2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 53–83. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Sets of admissible initial data for porous-medium equations with absorption ^{∗}

Emmanuel Chasseigne & Juan Luis Vazquez

Abstract In this article, we study a porous-medium equation with absorption in

ℝ^{N} × (0, T) or in Ω × (0, T):

u_{t} − Δu^{m} + u^{p} = 0 .

We give a rather complete qualitative picture of the initial trace problem in all the range m > 1, p ⩾ 0. We consider nonnegative Borel measures as initial data (not necessarily locally bounded) and discuss whether or not the Cauchy problem admits a solution. In the case of non-admissible data we prove the existence of some projection operators which map any Borel measure to an admissible measure for this equation.

1 Introduction

In this paper, we consider the equation

u_{t} − Δu^{m} + u^{p} = 0,  (1.1)

where m > 1 and p > 0. In particular, we look for nonnegative weak solutions u = u(x, t) defined in Q_{T} = ℝ^{N} × (0, T) for some T ∈ (0, ∞]. We aim at describing the sets of nonnegative initial data for which there exists a solution of the Cauchy problem. We call these sets admissible sets; they can be quite different depending on the value of the exponents. For convenience we think of m as fixed and p as a variable parameter, hence we denote the admissible set by A^{+}(p).

1.1 Measures in pure diffusion

Our equation can be seen as a perturbation of the heat equation u_{t} = Δu, and it will be convenient to review the situation for this equation in order to present the main ideas. It is well-known [1] that any nonnegative distributional solution u in ℝ^{N} × (0, T) of the heat equation has an initial trace μ which is a Radon measure (i.e., a locally finite measure). This means that for any φ ∈ C_{0}(ℝ^{N}), (i.e. continuous and compactly supported in ℝ^{N}) we have

∗Mathematics Subject Classifications: 35K55, 35K65.

Key words: Initial trace, Borel measures, initial projection, singular solution.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 21, 2002.

∫_{ℝ^{N}} u(x, t)φ(x)dx → ∫_{ℝ^{N}} φ(x)dμ(x) as t → 0.  (1.2)

Moreover, the initial trace satisfies the following growth property:

∫_{ℝ^{N}} e^{−|x|^{2}/4T} dμ(x) < +∞.  (1.3)

On the other hand, given a nonnegative measure μ which satisfies the above condition, there exists a unique nonnegative solution, so that the nonnegative admissible data are measures which are characterized by (1.3).

We have a similar result for the the porous-medium equation u_{t} = Δu^{m}, m > 1 [2, 4], where the admissible non-negative data are non-negative Radon measures which satisfy

sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} dμ < +∞,  (1.4)

where − ∫_{E} f denotes the average of f on E: − ∫_{E} f = |E|^{−1} ∫_{E} f(x)dx. On the other hand, the case m < 1 usually called fast diffusion, leads to difficulties that is better to postpone.

1.2 Complete equation and Borel measures

If we add an absorption term u^{p}, we get equation (1.1) and then there is a big extension of the class of initial data. Namely, we may have Borel measures, i.e., not necessarily locally finite measures as initial data of standard weak solutions, even locally bounded and continuous.

A significant example of such a situation is well-known and goes back to the works of [5] and [13]. It concerns the heat equation with absorption

u_{t} − Δu + u^{p} = 0, p > 1.  (1.5)

For any c > 0, this equation admits a solution u_{c} with initial data u_{c}(0) = cδ_{0}, so-called fundamental solution with mass c > 0. Now, letting c → ∞, one obtains a new kind of singular solution called Very Singular Solution (VSS for short) which is continuous (and locally bounded in Q_{T}), although it takes on the initial data “+∞ · δ_{0}”. Such a measure is of course not locally bounded and is a basic example of Borel measure, that we briefly recall below. Later on, Marcus and Véron [16] proved that any Borel measure is admissible as initial data for this equation, under some capacity condition if p ⩾ 1 + 2/N.

We denote by B^{+}(ℝ^{N}) the set of Borel measures. Let us recall [16, 8] that any ν ∈ B^{+}(ℝ^{N}) may be written as ν = (S, μ), where S is a closed subset of ℝ^{N} and μ ⩾ 0 is a Radon measure on R = ℝ^{N} \ S. The set S is defined as follows:

S = {x ∈ ℝ^{N} : ∀r > 0, ν(B_{r}(x)) = +∞}.

Thus, a Radon measure is a Borel measure with S = ∅. Let us now define what kind of solutions we consider.

Definition. By a solution we mean a function u ∈ C^{0}(Q_{T}), u ⩾ 0 such that (1.1) holds in the sense of distributions. We also consider u ≡ +∞ as a solution to simplify the statements of our results.

It is important to notice that since u is continuous in Q_{T}, then it is bounded on every compact subset of Q_{T}, and thus it is the uniquely determined on such compact sets [8]. This proves that u can be viewed as a limit solution, i.e. as the limit of a sequence of smooth solutions.

We prove that the solutions we consider will take on initial data in the set of nonnegative Borel measures, B^{+}(ℝ^{N}). Now it means that there exists a measure ν ∈ B^{+}(ℝ^{N}) such that for any φ ∈ C_{0}(ℝ^{N}), (i.e. continuous and compactly supported in ℝ^{N}), φ ⩾ 0, we have

∫_{ℝ^{N}} u(x, t)φ(x)dx → ∫_{ℝ^{N}} φ(x)dν(x) as t → 0.

Since ν = (S, μ) may have nonempty singular set S, the right-hand may be infinite (it is well-defined in ℝ_{+} ∪ {+∞} since φ is nonnegative). The measure ν will be called the initial trace of u at t = 0, and we note tr_{ℝ^{N}} (u) = ν.

Definition. Let ν ∈ B^{+}(ℝ^{N}). Then ν is said to be admissible and we note ν ∈ A^{+}(p) if there exists a solution u of (1.1) such that tr_{ℝ^{N}} (u) = ν. We will consider that +∞ is allowed as initial data, associated to the special solution u ≡ +∞.

For some values of p, the set of admissible initial data is known:

• The case p = 1 is easily reduced to the pure diffusion case u_{t} = Δu^{m} by the transformation v(x, τ) = e^{t}u(x, t), dτ/dt = e^{−(m−1)t}, τ(0) = 0 so that A^{+}(1) is given by (1.4) for m > 1, and (1.3) for m = 1, with +∞ allowed for u ≡ +∞.

• By [8] (and [16] in the case m = 1), it is known that if m < p < m + 2/N, any Borel measure is admissible as initial data, so that

A^{+}(p) = B^{+}(ℝ^{N}).

• The case p = m. It was shown in [9] the following exact condition

A^{+}(m) = \{+∞\} ∪ \{∫_{ℝ^{N}} \frac{e^{−|x|}}{1 + |x|^{\frac{N−1}{2}}} dμ(x) < +∞\} ,

and +∞ is associated with the flat solution u(x, t) = c_{∗}(m)t^{−1/(m−1)},

c_{∗} = (m − 1)^{−1/(m−1)}, and also with u ≡ +∞.

Contents:

1. Preliminaries 1.1 Some known results 1.2 Initial Trace

2. Very Weak Absorption (0 ⩽ p ⩽ 1)

2.1 Harnack inequality and admissibility 2.2 Optimal uniqueness

3. Weak Absorption (1 < p ⩽ m)

3.1 The projection operator ℙ_{p}

3.2 Admissibility sets

4. Super-Critical Absorption (p > m + 2/N)

4.1 The operator ℙ_{p} 4.2 Admissibility sets

5. Adaptations to the Case Ω Bounded

2 Main Results

Our results can be grouped in two areas: admissibility and the study of the projection operator.

Admissibility. In the case of very weak absorption, 0 ⩽ p ⩽ 1, we prove that the admissible initial data are exactly the same as in the case of the purely diffusive equation:

A^{+}(p) = \{μ ∈ B^{+}(ℝ^{N}) : sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} dμ < +∞\} ∪ \{+∞\},

where the measure ν = +∞ is associated to the “special solution” u ≡ +∞. Moreover, we prove uniqueness of continuous weak solutions.

The situation is different in the case of weak absorption 1 < p < m, because for example the infinite initial data is associated to a nontrivial solution, the flat solution

u(x, t) = c_{∗}t^{−1/(p−1)}, c_{∗} = (p − 1)^{−1/(p−1)}.

We do not have the exact description of the admissible initial traces in this range, but we can prove that the mapping

p ↦ A^{+}(p)  (2.1)

is strictly increasing in [1, m]. At least, it is known that the following rate is critical for functions [14]:

u_{0}(x) ∽_{+∞} c_{0}|x|^{\frac{2}{m−p}} , c_{0}(m, p, N).

Finally, in the case of super-critical absorption p ⩾ m + 2/N, we prove that the mapping (2.1) is non-increasing with p. Here also, we do not have the exact characterization of A^{+}(p), which is a question of local regularity of the measure (in terms of capacity - see [7]), however, we give some qualitative properties of this set, and some examples of admissible data.

Projections. Another important question that we investigate is what happens to non-admissible initial data when we perform an approximation process and we pass to the limit. We prove the following results:

(i) In the range 0 ⩽ p ⩽ m, there exists a projection operator

ℙ_{p} : B^{+}(ℝ^{N}) → A^{+}(p),

which maps any Borel measure to an admissible initial data. For ν ∈ B^{+}(ℝ^{N}), ℙ_{p}(ν) is defined as the initial trace of the limit solution u[ν] obtained by approximation of ν with compactly supported data. It satisfies

ℙ_{p} ∘ ℙ_{p} = ℙ_{p}, ℙ_{p}(ν) ⩾ ν,

that is, ℙ_{p} is a nondecreasing projection. More precisely, we have

Case 0 ⩽ p ⩽ 1 : if ν ∉ A^{+}(p), then ℙ_{p}(ν) = +∞ and u[ν] ≡ +∞. Case 1 < p ⩽ m : if ν ∉ A^{+}(p), then ℙ_{p}(ν) = +∞ and u[ν](x, t) = c_{∗}t^{−1/(p−1)}. We can call the first case projection by complete blow-up and the second projection by instantaneous blow-up. (ii) In the case m < p < m + 2/N, ℙ_{p} is the identity operator in B^{+}(ℝ^{N}) since any Borel measure is admissible and the problem is well-posed. (iii) In the case of super-critical absorption, we cannot prove that a projection operator is well-defined. This comes from the fact that for 0 ⩽ p ⩽ m, admissibility is just a question of growth at infinity, hence it can be handled by monotonicity techniques (although we do not know explicitly A^{+}(p) for 1 < p < m), whereas in the case p ⩾ m + 2/N, the local regularity of the initial data is involved. However, we prove that given an approximation process of ν by smooth function, the limit solution will have a trace ν′ ⩽ ν. Hence, this may be viewed as a projection down towards an admissible initial data. We can say at least that ν′ does not contain the part of ν which is singular with respect to the capacity in the Besov space B_{q/(q−m),q/(q−1)}^{2/q}. We refer to [7] for proofs of this latter fact.

3 Preliminaries

In this section, we recall some results which will be useful in the sequel.

3.1 Some Known Results

Let us begin with the following basic existence and uniqueness result taken from [8]:

Theorem 3.1 Let m > 1 and ⩽ p < m + 2/N and μ ⩾ 0 be a compactly supported Radon measure in Ω. Then there exists a unique weak solution u with initial trace μ such that u ∈ C^{0}((0, T) × \overline{Ω}), with u = 0 on ∂Ω × (0, T). The same results holds when Ω = ℝ^{N} (with no boundary conditions).

The following local estimate is taken from [19].

Lemma 3.2 Let u be a smooth solution of (1.1) such that

sup_{0<t<T} ∫_{B_{R}} u(x, t)dx ⩽ C(R),

then there exists a constant C(R, α) such that

∫_{0}^{T} ∫_{B_{R}} u(x, t)^{m+2/N−α}dxdt ⩽ C(R, α),

for every 0 < α < min{1/N; m − p + 2/N}.

We will now study the limits of fundamental solutions when 0 ⩽ p ⩽ m. In fact, when p ⩾ m + 2/N, those solutions do not exist, and we know that in the range m < p < m + 2/N, the limit of such solutions is the Very Singular Solution v_{∞}, which takes on the initial trace ν = +∞δ_{0}. We thus note v_{c} the fundamental solution of (1.1) with initial data cδ_{0}, in ℝ^{N}. We refer to [14] for the following result:

Theorem 3.3 Let Ω = ℝ^{N}.

• If 0 ⩽ p ⩽ 1, then v_{c} → +∞ uniformly in ℝ^{N} × (0, T).

• If 1 < p ⩽ m, then v_{c} → c(p)t^{−1/(p−1)} uniformly in ℝ^{N} × (0, T).

3.2 Initial Trace

We will now prove that we can define an initial trace for continuous weak solutions of (1.1). In fact this result was proved in [8, 9] for the cases 1 < m < p and 1 < m = p respectively, and we first give below a localization Lemma valid for the range 0 ⩽ p ⩽ m.

Lemma 3.4 Let 0 ⩽ p ⩽ m and u ⩾ 0 be a solution of (1.1). Let U be a subset of ℝ^{N}. Then the following alternative holds:

(i) If ∫_{U} u(x, s)dx remains bounded when s decreases to zero, then u^{m} ∈ L^{1}(U × (0, T)).

(ii) If u^{m} ∈ L^{1}(U × (0, T)), then for every ζ ∈ C_{0}^{2}(U), the following limit

exists:

lim_{t→0} ∫_{U} u(x, t)ζ(x) dx = ℓ(ζ).

Proof. Since u is the limit of smooth solutions u_{n}, (i) is a direct consequence of Lemma 3.2. Indeed, we can assume with no restriction that the u_{n} satisfy

∫_{U} u_{n}(x, s)ds ⩽ 2 ∫_{U} u(x, s)ds,

which remains bounded when s → 0. Hence

∫_{0}^{T} ∫_{U} u_{n}^{m+2/N−α} ⩽ C(R, α),

and in the limit, the same holds for u, so that in particular

∫_{0}^{T} ∫_{U} u^{m} < ∞.

The proof of (ii) is done as in [9, lemma 4.1]. Indeed, we can show an integral version of the equation with function test ζ :

∫_{U} u(t)ζ − ∫_{U} u(s)ζ − ∫_{s}^{t} ∫_{U} u^{m}Δζ + ∫_{s}^{t} ∫_{U} u^{p}ζ = 0.

Moreover, there exists two constants c_{1} and c_{2} such that

|ζ| ⩽ c_{1},
|Δζ| ⩽ c_{2},

and thus

∫_{0}^{t} ∫_{U} u^{m}|Δζ|, ∫_{0}^{t} ∫_{U} u^{m}|ζ|, ∫_{0}^{t} ∫_{U} u^{p}ζ < ∞,

so that if we let s decrease to zero,

lim_{s→0} ∫_{U} u(s)ζ = ℓ(ζ) < ∞.

□

Thanks to this Lemma, we obtain the main initial trace result.

Theorem 3.5 Let m > 1, p ⩾ 0 and u ⩾ 0 be a solution of u_{t} − Δu^{m} + u^{p} = 0. Then u(t) has an initial trace when t decreases to zero which is a Borel measure ν ⩾ 0 in ℝ^{N}, in the following sense: for any continuous, compactly supported φ ∈ C_{0}(ℝ^{N}), φ ⩾ 0,

∫_{ℝ^{N}} u(x, t)φ(x)dx ⟶_{t→0} ∫_{ℝ^{N}} φ(x)dν(x),  (3.1)

whether the last integral is finite or not.

Proof. As was said, we already know the result if 1 < p < m, so let us assume that 0 ⩽ p ⩽ m and for u as above, let us define

S = {x ∈ ℝ^{N} : ∀U neighborhood of x, ∫_{0}^{T} ∫_{U} u^{m} = ∞}.

Then S is a closed set, and for every x ∈ S, U neighborhood of x, it is clear that

∫_{U} u(x, t)dx ⟶_{t→0} +∞,

because if the above integral remains bounded for some U, then by Lemma 3.4,

∫_{0}^{T} ∫_{K} u^{m} < ∞,

for every K ⊂ U, which leads to a contradiction. On the other hand, if x ∈ R = ℝ^{N} \ S, then there exists a neighborhood U of x such that

∫_{0}^{T} ∫_{U} u^{m} < ∞,

and again by Lemma 3.4, there exists a non negative linear functional ℓ_{U} defined on C_{0}^{2}(U).

As was done in [8], there exists a linear non negative functional ℓ on C_{0}(R), which can then be represented by a Radon measure μ on R. If we set for every Borel subset E of ℝ^{N},

ν(E) = \{ \begin{matrix}μ(E) & if E ⊂ R, \\ +∞ & if E ∩ S ≠ ∅, \end{matrix}

then ν is a Borel measure in ℝ^{N} and u(t) → ν when t → 0 in the following sense: for every φ ∈ C_{0}(ℝ^{N}), such that supp(φ) ⊂ R,

∫_{ℝ^{N}} u(t)φ ⟶_{t→0} ∫_{ℝ^{N}} φdμ,

and for every U such that U ∩ S ≠ ∅,

∫_{U} u(t) ⟶_{t→0} +∞.

Then (3.1) follows for nonnegative continuous φ’s with compact support in ℝ^{N}. □

By convention, we define the initial trace of the infinite solution by:

tr_{ℝ^{N}} (u ≡ +∞) = +∞.

Theorem 3.6 The following results hold:

i) If 0 ⩽ p ⩽ 1 the initial trace of any solution u ≢ +∞ is a Radon measure in ℝ^{N}

ii) If 1 < p ⩽ m the initial trace of any solution u ≢ +∞ is either a Radon measure in ℝ^{N}, or +∞, and in this latter case, we have

u(x, t) ⩾ c(p)t^{−1/(1−p)}.

Proof. This result relies upon comparison with fundamental solution, by using Theorem 3.3. Since such an argument was already used in [9], we only outline the proof.

If the trace of u contains a nonempty singular set S, let y ∈ S and fix c > 0. Then there exists sequences t_{n} → 0 and r_{n} → 0 such that

∀n ∈ ℕ, ∫_{B_{rn} (y)} u(x, t_{n})dx = c,

otherwise the integrals of u would remain bounded near y, which could not be a singular point. Now by comparison, this implies that u is not less than the solution u_{n} in (t_{n}, T) × ℝ^{N} with initial data u_{n}(t_{n}) = u(t_{n})χ_{n}, where χ_{n} is the characteristic function of B_{r_{n}}(y). Then by concentration, u_{n} converges to the fundamental solution v_{c,y} with initial data cδ_{y}, so that u ⩾ v_{c,δ_{y}}. Since c is arbitrary, we let c → ∞, thanks to Theorem 3.3, which yields that u is either +∞ everywhere or the flat solution whether p is less or greater than 1. □

4 Very Weak Absorption: 0 ⩽ p ⩽ 1

We consider equation (1.1) in the range 0 ⩽ p ⩽ 1. We will see that in terms of initial data, the admissiblity condition is the same as for the diffusive case u_{t} = Δu^{m} (see [2, 4]). Recall that in this range, solutions are only local a priori, as it is the case for the diffusive equation. A solution in Q_{T} is thus understood to be defined up to t = T (see remark after Theorem 4.4).

4.1 Harnack Inequality and Admissibility

We prove now that when p < 1, A^{+}(p) = A^{+}(1), in other terms, in this case, the absorption has no effect on admissibility of initial traces, compared with the purely diffusive case u_{t} = Δu^{m} (we showed above that the case p = 1 can be reduced to the diffusive equation with a suitable change of variables and functions).

The following lemma reduces our study to the case p = 0.

Lemma 4.1 Let μ be a Radon measure. Then for every p ∈ [0, 1], the following inclusion holds:

A^{+}(1) ⊂ A^{+}(p) ⊂ A^{+}(0).

Proof. The first inclusion is obvious since A^{+}(1) is related to equation E_{1} : u_{t} − Δu^{m} + u = 0, and this equation has the same admissibility set than the purely diffusive equation u_{t} = Δu^{m}. Indeed, there exists a change of time variable which maps solutions of u_{t} = Δu^{m} to E_{1} and preserves the initial data. So if a trace is admissible for E_{1}, it is also for the diffusive equation, and then it is admissible if we add any absorption term. We are left to prove the second inclusion.

Let p ∈ [0, 1], and μ ∈ A^{+}(p). Then we have u^{p} ⩽ u + 1, and thus

u_{t} − Δu^{m} + u + 1 ⩾ 0.

Now we use the same change of time variable (and of function) that maps E_{1} to u_{t} = Δu^{m} : we get a function w which has initial data μ and

w_{t} − Δw^{m} + 1 ⩾ 0,

that is w is a super-solution of E_{0}. Now it is easy to construct a solution with initial data μ : let μ_{n} be a sequence of bounded measures converging monotonically to μ and v_{n} the sequence of minimal solutions of E_{0} with initial data μ_{n}. Then v_{n} increases to some function v and since w is an upper bound for v, we know that v has locally finite initial trace. Then by monotonicity, it is obvious that tr_{ℝ^{N}} (v) = μ (we have already proved this thing in the previous theorems). Thus μ ∈ A^{+}(0) and the result is proved. □

We now consider the case p = 0, and more generally, we deal with nonnegative solutions of the equation

u_{t} = Δ(u^{m}) − aχ(u > 0),  a ⩾ 0,  (4.1)

defined in Q_{T} = ℝ^{N} × (0, T). We obtain a Harnack inequality for this kind of equations, which includes in particular the porous medium equation when a = 0. In this case, it was already obtained by Aronson and Caffarelli [2], but our method is new for a = 0 and quite simple.

Lemma 4.2 Let u ∈ C^{0}(ℝ^{N} × [0, T]) be a nonnegative solution of (4.1) in p_{1} with 0 ⩽ a ⩽ A. Let

M = ∫_{B_{1}} u(x, 0) dx.  (4.2)

There exist positive constants M_{0} = M_{0}(N, m, A) and k = k(N, m, A) such that for M ⩾ M_{0}

u(0, 1) ⩾ k M^{2λ}, λ = (N(m − 1) + 2)^{−1}.  (4.3)

Proof. It is the combination of several steps. The letter C will denote different positive constants that depend only on N and m.

• By comparison we may assume that u_{0} is supported in the unit ball B_{1}. Indeed, for general u_{0}, then u_{0} is greater than u_{0}η, η being a suitable cut-off function compactly supported in B_{1} and less than one. Thus if v is the solution with initial data u_{0}η (existence and uniqueness are well-known in this case), we obtain

∫_{B_{1}} u(x, 0)dx ⩾ ∫_{B_{1}} u_{0}η = M,

and if the lemma holds true for v, then

u(0, 1) ⩾ v(0, 1) ⩾ kM^{2λ}.

We may then take the domain of definition as p = ℝ^{N} × (0, ∞).

• By comparison with the porous medium equation without absorption, we know the a priori estimate for the solution [4, p. 54]

0 ⩽ u(x, t) ⩽ C M^{2λ} t^{−Nλ}  (4.4)

and the a priori estimate for the support at time t,

supp u(·, t) ⊂ B_{R}(t), R(t) = C M^{(m−1)λ} t^{λ}.  (4.5)

Hence, if M is large this radius is much larger than 1 at t = 1.

• The reflection argument of Aleksandrov used in Lemma 2.2 of [2] means that for |x| ⩾ 2 we have

u(0, t) ⩾ u(x, t).  (4.6)

• Let us now estimate the mass at time t

∫ u(x, t) dx = ∫ u_{0}(x) dx − a ∫_{0}^{t} ∫ χ(u > 0) dxdt.  (4.7)

The last term is bounded above by C a t R(t)^{N}, while the first member can be split into the integrals

∫_{|x|⩾2} u(x, t) dx + ∫_{|x|⩽2} u(x, t) dx,

and the last term can be estimated by

C M^{2λ} t^{−Nλ}2^{N}.

We conclude that

Cu(0, t) (R(t)^{N} − 2^{N}) ⩾ ∫_{|x|≥2} u(x, t) dx ⩾ M − C a t R(t)^{N} − C M^{2λ} t^{−Nλ}2^{N},

hence for t = 1,

C u(0, 1) (M^{N(m−1)λ} − 2^{N}) ⩾ M − C a M^{N(m−1)λ} − C M^{2λ},

so that there are three constants c_{1}, c_{2}, c_{3}(m, N) such that

u(0, 1) ⩾ c_{1}M^{2λ} − ac_{2} − c_{3}M^{γλ}, γ = 2 − (m − 1)N.

Since γ < 2, there exists some constants M_{0} and k such that

c_{1}M^{2λ} − ac_{2} − c_{3}M^{γλ} ⩾ kM^{2λ}

holds for every M ⩾ M_{0}, and this proves the Lemma. □ Now we give the Harnack-type inequality.

Lemma 4.3 The estimate of the form

∫_{B_{r}(x_{0})} u(x, t) dx ⩽ C (r^{1/λ(m−1)}T^{−1/(m−1)} + T^{N/2}u^{1/2λ}(x_{0}, T))  (4.8)

holds for all nonnegative solutions if r ⩾ T^{m/2} and t ⩽ T.

Proof. We can use the previous lemma on (t, T) since u ∈ C^{0}(Q_{T}), perform the transformation

u^{∗}(x, t) = r^{−2/(m−1)}T^{1/(m−1)}u(rx, T t)  (4.9)

as in [2, p. 361], and look at the equation satisfied by u^{∗}. Now it is the same (4.1) but for the constant a which becomes

a′ = a r^{−2/(m−1)} T^{m/(m−1)}.  (4.10)

Hence, in order to apply the previous lemma we need to impose the condition r ⩾ T^{m/2}. □

Remark. The previous lemma gives a direct proof of the existence of the initial trace for E_{0}, which is a Radon measure.

Theorem 4.4 For every p ∈ [0, 1], we have that following characterization:

μ ∈ A^{+}(p) ⇔ sup_{R⩾1} R^{− \frac{2}{m−1} −N} ∫_{B_{R}} dμ(x) < ∞.

In other words, the admissibility condition is the same as in the purely diffusive equation.

Proof. By Lemma 4.1, we have only to prove the converse inclusion

A^{+}(0) ⊂ A^{+}(1).

If μ is admissible for E_{0}, there exists a minimal solution u. Hence by (4.8), we get

r^{1/λ(m−1)} ∫_{B_{r}(x_{0})} dμ ⩽ C (T^{−1/(m−1)} + r^{−1/λ(m−1)}T^{N/2}u^{1/2λ}(x_{0}, T)) , (4.11)

and since (λ(m − 1))^{−1} = N + 2/(m − 1), this implies that

sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} dμ < ∞,

thus μ is admissible for E_{1}, and the theorem is proved. □

Remark. The existence of a solution of E_{p} with initial data μ is only valid up a time T_{p}(μ) in this range. It is obvious that T_{p}(μ) is not less that the blow-up time T (μ) in the case of the purely diffusive equation [4]:

T_{p}(μ) ⩾ T (μ) ⩾ C(m, N)/ℓ(μ)^{m−1},

where

ℓ(μ) = lim_{r→∞} sup_{R⩾r} R^{− \frac{2}{m−1}} − ∫_{B_{R}} dμ.

In fact, in the case p = 1, the blow-up time can be computed thanks to the exponential change of time variable, and we find:

T_{1}(μ) ⩾ \frac{1}{m − 1} exp ((m − 1)\frac{C(m, N)}{ℓ(μ)}),

which is greater than T (μ).

4.2 Optimal Uniqueness

We have seen that in the case 0 ⩽ p ⩽ 1, the admissibility condition for E_{p} was the same as for the case u_{t} = Δu^{m}, and since uniqueness holds for this diffusive equation with no growth restriction [11], one can reasonably think that the same holds for E_{p}. We prove here that it is indeed the case.

The following Lemma shows some a priori estimates for solutions of E_{p}, similar to the one satisfied by the solutions of the diffusive equation. For every α > 0, we note

ρ_{α}(x) = [1 + |x|^{2}]^{α}.

Lemma 4.5 Let u be a solution of E_{p} with initial data μ ∈ A^{+}(p), i.e.

sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} dμ < ∞.

Then the following bound holds:

u(x, t) ⩽ C(t)ρ_{\frac{1}{m−1}} (x) in \{|x| ⩾ 1\} × (0, T),

where C(·) ∈ L_{loc}^{∞}(0, T). Moreover,

∫_{ℝ^{N}} u(x, t)ρ_{α}dx ⟶_{t→0} ∫_{ℝ^{N}} ρ_{α}(x)dμ(x),  (4.12)

for every α > 1 + \frac{N}{2} + \frac{1}{m−1}.

Proof. Let s > 0. Since u is a solution of E_{p} on (s, T), necessarily,

sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} u(s) < ∞,

and thus u(s) is also an admissible initial data for u_{t} − Δu^{m} = 0. We call v_{s} the solution associated with v_{s}(0) = u(s) for the diffusive equation (v_{s} is unique by the results of [11]). Moreover, since u is the limit of a sequence of smooth solutions on (s, T), we can compare u with v_{s}:

u(x, t) ⩽ v_{s}(x, t) in ℝ^{N} × (s, t).

By the estimates on v_{s} [4, Rem. 3], we know that

v_{s}(x, t) ⩽ \frac{c(s)}{t^{λ}}[1 + |x|^{2}]^{\frac{1}{m−1}} in \{|x| ⩾ 1\} × (s, T),

where λ = \frac{N}{N(m−1)+2} and

c(s) ⩽ c(N, m)[ sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} u(s)]^{2λ/N}.

The function c(s) remains bounded when s decreases to zero (this is a consequence of the Harnack inequality (4.8)), we find that for some function C(·) ∈ L_{loc}^{∞}(0, T),

u(x, t) ⩽ C(t)[1 + |x|^{2}]^{\frac{1}{m−1}} in \{|x| ⩾ 1\} × (0, T).

Moreover, these techniques show that

u(x, t) ⩽ v(x, t) in Q_{T} ,

where v is the unique solution of the diffusive equation u_{t} = Δu^{m} with initial data μ. But the convergence property (4.12) holds for v (see [4, p. 81]), so that it also holds for u. Indeed, (all integrals are taken over ℝ^{N})

∫ u(t)ρ_{α} − ∫ ρ_{a}dμ = ∫ \underbrace{(u − v)}_{⩽0}(t)ρ_{a} + ∫ v(t)ρ_{α} − ∫ ρ_{α}dμ,

so that

sup lim_{t→0} ∫ u(t)ρ_{α} ⩽ ∫ ρ_{a}dμ,

and since u(t) → μ weakly in measure, clearly

∫ u(t)ρ_{α} ⟶_{t→0} ∫ ρ_{a}dμ.

□

Theorem 4.6 Let 0 ⩽ p ⩽ 1 and ν ∈ A^{+}(p), i.e., ν satisfies

sup_{R⩾1} R^{− \frac{2}{m−1}} − ∫_{B_{R}} dν < ∞.

Then there exists a unique solution u to(1.1) such that tr_{ℝ^{N}} (u) = ν.

Proof. Thanks to the previous a priori estimate, we can use the same techniques as in [4, Prop. 2.1], which consists in solving the dual problem. Before this, we need to construct a minimal solution, which can be obtained as in [8, Sec. 4.2]: if u is any solution, let u_{R,τ} be the unique solution of the problem

∂_{t}u_{R,τ} − Δu_{R,τ}^{m} + u_{R,τ}^{p} = 0 in B_{R} × (τ, T),
u_{R,τ} (x, t) = 0 on ∂B_{R} × (τ, T),
u_{R,τ} (τ) = u(τ) in B_{R}.

By comparison in this set, since both solutions are bounded,

u_{R,τ} (x, t) ⩽ u(x, t) in B_{R} × (τ, T).

Then if we let τ decrease to zero, we see that u_{R,τ} converges locally uniformly to a solution u_{R} with initial data μ/_{B_{R}} and zero lateral data on ∂B_{R} × (0, T). Moreover, u_{R} is uniquely determined, as was proved in [8, Theorem 6.2], so that it can be constructed independently of any solution, and in the limit,

u_{R}(x, t) ⩽ u(x, t) in B_{R} × (0, T).

Finally, when R increase to +∞, u_{R} increases to some solution \underline{u} with initial data μ, which is the (unique) minimal solution since \underline{u}(x, t) ⩽ u(x, t) in Q_{T}.

Now if u is any solution with initial data μ, we will prove that u ≡ \underline{u}, hence uniqueness since \underline{u} can be constructed independently of any solution. Let us first fix s > 0 and t ∈ (s, T). Since u ⩾ \underline{u}, we have

(u − \underline{u})_{t} − Δ(u^{m} − \underline{u}) = \underline{u}^{p} − u^{p} ⩽ 0,

and thanks to the a priori estimate given by Lemma 4.5,

uρ_{\frac{1}{m−1}} , \underline{u}ρ_{\frac{1}{m−1}} ∈ L^{∞}(ℝ^{N} × (s, t)).

Then the techniques of [4, Prop. 2.1] apply verbatim and give

∫_{ℝ^{N}} (u − \underline{u})(t)θ ⩽ ∫_{ℝ^{N}} |u − u\underline{u}|(s)ρ_{β},

where θ ∈ C_{0}^{∞}(ℝ^{N}), 0 ⩽ θ ⩽ 1 is arbitrary and β > \frac{N−1}{2} + \frac{m}{m−1} can also be chosen freely. In [4], since both solutions have the same value at s = 0 in L_{loc}^{1}, the conclusion is that the solutions coincide everywhere. But here we have to let s decrease to zero, so we use that fact that u[ν] is minimal:

∫ |u − \underline{u}|(s)ρ_{β} = ∫ u(s)ρ_{β} − ∫ \underline{u}(s)ρ_{β},

which both converge to the same value when s goes to zero, thanks to (4.12) with a suitable β. Hence in the limit,

∫_{ℝ^{N}} (u − \underline{u})(t)θ ⩽ 0,

which proves that u ≡ \underline{u} since θ ⩾ 0 and t > 0 are arbitrary. □

5 Weak Absorption: 1 ⩽ p ⩽ m

5.1 The Projection Operator

We now construct a projection operator ℙ_{p} : B^{+}(Ω) → B^{+}(Ω). Actually, the construction remains valid in the range 0 ⩽ p ⩽ 1 previously studied, so that we give it in its full generality below. It is based on the following Lemma:

Lemma 5.1 Let 0 ⩽ p ⩽ m, and ν ∈ B^{+}(ℝ^{N}). Let μ_{n} ⩾ 0 be a sequence of compactly supported measures converging monotonically to ν and u_{n} the associated sequence of solutions. Then u_{n} converges locally uniformly to the minimal solution u[ν] with initial data ν. Moreover, if tr_{ℝ^{N}} (u[ν]) is not locally finite, then

\begin{aligned}0 ⩽ p ⩽ 1 & ⇒ u[ν] ≡ +∞, \\ 1 < p ⩽ m & ⇒ u[ν] = c(p)t^{−1/(p−1)}. \end{aligned}  (5.1)

Proof. Let ν ∈ B^{+}(ℝ^{N}). We will proceed in several steps: Step 1 Let us assume that 1 < p ⩽ m. We construct a minimal solution u[ν] by the following procedure: let {μ_{n}}_{n∈ℕ} be a sequence of bounded measures converging monotonically to ν, and {u_{n}}_{n∈ℕ} the associated sequence of solutions of (1.1). Then u_{n} also converges monotonically to some function u ∈ C^{0}(ℝ^{N} × (0, T)). Indeed, we have the universal bound u_{n} ⩽ c(p)t^{−1/(p−1)}. We will prove later the minimality of u, so that we are left to show that tr_{ℝ^{N}} (u) = ν. In fact, we know that either tr_{ℝ^{N}} (u) = ν′ is locally finite, or tr_{ℝ^{N}} (u) = +∞.

We first assume that tr_{ℝ^{N}} (u) = ν′ is locally finite. For every φ ∈ C_{0}^{2}(ℝ^{N}), we have both

∫_{ℝ^{N}} u_{n}(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ = ∫_{ℝ^{N}} φdμ_{n},  (5.2)

and

∫_{ℝ^{N}} u(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} u^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u^{p}φ = ∫_{ℝ^{N}} φdν′.

But since u_{n} ⩽ u and u ∈ L^{m}(0, T; L_{loc}^{m}(ℝ^{N})) (because ν′ is locally finite - see Lemma 3.4), then passing to the limit in the first equation and using the second leads to

∫_{ℝ^{N}} φdν′ = ∫_{ℝ^{N}} φdν,

which proves that ν′ = ν, hence u is a solution with initial data ν. Now if tr_{ℝ^{N}} (u) = +∞, then u_{n} converges to a solution u which has initial trace +∞. Indeed, if this were not true, then u_{n} would converge to a function u which has a locally finite initial trace and it would thus enjoy the following property:

u ∈ L^{m}(0, T; L_{loc}^{m}(ℝ^{N})).

Then choosing φ ∈ C_{0}^{2}(ℝ^{N})^{+}, and passing to the limit in equation (5.2) we would get (recall that μ_{n} converges to +∞)

∫_{ℝ^{N}} u(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} u^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u^{p}φ = +∞,

which contradicts the fact that the trace of u is locally finite. Step 2. Let us assume now that 0 ⩽ p ⩽ 1. In this case, we prove that u ≡ +∞. In fact, the limit u of the u_{n} cannot be locally bounded in Q_{T} otherwise we would obtain a solution with trace ν (same proof as Step 1). Actually the exact condition for ν to be admissible is known and we can write (4.11) for the u_{n} under the form:

r^{1/λ(m−1)} ∫_{B_{r}(x_{0})} dμ_{n} ⩽ C(t_{0}^{−1/(m−1)} + t_{0}^{N/2}r^{−1/λ(m−1)}u_{n}^{1/2λ}(x_{0}, t_{0})).

Thus if we assume that there exists a point (x_{0}, t_{0}) such that u(x_{0}, t_{0}) < ∞, we get a bound for r^{1/λ(m−1)} ∫_{B_{r}(x_{0})} dμ_{n} as n → +∞, which says that ν satisfies the same bound. Then it is admissible by Theorem 4.4, and there exists a local solution u with trace ν. Thus we reach a contradiction, so that u ≡ +∞ in Q_{T}. Step 3. The minimality of u follows from uniqueness for finite measures (by Theorem 3.1). Thus, u ⩾ u_{n} follows from uniqueness of u_{n} and the fact that μ_{n} ⩽ μ (this kind of argument was used extensively in [8, 9], so that we omit the details). Passing to the limit in n gives the minimality of u. □

Note that the minimality of u[ν] guaranties that whatever the approximations μ_{n} are, we obtain always the same limit, so that the following definition makes sense.

Theorem 5.2 For every 0 ⩽ p ⩽ m, we define a mapping ℙ_{p} : B^{+}(ℝ^{N}) → B^{+}(ℝ^{N}) by

ℙ_{p}(ν) = tr_{ℝ^{N}} (u[ν]),

with the convention that tr_{ℝ^{N}} (+∞) = +∞. Then ℙ_{p}(B^{+}(ℝ^{N})) = A^{+}(ℝ^{N}), and ℙ_{p} is a projection, i.e., ℙ_{p} ∘ ℙ_{p} = ℙ_{p}. Moreover, we have the characterization:

ν ∈ A^{+}(p) ⇒ ℙ_{p}(ν) = ν,
ν ∉ A^{+}(p) ⇒ ℙ_{p}(ν) = +∞,

and in this latter case, (5.1) holds.

Proof. As we noticed, ℙ_{p} is well-defined since the minimal solution does not depend on the approximations used. By construction, it is clear that ℙ_{p}(ν) is an admissible measure for any ν ∈ B^{+}(ℝ^{N}). Claim: ν ∈ A^{+}(p) ⇒ ℙ_{p}(ν) = ν. Let us first assume that 1 < p ⩽ m and let μ_{n} be a sequence of nonnegative compactly supported measures converging to ν. Let also u_{n} be the associated sequences of solutions. Then u_{n} converges to some function u′ with trace ℙ_{p}(ν) = ν′ ⩽ ν. Then arguing as in Lemma 5.1 (Step 1), yields

ν′ = ν, i.e., ℙ_{p}(ν) = ν.

Now if 0 ⩽ p ⩽ 1, we use exactly the same argument if ν is locally finite. If ν ≡ +∞, then for any compact set K,

∫_{K} dμ_{n} ⟶_{n→∞} +∞,

so that the Harnack Inequality (4.11) implies that u_{n} → +∞ everywhere. Here also, the argument follows Lemma 5.1 (Step 2).

This result proves several things: first that ℙ_{p}(B^{+}(ℝ^{N})) = A^{+}(ℝ^{N}), since any admissible data ν may be written under the form ν = ℙ_{p}(ν). Second, ℙ_{p} is a projector:

∀ν ∈ B^{+}(ℝ^{N}), ℙ_{p}(ℙ_{p}(ν)) = ℙ_{p}(ν), since ℙ_{p}(ν) ∈ A^{+}(ℝ^{N}).

Third, we have

ℙ_{p}(ν) = ν ⟺ ν ∈A^{+}(ℝ^{N}).

It remains to prove that if ν ∉ A^{+}(ℝ^{N}), then ℙ_{p}(ν) = +∞. In the case 0 ⩽ p ⩽ 1, we have the exact characterization, so that clearly, if ν ∉ A^{+}(ℝ^{N}), when we approximate it, the left-hand side of (4.11) goes to +∞, so that u_{n} → +∞ everywhere, x_{0} and T being arbitrary. Thus indeed, ℙ_{p}(ν) = +∞ since u[ν] ≡ +∞. Now in the case 1 < p ⩽ m, u_{n} will always converge monotonically to some solution u[ν]. And we saw that tr_{ℝ^{N}} (u) cannot be locally finite, otherwise monotonicity would imply that tr_{ℝ^{N}} (u) = ν (see Lemma 5.1). Thus tr_{ℝ^{N}} (u) = +∞, that is, ℙ_{p}(ν) = +∞ and the Theorem is completed. □

5.2 Admissibility Sets

We have seen that when p ∈ [0, 1], the absorption term has no effect on admissibility of measures. Moreover, when p = m, we know the exact characterization of admissible data. Unfortunately, such a characterization is not known so far for measures when 1 < p < m, but we have at least the following result which proves that the set of admissible data is increasing with p in this range.

Theorem 5.3 For every p, p′ ∈ [1, m] such that p < p′, we have

A^{+}(p) ⊊ A^{+}(p′).

In other words, the function p ↦ A^{+}(p) is increasing in [1, m].

Proof. Recall that thanks to Lemma 5.1, we can work with minimal solutions, hence limit solutions. We can thus use smooth solutions, the passage to the limit being automatic.

Let us assume that p′ > p and that μ ∈ A^{+}(p). Then we shall show that μ is also an admissible data for E_{p′}, that is, μ ∈ A^{+}(p′). In fact, we will first construct a super-solution for equation E_{p′} on some small interval (0, t_{0}), with initial data μ, and then we will easily show that there exists a solution with initial data μ on (0, T).

Let u be the minimal solution associated with μ, and let

v(x, t) = (1 + ct) · u(x, t), (x, t) ∈ ℝ^{N} × (0, t_{0}),

for some parameters c, t_{0} > 0 which will be specified later on. Then a straightforward calculation gives

v_{t} = (1 + ct)^{1−m}Δv^{m} + c(1 + ct)^{−1}v − (1 + ct)^{1−p}v^{p},

and changing the time variable in τ ∈ (0, τ_{0}(t_{0})) such that

\frac{dτ}{dt} = (1 + ct)^{1−m},

we get, setting v(x, t) = w(x, τ),

w_{τ} = Δw^{m} − w^{p}(1 + ct)^{m−p} + cw(1 + ct)^{m−2}.

Note that if we work with classical solutions,

w(x, 0) = v(x, 0) = u(x, 0),

and this is again the case if the initial data is a measure, the above equality being understood in the sense of initial traces. Moreover, for a good choice of t_{0} and c, we show that w is a super-solution of equation E_{p′} with initial data μ : let c be a free parameter for the moment and put t_{0} = 1/c, then for every t ∈ (0, t_{0}),

min{1; 2^{m−2}} < (1 + ct)^{m−p}, (1 + ct)^{m−2} < 2^{m}.  (5.3)

Now since p′ > p, there exists a constant k(m, p, p′),such that for w ⩾ k,

w^{p′} ⩾ 2^{m}w^{p},

and for w ⩽ k (fixed above), thanks to (5.3) there exists some c(m, p, p′) (maybe big but finite), such that

cw(1 + ct)^{m−2} ⩾ w^{p}(1 + ct)^{m−p}.

Thus, we have obtained that whatever the value of w,

w^{p′} + cw(1 + ct)^{m−2} ⩾ (1 + ct)^{m−p}w^{p}, on (0, t_{0}),

so that there exists some τ_{0} > 0 (only depending on m, p and p′ through c and k) such that for τ ∈ (0, τ_{0})

w_{τ} ⩾ Δw^{m} − w^{p′},

that is, w is a super-solution of E_{p′} with initial data μ.

Now we can construct a solution of E_{p′} with initial data μ : let μ_{n} be a sequence of bounded measures converging to μ monotonically and u_{n} be the associated sequence of minimal solutions. Then u_{n} also increases to some distributional solution u ∈ C^{0}(ℝ^{N} × (0, T)), and u_{n} has an initial trace ν ∈ B^{+}(ℝ^{N}). We also construct the associated sequence of functions w_{n} on (0, τ_{0}) by the same process as above, taking the initial data μ_{n}. By construction, we have on (0, τ_{0}),

u_{n}(t) ⩽ w_{n}(t) ⩽ w(t),

and since u ∈ L^{m}(0, T; L_{loc}^{m}(ℝ^{N})), then so is w, so that the {u_{n}} remain uniformly bounded by w which is in L^{m}(0, T; L_{loc}^{m}(ℝ^{N})), and in L_{loc}^{1}(ℝ^{N}) for every τ ∈ (0, τ_{0}). As we already seen, this argument allows us to pass to the limit in the following equation, where φ ∈ C_{0}^{2}(ℝ^{N}):

∫_{ℝ^{N}} u_{n}(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ = ∫_{ℝ^{N}} φdμ_{n},

and thus we get

∫_{ℝ^{N}} u(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} u^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u^{p}φ = ∫_{ℝ^{N}} φdμ,

hence initial trace of u is μ. Thus μ is an admissible data for E_{p′}, and the theorem is proved. Finally, for p < p′ in (1, m), the inclusion A^{+}(p) ⊂ A^{+}(p′) is strict, this can be easily seen on function initial data [15], and the limit cases p = 1 and p = m where discussed above. □

6 Super-Critical Absorption: p ⩾ m + 2/N

We will now concentrate on the case when the absorption exponent is bigger than the critical value p_{c} = m + 2/N. The operator is defined differently from the cases 0 ⩽ p ⩽ m since we cannot use monotonicity properties here.

6.1 The Operator ℙ_{p}

In this section, we will see that given an approximation process of the initial data ν ∈ B^{+}(Ω) by smooth functions, we obtain in the limit a solution which may have a smaller initial trace ν′ ⩽ ν. As for the case 1 < p ⩽ m, we will define ℙ_{p} as follows: ℙ_{p}(ν) = ν′, but are not able to prove that ℙ_{p} is a projector in B^{+}(ℝ^{N}). To fix ideas, we will use a sequence {ρ_{n}} of nonnegative functions which approximate δ_{0}, with support supp(ρ_{n}) = B_{1/n}(0), but any other approximation process would give the same property. We point out however, that the value of ℙ_{p}(ν) may depend on the approximation process.

The definition of ℙ_{p} will relies on the two following Lemmas.

Lemma 6.1 Let p > m and {u_{n}}_{n∈ℕ} be a sequence of solution of (1.1) in ℝ^{N} × (0, T) with initial trace ν_{n} such that u_{n} is increasing and converges locally uniformly to some u. Then u is a solution of (1.1) with initial trace

tr_{Ω}(u) = lim ν_{n}.

Proof. Let us first notice that ν_{n} = tr_{Ω}(u_{n}) is well-defined since u_{n} is a weak solution, as well as {tr_{Ω}(u) = (S, μ). Let us call ν = lim ν_{n}, which is well-defined by monotonicity. Now for any φ ∈ C_{0}(R), and n big enough, we may write

∫_{ℝ^{N}} u_{n}(t)φ(t) + ∫_{0}^{t} ∫_{ℝ^{N}} {−u_{n}φ_{t} − u_{n}^{m}Δφ + u_{n}^{p}φ} = ∫_{ℝ^{N}} φdν_{n} < ∞. (6.1)

Indeed, by monotonicity of u_{n}, the sequence of Borel measures ν_{n} = (S^{n}, μ_{n}) is also monotone so that φ has compact support in ℝ^{N} \ S^{n}, for n sufficiently. To get some bounds, let us take φ(x, t) = φ(x) ∈ C_{0}^{∞}(ℝ^{N}), 0 ⩽ φ ⩽ 1, with support in some ball B_{r} ⊂ R, and let us define

X_{n}(t) = ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}φ, Y_{n}(t) = ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ.

Then for a good choice of φ (see [8]), we have

∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{m}|Δφ| ⩽ c(φ)Y_{n}(t)^{m/p},

so that we arrive at the following differential inequality, where C = ν(B_{r}) < ∞ :

\frac{dX_{n}(t)}{dt} − c(φ)Y_{n}(t)^{m/p} + Y_{n}(t) ⩽ C.

Then clearly, we get a uniform bound for dX_{n}/dt = ∫_{ℝ^{N}} u_{n}(t)φ(t), so that now we can use Lemma 3.2: it yields that u_{n} is uniformly bounded in L^{q}(0, T; L_{loc}^{q}(R)), for any q < m + 2/N, so that

∫_{0}^{t} ∫_{ℝ^{N}} u_{n}φ_{t} ⟶_{n→∞} ∫_{0}^{t} ∫_{ℝ^{N}} uφ_{t}, ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{m}Δφ ⟶_{n→∞} ∫_{0}^{t} ∫_{ℝ^{N}} u^{m}Δφ.

If m < p < m + 2/N, we can use the same convergence for the absorption term, but for p ⩾ m + 2/N, we use the monotonicity to pass to the limit:

∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ ⟶_{n→∞} ∫_{0}^{t} ∫_{ℝ^{N}} u^{p}φ.

Passing to the limit in (6.1) yields

∫_{ℝ^{N}} u(t)φ(t) + ∫_{0}^{t} ∫_{ℝ^{N}} {−uφ_{t} − u^{m}Δφ + u^{p}φ} = ∫_{ℝ^{N}} φdν < ∞.

¿From this we deduce that on ℝ^{N} \ S, μ = ν. It remains to show that the blow-up set of ν is exactly S, which is easy: for any y in the blow-up set of ν, and any r > 0,

∫_{B_{r}(y)} u(x, t)dx ⩾ ∫_{B_{r}(y)} u_{n}(x, t)dx,

so that as t → 0,

lim_{t→0} ∫_{B_{r}(y)} u(x, t)dx ⩾ ν_{n}(B_{r}(y)).

Thus when n → ∞, we find

lim_{t→0} ∫_{B_{r}(y)} u(x, t)dx = +∞,

which means that y ∈ S. On the other hand, if we assume that y ∈ S, there exists some r > 0 such that ν(B_{r}(y)) < ∞. This implies that for any n ∈ ℕ,

ν_{n}(B_{r}(y)) ⩽ ν(B_{r}(y)) < ∞.

Thus by Lemma 3.2, we have a uniform bound for u_{n}^{m} in L^{1}((0, T)×B_{r}(y)), and thus also for u_{n}^{p} (this derives from (6.1)), so that in the limit, u^{m} and u^{p} will be locally integrable near y. This proves that y ∉ S, and we reach a contradiction. □

Lemma 6.2 Let 1 < m < p. Then for any ν_{n} = ρ_{n} ⋆ ν as above, there exists a solution u_{n} of (1.1) which takes on the initial trace ν_{n}.

Proof. Let us decompose ν as (S, μ), where S ⊂ ℝ^{N} is closed and μ is a Radon measure on R = ℝ^{N} \ S. Then ν_{n} = ρ_{n} ⋆ ν has the decomposition (S^{n}, μ_{n}), where μ_{n} is smooth in R^{n} = ℝ^{N} \ S^{n}, and

S^{n} = {x ∈ ℝ^{N} : dist(x, S) ⩽ 1/n}.

This small expansion of S allows us to construct the following initial data:

u_{n,c}(0, x) = \begin{cases}min\{c; μ_{n}(x)\} & on R^{n}, \\ c & on S^{n}, \\ 0 & otherwise, \end{cases}

which clearly converges monotonically to (S^{n}, μ_{n}) as c increases to +∞. Then the associated sequence of solutions u_{n,c} will also converge monotonically and locally uniformly to some function u, and monotonicity insures that the initial trace of u is exactly (S^{n}, μ_{n}). □

Now we can define ℙ_{p} as follows: for any ν ∈ B^{+}(ℝ^{N}), and ν_{n} = ρ_{n} ⋆ ν, the sequence u_{n} of Lemma 6.2 converges locally uniformly to some weak solution u[ν] in ℝ^{N} × (0, T). Thus setting

ℙ_{p}(ν) = tr_{ℝ^{N}} (u[ν]),

we have the following theorem:

Theorem 6.3 For any p ⩾ m + 2/N, the operator ℙ_{p} defined above satisfies ℙ_{p}(B^{+}(ℝ^{N})) = A^{+}(ℝ^{N}), and for any ν ∈ B^{+}(ℝ^{N}),

ℙ_{p}(ν) = ν′ ⩽ ν.

Proof. We note ν = (S, μ), ν′ = (S′, μ′) and test functions φ ⩾ 0, with compact support in R. Let also ν_{n} = ρ_{n}ν. Since ∫ φdν_{n} remains bounded (for n big enough), we get a bound for u in L_{loc}^{q}((ℝ^{N} \ S) × (0, T)), for any 1 < q < m + 2/N. Thus, in (6.1), we have convergence of u in L_{loc}^{1} and L_{loc}^{m} of the same set. This proves first that S′ ⊂ S, and moreover using Fatou’s Lemma for the term

∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ,

one gets in the limit

∫_{ℝ^{N}} u(t)φ(t) + ∫_{0}^{t} ∫_{ℝ^{N}} {−uφ_{t} − u^{m}Δφ + u^{p}φ} ⩽ ∫_{ℝ^{N}} φdν,

which proves that on ℝ^{N} \ S, tr_{Ω}(u) ⩽ ν. If p < m + 2/N, then the absorption term also converges, and thus we pass to the limit with equality. The same happens if we assume monotonicity, by monotone convergence of the absorption term. The case when φ is not assumed to be nonnegative easily follows. □

6.2 Admissibility Sets

As we said in the introduction, in the super-critical case, the operator ℙ_{p} is not well-defined. However, we prove here that the sets of admissible initial data do not increase with p in this range, and we give some properties of A^{+}(p) further.

Let us recall [8, Sec. 4.2] that uniqueness holds for locally finite measures also in the super-critical case. The proof was made under the assumption 1 < m < p < m + 2/N, but the same arguments remain true if p is super-critical with no adaptations needed. In fact, this was stated in the introduction of [8]. Uniqueness is not known so far for general Borel measures, but we refer to [7] for further results.

Here is now the monotonicity property of the sets A^{+}(p).

Theorem 6.4 Let m + 2/N ⩽ p < p′. Then A^{+}(p′) ⊂ A^{+}(p).

Proof. Let ν ∈ A(p′), where p′ > p, and v be the solution associated with equation E_{p′}. We would like to show that there exists a solution u of equation E_{p}. Let us first remark that using the same technique as in Theorem 5.3, we can easily show that there exist some constants t_{0}, c > 0 such that the function

w(x, τ) = (1 − ct) · v(x, t), \frac{dτ}{dt} = (1 − ct)^{1−m},

satisfies

w_{τ} − Δw^{m} + w^{p} ⩽ 0 in ℝ^{N} × (0, τ_{0}),

where τ_{0} > 0 depends on t_{0} by the change of variable. Hence w is a sub-solution of equation E_{p} in ℝ^{N} × (0, τ_{0}), and remembering that the constants t_{0} and c depend only on m, p, p′, we take n big enough so that 1/n < τ_{0}. Now let u_{n} be the solution of the following problem:

∂_{t}u_{n} − Δu_{n}^{m} + u_{n}^{p} = 0 in Q_{T} ,
u_{n}(1/n) = w(1/n) in ℝ^{N}.

It is well-known that u_{n} exists and is unique since w(1/n) is continuous in ℝ^{N} (see [18] or [8]). Using the same arguments as in the previous Subsection, it is clear that the u_{n} will converge (up to extraction) to a solution u such that

tr_{ℝ^{N}} (u) ⩽ ν.

But both u_{n} and v (thus w) are limits of regular solutions in ℝ^{N} × (1/n, T), and thus they can be compared in ℝ^{N} × (1/n, τ_{0}), which gives that

u_{n}(x, τ) ⩾ w(x, τ) in ℝ^{N} × (1/n, τ_{0}).

Thus in the limit,

u(x, τ) ⩾ w(x, τ) in ℝ^{N} × (0, τ_{0}),

and this proves that the initial trace of u is controlled (from below) by w. Hence we have necessarily

tr_{ℝ^{N}} (u) = ν,

so that ν ∈ A^{+}(p), and the result is proved. □

We now give some qualitative properties for the sets A^{+}(p) in the supercritical range. The first result is a consequence Lemma 6.1, concerning monotone convergence.

Lemma 6.5 Let p ⩾ m + 2/N, and ν_{n} be an increasing sequence of admissible Borel measures such that μ_{n} converges in the sense of Borel measures to a measure ν ∈ B^{+}(ℝ^{N}). Then ν ∈ A^{+}(p), i.e., ν is also admissible.

Proof. Let ν = (S, μ) and u_{n} be the sequence associated with ν_{n}. Then u_{n} converges monotonically and locally uniformly to a solution u of equation E_{p}. By Lemma 6.1, we know that

tr_{ℝ^{N}} (u) ⩽ ν,

but here we can use monotone convergence instead of Fatou’s lemma in the proof of Lemma 6.1, which gives that on ℝ^{N} \ S, the trace of u is exactly μ. It remains to prove that u blows up on S. This is easy since for every open set U such that U ∩ S ≠ ∅,

∫_{U} u(x, t)dx ⩾ ∫_{U} u_{n}(x, t)dx,

and thus for every n ∈ ℕ,

inf lim_{t→0} ∫_{U} u(x, t)dx ⩾ ν_{n}(U ∩ S),

which blows up when n goes to infinity, because ν_{n} converges to +∞ on S. Thus we have obtained that tr_{ℝ^{N}} (u) = ν, hence ν ∈ A^{+}(p). □

Let us give now some basic examples of admissible measures, and admissible singular sets (see definition just below).

Definition. A closed set S ⊂ ℝ^{N} is said to be admissible as a singular set if the measure ν = (S, 0) is admissible as initial data.

Proposition 6.6 Every closed ball of positive radius is admissible as a singular set.

Proof. Let \overline{B}_{r} be such a ball, and χ be the characteristic function of this ball. Let u_{n} be the unique solution of equation E_{p} with initial data

u_{n}(0) = n · χ.

Then u_{n} is not zero since χ is not zero almost everywhere, and it is increasing, and by a direct application of Lemma 6.5, we obtain the result: u_{n} converges to a solution u with initial trace +∞ · χ, which is thus an admissible data. Of course, +∞ · χ is the Borel measure which is represented by

ν = (\overline{B}_{r}, 0),

hence the closed balls are admissible sets. □

Remarks. 1. If we want to do the same with a set of zero Lebesgue measure, this method cannot work since u_{n}(0) = 0 almost everywhere, and then u_{n} ≡ 0. In fact it is clear that in the super-critical case, the singular set has to be dense enough to insure existence of a non trivial solution, since for instance a point is not admissible as a singular set (non existence of the V.S.S.). Obviously, balls are dense enough.

2. The same result holds true if we consider the Borel measure ν = (\overline{B}_{r}, μ), where μ is an admissible Radon measure. The modification is the following: let v be the solution of E_{p} with initial data μ. Then if we put

u_{n}(0) = n · χ + v(1/n),

it is obvious that u_{n} will converge locally uniformly to a solution u with initial trace +∞ on \overline{B}_{r}, and to check that the initial trace is not lost on the complement of \overline{B}_{r}, we use the fact that u ⩾ v (by construction), so that finally, tr_{ℝ^{N}} (u) = ν. In the same spirit, one can easily prove the following results:

Proposition 6.7 Let S be a closed set in ℝ^{N} such that S is equal to the closure of its interior. Then S is admissible as a singular set.

Proof. Let u_{n} be defined as in the case of a ball as the solution of E_{p} with initial data

u_{n}(0) = n · χ,

where χ is the indicator function of int(S). In particular, for every y ∈ int(S), there exists a ball B_{y} of positive Radius centered at y such that B_{y} ⊂ int(S), and if χ_{y} is the indicator function of B_{y},

u_{n}(0) ⩾ n · χ_{y}.

Thus when n increases, u_{n} also increases to a limit solution u with initial trace ν, and clearly for every y ∈ int(S),ν = +∞ on B_{y}. Then it is obvious that the singular set of ν contains the closure of int(S), i.e. S itself. But on the complement of S, it is also obvious that ν = 0, by using exactly the same method as in Proposition 6.6 (or using directly Lemma 6.1). □

Theorem 6.8 Assume that ν = (S, μ), where μ ∈ A^{+}(p) and S is admissible as a singular set, i.e., there exists a solution V_{S} of E_{p} such that

tr_{ℝ^{N}} (V_{S}) = (S, 0).

Then ν ∈ A^{+}(p).

Proof. Since μ is admissible, there exists a solution v with initial data μ, and if we take the solution u_{n} of E_{p} with initial data

u_{n}(0) = V_{S}(1/n) + v(1/n),

then obviously, u_{n} ⩾ max{V_{S}; v} on ℝ^{N} × (1/n, T), so that u_{n} converges to a limit solution u locally uniformly, and tr_{ℝ^{N}} (u) is exactly (S, μ). Indeed, it is obvious that u will blow-up on S since it is greater than V_{S}, and on the regular set, there is no loss of initial trace because u ⩾ v, which takes on the initial data μ. □

Remark. The Radon measure μ is defined on the complement of S, hence it may not be a Radon measure in ℝ^{N}. Indeed, one may have μ(∂S) = +∞. Hence in this result, we say that μ ∈ A^{+}(p) if μ, extended by zero on S is a Borel measure in ℝ^{N} which is admissible. Note also that by Lemma 6.5, we know that if μ/_{K} is admissible for every compact set K ⊂ ℝ^{N} \ S, then μ is admissible.

Proposition 6.9 Let p ⩾ m + 2/N and μ ∈ A^{+}(p), locally finite. Then for every μ ∈ B^{+}(ℝ^{N}),

0 ⩽ μ′ ⩽ μ ⇒ μ′ ∈ A^{+}(p).

Proof. Let μ and μ′ as above (μ′ ⩽ μ implies that μ′ is also locally finite). Since μ ∈ A^{+}(p), here exists a solution u of E_{p} such that tr_{ℝ^{N}} (u) = μ, and by the uniqueness result in [8, Sec. 4.2], recalled at the beginning of this section, u is unique. Thus the solution u can be viewed as the limit of the u_{n}, where u_{n} is the unique solution of E_{p} with initial data

μ_{n} = ρ_{n} ⋆ μ,

ρ_{n} being a convolution kernel in ℝ^{N}. And if we call v_{n} the solution with initial data

μ′_{n} = ρ_{n} ⋆ μ′,

then by construction, v_{n} ⩽ u_{n} in Q_{T}, and v_{n} will converge to a solution v of E_{p}. The problem is to check that v takes on the initial data μ′, and to achieve this, it is sufficient to prove that the sequence v_{n} converges in L^{p}(0, T; L_{loc}^{p}(ℝ^{N})). Indeed, in this case, we can pass to the limit in the following weak formulation:

∫_{ℝ^{N}} v_{n}(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} v_{n}^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} v_{n}^{p}φ = ∫_{ℝ^{N}} φdμ′_{n},

which gives

∫_{ℝ^{N}} v(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} v^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} v^{p}φ = ∫_{ℝ^{N}} φdμ′,

hence tr_{ℝ^{N}} (v) = μ′. But since v_{n} ⩽ u_{n}, it is sufficient to prove that the u_{n} converge in L^{p}(0, T; L_{loc}^{p}(ℝ^{N})). So let us use the weak formulation for the u_{n}, where φ ∈ C^{2}(ℝ^{N}), supp(φ(t)) ⊂ K fixed:

∫_{ℝ^{N}} u_{n}(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}φ_{t} − ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ = ∫_{ℝ^{N}} φdμ_{n},

and in the limit, u is a solution with initial trace μ :

∫_{ℝ^{N}} u(t)φ − ∫_{0}^{t} ∫_{ℝ^{N}} uφ_{t} − ∫_{0}^{t} ∫_{ℝ^{N}} u^{m}Δφ + ∫_{0}^{t} ∫_{ℝ^{N}} u^{p}φ = ∫_{ℝ^{N}} φdμ.

Since u_{n}, u_{n}^{m} converge in L^{1}(0, T; L_{loc}^{1}(ℝ^{N})) (from equi-integrability given by Lemma 3.2), then

∫_{0}^{t} ∫_{ℝ^{N}} u_{n}^{p}φ ⟶_{n→∞} ∫_{0}^{t} ∫_{ℝ^{N}} u^{p}φ.  (6.2)

Then it is easy to see that u_{n} converges in L^{p}(0, T; L_{loc}^{p}(ℝ^{N})) : if we take φ ⩾ 0,

∫_{0}^{t} ∫_{ℝ^{N}} (u^{p} − u_{n}^{p})_{+}φ ⟶_{n→∞} 0

by dominated convergence, and combining this with (6.2), we obtain the same for (u^{p} − u_{n}^{p})_{−}, which in turn implies the strong convergence:

∫_{0}^{t} ∫_{ℝ^{N}} |u^{p} − u_{n}^{p}|φ ⟶_{n→∞} 0.

Thus v is indeed a solution of E_{p} with initial data μ′ (and in fact it is unique), hence μ′ ∈ A^{+}(p). □

Remark. In this result, the restriction to Radon measures is essential. Indeed, we have for instance (χ being the characteristic function of the ball B_{1}(0)):

0 ∈ A^{+}(p), δ_{0} ∉ A^{+}(p), +∞ · χ ∈ A^{+}(p),

although these measures are ordered.

7 Adaptations to the Case Ω Bounded

We give below the adapations of our results to the case when equation (1.1) is considered in Q_{T} = Ω × (0, T), with Ω open, bounded and regular. Actually, the main difference is the existence of the so-called Friendly Giant, which was studied first by Dahlberg and Kenig [12] for the purely diffusive equation. We will use the same definition of weak solution with zero boundary data: u ∈ C^{0}(0, T; C^{0}(\overline{Ω})) ∩ L_{loc}^{1}(0, T; L^{1}(Ω, δ)), where δ(x) = dist(x, ∂Ω), and for any φ ∈ C^{0}((0, T) × \overline{Ω}), with compact support in some time interval I ⊂ (0, T),

∫∫_{Q_{T}} {−uφ_{t} − u^{m}Δφ + u^{p}φ} = 0.

The limit of fundamental solutions in the range 0 ⩽ p ⩽ m is thus the Friendly Giant:

Theorem 7.1 Let Ω ⊂ ℝ^{N} be open, bounded, regular and 0 ⩽ p ⩽ m. Then the following limit holds:

v_{c} → V_{p} locally uniformly in \overline{Ω} × (0, T),

where V_{p} is the so-called “Friendly Giant”, which has the following properties: V_{p} takes on the initial trace ν = +∞ in Ω, V_{p} ∈ C^{0}((0, T) × \overline{Ω}), V_{p} = 0 on ∂Ω × (0, T).

Proof. It is clear by uniqueness of the v_{c}’s we have comparison with the Friendly Giant for u_{t} = Δu^{m}, that is:

v_{c}(x, t) ⩽ t^{− \frac{1}{m−1}} f(x),

where f satisfies −Δf − 1/(m − 1)f = 0 with zero boundary conditions on ∂Ω. This gives a universal bound up to the boundary, hence by equi-continuity results [6], v_{c} will converge in C^{0}((0, T) × \overline{Ω}) to some V_{p}. Now, we are left to show that the initial trace of V_{p} is +∞. We argue as in [9], with a little modification. For k > 1, let T_{k} be the transformation

T_{k}(u)(x, t) = k^{1/(m−1)}u(x, kt).

The set Ω is invariant under T_{k}, and T_{k} maps v_{c} into a sub-solution with initial data ck^{1/(m−1)}δ_{y}. Indeed, since p ⩽ m and k > 1, we have

\frac{∂}{∂t}T_{k}(v_{c}) − ΔT_{k}(v_{c})^{m} + T_{k}(v_{c})^{p} = k^{m/(m−1)}u^{p} − k^{p/(m−1)}u^{p} ⩽ 0.

By uniqueness results in Ω and zero lateral data, we know that v_{ck^{1/(m−1)}} ⩾ T_{k}(v_{c}), and thus in the limit as c → ∞, we obtain

V_{p}(x, t) ⩾ k^{1(m−1)}V_{p}(x, kt).

So taking k = t^{−1}, which is greater than 1 for small t, it turns out that

V_{p}(x, t) ⩾ t^{−1/(m−1)}V_{p}(x, 1).

In fact, it is clear by positivity properties that V_{p}(x, 1) > 0 at least on some open set B_{η}(0) ⊂ Ω, which proves that the singular set of the initial trace of V_{p} contains B_{η}(0). Now for any z ∈ B_{η}(0), we can easily prove by comparison that for any c > 0,

V_{p}(x, t) ⩾ v_{c}(x − y, t),

so that V_{p}(x, t) ⩾ V_{p}(x − y, t). Hence, the singular set of tr_{Ω}(V_{p}) contains also B_{2η}(0) ∩ Ω, and by induction, we deduce that finally tr_{Ω}(V_{p}) = Ω. □

The trace that we defined in ℝ^{N} is based on purely local arguments, so that it is also valid in Ω bounded. Our solutions are clearly comparable with the solutions of the diffusive equation u_{t} = Δu^{m}, so that by the results of [12], the following result holds:

Theorem 7.2 If 0 ⩽ p ⩽ m, the initial trace of any solution u ≢ V_{p} in Ω is a Radon measure μ in Ω which satisfies

∫_{Ω} dist(x, ∂Ω) dμ(x) < ∞

For 0 ⩽ p ⩽ m, the operator ℙ_{p} is constructed as in Ω = ℝ^{N}.

Theorem 7.3 Let 0 ⩽ p ⩽ m. Then the operator ℙ_{p} is a projection in B^{+}(Ω), i.e. ℙ_{p} ∘ ℙ_{p} = ℙ_{p}. Moreover,

• ℙ_{p}(ν) = ν if and only if ν is admissible, and then u[ν] is the unique solution with trace ν,

• ℙ_{p}(ν) = +∞ if ν is not admissible and then u[ν] = V_{p} (the Friendly Giant).

Therefore, for bounded Ω’s, we do not find any case of projection by complete blow-up.

The situation for p > m is similar to the case Ω = ℝ^{N}. Actually, the Friendly Giant gives a bound for t > 0, up to the boundary which allows to pass to the limit up to the boundary. Thus the limits are still weak solutions and the problem of the initial trace is exactly the same as for Ω = ℝ^{N}. Thus, all the qualitative aspects of the sets A^{+}(p) for p ⩾ m + 2/N remain valid in Ω. It is important to notice that contrary to what happens in the fast diffusion case (m < 1) [10], the initial Borel measure can have singular points at the boundary.

8 Some open problems and extensions

We list some of the questions that have not been solved in the preceding discussion.

• To characterize the admissibility set for 1 < p < m.

• The same in the super-critical case p ⩾ m + 2/N and to decide whether there is a projection operator.

• To describe the domains for which there exists a friendly giant and the results of the preceding section apply. For instance a domain bounded in one direction behaves like a bounded domain.

• A more difficult question is to describe admissibility sets for solutions with changing sign.

Concerning extensions to other equations, maybe the closest example is the p-Laplacian equation with absorption

u_{t} = ∇ · (|∇u|^{p−2}∇u) − u^{q}.

We refer to [3] for a partial study. The super-critical exponent is now q_{c} = p−1+p/N. Different types of nonlinear equations involving absorption, reaction or convection terms come next to mind.

References

[1] D. G. Aronson, Widder’s inversion theorem and the initial distribution problem, SIAM J. Math. Anal. 12 (1981), 639-651.

[2] D. G. Aronson, L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), 351-366.

[3] M.-F. Bidaut-Véron, E. Chasseigne, L. Véron, Initial trace of solutions of some quasilinear parabolic equations with absorption, submitted (2001).

[4] P. Bénilan, M.G. Crandall, M. Pierre, Solutions of the PME in ℝ^{N} with optimal conditions on the initial data, Indiana Univ. Math. Jl. 33 (1985), 51-87.

[5] H. Brezis, L. A. Peletier, D. Terman, A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal. 99 (1986), 185–209.

[6] E. DiBenedetto, A boundary modulus of continuity for a class of singular parabolic equations, J. Diff. Eq., 63 (1986), 418-447.

[7] E. Chasseigne, Thesis, Univ. Tours, France (2000).

[8] E. Chasseigne, Initial trace for a porous medium equation: I. The strong absorption case, to appear in Annali di Matematica (2001).

[9] E. Chasseigne, Initial trace for a porous medium equation: II. The critical absorption term, Asymptotic Analysis 24 (2000), 37-72.

[10] E. Chasseigne, J.L. Vazquez, Theory of Weak and Limit Solutions for Fast Diffusion Equations in Optimal Classes of Data. The Cauchy-Dirichlet Problem, submitted (2000).

[11] B. E. J. Dahlberg, C.E. Kenig, Non-negative solutions of the porous medium equation, Comm. Partial Diff. Eqs. 9 (1984), 409-437.

[12] B. E. J. Dahlberg, C.E. Kenig, Non negative solutions of the initial Dirichlet problem for generalized porous medium equations in cylinders, J.. Amer. Math. Soc. 1 (1988), 401-412.

[13] V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii, On the asymptotic “eigenfunctions” of the Cauchy problem for some nonlinear parabolic equation, Mat. Sb. 126 (1985), 435–472 (in Russian), english translation Mat. USSR Sbornik 54 (1986), 421–455.

[14] S. Kamin, L. A. Peletier, J. L. Vazquez, Classification of singular solutions of a nonlinear heat equation, Duke Math. J. 58 (1989), 243-263.

[15] S. Kamin, L. A. Peletier, J. L. Vazquez, A nonlinear diffusionabsorption equation with unbounded data, Nonlinear Diffusion Equations and their Equilibrium States 3, 243-63.

[16] M. Marcus, L. Véron, Initial trace of positive solutions of some nonlinear parabolic equation, Comm. in P.D.E. 24 (1999), 1445-1499.

[17] J. L Vazquez, L. Véron, Different kinds of singular solutions of nonlinear parabolic equations, in Nonlinear Problems in Applied Mathematics (SIAM), (1996), 240-249.

[18] J. L. Vazquez, M. Walias, Existence and uniqueness of solutions of diffusion-absorption equations with general initial data, Diff. Int. Eq. 7 (1994), 15-36.

[19] Zhao Junning, Liu Huizhao, The Cauchy problem of the porous medium equation with absorption, J. Partial Diff. Eqs 7 (1994), 231-247.

Emmanuel Chasseigne: Laboratoire de Mathématiques et Physique Théorique, Université de Tours Parc de Grandmont, 37200 Tours, France.

E-mail: echasseigne@univ-tours.fr

Juan Luis Vazquez: Departamento de Matemáticas, Universidad Autónoma de Madrid 28046 Madrid, Spain.

E-mail: juanluis.vazquez@uam.es