Tai Nguyen

Phuoc Laurent Véron

Initial trace of positive solutions of a class of degenerate heat equation with absorption

We study the initial value problem with unbounded nonnegative functions or measures for the equation

and f is a continuous, nondecreasing nonnegative function such that f (0) = 0. In the case p > 2N N +1 , we provide a sufficient condition on f for existence and uniqueness of the solutions satisfying the initial data kδ 0 and we study their limit when k → ∞, according f -1 and F -1/p are integrable or not at infinity, where F (s) = s 0 f (σ)dσ. We also give new results dealing with non uniqueness for the initial value problem with unbounded initial data. If p > 2, we prove that, for a 1 large class of nonlinearities f , any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case f (u) = u α ln β (u + 1), where α > 0 and β ≥ 0.

Introduction

The aim of this article is to study some qualitative properties of the positive solutions of

∂ t u -∆ p u + f (u) = 0 (1.1)
in Q ∞ := R N × (0, ∞) where p > 1, ∆ p u = div(|∇u| p-2 ∇u) and f is a continuous, nondecreasing function such that f (0) = 0 = f -1 (0). The properties we are interested in are mainly: (a) the existence of fundamental solutions i.e. solutions with kδ 0 as initial data and the behaviour of these solutions when k → ∞; (b) the existence of an initial trace and its properties; (c) uniqueness and non-uniqueness results for the Cauchy problem. This type of questions have been considered in a previous paper of the authors [START_REF] Nguyen Phuoc | Local and global properties of solutions of heat equation with superlinear absorption[END_REF] in the semilinear case p = 2. The breadcrumbs of this study lies in the existence of two types of specific solutions of (1.1). The first ones are the solutions φ := φ a of the ODE

φ ′ + f (φ) = 0 (1.2)
defined on [0, ∞) and subject to φ(0) = a ≥ 0; it is given by a φ(t) ds f (s) .

(1.

3)

The second ones are the solutions of the elliptic equation -∆ p w + f (w) = 0, (1.4) defined in R N or in R N \ {0}. It is well-known that the structure of the set of solutions of (1.2) depends whether the following quantity

J := ∞ 1 ds f (s) (1.5)
is finite or infinite. If J < ∞ there exists a maximal solution φ ∞ to (1.2) defined on (0, ∞) while no such solution exists if J = ∞ since lim a→∞ φ a (t) = ∞. This maximal solution plays an important role since, by the maximum principle, it dominates any solution u of (1.1) which satisfies lim |x|→∞ u(x, t) = 0 (1.6) for all t > 0, locally uniformly on (0, ∞). Concerning (1.4) we associate the quantity

K := ∞ 1 ds F (s) 1/p . (1.7)
It is a consequence of the Vázquez's extension of the Keller-Osserman condition (see [START_REF] Vazquez | An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion[END_REF], [START_REF] Keller | On solutions of ∆u = f (u)[END_REF]) that if K < ∞, equation (1.4) admits a maximal solution W R N * in R N \ {0}. This solution is constructed as the limit, when R → ∞ and ǫ → 0 of the solution W := W ǫ,R of (1.4) in Γ ǫ,R := B R \ B ǫ , subject to the conditions lim |x|↓ǫ W ǫ,R (x) = ∞ and lim |x|↑R W ǫ,R (x) = ∞. On the contrary, if K = ∞, such functions W ǫ,R and W R N * do not exist, a situation which will be exploited in Section 3 for proving existence of global solutions of (1.4) in R N . An additional natural growth assumption of f that will be often made is the super-additivity

f (s + s ′ ) ≥ f (s) + f (s ′ ) ∀s, s ′ ≥ 0, (1.8)
which, combined with the monotonicity of f , implies a minimal linear growth at infinity lim inf s→∞ f (s) s > 0.

(1.9)

If p ≥ 2, K < ∞ jointly with (1.8) implies J < ∞, but this does not hold when 1 < p < 2. When p > 2 and f satisfies J < ∞ and K < ∞, Kamin and Vázquez proved universal estimates for solutions which vanish on R N × {0} \ {(0, 0)} (see [START_REF] Kamin | Singular solutions of of some nonlinear parabolic equations[END_REF]). By a slight modification of the proof in [15, Proposition 2.3 and Proposition 2.6], it is possible to extend their result to the case p > 1.

Proposition (Universal estimates) Assume p > 1 and f satisfies K < ∞. Let u ∈ C(Q ∞ \ {(0, 0)}) be a solution of (1.1) in Q ∞ , which vanishes on R N × {0} \ {(0, 0)}.Then

u(x, t) ≤ W R N * (x) ∀(x, t) ∈ Q ∞ .
(1.10)

If we suppose moreover J < ∞ and that (1.8) holds, then

u(x, t) ≤ min φ ∞ (t), W R N * (x) ∀(x, t) ∈ Q ∞ . (1.11)
When K = ∞, no such estimate exists since the function w a solution of (1. [START_REF] Segura De Leon | Regularity for entropy solutions of parabolic p-Laplacian type equations[END_REF]) is a stationnary solution of (1.1) with unbounded initial data.

In Section 2 we study the existence of the fundamental solutions u k and their behaviour when k → ∞. Kamin and Vázquez proved in [11, Lemma 2.3 and Lemma 2.4], that if p > 2 and

∞ 1 s -p-p N f (s)ds < ∞, (1.12)
then for any k > 0, there exists a unique positive solution u := u k to problem

∂ t u -∆ p u + f (u) = 0 in Q ∞ u(., 0) = kδ 0 in R N . (1.13)
Furthermore the mapping k → u k is increasing. Their existence proof heavily relies on the fact that, if we denote by v := v k the fundamental (or Barenblattt-Prattle) solution of

∂ t v -∆ p v = 0 in Q ∞ v(., 0) = kδ 0 in R N , (1.14)
then v k (., t) is compactly supported in some ball B δ k (t) , where δ k (t) is explicit. Since v k is a natural supersolution for (1.13), condition (1.12) states that f (v k ) ∈ L 1 loc (Q ∞ ). When 2N/(N + 1) < p ≤ 2, v k (x, t) > 0 for all (x, t) ∈ Q ∞ . It is already proved in [START_REF] Marcus | Initial trace of positive solutions to semilinear parabolic inequalities[END_REF] that, when p = 2, condition (1.12) yields to f (v k ) ∈ L 1 (Q T ). We prove here that this result also holds when 2N/(N + 1) < p ≤ 2 and more precisely, Theorem 1.1 Assume p > 2N N +1 and f satisfies (1.12). Then there exists a unique positive solution u := u k to problem (1.13).

In view of this result and the a priori estimates (1.10) and (1.11), it is natural to study the limit of u k when k → ∞. We denote by U 0 the set of positive u ∈ C(Q ∞ \ {(0, 0)}) which are solutions of (1.1) in Q ∞ , vanishes on the set {(x, 0) : x = 0} and satisfies lim t→0 Bǫ u(x, t)dx = ∞ ∀ǫ > 0.

Theorem 1.2 Assume p > 2N/(N + 1), J < ∞, K < ∞ and (1.12) holds. Then U = lim k→∞ u k exists and it is the smallest element of U 0 .

When one, at least, of the above properties on J and K fails, the situation is much more complicated and fairly well understood only in the case where f has a power-like or a logarithmic-power-like growth. We first note that

(A) If f (s) ∼ s α (α > 0), then J < ∞ if and only if α > 1, while K < ∞ if and only if α > p -1. Moreover (1.12) holds if and only if α < p(1 + 1 N ) -1. (B) If f (s) ∼ s α ln β (s + 1) (α, β > 0), then J < ∞ if
and only if α > 1 and β > 0, or α = 1 and β > 1 while K < ∞ if and only if α > p -1 and β > 0, or α = p -1 and β > p. Moreover (1.12) holds if and only if α < p(1 + 1 N ) -1 and β > 0.

Theorem 1.3 Assume p > 2 and f (s) = s α ln β (s + 1) where α ∈ (1, p -1) and β > 0.

Let u k be the solution of (1.13). Then lim

k→∞ u k (x, t) = φ ∞ (t) for every (x, t) ∈ Q ∞ .
When α = 1 the following phenomenon occurs.

Theorem 1.4 Assume p > 2 and f (s) = s ln β (s + 1) with β > 0. Let u k be the solution of (1.13).

Then (i) If β > 1 then lim k→∞ u k (x, t) = φ ∞ (t) for every (x, t) ∈ Q ∞ , (ii) If 0 < β ≤ 1 then lim k→∞ u k (x, t) = ∞ for every (x, t) ∈ Q ∞ .
Section 3 is devoted to study non-uniqueness of solutions of (1.1) with unbounded initial data. The starting observation is the following global existence result for solutions of (1.4): Theorem 1.5 Assume p > 1, f is locally Lipschitz continuous and K = ∞. Then for any a > 0, there exists a unique solution w := w a to the problem

-(r N -1 |w r | p-2 w r ) r + r N -1 f (w) = 0 (1.15)
defined on [0, ∞) and satisfying w(0) = a, w r (0) = 0. It is given by

w a (r) = a + r 0 H p s 1-N s 0 τ N -1 f (w a (τ ))dτ ds (1.16)
where H p is the inverse function of t → |t| p-2 t.

This result extends to the general case p > 1 a previous theorem of Vázquez and Véron [START_REF] Vazquez | Isolated singularities of some semilinear elliptic equations[END_REF] obtained in the case p = 2. The next theorem extends to the case p = 2 a previous result of the authors in the case p = 2.

Theorem 1.6 Assume p > 2N/(N + 1), f is locally Lipschitz continuous, J < ∞ and

K = ∞. For any function u 0 ∈ C(Q ∞ ) which satisfies w a (|x|) ≤ u 0 (x) ≤ w b (|x|) ∀x ∈ R N (1.17)
for some 0 < a < b, there exist at least two solutions u, u ∈ C(Q ∞ ) of (1.1) with initial value u 0 . They satisfy respectively

0 ≤ u(x, t) ≤ min{w b (|x|), φ ∞ (t)} ∀(x, t) ∈ Q ∞ ,
thus lim t→∞ u(x, t) = 0, uniformly with respect to x ∈ R N , and

w a (|x|) ≤ u(x, t) ≤ w b (|x|) ∀(x, t) ∈ Q ∞ thus lim |x|→∞ u(x, t) = ∞, uniformly with respect to t ≥ 0.
In section 4 we prove an existence and stability result for the initial value problem

∂ t u -∆ p u + f (u) = 0 in Q ∞ u(., 0) = µ in R N (1.18)
where µ ∈ M b + (R N ), the set of positive and bounded Radon measures in R N .

Theorem 1.7 Assume p > 2N N +1 and f satisfies (1.12). Then for any

µ ∈ M b + (R N ) the problem (1.18) admits a weak solution u µ . Moreover, if {µ n } is a sequence of functions in L 1 + (R N ) with compact support, which converges to µ ∈ M b + (R N )
in the weak sense of measures, then the corresponding solutions {u µn } of (1.18) with initial data µ n converge to some solution u µ of (1.18), strongly in L 1 loc (Q T ) and locally uniformly in

Q T := R N ×(0, T ). Furthermore {f (u µn )} converges strongly to f (u µ ) in L 1 loc (Q T ).
In Section 5, we discuss the initial trace of positive weak solution of (1.1). The power case f (u) = u q with q > 0 was investigated by Bidaut-Véron, Chasseigne and Véron in [START_REF] Bidaut-Véron | Initial trace of solution of some quasilinear parabolic equations with absorption[END_REF]. They proved the existence of an initial trace in the class of positive Borel measures according to the different values of p-1 and q. Accordingly they studied the corresponding Cauchy problem with a given Borel measure as initial data. However their method was strongly based upon the fact that the nonlinearity was a power, which enabled to use Hölder inequality in order to show the domination of the absorption term over the other terms. In the present paper, we combine the ideas in [START_REF] Bidaut-Véron | Initial trace of solution of some quasilinear parabolic equations with absorption[END_REF] and [START_REF] Nguyen Phuoc | Local and global properties of solutions of heat equation with superlinear absorption[END_REF] with a stability result for the Cauchy problem and Harnack's inequality in the form of [START_REF] Dibenedetto | Degenerate Parabolic Equations[END_REF] to establish the following dichotomy result which is new even in the case p = 2.

Theorem 1.8 Assume p ≥ 2 and (1.12) holds. Let u ∈ C(Q T ) be a positive weak solution of (1.1) in Q T . Then for any y ∈ R N the following alternative holds

(i) either u(x, t) ≥ lim k→∞ u k (x -y, t) ∀(x, t) ∈ Q T , (1.19
µ ∈ M + (R(u)) such that lim t→0 R(u) u(x, t)ζ(x)dx = R(u) ζdµ ∀ζ ∈ C c (R(u)).
(1.22)

Owing to the above result we define the initial trace of a positive solution u (1.1) in Q T as the couple (S(u), µ) for which (1.20) and (1.21) holds and we denote it by tr R N (u). The set S(u) is the set of singular points of tr R N (u), while µ is the regular part of tr R N (u). It is classical that any ν ∈ B reg (R N ), the set of positive outer regular Borel measures in R N , can be represented by a couple (S, µ) where S is a closed subset of R N and µ ∈ M + (R), where R = R N \ S, in the following way

ν(A) = ∞ if A ∩ S = ∅, µ(A) if A ⊂ R, ∀A Borel.
Therefore Theorem 1.8 means that tr R N (u) ∈ B reg (R N ). The initial trace can be made more precise when the Keller-Osserman-Vázquez condition does not hold, and if we know whether lim k→∞ u k is equal to φ ∞ or is infinite. As a consequence of I, there exist infinitely many positive solutions u of (1.1) in Q ∞ such that tr R N (u) = ν ∞ . By Theorem 1.3, Theorem 1.4, the previous results apply in particular if f (s) = s α ln β (s + 1).

Isolated singularities

Throughout the article c i denote positive constants depending on N , p, f and sometimes other quantities such as test functions or particular exponents, the value of which may change from one occurrence to another.

The semigroup approach

We refer to [9, p 117] for the detail of the Banach space framework for the construction of solutions of (1.1) in Q ∞ with initial data in L 1 (R N ) ∩ L ∞ (R N ). We set

J(u) = R N 1 p |∇u| p + F (u) dx (2.1)
when u belongs to the domain D(J) of J which is the set of u ∈ L 2 (R N ) such that ∇u ∈ L p (R N ) and F (u) ∈ L 1 (R N ), and

J(u) = ∞ if u / ∈ D(J). Then J is a proper convex lower semicontinuous function in L 2 (R N ). Its sub-differential A is defined by its domain D(A) which is the set of u ∈ L 2 (R N ) such that ∇u ∈ L p (R N ) and F (u) ∈ L 1 (R N ) with the property that -∆ p u + f (u) ∈ L 2 (R N ) and - R N v∆ p udx = R N |∇u| p-2 ∇u.∇vdx ∀v ∈ D(J), (2.2)
and by its expression

Au = -∆ p u + f (u) ∀u ∈ D(A). (2.3) Notice that (2.2) implies that vf (u) ∈ L 1 (R N ) for all v ∈ D(J). The restriction of the operator A is accretive in L 1 (R N ) and in L ∞ (R N ), hence in every L q (R N ). The operator A q defined in L q (R N ) is the closure in L q (R N ) of the restriction of A to L q (R N ). It is a m-accretive operator, with domain D(A q ). Since C ∞ 0 (R N ) ⊂ D(A q ), D(A q ) is dense in L q (R N ). If u 0 ∈ L q the generalized solution u to    du dt + A q u = 0 in (0, ∞) u(0) = u 0 (2.4)
is obtained by the Crandall-Liggett scheme

u i -u i-1 h + A q u i = 0 in i = 0, 1, ... (2.5)
when we let h → 0, in the sense that the continuous piecewise linear function U h defined by U h (ih) = u i converges to u in the C([0, T ], L q (R N ))-topology, for every T > 0. Furthermore, if q = 2 and [START_REF] Véron | Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces[END_REF]. We shall denote by {S Aq (t)} t>0 the semigroup of contractions of L q (R N ) generated by -A q thru the Crandall-Liggett Theorem [START_REF] Crandall | Generation of seigroups of nonlinear transformations in general Banach spaces[END_REF].

u 0 ∈ D(A 2 ) (resp. u 0 ∈ L 2 (R N )), then dU h dt converges to du dt in L 2 ([0, T ], L 2 (R N )) (resp. L 2 ([0, T ], L 2 (R N ); tdt)), see
An important property [9, Lemma 2] is that if w ∈ L 1 (R N ) satisfies A 1 w + σw = h (2.6)
where σ > 0 and h ∈ L 1 (R N ), then

R N A 1 wdx = 0. (2.7) Definition 2.1 (i) A function u ∈ C([δ, ∞); L 1 (R N )) where δ ≥ 0 is a semigroup solution (1.1) on (δ, ∞) if for any t ≥ δ there holds u(., t) = S A 1 (t -δ)[u(., δ)]. (ii) A function u ∈ C((δ, ∞); L 1 (R N )) is an extended semigroup solution of (1.1) on (δ, ∞) if for any t ≥ τ > δ, there holds u(., t) = S A 1 (t -τ )[u(., τ )].

The Barenblatt-Prattle solutions

We recall the explicit expression, due to Barenblatt and Prattle, of the solution

v = v k of problem (1.14). If p = 2 v k (x, t) = k(4πt) -N 2 e -|x| 2 4t , (2.8
)

and if 2N N +1 < p = 2, v k (x, t) = t -λ V x t λ N
, where

V (ξ) = C k -d|ξ| p p-1 p-1 p-2 + (2.9) with λ = N N (p -2) + p and d = p -2 p λ N 1 p-1 , (2.10)
and where C k is connected to the mass k by

C k = c(N, p)k ℓ with ℓ = p(p -2)λ (p -1)N . (2.11)
The condition p > 2N N +1 appears in order λ be positive. Notice that, if p > 2 then

d > 0, therefore the support of v k (., t) is the ball B δ k (t) where δ k (t) = C k d p-1 p t λ N , while v k (x, t) > 0 for all (x, t) ∈ Q ∞ if 2N N +1 < p < 2 (and also p = 2 although the expression of v k is different). Furthermore, if 2N N +1 < p < 2, the limit of v k when k → ∞ is explicit v ∞ (x, t) = Λ N t |x| p 1 2-p , (2.12)
where Λ N = (-d)

p-1 p-2 . This type of singular solution which is singular on the whole axis (0, t) ⊂ Q ∞ , is called a razor blade (see [START_REF] Vazquez | Different kinds of singular solutions of nonlinear parabolic equations[END_REF] for some examples). To this solution corresponds a universal estimate.

Lemma 2.2 Assume 1 < p < 2 and let v ∈ C(Q ∞ \ B R 0 × {0}) be a semigroup solution positive of (1.1) ∂ t v -∆ p v = 0 in Q ∞ (2.13)
which satisfies

lim t→0 K v(x, t)dx = 0, (2.14) for any compact set K ⊂ R N \ B R 0 . Then there exists c 1 = c 1 (N, p) > 0 such that sup 0≤τ ≤t {x:|x|>R} v(x, τ )dx ≤ c 1 t (R -R 0 ) N λ 1 2-p ∀R > R 0 , t > 0.
(2.15)

If we assume moreover that lim |x|→∞ v(t, x) = 0 locally uniformly with respect to t ≥ 0, then

v(x, t) ≤ Λ 1 t (|x| -R 0 ) p 1 2-p ∀(x, t) ∈ Q ∞ , |x| > R 0 , (2.16)
where Λ 1 is the value of the constant in (2.12) when N = 1.

Proof. The first estimate is a consequence of sup

0≤τ ≤t Bρ(a) v(x, t)dx ≤ c 2   B 2ρ (a) v(x, 0)dx + t ρ N λ 1 2-p   (2.17)
in [START_REF] Dibenedetto | Non-negative Solutions of the Evolution p-Laplacian Equation. Initial Traces and Cauchy Problem when 1 < p < 2[END_REF]Lemma III.3.1] under the assumption that v(., 0) is continuous with compact support. Actually this assumption is not used. In this proof the first step is the following estimate obtained by a suitable choice of test function:

sup 0≤τ ≤t B R (a) v(x, t)dx ≤ B 2R (a) v(x, 0)dx + c 3 R t 0 B R (a) |∇v| p-1 dx dτ (2.18)
valid for any a ∈ R N \ {(0, 0)} and R ≤ |a|/2. The second step to get (2.17) is to estimate the integral on the right-hand side by relation (I.4.2) in [6, Lemma I.4.1] with the same choice of ǫ. We apply estimate (2.17) with a sequence of points in a fixed direction e (with

|e| = 1) a = a k = 2 k (R -R 0 ) + R 0 e and ρ = ρ k = 2 k-1 (R -R 0 ) (actually we start with ρ < ρ k and let it grow up to ρ k ). Then we get sup 0≤τ ≤t Bρ k (a k ) v(x, t)dx ≤ c 4 2 - N(k-1) λ(2-p) t (R -R 0 ) N λ 1 2-p . (2.19)
Since the ball B ρ k (a k ) and B ρ k+1 (a k+1 ) are overlapping there exist a finite number of points {e j } d 1 j=1 and {e ′ j } d 2 j=1 (d 1 and d 2 depend only on N ) on the unit sphere such that

x ∈ R N : |x| ≥ R ⊂   d 1 j=1 ∞ k=1 B ρ k (2ρ k e j )     d 2 j=1 B R-R 0 2 (Re ′ j )   . Therefore sup 0≤τ ≤t {x:|x|>R} v(x, τ )dx ≤ c 4 d 1 ∞ k=0 2 -Nk λ(2-p) + d 2 2 N λ(2-p) t (R -R 0 ) N λ 1 2-p which is (2.15). Estimate (2.16) follows from comparison with the 1-dim form of v ∞ v ∞ (s, t) = Λ 1 t s p 1 2-p ∀s, t > 0.
(2.20)

For ǫ > 0, the function (x, t) → v ∞ (x 1 -R 0 -ǫ, t) + ǫ where x = (x 1 , ..., x N ) = (x 1 , x ′ ), is a solution of (2.13) in H 1,R 0 +ǫ × (0, ∞) where H 1,m = {x ∈ R N : x 1 > m}. For R large enough v(x, t) ≤ v ∞ (x 1 + R 0 + ǫ, t) + ǫ on the set ((H 1,R 0 +ǫ ∩ ∂B R ) ∪ (∂H 1,R 0 +ǫ ∩ B R )) × [0, T ]
for any T > 0, and for t = 0. By the maximum principle v(x, t)

≤ v ∞ (x 1 -R 0 -ǫ, t) + ǫ in (H 1,R 0 +ǫ ∩ B R ) × (0, T ].
Letting successively R → ∞, T → ∞ and ǫ → 0 and using the invariance of the equation by rotation implies (2.16).

Proposition 2.3 Let p > 2N N +1 and {v n } ⊂ C([0, ∞); L 1 (R N )) be a sequence of positive semigroup solutions of (2.13) on (0, ∞) such that v n (., 0) has support in B ǫn where ǫ n → 0. If R N v n (x, 0)dx = k n → k as n → ∞ then v n → v k locally uniformly in Q ∞ .
Proof. We first give the proof in the case 2N N +1 < p < 2. By a priori estimates, up to a subsequence v n converges locally uniformly in

Q ∞ to a solution v of (2.13) in Q ∞ . By Herrero-Vazquez mass conservation property [9, Theorem 2] (valid if p > 2N N +1 ) R N v n (x, t)dx = R N v n (x, 0)dx = k n .
By (2.16)

v n (x, t) ≤ Λ 1 t (|x| -ǫ n ) p 1 2-p ∀t > 0, ∀|x| > ǫ n . Since p 2-p > N , the function x → t (|x| -ǫ n ) p 1 2-p belongs to L 1 (R N \ B δ ), for any δ > ǫ n . Since v n (x, t) → v(x, t) uniformly in B δ , it follows by the dominated convergence theorem lim n→∞ R N v n (x, t)dx = R N v(x, t)dx = k. (2.21)
Because v is a positive solution with isolated singularity at (0, 0), it follows from [START_REF] Chen | Singular solution of the parabolic p-Laplacian with absorption[END_REF] that v = v k , solution of (1.14).

When p ≥ 2, the function v k (., t) has a compact support D kn (t) for any t > 0 and

D kn (t) ⊂ B Rn(t) where R n (t) = ǫ n + c 5 k p-2 p n t 1 N(p-2)+p ≤ ǫ * + c 5 k p-2 p * t 1 N(p-2)+p

Fundamental solutions

The following lemma is fundamental.

Lemma 2.4 Assume p > 2N

N +1 and f is a continuous nondecreasing function defined on R such that f (0) = 0. Then, for any k, R, T > 0,

∞ 1 f (s)s -p(N+1) N ds < ∞ =⇒ f (v k ) ∈ L 1 (B R × (0, T )).
(2.23)

Proof. The result is already proved in [START_REF] Kamin | Fundamental solutions and asymptotic behaviour for the p-Laplacian equation[END_REF] in the case p > 2. It is probably known in the case p = 2, but we have not found any reference. It appears to be new in the case

2N N +1 < p < 2.
Without any loss of generality we can assume R = T = 1. Case 1: p = 2. By linearity we can assume that k = (4π)

N 2 . Let I = B 1 ×(0,1) f (v k )dx dt = ω N 1 0 1 0 f t -N 2 e -r 2 4t r N -1 dr dt. Set s = t -N 2 e -r 2 4t
, then

I = 2 N -1 ω N 1 0 t -N 2 t -N 2 e -1 4t -ln s -ln t N 2 N-2 2 f (s)s -1 ds t N 2 dt ≤ 2 N -1 ω N 1 0 t -N 2 e -1 4t -ln s -ln t N 2 N-2 2 f (s)s -1 ds t N 2 dt ≤ 2 N -1 ω N (I 1 + I 2 )
where,

I 1 = e -1 4 0 -1 4 ln s 0 -ln s -ln t N 2 N-2 2 t N 2 dt s -1 f (s)ds = 2 N e -1 4 0 s (-4 ln s) N 2 0 (-ln τ ) N-2 2 τ 2 N dτ s -2-2 N f (s)ds, by setting τ = st N 2 . But s (-4 ln s) N 2 0 (-ln τ ) N-2 2 τ 2 N dτ ≤ c 6 (-ln τ ) N-2 2 τ N+2 N s (-4 ln s) N 2 0 ≤ c 6 s 1+ 2 N (-ln s) -2 1 + N 2 ln(-4 ln s) -ln s N-2 2 ≤ c 7 s 1+ 2 N (-ln s) -2 , thus I 1 ≤ c 8 e -1 4 0 s -1 (-ln s) -2 f (s)ds < ∞ by Duhamel's rule. Further I 2 ≤ ∞ e -1 4 s -2 N 0 -ln s -ln t N 2 N-2 2 t N 2 dt s -1 f (s)ds ≤ 2 N ∞ e -1 4 1 0 (-ln τ ) N-2 2 τ 2 N dτ s -2-2 N f (s)ds ≤ c 9 ∞ e -1 4 s -2-2 N f (s)ds,
for some c 9 = c 9 (N ) > 0. This implies the claim when p = 2. Case 2: 2N N +1 < p < 2. We set d * = -d. By rescaling we can assume that

C k = d * = 1. Therefore I = B 1 ×(0,1) f (v k )dxdt = ω N 1 0 1 0 f   t -λ 1 + r t λ N p p-1 p-1 p-2   r N -1 dr dt. Set s = t -λ 1 + r t λ N p p-1 p-1 p-2 , then r = t λ N (t λ s) p-2 p-1 -1 p-1 p and I = 2-p p ω N 1 0 t -λ t -λ (1+t - λp p-1 ) p-1 p-2 (t λ s) -1 p-1 t λ s p-2 p-1 -1 N(p-1) p -1 f (s)ds t 2λ dt = 2-p p ω N (I 1 + I 2 )
where

I 1 = 1 2 p-1 p-2 1 a(s) (t λ s) -1 p-1 t λ s p-2 p-1 -1 N(p-1) p -1
t 2λ dt f (s)ds 1) .

I 2 = ∞ 1 s -1 λ a(s) (t λ s) -1 p-1 t λ s p-2 p-1 -1 N(p-1) p -1 t 2λ dt f (s) and a(s) is the inverse function of t → t -λ (1 + t -λp p-1 ) p-1 p-2 . Clearly t -λ (1 + t -λp p-1 ) p-1 p-2 ≤ t 2λ(p-1) 2-p =⇒ a(s) ≥ s 2-p 2λ(p-
Therefore

I 1 ≤ 1 2 p-1 p-2 1 s 2-p 2λ(p-1) (t λ s) -1 p-1 t λ s p-2 p-1 -1 N(p-1) p -1 t 2λ dt f (s)ds ≤ 1 λ 1 2 p-1 p-2 s s p 2(p-1) (1 -τ 2-p p-1 ) N(p-1) p -1 τ 1 λ + N(p-2) p dτ s -2-1 λ f (s)ds. Since 1 λ + N (p-2) p > -1 and N (p-1) p -1 > -1, s s p 2(p-1) (1 -τ 2-p p-1 ) N(p-1) p -1 τ 1 λ + N(p-2) p dτ < 1 0 (1 -τ 2-p p-1 ) N(p-1) p -1 τ 1 λ + N(p-2) p dτ < ∞. Furthermore -2 -1 λ = -p -p N thus I 1 ≤ c 10 1 2 p-1 p-2 f (s)s -p(N+1) N ds.
We perform the same change of variable with I 2

I 2 ≤ ∞ 1 s -1 λ s 2-p 2λ(p-1) (t λ s) -1 p-1 t λ s p-2 p-1 -1 N(p-1) p -1 t 2λ dt f (s)ds ≤ 1 λ ∞ 1 1 s p 2(p-1) (1 -τ 2-p p-1 ) N(p-1) p -1 τ 1 λ + N(p-2) p dτ s -2-1 λ f (s)ds. Again 1 s p 2(p-1) (1 -τ 2-p p-1 ) N(p-1) p -1 τ 1+ N(p-2) p dτ < 1 0 (1 -τ 2-p p-1 ) N(p-1) p -1 τ 1 λ + N(p-2) p dτ < ∞, then I 2 ≤ c 11 ∞ 1 f (s)s -p(N+1) N ds.
Therefore (2.23) holds.

Notice that the assumption implies that

v k ∈ C(Q ∞ ) ∩ L ∞ (δ, ∞; L 1 (R N ) ∩ L ∞ (R N )) for every δ > 0. Proof of Theorem 1.1. Existence. Let ǫ > 0, Q ǫ,∞ = R N × (ǫ, ∞) and denote by u ǫ the solution of ∂ t u -∆ p u + f (u) = 0 in Q ǫ,∞ u(., ǫ) = v k (., ǫ) in R N . (2.24)
Since v k (., ǫ) is a smooth positive function belonging to L 1 (R N ) the function u ǫ is constructed by truncation. By the maximum principle

u ǫ (x, t + ǫ) ≤ v k (x, t + ǫ) ∀(x, t) ∈ Q ǫ,∞ . (2.25) For 0 < ǫ ′ < ǫ, u ǫ ′ (x, ǫ) ≤ v k (x, ǫ) = u ǫ (x, ǫ), thus u ǫ ′ (x, t + ǫ) ≤ u ǫ (x, t + ǫ) in Q ǫ,∞ . Set ũ = lim ǫ→0 u ǫ , then ũ ≤ v k in Q ∞ .
By the standard local regularity theory for degenerate equations, ∇u ǫ remains locally compact in (C 1 loc (Q ∞ )) N , thus ũ satisfies (1.1) in Q ∞ . In order to prove that

d dt R N u ǫ (x, s)dx + R N f (u ǫ (x, s))dx = 0
we recall that u ǫ can be obtained as the limit of thru the iterative implicit scheme (2.4)

with q ∈ [1, ∞] is arbitrary since u ǫ,0 ∈ L 1 (R N ) ∩ L ∞ (R N ).
For h > 0 we can write it under the form u ǫ,i -h∆ p u ǫ,i = -hf (u ǫ,i ) + u ǫ,i-1 .

By (2.7), and denoting by Ũǫ,h the piecewise constant function such that Ũǫ,h (jh) = u ǫ,j , we obtain since

u ǫ,0 = v k (ǫ) R N (u ǫ,i -v k (ǫ))(x)dx = - ih ǫ R N f ( Ũǫ,h (x))dxdt. ( 2

.26)

Letting h → 0 and i → ∞ such that ih = t > ǫ and using the uniform convergence, we obtain

R N u ǫ (x, t)dx - R N v k (x, ǫ)dx = - t ǫ R N f (u ǫ (x, s))dxdt.
(2.27) Since 0 ≤ u ǫ ≤ v k and v k (., t) has constant mass equal to k, we derive

R N u ǫ (x, t)dx -k ≤ t ǫ R N f (v k (x, s))dxdt. (2.28) Because f (v k ) ∈ L 1 (R N × (0, T ))
, we can let ǫ → 0, using the monotone convergence theorem, in order to get

R N u(x, t)dx -k ≤ t 0 R N f (v k (x, s))dxdt. (2.29)
This implies that

lim t→0 R N u(x, t)dx = k. (2.30) If φ ∈ C c (R N ), let ζ ∈ C ∞ c (R N ) such that 0 ≤ ζ ≤ 1, ζ = 1 on the support of φ and ζ(0) = 1. Then R N u(x, t)φ(x)dx = R N u(x, t)φ(x)ζ(x)dx = φ(0) R N u(x, t)dx + R N u(x, t)(φ(x)ζ(x) -φ(0))dx. Thus R N u(x, t)φ(x)dx -φ(0) R N u(x, t)dx ≤ R N v k (x, t) |φ(x)ζ(x) -φ(0)| dx.
Because |φ(x)ζ(x)φ(0)| is continuous and vanishes at zero and v k (., 0) = kδ 0 , it follows from (2.30)

lim t→0 R N u(x, t)φ(x)dx = kφ(0).
(2.31)

Uniqueness. The proof uses some ideas from [START_REF] Kamin | Fundamental solutions and asymptotic behaviour for the p-Laplacian equation[END_REF]Th 2.4]. Assume ũ is any nonnegative solution of problem (1.13), then, for any ǫ > 0 we denote by ṽǫ the solution of

∂ t v -∆ p v = 0 in Q ǫ,∞ v(., ǫ) = ũ(., ǫ) in R N . (2.32)
By the maximum principle ṽǫ ≥ ũ in Q ǫ,∞ . When ǫ → 0, ṽǫ converges locally uniformly to a solution ṽ of the same equation in Q ∞ . Furthermore, using again [9, Lemma 2],

R N ṽǫ (x, t + ǫ)dx = R N ũ(x, ǫ)dx.
By Fatou's Lemma and using the fact that

lim ǫ→0 R N ũ(x, ǫ)dx = k, we derive R N ṽ(x, t)dx ≤ k. (2.33)
Since ṽ ≥ ũ, equality holds in (2.33). Since the fundamental solution is unique [3, Th 4.1], it implies ṽ = v k and ũ ≤ v k . We end the proof as in [3, Th 4.1], using the L 1 -contraction mapping principle and the fact that any solution of (1.13) is smaller than v k : for t > s > 0, there holds

R N |u(x, t) -ũ(x, t)| dx ≤ R N |u(x, s) -ũ(x, s)| dx ≤ R N |u(x, s) -v k (x, s)| dx + R N |v k (x, s) -ũ(x, s)| dx ≤ R N (v k (x, s) -u(x, s))dx + R N (v k (x, s) -ũ(x, s))dx.
(2.34) When s → 0 the right-hand side of the last line goes to 0. This implies the claim.

The next result shows some geometric properties of the u k . Proof. It is sufficient to prove the result with the approximation u ǫ (., t). By (2.9), v k (., t) is radial and decreasing, therefore u ǫ (., t) is radial too by uniqueness. We notice that u ǫ is the increasing limit, when R → ∞, of the solution u ǫ,R of

   ∂ t u -∆ p u + f (u) = 0 in Q B R ǫ,∞ u = 0 in ∂B R × (ǫ, ∞) u(., ǫ) = v k (., ǫ) in B R . (2.35) For λ ∈ (0, R), we set Σ λ = B R ∩ {x = (2λ -x 1 , x ′ ) : x 1 > λ} ∩ B R and define w λ by w λ (x, t) = w λ (x 1 , x ′ , t) := u λ,ǫ,R (x) -u ǫ,R (x) = u ǫ,R (2λ -x 1 , x ′ , t) -u ǫ,R (x 1 , x ′ , t). If Q Σ λ ǫ,∞ = Σ λ × (ǫ, ∞), there holds    ∂ t w λ + Aw λ + d(x)w λ = 0 in Q Σ λ ǫ,∞ w λ ≥ 0 in ∂Σ λ × (ǫ, ∞) w λ (., ǫ) ≥ 0 in Σ λ .
(2.36)

where

d(x) = f (u λ,ǫ,R )-f (u ǫ,R ) u λ,ǫ,R -u ǫ,R if u λ,ǫ,R = u ǫ,R 0 if u λ,ǫ,R = u ǫ,R and Aw λ = -∆ p u λ,ǫ,R + ∆ p u ǫ,R .
Notice that d ≥ 0 since f is nondecreasing and A is elliptic [START_REF] Friedman | Singular Solutions of Some Quasilinear Elliptic Equations[END_REF]Lemma 1.3]. Furthermore the boundary data are continuous, therefore w λ ≥ 0. Letting λ → 0, changing λ in -λ and replacing the x 1 direction, by any direction going thru 0, we derive that u ǫ,R (., t) is radially decreasing. Letting R → ∞ yields to u ǫ (., t) is radially decreasing too.

In the next result we characterize positive solutions of (1.1) with an isolated singularity at t = 0 Proposition 2.6 Assume p > 2N N +1 and f is continuous nondecreasing function vanishing only at 0 and satisfying (1.12).

If u ∈ C(Q ∞ \ {(0, 0)}) is a positive semigroup solution of (1.1) in Q ∞ such that u(x, 0) = 0, for all x = 0 and lim t→0 R N u(x, t)dx < ∞, then there exists k ≥ 0 such that u = u k .
Proof. Using [11, Lemma 2.2 ] when p ≥ 2, or the proof of Theorem 1.1 when 2N N +1 < p < 2 jointly with the fact that

t → R N u(x, t)dx
is decreasing, we derive that u ≤ v m for some m ≥ 0 and there exists k ≥ 0 such that

lim t→0 R N u(x, t)dx = k. Since u(., 0) vanishes if x = 0, it implies lim t→0 R N u(x, t)φ(x)dx = kφ(0) ∀φ ∈ C c (R N ).
Therefore u satisfies (1.13). By uniqueness, u = u k .

Strong singularities

This section is devoted to study the limit of the sequence of the solutions u k to (1.13) as k → ∞ with f (s) = s α ln β (s + 1) where p > 2, α ∈ [1, p -1) and β > 0.

Proof of Theorem 1.3. By the comparison principle,

u k (x, t) ≤ v k (x, t) ≤ c 12 k (p-1)ℓ p-2 t -λ ,
where v k is the solution of (1.14) in Q ∞ and c 12 = c 12 (N, p) > 0 in (2.11). We set

θ k (t) = c α-1 12 k ℓ(α-1)(p-1) p-2 t -λ(α-1) ln β (c 12 k (p-1)ℓ p-2 t -λ + 1) (2.37) then ∂ t u k -∆ p u k + u k θ k (t) ≥ 0. ( 2
2-p (t)s ′ (t)∂ s w k (x, s) -∆ p w k (x, s) + b 1-p [b ′ (t) + b(t)θ k (t)]w k (x, s) ≥ 0. (2.39)
We choose the functions b and s such that

w k ≥ v k in R N × (0, s k,0 ). Hence u k (x, t) ≥ b(t)v k (x, s) = exp - t 0 θ k (τ )dτ s -λ c 13 k ℓ -c 14 s -pλ (p-1)N |x| p p-1 p-1 p-2 + . (2.41) Let δ 1 > ℓ(α-1)(p-1)
p-2 and 0 < δ 2 < 1λ(α -1). Using (2.37) there exists t 0 > 0 depending on δ 1 , δ 2 and k large enough, such that, for any t ∈ (0, t 0 ) there holds In order to obtain (2.47), it is sufficient to choose γ such that

t 0 θ k (τ )dτ ≤ c 15 k δ 1 t δ 2 ∀t ∈ (0, t 0 ) (2.
1 p -2 < γ < δ 2 δ 1 . (2.48)
Indeed, since α < p -1, we may choose δ 1 and δ 2 close enough ℓ(α-1)(p-1) p-2 and 1λ(α -1) respectively such that (2.48) holds true. When t has the form (2.46) where γ satisfies (2.48), from (2.41), (2.42)-(2.44) and the fact that U ≥ u k in Q ∞ , we deduce that U (x, t) ≥ c 16 t -λ exp c 17 ln(t -1 )c 15 t δ 2 -δ 1 γ

(2.49)

for every (x, t) ∈ B 1 × (0, t 0 ) with t 0 small enough and c 16 = c 16 (N, p), c 17 = c 17 (N, p, γ). Since γ satisfies (2.48), c 17 ln(t -1 )c 15 t δ 2 -δ 1 γ > 0 for every t ∈ (0, t 0 ). Therefore lim t→0 U (x, t) = ∞ uniformly with respect to x ∈ B 1 .

We next proceed as in [START_REF] Vazquez | Different kinds of singular solutions of nonlinear parabolic equations[END_REF]Lemma 3.1] to deduce that U (x, t) is independent of x and therefore it is a solution of (1.2). Since

J < ∞, U (x, t) = φ ∞ (t) for every (x, t) ∈ Q ∞ .
Theorem 1.4 is proved by the same arguments as Theorem 1.3, using the fact that U (x, t) is independent of x.

Non-Uniqueness

The next result shows that K = ∞ is the necessary and sufficent condition so that a local solution of (r

N -1 w ′ p-2 w ′ ) ′ = r N -1 f (w) ( 3
∞ α ds (F (s)) 1/p = ∞ (3.2)
for any α > 0.

Proof. The proof is is an extension to the case p = 2 of the one of [18, Lemma 2.1] for the case p = 2.

Step 1. We first assume that w is defined on a maximal interval [a, a * ) with a * < ∞ and lim r→a * w(r) = +∞. Since w is a nondecreasing function, w ′ ≥ 0. And hence we may write (3.1) under the following form

N -1 r (w ′ ) p-1 + (p -1)(w ′ ) p-2 w ′′ = f (w), which implies that (p -1)(w ′ ) p-2 w ′′ ≤ f (w) (3.3) and hence p -1 p (w ′ p ) ′ ≤ (F (w)) ′ .
Taking the integral over [a, r], we get

p -1 p [(w ′ ) p (r) -(w ′ ) p (a)] ≤ F (w(r)) -F (w(a)) ≤ F (w(r)).
Since f is positive on (0, ∞), F (s) → ∞ when s → ∞, thus there exists ã ∈ (a, a * ) such that

0 < w ′ (r)) p ≤ p 2(p -1) F (w(r)) =⇒ w ′ (r) F (w(r)) 1/p ≤ p 2(p -1) 1/p ∀r ∈ [ã, a * ).
Taking the integral over [ã, r], we obtain

w(r) w(ã) ds F (s) 1/p ≤ p 2(p -1) 1/p (r -ã). Letting r → a * yields to ∞ w(ã) ds F (s) 1/p ≤ p 2(p -1) 1/p (a * -ã) < ∞ and (3.2) is not satisfied. Step 2. We assume that ∞ α ds F (s) 1/p < ∞
for some α > 0, and we fix A > a. By [17, Theorem 1] there exists a function γ defined on (a, A) such that w(r) < γ(r) ∀r ∈ (a, A)

for any solution of (3.1) on (a, A). Moreover, γ can be assumed convex, and lim and γ ′ (a) > 0, it is clear that w(r * ) = γ(r * ) for some r * < A and w(r) > γ(r) for r ∈ (r * , r * + ǫ), so w can not be defined on the whole (a, A), and there exists a * < A such that lim r→a * w(r) = ∞. Proof of Theorem 1.5 By the Picard-Lipschitz fixed point theorem in the case 1 < p < 2 and [8, Th 5.2] in the case p ≥ 2, there exists a unique solution w a to (1.16) defined on a maximal interval [0, r a ) and w a is an increasing function. Since Keller-Osserman estimate does not hold, by Lemma 3.1, the solution can be continued on the whole [0, +∞) and global uniqueness follows from the local uniqueness. The function r → w a (r) is increasing and

w a (r) ≥ a + p -1 p f (a) N 1 p-1 r p p-1 and w ′ a (r) ≥ f (a) N 1 p-1 r 1 p-1
for any r > 0.

Proposition 3.2 Assume p > 2N/(N + 1), f is locally Lipschitz continuous and K = ∞ hold. For any positive function

u 0 ∈ C(Q ∞ ) which satisfies w a (|x|) ≤ u 0 (x) ≤ w b (|x|) ∀x ∈ R N (3.4)
for some 0 < a < b, there exists a positive function

u ∈ C(Q ∞ ) solution of (1.1) in Q ∞ and satisfying u(., 0) = u 0 in R N . Furthermore w a (|x|) ≤ u(x, t) ≤ w b (|x|) ∀(x, t) ∈ Q ∞ . (3.5)
Proof. Clearly w a and w b are ordered solutions of (1.1). We denote by u n the solution to the initial-boundary problem

   ∂ t u n -∆ p u n + f (u n ) = 0 in Q n := B n × (0, ∞) u n (x, 0) = u 0 (x) in B n u n (x, t) = (w a (|x|) + w b (|x|))/2 in ∂B n × (0, ∞).
0 ≤ u 0 (x) ≤ w b (|x|) ∀x ∈ R N (3.7)
for some 0 < b, there exists a positive solution

u ∈ C(Q ∞ ) of (1.1) in Q ∞ satisfying u(., 0) = u 0 in R N and u(x, t) ≤ min{w b (|x|), φ ∞ (t)} ∀(x, t) ∈ Q ∞ . (3.8)
Proof. For any R > 0, let u R be the solution of

∂ t u R -∆ p u R + f (u R ) = 0 in Q ∞ u R (x, 0) = u 0 (x)χ B R (x) in R N .
The functions φ ∞ and w b are solutions of (1.1) in Q ∞ , which dominate u R at t = 0, therefore, by the maximum principle,

min{φ ∞ (t), w b (|x|)} ≥ u R (x, t) ∀(x, t) ∈ Q ∞ . (3.9)
The mapping R → u R is increasing, jointly with (3.9) it implies that there exists a solution u := lim

R→∞ u R of (1.1) in Q ∞ which satisfies u(x, 0) = u 0 (x) in R N . Letting R → ∞ in (3.9) yields to (3.8).
Proof of Theorem 1.6. Combining Proposition 3.2 and Proposition 3.3 we see that there exist two solutions u and u with the same initial data u 0 , which are ordered and different since lim

|x|→∞ u(x, t) = ∞ and lim |x|→∞ u(x, t) ≤ φ ∞ (t) < ∞ for all t > 0.

Estimate and stability

In this section we assume that Ω is a domain in R N , possibly unbounded, 0 < T ≤ ∞ and set Q Ω T := Ω × (0, T ) and Q T := R N × (0, T ). We denote by M(Ω) the set of Radon measures in Ω and by M + (Ω) its positive cone.

Definition 4.1 A nonnegative function u is called a weak solution of (1.1) in Q Ω T if u, |∇u| p , f (u) ∈ L 1 loc (Q Ω T ) and T 0 Ω -G(u)∂ t ϕ + |∇u| p-2 ∇u.∇(g(u)ϕ) + f (u)g(u)ϕ dxdt = 0 (4.1)
for any ϕ ∈ C ∞ c (Q Ω T ) and any function g ∈ C(R) ∩ W 1,∞ (R) where G ′ (r) = g(r)
. The next results are obtained by adapting the proofs in [START_REF] Bidaut-Véron | Initial trace of solution of some quasilinear parabolic equations with absorption[END_REF].

Regularity Properties

The following integral estimates are essentially [2, Prop 2.1] with u q replaced by f (u).

Proposition 4.2 Assume p > 1. Let δ < 0, δ = -1 and 0 < t < θ < T . Let u be a nonnegative weak solution of (1.1) in Q Ω T . For any nonnegative function ζ ∈ C ∞ c (Ω) and τ > p, 1 δ + 1 Ω (1 + u(x, t)) 1+δ ζ τ (x)dx + |δ| 2 θ t Ω (1 + u) δ-1 ζ τ |∇u| p dx dt ≤ 1 δ + 1 Ω (1 + u(x, θ)) 1+δ ζ τ (x)dx + θ t Ω (1 + u) δ f (u)ζ τ dx dt + c 18 θ t Ω (1 + u) δ+p-1 ζ τ -p |∇ζ| p dx dt. (4.2)
and

Ω (1 + u(x, t))ζ τ (x)dx ≤ Ω (1 + u(x, θ))ζ τ (x)dx + θ t Ω f (u)ζ τ dx dt + τ θ t Ω (1 + u) δ-1 ζ τ |∇u| p dx dt + τ θ t Ω (1 + u) (1-δ)(p-1) ζ τ -p |∇ζ| p dx dt. (4.3)
Conversely,

1 4 Ω u(x, θ)ζ τ (x)dx + 1 2 θ t Ω f (u)ζ τ dx dt ≤ c 19 + Ω u(x, t)ζ τ (x)dx + τ θ t Ω ζ τ -1 |∇u| p-1 |∇ζ| dx dt (4.4)
and

1 4 Ω (1 + u(x, θ))ζ τ (x)dx + 1 2 θ t Ω f (u)ζ τ dx dt ≤ Ω (1 + u(x, t))ζ τ (x)dx + τ θ t Ω (1 + u) δ-1 ζ τ |∇u| p dx dt + τ θ t Ω (1 + u) (1-δ)(p-1) ζ τ -p |∇ζ| p dx dt + c 20 (4.5
) [START_REF] Vazquez | Isolated singularities of some semilinear elliptic equations[END_REF][START_REF] Vazquez | Different kinds of singular solutions of nonlinear parabolic equations[END_REF][START_REF] Véron | Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces[END_REF].

where c i = c i (p, f, δ, τ ) (i =
The next result is the keystone for the existence of an initial trace in the class of Radon measures. It is essentially [2, Prop 2.2] with u q replaced by f (u), but we shall sketch its proof for the sake of completeness.

sup t∈(0,θ] U u(x, t)dx < ∞, (4.6)
θ 0 U f (u)dx dt < ∞, (4.7)
θ 0 U |∇u| p-1 dx dt < ∞. (4.8)
Then the third one holds for any U ⊂⊂ Ω. Moreover,

θ 0 U u σ dx dt < ∞ ∀σ ∈ (0, q c ) (4.9
)

and θ 0 U |∇u| r dx dt < ∞ ∀r ∈ (0, N N + 1 q c ) (4.10)
where q c = p -1 + p/N . Finally, there exists a Radon measure µ ∈ M + (Ω) such that for

any ζ ∈ C c (Ω), lim t→0 Ω u(x, t)ζ(x)dx = Ω ζ(x)dµ (4.11)
and u satisfies

θ 0 Ω (-u∂ t ϕ + |∇u| p-2 ∇u.∇ϕ + f (u)ϕ)dx dt = Ω ϕ(x, 0)dµ - Ω u(x, θ)ϕ(x, θ)dx (4.12
)

for any 0 < θ < T and ϕ ∈ C ∞ c (Ω × [0, T )). Proof.
Step 1: Assume (4.6) and (4.8) holds. Let ζ and τ as in Proposition 4.2, there holds

Ω (1 + u(x, t))ζ τ dx = Ω (1 + u(x, θ))ζ τ dx + θ t Ω f (u)ζ τ dxdt + τ θ t Ω ζ τ -1 |∇u| p-2 ∇u.∇ζdxdt. (4.13)
It follows that f (u) ∈ L 1 ((0, θ), L 1 loc (Ω)).

Step 2: Assume that (4.7) and (4.8) hold. Then (4.6) follows from (4.13).

Step 3: Assume that (4.6) and (4.7) hold. Let δ ∈ (max(1p, -1), 0) be fixed. From (4.2), we get for any 0 < t < θ,

|δ| 2 θ t Ω (1 + u) δ-1 |∇u| p ζ τ dx dt ≤ 1 δ + 1 Ω (1 + u(x, θ)) δ+1 ζ τ dx + θ t Ω (1 + u) δ f (u)ζ τ dx dt + c 18 θ t Ω (1 + u) δ+p-1 ζ τ -p |∇ζ| p dx dt. (4.14)
If p ≤ 2, then (1 + u) δ+p-1 ≤ 1 + u. Consequently, by (4.6),

θ 0 Ω (1 + u) δ+p-1 ζ τ -p |∇ζ| p dx dt < θ 0 Ω (1 + u)ζ τ -p |∇ζ| p dx dt < ∞, (4.15)
which, along with (4.7) and (4.14), implies that

θ t Ω (1 + u) δ-1 |∇u| p ζ τ dx dt < c 21 . (4.16)
If p > 2, we choose δ ∈ (1p, 2p), δ = -1, ζ and τ as in Proposition 4.2, then (4.2) remains valid. From the inequality (1 + u) 1+δ < 1 + u and (4.6), we find that

1 |δ + 1| Ω (1 + u(x, t)) 1+δ ζ τ (x)dx < 1 |δ + 1| Ω (1 + u(x, t))ζ τ (x)dx < c 22 .
Hence, by (4.2), (1 + u(x, t)) σ dx < c 23 for any σ ∈ (0, q c ) with q c = p -1 + p/N , which leads to (4.9). Next for 0 < r < p, and any δ < 0, we find

|δ| 2 θ t Ω (1 + u) δ-1 ζ τ |∇u| p dx dt ≤ 1 δ + 1 Ω (1 + u(x, θ)) δ+1 ζ τ dx + θ t Ω (1 + u) δ f (u)ζ τ dx dt + c 18 θ t Ω (1 + u) δ+p-1 ζ τ -p |∇ζ| p dx dt + c 22 . (4.17) Since δ < 2 -p, δ + p -1 < 1, hence (1 + u) δ+p-1 ≤ 1 + u.
θ 0 U |∇u| r dx ≤ θ 0 U (1 + u) δ-1 |∇u| p dx dt r p × θ 0 U (1 + u) (1-δ)r p-r dx dt p-r p . (4.18)
Thus, if r ∈ (0, N q c /(N + 1)), this proves (4.10); furthermore, since p -1 < N q c /(N + 1), we obtain (4.8).

Step 4: End of the proof. Now we use (4.1) with g = 1, for any ζ ∈ C ∞ c (Ω) and any 0 < t < θ < T ,

Ω u(x, t)ζ(x)dx = Ω u(x, θ)ζ(x)dx + θ t Ω |∇u| p-2 ∇u.∇ζ + f (u)ζ dx dt. (4.19)
Because the right-hand side of (4.19) has a finite limit when t → 0, the same holds with t → Thus, letting t → 0 in (4.20) implies (4.12).

Next we consider the the following problems where µ ∈ M + (Ω). The solutions are considered in the entropy sense (see [START_REF] Segura De Leon | Regularity for entropy solutions of parabolic p-Laplacian type equations[END_REF] and [START_REF] Li | Regularity for entropy solutions of a class of parabolic equations with irregular data[END_REF]).

   ∂ t u -∆ p u + f (u) = 0 in Q Ω T , u = 0 in ∂Ω × (0, T ) u(., 0) = µ in Ω.
We recall that for q ≥ 1 and Θ ⊂ R d open, the Marcinkiewicz space (or weak Lebesgue space) M q (Θ) is the set of all locally integrable functions u : Θ → R such there exists C ≥ 0 with the property that for any measurable set E ⊂ Θ,

E |u|dy ≤ C|E| 1-1 q . (4.22)
The norm of u in M q (Θ) is the smallest constant such that (4.22) holds for any measurable set E (see [START_REF] Segura De Leon | Regularity for entropy solutions of parabolic p-Laplacian type equations[END_REF], [START_REF] Li | Regularity for entropy solutions of a class of parabolic equations with irregular data[END_REF] for more details). Here dy denotes the Lebesgue measure in R d , although any positive Borel measure can be used. We recall the following result of Segura de Leon and Toledo [START_REF] Segura De Leon | Regularity for entropy solutions of parabolic p-Laplacian type equations[END_REF]Th 2] and Li [13, Th 1.1] dealing with entropy solutions with initial data in L 1 . However such solutions coincide with the semi-group solutions because of uniqueness.

Proposition 4.4 Assume p > 2N N +1 , Ω ⊂ R N is any open subset and, h ∈ L 1 (Q Ω T )and µ ∈ L 1 + (Ω). Let v ∈ C([0, T ; L 1 (Ω)) be the entropy solution to problem    ∂ t v -∆ p v = h in Q Ω T v = 0 in ∂Ω × (0, ∞) v(., 0) = µ in Ω. (4.23) Then v ∈ M p-1+ p N (Q Ω T ), ∇v ∈ M p-N N+1 (Q Ω T )
and there holds

v M p-1+ p N (Q Ω T ) + ∇v M p-N N+1 (Q Ω T ) ≤ c 25 , (4.24)
for some c 25 > 0 depending on p, N , µ L 1 (Ω) and h L 1 (Q Ω T ) .

Stability

Let {µ n } ⊂ L 1 + (R N ) be a sequence converging to µ in weak sense of measures, then

µ n L 1 (R N ) ≤ c * ,
v µn (., t) L ∞ (R N ) ≤ c 26 t -N N(p-2)+p µ n p N(p-2)+p L 1 (R N ) ∀t > 0,
where c 26 = c 26 (N, p) > 0. Thus It follows from (4.24) and (4.25) that

u µn (., t) L ∞ (R N ) ≤ c 26 t -N N(p-2)+p µ n p N(p-2)+p L 1 (R N ) ≤ c 27 t -N N(p-2)+p
u µn M p-1+p/N (Q T ) ≤ c 25 µ n p+N 1+p(N-1) L 1 (R N ) ≤ c 28 (N, p, c * ).
(4.27) By (4.26) and the regularity theory of degenerate parabolic equations [START_REF] Dibenedetto | Degenerate Parabolic Equations[END_REF], we derive that the sequence {u µn } is equicontinuous in any compact subset of Q T . As a consequence, there exist a subsequence, still denoted by {u µn } and a function u such that {u µn } converges to u locally uniformly in Q T .

Lemma 4.5 The sequence f (u µn ) converges strongly to f (u) in L 1 (Q T ). Furthermore, {u n } converges strongly to u in L q loc (Q T ) for every 1 ≤ q < q c . Proof. Since u µ → u a.e in Q T , by Vitali's theorem, it is sufficient to show that the sequence {f (u µn )} is uniformly integrable. Let E be a Borel subset of Q T and let R > 0.

Then, since f is increasing,

E f (u µn )dx dt = E∩{u µn≤R } f (u n )dx dt + E∩{uµ n >R} f (u µn )dx dt ≤ f (R) E dx dt + E∩{uµ n >R} f (u µn )dx dt. For λ ≥ 0, we set B n (λ) = {(x, t) ∈ Q T ) : u µn > λ} and a n (λ) = Bn(λ) dx dt. Then E∩{uµ n >R} f (u µn )dx dt ≤ {uµ n ≥R} f (u µn )dx dt = - ∞ R f (λ)da n (λ) (4.28) and - ∞ R f (λ)da n (λ) ≤ f (R)a n (R) + ∞ R a n (λ)df (λ).
It follows from (4.27) that

a n (λ) ≤ c 25 µ n p+N 1+p(N-1) M + (R N ) λ -(p-1+ p N ) ≤ c 29 λ -(p-1+ p N ) .
and the proof is similar to the one of [2, Th 5.

∇u µn → D strongly in (L s loc (Q T )) N , ∀s ∈ [1, s c ), {|∇u µn | p-2 ∇u µn } → |D| p-2 D strongly in (L σ loc (Q T )) N , ∀σ ∈ [1, σ c ). (4.32)
which implies ∇u = D and the conclusion of the lemma follows.

Proof of Theorem 1.7.

Step 1. For any ζ ∈ C ∞ c (R N ) and t > 0, we have

R N u µn (x, t)ζ(x)dx + t 0 R N (|∇u µn | p-2 ∇u µn ∇ζ + f (u µn )ζ)dx dt = R N µ n (x)ζ(x)dx
By Lemma 4.5 and Lemma 4.6, up to the extraction of a subsequence, we can pass to the limit in each term and get

R N u(x, t)ζ(x)dx + t 0 R N (|∇u| p-2 ∇u∇ζ + f (u)ζ)dx dt = R N ζdµ. Letting t → 0 yields lim t→0 R N u(x, t)ζ(x)dx = R N ζ(x). (4.33) For any ϕ ∈ C ∞ c (R N × [0, ∞)) and θ > 0, we have θ 0 R N (-u µn ∂ t ϕ + |∇u µn | p-2 ∇u µn .∇ϕ + f (u µn )ϕ)dx dt = R N ϕ(0, x)µ n (x)dx - R N u µn (x, θ)ϕ(x, θ)dx.
(4.34)

By the previous convergence results, we can pass to the limit in (4.34) to obtain

θ 0 R N (-u∂ t ϕ + |∇u| p-2 ∇u∇ϕ + f (u)ϕ)dx dt = R N ϕ(., 0)dµ - R N u(., θ)ϕ(., θ)dx. (4.35)
Step 2: u is a weak solution. By (4.26)

sup{ u µn (., t) L ∞ (R N ) , u(., t) L ∞ (R N ) } ≤ c 27 t - N N(p-2)+p ∀t ∈ (0, T ]. Let ζ ∈ C ∞ c (R N ).
Since {u µn (., θ)} converges locally uniformly to u(., θ) in R N , for any θ > 0, there holds

1 2 R N (u µn -u µm ) 2 (., T )ζdx dt + T θ R N (f (u µn ) -f (u µm ))(u µn -u µm )ζdx dt + T θ R N (|∇u µn | p-2 ∇u µn -|∇u µm | p-2 ∇u µm ).∇(u µm -u µn )ζdx dt ≤ 1 2 R N (u µn -u µm ) 2 (., θ)ζdx dt + T θ R N |∇u µn | p-
2 R N (u µn -u) 2 (., T )ζdx dt + T θ R N (f (u µn ) -f (u))(u µn -u)ζdx dt + T θ R N (|∇u µn | p-2 ∇u µn -|∇u| p-2 ∇u).∇(u µn -u)ζdx dt ≤ 1 2 R N (u µn -u) 2 (., θ)ζdx dt + T θ R N |∇u µn | p-
∈ C ∞ c (Q T ) and consider 0 < θ < T and U ⊂⊂ R N such that supp ϕ ⊂ (θ, T ) × U . Let g ∈ C(R N ) ∩ W 1,∞ (R N ) where G ′ (r) = g(r). Multiplying the equation in (4.21) (with initial data µ = µ n ) by g(u µn )ϕ, we obtain T 0 R N (-G(u µn )∂ t ϕ + |∇u µn | p g ′ (u µn )ϕ)dx dt + g(u µn ) |∇u µn | p-2 ∇u µn .∇ϕ + T 0 R N g(u µn )f (u µn )dx dt = 0. (4.40)
By Lemma 4.5 and (4.37), we can pass to the limit in each term. As a consequence, u is a weak solution.

Step 3: Stability. Assume that {µ n } is a sequence of functions in L 1 + (R N ) with compact support, which converges to µ ∈ M b + (R N ) in the dual sense of C(R N ), then µ n L 1 (R N ) is bounded independently of n. By the same argument as in step 1 and step 2, we can pass to the limit in each term of (4.40), hence the conclusion follows. Proof. By contradiction we assume that (4.42) does not hold. Then there exist

A 1 > 0 such that sup τ ∈(0,T ) B 8r u(x, τ ) = A 1 . (4.43)
Step 1: We claim that u ∈ L ∞ (Q B 2r T ). Since u is a positive subsolution of the equation in (2.13), by [5, Theorem 4.2, Chapter V], there exists a constant c 30 = c 30 (N, p) such that for every x 0 ∈ R N , 0 < θ ≤ t 0 < T and σ ∈ (0, 1), there holds sup Kσρ×(t 0 -σθ,t 0 ) u ≤ c 30 θ where K ρ (x 0 ) is the cube centered at x 0 and wedge 2ρ, i.e., K ρ (x 0 ) = {x ∈ R N : max 1≤i≤N

x ix i 0 < ρ}.

We choose x 0 = 0, t 0 = θ = t, σ = 1/2 and ρ = 4r, then (4.44) becomes

in Q ∞ .
Proof. By translation we may suppose that y = 0. Since 0 ∈ S(u), for any η > 0 small enough lim t→0 Bη u(x, t)dx = ∞. for some bounded open subset G ⊂ R N , then u(x, t) ≥ φ ∞ (t).

Theorem 1 . 9

 19 Assume p > 2 and (1.12) holds and u is a positive solution of (1.1) in Q ∞ . I-If J < ∞ and K = ∞ are verified and lim k→∞ u k = φ ∞ . Then either tr R N (u) is the Borel measure infinity ν ∞ which satisfies ν ∞ (O) = ∞ for any non-empty open subset O ⊂ R N , or is a positive Radon measure µ on R N . II-If J = ∞ and K = ∞ are verified and lim k→∞ u k = ∞. Then tr R N (u) is a positive Radon measure µ on R N

( 2

 2 .22) where c 5 = c 5 (N, p) > 0, ǫ * = sup{ǫ n ; n ∈ N} and k * = sup{k n ; n ∈ N}. Using Lebesgue dominating theorem we obtain again (2.21).

Proposition 2 . 5

 25 The solution u = u k of problem (1.15) is radial and nonincreasing with respect to |x|.

  b

b 2 -

 2 p (t)s ′ (t) = 1 and b ′ (t) + b(t)θ k (t) ∂ s w k -∆ p w k ≥ 0 in R N × (0, s k,0) with some s k,0 > 0 and w k (., 0) = kδ 0 . It follows by comparison principle that

. 1 )Lemma 3 . 1

 131 can be continued as a global solution. More precisely, Every positive and increasing solution of (3.1) defined in an interval [a, a * ] to the right of a > 0 can be continued as a solution of (3.1) on [a, +∞) if and only if f satisfies

  ) = +∞. If w is a solution of (3.1) on (a, a + ǫ) such that w(a) > min a<r<A γ(r)

( 3 . 6 )

 36 By the maximum principle, u n satisfies (3.5) in Q n . Using locally parabolic equation regularity [5, Th 1.1, chap III] if p ≥ 2 or [5, Th 1.1, chap IV] if 1 < p < 2, we derive that the set of functions {u n } is eventually equicontinuous on any compact subset of Q ∞ . Using a diagonal sequence, combined with Proposition 4.4, we conclude that there exists a subsequence {u n k } which converges locally uniformly in Q ∞ to some weak solution u ∈ C(Q ∞ ) which has the desired properties. Proposition 3.3 Assume p > 2N/(N + 1), f is locally Lipschitz continuous and J = ∞ and K = ∞ hold. Then for any u 0 ∈ C(R N ) which satisfies

Proposition 4 . 3

 43 Let u be a nonnegative solution of (1.1) in Q Ω T . Let 0 < θ < T . Assume that two of the three following conditions holds, for any open set U ⊂⊂ Ω:

Ω

  u(x, t)ζ(x)dx. The mapping ζ → lim t→0 Ω u(x, t)ζ(x)dx is a positive linear functional ℓ Ω on the space C ∞ c (Ω). By a partition of unity it can be extended in a unique way as a Radon measure µ ∈ M + (Ω) and (4.11) holds.

Finally, let 0

 0 < t < θ be fixed, g = 1 and ϕ ∈ C ∞ c (Q Ω T ), thus θ t Ω (-u∂ t ϕ + |∇u| p-2 ∇u.∇ϕ + f (u)ϕ)dx dτ = Ω u(x, t)ϕ(x, 0)dx -Ω u(x, θ)ϕ(x, θ)dx.

  , t)(ϕ(x, t)ϕ(x, 0))dx ≤ c 24 t Ω u(x, t)dx. By (4.11), letting t → 0 yields Ω u(x, t)ϕ(x, t)dx → Ω ϕ(x, 0)dµ.

  where c * depends only on N, p and µ M(R N ) . Denote by u µn (resp. v µn ) the solution to problem (4.21) (resp. (4.23) with h ≡ 0) with the initial data µ n . Then the following estimate holds 0 ≤ u µn ≤ v µn .

( 4 .

 4 26)for every t > 0, where c 27 = c 27 (N, p, c * ) > 0.

Lemma 4 . 7 0

 470 Assume p > 2. Let u ∈ C(Q T ) be a positive weak solution of (1.1) in Q T . Assume that there exists r > 0 such that T Br |∇u| p-1 dx dt = ∞. (4.41) Then sup τ ∈(0,T ) B 8r u(x, τ ) = ∞. (4.42)

sup K 2r ×( t 2 ,t) u ≤ 2 N 2 NStep 2 :J 1 ( 2 p

 22212 B 2r ⊂ K 2r and K 4r ⊂ B 8r , from (4.43) and (4.45), we obtain that supB 2r ×(0,T ) u ≤ Let ζ ∈ C ∞ c (R N ) such that ζ ≥ 0 in R N , ζ = 1 in B r and |∇ζ| ≤ 1/r. We show that ζ p |∇u| p dx dτ < ∞,

For ǫ > 0 Lemma 5 . 2

 052 , denote M ǫ,η = Bη u(x, ǫ)dx. For any m > m η = inf σ>0 M σ,η there exists ǫ = ǫ(m, η) such that m = M ǫ,η and lim η→0 ǫ(m, η) = 0. Let ũη be the solution to the problem∂ t ũη -∆ p ũη + f (ũ η ) = 0 in Q ∞ ũη (x, 0) = u(x, ǫ)χ Bη in R Nwhere χ Bη is the characteristic function of B η . By the maximum principle ũη ≤ u in R N × (ǫ, ∞). By Theorem 1.7 v η converges to u k when η goes to zero. Letting m go to infinity yields (5.1).Proof of Theorem 1.2 The conclusion follows directly from Proposition 5.1.5.2The Keller-Osserman condition does not hold Assume p > 2, (1.12) and J < ∞ are satisfied and limk→∞ u k = φ ∞ . If u is a positive solution of (1.1) in Q ∞ which satisfies lim sup t→0 G u(x, t)dx = ∞,(5.2)

  ∞ holds, there exists the solution φ ∞ of (1.2). The sequence {u k } is increasing and is bounded from above by φ ∞ , then the function U (x, t) := lim

	Since J < -pλ (p-1)N >	1 2	c 13 k ℓ ⇐⇒ k >	2c 14 c 13	1 ℓ	s(t) -1 p-2 .	(2.44)
	By (2.43), we only need to choose t such that
	k ≥	2c 14 c 13	1 ℓ	t	-1 p-2 exp c 15 k δ 1 t δ 2 .	(2.45)
	We choose t under the form	t = k	-1 γ with γ > 0,	(2.46)
	then (2.45) becomes
	t -γ ≥	2c 14 c 13			1 ℓ	t	-1 p-2 exp c 15 t δ 2 -δ 1 γ .	(2.47)
								(2.43)

42) with c 15 = c 15 (c i , α, β, p, N ) > 0. It follows from (2.40) and (2.42) that

t exp -(p -2)c 15 k δ 1 t δ 2 ≤ s(t) ≤ t. k→∞ u k (x, t) satisfies U (x, t) ≤ φ ∞ (t) for every (x, t) ∈ Q ∞ .

We restrict x ∈ B 1 and we choose t such that c 13 k ℓc 14 s(t)

  1-step2]. Up to the extraction of a subsequence, {∇u µn } converges a.e. to some D = (D 1 , ..., D N ) in Q T . Consequently, {|∇u µn | p-2 ∇u µn } converges a.e. to |D| p-2 D in Q T and, by Vitali's theorem,

  2 ∇u µn -|∇u µm | p-2 ∇u µm |u µmu µn | |∇ζ| dx dt.

			(4.36)
	This implies directly	∇u µn → ∇u in L p loc (Q T ),	(4.37)
	by Lemma 4.6 when p ≥ 2. When 1 < p < 2, we derive by Fatou's lemma
	1

  2 ∇u µn -|∇u| p-2 ∇u |u µm -u| |∇ζ| dx dt.

										(4.38)
	Using again Lemma 4.6, it implies
	lim n→∞	θ	T	R N	|∇u µn | p ζdx dt =	θ	T	R N	|∇u| p ζdx dt.	(4.39)

Since ∇u µn ⇀ ∇u weakly in L p loc (Q T ), it implies again that (4.37) holds true. At end, let ϕ

  ζ p and then integrating on R N × [ǫ, t] with 0 < ǫ < t, we get The dichotomy result Theorem 1.8 is a consequence of Proposition 4.3 and Lemma 4.7. Proof of Theorem 1.8 By translation we may suppose that y = 0. Case 1: there exists an open neighborhood U of 0 such that (4.7) and (4.8) hold true. Then the statement (ii) follows from Proposition 4.3.Case 2: for any open neighborhood U of 0, (4.7) or (4.8) does not holds. We first suppose that (4.8) does not hold. We can choose r > 0 such that B 8r ⊂ U and (4.41) holds. Then the statement (i) follows from Lemma 4.7. Suppose next that (4.8) holds but (4.7) does not hold, then Proposition 4.3 implies that (4.6) does not hold and the statement (i) follows.Proposition 5.1 Assume p > 2 and f is nondecreasing and satisfies (1.12). Let u is a positive weak solution of (1.1) in Q ∞ with initial trace (S, µ). Then for every y ∈ S, U y (x, t) := U (xy, t) ≤ u(x, t)(5.1)

	It follows from (4.48) and (4.49) that
	2(p-1) p p ζ p |∇u| p dx dτ -2 (u(x, ǫ) + 1) 2(p-1) p ζ p dx dx dτ < ∞. 2(p -1) B 2r (u + 1) ǫ B 2r t ≤ p + p 2p 2 p-1 t (u + 1) p 2 -2 p p -2 0 B 2r p-2 Multiplying (1.1) by (u + 1) p -2 2p By (4.46),	|∇ζ| p dx dτ.	(4.47) (4.50)
	and	p 2(p -1) B 2r ǫ∈(0,T ) B 2r (u(x, t) + 1) sup (u(x, ǫ) + 1) 2(p-1) p ζ p dx + 2(p-1) p -2 p p ζ p dx ≤ c 31 (N, p, r, ζ, A 2 ) t ǫ B 2r (u + 1) -2 p ζ p |∇u| p dx dτ
		0 B 2r t	+ (u + 1) ζ p dx 0 B 2r 2(p-1) p ǫ B 2r t t |∇ζ| p dx dτ ≤ r -p 2(p -1) B 2r p (u(x, ǫ) + 1) (u + 1) = p 2 -2 p	(u + 1)	p-2 p f (u)ζ p dx dτ
					t
				-p		(u + 1)
					ǫ B 2r
	which implies that
		p -2 p	t ǫ B 2r ≤ 2(p -1) B 2r (u + 1) -2 p ζ p |∇u| p dx dτ p (u(x, ǫ) + 1) 2(p-1) p -p t (u + 1) p-2	ζ p dx	p-1 p (J 2 (t))	1 p .	(4.48)
	By step 2, we deduce that	ǫ B 2r
	By Young's inequality, p t (u + 1) 0 T	p-2 B 2r p ζ p-1 |∇u| p-1 |∇ζ| dx dtτ |∇u| p-1 ζ p-1 dx dt < c 35 (N, p, r, T, ζ),	(4.53)
	ǫ B 2r ≤ which contradicts (4.41). p -2 2p 5 Initial trace	t ǫ B 2r + p	(u + 1) 2p 2 p -2	-2 p ζ p |∇u| p dx dτ ǫ B 2r p-1 t (u + 1)	p 2 -2 p	|∇ζ| p dx dτ.	(4.49)
	5.1 The dichotomy theorem

p p-2 p ζ p-1 |∇u| p-2 ∇u∇ζdx dτ, p ζ p-1 |∇u| p-2 ∇u∇ζdx dτ.

p 2 -2 p dx dt ≤ c 32 (N, p, r, T, A 2 ).

Combining the previous two estimates with (4.50) yields

J 1 (t) ≤ c 33 (N, p, r, T, ζ), ∀t ∈ (0, T ). (

4.51)

By (4.46), we also find that

J 2 (t) ≤ c 34 (N, p, r, T, A 2 ).

(4.52)

Step 3: End of proof. By Hölder's inequality, we get t 0 B 2r |∇u| p-1 ζ p-1 dx dτ ≤ (J 1 (t))

Plugging these estimates into (4.28) yields

for given ǫ > 0,we can choose R > 0 large enough such that

which proves the uniform integrability of the sequence {f (u µn )}. The last assertion follows from the fact that u µn is bounded in M qc (Q T ) (remember that q c = p -1 + p/N ) and M qc (Q T ) ⊂ L q loc (Q T ) with continuous imbedding, for any q < q c . The conclusion follows again by Vitali's theorem.

Lemma 4.6 Assume p > 2N

N +1 , then for any U ⊂⊂ R N , the sequence {∇u µn } converges strongly to ∇u in (L s (Q T )) N for every 1 ≤ s < s c := p -N N +1 .

Proof. We set h n = -f (u µn ) and write the equation under the form

We already know from the L 1 -contraction principle and Proposition 4.4 that

Furthermore there exists a unique a ∈ ∩ k G k . We set

Since lim n→∞ M n,k = ∞, we claim that for any m > 0 and any k, there exists

By induction, we define n(1) as the smallest integer n such that M n,1 ≥ m. This is always possible. Then we define n(2) as the smallest integer larger than n( 1)

Next, for any k, there exists ℓ = ℓ(k) such that

and we set Ûk (x) = inf{u(x, t n(k) ); ℓ}χ G k (x).

Let ûk = u be the unique bounded solution of

Since ûk (x, 0) ≤ u(x, t n(k) ), we derive Proof. If we assume that such a u exists, we proceed as in the proof of the previous lemma. Since Theorem 1.7 holds, we derive that u ≥ u mδa for any m. Since lim m→∞ u mδa (x, t) = ∞ for all (x, t) ∈ Q ∞ , we are led to a contradiction.

Thanks to these results, we can characterize the initial trace of positive solutions of (1.1) when the Keller-Osserman condition does not hold.

Using the argument as in Theorem 1.9 and because of Lemma 5.3,

Corollary 5.4 Assume p > 2. If f is convex and satisfies (1.12), J < ∞ and K = ∞, there exist infinitely many different positive solutions u of (1.1)

and |∇V | p are locally integrable in Q T ; actually V is locally Lipschitz continuous. Let ǫ > 0 and ρ ǫ be a smooth approximation defined by

Letting ǫ → 0 implies

Thus V is a subsolution, smaller than U . Therefore there exists a solution u b such that V ≤ u ≤ U . This implies that tr


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Denise versus Jean-Jacques: homonymies, homologies and tectonic faults between psychological contract and social contract.

Rémi Jardat

To cite this version:

Rémi Jardat. Denise versus Jean-Jacques: homonymies, homologies and tectonic faults between psychological contract and social contract.. 2011. hal-00561248

Denise versus Jean-Jacques: homonymies, homologies and tectonic faults between psychological contract and social contract.

Rémi JARDAT, directeur de la recherche, ISTEC Ecole supérieure de commerce et de marketing, Paris, France.

Abstract

Making any comparison between Jean-Jacques' and Denise's thoughts of contract may appear as a foolish project. Whereas the Swiss-French author cared two centuries ago about politics and states architecture and tried to build a normative utopia dedicated to the defence of liberty and -as a necessary condition for the latter -equality, the American one focuses today on organizations -not on political regimes, endorses a descriptive purpose -not a normative one, and cares about efficiency and common believes-not about transcendent values and absolute truth. Moreover, Jean-Jacques does not believe that his proposals could be concretely implemented, due to permanent flaws of human societies, whereas Denise observes trends in social contracts and shows more transformative position. Should then each of both contracts be a pure homonymy of the other, as well as both authors' names?

When more carefully looked at, their works show disturbing similarities. Firstly conditions of existence of what they study -the contract-are quite the same, although implemented into different contexts. Indeed, Jean-Jacques argues that monarchy could not rely on any contract based on the law of war, because in a situation of war people have no choice, which means that no contract be possible. Denise translates exactly the same choice argument into the sphere of labour market, explaining that a job contract in high unemployment times is not any more a true contract. Secondly, though differently named, some concepts around contracts prove quite close. For example, Denise' agency problem and Jean-Jacques' representatives' key problem are quite similar, as well as Denise' mutuality raises stakes that are not so far from Jean-Jacques' elaborated computations about balance of powers and interests. Fourthly, although human nature may look substantially different from one thought to another, it raises quite analogous transparency/obstacle (Starobinsky, 1957) anthropological tension. According to Jean-Jacques, the obstacle is of moral nature (due to moral imperfection, human beings care for self-interest instead of general interest), whereas, from Denise point of view, this obstacle is essentially cognitive (people don't know what other people actually think or are keen on doing, this limited frame of references is due the limited cognitive capacities of human beings). Whatever its nature may be, moral or cognitive, the limited capacity of human beings compared with transcendent principles (like general interest or transparency) belongs to a common epistemic framework that emerged at the end of XVIIIth Century and that we call, following the structuralist philosopher Michel Foucault (1966) "analytic of finitude". An illustration of this common analytic of finitude underpinning generation of analogous concepts by Jean-Jacques and Denise is related to temporality. Both of them attempted to address the subtle problem of temporal continuous change, Denise with the contract drift and Jean-Jacques with the degeneration of political regimes. Similarly, both recommend trade-offs between the abstract principles and their concrete field of implementation. Whereas, following Montesquieu, Jean-Jacques recommends to choose a political regime that be adapted to the particular people it is supposed to rule, Denise explains that contracts must be coherent with business conditions and corporate strategy. Does all this mean that, finally, Jean-Jacques and Denise develop, with an interval of two centuries, two native languages and 6000 km, quite structurally same fundamental anthropologies? A deeper inquiry into both structures of thought unveils us irreducible differences below previously raised similarities. Jean-Jacques's and Denise's grammars of contracts are separated by a sort of tectonic fault that we can describe through Greimas' Saussurian Semiotics. Whereas Jean-Jacques's contract is ruled by the principle of sovereignty, Denise's contract is related to transactions, exchanges, or maybe gifts and counter gifts. From a fundamental anthropological point of view, these are totally incommensurable universes. Whilst the latter has to do with reciprocity and mutuality, sovereignty is a paradoxical non-reciprocal gift with no loss for the donator. Such a difference separates two ways of shifting objects and values within human communities. On the one side, there is the quite common closed system of values, in which all what is removed from any human subject is transferred to another one. This means that mutuality is then underpinned by exchange, counter gift or transaction. On the other side, there is the participative communication, where the donator does not look loosing what he makes a gift of. In this case mutuality presupposes nothing but maybe reciprocal acknowledgement.

As a conclusion we may envisage crossing the tectonic fault by prolonging Denise's work thanks to Jean-Jacques concept of sovereignty and by shifting from descriptive to quite normative purpose. Indeed, Denise's contract is relevant insofar as organizations are considered from a transaction's point of view. This point of view is consistent with a world where most of products and services are supplied by firms and their managerial system of power. Psychological contracts in organizations are becoming all the more relevant than the world as it is currently ruled tends to a pervasive "firmilization" where even public services dedicated to state sovereignty (like military forces) are operated by the form of private firms. But what about imagining another world where production itself would be processed by nonmanagerial organization ruled by the principle of sovereignty? May we imagine that every organization, like coops, could be governed with workers and / or customers that would be collectively at the same time the sovereign body of this organization? What about situations then, where Denise's contract and Jean-Jacques' contract would tightly intertwine?


A F H Prins

Stalenhoef

Dr E F H Van Bommel
email: e.f.h.vanbommel@asz.nl

PhD L J H Van Tits

PhD E A Van Den Berg

MD J Prins

PhD A F H Stalenhoef

Autoantibodies against oxidized low-density lipoprotein and lipid profile in patients with chronic periaortitis: case-control study

Keywords: Chronic periaortitis, Autoantibodies, Oxidized low-density lipoprotein, Apolipoprotein B, Atherosclerosis

Autoantibodies against oxidized low-density lipoprotein

and lipid profile in patients with chronic periaortitis: case-control study

Introduction

The term chronic periaortitis refers to a group of rare chronic inflammatory disorders sharing similar clinical and histopathological findings [START_REF] Mitchinson | Retroperitoneal fibrosis revisited[END_REF][START_REF] Parums | The spectrum of chronic periaortitis[END_REF]. These include idiopathic retroperitoneal and mediastinal fibrosis, peri-aneurysmal fibrosis and inflammatory abdominal aortic aneurysm [START_REF] Mitchinson | Retroperitoneal fibrosis revisited[END_REF][START_REF] Parums | The spectrum of chronic periaortitis[END_REF]. All show severe adventitial lymphocyte and plasma cell infiltration with fibrosis around atherosclerotic plaques. This periaortic inflammation is found only where advanced atherosclerosis breaches or attenuates the media, thereby allowing contact of atheromatous material with the adventitia [START_REF] Mitchinson | Retroperitoneal fibrosis revisited[END_REF][START_REF] Parums | The spectrum of chronic periaortitis[END_REF][START_REF] Parums | Characterisation of inflammatory cells associated with 'idiopathic retroperitoneal fibrosis[END_REF].

These findings led to the hypothesis that chronic periaortitis may be due to an autoallergic reaction to a component of the atherosclerotic plaque [START_REF] Mitchinson | Retroperitoneal fibrosis revisited[END_REF][START_REF] Parums | The spectrum of chronic periaortitis[END_REF][START_REF] Parums | Characterisation of inflammatory cells associated with 'idiopathic retroperitoneal fibrosis[END_REF]. The possibility that ceroid, an oxidized lipoprotein byproduct, might be antigenic was first suspected when human immunoglobulin IgG and to lesser extent IgM was found lo localize to ceroid in atherosclerotic plaques from patients with chronic periaortitis [START_REF] Parums | The localisation of immunoglobulin in chronic periaortitis[END_REF]. Subsequently, circulating autoantibodies to ceroid and oxidized low-density lipoprotein (OxLDL) were demonstrated in patients with chronic periaortitis but also, albeit to lesser extent, in elderly persons with minimal atherosclerosis and in patients with ischemic heart disease [START_REF] Parums | Serum antibodies to oxidized low-density lipoprotein and ceroid in chronic periaortitis[END_REF]. Ceroid and OxLDL showed strong cross-reactivity [START_REF] Parums | Serum antibodies to oxidized low-density lipoprotein and ceroid in chronic periaortitis[END_REF]. To date, these preliminary findings have not been verified.

Although the presence of these antibodies may be an epiphenomenon of little pathogenetic significance, the postulated pathogenesis of chronic periaortitis as being the result of an autoallergic reaction to OxLDL has since been adopted by many investigators [START_REF] Bullock | Idiopathic retroperitoneal fibrosis[END_REF][START_REF] Baker | Auto-allergic periaortitis (idiopathic retroperitoneal fibrosis)[END_REF][START_REF] Meier | Rethinking the triggering inflammatory processes of chronic periaortitis[END_REF]. The described association with autoimmune disorders and the often observed beneficial response to immunosupressive treatment would further fit the proposed hypothesis [START_REF] Haug | Inflammatory aortic aneurysm is associated with increased incidence of autoimmune disease[END_REF][START_REF] Vaglio | Evidence of autoimmunity in chronic periaortitis: a prospective study[END_REF]. However, atherosclerosis per se is considered a chronic inflammatory process of the arterial wall associated with systemic and local immune responses to various antigens, notably OxLDL [START_REF] Galkina | Immune and inflammatory mechanisms of atherosclerosis[END_REF]. No specific data exist on lipid profile and atherosclerotic biomarkers in patients with chronic periaortitis. In addition, aneurysm exclusion does not always lead to regression of periaortic fibrosis in aneurysmal chronic periaortitis [START_REF] Van Bommel | Persistent chronic periaortitis ('inflammatory aneurysm') after abdominal aortic aneurysm repair: systematic review of the literature[END_REF].

We performed a case-control study comparing serum aOxLDL antibodies and lipid profile in patients with chronic periaortitis with those in age-and sex-matched control subjects.

Material and methods

Subjects

In a cross-sectional case-control study, 20 study patients were randomly selected from our cohort of patients with a definite diagnosis of chronic periaortitis followed at the out-patient department of internal medicine of the Albert Schweitzer hospital, Dordrecht. Chronic periaortitis was diagnosed on the basis of clinicoradiological and/or histopathological criteria and encompasses patients with idiopathic RPF and patients with peri-aneurysmal fibrosis ('inflammatory aneurysm') [START_REF] Mitchinson | Retroperitoneal fibrosis revisited[END_REF][START_REF] Vaglio | Evidence of autoimmunity in chronic periaortitis: a prospective study[END_REF][START_REF] Van Bommel | Persistent chronic periaortitis ('inflammatory aneurysm') after abdominal aortic aneurysm repair: systematic review of the literature[END_REF][START_REF] Van Bommel | Long-term renal and patient outcome in idiopathic retroperitoneal fibrosis treated with prednisone[END_REF]. Median time interval since diagnosis of chronic periaortitis was 19.5 (IQR 10-33) months. Except for 4 patients who were treated with immunosuppressants, first-line treatment consisted of tamoxifen 20 mg two-times daily.

Study patients were compared to 20 age-and sex-matched control subjects. These were predominantly recruited from partners (n = 18), thereby eliminating possible confounding by environmental factors. Written informed consent was obtained from all participants. The study was approved by the ethics committee of our institution. Information of smoking habits, history of cardiovascular disease and use of antihypertensive and/or lipid-lowering medication was obtained from all subjects. Clinical examination included measurement of weight and height for calculation of body mass index and measurement of resting arm blood pressure.

Hypertension was defined as on-going use of antihypertensive medication and/or a resting arm blood pressure of > 140 mm Hg systolic and/or > 90 mm Hg diastolic. Cardiovascular disease was defined as established coronary vascular and/or cerebrovascular disease and/or peripheral arterial occlusive disease requiring intervention. Life-time amount of smoking was registered in pack-years by dividing the life-time number of cigarettes by 20 x 365. Blood samples from patients and controls were obtained under fasting conditions and placed in K 2 ethylenediaminetetra-acetate(1.8 mg/ml)-containing vials and in vials without anticoagulant at 4 o C. Serum samples were prepared from blood allowed to clot by centrifugation, and lipids were determined freshly. Plasma samples were prepared within 30 minutes of collection, supplemented with sacharose (0.6%, w/v) to prevent for mation of LDL aggregates, and stored at -80 o C until analysis of OxLDL.

Antibodies against oxidized LDL

Circulating autoantibodies to OxLDL, differentiated by IgG and IgM subclass, were measured as described previously [START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF]. In short, samples were incubated in wells of microtiter plates precoated with native or oxidized LDL, and bound antibodies were detected using peroxidase-conjugated antibodies from goat specific for human IgG or IgM (Sigma-Aldrich, Steinheim, Germany). The results were expressed as the mean optical density values at 450 nm from duplicate measurements, and the antibody titer to oxidized LDL was calculated by subtracting the binding to native LDL from the binding to oxidized LDL. With this subtraction method the assay not only corrects for specific binding to native LDL but also for nonspecific binding of each sample to the microtiter wells. All determinations of autoantibodies to oxidized LDL were performed on 1 day using the same batch of native and oxidized LDL and the same batch of reagents.

Other methods

For measurement of OxLDL we used a commercially available non-competitive ELISA (Mercodia, Uppsala, Sweden). In this assay the immobilized monoclonal antibody 4E6 specifically captures oxidized apolipoprotein B from the sample, which is subsequently detected with an antibody to apolipoprotein B. Intra-and interassay coefficients of variation amounted to 6.0% and 7.0%, respectively. Total plasma apoB concentrations were determined by immunonephelometry. Intra-and interassay coefficients of variation amounted to 2.4 and 6.0%, respectively. High-sensitivity C-reactive protein (hsCRP) was measured using a commercially available enzyme immunoassay according to the instructions of the manufacturer (Hyphen BioMed, Neuville-sur-Oise, France; and Dako, Glastrup, Denmark, respectively). Intra-and interassay coefficients of variation amounted to 1.4% and to 3.8% and to 6.0% and 9.7%, respectively. Plasma cholesterol and triglycerides were measured with an enzymatic colorimetric assay system (Boerhinger-Mannheim automatic analyses for Hitachi 717; Diagnostica). High-density lipoprotein (HDL)-cholesterol was measured after lipoproteins that contained apoB were precipitated with phosphotungstate-magnesium chloride. Low-density lipoprotein (LDL)-cholesterol was calculated using the Friedewald formula.

Statistical analysis

Because of skewed distribution of several variables, all continuous variables were reported as median and interquartile range (25th to 75th percentile). In addition, all analyses were done with non-parametric tests. Differences between continuous variables were analyzed by using Mann-Whitney test. Categorical variables were expressed as proportions and compared with Fisher exact test. Spearman rank correlation coefficient was used to test associations between variables. All reported P-values are two-sided. A P-value of < 0.05 was considered significant. All statistical analyses were performed with SPSS software (version 11.0.1; SPSS inc., Chicago, IL).

Results

Demographic and clinical characteristics of study patients and control subjects are shown in

Table 1. There were only 3 non-smokers in the patient group (15%) and 4 non-smokers (20%) in the control group, but patients tended to have more smoking pack-years. Use of antihypertensive medication was more frequent among study patients than among control subjects. Otherwise, there were no significant differerences in demographic or clinical characteristics between groups. Lipid profile and hsCRP levels of study patients and control subjects are also shown in Table 1. Study patients had significantly lower LDL-cholesterol levels than control subjects. As plasma apoB concentrations were similar, this probably reflects higher VLDL levels in study patients. HsCRP levels tended to be higher in study patients. Titres of aOxLDL-IgM and IgG, OxLDL and ratio OxLDL/apoB of study patients and control subjects are shown in Table 2. Titres of aOxLDL-IgM and IgG did not differ significantly between study patients and control subjects. When cases and controls were analyzed together (n = 40), aOxLDL-IgM titres were significantly higher in females than in males (Fig 1). This difference in aOxLDL-IgM titres between females and males was observed in the patient group as well as in the control group, albeit only statistically significant in the patient group. Titres of aOxLDL-IgG did not differ between males and females in the total group nor in the subgroup of patients or control subjects (Fig 1).

Compared to male patients, female patients had higher levels of total cholesterol and HDLcholesterol. Otherwise, there were no significant differences between males and females in the patient group (Table 3). In study patients, aOxLDL-IgM titres were significantly lower in those with established cardiovascular disease compared to those without cardiovascular disease (Fig 2). This difference could not be explained by differences in the number of males among study patients with or without cardiovascular disease (5/8 males [62%] vs. 7/12 males [58%]). In study patients, aOxLDL-IgM titres correlated inversely with diastolic blood pressure and with the OxLDL/apoB ratio (Fig 3). aOxLDL-IgM titres in patients also correlated positively with total cholesterol (r = 0.56, 95%CI 0.14 to 0.80; P = 0.009), LDLcholesterol (r = 0.64, 95%CI 0.25 to 0.85; P = 0.003) and HDL-cholesterol (Fig 3 ), respectively. No correlation was found between titres of aOxLDL-IgM in patients and other variables (data not shown). In control subjects, no correlations were found between aOxLDL-IgM titres and other variables (data not shown). With respect to titres of aOxLDL-IgG, no correlation with other variables was found in both the patient group and control group (data not shown). There were no differences between study patients and control subjects in OxLDL levels and OxLDL/apoB ratio (Table 2). Circulating plasma OxLDL levels correlated significantly with total cholesterol, LDL-cholesterol and apoB levels in both the patient group and the control group (data not shown). In study patients but not in control subjects, plasma OxLDL concentrations correlated negatively with age (Spearman r = -0.62 [95%CI -0.84 to -0.24]; P < 0.01) and positively with smoking pack-years (Spearman r = 0.58 (95%CI 0.17 to 0.81]; P < 0.01).

Discussion

This study was undertaken to evaluate the role of aOxLDL antibodies in patients with chronic periaortitis and to distinguish them from age-and sex-matched controls. Chronic periaortitis typically occurs in patients at increased cardiovascular risk [START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF]. In addition, aOxLDL antibodies are prevalent among patients with atherosclerotic vascular disease [START_REF] Tits | Effects of alfa-Tocopherol on superoxide production and plasma intercellular adhesion molecule-1 and antibodies to oxidized LDL in chronic smokers[END_REF][START_REF] Virella | Lipoprotein autoantibodies: measurement and significance[END_REF][START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF]. Except for a tendency of more smoking pack-years and higher hs-CRP levels, patients and controls were comparable in terms of cardiovascular risk. We observed no significant differences in aOxLDL IgG or IgM antibody levels between patients and controls.

Parums et al found aOxLDL antibodies, undifferentiated by subclass IgG or IgM, to be more frequently present in patients with chronic periaortitis compared to elderly 'controls' (i.e., subjects without chronic periaortitis and only minimal atherosclerosis) and compared to patients with angina pectoris (5). Data were not adjusted for age and sex. Indeed, elderly 'controls' were older (mean age 66.7 y vs. 62.7 y) and more often of female sex (40% vs. 15%). aOxLDL antibody levels are inversely proportional to aging of people [START_REF] Tinahones | Influence of age and sex on levels of anti-oxidized LDL antibodies and anti-LDL immune complexes in the general population[END_REF]. Moreover, aOxLDL antibodies in elderly individuals were determined in stored serum samples obtained before death. It may be that age of serum samples and storage conditions influence aOxLDL antibody levels. All these factors may have biased results. In addition, complete absence of clinical and laboratory data, including medication use, in this study makes direct comparison of patients and 'controls' difficult [START_REF] Parums | Serum antibodies to oxidized low-density lipoprotein and ceroid in chronic periaortitis[END_REF].

The frequent presence of aOxLDL antibodies in subjects without chronic periaortitis in

Parum's study and the non-significant difference in aOxLDL antibody titres between study patients and controls may suggest that antigenic exposure with subsequent formation of aOxLDL antibodies is common, whereas clinical periaortitis disease is rare. This may reflect inter-individual differences in response to this antigenic stimulus, or that antibody formation is no more than an epiphenomenon seen in patients with (subclinical) atherosclerosis of no clinical significance. Of note, most patients with chronic periaortitis in our study were (being) treated with tamoxifen. Tamoxifen is associated with beneficial effects on lipid profile, C-reactive protein levels and on atherosclerotic lesions [START_REF] Bonanni | Effect of tamoxifen at low doses on ultrasensitive C-reactive protein in healthy women[END_REF][START_REF] Medina | Tamoxifen is a potent inhibitor of cholesterol esterification and prevents the formation of foam cells[END_REF][START_REF] Grainger | Tamoxifen for the prevention of mycardial infarction in humans: preclinical and clinical evidence[END_REF], and as such, may reduce aOxLDL antibody levels. Although one can not exclude the possibility that a significant difference in aOxLDL antibody titres between patients and controls is masked by suc h mechanisms, no data exist substantiating this possible confounding effect.

Many studies have been conducted to understand the relevance of aOxLDL antibodies in atherosclerotic vascular disease, but its clinical significance is not clear [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF][START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF][START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF]. aOxLDL antibodies are also present in various chronic inflammatory and auto-immune disorders such as systemic lupus erythematosus, gout, rheumatoid arthritis and Behcet's disease as well as in other conditions such as diabetes mellitus and pre-eclampsia [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF][START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF][START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF][START_REF] Gomez-Zumaquero | Association of biological markers of activity of lupus erythematosus with levels of antioxidized low-density lipoprotein antibodies[END_REF][START_REF] Tsutsumi | Oxidized low-density lipoprotein autoantibodies in patients with primary gout: effect of urate-lowering therapy[END_REF][START_REF] Winyard | Presence of foam cells containing oxidised low density lipoprotein in the synovial membrane from patient with rheumatoid arthritis[END_REF]. One thing they have in common is that all these diseases are thought to be prone for atherosclerosis development [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF][START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF][START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF]. The same holds true for patients with chronic periaortitis [START_REF] Mitchinson | Retroperitoneal fibrosis revisited[END_REF][START_REF] Parums | The spectrum of chronic periaortitis[END_REF][START_REF] Virella | Lipoprotein autoantibodies: measurement and significance[END_REF].

IgM and IgG antibodies to OxLDL may play a different part in the pathogenesis of atherosclerotic vascular disease, with aOxLDL-IgM having a protective role whereas aOxLDL-IgG enhances plaque formation [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF][START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF][START_REF] Dotevall | Autoantibodies against oxidized low-density lipoprotein and C-reactive protein are associated with diabetes and myocardial infarction in women[END_REF]. aOxLDL antibody level may decline over the years, thereby possibly explaning the higher incidence of cardiovascular disease at older age [START_REF] Tinahones | Influence of age and sex on levels of anti-oxidized LDL antibodies and anti-LDL immune complexes in the general population[END_REF]. In hypertensive subjects, high aOxLDL-IgM levels predicted a favorable outcome in the development of carotid atherosclerosis [START_REF] Georgiades | Antibodies of IgM subclass to phosphorycholine and oxidized LDL are protective factors for atherosclerosis in patients with hypertension[END_REF]. In women, low aOxLDL-IgM titres were associated with the presence of diabetes mellitus and/or myocardial infarction [START_REF] Dotevall | Autoantibodies against oxidized low-density lipoprotein and C-reactive protein are associated with diabetes and myocardial infarction in women[END_REF]. Low aOxLDL-IgM titres were associated with the presence of established CVD in our patients.

These findings might further suggest a protective role of aOxLDL-IgM. Increased consumption of aOxLDL-IgM in advancing atherosclerosis may be an alternative explanation [START_REF] Orchard | Antibodies to oxidized LDL predict coronary artery disease in type 1 diabetes: a nested case-control study from the Pittsburgh Epidemiology of Diabetes Complications Study[END_REF].

Interestingly, patients with low aOxLDL-IgM titres tended to have a high diastolic blood pressure, low HDL-cholesterol level and high OxLDL/apoB ratio. A higher OxLDL/apoB ratio indicates increased oxidative stress in the vascular wall [START_REF] Van Tits | Proportion of oxidized LDL relative to plasma apolipoprotein B does not change during statin therapy in patients with heterozygous familial hypercholesterolemia[END_REF][START_REF] Van Tits | Increased levels of low-density lipoprotein oxidation in patients with familial hypercholesterolemia and in end-stage renal disease patients on hemodialysis[END_REF]. All these factors may contribute to the increased risk of CVD in patients with low aOxLDL-IgM.

As we tested sera from patients with stable chronic periaortitis, aOxLDL antibody titres reflect basal values. This may account for the noted associations. In various states of active vascular inflammation, aOxLDL antibody titres may be temporarily reduced or increased and can therefore not be used to ascertain basal values [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF][START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF][START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF][START_REF] Gomez-Zumaquero | Association of biological markers of activity of lupus erythematosus with levels of antioxidized low-density lipoprotein antibodies[END_REF][START_REF] Tsutsumi | Oxidized low-density lipoprotein autoantibodies in patients with primary gout: effect of urate-lowering therapy[END_REF][START_REF] Swets | Patients with systemic vasculitis have increased levels of autoantibodies against oxidized LDL[END_REF]. However, it is remarkable that aOxLDL-IgM but not aOxLDL-IgG antibody titres were associated with several other cardiovascular variables in our study. Contrary to aOxLDL-IgG, aOxLDL-IgM is probably largely confined to the intravascular compartment [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF]. Therefore, one might speculate that these serum levels are of more clinical relevance than serum levels of aOxLDL-IgG, which is in large part confined to the vascular wall [START_REF] Parums | The localisation of immunoglobulin in chronic periaortitis[END_REF][START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF].

As in other studies [START_REF] Dotevall | Autoantibodies against oxidized low-density lipoprotein and C-reactive protein are associated with diabetes and myocardial infarction in women[END_REF][START_REF] Georgiades | Antibodies of IgM subclass to phosphorycholine and oxidized LDL are protective factors for atherosclerosis in patients with hypertension[END_REF], females tended to have higher aOxLDL-IgM titres than males in our study. A possible explanation is that postmenopausal women have lower estrogen levels, which may lead to uninhibited oxidation of LDL [START_REF] Bureau | No antioxidant effect of combined HRT on LDL oxidizability and oxidative stress biomarkers in treated post-menopausal women[END_REF][START_REF] Hoogerbrugge | Estrogen replacement decreased the levels of antibodies against oxidized low density lipoprotein in postmenopausal women with coronary heart disease[END_REF]. The risk of a primary immune response increases with the formation of aOxLDL antibodies. Postmenopausal hormone replacement therapy may [START_REF] Heikkinen | Postmenopausal hormone replacement therapy and autoantibodies against oxidized LDL[END_REF] or may not reduce aOxLDL levels [START_REF] Bureau | No antioxidant effect of combined HRT on LDL oxidizability and oxidative stress biomarkers in treated post-menopausal women[END_REF][START_REF] Uint | Hormone replacement therapy increases levels of antibodies against heat shock protein 65 and certain species of oxidized low density lipoprotein[END_REF]. However, none of our female patients or controls used hormonal replacement therapy.

Circulating OxLDL was not associated with aOxLDL-IgG or IgM antibody titres.

Consistent with the literature [START_REF] Shoenfeld | Are anti-oxidized low-density lipoprotein antibodies pathogenic or protective?[END_REF][START_REF] Tsimikas | C-reactive protein and other emerging blood biomarkers to optimize risk stratification of vulnerable patients[END_REF][START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF][START_REF] Van Tits | Proportion of oxidized LDL relative to plasma apolipoprotein B does not change during statin therapy in patients with heterozygous familial hypercholesterolemia[END_REF][START_REF] Meisinger | Plasma oxidized lowdensity lipoprotein, a strong predictor for acute coronary heart disease events in apparently healthy, middle-aged men from the general population[END_REF][START_REF] Hulthe | Circulating oxidized LDL is associated with subclinical atherosclerosis development and inflammatory cytokines (AIR study)[END_REF], OxLDL levels were positively associated with total cholesterol, LDL-cholesterol and apoB levels in patients and controls. Smoking was not associated with aOxLDL antibody titres in this and other studies [START_REF] Nitecki | Inflammatory abdominal aneurysms: a case-control study[END_REF][START_REF] Frostegard | Circulating oxidized low-density lipoprotein is increased in hypertension[END_REF]. However, smoking was positively correlated with OxLDL levels in patients but not in controls. Others also noted no correlation between OxLDL levels and smoking in clinically healthy, middle-aged men from the general population [START_REF] Meisinger | Plasma oxidized lowdensity lipoprotein, a strong predictor for acute coronary heart disease events in apparently healthy, middle-aged men from the general population[END_REF]. Smoking is a well-known risk factor for atherosclerotic aortic aneurysm, and an even stronger risk factor for the inflammatory variant [START_REF] Haug | Inflammatory aortic aneurysm is associated with increased incidence of autoimmune disease[END_REF][START_REF] Nitecki | Inflammatory abdominal aneurysms: a case-control study[END_REF]. We recently observed an independent association between smoking and infrarenal aortic diameter in patients with chronic periaortitis [START_REF] Van Bommel | Idiopathic retroperitoneal fibrosis: prospective evaluation of incidence and clinicoradiological presentation[END_REF]. Cessation of smoking should therefore be strongly encouraged in all cases of chronic periaortitis, irrespective of the presence of aneurysmal dilation.

The exact pathway by which chronic periaortitis occurs remains unknown. Although it seems clear that not aneurysm formation per se but advanced atherosclerosis is the prerequisite for development of typical chronic periaortitis, it remains an enigma why so few patients with advanced atherosclerosis develop this chronic fibrotic inflammatory disorder. Indeed, only 5% of atherosclerotic abdominal aortic aneurysms are of the inflammatory variant [START_REF] Van Bommel | Persistent chronic periaortitis ('inflammatory aneurysm') after abdominal aortic aneurysm repair: systematic review of the literature[END_REF][START_REF] Nitecki | Inflammatory abdominal aneurysms: a case-control study[END_REF]. Deranged immunological self-tolerance, possibly by genetic predisposition, may play a role [START_REF] Meier | Rethinking the triggering inflammatory processes of chronic periaortitis[END_REF][START_REF] Vaglio | Evidence of autoimmunity in chronic periaortitis: a prospective study[END_REF][START_REF] Virella | Lipoprotein autoantibodies: measurement and significance[END_REF]). An association between chronic periaortitis and HLA-DRB1*03, an allele linked to several autoimmune conditions, was recently noted [START_REF] Martorana | Chronic periaortitis and HLA-DRB1*03: another clue to an autoimmune origin[END_REF]. This might explain the simultaneous occurrence of autoimmune disorders (e.g., lupus erythematosus, Wegener's granulomatotis, polymyalgia rheumatica) in some patients with chronic periaortitis [START_REF] Haug | Inflammatory aortic aneurysm is associated with increased incidence of autoimmune disease[END_REF][START_REF] Vaglio | Evidence of autoimmunity in chronic periaortitis: a prospective study[END_REF][START_REF] Virella | Lipoprotein autoantibodies: measurement and significance[END_REF]) and the often observed beneficial response to immunosuppressive treatment [START_REF] Van Bommel | Brief communication: tamoxifen therapy for idiopathic retroperitoneal fibrosis[END_REF]. One might speculate that in predisposed patients, an exaggerated and persistent inflammatory response with subsequent development of chronic periaortitis results from the same stimuli and mechanisms that are involved in atherosclerosis development per se. In addition, as this disorder may occur in patients without atherosclerosis, chronic periaortitis as a primary autoimmune disease should also be considered [START_REF] Vaglio | Evidence of autoimmunity in chronic periaortitis: a prospective study[END_REF].

Our study also has limitations. As sample size was rather small, our study may not have enough power to detect significant differences. However, given the low annual incidence of chronic periaortitis of 1.3 per 100.000 [START_REF] Virella | Lipoprotein autoantibodies: measurement and significance[END_REF], it will be difficult to recruit a large number of patients. As noted, the cross-sectional nature of our study may have influenced results as some patients were still receiving tamoxifen. Unfortunately, we did not have serum samples of study patients before the start of treatment for chronic periaortitis to determine sequential aOxLDL antibody titres before and during treatment.

In conclusion, this first case-control study showed no significant difference in aOxLDL antibody titres between patients with stable chronic periaortitis and control subjects. However, distinctive differences and associations of aOxLDL-IgM antibodies and of circulating OxLDL levels were observed within the study but not within the control group. Data suggest a differing innate immune response to OxLDL in patients with chronic periaortitis compared to controls.

Whether this response is causally related to chronic periaortitis development remains to be clarified. aOxLDL antibody titres, differentiated by subclass IgM and IgG, according to sex in the overall group, patients and controls. Female patients had higher aOxLDL titres than male patients. A similar trend, albeit not significant, was seen for aOxLDL-IgM in the control group.

Legends to the figures

Figure 1 :

 1 Figure 1:

Figure 2 :

 2 Figure 2: aOxLDL-IgM titres in study patients with established cardiovascular disease (CVD) and in study patients with no established CVD. aOxLDL-IgM titres were significantly lower in patients with established CVD.

Figure 3 :

 3 Figure 3:aOxLDL-IgM titres in study patients were inversely correlated with diastolic blood pressure (Spearman r = -0.46, 95%CI -0.75 to -0.01) and ratio OxLDL/apoB (Spearman r = -0.41, 95%CI -0.73 to 0.04), and positively correlated with HDL-cholesterol levels (Spearman r = 0.47, 95%CI 0.02 to 0.69). Lines depict regression line of y on x and 95% confidence intervals for regression line.

Table 1 . Demographics, Clinical Characteristics, Lipid Profile and High-Sensitivity C- reactive Protein Level of Study Patients and Control Subjects.

 1

Acknowledgement:

We thank Heidi Lemmers for technical assistance.