Marlène Gamelon
email: marlene.gamelon@gmail.com

Stefano Focardi

Jean-Michel Gaillard

Olivier Gimenez

Christophe Bonenfant

Barbara Franzetti

Rémi Choquet

Francesca Ronchi

Eric Baubet

Jean-François Lemaître

Do age-specific survival patterns of wild boar fit current evolutionary theories of senescence?

Keywords: ageing, capture-mark-recapture, fertility, life history, mortality, Sus scrofa

Actuarial senescence is an inevitable outcome of evolution in age-structured populations. In growing populations, the progressive decline of Hamiltonian forces of selection with age leads to decreasing survival. As actuarial senescence is over-compensated by a high fertility, actuarial senescence should be more intense in species with high reproductive effort, a theoretical prediction that has not been yet explicitly tested across species. Wild boar (Sus scrofa) has an unusual life-history strategy among large mammals by associating both early and high reproductive effort with potentially long lifespan. Therefore, wild boar should show stronger actuarial senescence than similar-sized related mammals. Moreover, being polygynous and much larger than females, males should display higher senescence rates than females. From a long-term monitoring of 1,783 individuals, we tested these predictions. We provided clear evidence of actuarial senescence in both sexes. Wild boar females had earlier but not stronger actuarial senescence than similar-sized ungulates and both sexes displayed similar senescence rates. Our study indicates that the timing of senescence, not the tempo, is associated with the magnitude of fertility in ungulates. This demonstrates the importance of including the timing of senescence in addition to its rate to understand variation in senescence patterns in wild populations.

Actuarial senescence (hereafter senescence) defined as the decline of survival with increasing age, has been repeatedly documented in wild species of vertebrates (see [START_REF] Nussey | Senescence in natural populations of animals: Widespread evidence and its implications for biogerontology[END_REF] for a review). However, empirical evidence has increased only recently, thanks to the development of long-term individual monitoring of known-aged animals [START_REF] Clutton-Brock | Individuals and populations: the role of longterm, individual-based studies of animals in ecology and evolutionary biology[END_REF]. William D. Hamilton already demonstrated nearly fiftyyears ago [START_REF] Hamilton | The moulding of senescence by natural selection[END_REF]) that senescence was an inevitable outcome of evolution in agestructured populations. More specifically, based on the decline of the forces of natural selection with age in growing age-structured populations [START_REF] Medawar | An Unsolved Problem of Biology[END_REF], Hamilton predicted that an increase in fertility should be selected at the costs of a survival decrease at old ages. In other words, high fertility should be associated with high rates of senescence [START_REF] Hamilton | The moulding of senescence by natural selection[END_REF]. Until now, intraspecific studies have provided good evidence for a relationship between early allocation to reproduction and fitness-related traits, such as survival [START_REF] Hammers | The impact of reproductive investment and early-life environmental conditions on senescence: support for the disposable soma hypothesis[END_REF]) and components of reproductive success [START_REF] Nussey | The rate of senescence in maternal performance increases with early-life fecundity in red deer[END_REF] in late life, in line with the disposable soma theory of ageing [START_REF] Kirkwood | Evolution of ageing[END_REF][START_REF] Kirkwood | Why do we age?[END_REF]. On the other hand, while some recent comparative studies have revealed an association between intensity of actuarial senescence and pace of life across species within and among classes of vertebrates [START_REF] Jones | Senescence rates are determined by ranking on the fast-slow life-history continuum[END_REF][START_REF] Péron | Age at onset of senescence in birds and mammals is predicted by early-life performance[END_REF], Lemaître and Gaillard 2013), no study has tested yet the exact Hamilton's prediction of a direct link between fecundity and intensity of actuarial senescence across species. Moreover, while empirical evidence that females live much longer than males in most mammals studied so far is compelling [START_REF] Carey | Longevity records: life spans of mammals, birds, amphibians, reptiles and fish[END_REF], whether and how age-specific trajectories differ between sexes remains an open question (Nussey et al. 2013, Maklakov and[START_REF] Maklakov | Evolution of sex differences in lifespan and aging: causes and constraints[END_REF], although the theoretical basis for such sex differences has been established for a long time [START_REF] Williams | Pleiotropy, natural selection, and the evolution of senescence[END_REF]. Hamiltonian forces of selection again provide the explanation for more intense senescence in males when these latter suffer from higher mortality during the prime-age stage. Starvation and antagonist contests during the mating season are likely to cause increased mortality rates in males compared to females in polygynous species [START_REF] Bonduriansky | Sexual selection, sexual conflict and the evolution of ageing and life span[END_REF], especially when environmental conditions are harsh [START_REF] Toïgo | Causes of sex-biased adult survival in ungulates: sexual size dimorphism, mating tactic or environment harshness?[END_REF]. Thus, under current life history theory, males from highly polygynous and dimorphic species should exhibit a 'live fast, die young' life history strategy [START_REF] Vinogradov | Male reproductive strategy and decrease longevity[END_REF][START_REF] Bonduriansky | Sexual selection, sexual conflict and the evolution of ageing and life span[END_REF][START_REF] Festa-Bianchet | Human-induced reversal of fortunes for migratory ungulates?[END_REF] and should therefore have higher senescence rate than females. Using a 17 year-long monitoring of an Italian population of wild boar (Sus scrofa), we filled these gaps by investigating the pattern of actuarial senescence for this large mammal characterized by a unusual association between high fertility and high sexual size dimorphism [START_REF] Carranza | Sexual selection for male body mass and the evolution of litter size in mammals[END_REF]. Indeed, compared to similar-sized related species, wild boar females give birth for the first time earlier (1 vs. 2-3 years of age, [START_REF] Servanty | Pulsed resources and climate-induced variation in the reproductive traits of wild boar under high hunting pressure[END_REF], at a lower body mass (40% vs. 80% of adult body mass, [START_REF] Servanty | Pulsed resources and climate-induced variation in the reproductive traits of wild boar under high hunting pressure[END_REF], and produce larger litters (>3 vs. 1 or 2, [START_REF] Gamelon | The relationship between phenotypic variation among offspring and mother body mass in wild boar: Evidence of coin-flipping?[END_REF]. By allocating early and heavily to reproduction, wild boar females have an unusual life-history strategy among large mammals [START_REF] Focardi | Survival of wild boars in a variable environment: unexpected life-history variation in an unusual ungulate[END_REF].

Therefore, in accordance with Hamilton's theory, wild boar females should show faster senescence than reported for similar-sized ungulates. Moreover, as wild boar males are markedly heavier than females (90 vs. 70 kg for males and females, respectively, [START_REF] Gaillard | Sex differentiation in postnatal growth rate: a test in a wild boar population[END_REF]) and are highly polygynous [START_REF] Delgado | Paternity assessment in free-ranging wild boar (Sus scrofa) -Are littermates full-sibs?[END_REF], they should suffer from more intense senescence than females.

Materials and methods

STUDY SITE AND DATA COLLECTION

We studied a weakly hunted and growing wild boar population in the fenced 6,000 ha preserve of Castelporziano (41°44'N, 12°24'E) in Italy [START_REF] Focardi | Survival of wild boars in a variable environment: unexpected life-history variation in an unusual ungulate[END_REF]. From 1995 to 2012, a total of 1,783 juvenile (< 1 year of age) and yearling (1 to 2 years old) wild boars were marked with numbered and colored ear tags and released after handling. Between July and September, wild boars were artificially fed three to six times per week on 80-88 feeding sites [START_REF] Franzetti | Nocturnal line transect sampling of wild boar (Sus scrofa) in a Mediterranean forest: long-term comparison with capture-mark-resight population estimates[END_REF] to allow counts and identification of marked animals from year to year.

ESTIMATING AGE-SPECIFIC SURVIVAL, RATE AND ONSET OF SENESCENCE IN MALE AND FEMALE WILD BOAR

We estimated natural survival rates of marked individuals using Capture-Mark-Recapture analyses [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies[END_REF]) of both recaptures and resightings. Animals removed under population control plans were right-censored to allow estimating natural survival. As the preserve was fenced, no dispersal occurred so that true survival was estimated.

We first tested the goodness-of-fit (GOF) [START_REF] Pradel | Principles and interest of GOF tests for multistate capture-recapture models[END_REF] of the survival models using U-CARE (Choquet et al. 2009a). We accounted for heterogeneity in the detection process of juveniles (see Results) by modelling two different recapture probabilities according to whether an individual was previously detected (trap-dependence: [START_REF] Gimenez | Parameter redundancy in multistate capture-recapture models[END_REF][START_REF] Pradel | Modeling trap-awareness and related phenomena in capture-recapture studies[END_REF]. As capture effort varied among years [START_REF] Focardi | Survival of wild boars in a variable environment: unexpected life-history variation in an unusual ungulate[END_REF], we included time-dependence in recapture probabilities. For survival probabilities, we fitted both a model without any effect of age and the full age-dependent model. From the latter, the rate of senescence was measured as the slope of the regression of log-transformed age-specific mortality rates on age to compare with senescence rates previously reported in ungulates [START_REF] Gaillard | Ecological correlates of lifespan in populations of large herbivorous mammals[END_REF]. To avoid confounding effects of growth or reproduction, we analyzed mortality rates from 2 years of age onward, when growth has been completed.

To estimate the age at the onset of senescence, we constrained survival to vary linearly on the logit scale over different ranges of ages (ages 1, 2, 3, 4, 5, or 6; see [START_REF] Nichols | Tests for senescent decline in annual survival probabilities of common pochards, Aythya ferina[END_REF] for a similar approach and table 1). The age at onset of senescence was defined as the age when survival started to decline as assessed from the selected model. We thus fitted agespecific survival models while testing for between-sex differences in age-specific survival and rate of senescence. We used the Akaike's Information Criterion for model selection [START_REF] Burnham | Model selection and multimodel inference: a practical information-theoretic approach[END_REF], selecting the model with the lowest AIC. All analyses were implemented in E-SURGE (Choquet et al. 2009b).

SURVIVAL PATTERNS OF WILD BOAR COMPARED TO SIMILAR-SIZED UNGULATES

To test whether wild boar females show faster and/or earlier senescence than reported for similar-sized ungulates, we accounted for the confounding effect of body mass [START_REF] Calder | Size, function, and life history[END_REF]. Indeed, it is well-established that the pace of life decreases with increasing mass [START_REF] Gaillard | Generation time: a reliable metric to measure life-history variation among mammalian populations[END_REF], leading large species to exhibit later onset and smaller rate of senescence than small ones [START_REF] Jones | Senescence rates are determined by ranking on the fast-slow life-history continuum[END_REF][START_REF] Péron | Age at onset of senescence in birds and mammals is predicted by early-life performance[END_REF][START_REF] Ricklefs | Life-history connections to rates of aging in terrestrial vertebrates[END_REF]). We thus assessed the relationship between senescence rate (on a log-scale) and female body mass (on a log-scale) as well as the relationship between onset of senescence (on a log-scale) and female body mass (on a log-scale) of similar-sized ungulates. (2010)'s study.

Patterns of senescence across species might be not independent as a result of common ancestry [START_REF] Harvey | The Comparative Method in Evolutionary Biology[END_REF]. We thus controlled for this non-independence by using Phylogenetic Generalized Least-Squares models (PGLS) that provide an index of the phylogenetic signal called λ, varying in most of the cases between 0 (corresponding to the complete absence of phylogenetic structure) and 1 (observed pattern predicted by the phylogeny; see [START_REF] Freckleton | Phylogenetic analysis and comparative data: A test and review of evidence[END_REF]. This parameter λ was then introduced in the model to control for the phylogenetic effect [START_REF] Freckleton | Phylogenetic analysis and comparative data: A test and review of evidence[END_REF]. We used the phylogeny derived from the phylogenetic supertree of mammals, which provides information on both topology and branch length (Bininda-Emonds et al. 2007). However, in the models linking rate and onset of senescence (on a log-scale) to body mass (on a log-scale) in similar-sized ungulates, the phylogenetic signal (λ) did not differ from 0 (λ rate < 0.0001 and λ onset < 0.0001), meaning that there was no detectable phylogenetic signal in both rate and onset of senescence.

Therefore, we conducted inter-specific analyses linking senescence rate and onset of senescence to female body mass, using standard linear models (i.e. without any correction for phylogeny) to avoid biased estimates [START_REF] Revell | Phylogenetic signal and linear regression on species data[END_REF]. From these linear models, expected rate and onset of senescence of wild boar females were estimated and compared with rates and onsets found in populations of similar-sized ungulates. All analyses were performed with the R software (R Development Core Team 2012).

Results

ESTIMATING AGE-SPECIFIC SURVIVAL, RATE AND ONSET OF SENESCENCE IN MALE AND FEMALE WILD BOAR

By only including individuals first captured as juveniles, the GOF test indicated some lack of fit for both sexes (global test: for male juveniles: χ 2 (61) = 102.743, P < 0.0001; for female juveniles: χ 2 (72) = 128.411, P < 0.0001). More precisely, trap-dependence was detected for both sexes and accounted for in the model. On the contrary, the GOF test indicated no lack of fit for individuals first captured as yearlings (global test: for males: χ 2 (27) = 31.273, P = 0.260; for females: χ 2 (33) = 27.575, P = 0.734). Tested models thus satisfactorily fitted data when trap-dependence was included in capture probabilities of juveniles.

The best model included marked between-sex differences at age 2 (Fig. 1), a linear relationship between survival and age from 3 years old onwards, and year-dependent capture probabilities (table 1, appendix S1). Males and females had same survival at age 1 (Φ age1 = 0.68 [0.64; 0.72]), but males survived a bit less than females at 2 years old (Φ age2 male = 0.57 [0.50; 0.64], Φ age2 female = 0.85 [0.77; 0.91]). From 3 years old onwards, males consistently had lower survival than females (Fig. 1) but both sexes had similar senescence rates (slope on the logit scale = -0.15 [-0.23; -0.08]).

To compare senescence rates between wild boar and similar-sized ungulates, senescence rates were re-estimated using the linear regression between mortality rates (on a log scale) and age. These rates exactly correspond to the Gompertz rates (i.e. the exponential rate of increasing mortality with age) and slightly differ from the rates estimated directly from analyses provided above (which do not exactly correspond to Gompertz rates, [START_REF] Loison | Agespecific survival in five populations of ungulates: evidence of senescence[END_REF]. After removing survival rates at 10 and 12 years old for males and females, respectively, estimated on the bounds of the interval [0, 1] (see Fig. 1), we found evidence of actuarial senescence in both sexes (rates of 0.11 (SE: 0.04) for males (Wald test: 9.7, df = 1, p-value = 0.002) and 0.14 (SE: 0.04) for females (Wald test: 16.0, df = 1, p-value < 0.0001), Fig. 2).

SURVIVAL PATTERNS OF WILD BOAR COMPARED TO SIMILAR-SIZED UNGULATES

As expected, across ungulate species, the senescence rate tended to decrease (slope: -0.11, SE: 0.18) and the onset of senescence to be delayed (slope: 0.17, SE: 0.12) when body mass increased (Fig. 3). From the relationship between senescence rate and body mass (on a log-log scale) performed using 9 populations belonging to 7 different species, the expected rate of senescence for wild boar females was similar to the expectation (expected value of 0.16 vs. observed value of 0.14, Fig. 3a). On the other hand, the onset of senescence was much earlier than expected. From the relationship between the onset of senescence and body mass (on a log-log scale), the expected value was 8.56 years, whereas it was estimated to only 3 years in our analysis (Fig. 3b). Wild boar females had thus markedly earlier, but not stronger senescence (Fig. 3) for their body size compared to other ungulates.

Discussion

Our results provide clear evidence for actuarial senescence in male and female wild boar and add to the spate of studies that have recently found senescence in free-living vertebrates [START_REF] Nussey | Senescence in natural populations of animals: Widespread evidence and its implications for biogerontology[END_REF]). More importantly, using a pioneering inter-specific test of the Hamilton's prediction of a link between fecundity and intensity of actuarial senescence, our findings show that wild boar females have earlier, but not stronger senescence than related species once the confounding effects of body size has been accounted for. Indeed, our findings emphasize that the high reproductive output of wild boars (with litter size up to 14, [START_REF] Servanty | Litter size and fetal sex ratio adjustment in a highly polytocous species: the wild boar[END_REF]) is associated with earlier, but not stronger decline in survival with age.

As Hamilton's theory was strictly focused on senescence rates, our study did not provide a direct support for it. However, we show that actuarial senescence occurred much earlier in wild boar females than in other ungulates. High and early fertility in wild boar females during the prime-age stage might advance the decline of the forces of natural selection, which indicates that the timing of senescence, in addition to the rate, constitutes a key component of Hamilton's theory of senescence. We thus propose to expand Hamilton's theory to include potential responses of the timing of senescence to increased fertility, by considering both the timing and the intensity of senescence patterns to test reliably evolutionary theories of ageing.

When looking at sex-differences in senescence patterns, we did not detect any difference in the rate of senescence between male and female wild boar. This finding contrasts with a few studies that reported greater actuarial senescence in males compared to females in large herbivores [START_REF] Gaillard | Ecological correlates of lifespan in populations of large herbivorous mammals[END_REF][START_REF] Lemaître | Male survival patterns do not depend on male allocation to sexual competition in large herbivores[END_REF], a pattern that is generally attributed to the polygynous mating system of these species (Clutton-Brock and Isvaran 2007, [START_REF] Bonduriansky | Sexual selection, sexual conflict and the evolution of ageing and life span[END_REF]. The red deer (Cervus elaphus) case study provides a striking example of higher senescence rates (but not earlier onset) in males than in females [START_REF] Catchpole | Dimorphism, survival and dispersal in red deer[END_REF]). Our results are thus particularly surprising from both empirical (Loison et al. 1999, Lemaître andGaillard 2013) and theoretical [START_REF] Williams | Pleiotropy, natural selection, and the evolution of senescence[END_REF] viewpoints. Indeed, because males had higher mortality rates at all ages from 2 years old onwards than females, they were expected to show greater actuarial senescence [START_REF] Williams | Pleiotropy, natural selection, and the evolution of senescence[END_REF][START_REF] Hamilton | The moulding of senescence by natural selection[END_REF].

The higher mortality rates of male wild boar during the whole adult stage might be due to sexspecific reproductive tactics and associated mortality risks [START_REF] Bonduriansky | Sexual selection, sexual conflict and the evolution of ageing and life span[END_REF][START_REF] Festa-Bianchet | Human-induced reversal of fortunes for migratory ungulates?[END_REF] or to the occurrence of harsh environmental conditions in the study site.

Indeed, harsh environments are more likely to influence male than female survival [START_REF] Toïgo | Causes of sex-biased adult survival in ungulates: sexual size dimorphism, mating tactic or environment harshness?[END_REF]. In wild boar, the differential mortality suffered by males do not translate into stronger senescence. Although the causes of similar senescence rates in both sexes despite higher mortality rates in male wild boar remain unknown, it is interesting to notice that similar patterns have been reported in humans [START_REF] Austad | The delayed rise of present-day mammals[END_REF]. In both wild boar and humans, the between-sex differences in adult mortality was quite low (about 10%), which might be not enough to trigger substantial between-sex differences in the decline of Hamilton's forces of selection with age. Simulation studies will be required to test this hypothesis.

Alternatively, differential sources of mortality between sexes could account for the absence of sex differences in actuarial senescence we reported in wild boar. Indeed, recent laboratory experiments have demonstrated that the cause of mortality at early ages can modulate senescence patterns (Chen andMaklakov 2012, Chen et al. 2013). Basically, when the source of mortality is random, all animals in the population have the same probability of dying, independently of their individual phenotype. On the contrary, when mortality is condition-dependent, the probability of dying acts as a filter in the population by removing weakest individuals, which ultimately decreases the intensity of senescence at the population level (Vaupel et al. 1979). In support, an experiment on wild type derived nematodes (Caenorhabditis remanei) has shown high rates of condition-dependent mortality promotes a deceleration of ageing, independently of sex, within only 12 generations [START_REF] Chen | Why ageing stops: heterogeneity explains late-life mortality deceleration in nematodes[END_REF]. In wild boar, differential sources of mortality between sexes could explain our findings in wild boar if condition-dependent mortality only occurs in males. The twofold higher mortality rate of male wild boar at 2 years compared to females might provide the potential filter to keep the best phenotypes at old ages. However, whether mortality at 2 years of age is conditiondependent in males remains to be explored.

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. Relationship between log-transformed mortality rates and age in the wild boar

Figure 3 .

 3 Figure 3. Relationships (a) between senescence rates and body mass (in g) (on a log-log

  Senescence rates were extracted from Gaillard et al. (2003)'s study and onsets of senescence from Péron et al.

	Females of 9 intensively monitored
	populations belonging to 7 different ungulate species were included in the analysis, namely

pronghorn (Antilocapra americana), Alpine ibex (Capra ibex), roe deer (Capreolus capreolus), mountain goat (Oreamnos americanus), bighorn sheep (Ovis canadensis), red deer (Cervus elaphus) and Pyrenean chamois (Rupicapra pyrenaica).

Table 1 .

 1 Model selection for age-specific survival patterns in the wild boar population of Castelporziano, Italy. Φ a(x-13) corresponds to a linear relationship (on a logit-scale) between age and survival from x year of age. Displayed are the number of parameters (Np), the AIC of the 10 best models and the difference between each tested model and the best one (ΔAIC).

	Model notation	Np	AIC	ΔAIC

ACKNOWLEDGMENTS

We thank the Segretariato per la Presidenza della Repubblica for allowing the study at Castelporziano, for its financial support and the data availability.

LITERATURE CITED


Thierry Gallay
email: thierry.gallay@ujf-grenoble.fr

Philippe Laurençot
email: laurenco@mip.ups-tlse.fr

Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent

Keywords: MSC 2000 35B33, 35B40, 35K55, 37L25 diffusive Hamilton-Jacobi equation, large time behavior, critical exponent, absorption, invariant manifold, self-similarity

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation ∂ t u -∆u + |∇u| q = 0 in (0, ∞) × R N is investigated for the critical exponent q = (N + 2)/(N + 1). Convergence towards a rescaled self-similar solution to the linear heat equation is shown, the rescaling factor being (ln t) -(N +1) . The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.

Introduction

The dynamics of integrable non-negative solutions to the viscous Hamilton-Jacobi equation

∂ t u -∆u + |∇u| q = 0 , (t, x) ∈ (0, ∞) × R N , (1)
depends strongly on the value of the parameter q ∈ (0, ∞) and results from the competition between the linear diffusion term ∆u and the nonlinear absorption term |∇u| q . An important issue is therefore to determine which mechanism (diffusion or absorption) is dominant for large times. A first indication is given by the behavior of the L 1 norm u(t) L 1 , which is time-independent for non-negative solutions of the heat equation and strictly decreasing for nontrivial non-negative solutions of [START_REF] Amour | Global existence and decay for viscous Hamilton-Jacobi equations[END_REF]. For such solutions, it is proved in [START_REF] Amour | Global existence and decay for viscous Hamilton-Jacobi equations[END_REF][START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF][START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] that

I ∞ := lim t→∞ u(t) L 1      > 0 if q > q ⋆ := N + 2 N + 1 , = 0 if q ∈ (0, q ⋆ ] . (2)
This suggests that diffusion dominates the large time behavior when q > q ⋆ , whereas absorption becomes effective for q ≤ q ⋆ . As a matter of fact, if q > q ⋆ it is shown in [START_REF] Benachour | Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations[END_REF][START_REF] Biler | Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation[END_REF] that the nonlinear term |∇u| q becomes negligible for large times, and that the solution of (1) behaves as t → ∞ like the self-similar solution I ∞ g of the linear heat equation, where

g(t, x) = 1 t N/2 G x t 1/2
and G(ξ) = 1 (4π) N/2 exp -|ξ| 2

4

.

On the other hand, if q ∈ (1, q ⋆ ), both diffusion and absorption play a role in the large time asymptotics. Indeed, if u(0, x) decays faster than |x| -α as |x| → ∞ with α = (2-q)/(q-1) > N, it is proved in [START_REF] Benachour | Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations[END_REF] that the solution u(t) converges as t → ∞ to the so-called very singular solution, a self-similar solution of (1) whose existence and uniqueness have been established in [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. II -Uniqueness[END_REF][START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I. Existence[END_REF][START_REF] Qi | The self-similar profiles of generalized KPZ equation[END_REF]. In that case, the L 1 -norm of u(t) decays to zero like t -(α-N )/2 as t → ∞.

Finally, the influence of the absorption term |∇u| q is much stronger for q ∈ (0, 1]: depending on the initial data, one might have exponential decay of the solution as t → ∞ [START_REF] Amour | Global existence and decay for viscous Hamilton-Jacobi equations[END_REF][START_REF] Benachour | Solutions fondamentales de u t -1 2 u xx = ±|u x[END_REF][START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF][START_REF] Gilding | The Cauchy problem for u t = ∆u + |∇u| q , large-time behaviour[END_REF], or even extinction in finite time if q ∈ (0, 1) [START_REF] Benachour | Extinction and decay estimates for viscous Hamilton-Jacobi equations in R N[END_REF][START_REF] Benachour | Extinction and nonextinction for viscous Hamilton-Jacobi equations in R N , Asymptot[END_REF][START_REF] Gilding | The Cauchy problem for u t = ∆u + |∇u| q , large-time behaviour[END_REF]. For such values of the parameter, it is the diffusion term which is expected to be negligible for large times.

To summarize, precise asymptotic expansions show that the large time behavior of nonnegative solutions to [START_REF] Amour | Global existence and decay for viscous Hamilton-Jacobi equations[END_REF] with sufficiently localized initial data is determined by the sole diffusion if q > q ⋆ , whereas absorption plays an important role if q < q ⋆ . With this perspective in mind, it is interesting to investigate the critical case q = q ⋆ = (N + 2)/(N + 1) where a transition between both regimes is expected to occur. Very few results are available in this situation: we only know that u(t) L 1 → 0 as t → ∞, as already stated in [START_REF] Andreucci | The Cauchy problem for degenerate parabolic equations with source and damping[END_REF], and that u(t) L 1 cannot decay faster than (ln t) -(N +1) for large times [START_REF] Benachour | Extinction and decay estimates for viscous Hamilton-Jacobi equations in R N[END_REF]Proposition 3]. The purpose of this work is to fill this gap and to give an accurate description of the large time behavior of the non-negative solutions to

∂ t u -∆u + |∇u| q⋆ = 0 , (t, x) ∈ (0, ∞) × R N , (4) u(0) = u 0 , x ∈ R N , (5)
when the initial data u 0 (x) decay to zero sufficiently rapidly as |x| → ∞. More precisely, we assume that u 0 ≥ 0 belongs to the weighted L 2 space

L 2 m (R N ) = u ∈ L 2 (R N ) |u| m := (1+|x| 2m ) 1/2 u L 2 < ∞ , (6)
for some m > N/2. Then (by Hölder's inequality)

u 0 ∈ L 1 (R N ) ∩ L 2 (R N
) and it follows from [START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF][START_REF] Ben-Artzi | The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces[END_REF][START_REF] Gilding | The Cauchy problem for u t = ∆u + |∇u| q[END_REF] that the Cauchy problem (4), ( 5) has a unique global solution

u ∈ C([0, ∞); L 1 (R N )) ∩ C((0, ∞); W 1,∞ (R N )).
Our main result describes the large time behavior of this solution:

Theorem 1 Assume that the initial condition u 0 is non-negative, not identically zero, and belongs to L 2 m (R N ) for some m > N/2. Then the (unique) solution u to ( 4), [START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF] satisfies, for all p ∈ [1, ∞],

lim t→∞ t N 2 (1-1 p ) (ln t) N +1 u(t) - M ⋆ (ln t) N +1 g(t) L p = 0 , (7)
where

M ⋆ = (N + 1) N +1 ∇G -(N +2) L q⋆
and g(t, x), G(ξ) are defined in [START_REF] Benachour | Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations[END_REF].

In other words, if the initial condition decays sufficiently rapidly at infinity, the solution u to (4) behaves asymptotically like a particular self-similar solution M ⋆ g of the linear heat equation, with an extra logarithmic factor due to the effect of the absorption term. Such a logarithmic correction also appears in other parabolic equations with critical nonlinearity, for instance in the nonlinear diffusion equation ∂ t u-∆u m +u m+(2/N ) = 0 with m ≥ 1, see e.g. [START_REF] Galaktionov | Spectra of critical exponents in nonlinear heat equations with absorption[END_REF][START_REF] Galaktionov | Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach[END_REF][START_REF] Herraiz | Asymptotic behaviour of solutions of some semilinear parabolic problems[END_REF] and the references therein. It is interesting to observe that, in both situations, the leading order term in the long-time asymptotics is completely independent of the initial conditions. In the case of the viscous Hamilton-Jacobi equation (4), Theorem 1 indeed shows that neither the asymptotic profile g(t, x) nor the prefactor M ⋆ (ln t) -(N +1) depends on u 0 . This universality property reflects the important role played here by the nonlinearity: when the large time behavior is driven by the linear part of the system, which is the case for (1) when q > q ⋆ , the leading term in the asymptotics does depend on the initial condition.

Remark 2 As g(t) L 1 = 1 for all t > 0, the L 1 -norm of the solution u(t) behaves exactly like M ⋆ (ln t) -(N +1) for large times under the assumptions of Theorem 1. This has to be compared with [START_REF] Benachour | Extinction and decay estimates for viscous Hamilton-Jacobi equations in R N[END_REF]Proposition 3], where it is shown that there is no nontrivial non-negative solution of (4) such that u(t) L 1 ≤ C(ln t) -γ for γ > N + 1. In fact, using Theorem 1 and a comparison argument, it is straightforward to verify that, for all nontrivial non-negative integrable data, the solution of (4) satisfies

lim inf t→∞ (ln t) N +1 u(t) L 1 ≥ M ⋆ .
Our analysis of the large time behavior of solutions to (4), [START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF] relies on an alternative formulation of (4) in terms of the so-called "scaling variables" or "similarity variables"

ξ = x (1 + t) 1/2 and τ = ln (1 + t) . (8)
Introducing the new unknown function v defined by

u(t, x) = 1 (1 + t) N/2 v ln (1 + t), x (1 + t) 1/2 , (t, x) ∈ [0, ∞) × R N , (9)
we deduce from (4), (5) that v(τ, ξ) solves the initial-value problem

∂ τ v = Lv -|∇v| q⋆ , (τ, ξ) ∈ (0, ∞) × R N , ( 10
) v(0) = u 0 , ξ ∈ R N , (11)
where the linear operator L is given by

Lv(ξ) = ∆v(ξ) + 1 2 ξ • ∇v(ξ) + N 2 v(ξ) , ξ ∈ R N . ( 12
)
Observe that equation ( 10) is still autonomous, although it was obtained from (4) through the time-dependent transformation [START_REF] Benachour | Solutions fondamentales de u t -1 2 u xx = ±|u x[END_REF]. This crucial property follows from the fact that (4) is invariant under the rescaling u(t, x) → λ N u(λ 2 t, λx), because q ⋆ = (N + 2)/(N + 1). Remark also that LG = 0, where G is defined in [START_REF] Benachour | Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations[END_REF]. At this stage, we follow the approach of [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF][START_REF] Wayne | Invariant manifolds for parabolic partial differential equations on unbounded domains[END_REF] and prove that the large time behavior of the solutions of [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF], [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] is governed, up to exponentially decaying terms, by an ordinary differential equation which results from restricting the dynamics of (10) to a one-dimensional invariant manifold. This manifold is tangent at the origin to the kernel RG of L, and solutions to [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF] which lie on this manifold satisfy v(τ, ξ) ≈ M(τ ) G(ξ) for large times. Inserting this ansatz into [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF] and integrating over R N we obtain the ordinary differential equation dM/dτ + ∇G q⋆ L q⋆ M q⋆ = 0 for M(τ ), from which we deduce that M(τ ) ≈ M ⋆ τ -(N +1) for large times. Returning to the original variables (t, x), we then conclude that u(t) ≈ M ⋆ (ln t) -(N +1) g(t) as t → ∞, and Theorem 1 follows.

To construct the center manifold, it is necessary to assume that the solutions we consider decay a little bit faster as |x| → ∞ than what is needed to be integrable. Indeed, using the results of [17, Appendix A], it is easy to see that the spectrum of the linear operator L in L 1 (R N ) is just the left half-plane {z ∈ C | ℜ(z) ≤ 0} (no spectral gap). In contrast, the spectrum of the same operator in

L 2 m (R N ) = L 2 (R N ; (1 + |ξ| 2m ) dξ) is given by σ(L; L 2 m (R N )) = z ∈ C ℜ(z) ≤ N 4 - m 2 - k 2 k ∈ N ,
see [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF]Theorem A.1]. Thus, if m > N/2, the operator L has a simple isolated eigenvalue at the origin and the rest of the spectrum is strictly contained in the interior of the left halfplane, a spectral configuration which allows to construct the center manifold. This explains the choice of the weighted Lebesgue space L 2 m (R N ) in Theorem 1. In fact, since the nonlinearity in (4) involves the gradient of the solution u, we shall rather use the corresponding Sobolev space H 1 m (R N ) (defined in ( 16) below) in the proof. The rest of this paper is organized as follows. In the next section, we recall existence and uniqueness results for [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF], [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] and establish the convergence to zero of the solution v(τ ) in H 1 m (R N ) as τ → ∞. In Section 3, we study a suitable truncated version of ( 10), ( 11) to which we can apply an abstract result of [START_REF] Chen | Invariant foliations for C 1 semigroups in Banach spaces[END_REF] to construct the invariant manifold. The proof of Theorem 1 is then performed in the final section.

Global existence and convergence to zero

In this section we briefly discuss the Cauchy problem for the rescaled equation ( 10) and we show that the solutions converge to zero as τ → ∞. We first consider initial data in the Lebesgue space 10), [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] have a unique non-negative (mild) solution

L 1 (R N ) ∩ L 2 (R N ). Proposition 3 Let u 0 be a non-negative function in L 1 (R N ) ∩ L 2 (R N ). Then (
v ∈ C([0, ∞); L 1 (R N ) ∩ L 2 (R N )) ∩ L ∞ loc ((0, ∞); W 1,∞ (R N )) ,
which moreover satisfies

lim τ →∞ { v(τ ) L 1 + v(τ ) L ∞ + ∇v(τ ) L ∞ } = 0 . (13)
Proof: For such initial data u 0 , the results of [START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF][START_REF] Ben-Artzi | The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces[END_REF][START_REF] Gilding | The Cauchy problem for u t = ∆u + |∇u| q[END_REF] imply that the original system (4), (5) has a unique (mild) solution

u ∈ C([0, ∞); L 1 (R N ) ∩ L 2 (R N )) ∩ C((0, ∞); W 1,∞ (R N )) .
For all t > 0, the function u(t, x) is C 1 with respect to t, C 2 with respect to x, and ( 4) is satisfied in the classical sense. In addition, the following bounds hold for all t > 0:

u(t) L 1 + t N/2 u(t) L ∞ + t (N +1)/2 ∇u(t) L ∞ ≤ C u(t/2) L 1 ≤ C u 0 L 1 . (14)
Since u(t) L 1 → 0 as t → ∞ by [START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF][START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF], we deduce from ( 14) that

lim t→∞ u(t) L 1 + t N/2 u(t) L ∞ + t (N +1)/2 ∇u(t) L ∞ = 0 . ( 15
)
The conclusions of Proposition 3 are straightforward consequences of these results, since [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF] is obtained from (4) via the simple transformation [START_REF] Benachour | Solutions fondamentales de u t -1 2 u xx = ±|u x[END_REF]. In particular, v(τ

) L 1 = u(t) L 1 , v(τ ) L ∞ = (1+t) N/2 u(t) L ∞ , ∇v(τ ) L ∞ = (1+t) (N +1)/2 ∇u(t) L ∞ , hence (13)
follows immediately from [START_REF] Galaktionov | Spectra of critical exponents in nonlinear heat equations with absorption[END_REF]. Finally, as the transformation (9) involves a time-dependent dilation which is not continuous in

L ∞ (R N ), the fact that u ∈ C((0, ∞); W 1,∞ (R N )) only implies that v ∈ L ∞ loc ((0, ∞); W 1,∞ (R N )).
We next study the properties of solutions to [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF], [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] in the weighted Lebesgue space L 2 m (R N ) defined in [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I. Existence[END_REF], and in the corresponding Sobolev space

H 1 m (R N ) = v ∈ H 1 (R N ) v m := (|v| 2 m + |∇v| 2 m ) 1/2 < ∞ , (16)
where

|v| m = R N 1 + |ξ| 2m |v(ξ)| 2 dξ 1/2 . Proposition 4 Let u 0 be a non-negative function in L 2 m (R N ) for some m > N/2.
Then the solution v to [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF], [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] given by Proposition 3 satisfies

v ∈ C([0, ∞); L 2 m (R N )) ∩ C((0, ∞); H 1 m (R N )) ,
and lim

τ →∞ v(τ ) m = 0 . ( 17
)
Proof: The fact that v ∈ C([0, T ); L 2 m (R N )) ∩ C((0, T ); H 1 m (R N ))
for some T > 0 can be established by a classical fixed point argument, which will be implemented in Section 3 for a truncated version of [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF]. Here we just obtain differential inequalities for the norms |v(τ )| m and |∇v(τ )| m which imply (in view of the local existence theory) that T > 0 can be chosen arbitrarily large and that (17) holds.

We first multiply (10) by v and integrate over R N . Using the non-negativity of v and integrating by parts, we find 1 2

d dτ R N v 2 dξ = R N v ∂ τ v dξ ≤ R N vLv dξ = - R N |∇v| 2 dξ + N 4 R N v 2 dξ . (18)
Similarly, multiplying [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF] by |ξ| 2m v, we obtain 1 2

d dτ R N |ξ| 2m v 2 dξ ≤ R N |ξ| 2m v ∆v + 1 2 ξ • ∇v + N 2 v dξ (19) = - R N |ξ| 2m |∇v| 2 dξ -2m R N |ξ| 2m-2 v ξ • ∇v dξ - 2m-N 4 R N |ξ| 2m v 2 dξ .
The only difficulty is to bound the integral involving ξ • ∇v. If m ≥ 1, we have by Young's inequality

2m|ξ| 2m-1 v|∇v| ≤ 1 2 |ξ| 2m |∇v| 2 + 2m 2 |ξ| 2m-2 v 2 , and |ξ| 2m-2 ≤ ε|ξ| 2m + C(ε) ,
where ε > 0 is arbitrary. If 1/2 < m < 1 (which is possible only if N = 1) we find similarly

2m|ξ| 2m-1 v|∇v| ≤ 1 2 |∇v| 2 + 2m 2 |ξ| 4m-2 v 2 , and |ξ| 4m-2 ≤ ε|ξ| 2m + C(ε) .
In both cases, summing up ( 18) and ( 19), we obtain the inequality

d dτ |v(τ )| 2 m + |∇v(τ )| 2 m + 2m-N-8m 2 ε 2 |v(τ )| 2 m ≤ (m + 4m 2 C(ε)) v(τ ) 2 L 2 . ( 20
)
We now choose ε > 0 sufficiently small so that 2m -N -8m 2 ε > 0. Since v(τ ) L 2 → 0 by [START_REF] Biler | Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation[END_REF], the differential inequality [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF] implies that |v(τ )| m → 0 as τ → ∞.

We next control the evolution of ∇v. Multiplying (10) by -∆v and integrating by parts, we obtain

1 2 d dτ R N |∇v| 2 dξ = - R N |∆v| 2 dξ + N+2 4 R N |∇v| 2 dξ + R N ∆v |∇v| q⋆ dξ . (21)
Similarly,

1 2 d dτ R N |ξ| 2m |∇v| 2 dξ = - R N |ξ| 2m |∆v| 2 dξ + N+2 -2m 4 R N |ξ| 2m |∇v| 2 dξ -2m R N |ξ| 2m-2 ∆v ξ • ∇v dξ + R N |ξ| 2m ∆v + 2m|ξ| 2m-2 ξ • ∇v |∇v| q⋆ dξ . ( 22
)
Using the crude estimate

|ξ| 2m-1 ≤ 1 + |ξ| 2m , we find 2m R N |ξ| 2m-1 |∆v| |∇v| dξ ≤ 1 4 |∆v| 2 m + C|∇v| 2 m .
Moreover, as ∇v L ∞ is uniformly bounded for all τ ≥ 1 by (13), we have for such times

R N (1+|ξ| 2m ) |∆v| |∇v| q⋆ dξ ≤ 1 4 |∆v| 2 m + C|∇v| 2 m , 2m R N |ξ| 2m-1 |∇v| |∇v| q⋆ dξ ≤ C|∇v| 2 m .
Thus adding up ( 21) and ( 22), we obtain

d dτ |∇v(τ )| 2 m + |∆v(τ )| 2 m + |∇v(τ )| 2 m ≤ K |∇v(τ )| 2 m , τ ≥ 1 , (23)
for some K > 0. Now, if we combine ( 20) and ( 23), we see that h(τ

) := K|v(τ )| 2 m + |∇v(τ )| 2 m
satisfies a differential inequality of the form

h ′ (τ ) + ε 0 h(τ ) ≤ C v(τ ) 2 L 2 , τ ≥ 1 ,
for some positive constants ε 0 and C. Thus h(τ ) → 0 as τ → ∞, and ( 17) follows. This concludes the proof of Proposition 4.

Construction of the center manifold

We now proceed along the lines of [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF]Section 3] to describe the large time behavior of the non-negative solutions to [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF], [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF] in the space L 2 m (R N ). By Proposition 4 these solutions converge to zero in H 1 m (R N ) as τ → ∞, hence the large time asymptotics remain unchanged if we truncate the nonlinearity in [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF] outside a neighborhood of the origin. This modification will allow us to apply the center manifold theorem as stated in [START_REF] Chen | Invariant foliations for C 1 semigroups in Banach spaces[END_REF].

The goal of the present section is to verify that our problem fits into the general framework considered in [START_REF] Chen | Invariant foliations for C 1 semigroups in Banach spaces[END_REF]. First, we introduce a truncated version of system [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF] and we show that it generates a C 1 -smooth and globally Lipschitz continuous semiflow (ϕ τ ) τ ≥0 in H 1 m (R N ) (Proposition 5). Using [14, Theorem 1.1], we then prove the existence of a one-dimensional center manifold W c ⊂ H 1 m (R N ), which is tangent at the origin to the kernel of the linear operator L, and which attracts all trajectories of the semiflow (ϕ τ ) τ ≥0 as τ → ∞ (Theorem 9). Thanks to this construction, proving Theorem 1 is reduced to determining the large time behavior of the solutions on the center manifold, a relatively simple task that is postponed to Section 4. The reader who is mainly interested in computing the large time asymptotics may just read the beginning of Sections 3.1 and 3.2, including the statements of Proposition 5 and Theorem 9, and then proceed directly to Section 4.

The semiflow of a truncated system

Throughout this section, we fix a function χ

∈ C ∞ ([0, ∞)) such that 0 ≤ χ ≤ 1, χ(r) = 0 if r ≥ 4 and χ(r) = 1 if r ≤ 1.
For ̺ > 0 and r ≥ 0, we denote χ ̺ (r) = χ(r/̺ 2 ). Given ̺ ∈ (0, 1) and m > N/2, we consider the initial-value problem

∂ τ v = Lv -F ̺ (v) , (τ, ξ) ∈ (0, ∞) × R N , (24) v(0) = v 0 ∈ H 1 m (R N ) , ξ ∈ R N , ( 25
)
where L is the linear operator [START_REF] Ben-Artzi | The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces[END_REF] and F ̺ is the truncated nonlinearity

F ̺ (v) = χ ̺ v 2 m |∇v| q⋆ , v ∈ H 1 m (R N ) . (26)
We first establish the well-posedness of (24), (25) and show that this system generates a

C 1 -smooth semiflow in H 1 m (R N ). Proposition 5 Fix ̺ ∈ (0, 1) and m > N/2. For each v 0 ∈ H 1 m (R N ), the initial-value problem (24), (25) has a unique global solution v ∈ C([0, ∞); H 1 m (R N )). Moreover, the map ϕ τ : H 1 m (R N ) → H 1 m (R N ) defined for τ ≥ 0 by ϕ τ (v 0 ) = v(τ ) is globally Lipschitz continuous, uniformly in τ on compact intervals. Finally ϕ τ is C 1 -smooth for all τ ≥ 0, so that the family (ϕ τ ) τ ≥0 defines a C 1 semiflow in H 1 m (R N ).
Before proving Proposition 5, we recall that the linear operator L defined by [START_REF] Ben-Artzi | The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces[END_REF] is the generator of a strongly continuous semigroup

e τ L τ ≥0 in L 2 m (R N ), see e.g. [17, Appendix A].
If m > N/2, this semigroup is uniformly bounded, i.e. there exists C 1 > 0 such that, for all

w ∈ L 2 m (R N ), e τ L w m ≤ C 1 |w| m , τ ≥ 0 . (27)
More generally, let b(ξ) = (1 + |ξ| 2 ) 1/2 and assume that b m w ∈ L p (R N ) for some p ∈ [START_REF] Amour | Global existence and decay for viscous Hamilton-Jacobi equations[END_REF][START_REF] Andreucci | The Cauchy problem for degenerate parabolic equations with source and damping[END_REF]. Then e τ L w ∈ L 2 m (R N ) for all τ > 0, and there exists C 2 > 0 such that

e τ L w m ≤ C 2 a(τ ) N 2 ( 1 p -1 2 ) b m w L p , τ > 0 , (28)
where a(τ

) = 1 -e -τ ,
(R N ). In addition, if b m w ∈ L p (R N ) for some p ∈ [1, 2],
we have the analog of (28):

∇e τ L w m ≤ C 3 a(τ ) N 2 ( 1 p -1 2 )+ 1 2 b m w L p , τ > 0 . (29)
In the rest of this section, we fix p ∈ (1, 2) such that

2(N + 1) N + 3 < p < 2(N + 1) N + 2 . ( 30
)
Given T > 0 and w ∈ C([0, T ]; H 1 m (R N )), we define

(N ̺ w)(τ ) = τ 0 e (τ -s)L F ̺ (w(s)) ds , τ ∈ [0, T ] . (31)
Then N ̺ w belongs to C([0, T ]; H 1 m (R N )) and enjoys the following property:

Lemma 6 There exists a constant C 4 > 0 such that, for all T > 0, all ̺ ∈ (0, 1), and all w 1 , w 2 ∈ C([0, T ]; H 1 m (R N )), the following inequality holds:

(N ̺ w 1 -N ̺ w 2 ) (τ ) m ≤ C 4 Z p (τ ) ̺ q⋆-1 sup s∈[0,τ ] (w 1 -w 2 )(s) m , τ ∈ [0, T ] ,
where

Z p (τ ) = τ 0 1 a(s) N 2 ( 1 p -1 2 )
1 + 1 a(s) 1/2 ds .

Proof: We first observe that our choice of p in (30) guarantees that N 2 ( 1 p -1 2 ) < 1 2 , so that Z p (τ ) is well-defined and finite for every τ ≥ 0. The following inequality will also be useful:

if f, g ∈ L 2 m (R N ), then b m |f | q⋆-1 g ∈ L p (R N ) and b m |f | q⋆-1 g L p ≤ C 5 |f | q⋆-1 m |g| m . (32)
Indeed, by Hölder's inequality,

b m |f | q⋆-1 g L p ≤ b m g L 2 f q⋆-1 L r
, where r = (q ⋆ -1) 2p 2p .

But b m g L 2 ≤ C|g| m , and 1 < r < 2 by (30), hence

f L r ≤ C( f L 1 + f L 2 ) ≤ C|f | m because L 2 m (R N ) ֒→ L 1 (R N ) ∩ L 2 (R N ) for m > N/2.
This proves (32). Now, let T > 0, ̺ ∈ (0, 1), and w 1 , w 2 ∈ C([0, T ]; H 1 m (R N )). For all τ ∈ [0, T ] we have by (31), ( 28)

|(N ̺ w 1 -N ̺ w 2 ) (τ )| m ≤ τ 0 e (τ -s)L (F ̺ (w 1 (s)) -F ̺ (w 2 (s))) m ds ≤ τ 0 C 2 a(τ -s) N 2 ( 1 p -1 2 ) b m (F ̺ (w 1 (s)) -F ̺ (w 2 (s))) L p ds . ( 33
)
For a fixed s ∈ [0, τ ], we can assume for instance that w 1 (s

) m ≥ w 2 (s) m . Then b m (F ̺ (w 1 (s)) -F ̺ (w 2 (s))) L p ≤ χ ̺ w 1 (s) 2 m -χ ̺ w 2 (s) 2 m b m |∇w 2 (s)| q⋆ L p + χ ̺ w 1 (s) 2 m b m (|∇w 1 (s)| q⋆ -|∇w 2 (s)| q⋆ ) L p .
Obviously, the right-hand side vanishes if w 2 (s) m ≥ 2̺, hence we can suppose that w 2 (s) m ≤ 2̺. To bound the first term, we apply (32

) with f = g = |∇w 2 | and obtain b m |∇w 2 (s)| q⋆ L p ≤ C 5 |∇w 2 (s)| q⋆ m ≤ C̺ q⋆ . Moreover, if w 1 (s) m ≤ 4̺, we have χ ̺ w 1 (s) 2 m -χ ̺ w 2 (s) 2 m ≤ C ̺ 2 w 1 (s) 2 m -w 2 (s) 2 m ≤ C ̺ (w 1 -w 2 )(s) m ,
and the same estimate holds if w 1 (s) m ≥ 4̺ because (w 1w 2 )(s) m ≥ 2̺ in that case. Thus

χ ̺ w 1 (s) 2 m -χ ̺ w 2 (s) 2 m b m |∇w 2 (s)| q⋆ L p ≤ C ̺ q⋆-1 (w 1 -w 2 )(s) m .
On the other hand, using (32) and the inequality | |y| q⋆ -|z| q⋆ | ≤ q ⋆ (|y| q⋆-1 + |z| q⋆-1 )|y -z|, we obtain

χ ̺ w 1 (s) 2 m b m (|∇w 1 (s)| q⋆ -|∇w 2 (s)| q⋆ ) L p ≤ C χ ̺ w 1 (s) 2 m |∇w 1 (s)| q⋆-1 m + |∇w 2 (s)| q⋆-1 m |∇(w 1 -w 2 )(s)| m ≤ C χ ̺ w 1 (s) 2 m w 1 (s) q⋆-1 m (w 1 -w 2 )(s) m ≤ C ̺ q⋆-1 (w 1 -w 2 )(s) m . Therefore b m (F ̺ (w 1 (s))-F ̺ (w 2 (s))) L p ≤ C ̺ q⋆-1 (w 1 -w 2 )(s) m ,
and inserting this bound into (33) we conclude that

|(N ̺ w 1 -N ̺ w 2 ) (τ )| m ≤ C ̺ q⋆-1 τ 0 1 a(τ -s) N 2 ( 1 p -1 2 ) (w 1 -w 2 )(s) m ds . ( 34
)
Finally, using (29) and proceeding in the same way, we also obtain

|∇ (N ̺ w 1 -N ̺ w 2 ) (τ )| m ≤ τ 0 C 3 a(τ -s) N 2 ( 1 p -1 2 )+ 1 2 b m (F ̺ (w 1 (s)) -F ̺ (w 2 (s))) L p ds ≤ C ̺ q⋆-1 τ 0 1 a(τ -s) N 2 ( 1 p -1 2 )+ 1 2 (w 1 -w 2 )(s) m ds . ( 35
)
Lemma 6 is now a immediate consequence of (34) and (35).

Proof of Proposition 5: Given v 0 ∈ H 1 m (R N ), we choose K > 2C 1 v 0 m and T > 0 sufficiently small so that

C 1 v 0 m e T /2 ≤ K 2 , and
C 4 ̺ q⋆-1 Z p (T ) ≤ 1 2 , ( 36
)
where C 1 is as in ( 27) and C 4 , Z p are defined in Lemma 6. We introduce the set

X K,T = w ∈ C([0, T ]; H 1 m (R N )) sup τ ∈[0,T ] w(τ ) m ≤ K ,
which is a complete metric space for the distance d T defined by

d T (w 1 , w 2 ) = sup τ ∈[0,T ] (w 1 -w 2 )(τ ) m , (w 1 , w 2 ) ∈ X K,T × X K,T .
Using ( 27) and Lemma 6 it is straightforward to verify that, if w ∈ X K,T , then the function

T ̺ w : [0, T ] → H 1 m (R N ) defined by (T ̺ w)(τ ) = e τ L v 0 -(N ̺ w)(τ ) , τ ∈ [0, T ] ,
belongs to X K,T , and that the map w → T ̺ w is a strict contraction in X K,T . By the Banach fixed point theorem, T ̺ has thus a unique fixed point v in X K,T . This proves that the Cauchy problem (24), ( 25) is locally well-posed in H 1 m (R N ). Let T * (v 0 ) ∈ (0, ∞] be the maximal existence time for the solution of ( 24), ( 25) in H 1 m (R N ). For all τ < T * (v 0 ), it follows from ( 27), (34), and (35) (with

w 1 = v and w 2 = 0) that v(τ ) m ≤ C 1 e τ /2 v 0 m + C 4 ̺ q⋆-1 τ 0 v(s) m a(τ -s) N 2 ( 1 p -1 2 )+ 1 2 ds .
Using a version of Gronwall's lemma (see e.g. [20, Lemma 7.1.1]), we deduce that v(τ

) m cannot blow up in finite time, hence T * (v 0 ) = ∞. Thus (24) has a unique global solution v ∈ C([0, ∞); H 1 m (R N )) for all v 0 ∈ H 1 m (R N
), and we may define a semiflow (ϕ τ ) τ ≥0 by the relation ϕ τ (v 0 ) = v(τ ) for τ ≥ 0.

By construction, the map v 0 → ϕ τ (v 0 ) is globally Lipschitz continuous, uniformly in time on compact intervals: for each T > 0, there exists L(T ) > 0 such that

ϕ τ (v 0 ) -ϕ τ (v 0 ) m ≤ L(T ) v 0 -v0 m , (37)
for all τ ∈ [0, T ] and all (v 0 , v0 )

∈ H 1 m (R N ) × H 1 m (R N )
. Indeed, by the semigroup property, it is sufficient to prove (37) for a T > 0 satisfying (36), in which case (37) follows immediately from the fixed point argument above, with L(T ) = 2C 1 e T /2 . This proof also shows that L(T ) can be chosen independent of ̺ if ̺ ∈ (0, 1). Finally, the fact that the map ϕ τ is C 1 for each τ ≥ 0 is obtained by classical arguments which we omit here. We only mention that, given v 0 ∈ H 1 m (R N ), τ ≥ 0, and h ∈ H 1 m (R N ), the differential Dϕ τ (v 0 )h of ϕ τ at v 0 applied to h is equal to V (τ ), where V denotes the solution of the linear non-autonomous equation

∂ τ V = LV -q ⋆ χ ̺ v 2 m |∇v| q⋆-2 ∇v • ∇V -2χ ′ ̺ v 2 m |∇v| q⋆ ≪ v, V ≫ m , V (0) = h . Here v(τ ) = ϕ τ (v 0 ) and ≪ •, • ≫ m denotes the scalar product in H 1 m (R N ).
In particular, since ϕ τ (0) = 0 for all τ ≥ 0, this formula shows that Dϕ τ (0) = e τ L for each τ ≥ 0.

Remark 7 It can actually be shown that the differential Dϕ

τ : H 1 m (R N ) → L (H 1 m (R N )
) is Hölder continuous with exponent q ⋆ -1 for any τ ≥ 0.

For later use, we also point out the following properties of the time-one map ϕ 1 :

Corollary 8 The map R = ϕ 1 -e L belongs to C 1 (H 1 m (R N ); H 1 m (R N ))
and satisfies R(0) = 0, DR(0) = 0. Moreover R is globally Lipschitz continuous and there exists C 6 > 0 (independent of ̺) such that its Lipschitz constant satisfies Lip(R) ≤ C 6 ̺ q⋆-1 .

Proof: We know from Proposition 5 that R is indeed a C 1 -map from H 1 m (R N ) into itself, and it was observed at the end of the proof that ϕ 1 (0) = 0 and Dϕ 1 (0) = e L , hence R(0) = 0 and DR(0) = 0. Now, given v 0 , v0 in H 1 m (R N ) we define v(τ ) = ϕ τ (v 0 ) and v(τ ) = ϕ τ (v 0 ) for τ ≥ 0. Using Lemma 6 and estimate (37) we find

R(v 0 ) -R (v 0 ) m = (N ̺ v)(1) -(N ̺ v)(1) m ≤ C 4 ̺ q⋆-1 sup s∈[0,1] (v -v) (s) m ≤ C 4 L(1) ̺ q⋆-1 v 0 -v0 m ,
which is the desired bound.

Existence of the center manifold

Having associated a C 1 -semiflow to the truncated system (24), we now turn to the construction of a center manifold for this semiflow at the origin. If m > N/2, we can decompose

H 1 m (R N ) = E c ⊕ E s
, where E c = {αG | α ∈ R} is the kernel of the operator L and

E s = w ∈ H 1 m (R N ) R N w(ξ) dξ = 0 . ( 38
)
We recall that G is the Gaussian function defined in [START_REF] Benachour | Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations[END_REF]. Let P 0 be the continuous projection onto E c along E s , namely

P 0 w = R N w(ξ) dξ G , w ∈ H 1 m (R N ) ,
and let Q 0 = 1-P 0 . It is easily verified that P 0 and Q 0 commute with L, so that the subspaces E c and E s are invariant under the action of L. Moreover, we know from [17, Appendix A] that the spectrum of the restriction of L to the invariant subspace E s is strictly contained in the left-half plane in C, because the associated semigroup e τ L decreases exponentially in E s . More precisely, if µ 0 ∈ (0, 1/2) satisfies 2µ 0 < m -(N/2), there exists C 7 > 0 such that

e τ L Q 0 w m + a(τ ) 1/2 ∇e τ L Q 0 w m ≤ C 7 e -µ 0 τ |w| m , (39)
for all w ∈ L 2 m (R N ) and all τ > 0, see [START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF]Proposition A.2]. We are now in a position to apply the invariant manifold theorem as stated in [START_REF] Chen | Invariant foliations for C 1 semigroups in Banach spaces[END_REF]Theorem 1.1]. The main result of this section reads:

Theorem 9 Fix µ ∈ (0, 1/2) such that 2µ < m -(N/2). If ̺ > 0 is sufficiently small, there exists a globally Lipschitz continuous map f ∈ C 1 (E c ; E s ) with f (0) = 0 and Df (0) = 0 such that the submanifold W c = {αG + f (αG) | α ∈ R} ⊂ H 1 m (R N )
Proof: Theorem 9 readily follows from [14, Theorem 1.1] once we have checked that the assumptions (H.1)-(H.4) of [START_REF] Chen | Invariant foliations for C 1 semigroups in Banach spaces[END_REF] are fulfilled in our case. By Proposition 5, (ϕ τ ) τ ≥0 is a C 1 semiflow in H 1 m (R N ) and ϕ τ is globally Lipschitz continuous, uniformly for τ ∈ [0, 1]. Therefore, [14, (H.1)] is verified. Next, assumption [14, (H.2)] is nothing but the decomposition ϕ 1 = e L + R described in Corollary 8. To check [14, (H.3)], we remark that P 0 e L P 0 = 1, hence P 0 e L P 0 -k P 0

L (Ec) = P 0 L (Ec) , for all k ∈ N .
On the other hand, if we choose µ 0 ∈ (µ, 1/2) such that 2µ 0 < m-(N/2), it follows from (39) that e kL Q 0 w m ≤ C e -kµ 0 w m for all k ∈ N. Since Q 0 and e L commute, this inequality is equivalent to

Q 0 e L Q 0 k Q 0 L (Es)
≤ C e -kµ 0 , for all k ∈ N .

As e -µ 0 < 1, we have thus checked that [14, (H.3)] is fulfilled. Finally [14, (H.4)] is automatically satisfied if the Lipschitz constant of R is sufficiently small. By Corollary 8, this is easily achieved by choosing ̺ appropriately small. Therefore, by [14, Theorem 1.1], there exist µ 1 ∈ (0, µ 0 ) and a globally Lipschitz continuous map f ∈ C 1 (E c ; E s ) such that the submanifold

W c = {αG + f (αG) | α ∈ R} ⊂ H 1 m (R N )
enjoys the following properties:

Invariance: ϕ τ (W c ) = W c for all τ ≥ 0, and the restriction to W c of the semiflow (ϕ τ ) τ ≥0 can be extended to a Lipschitz continuous flow on W c .

Invariant foliation: There is a continuous map h :

H 1 m (R N ) × E s → E c such that, for each v ∈ W c , one has h(v, Q 0 v) = P 0 v and the manifold M v = {h(v, w) + w | w ∈ E s } ⊂ H 1 m (R N ) passing through v satisfies ϕ τ (M v ) ⊂ M ϕτ (v)
for τ ≥ 0 and is characterized by

M v = w ∈ H 1 m (R N ) lim sup τ →∞ 1 τ ln ( ϕ τ (w) -ϕ τ (v) m ) ≤ -µ 1 . Completeness: For every v ∈ W c , M v ∩ W c = {v}. In particular, M v ∩ M w = ∅ if (v, w) ∈ W c × W c and v = w, and H 1 m (R N ) = ∪ v∈Wc M v . Moreover, we can assume that µ 1 ∈ (µ, µ 0 ) if ̺ > 0 is sufficiently small.
We can now conclude the proof of Theorem 9. Assertion (a) is nothing but the invariance property of W c . To prove (b), let v 0 ∈ H 1 m (R N ). By the completeness property of W c , there is a unique w 0 ∈ W c such that v 0 ∈ M w 0 . Since µ < µ 1 , we deduce from the invariant foliation property of W c that there is τ 0 > 0 such that ϕ τ (v 0 )ϕ τ (w 0 ) m ≤ e -µτ , for all τ ≥ τ 0 .

Using in addition (37), we obtain (40).

Large time behavior

This final section is entirely devoted to the proof of Theorem 1. Assume that u 0 is a non-negative function in L 2 m (R N ), m > N/2, such that u 0 L 1 > 0. Let u(t, x) be the corresponding solution of (4), (5) and v(τ, ξ) the corresponding solution of ( 10), [START_REF] Ben-Artzi | Decay of mass for a semilinear parabolic equation[END_REF]. By the strong maximum principle [19, Corollary 4.2], we know that u(t, x) > 0 for all t > 0 and all x ∈ R N . Choose µ ∈ (0, 1/2) such that 2µ < m -(N/2) and ̺ ∈ (0, 1) sufficiently small so that Theorem 9 applies.

By Proposition 4, the solution v of (10) converges to zero in H 1 m (R N ) as τ → ∞, hence there exists τ 0 ≥ 0 such that v(τ ) m ≤ ̺ for all τ ≥ τ 0 . Setting v 0 = v(τ 0 ) and v(τ ) = v(τ +τ 0 ), we obtain a solution v(τ ) of [START_REF] Benachour | Asymptotic estimates of solutions of u t -1 2 ∆u = -|∇u| in R + × R d , d ≥ 2[END_REF]

with initial condition v 0 ∈ H 1 m (R N ) which satisfies v(τ ) m ≤ ̺ for all τ ≥ 0 . (41)
Using the notations of Section 3, it follows that v(τ ) = ϕ τ (v 0 ) for τ ≥ 0, because (41) implies that χ ̺ ( v(τ ) 2 m ) = 1. Thus, in view of Theorem 9, there exist w 0 ∈ W c and

C 9 > 0 such that v(τ ) -ϕ τ (w 0 ) m ≤ C 9 e -µτ , τ ≥ 0 . (42)
To simplify the notations, we set w(τ ) = ϕ τ (w 0 ) and

M(τ ) = R N w(τ, ξ) dξ , τ ≥ 0 .
We claim that M(τ ) > 0 for all τ ≥ 0 , and lim

τ →∞ M(τ ) = 0 . (43) Indeed, since H 1 m (R N ) ֒→ L 1 (R N ), it follows from (42) that R N v(τ, ξ) dξ -M(τ ) ≤ C v(τ ) -w(τ ) m ≤ C 10 e -µτ , (44)
for all τ ≥ 0. Assume by contradiction that there exists τ 1 ≥ 0 such that M(τ 1 ) ≤ 0. Since w is a solution of (24), (25) and F ̺ ≥ 0, it is clear that τ → M(τ ) is non-increasing, hence M(τ ) ≤ M(τ 1 ) ≤ 0 for τ ≥ τ 1 . Using (44) and recalling that v is non-negative, we thus find v(τ ) L 1 = R N v(τ, ξ) dξ ≤ M(τ ) + C 10 e -µτ ≤ C 10 e -µτ , for τ ≥ τ 1 . As a consequence, u(t) L 1 = v(ln(1+t)τ 0 ) L 1 ≤ C 10 e µτ 0 (1 + t) -µ for t ≥ e τ 1 +τ 0 -1 .

By [START_REF] Chen | Invariant foliations for C 1 semigroups in Banach spaces[END_REF], we also have u(t) L ∞ ≤ C t -µ-(N/2) for t sufficiently large, a property which implies that u ≡ 0 by [7, Proposition 3] and [START_REF] Gilding | The Cauchy problem for u t = ∆u + |∇u| q[END_REF]Corollary 4.2]. This contradicts the fact that u(t, x) > 0 for t > 0, hence we have proved the first assertion in (43). As for the second claim, it is a straightforward consequence of ( 13) and (44). Now, since v(τ ) m → 0 as τ → ∞, it follows from (42) that there exists τ 2 ≥ 0 such that w(τ ) m ≤ ̺ for all τ ≥ τ 2 . On the other hand, as w(τ ) ∈ W c for each τ ≥ 0, we have w(τ, ξ) = M(τ ) G(ξ) + f (M(τ ) G(ξ)) for (τ, ξ) ∈ [0, ∞) × R N , where f is as in Theorem 9. In view of (24) and (26) we deduce that, for τ ≥ τ

where M ⋆ is as in Theorem 1. As w(τ, ξ) = M(τ ) G(ξ) + f (M(τ ) G(ξ)), we deduce from (43), (47) and the properties of f that τ N +1 w(τ ) -M(τ ) G m → 0 as τ → ∞. Combining this result with (42), (47), we arrive at lim

τ →∞ τ N +1 v(τ ) - M ⋆ τ N +1 G L 1 = 0 . ( 48
)
Of course, the same result holds for v(τ ) = v(τ -τ 0 ). If we now return to the original function u(t, x) via the transformation (9), we obtain exactly [START_REF] Benachour | Extinction and decay estimates for viscous Hamilton-Jacobi equations in R N[END_REF] for p = 1. The case p ∈ (1, ∞] then follows from ( 14) by a classical interpolation argument.

  enjoys the following properties:(a) ϕ τ (W c ) = W c for every τ ≥ 0, (b) for every v 0 ∈ H 1 m (R N) there exist a unique w 0 ∈ W c and a positive constant C 8 (v 0 ) such that ϕ τ (v 0 )ϕ τ (w 0 ) m ≤ C 8 (v 0 ) e -µτ for τ ≥ 0 .

+ 2 , 1 L 2 b

 212 ξ)| q⋆ dξ = -∇G q⋆ L q⋆ M(τ ) q⋆ω(τ ) , ξ)| q⋆ -M(τ ) q⋆ |∇G(ξ)| q⋆ dξ .To bound ω(τ ), we remark that | |y + z| q⋆ -|y| q⋆ | ≤ q ⋆ (|y| + |z|) q⋆-1 |z| for all y, z ∈ R.it follows from Hölder's inequality that, for all g, h ∈ L 2 m (R N ),|h| q⋆-1 g L 1 ≤ b m h q⋆-m g L 2 b -mq⋆ L 2(N+1)/N ≤ C |h| q⋆-1 m |g| m , where b(ξ) = (1 + |ξ| 2 ) 1/2 . Thus |ω(τ )| ≤ q ⋆ R N M(τ ) |∇G(ξ)| + |∇f (M(τ ) G(ξ))| q⋆-1 |∇f (M(τ ) G(ξ))| dξ ≤ C (M(τ ) G m + f (M(τ ) G) m ) q⋆-1 f (M(τ ) G) m ≤ C M(τ ) q⋆-1 f (M(τ ) G) m ,where in the last inequality we have used the fact that f (0) = 0 and f is globally Lipschitz continuous. Since f ∈ C 1 (E c ; E s ) with f (0) = 0 and Df (0) = 0, the above inequality and (43) imply that limτ →∞ ω(τ ) M(τ ) q⋆ = 0 .(46)Combining (43), (45), (46), and recalling that q ⋆ -1 = 1/(N+1), we conclude that lim τ →∞ τ M(τ ) q⋆-1 = 1 (q ⋆ -1) ∇G q⋆ L q⋆ , i.e. lim τ →∞ τ N +1 M(τ ) = M ⋆ ,

  see[START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF] Proposition A.5] and[START_REF] Th | Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF] Proposition A.2]. Similar estimates hold for the spatial derivatives of e τ L w. For instance, as ∇e τ L = e τ /2 e τ L ∇, it follows from (27) that |∇e τ L w| m ≤ C 1 e τ /2 |∇w| m for all w ∈ H 1 m