REGULARITY FOR ENTROPY SOLUTIONS OF PARABOLIC p-LAPLACIAN TYPE EQUATIONS

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut − div ap(x,∇u) = f in ]0, T [×Ω with initial datum in L1(Ω) and assuming Dirichlet’s boundary condition, where ap(., .) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1(]0, T [×Ω) and Ω is a domain in RN . We find spaces of type Lr(0, T ;Mq(Ω)) containing the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian equation is considered. 1991 Mathematics subject classifications: 35K65, 47H20. This research has been supported by the Spanish DGICYT, Proyecto PB94-0960.


Introduction
Let Ω be a domain in R N (bounded or not) and let 1 < p < N. Let a p : Ω × R N → R N be a Carathéodory function satisfying the classical Leray-Lions conditions in such a way that div a p (x, ∇u(x)) defines an operator from W 1,p 0 (Ω) onto W −1,p (Ω) (see [10]).The model example of such function is a p (x, ξ) = |ξ| p−2 ξ which determines the p-Laplacian operator ∆ p u = div(|∇u| p−2 ∇u).
We are interested in the regularity of entropy solutions (see Definition 2.4 below) of the following parabolic problem: where u 0 ∈ L 1 (Ω) and f ∈ L 1 (Q T ).
A distributional solution of this problem was found in [9].The notion of entropy solution of (P) is used in [1] and [11] to obtain uniqueness.(An equivalent concept, in terms of renormalized solution, can be found in [7], [13], [14] and [16].)We remark that in [1] it is only considered the case f = 0, although the proof of the general case can be easily obtained following the same steps and taking into account the arguments of [6,Chapter 4].On the other hand, the hypotheses in [11] and Ω bounded.In both articles some summability results on the entropy solution u and its gradient are given; more precisely, u ∈ M (N +1)p−N N (Q T ) and |∇u| ∈ M p− N N +1 (Q T ) (see [1]), and |∇u| ∈ ∩ q<p− N N +1 L q (Q T ) (see [11] or [9]).These summability results are obtained jointly for time and space variables; thus, as a consequence of these papers, it is not possible to get optimal regularity when both variables are separately considered.
The purpose of this note is to give a precise summability result of the entropy solution and its gradient with respect to space and time.This was studied in [8] for weak solutions of (P) in the framework of Lebesgue and Sobolev spaces when p ≥ 2, u 0 = 0 and Ω is bounded.The main result of [8] states that there exists a weak solution u of (P) such that u ∈ L r (0, T ; W 1,q 0 (Ω)) for p/2 ≤ q < N(p − 1)/(N − 1) and 1 ≤ r < q((N + 1)p − 2N )/((N + 1)q − N ), and for 1 ≤ q < p/2 and r = p.Some remarks on their assumptions are as follows.Firstly, it is possible to get a similar result if 2 − 1/(N + 1) < p < 2, as it is observed in [8,Remark 1.7].Moreover, analogous results also hold for other initial data u 0 = 0, as it is pointed out in [12].Nevertheless, their hypothesis of Ω bounded cannot be removed if their arguments are to be followed, in fact it is needed from the very begining when they obtain the a priori estimates in [8,Lemma 2.2].We want to show in this paper what the situation looks like in the framework of Marcinkiewicz spaces (see Definition 2.1 below) since we think they should be more suitable for L 1 -data (see [3] for the elliptic case).We obtain an improvement of the results of [8] since our arguments work for small p > 1 and unbounded domains.

Theorem 1. Let Ω be a bounded domain and let u be the entropy solution of problem (P). (1) If
for q = p 2 and r < p, and for q < p 2 and r = p. (2 Remark 1.1.When Ω is bounded, one can replace M q (Ω) by L q (Ω) due to the relations between Marcinkiewicz and L q spaces in bounded domains, and to the fact that q satisfies strict inequalities.

Theorem 2.
Let Ω be an unbounded domain and let u be the entropy solution of problem (P). (1) and r < q (N + 1)p − 2N (N + 1)q − N .
Remark 1.2.For p = 2, the bound on r is exactly the same one that can be computed for the fundamental solution of the Heat equation.
We improve these regularity results when f = 0 and the p-Laplacian operator is considered.

Theorem 3.
Let Ω be a bounded or unbounded domain, 1 < p < N and let u be the entropy solution of problem (P) with the p-Laplacian operator and f = 0, then This paper is divided into five sections.The next one is on preliminaries: we include the definitions of Marcinkiewicz spaces and entropy solutions of problem (P) and the first estimates on the solution and its derivative.Section 3 is devoted to prove Theorem 1 (Theorems 3.1 and 3.2), while in section 4 we prove Theorem 2 (Theorems 4.1 and 4.2).Finally, in section 5, we prove Theorem 3, and we also give other regularity results when f = 0 and the initial datum

Preliminaries
Throughout this note the Lebesgue measure on Ω will be denoted by µ N .Definition 2.1.For 0 < q < ∞, the set of all measurable functions u : Ω → R such that the functional [u] q := sup k>0 kµ N {|u| > k} 1/q is finite is called a Marcinkiewicz space and is denoted by M q (Ω).
It is straightforward that, for bounded Ω, we have M q (Ω) ⊂ M r (Ω) for r < q.The connection between Marcinkiewicz and Lebesgue spaces is easy: L q (Ω) ⊂ M q (Ω) ⊂ L r loc (Ω) for r < q (see, for instance, [17]); let us point out that Marcinkiewicz spaces are also known as weak-Lebesgue spaces.
When q > 1, the Marcinkiewicz space M q (Ω) is a Banach space with the norm defined by u q = sup t>0 t 1−q q t 0 u * (τ ) dτ , where u * (τ ) = inf k > 0 : µ N {|u| > k} ≤ τ defines the non-increasing rearrangement of u (see, for instance, [17,Definition 1.8.6]).As a consequence of this definition one has that K |u| ≤ µ N {K} q−1 q u q for any K ⊂ Ω with finite measure.Note that, endowed with this norm, if u, v ∈ M q (Ω), then q dt is finite.
The following result is well known; a proof may be seen, for instance, in [2, Lemma 1.3].

Lemma 2.3. Let I ⊂ R be an interval and denote by Λ the set of all measurable functions
Before introducing the concept of entropy solution, we will define the following functional spaces (see [3]).Given k > 0, define the truncature operator by , a derivative ∇u can be defined as the unique measurable function such that ∇T k u = ∇u•χ {|u|<k} for all k > 0 (see [3,Lemma 2.1]).Definition 2.4.We will say that a function u ∈ C [0, T ]; L 1 (Ω) is an entropy solution of (P) if u(t) ∈ T 1,p 0 (Ω) for almost all t, ∇T k (u) ∈ L p (Q T ) for all k > 0 and Taking ϕ = 0 in the formulation of entropy solution it follows that (Recall that one of the Leray-Lions assumptions asserts that there exists α > 0 satisfying α|ξ| p ≤ a p (x, ξ), ξ ; we consider α = 1 since there is not loss of generality in doing so.)From now on u will denote the entropy solution of (P).We will finish this section giving the basic estimates in order to prove our regularity results.We will use the fact that u ∈ C([0, T ], L 1 (Ω)) and the estimate given in (2.1).
Observe that there is no loss of generality in taking g(t) ≥ M N −p N as we will do in our following estimates.
Proof: Proposition 2.5 and Sobolev's inequality imply two facts: on the one hand, for all k ≥ 1 and almost all t, and, on the other hand, for all k ≤ 1 and almost all t, from where the result follows.
Proposition 2.7.Let δ ∈]0, p−1[.Then there exists a constant C > 0 such that for almost all t ∈ [0, T ], sup Proof: A similar argument to the one in [3, Lemma 4.2] can be applied.We just prove (1) and ( 2), since the other assertions can be proved in an analogous manner.
Before showing (1), let us point out that the operator associated with the elliptic term − div a p (x, ∇u) + the Dirichlet boundary condition, is accretive in L 1 (Ω) (see [3,Theorem 7.1] or [1]); thus, u(t) 1 ≤ u 0 1 + f 1 for all t ∈ [0, T ] and so This fact and Proposition 2.5 imply the following inequalities.

Case
Proof: Let 2N N +1 < p < N and q be as above and take δ > 0 such that p > N (2+δ) N +1 and 1 < q < N (p−1−δ) . It will be enough to show that u ∈ L r 0, T ; M q (Ω) for r = p(N +1)−N (2+δ) N q q−1 .First observe that the above inequalities imply that r > p−1−δ; thus, for k ≥ 1, we obtain that On the one hand, applying Proposition 2.6, g being an integrable function.
Proof: Assume first that 2N N +1 < p < N; let 1 < q < N (p−1) . We are going to prove that . By Proposition 2.6 and (2.2), we obtain that and thus sup k>0 kµ N {|u(t)| > k} 1/q defines a function that belongs to L r (0, T ), with r the minimun of In the case 1 < p < 2N N +1 , consider δ > 0 small enough such that for q strictly between p 2 and N (p−1) N −1 , and r < q (N +1)p−2N (N +1)q−N .

Other estimates
To finish this note we show what happens when we take into account another estimate instead of that of Proposition 2.5.
Taking this estimate and following the steps given in this paper, one will obtain, besides Theorem 1 and 2, the following improvement.
for r < p.
Let us point out that, in particular, one has u ∈ L r (0, T ; M N −1 (Ω)) for any r < p−1, so that this argument gives us the same type of regularity obtained in [3] for the stationary problem associated to (P).

More regular initial data.
We point out that the summability on the solution and its derivative improves when L 1 -data are replaced by more regular ones (see [12] and [15] where data u 0 ∈ L s (Ω), with 1 < s < 2, are considered).Following the arguments of the above sections, we will show what is the precise summability assuming that f = 0 and the initial datum u 0 belongs either to a Lebesgue space or to a Marcinkiewicz one.Finally, we discuss an example which proves that these estimates are sharp.
Whether u 0 lives in L s (Ω) or in M s (Ω), our computations imply u(t) ≤ u 0 for all t > 0. This fact holds true as a consequence of being these spaces normal and the operator associated to the elliptic term: − div a p (x, ∇u) + Dirichlet boundary condition, completely accretive (see [5] for the definition and [3, Theorem 7.1]).In order to use the arguments of the above sections, we need another fact which is stated in the following result.Lemma 5.1.Let u be the entropy solution of (P) and let 1 < s < 2.
Proof: (1) As in the proof of Proposition 2.5, we only consider k ≥ 1, the other case is similar.
Using the arguments of the above sections, from Lemma 5.1 the following results yield.