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 $ \prt_tu-\Gd_p u+f(u)=0$ in $\BBR^N\ti(0,\infty)$ where $p>1$, $\Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u)$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $k\gd_0$ and we study their limit when $k\to\infty$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)=\int_0^s f(\gs)d\gs$. 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 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 $f(u)=u^\ga \ln^\gb(u+1)$, where $\ga>0$ and $\gb\geq 0$.

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 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 [15] 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 defined on [0, ∞) and subject to φ(0) = a ≥ 0; it is given by It is a consequence of the Vázquez's extension of the Keller-Osserman condition (see [17], [12]) 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 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 [11]). 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.
In Section 2 we study the existence of the fundamental solutions u k and their behaviour when k → ∞. Kamin  then for any k > 0, there exists a unique positive solution u := u k to problem 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 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 [14] 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, 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 , 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 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 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 defined on [0, ∞) and satisfying w(0) = a, w r (0) = 0. It is given by 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 [18] 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 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 thus lim t→∞ u(x, t) = 0, uniformly with respect to x ∈ R N , and In section 4 we prove an existence and stability result for the initial value problem , the set of positive and bounded Radon measures in R N .
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 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 [2]. 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 [2] and [15] with a stability result for the Cauchy problem and Harnack's inequality in the form of [5] 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  The set of points y such that (1.20 ) (resp. (1.21 )) holds is clearly open (resp. closed) and denoted by R(u) (resp (S(u)). Using a partition of unity, there exists a unique Radon measure µ ∈ M + (R(u)) such that 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 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.

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 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 and by its expression is obtained by the Crandall-Liggett scheme 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 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 [20]. 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 [4].
An important property [9, Lemma 2] is that if w ∈ L 1 (R N ) satisfies where σ > 0 and h ∈ L 1 (R N ), then

The Barenblatt-Prattle solutions
We recall the explicit expression, due to Barenblatt and Prattle, of the solution v = v k of problem (1.14 ).
and where C k is connected to the mass k by The condition p > 2N N +1 appears in order λ be positive. Notice that, if p > 2 then This type of singular solution which is singular on the whole axis (0, t) ⊂ Q ∞ , is called a razor blade (see [19] for some examples). To this solution corresponds a universal estimate.
If we assume moreover that lim where Λ 1 is the value of the constant in (2.12 ) when N = 1.
Proof. The first estimate is a consequence of in [6, 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: 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 (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 T ] for any T > 0, and for t = 0. By the Letting successively R → ∞, T → ∞ and ǫ → 0 and using the invariance of the equation by rotation implies (2.16 ).
) 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 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, Because v is a positive solution with isolated singularity at (0, 0), it follows from [3] that v = v k , solution of (1.14 ).

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, Proof. The result is already proved in [10] 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) where, 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 where Therefore We perform the same change of variable with I 2 Therefore (2.23 ) holds.
Notice that the assumption implies that 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 By the standard local regularity theory for degenerate equations, ∇u ǫ remains locally compact in (C 1 loc (Q ∞ )) N , thusũ satisfies (1.1 ) in Q ∞ . In order to prove that we recall that u ǫ can be obtained as the limit of thru the iterative implicit scheme By (2.7 ), and denoting byŨ ǫ,h the piecewise constant function such thatŨ ǫ,h (jh) = u ǫ,j , we obtain since u ǫ, Letting h → 0 and i → ∞ such that ih = t > ǫ and using the uniform convergence, we obtain Since 0 ≤ u ǫ ≤ v k and v k (., t) has constant mass equal to k, we derive T )), we can let ǫ → 0, using the monotone convergence theorem, in order to get Because |φ(x)ζ(x) − φ(0)| is continuous and vanishes at zero and v k (., 0) = kδ 0 , it follows from (2.30 ) Uniqueness. The proof uses some ideas from [10,Th 2.4]. Assumeũ is any nonnegative solution of problem (1.13 ), then, for any ǫ > 0 we denote byṽ ǫ the solution of By the maximum principleṽ ǫ ≥ũ in Q ǫ,∞ . When ǫ → 0,ṽ ǫ converges locally uniformly to a solutionṽ of the same equation in Q ∞ . Furthermore, using again [9, Lemma 2], By Fatou's Lemma and using the fact that Sinceṽ ≥ũ, equality holds in (2.33 ). Since the fundamental solution is unique [3, Th 4.1], it impliesṽ = v k andũ ≤ v k . We end the proof as in [3, Th 4.1], using the L 1contraction mapping principle and the fact that any solution of (1.13 ) is smaller than v k : for t > s > 0, there holds (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 Notice that d ≥ 0 since f is nondecreasing and A is elliptic [7, 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 ).
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 is decreasing, we derive that u ≤ v m for some m ≥ 0 and there exists k ≥ 0 such that 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, where v k is the solution of (1.14 ) in Q ∞ and c 12 = c 12 (N, p) > 0 in (2.11 ). We set Next we write u k (x, t) = b k (t)w k (x, s k (t)) (the functions b k and s k will be defined later). For simplicity, we drop the subscript k in b k and s k . Inserting in (2.38 ), we get We choose the functions b and s such that Then ∂ 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 w k ≥ v k in R N × (0, s k,0 ). Hence p−2 and 0 < δ 2 < 1 − λ(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 Since J < ∞ 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 We restrict x ∈ B 1 and we choose t such that In order to obtain (2.47 ), it is sufficient to choose γ such that Indeed, since α < p − 1, we may choose δ 1 and δ 2 close enough ℓ(α−1)(p−1) p−2 and 1 − λ(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 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 ), 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 [19, 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 can be continued as a global solution. More precisely, 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 Taking the integral over [a, r], we get Since f is positive on (0, ∞), F (s) → ∞ when s → ∞, thus there existsã ∈ (a, a * ) such that Taking the integral over [ã, r], we obtain w(r)

and (3.2 ) is not satisfied.
Step 2. We assume that 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) = ∞.
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 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 in B n u n (x, t) = (w a (|x|) + w b (|x|))/2 in ∂B n × (0, ∞). 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.
Proof. For any R > 0, let u R be the solution of 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, 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

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
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 [2].

Regularity Properties
The following integral estimates are essentially [2, Prop 2.1] with u q replaced by f (u).
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. Proposition 4.3 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: Then the third one holds for any U ⊂⊂ Ω. Moreover, 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), and u satisfies for any 0 < θ < T and ϕ ∈ C ∞ c (Ω × [0, T )). Proof.
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 , Because the right-hand side of (4.19 ) has a finite limit when t → 0, the same holds . By a partition of unity it can be extended in a unique way as a Radon measure µ ∈ M + (Ω) and (4.11 ) holds.
Next we consider the the following problems in Ω. where µ ∈ M + (Ω). The solutions are considered in the entropy sense (see [16] and [13]).
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, The norm of u in M q (Θ) is the smallest constant such that (4.22 ) holds for any measurable set E (see [16], [13] 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 [16, 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.

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 * , 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 (4.25) where c 26 = c 26 (N, p) > 0. Thus (4.27) By (4.26 ) and the regularity theory of degenerate parabolic equations [5], 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, For λ ≥ 0, we set B n (λ) = {(x, t) ∈ Q T ) : u µn > λ} and a n (λ) = Bn(λ) dx dt. Then It follows from (4.27 ) that a n (λ) ≤ c 25 µ n p+N 1+p(N−1) 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 and u µn → u in L q loc (Q T ) for every q ∈ [1, q c ) and |∇u µn | is bounded in L s loc (Q T ) for every 1 ≤ s < s c . Thus |∇u µn | p−1 remains bounded in bounded in L σ loc (Q T ) for every 1 ≤ σ < σ c := 1 + (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 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 For any ϕ ∈ C ∞ c (R N × [0, ∞)) and θ > 0, we have (4.34) By the previous convergence results, we can pass to the limit in (4.34 ) to obtain (4.35) Step 2: u is a weak solution. By (4.26 ) 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 by Lemma 4.6 when p ≥ 2. When 1 < p < 2, we derive by Fatou's lemma |∇u µn | p−2 ∇u µn − |∇u| p−2 ∇u |u µm − u| |∇ζ| dx dt. Since ∇u µn ⇀ ∇u weakly in L p loc (Q T ), it implies again that (4.37 ) holds true. At end, let ϕ ∈ C ∞ c (Q T ) and consider 0 < θ < T and U ⊂⊂ R N such that supp ϕ ⊂ (θ, . Multiplying the equation in (4.21 ) (with initial data µ = µ n ) by g(u µn )ϕ, we obtain (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 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 where K ρ (x 0 ) is the cube centered at x 0 and wedge 2ρ, i.e., We choose x 0 = 0, t 0 = θ = t, σ = 1/2 and ρ = 4r, then (4.44 ) becomes (4.45) Since B 2r ⊂ K 2r and K 4r ⊂ B 8r , from (4.43 ) and (4.45 ), we obtain that which implies the claim.
It follows from (4.48 ) and (4.49 ) that (u(x, ǫ) + 1) Combining the previous two estimates with (4.50 ) yields 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 By step 2, we deduce that  Proof of Theorem 1.8 By translation we may suppose that y = 0. 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, 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 = ∞. where χ Bη is the characteristic function of B η . By the maximum principleũ η ≤ 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. for some bounded open subset G ⊂ R N , then u(x, t) ≥ φ ∞ (t).

The Keller-Osserman condition does not hold
Proof. By assumption, there exists a sequence {t n } decreasing to 0 such that lim n→∞ G u(x, t n )dx = ∞. Furthermore there exists a unique a ∈ ∩ k G k . We set G k u(x, t n )dx = M n,k .
Since lim n→∞ M n,k = ∞, we claim that for any m > 0 and any k, there exists n = n(k) ∈ N such that G k u(x, t n(k) )dx ≥ m. (5.5) 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) such that M n,2 ≥ m. By induction, n(k) is the smallest integer n larger than n(k − 1) such that M n,k ≥ m. Next, for any k, there exists ℓ = ℓ(k) such that G k inf{u(x, t n(k) ); ℓ}dx = m (5.6) and we setÛ k (x) = inf{u(x, t n(k) ); ℓ}χ G k (x). Letû k = u be the unique bounded solution of Sinceû k (x, 0) ≤ u(x, t n(k) ), we derive u(x, t + t n(k) ) ≥û k (x, t) ∀(x, t) ∈ Q ∞ . (5.8) When k → ∞,Û k → mδ a , thusû k → u mδa by Theorem 1.7. Therefore u ≥ u mδa . Since m is arbitrary and u mδa → φ ∞ when m → ∞, it follows that u ≥ φ ∞ . 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.
Proof of Theorem 1.4. (i) If S(u) = ∅, there exists y ∈ S(u) and an open neighborhood G of y such that (5.2 ) holds. By Lemma 5.2, u ≥ φ ∞ and the initial trace of u is the Borel measure ν ∞ . Otherwise, R(u) = R N and T r R N (u) ∈ M + (R N ). (ii) Using the argument as in Theorem 1.9 and because of Lemma 5.3, S(u) = ∅. Therefore R(u) = R N and T r R N (u) ∈ M + (R N ).
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 ) such that tr R N (u) = ν ∞ .
Proof. Let b > 0 be fixed. Since f is increasing, (x, t) → U (x, t) = w b (x) + φ ∞(t) is a supersolution for (1.1 ). Let V (x, t) = max{w b (x), φ ∞ (t)} then V , f (V ) 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 Thus V is a subsolution, smaller than U . Therefore there exists a solution u b such that V ≤ u ≤ U . This implies that tr R N (u b ) = ν ∞ . If b ′ > b we construct u b ′ with tr R N (u b ′ ) = ν ∞ and lim t→∞ (u b ′ (0, t) − u b (0, t)) > 0.