Baptiste Fix
email: baptiste.fix@onera.fr

Benjamin Vest

Julien Jaeck

Riad Hadar

Competition between sub-bandgap linear detection and degenerate two-photon absorption in gallium arsenide photodiodes

come

Competition between sub-bandgap linear detection and degenerate two-photon absorption in gallium arsenide photodiodes (Orale)

Introduction

Room temperature infrared quantum detection is still a challenge today. Indeed, the low bandgap semiconductors used to achieve direct infrared detection suffer from an important dark current related noise. This issue is usually solved by cooling down the material to cryogenic temperature thus reducing the thermal agitation. We propose to investigate a strategy to mitigate this problem by making use of two-photon absorption inside wide gap semiconductors from the visible sector such as GaAs and InP.

Infrared detection through two photon absorption processes implies the detection of low optical fluxes delivered by CW sources, which proves to be quite challenging. Indeed in the low flux regime, a linear dependence of the photocurrent to the subbandgap light intensity has been previously observed in silicon devices clearly showing that another photocurrent generation process can prevail over the two photon photocurrent (TPC). This process called photo-assisted Shockley-Read-Hall (PASRH) has been described in silicon [1].

We observe and investigate the competition between those two processes in a commercial GaAs PIN diode at 1.55µm at very low flux for two temperatures. We also study the transitions for various temporal regimes from ns to µs allowing to determine the two-photon absorption coefficient and the trap densities in the photodiode material.

Competition between two transition processes

The experimental set-up consists of a pulsed light source focused on a commercial GaAs PIN photodiode. We acquire the current vs incident power characteristics represented in Fig. 1a. Two distinct regimes can be clearly identified : the quadratic regime above 1 mW of average incident optical power and a linear behaviour below 100 µW. Moreover, as shown by the current vs incident power characteristic on Fig. 1b, the linear regime disappears when the diode is cooled down to cryogenic temperature.

The quadratic regime is characteristic of TPC. It is a third order non-linear process which can be described as the quasi simultaneous absorption of a pair of photons as shown on the inset of Fig. 1b. Indeed, TPC in semiconductors using sub-bandgap photons can be modeled by a two-step process: a photon of energy E < E g can promote an electron from the valence band to a virtual state (meaning a non stationary state) in the gap. During the lifetime of this state (given by the second Heisenberg principle ∆τ ≥ transition, hence creating an electron-hole pair and generating a photocurrent, namely a two-photon current. The very short lifetime of this intermediate state in visible gap material (in the fs range)is responsible for the intrinsically low efficiency of this process, as it requires the quasi simultaneous occurrence of two photons in this time interval.

The linear regime has been previously observed in silicon devices and has been described as a photo-assisted Shockley-Read-Hall (PASRH) effect. It is based on the ionization of traps from deep levels in the bandgap to the conduction band. Those deep levels are delocalized real states, meaning that they can be reached at any time by any photon, thus leading to an appreciable efficiency for the process. The inset of Fig. 2b schematically represent the transition.

We model the total photocurrent as the sum of both contributions in order to derive their respective parameters: the two-photon absorption coefficient β and the the quantum efficiency of the PASRH η.

< J phot >=< J 1ph > + < J 2ph >= K 1ph < P opt > +K 2ph < P opt > 2
(1)

β = 2hν e S L < J 2ph > /R c < P opt > /R c 2 = 2hν e S L K 2ph R c , η = hν e < J 1ph > < P opt > = hν e K 1ph (2)
We derived from our measurements the values of β = 65.1 ± 1.7 cm/GW and η = [6.1 ± 0.6].10 -8

Conclusion

To windup, we observe that a real competition appears in terms of photocurrent generation in our device, dramatically limiting the capability to detect subbandgap photon pairs at low optical powers. Nevertheless, several strategies are accessible to allow a clear predomination of the TPC signal: on one hand, PASRH level could be decreased, either by cooling the detector [1] or improving the active medium quality in term of volumic trap density. On the other hand, TPC could be increased through enhancing the non-linear process thanks to adequate devices, such as resonant nanostructures [2] or quantum wells.

h 2 (Fig 1 .

 21 photocourrentu(A)


Clara Barbanson
email: clara.barbanson@onera.fr

Andrés Almansa

Yann Ferrec

Pascal Monasse

Relief computation from images of a Fourier transform spectrometer for interferogram correction

A method to extract digital elevation models of the ground from the images of an airborne Fourier transform spectrometer is presented. Such information allows to construct accurate interferograms in non-flat areas, where they are otherwise flawed.

Introduction

Airborne imaging Fourier transform spectrometers possess the singularity that spectrum acquisition is not snapshot, but temporal. When such an instrument, onboard an aircraft, is aimed towards the nadir, artefacts may appear due to changes in the view angle during acquisition. This is true in the case of Sieleters, an infrared imaging Fourier transform spectrometer developed by ONERA, the French Aerospace Lab [START_REF] Coudrain | SIELETERS, an airborne infrared dual-band spectro-imaging system for measurement of scene spectral signatures[END_REF]. Its interferometer is static : a Michelson interferometer is placed in front of the imaging optics, and outputs an image of the overflown scene on which interference fringes are superimposed, as can be seen in figure 1. The fringes are still and oriented perpendicularly to the linear flight path. Dense sequences of images, where each is translated by 1 pixel, are obtained. Thus, any point in the scene is seen on several images with different optical paths values. Combining all the values taken by each point of the scene leads to construction of an interferogram. The spectrum is then derived by Fourier transform. However, when the scene is not flat (containing for instance buildings or natural slopes), if relief is not taken into account, the position of an elevated object across the sequence of images is miscalculated. As a consequence, the scene element is wrongly matched on the images and the interfergram thus computed contains erroneous values. An example is shown in figure 2.

As a result, an accurate estimation of ground elevation is needed in order to identify flawed interferograms in non-flat areas and correct them.

Methods

Fringe removal

Before proceeding with relief estimation, the fringes need to be removed from the images. Indeed, while they are paramount to spectrum reconstruction, the fact that they remain at the same position on all images interferes with relief retrieval algorithms, which exploit the changes between images taken from different positions. Moreover, the fringes induce variations in radiometry that are uncorrelated to relief.

It was therefore decided to remove the fringes, using the fact that they are mostly band-limited in the Fourier space. We call Ω the area containing the frequencies of the fringes under homogeneous illumination. In addition, we adopt an image model inspired in [START_REF] Osher | Image Decomposition and Restoration Using Total Variation Minimization and the H1[END_REF], where w, the mesured image (with fringes), is the sum of u, the image of the overflown scene, and v, an image containing only fringes. Our aim is to find u by minimising a function called Total Variation (TV) on u. Total Variation is an operator that exploits the gradient of the image : TV (u) = ∑ i∈N ∑ j∈M ||∇(u)||, where N and M are the height and width of the image. It allows smoothing an image without loosing discontinuities.

The problem can be expressed as following, where û and ŵ denote the Fourier transforms of u and w :

min u,v TV (u) subject to û| Ω c = ŵ| Ω c and w = u + v
This step of fringe removal gives satisfactory results, even if some artefacts may still be visible in areas of high radiance gradients.

Depth computation

The method used to compute depth estimations is that of the Epipolar-Planes Images (EPI) [START_REF] Criminisi | Extracting Layers and Analyzing their Specular Properties Using Epipolar-Plane-Image Analysis[END_REF]. It requires that the aircraft follow a linear path, and that the image acquisition be dense, which is a given in the case of imaging Fourier transform spectrometers, where scene elements can be traced on more than 400 images.

To build Epipolar-Plane Images, all images first need to be put in epipolar geometry, which consists in having all images aligned so that the same scene elements are observed in the same column on all images. This setup can be observed on figure 3.

Then, we construct new images, called EPIs, one for each column of the Sieleters image. These images contain only one column from each original image, put one after the other in chronological order. Thus, the vertical axis of the EPI describes a spatial unit, and the horizontal axis shows time (or more precisely image number). Figure 3 shows an EPI extracted from a sequence of pictures.

In an EPI, an element of the scene is translated into a line, corresponding to the path followed by the object with regard to the first image of the sequence. Moreover, the slope of those paths is proportional to the distance between the scene element and the camera, providing a depth value. Our EPIs show only little variations on slopes between different paths. This is due to the fact that the altitude of the aircraft is much greater that the elevation differences observed in the scene. A precise line detection algorithm needs to be used to determine the depth of elements in the scene. For this purpose we chose a robust algorithm based on gradient orientation of pixels [START_REF] Desolneux | Meaningful Alignments[END_REF].

Conclusion

The procedure presented here outputs a sparse elevation map, as seen in figure 4, computed from the images of an imaging Fourier transform spectrometer. Such information allows to identify flawed interfograms. It can also be densified and used as a digital elevation model for all scene points, to allow reliable matching of scene elements through all images. Thanks to this, accurate interferograms can be computed for non-flat areas. Further work will study precise interpolation of interferograms on irregular sampling grids.

Fig. 1 .

 1 Fig. 1. An image from Sieleters. On the left, a magnified detail shows contrasted interference fringes.

Fig. 2 .

 2 Fig. 2. On the left, two images of a sequence acquired by Sieleters are shown. Values from matching scene elements are used to construct the interferogram (on the right). The red crosses on the images correspond to matches made without taking the height and geometry of the building into account. Hence, part of the rooftop and part of the wall are used to construct a flawed interferogram. The green crosses are also mismatched, but are contained in an homogeneous area, thus providing a correct interferogram.

Fig. 3 .

 3 Fig. 3. On the left, images of a sequence of images from Sieleters are shown. They are in epipolar geometry, where columns contain the same set of scene elements where both images overlap. On the right, the EPI corresponding to the columns with index 510. Identical columns are shown in color.

Fig. 4 .

 4 Fig. 4. Sparse elevation map with the corresponding Sieleters image as background in gray values. Height is coded in color, from red for the highest points to blue for the lowest.


T Messaoudi
email: tommy.messaoudi@map.cnrs.fr

L De Luca

P Véron

Towards an Ontology for Annotating Degradation Phenomena

Keywords: Conservation, Ontology, Data management, digital modeling, semantic annotation, knowledge, cultural heritage

In the field of built heritage, a huge amount of data describes the state of monuments: documentary data (texts, sounds, images) as well as analytic data from sensors, provides historical, archeological and constructive information. These data, produced by experts coming from several fields, are the foundation for the creation of new information through scientific observations. The problem is that these data are not related nor spatialized. Nowadays we are able to generate extreme accurate 3D model with throughout images-based or laser scanner acquisition. However these 3D models do not carry information regarding their morphological complexities. The design of an ontology for the conservation domain seems to be the best solution in order to obtain understandable entities thanks to their own data related between them. This article present the first attempt of a development of a web information system based on spatialized images semantic annotations tool, related to a domain ontology describing knowledge regarding stone degradation phenomena. Our approach is to produce a domain ontology able to document and therefore provide a framework able to help the decisionmaking process of experts in the cultural heritage conservation domain. The information annotated with the use of this ontology can enrich not only the scientific observations, but also to help to create new knowledge. In this way, it is possible to link and gather quantitative and qualitative aspect into only one information system. We will propose several example and queries able to exploit the reasoning power of the above information system.

I. INTRODUCTION

The analysis of cultural data is one of the biggest challenges in the cultural heritage field. Masonry structures, which is our object of analysis, are documented by a lots of heterogeneous data, like documentary (text, sound, historical images, etc.) or analytic data from sensors. These data provides constructive, historical and archeological information about the studied building. These are also produced by a lot of various experts coming from different domain (architecture, mechanics, computer science, etc.). According to this context, we can conclude there are various digital information acquisition. Nowadays it is possible to produce accurate 3D model but it is not sufficient to completely analyze its morphological complexity. This is the reason why we want to create and define a novel information processing chain from data collection to tool validation. On the one hand conservation experts use conventional technic to describe building morphology by using simple images or even translucent paper to create cartography documentation. On the other hand, different experts realize a lot of acquisition like point cloud from laser scanning, scientific imagery (IR thermography, UV imagery), a structured modeling to make mechanical behavior analysis. These two things respectively represented by qualitative and quantitative aspects can be related around a 3D multilayered analysis support with different level of complexity. This 3D support based on imagebased modeling, allow us to realize annotations giving heterogeneous information about masonry building conservation from integrated and interpreted data by scientific observations. The goal is to integrate and structure different heterogeneous information into an understandable representation that take in consideration the multiple factors describing building conservation state. Then we introduce "informative continuum" notion thanks to the link between the whole of spatialized data to create information. Scientific observation is possible thank to domain ontology, formalized and integrated into an information system. This paper is divided into 4 part. A state of the art has been realized and I've selected two relevant example of them that will be presented in part 2. And then, our approach concerning the way to design our domain ontology dedicated to degradation phenomena annotation will be presented in part III. Finally, section IV will list on-going-works and bring some perspective for further development.

II. STATE OF THE ART

An ontology is used to describe, share and reuse knowledge and data between software and humans. Several ontologies emerged in several knowledge fields (medicine, mathematic, law etc.) in order to analyze and formalize terms. Our research is oriented toward heritage building monitoring, this is the reason why we have selected two relevant example to underlining our research orientation.

A. Domain ontology definition

"Ontology is defined as an explicit and formal specification of a shared conceptualization" [START_REF] Gruber | Toward Principles for the Design of Ontologies Used for Knowledge Sharing[END_REF]. The term is borrowed from Philosophy domain, where an Ontology is a systematic account of existence. This definition is probably difficult to understand but if we explain it term by term, we can easily understand its meaning. Conceptualization mean modeling an abstract phenomena in the world. "Explicit" refer to the concepts and constrains that need to be clearly defined. An ontology is "formal" because it is understandable and readable by a computer. And finally the most important for us, it can be "shared" that is to say it gather consensual knowledge agreed by a specific expert group. In other word, domain ontology is a shared knowledge formalization dedicated to a specific domain. Our domain is masonry building conservation domain.

B. In Museal domain

The first example is an information system dedicated to museum domain realized by the research group SeCo which stand for Semantic Computing [START_REF] Hyvönen | Publishing and Using Cultural Heritage Linked Data on the Semantic Web[END_REF]. Their ontology named Museoalan Ontologia structures a huge amount of heterogeneous data from several domain: like painting, building, cultural sites, archives, library, etc. Kulturisampo application, created by the same research group using this ontology, propose an interface for organizing and filtering a significant amount of data, giving to the users the possibility to overlap the several information stored in their knowledge base. For instance we can overlap a specific photograph showing an old map to a current map, scalable at the same time and in the same way. Each viewable element from database is linked with other various elements. The methodology is very interesting but this ontology don't manipulate 3D data type, but only basically 2D data.

C. MONDIS

MONDIS which stand for MONument Damage Information System dedicated to heritage building degradation also manages alteration phenomena. MONDIS is based on an automatic process, crossing cultural heritage alterations documentation, its diagnosis and interventions for potential restoration operations [START_REF] Cacciotti | Monument damage information system (MONDIS): An ontological approach to cultural heritage documentation[END_REF]. Several information types are connected like: "architectural component description" giving information about the cultural heritage object construction (physical and functional property), "event cultural heritage object" corresponds to changes of condition to the object studied. "Damage diagnosis and intervention" informs us about the relationships between alteration, active agent, and mechanisms and provides a way to intervene on one of them to stop the alteration. And then "risk and measure assessment" represents the prevention. This system is very interesting for us, however MONDIS doesn't use spatialized annotations, therefore it is mostly focused on the qualitative aspect. This is the reason why we would like to enhance it, using it as a starting point of our work.

III. MAIN APPROACH

The main aspects to analyze the building conservation state is in one hand acquisition that is to say a quantitative aspect. On the other hand the Human-based identification for the interpretation of these data to create information and knowledge. The main issue is the lack of relationship between these two aspects. Hence our hypothesis is: our Information system structured by an ontology allow us to relate quantitative and qualitative aspects to analyze cultural heritage conservation through Scientific Observation. The question is: how to design ontology? And how do ontology can manage that? Several method have been found to design ontology, and we have selected one of them. In addition a solution has been found in order to manage these different aspects.

A. Semantic annotation

Our work aims to define a domain ontology, allowing us to relate acquired data, with formalized shared knowledge in relation to our domain which is the heritage building conservation. Hence to combine qualitative and quantitative aspects. Our future web tool will mainly be based on referenced spatially photographs annotation [START_REF] Manuel | 2D/3D "Semantic Annotation Towards a Set of Spatially-Oriented Photographs[END_REF] produced by an automated image-based modeling. Thus architectural and state conservation analysis can be performed directly on images. The annotation process generates 3D information inherent to images. This means that a 2D/3D bijective link is introduced between pixels and their 3D coordinates. This relation allows us to transfer annotations between images. An image of the set is annotated and the 2Dto-3D relation extracts the corresponding 3D point cloud of the drawn area. The corresponding areas on other images of the set is then retrieved by using the 3D-to-2D relation "Figure 1". The latter relation conversely permits to retrieve positions in other images where 3D coordinates of the annotation are detected and highlight the corresponding areas [START_REF] Pérouse De Montclos | Architecture : méthode et vocabulaire[END_REF].

This method provides a simple and fast way to annotate images all at once instead of one by one. Annotations will be related with the data stored in database giving them a quantitative aspect. Indeed these regions will be used to extract dimensional values such as extension, position in 3D space, orientation, magnitude giving information about material loss, etc. And then to realize spatial distribution of same region type in order to extract similarities.

B. A domain ontology for annotating degradation phenomena on oriented images

Annotations related to data, will be linked with scientific observation and knowledge from a domain ontology. Thanks to the relationship between shared knowledge and scientific observation, annotation will be carrier of a qualitative aspect. Shared knowledge include the whole of concepts concerning stone degradation description and their links, retrieved from the Fig. 1. Transfer of annotation of a set using 2D/3D relation ICOMOS Glossary [START_REF] Bromblet | Illustrated glossary on stone deterioration patterns[END_REF]. Knowledge from this glossary and concepts from CIDOC-CRM scientific observation [START_REF] Doerr | The CIDOC CRM, an Ontological Approach to Schema Heterogeneity[END_REF] has been integrated. The latter is an ontological model able to document scientific practices in our domain and to manage both quantitative and qualitative aspects. In this case shared knowledge can be used and enriched by other case studies. The conservation experts 'goal is to provide a complete description of studied masonry building. Many observable information can be retrieved from the building. By crossing these information, it is possible to produce a line of reasoning induced to these links. Mainly, the expert need to obtain credible information and knowledge to undertake possible restoration operation. Annotations perform on spatialized images represents not only a shape but also the way to spatialize scientific observation and attach data. The user (expert) will have the possibility to annotate three thematic description layers: architectural components, materials and then alterations. Related to this different layers, we have consensually defined formal descriptor table allowing us to accurately characterize each them with their own specificities. Scientific observations will be done by using these data attached to layers with embedded information like environment (weather, building context for instance), acquisition (direct or indirect), temporal states, documentary resources, directly integrate into a dedicated data base. Thanks to these three linked layers, it will be possible to generate a lot of information about the studied building conservation state. The building's annotation set is representative of the whole structured knowledge concerning building conservation. To realize that it is important to exploit a relevant methodology in ontology design. The Lassila's method [START_REF] Hernandez | Ontologie de domaine pour la modélisation du contexte en recherche d'information[END_REF] [7] appears to be the best to fit our needs concerning its design method and therefore it was used to create our ontology. The above methodology required a domain expert to validate a controlled vocabulary which terms, as well as their meaning, will remain unchangeable during the whole process: by using an existing glossary. Following this step a thesaurus have to be built, in order to precisely define the complex and structured relationships between the terms and then obtain a taxonomy. At this step it's possible to create a lightweight ontology up to obtain a heavyweight ontology by adding more rules, constrains, and axioms. Two glossary types are exploited in our project: Illustrated glossary on stone deterioration patterns [START_REF] Bromblet | Illustrated glossary on stone deterioration patterns[END_REF] and architecture vocabulary [START_REF] Navigli | From Glossaries to Ontologies: Extracting Semantic Structure from Textual Definitions[END_REF]. Indeed expert can perform different annotations on layers "Fig. 2". We can count two annotations (red and white) giving us a quantitative aspect through extension, position, orientation (data) automatically extract from database or informed by the expert. In addition we have a qualitative aspect through a tag related to concept from ontology: buttress and alveolization. Thanks to the ontology reasoning, our system can inform us how the "alveolization", has been created. Our ontology is based on this follow structuration: "Masonry Building is composed of architectural component build with Material. And active agent acts on material and generates degradation mechanism inducing alteration".

According to domain ontology, to create this alteration, we need some components. This is a typical encountered alteration and it allows us to explain the domain ontology usefulness "Figure 3". A masonry like church is composed of several architectural entities such as buttress, built with porous stone containing soluble salts (chemical and physical properties). Due to the effect of the rain or underground circulation, water (active agent) seeps in stone by an alteration mechanism named rising damp. As a consequence a chemical reaction between water and soluble salts creates tensions because of salt crystallization. This internal pressure is cause of a typical feature by inducing material loss: an alteration named alveolization. We have used the Lassila's method in which the use of glossary is very important [START_REF] Hernandez | Ontologie de domaine pour la modélisation du contexte en recherche d'information[END_REF][7] [START_REF] Messaoudi | Laying the foundations for an information system dedicated to heritage building degradation monitoring based on the 2D/3D semantic annotation of photographs[END_REF]: therefore two glossary type are exploit: ICOMOS illustrated glossary on stone deterioration pattern [START_REF] Bromblet | Illustrated glossary on stone deterioration patterns[END_REF] to describe degradation and alteration phenomena and architecture vocabulary [START_REF] Navigli | From Glossaries to Ontologies: Extracting Semantic Structure from Textual Definitions[END_REF] to identify masonry structure architectural components. Protégé is the ontology editing tool used to integrate these two taxonomies and relationships between concepts to design our domain ontology.

C. Typical queries

This future web application will be able to answers to the following queries (query examples used come from our case study, the Caromb church (south of France)): Query 1:  Input: selected architectural element class: left buttress  Query: composing material and referenced alterations  Query results: left buttress; material => yellow stone, grey stone; alterations => alveolization, spalling Query 2:  Input: selected alteration type AND selected architectural element class => alveolization and left buttress  Query: composing material the most altered by selected alteration type  Query results: material => yellow stone "Figure 4" Query 3:  Input: selected mineralogical components list  Query: Some materials matching to this composition  Query results: yellow stone of Caromb, stone 2, stone 3 To conclude, thanks to annotation tool, it will be possible to cross several 3D regions and to generate information. Indeed, a scientific observation can identify a spatial region related to a concept like "alveolization". This concept will be linked with other regions that will allow us to realized similarities, to extract common orientations, extensions and other parameters that we are integrating into our future system. Our work is to make a link between scientific observations and shared knowledge, and use such information to create new knowledge. Moreover we can use these acquired knowledge in order to guide new observations. Then through scientific observation, we can enrich the data and provide information about studied building conservation state. The notion of temporal monitoring is still to be integrated. The main goal is to monitor masonry building degradation therefore other future acquisitions will be done to compare different conservation state. The difficulty is to readjust these acquisitions with the previous temporal state and compare two annotations analysis and generate other information. This is an on-going work and the ontology is still to be completed, because we are searching the most relevant methodology to combine our domain ontology with the CIDOC-CRM extension dedicated to scientific observation (CRMsci).

Fig. 2 .

 2 Fig. 2. Example of use linking scientific observation and domain ontology


Saïd Benachour
email: benachou@iecn.u-nancy.fr

Herbert Koch
email: koch@mathematik.uni-dortmund.de

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

VERY SINGULAR SOLUTIONS TO A NONLINEAR PARABOLIC EQUATION WITH ABSORPTION. II -UNIQUENESS

Keywords: viscous Hamilton-Jacobi equation, very singular solution, uniqueness

Very singular solutions to a nonlinear parabolic equation with absorptionII-Uniqueness

Said Benachour, Herbert Koch,

Introduction

The existence of a self-similar very singular solution at the origin to the following viscous Hamilton-Jacobi equation

u t -∆u + |∇u| p = 0 in (0, +∞) × R N , (1.1)
has been established in [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I -Existence[END_REF] and in [START_REF] Qi | The self-similar profiles of generalized KPZ equation[END_REF] by two different methods, when 1 < p < (N +2)/(N +1).

Recall that a very singular solution at the origin to (1.1) is a nonnegative solution to (1.1) which is smooth in (0, +∞) × R N and fulfils the following two conditions lim t→0 {|x|≤r} u(t, x) dx = +∞, lim t→0 {|x|≥r} u(t, x) dx = 0, for every r ∈ (0, +∞). In addition, a very singular solution u is self-similar if there is a smooth function f ∈ L 1 ((0, +∞); r N -1 dr) such that u(t, x) = t -a/2 f |x|t -1/2 , (t, x) ∈ (0, +∞) × R N ,

1
where a = (2 -p)/(p -1). In [START_REF] Qi | The self-similar profiles of generalized KPZ equation[END_REF], Qi & Wang show that there is one and only one self-similar very singular solution to (1.1). The purpose of this paper is to extend this uniqueness result without the self-similarity assumption. Before describing our results, let us mention that the name very singular solution has been introduced by Brezis, Peletier and Terman [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF] who proved the existence and uniqueness of a self-similar very singular solution to u t -∆u + u p = 0 in (0, +∞) × R N , (1.2) when 1 < p < 1 + 2/N . As self-similar very singular solutions to (1.2) satisfy an ordinary differential equation, the uniqueness proof in [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF] relies on ordinary differential equations techniques. The uniqueness of the very singular solution to (1.2) (without the self-similarity assumption) was subsequently obtained by Oswald [START_REF] Oswald | Isolated positive singularities for a nonlinear heat equation[END_REF]. Since then, the existence and uniqueness of nonnegative very singular solutions have been studied for other nonlinear parabolic equations with absorption such as

u t -∆u m + u p = 0 in (0, +∞) × R N , (1.3)
where m > (1 -2/N ) + , m = 1 [START_REF] Peletier | A very singular solution of the porous media equation with absorption[END_REF][START_REF] Kamin | Existence and uniqueness of the very singular solution of the porous media equation with absorption[END_REF][START_REF] Peletier | Source-type solutions of the porous media equation with absorption : the fast diffusion case[END_REF][START_REF] Leoni | A very singular solution for the porous media equation u t = ∆ (u m ) -u p when 0 < m < 1[END_REF][START_REF] Leoni | On very singular self-similar solutions for the porous media equation with absorption[END_REF], or

u t -div |∇u| m-2 ∇u + u p = 0 in (0, +∞) × R N , (1.4)
where m > 2N/(N + 1), m = 2 [START_REF] Peletier | A very singular solution of a quasilinear degenerate diffusion equation with absorption[END_REF][START_REF] Diaz | Uniqueness of very singular self-similar solution of a quasilinear degenerate parabolic equation with absorption[END_REF][START_REF] Kamin | Singular solutions of some nonlinear parabolic equations[END_REF][START_REF] Zhao | Source-type solutions of a quasilinear degenerate parabolic equation with absorption[END_REF]. Let us mention at this point that the uniqueness results obtained in the above mentioned papers are either restricted to self-similar very singular solutions or use the finite speed of propagation of the support of solutions to (1.3) and (1.4) when m > 1 and m > 2, respectively. ¿From another viewpoint, let us notice that a very singular solution u formally satisfies u(0, x) = 0 if x ∈ R N \ {0} and u(0, 0) = +∞ and thus belongs to the class of solutions having initial data taking infinite values on some subset of R N . Existence and uniqueness of such solutions have been investigated in [START_REF] Marcus | Initial trace of positive solutions of some nonlinear parabolic equations[END_REF] for (1.2) and in [START_REF] Crandall | Maximal solutions and universal bounds for some partial differential equations of evolution[END_REF] for (1.1) on a bounded open subset Ω of R N with homogeneous Dirichlet boundary conditions. In the latter work [START_REF] Crandall | Maximal solutions and universal bounds for some partial differential equations of evolution[END_REF] the initial data are required to take infinite values on a bounded subset of Ω with non-empty interior and thus do not include very singular solutions.

We now state our main result : we first make more precise the definition of a very singular solution to (1.1) we will use in this paper. Definition 1.1 A very singular solution to (1.1) is a function u ∈ C((0, +∞); L 1 (R N )) satisfying for each t ∈ (0, +∞) and τ ∈ (0, t): Here, G(t) denotes the linear heat semigroup in R N .

u(t) ≥ 0 a.e. in R N and u ∈ L p ((τ, t); W 1,p (R N )), (1.5)
u(t) = G(t -τ )u(τ ) - t τ G(t -σ) (|∇u(σ)| p ) dσ, (1.6) sup s∈(t,+∞) (s -t) N/2 u(s) L ∞ < ∞, (1.7) sup s∈(t,+∞) (s -t) (p(N +1)-N )/2p ∇u (p-1)/p (s) L ∞ < ∞, (1.8)
Our result then reads as follows.

Theorem 1.2 Assume that 1 < p < (N + 2)/(N + 1) and put a = (2 -p)/(p -1). There is one and only one very singular solution U to (1.1) in the sense of Definition 1.1. More precisely, there is a nonnegative and non-increasing function

f ∈ L 1 ((0, +∞); r N -1 dr) ∩ C ∞ ((0, +∞)) such that U (t, x) = t -a/2 f |x|t -1/2 , (t, x) ∈ (0, +∞) × R N , (1.11)
and f is a solution to the ordinary differential equation

f (r) + N -1 r + r 2 f (r) + a 2 f (r) -f (r) p = 0, r ∈ (0, +∞), (1.12)
with the boundary conditions

f (0) = 0 and lim r→+∞ r a f (r) = 0. (1.13)
As already mentioned the existence of a very singular solution to (1.1) which has the self-similar form (1.11) and with a profile f satisfying (1.12)-(1.13) has been proved in [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]. The main achievement of the present paper is the uniqueness part of Theorem 1.2 which we prove in the following way : we first proceed as in the proof of [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF]Theorem 2] to show that any very singular solution to (1.1) takes on the initial value zero uniformly on compact subsets of R N \ {0}. At this point a suitable modification of the proof of [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF]Theorem 2] is needed to handle the gradient term. This result then enables us to derive some estimates which are valid for every very singular solution to (1.1) and to prove that the very singular solution to (1.1) we constructed in [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I -Existence[END_REF] (denoted by U throughout the paper) is the minimal very singular solution to (1.1). The next section is devoted to the existence of a maximal very singular solution V to (1.1), following the approach of [START_REF] Kamin | Singular solutions of some nonlinear parabolic equations[END_REF]. Both minimal and maximal very singular solutions being self-similar with profiles satisfying (1.12)-(1.13), the conclusion U = V readily follows from [18, Theorem 2.1].

Let us finally mention that the very singular solutions to (1.2) (when they exist) play an important role in the description of the large time behaviour of the solutions to (1.2) (see, e.g., the survey paper [START_REF] Vázquez | Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents[END_REF] and the references therein). A similar result is expected for (1.1) and the above uniqueness result thus opens the path towards the study of the large time behaviour of the solutions to (1.1) when 1 < p < (N + 2)/(N + 1).

Preliminaries

We first recall the well-posedness of (1.1) in the space of nonnegative and bounded measures

M + b (R N ) [1, Theorems 1 & 3].
Theorem 2.1 Consider p ∈ (1, (N + 2)/(N + 1)) and u 0 ∈ M + b (R N ). There is a unique nonnegative function

u ∈ C((0, T ); L 1 (R N )) ∩ L p ((0, T ); W 1,p (R N )), T ∈ (0, +∞), satisfying u(t) = G(t -s)u(s) - t s G(t -σ) (|∇u(σ)| p ) dσ, 0 < s ≤ t, lim t→0 u(t, x) ψ(x) dx = ψ(x) du 0 (x), ψ ∈ BC(R N ), and      sup t∈(0,+∞) t N/2 u(t) L ∞ ≤ C H , sup t∈(0,+∞) t (p(N +1)-N )/2p ∇u (p-1)/p (t) L ∞ ≤ C H . (2.1)
Here BC(R N ) denotes the space of bounded and continuous functions in R N and C H is a positive real number depending only on N , p and u 0 M b . In addition, there holds sup t∈(0,+∞)

t 1/p ∇u (p-1)/p (t) L ∞ ≤ (p -1) 1-1/p p -1 := C HJ .
(2.

2)

It follows from Theorem 2.1 that, if u is a very singular solution to (1.1) in the sense of Definition 1.1, the conditions (1.6)-(1.8) imply that s → u(s+t) is the unique solution to (1.1) with initial datum u(t) given by Theorem 2.1 for each t ∈ (0, +∞). Therefore s → u(s + t) satisfies (2.2), i.e. sup s∈(0,+∞)

s 1/p ∇u (p-1)/p (s + t) L ∞ ≤ C HJ .
As this is valid for every t ∈ (0, +∞) we may let t → 0 in the above inequality and obtain the following result.

t 1/p ∇u (p-1)/p (t) L ∞ ≤ C HJ . (2.3)
We now recall additional estimates for solutions to (1.1) satisfying a growth condition for large values of x. For p ∈ (1, 2) we put

Γ p (r) = γ p r -a , r ∈ (0, +∞), (2.4)
where

γ p = (p -1) (p-2)/(p-1) (2 -p) -1 . Recall that a = (2 -p)/(p -1). The following result is proved in [2, Lemma 2.2 & Proposi- tion 2.4]. Lemma 2.3 Let p ∈ (1, (N +2)/(N +1)) and consider a nonnegative function u 0 in L 1 (R N ).
We define

R(u 0 ) := inf {R > 0 , |x| a u 0 (x) ≤ γ p a.e. in {|x| ≥ R}} ∈ [0, +∞],
and denote by u the nonnegative solution to (1.1) with initial datum u 0 given by Theorem 2.1.

If R(u 0 ) < +∞ and t ∈ (0, +∞) there holds

0 ≤ u(t, x) ≤ Γ p (|x| -R(u 0 )) , x ∈ R N , |x| > R(u 0 ).
In addition, there is a positive real number C 1 depending only on N and p such that

u(t) L 1 ≤ C 1 t -((N +2)-p(N +1))/2(p-1) , u(t) L ∞ ≤ C 1 t -a/2 , ∇u(t) L ∞ ≤ C 1 t -1/2(p-1) ,
for each t > τ (u 0 ), where

τ (u 0 ) = (N + 2) -p(N + 1) (N + 1)p -N 1-p R(u 0 ) 2 .

Some properties of very singular solutions

In this section we investigate the behaviour of the very singular solutions of (1.1) near t = 0 and in (0, +∞) × R N .

Proposition 3.1 Let u be a very singular solution to (1.1) and r ∈ (0, +∞).

Putting Ω r := {x ∈ R N ; |x| > r} we have u ∈ L ∞ ((0, 1) ; L 1 (Ω r )), ∇u ∈ L p ((0, 1) × Ω r ). (3.1)
The proof of this assertion follows step 2 of [3, Theorem 2].

Proposition 3.2 Let u be a very singular solution to (1.1). Then

u ∈ C 1,2 t,x ([0, +∞) × (R N \ {0})), (3.2)
and for every compact subset K of R N \ {0} there holds

lim t→0 u(t) C(K) = 0. (3.3)
Proof. We adapt step 3 of the proof of [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF]Theorem 2]. Let Ω be a bounded open subset of R N such that Ω ⊂ R N \ {0}. We define a function v on (-1, 1) × Ω by

v(t, x) =    u(t, x) if (t, x) ∈ (0, 1) × Ω, 0 if (t, x) ∈ (-1, 0) × Ω.
Then

∇v(t, x) =    ∇u(t, x) if (t, x) ∈ (0, 1) × Ω, 0 if (t, x) ∈ (-1, 0) × Ω, and (3.1) ensures that v ∈ L ∞ ((-1, 1) ; L 1 (Ω)), ∇v ∈ L p ((-1, 1) × Ω). (3.4) Also, v is a nonnegative function in (-1, 1) × Ω.
We may then proceed as in step 3 of [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF]Theorem 2] to show that

v t -∆v + |∇v| p = 0 in D ((-1, 1) × Ω). (3.5)
Since v satisfies (3.4) and is a subsolution to the heat equation, parabolic regularity theory then entails

v ∈ L ∞ loc ((-1, 1) × Ω). As Ω is an arbitrary open subset of R N with Ω ⊂ R N \ {0} we have shown that v ∈ L ∞ loc ((-1, 1) × (R N \ {0})). (3.6)
At this point we need to extend the argument of [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF] to obtain some more regularity on |∇v| p . Since v satisfies (3.5), (3.6) and is a nonnegative subsolution of the heat equation, classical arguments yield ∇v ∈ L 2 loc ((-1, 1) × (R N \ {0})).

(3.7)

On the other hand, we recall a well-known regularity result for a solution to the heat equation.

Lemma 3.3 [START_REF] Ladyženskaja | Linear and Quasilinear Equations of Parabolic Type[END_REF] Let Ω be a bounded open subset of R N , T > 0 and q ∈ (1, +∞). Consider w ∈ L q ((0, T ) ; W 1 q (Ω)) and f ∈ L q ((0, T ) × Ω) such that

w t -∆w = f in D ((0, T ) × Ω). Then, for each bounded open subset O of Ω such that O ⊂ O ⊂ Ω and ε ∈ (0, T /2), there holds w , w t , ∇w , D 2 w ∈ L q ((ε, T -ε) × O).
In addition

∇w ∈ L σ ((ε, T -ε) × O), where σ =        +∞ if q ≥ N + 2, q(N + 2) N + 2 -q if 1 < q < N + 2.
The last statement of Lemma 3.3 is a consequence of the embedding theorem for anisotropic Sobolev spaces [START_REF] Ladyženskaja | Linear and Quasilinear Equations of Parabolic Type[END_REF]Lemma II.3.3].

We continue with the proof of Proposition 3.2. By (3.6) and (3.7) we have

v ∈ L 2/p loc ((-1, 1) × (R N \ {0})), ∇v ∈ L 2/p loc ((-1, 1) × (R N \ {0})),
(recall that 1 < p < (N + 2)/(N + 1)). Thus, v satisfies (3.5). We may then apply Lemma 3.3 and obtain

∇v ∈ L q 1 loc ((-1, 1) × (R N \ {0})), q 1 = 2(N + 2) p(N + 2) -2 • (3.8)
Consequently, (3.6) and (3.8) yield

v ∈ L q 1 /p loc ((-1, 1) × (R N \ {0})), |∇v| p ∈ L q 1 /p loc ((-1, 1) × (R N \ {0})).
Noticing that q 1 > q 0 = 2 (since p < (N + 2)/(N + 1) < (N + 4)/(N + 2)) we have indeed a better regularity for |∇v| p . Applying again Lemma 3.3 we obtain

∇v ∈ L q 2 loc ((-1, 1) × (R N \ {0})), q 2 =        +∞ if q 1 ≥ p(N + 2) q 1 (N + 2) p(N + 2) -q 1 if q 1 < p(N + 2).
We then define, by induction, a sequence (q k ) by q 0 = 2 and

q k+1 =        +∞ if q k ≥ p(N + 2), q k (N + 2) p(N + 2) -q k if q k < p(N + 2).
We claim that q k ≥ 2, ∀k ≥ 0.

(3.9) Indeed, we have q 0 = 2. Assume that q k ≥ 2 for some k. Then either q k ≥ p(N + 2) and q k+1 = +∞. Or q k < p(N + 2) and

q k+1 = q k (N + 2) p(N + 2) -q k ≥ q k N + 2 p(N + 2) -2 • Since 1 < p < (N + 2)/(N + 1), we see that N + 2 p(N + 2) -2 > 1, (3.10)
whence q k+1 > q k > 2. Thus (3.9) holds true.

It then follows from (3.9) that

q k+1 ≥ q k (N + 2) p(N + 2) -2 > q k .
Therefore the sequence (q k ) k≥0 is increasing and (3.10) ensures that there is an integer k 0 > 2 such that q k < p(N + 2) for k ∈ {1, . . . , k 0 } and q k = +∞ if k > k 0 . Proceeding by induction we infer from Lemma 3.3 that

∇v ∈ L ∞ loc ((-1, 1) × (R N \ {0})).
Classical parabolic regularity results then entail

v ∈ C 1,2 t,x ((-1, 1) × (R N \ {0})),
whence (3.2) and (3.3).

Thanks to Proposition 3.2, we may now use comparison principle arguments to obtain additional information on the short time behaviour of the very singular solutions to (1.1). Lemma 3.4 Let u be a very singular solution to (1.1). Then

0 ≤ u(t, x) ≤ Γ p (|x|), (t, x) ∈ (0, +∞) × R N . (3.11)
Proof. Take r ∈ (0, +∞) and put Ω r := {x ∈ R N , |x| > r}. Then u and x → Γ p (|x| -r) are solutions to (1.1) on (0, +∞) × Ω r with

u(0, x) = 0 ≤ Γ p (|x| -r) if x ∈ Ω r , u(t, x) ≤ +∞ = Γ p (0) if (t, x) ∈ (0, +∞) × ∂Ω r .
The comparison principle then entails

u(t, x) ≤ Γ p (|x| -r), (t, x) ∈ (0, +∞) × Ω r . Now fix x 0 ∈ R N \ {0}. For r ∈ (0, |x 0 |), we have x 0 ∈ Ω r and u(t, x 0 ) ≤ Γ p (|x 0 | -r), t ∈ (0, +∞).
We then let r → 0 and obtain (3.11) for x 0 ∈ R N \ {0}. As Γ(0) = +∞, (3.11) also holds true for x = 0. Lemma 3.5 There is a constant K 1 depending only on p and N such that, if u is a very singular solution to (1.1) and t ∈ (0, +∞), there holds

u(t) L 1 ≤ K 1 t -(N +2-p(N +1))/2(p-1) , (3.12
)

u(t) L ∞ ≤ K 1 t -a/2 , (3.13) ∇u(t) L ∞ ≤ K 1 t -1/2(p-1) , (3.14)
with a = (2 -p)/(p -1).

Proof. Owing to Lemma 2.2 and Lemma 3.4, the proof of (3.12), (3.13) and (3.14) is similar to that of [2, Proposition 2.4] to which we refer.

The next lemma follows from [5, p. 186].

Lemma 3.6 For y ∈ R N and ρ > 0, we denote by α y,ρ the solution to -∆α y,ρ = 1 in B(y, ρ), α y,ρ = 0 on ∂B(y, ρ).

For every λ ∈ (0, +∞) there is C λ ∈ (0, +∞) such that, if u is a very singular solution to (1.1), y ∈ R N \ {0} and ρ ∈ (0, |y|), there holds

u(t, x) ≤ λe C λ t exp 1 α y,ρ (x) , (t, x) ∈ (0, +∞) × B(y, ρ).
The last lemma of this section will allow us to prove that the very singular solution we constructed in [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I -Existence[END_REF] is the minimal very singular solution. By a suitable truncation, it is possible to construct a sequence of nonnegative functions (u 0,k ) k≥k M such that

u 0,k (x) ≤ u(1/k, x), x ∈ R N , (3.15)
u 0,k || L 1 = M, (3.16)
for k ≥ k M , where k M is a sufficiently large integer. We denote by u k the unique nonnegative solution to (1.1) with initial datum u 0,k given by Theorem 2.1. Since (u 0,k ) is bounded in L 1 (R N ), we may use (2.1) to proceed as in the proof of [1, Theorem 3] and show that there are a subsequence of (u k ) (not relabeled) and a function u ∈ C((0, +∞), L 1 (R N )) such that, as k → +∞ :

u k → u in C((0, +∞) ; L 1 (R N )) ∩ L p ((s, t) ; W 1,p (R N )), (3.17) u(t) = G(t -s)u(s) - t s G(t -σ)(|∇u(σ)| p )dσ,
for every s ∈ (0, +∞) and t ∈ (s, +∞).

It also follows from (3.17) that u satisfies (2.1). It remains to identify the initial datum taken by u. Let ρ ∈ D(R N ), k ≥ k M and t ∈ (0, 1). By (1.1), (2.1) and (3.16) we have

u k (t, x)ρ(x)dx -u 0,k (x)ρ(x)dx ≤ ∆ρ L ∞ t 0 u k (σ, x)dxdσ + ρ L ∞ t 0 |∇u k (σ, x)| p dxdσ ≤ M t ∆ρ L ∞ + p p -1 p ρ L ∞ t 0 u k ∇u (p-1)/p k p dxdσ ≤ M t ∆ρ L ∞ + C(ρ, p, M, N ) t 0 σ (N -p(N +1))/2 dσ ≤ C(ρ, p, M, N ) t + t (N +2-p(N +1))/2 .
(3.18)

For r ∈ (0, +∞), we also have by (3.15) Thanks to (3.17) and (3.19) we may let k → +∞ in (3.18) and obtain

u 0,k (x)ρ(x)dx -M ρ(0) ≤ 2 ρ L ∞ {|x|≥r} u 1 k , x dx + {|x|≤r} u 0,k (x)dx sup {|x|≤r} |ρ(x) -ρ(0)| ≤ 2 ρ L ∞ {|x|≥r} u 1 k , x dx + M sup {|x|≤r} |ρ(x) -ρ(0)|.
u(t, x)ρ(x)dx -M ρ(0) ≤ C(ρ, p, M, N ) t + t (N +2-p(N +1))/2 . Consequently, for each ρ ∈ D(R N ), lim t→0 u(t, x)ρ(x)dx = M ρ(0) . (3.20)
As u is a subsolution to the heat equation, a comparison argument yields that (3.20) actually holds for every ρ ∈ BC(R N ). Summarizing, we have proved that u is a solution to (1.1) with initial datum M δ fulfilling all the requirements of Theorem 2.1. Such a solution being unique, we conclude that u = u M .

(3.21)

To complete the proof we only have to notice that the comparison principle and (3.15) entail that for

k ≥ k M , u k (t, x) ≤ u(t + 1/k, x), (t, x) ∈ (0, +∞) × R N .
We then use (3.17) and (3.21) to obtain Lemma 3.7.

We end up this section by recalling the main result of [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I -Existence[END_REF].

Theorem 3.8 The function

U (t, x) = sup {M >0} u M (t, x), (t, x) ∈ (0, +∞) × R N ,
is a very singular solution to (1.1), where u M denotes the solution to (1.1) with initial datum M δ. Moreover, there is a nonnegative and non-increasing function

f ∈ L 1 ((0, +∞), r N -1 dr) ∩ C ∞ ((0, +∞))
satisfying (1.12) and (1.13) and such that

U (t, x) = t -a/2 f (|x|t -1/2 ), (t, x) ∈ (0, +∞) × R N .
As a consequence of Lemma 3.7 and Theorem 3.8, we see that, if u is a very singular solution to (1.1), there holds u ≥ U.

Then U is the minimal very singular solution to (1.1), and it is the unique self-similar very singular solution to (

.

Existence of a maximal very singular solution

We denote by S the set of very singular solutions to (1.1) in the sense of Definition 1.1. Notice that, as the minimal very singular solution U (defined in Theorem 3.8) belongs to S, the set S is non-empty.

We now proceed as in [START_REF] Kamin | Singular solutions of some nonlinear parabolic equations[END_REF]Theorem 4.1] to prove that S has a maximal element. More precisely we put

V (t, x) = sup u∈S u(t, x), (t, x) ∈ (0, +∞) × R N . (4.1) Lemma 4.1 For each t ∈ (0, +∞), V (t) is a nonnegative function in L 1 (R N ) ∩ W 1,∞ (R N ) which satisfies t a/2 V (t) L ∞ + t 1/2(p-1) ∇V (t) L ∞ ≤ 2K 1 , (4.2)
t 1/p ∇V (p-1)/p (t) L ∞ ≤ C HJ , (4.3)
U (t, x) ≤ V (t, x) ≤ Γ p (|x|), x ∈ R N , (4.4)
where C HJ , Γ p and K 1 are defined in (2.2), (2.4) and Lemma 3.5, respectively.

Proof. Since U ∈ S, (4.4) is a straightforward consequence of (3.11) and (4.1). Next, (3.13) and (4.4) entail that 0 ≤ V (t, x) ≤ min (K 1 t -a/2 , Γ p (x)).

Consequently, for each t ∈ (0, +∞),

V (t) ∈ L 1 (R N ) ∩ L ∞ (R N ) and t a/2 V (t) L ∞ ≤ K 1 , t ∈ (0, +∞). It next follows from (3.14) that if u ∈ S and (x, y) ∈ R N × R N u(t, x) ≤ u(t, y) + K 1 t -1/2(p-1) |x -y| ≤ V (t, y) + K 1 t -1/2(p-1) |x -y|.
Therefore,

V (t, x) ≤ V (t, y) + K 1 t -1/2(p-1) |x -y|,
and V (t) is Lipschitz continuous with Lipschitz constant K 1 t -1/2(p-1) . We have thus shown (4.2). Since σ → σ (p-1)/p is non-decreasing on (0, +∞), a similar argument yields (4.3).

Lemma 4.2 V ∈ S.

Proof. Fix τ ∈ (0, +∞). As V (τ ) is a nonnegative function in L 1 (R N ) ∩ W 1,∞ (R N ), we denote by v τ the unique nonnegative solution to (1.1) in (τ, +∞) × R N with v τ (τ ) = V (τ )
given by Theorem 2.1. For u ∈ S it follows from (4.1) that

v τ (τ ) = V (τ ) ≥ u(τ ),
and the comparison principle entails

v τ (t) ≥ u(t), t ∈ [τ, +∞). Consequently v τ (t) ≥ V (t), t ∈ [τ, +∞). (4.5)
Next, on the one hand, it follows from (4.4) and the comparison principle that

v τ (t, x) ≤ Γ p (|x|), (t, x) ∈ [τ, +∞) × R N . (4.6)
On the other hand, by (2.2) we have sup t∈(τ,+∞)

(t -τ ) 1/p ∇(v τ ) (p-1)/p (t) L ∞ ≤ C HJ . (4.7)
We may therefore proceed as in [2, Proposition 2.4] to show that there is a positive constant K 2 depending only on p and N such that for each τ ∈ (0, +∞) and t ∈ (τ, +∞) there holds 1) . (4.10)

v τ (t) L 1 ≤ K 2 (t -τ ) -(N +2-p(N +1))/2(p-1) , (4.8) v τ (t) L ∞ ≤ K 2 (t -τ ) -a/2 , (4.9) ∇v τ (t) L ∞ ≤ K 2 (t -τ ) -1/2(p-
We next claim that

τ < σ ⇒ v σ (t, x) ≤ v τ (t, x), (t, x) ∈ [σ, +∞) × R N . (4.11)
Indeed, by (4.5) we have

v σ (σ) = V (σ) ≤ v τ (σ), σ ≥ τ,
and (4.11) follows from the comparison principle. We now define

W (t, x) = sup τ ∈(0,t/2)
v τ (t, x), (t, x) ∈ (0, +∞) × R N . (4.12)

Thanks to (4.8), (4.11) and the monotone convergence theorem, we realize that, for each t ∈ (0, +∞),

W (t) ∈ L 1 (R N ) and lim τ →0 W (t) -v τ (t) L 1 = 0. (4.13)
Next, owing to (4.8)-(4.10), we may proceed as in [START_REF] Benachour | Global solutions to viscous Hamilton-Jacobi equations with irregular initial data[END_REF]Section 3] to show that, for each t ∈ (0, +∞) and s ∈ (0, t), we have

W ∈ C((s, t) ; L 1 (R N )) ∩ L p ((s, t) ; W 1,p (R N )),
and W satisfies

W (t) = G(t -s)W (s) - t s G(t -σ)(|∇W (σ)| p )dσ. (4.14)
Also, if t 0 ∈ (0, +∞), it follows from (4.8) that

Y (t 0 ) := sup τ ∈(0,t 0 /2) v τ (t 0 ) L 1 < ∞.
We then infer from Theorem 2.1 that

(t -t 0 ) N/2 v τ (t) L ∞ + (t -t 0 ) (p(N +1)-N )/2p ∇(v τ ) (p-1)/p (t) L ∞ ≤ K 3 (t 0 ),
for t ∈ (t 0 , +∞) and τ ∈ (0, t 0 /2), where K 3 (t 0 ) only depends on N , p and Y (t 0 ). Owing to (4.13), we may let τ → 0 in the above estimate and see that

(t -t 0 ) N/2 W (t) L ∞ + (t -t 0 ) (p(N +1)-N )/2p ∇W (p-1)/p (t) L ∞ ≤ K 3 (t 0 ), (4.15)
for t ∈ (t 0 , +∞). We next infer from (4.4), (4.5), (4.6) and (4.12) that

U (t, x) ≤ V (t, x) ≤ W (t, x) ≤ Γ p (|x|), (t, x) ∈ (0, +∞) × R N . (4.16)
Since U is a very singular solution, we obtain lim t→0 {|x|≤r} W (t, x)dx = +∞, r ∈ (0, +∞). (4.17)

Finally, take r ∈ (0, +∞) and consider ρ

∈ C ∞ (R N ) such that 0 ≤ ρ ≤ 1, ρ(x) = 1 if |x| ≥ r and ρ(x) = 0 if |x| ≤ r 2 •
Since v τ is a solution to (1.1), we have for t ∈ (0, +∞) and τ ∈ (0, t)

ρ(x)v τ (t, x)dx + t τ ρ(x)|∇v τ (σ, x)| p dxdσ = ρ(x)v τ (τ, x)dx + t τ ∆ρ(x)v τ (σ, x)dσdx.
Recalling that v τ (τ, x) = V (τ, x), we find

ρ(x)v τ (t, x)dx ≤ ρ(x)V (τ, x)dx + t τ {r/2<|x|<r} |∆ρ|v τ (σ, x)dσdx. ( 4
(x)V (τ, x)dx ≤ {r/2<|x|<r} V (τ, x)dx + {|x|>R} Γ p (|x|)dx, (4.20)
for every R ∈ [2r, +∞). Since

K(r, R) := x ∈ R N , r/2 ≤ |x| ≤ R
is compact, there is a finite number of points (y i ) 1≤i≤k in K(r, R) such that

K(r, R) ⊂ k i=1
B (y i , r/8) .

Let i ≥ 1. Notice that, as |y i | ≥ r/2, we have |y i | > r/4. We then infer from Lemma 3.6 that, if u belong to S, there holds

u(t, x) ≤ λe C λ t exp 1 α y i ,r/4 (x) , (t, x) ∈ (0, +∞) × B (y i , r/4)
for every λ ∈ (0, +∞). The above pointwise estimate being true for every u in S, we deduce that

V (t, x) ≤ λe C λ t exp 1 α y i ,r/4 (x)
, (t, x) ∈ (0, +∞) × B (y i , r/4) .

Therefore, satisfying (1.12), (1.13) and such that V (t, x) = t -a/2 g (|x|t -1/2 ), (t, x) ∈ (0, +∞) × R N .

K(r,R) V (τ, x)dx ≤ λe C λ τ k i=1 B(y i ,r /
Proof. Consider u ∈ S and λ ∈ (0, +∞). It is straightforward to check that the function u defined by u(t, x) = λ a u(λ 2 t, λx), (t, x) ∈ (0, +∞) × R N , also belongs to S. The set S being invariant with respect to the above scaling transformation, we easily deduce that

V (t, x) = λ -a V t λ 2 ,
x λ , (t, x) ∈ (0, +∞) × R N , λ > 0. (4.23)

On the other hand, the equation (1.1) being rotationally invariant, it is clear that S is rotationally invariant and V (t, .) is therefore radially symmetric with respect to the space variable for every t ∈ (0, +∞). Putting g(r) = V (1, r, 0, . . . , 0), r ∈ (0, +∞), and using (4.23) yield V (t, x) = t -a/2 g (|x|t -1/2 ), (t, x) ∈ (0, +∞) × R N .

We then proceed as in [2, Section 3] to check that g has the required properties.

Proof of Theorem 1.2. Since f (cf. Theorem 3.8) and g (cf. Lemma 4.3) are solutions to (1.12), (1.13), it follows from [18, Theorem 2.1] that f = g, whence U = V . Now, if ϑ ∈ S, it follows from (3.22) and (4.1) that U ≤ ϑ ≤ V = U , whence ϑ = U . Therefore, S = {U } and Theorem 1.2 is proved.

  x) dx = 0, r ∈ (0, +∞).(1.10)

Lemma 2 . 2

 22 Let u be a very singular solution to (1.1) in the sense of Definition 1.1. Then sup t∈(0,+∞)

Lemma 3 . 7

 37 If u is a very singular solution to (1.1) and M ∈ (0, +∞) we have u M ≤ u, where u M denotes the solution to (1.1) with initial datum M δ. Proof. As u is a very singular solution to (1.1) we have lim t→0 u(t) L 1 = +∞.

  We let k → +∞ and use Definition 1.1 to obtain that lim supk→+∞ u 0,k (x)ρ(x)dx -M ρ(0) ≤ M sup {|x|≤r} |ρ(x) -ρ(0)|.Passing to the limit as r → 0 then yields lim k→+∞ u 0,k (x)ρ(x)dx = M ρ(0).(3.19)

VLemma 4 . 3

 43 (τ, x)dx ≤ λ k i=1 B(y i ,r/8) exp 1 α y i ,r/4 (x) dx.This inequality being valid for every λ ∈ (0, +∞), we finally obtainlim τ →0 K(r,R) V (τ, x)dx = 0. (4.21) We may then let τ → 0 in (4.18) and use (4.13), (4.19), (4.20) and (4.21) to obtain ρ(x)W (t, x)dx ≤ C(p, r, ρ)t + {|x|>R} Γ p (|x|)dx for every t ∈ (0, +∞) and R ≥ 2r. As Γ p ∈ L 1 (R N \ B(0, 1)), we may let R → +∞ in the above estimate to deduce that 0 ≤ {|x|≥r} W (t, x)dx ≤ C(p, r, ρ)t. from (4.14), (4.15), (4.17) and (4.22) that W is a very singular solution to (1.1) in the sense of Definition 1.1. Consequently, W ∈ S, whence W ≤ V by (4.1). Recalling (4.16), we realize that W = V and the proof of Lemma 4.2 is complete. There is a nonnegative and non-increasing function g ∈ L 1 ((0, +∞), r N -1 dr) ∩ C ∞ ((0, +∞)),