From the previous argument we obtain u = -∞ on ∂ p D.

Figure 2 : 2 4t+ 2 4t+

 222 Figure 2: The case with R(x) = R a positive constant, {x ∈ R| u 0 (x) > u m } = (a, b) and u 0 (•) ≥ u m on an interval [a, b]. Then Ω = ∪ d∈[a,b] {(x, t) | -|x-d| 2 4t + Rt + u 0 (d) ≥ u m }, the optimal trajectories

ψ(t) = t 0 -

 0 | γ(s)| 2 4+ R(γ(s)) ds + w(γ(0), 0),

Acknowledgements. The authors wish to thank the anonymous referee for his very careful reading of the first version of this article and his numerous suggestions to improve its readibility.

Partially supported by the National Science Foundation

is a subsolution of (60). Then using the standard comparison principle of viscosity solutions we obtain w(x, t) = ϕ(t)

and, since this is true for any C 1 -trajectory γ and any (x, t) ∈ C δ i [u 0 ], (59) follows. It remains to prove (60). Let φ ∈ C 1 ((0, t)) be a test function, assume that t is a strict minimum point of ϕ -φ and consider the function

which attains a local minimum at a point (y µ , t µ ) such that, as µ → 0,

Since w is a supersolution to (33), we have

It is immediate that

and, after letting µ → 0, we conclude using (61).

C The proof of (57)

We prove by induction on i that, for all

is the solution of (13). Moreover (54) yields

Therefore, for all h > h ≥ µ/a, we have

and, consequently,

where e t is the unit vector in the direction of time axis. Fix (x, t) ∈ C δ i+1 [u 0 ] ⊂ Ω δ i [u 0 ] and let γ be a C 1 -trajectory in Ω δ i [u 0 ] connecting (x, t) to a point (y, 0) with u 0 (y) > u m -δ. It follows from (55) that there exists z ∈ R d such that |z -y| < ρ δ,µ and u 0 (z) > u m -δ + µ. Without loss of generality we can take u 0 (z) ≥ u 0 (y).

The claim is that the trajectory γ : [0, t + h] → R d defined by

Indeed notice that the choice of h yields, for all h > h,

Consequently, it follows from (54) and the choice of z that the straight line connecting (y,

and, in particular, in

Moreover using (62) we find that (γ(s), s + h) ∈ Ω δ i [u 0 -µ] for all s ≥ 0. Hence, for all h < s, ( γ(s), s) ∈ Ω δ i [u 0 -µ], and we conclude that γ is included in

Next write

If this is true, then using (30), ( 64) and (65) we obtain, for all h > h and t > 0,

and we deduce (57).

It remains to prove (65). Since R ≥ a, in view of (63), we have u 0 (z) ≥ u 0 (y),


Laurence Danlos
email: danlos@linguist.jussieu.fr

Frédéric Meunier
email: meunier@watchsystance.com

Vanessa Combet Watchsystem

EasyText : an Operational NLG System

This paper introduces EasyText, a fully operational NLG System. This application processes numerical data (in tables) in order to generate specific analytical comments of these tables. We start by describing the context of this particular NLG application (communicative goal, user profiles, etc.). We then shortly present the theoretical background which underlies EasyText, before describing its implementation and realization.

Introduction

EasyText is a NLG system which is operational at Kantar Media, a French subsidiary company of TNS-Sofres1 . The company compiles numerical data for its customers on their advertising investments and sends to each customer seven tables every month, see Figure 1 for an example of a table. Before the existence of EasyText, these tables were presented with a general commentary written by a media analyst. Kantar decided to accompany these tables with specific charts and comments in order to make their reading easier. They survey 600 segments and there are 7 tables per segment: Manually writing these analytical comments was inconceivable. The idea of having an automatic system producing them naturally arose, but Kantar encountered major difficulties with the text generation task. Therefore, they subcontracted this project to Watch System Assistance. Figure 1 shows an example of an analytical comment generated by EasyText.

Section 2 describes the architecture of EasyText. Section 3 presents its implementation: EasyText is an instantiation for Kantar Media's needs of a readyto-use NLG framework, TextElaborator. Section 4 gives some details on the realization and evaluation of EasyText.

Architecture of EasyText

EasyText follows a standard architecture as described in [START_REF] Reiter | Building Natural Language Generation Systems[END_REF]. It includes a document planner for the content determination and document structuring tasks, and a tactical component.

The content determination task for a given table amounts to detecting the relevant cells of the table. This task was guided by business rules indicated by Kantar Media's analysts. These rules were hardcoded, i.e. without any reasoning module.

The content of a cell is transformed into the conceptual representation of an eventuality whose predicate is given by the column heading. This predicate subcategorizes two arguments, the first one corresponding to the line heading, the second one to the value of the cell. Therefore, the output of the content determination module can be seen as a conjunction of conceptual representations of eventualities.

The document structuring task consists in introducing rhetorical relations between the semantic content of the highlighted cells. For instance, if two opposite evolutions over a given period are observed (one decreasing, the other one increasing), the relation Contrast is introduced. On the contrary, would they have been going in the same direction, the relation P arallel would have been introduced, along with some hints to prepare an aggregation operation in the tactical component.

Évolution des investissements par Secteur / Variété

The discourse theory on which the document structuring module relies is SDRT (Segmented Discourse Representation Theory) [START_REF] Asher | Reference to Abstract Objects in Discourse[END_REF][START_REF] Asher | Indirect speech acts[END_REF], following [START_REF] Danlos | Document structuring à la SDRT[END_REF]. The output of the document structuring component is therefore consistent with a SDRS (SDRT structure), considered as a "conceptual" representation in which concepts (discourse relations, eventualities, entities) are embedded in a dependency structure (which is mathematically a Directed Acyclic Graph).

The tactical component (macro/micro-planner and surface realizer) is based on G-TAG formalism [START_REF] Danlos | G-TAG: A lexicalized formalism for text generation inspired from TAG[END_REF], the latter being itself founded on lexicalized Tree Adjoining Grammars (TAG), [START_REF] Joshi | Tree-adjoining grammars[END_REF].2 G-TAG deals with the How to say it? issue, understood as covering all (and only) linguistic decisions: segmentation of the text into sentences and linear ordering of these sentences 3 , choice of discourse connectives and other lexical items, syntactic constructions within sentences, aggregation operations, referring expressions, semantic and syntactic parallelism, etc.

The surface realizer is designed to use the syntactic and lexical information of a TAG grammar. This TAG grammar is extended to handle multi-sentential texts and not only isolated sentences 4 . Therefore, the macro/mico planner is designed as a TAG extension. More precisely, the architecture of G-TAG is outlined in Figure 2: • The output of the macro/mico planner is a "gderivation tree". In TAG, a derivation tree is [START_REF] Gardent | A symbolic approach to near-deterministic surface realisation using tree adjoining grammar[END_REF]. However, it is not in the scope of this paper to compare G-TAG with these other approaches.

3 These tasks are not considered as part of the document structuring component. This is why the term macro/micoplanner is used in Figure 2. 4 The idea of extending TAG to handle multi-sentential texts is also used in text interpretation, e.g. D-LTAG [START_REF] Webber | D-LTAG: extending lexicalized TAG to discourse[END_REF] and D-STAG [START_REF] Danlos | D-STAG: a formalism for discourse analysis based on SDRT and using synchronous TAG[END_REF]. not only seen as the history of the derivation but also as a linguistic representation, close to semantics, which can serve as a basis for a deeper semantic analysis [START_REF] Kallmeyer | Using an enriched tag derivation structure as basis for semantics[END_REF]. A g-derivation tree in G-TAG is closer to semantics than a derivation tree in TAG: it is a semantic dependency tree annotated with syntactic information. Moreover, a g-derivation tree represents a text while a derivation tree represents a unique sentence.

SDRS

The macro/micro planner relies on lexical databases associated with the various concepts (discourse relations, eventualities, entities) that are relevant for the NLG application. A lexical database for a given element records the lexemes lexicalizing it with their argument structure, and the mappings between the conceptual and semantic arguments. With such a lexicalized planner, the process for computing a gderivation tree relies upon a single type of operation: lexicalization, i.e. choice of a lexeme and its syntactic realization to convey an instance of a concept. Since all the main decisions are made during this process, G-TAG can be considered as a fully lexicalized formalism for text generation.

• Thanks to a TAG grammar (which specifies the mapping between the semantic and syntactic arguments), a g-derivation tree specifies a unique "g-derived tree", in the same way as a derivation tree specifies a unique derived tree. A g-derived tree is a syntactic tree annotated with morphological information.

• From a g-derived tree, a post-processing module computes a text by performing morphological computations5 and formatting operations.

Lexical databases for EasyText have been developed by a linguist, V. Combet, who was working in close collaboration with Kantar Media's analysts. Particular attention was paid to linguistic variation in order to avoid producing tiresome texts for Kantar Media's customers. This variation mainly concerns:

• the lexical choices: the databases associated to a given concept are as exhaustive as possible. For example, the concept INCREASE with a MAGNITUDE argument is lexicalized either with the verb augmenter, doubler or tripler or with the light verb construction être en hausse/augmentation or enregistrer une hausse/augmentation6 . Moreover, a verb can be modified with an adverb, e.g. faiblement, fortement, modéremment for augmenter and presque/pratiquement/plus que for doubler or tripler7 , while the noun in a light verb construction can be modified with a preposed adjective, e.g. faible/forte, or a postposed one, e.g. modérée8 .

• the order of the phrases: some phrases can appear more or less freely in different places in a sentence. This is the case for duration adverbials such as pendant le mois de mai (during May) and also for different prepositional phrases such as les investissements [pour la variété X] augmentent [pour la variété X] (ad spendings [for sector X] increase [for sector X]).

Implementation

A prototype of G-TAG was first implemented in Ada [START_REF] Meunier | Implémentation du formalisme G-TAG[END_REF]. G-TAG has been re-implemented as a ready-to-use framework, TextElaborator. Text-Elaborator is based on the Microsoft .Net framework. Particular attention was paid on functional and business issues while taking advantage of .Net for technical and non functional issues (persistence, reliability, scalability, etc.). We chose to rely on classical design patterns 9 , which garantee an effortless reusability of the different components.

Our main implementation effort for TextElaborator was to build an IDE (Integrated Development Environment) incorporating tools which facilitate the linguistic work, i.e. feeding, editing, debugging and testing the various lexical databases -tasks which 9 DAO (Data Access Object, http://en.wikipedia. org/wiki/Data_access_object) and DTO (Data Transfer Object, http://en.wikipedia.org/wiki/ Data_transfer_object) .

are crucial in G-TAG not only in the development but also the maintenance phases.

A screenshot of this IDE is shown in Figure 3. The left column gives the domain ontology hierarchized as Abstract Objects (discourse relations and eventualities) and Entities. When clicking on a concept of the domain ontology (e.g. Hausse in Figure 3), the tab LexicalPredicates indicates the G-TAG lexical database associated with this concept. When choosing an element of this database, the corresponding g-derivation tree is displayed, along with some corollary information.

EasyText is an instantiation of TextElaborator for Kantar Media's needs, constiting in an ontology and its corresponding lexical databases. TextElaborator is written in C# language and is built and run upon the Microsoft .Net framework. Thanks to .Net, its integration into Kantar Media's information system was easy. Generating a comment as the one shown in Figure 1 requires an average of 400ms. The development of EasyText took 7 mm (men month) altogether:

• 1 mm was dedicated to the linguist's training to TAG and G-TAG;

• 1 mm was dedicated to interviews with Kantar's media analysts;

• 3 mm to design TextElaborator and its IDE;

• 2 mm to fill the lexical databases.

During these 7 months, we were never in contact with Kantar's customers directly, but worked in closed interaction with the two departments involved in the project. On the one hand, we obviously interacted with Kantar's media analysts. They shared with us all their know-how on writing commentaries on Kantar's tables, enabling us to create lexical databases corresponding to their editorial habits.

On the other hand, EasyText was developed in close collaboration with Kantar's Information system department, so as to meet their technical requirements: performance and compatibility with the existing infrastructure.

When we released the first version of EasyText, Kantar decided to send the automatically generated commentaries to a couple of customers, without saying anything about the way they have been written.

These customers made some critics10 but gave Kantar Media the feedback that they were quite pleased with this offer. Therefore, Kantar Media decided in April 2010 to commercialize this new product and acknowledged that the commentaries were automatically generated. They keep on commercializing this service, which means that their clients are satisfied.

EasyText evaluation was made by Kantar's media analysts during several months. This evaluation was qualitative and concerned the relevance of the commentaries (the choice of the cells to comment) and their smoothness. We remind the reader (Section 1) that EasyText commentaries had never been handwritten. Therefore, we cannot make any comparison between the generated texts and handwritten ones. This situation seems to be quite usual, since NLG applications are likely to be commercialized when automatic writing doesn't replace hand writing 11 . Indeed, the few commercial NLG systems we are aware of are in the same situation. 12

Conclusion

We have presented an operational system and, while many NLG prototypes exist, not many are commercialized, eventhough NLG technology is mature.

EasyText is an instantiation of a ready-to-use framework, TextElaborator, which is based on solid scientific basis concerning not only its architecture -the standard one [START_REF] Reiter | Building Natural Language Generation Systems[END_REF] -but also the particular instantiation of this architecture with well-established analysis formalisms (SDRT and TAG) which have been adapted to text generation.

It is foreseen that TextElaborator will be used for other applications and will produce texts in other languages than French, since it was developed as a ready-to-use framework. For a new application, the domain ontology has to be adapted and the G-TAG lexical databases associated with the concepts have to be filled. When moving to another language, only the lexical databases will have to be changed, hopefully.

A demonstration of EasyText will be presented during the conference.

Macro

  Figure 2: G-TAG tactical component

Figure 3 :

 3 Figure 3: Sceenshot of TextElaborator's integrated development environment

  Par ailleurs, les investissements pour la variété CREDIT PERSONNEL O.F marquent une progression de 6% pour le cumul à date étudié. Au contraire, pour la variété MULTIPROD.ORG.FINANCIERS, ils voient leur volume diminuer (-3%) sur la même période.

	Investissements publicitaires plurimedia -Tri décroissant sur le cumul de l'année en cours -En k€
		Mai 2008	Mai 2009	Evol%	Cumul 2008 janvier à mai	Cumul 2009 janvier à mai	Evol%
	ORGANISMES FINANCIERS	16 587	26 312	59 %	216 948	177 353	-18 %
	CREDIT PERSONNEL O.F	5 868	11 227	91 %	50 610	53 772	6 %
	~MULTIPROD.ORG.FINANCIERS	3 243	7 463	130 %	53 191	51 718	-3 %
	CREDIT RENOUVELABLE O.F	3 930	1 994	-49 %	60 094	34 987	-42 %
	INTERNET TELEMATIQUE 583	2 648	4 687	77 %	16 460	27 613	68 %
	RACHAT DE CREDITS O.F	777	732	-6 %	15 817	5 637	-64 %
	CREDIT AUTO MOTO O.F	79	110	39 %	5 638	993	-82 %
	CREDIT TRAVAUX O.F		86		535	797	49 %
	PARRAINAGE MECENAT O.F				80	0	-100 %
	Dans votre univers, les investissements marquent une très forte progression (+59%) dans le secteur ORGANISMES
	FINANCIERS en mai 2008 par rapport à mai 2007. Toutefois, pour le cumul à date de l'étude, ils connaissent une baisse de 18%.
	Dans ce secteur, les investissements ont doublé (+130%) pour la variété MULTIPROD.ORG.FINANCIERS en mai 2008 par
	rapport à mai 2007.

(Within your business area, ad spending ramps up (+59%) for sector ORGANISMES FINANCIERS in May 2008 compared with May 2007. However, year to date, it falls 18%. Within this sector, ad spending doubles (+130%) for segment MULTI -PROD.ORG.FINANCIERS in May 2008 compared to May 2007. Furthermore, ad spending for segment CREDIT PERSONNEL OF increases of 6% year to date. On the contrary, for segment MULTIPROD.ORG.FINANCIERS, it decreases (-3%) over the same period.) Figure 1: Example of a table and its automatically generated comment

Kantar Media is one of the leaders in advertising expenditure monitoring, exploring all existing media (radio, internet, mobile telephony, etc.).

Since it was put forward by A. Joshi that TAG is an especially well suited grammatical theory for text generation, adapting TAG for generation has been widely explored, among many others, let us cite[START_REF] Stone | Sentence planning as description using tree adjoining grammar[END_REF] and(Gardent and

Morphological operations include elisions (la augmentation → l'augmentation) and contractions (de le mois → du mois).

 6 In English, verbs increase, double or triple and light verb constructions be on the increase or record an

increase.7 In English, adverbs slightly, seriously, moderately for increase and almost, nearly, more than for

double or triple.8 In English, an adjective is always preposed.

The main critic concerned the laying-out of these commentaries.