SOME DEGENERATE PARABOLIC PROBLEMS: EXISTENCE AND DECAY PROPERTIES

. We study the existence of solutions u belonging to L 1 (0 ,T ; 1 , ∩ L ∞ (0 ,T ; L 2 (Ω)) of a class of nonlinear problems whose prototype is the following . (1) We investigate also the asymptotic estimates satisﬁed by distributional solu- tions that we ﬁnd and

in Ω. (1) We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.
1. Introduction. In this paper we study the following class of nonlinear parabolic problems    u t − div (a(x, t, u)∇u) = 0, in Ω T ; u = 0, on ∂Ω × (0, T ); u(x, 0) = u 0 (x), in Ω; (2) where u 0 (x) belongs to some Lebesgue space L m (Ω), Ω is a bounded open set of IR N , N ≥ 2 and Ω T = Ω × (0, T ), T > 0. Here a(x, t, ρ) : Ω × (0, T ) × IR → IR is a Caratheodory function satisfying the following structure assumptions α (1 + |ρ|) γ ≤ a(x, t, ρ) ≤ β, where α, β and γ are positive constants. The simplest example is given by the parabolic problem (1). The feature of our existence results is that the solutions only belongs to the non reflexive space L 1 (0, T ; W 1,1 0 (Ω)) and the main difficulty of the problem is due to the principal part of the operator: it can degenerate for ρ large; that is when the solution is unbounded. Hence a slow diffusion can appear for large value of the solution.
In the elliptic case, this type of problems, under the assumption α (1 + |ρ|) γ ≤ a(x, ρ) ≤ β, 618 LUCIO BOCCARDO AND MARIA MICHAELA PORZIO was introduced in [3] and developed in [1]. In the last paper, the impact of a lower order term of order zero is considered div(a(x, u)∇u) + u = f ∈ L m (Ω), in Ω; u = 0, on ∂Ω.
In these papers the authors study the existence of weak or distributional or entropy solutions (definited using the truncature, useful if the gradient of the solution does not belong to L 1 , see [4]); the paper [16] deals with uniqueness. In particular in [1], thanks to the presence of the lower order term (regularizing effect), some results of [3] are improved: it is proved for the boundary value problem (5) • the existence of finite energy solutions, if m ≥ γ + 2; • the existence of bounded weak solutions, if m > γ N 2 and γ > 1. The paper [2] covers the borderline case m = γ = 2 of (5), where the distributional solutions belong W 1,1 0 (Ω). Further existence and regularity results can be found in [10], [11] and [23].
In the parabolic case, the problem (2) was studied in [20] with u 0 (x) ∈ L m (Ω), m ≥ 1. In particular the existence of solutions (which become distributional solutions only for suitable values of γ and m) is proved for every value of γ > 0 and m ≥ 1. Moreover existence results when u 0 is a Radon measure and γ > 1 are proved in [21] and in [22] where a new notion of solution (measure-valued solutions) is given to study the cases of measure data. Notice that in all these last cases the solutions constructed are very weak and no information on the summability of the gradients of these solutions is given.
In this paper we prove the existence of distributional solutions of the problem (2) in the borderline case u 0 (x) ∈ L 2 (Ω), The distinctive aspect of our existence results is that the gradient of the solutions only belongs to L 1 . Moreover we study the behavior of the solution constructed for t large and we prove that decay estimates hold. In detail we show that if N ≥ 3, for every p < m the L p (Ω)-norm of such a solution decays in time according to the estimate (25). If N = 2 a faster (exponential) decay occurs (see Theorem 3.2).
Finally we prove that if the function a in (2) is Lipschitz continuous with respect to ρ, then the solution obtained by approximation is unique.
The plan of the paper is the following: in section 2 we enounce and prove the existence results. In section 3 we investigate on decay estimates. Finally in the last section we study the uniqueness of the solutions.
Step 2 -Estimates. The following estimate holds true, ∀ 0 < t ≤ T, k ≥ 0, taking G k (u n ) as test function in (8). Indeed In particular we have (with k = 0) Moreover (11) implies Then, using estimates (12), (13) we deduce ΩT ∩{|un|≥k} In particular we have (k = 0) Now we prove that, for every measurable subset A ⊂ Ω T , we have Indeed, we choose T k (u n ) as test function in (8) and we use that Then thanks to assumption (3) we obtain By the previous inequality we deduce that, for every measurable subset A ⊂ Ω T , we have that is (16).
Step 3 -Convergences. Now we prove that the approximating solutions u n converge to a solution u of (2). In (15), we proved the boundedness of ). Hence we can apply Corollary 4 of [24], obtaining that there exists a measurable function u such that (up to subsequences) u n converges to u in L 1 (Ω T ) and a.e. in Ω T .
Now we prove that the following convergence holds true We follow a technique of [2]. We start proving that there exists a function Y ∈ (L 1 (Ω T )) N such that the following convergence holds true The sequence {∇u n } n∈I N is bounded in (L 1 (Ω T )) N . Hence by Dunford-Pettis' Theorem the assertion (19) follows if we show that for every positive ε there exists a positive constant δ such that for every measurable A ⊂ Ω T satisfying meas(A) < δ it results We have, using (14) and (16) , which implies (20). Hence, to conclude the proof, it remains to prove that Y is equal to ∇u; this is a consequence of the definition of weak gradient and the convergences (17) and (18).
Step 4 -Pass to the limit. Note that, thanks to (17), (18), (20) and to the inequality it is easy to pass to the limit in the weak formulation of (8), that is It follows that u ∈ L 1 (0, T ; W 1,1 0 (Ω)) and it solves (2). Notice that by (12) and again by (17) we obtain (thanks to Fatou's Lemma) that That is u belongs also to L ∞ (0, T ; L 2 (Ω)).
Remark 1. Our problems, in particular the model case (1), thanks to the change z = u 1 + |u| , are formally related to the singular problem in Ω.
Remark 2. Notice that under the assumption (3) with γ > 0, if the summability coefficient m of the initial datum u 0 satisfies the following condition then there exists a weak solution of (2) in L ∞ (0, T ; L m (Ω)) ∩ L h (0, T ; W 1,h 0 (Ω)), where h > 1 depends on γ, m and N (see formula (2.7) and Theorem 2.9 in [20]). Notice that if m = γ = 2, that is the case considered here, the previous condition becomes since the assumption N > 2 is equivalent to require 3N N +1 > 2, and hence it is not satisfied. Thus, as said before, we study here a limit case not considered in [20] and we prove, as stated in Theorem 2.1 above, the existence of a solution with summable gradient.
3. Decay. We show the following decay estimate.

Remark 3. In particular, it results
, if u 0 ∈ L m (Ω), Remark 4. Notice that by (25) it follows that also the L 1 (Ω) norm of u(t) decays since, Ω being a bounded open set, we have that , ∀ p > 1.

Remark 5.
We recall that if the initial datum satisfies then there exists a solution of problem (2) that becomes immediately bounded and satisfies the following decay estimate where K 0 , K and ν are positive constants depending only on the data (see Theorem 2.15 in [20]). The above kind of estimates (28) are known in literature also as ultracontractive estimates or decay estimates to underline that they imply that the solutions go to zero (i.e. decay) when t goes to infinity. There is a wide literature on this interesting phenomenon which is known to appear for a large class of parabolic problems (see for example [25], [7], [6], [8], [20], [17], [18] and the references cited therein).
Notice that the assumption (27) is optimal to obtain instantaneous boundedness (see section 6 in [20]).
We observe explicitly that in the limit case we study here such assumption is not satisfied, since being γ = 2 it becomes m > N, and hence the instantaneous boundedness phenomenon does not take place although there exists a solution that continue to decay in time.
We notice that decay estimates for solutions that do no become immediately bounded are less known in literature; some interesting examples can be found for other parabolic problems in [12], [13], [19] and [18].
Finally, in the particular case N = 2 we have the following faster decay result. where , and S is the Sobolev constant (hence a constant depending only on N ).
Proof of Theorem 3.1. We start proving that for every p > 1 arbitrarily fixed it results where all the terms in (30) are finite for a.e. t ∈ (0, T ).

LUCIO BOCCARDO AND MARIA MICHAELA PORZIO
Hence we can pass to the limit in (31) getting a(x, t, T n (u n ))|∇u n | 2 |u n | p−2 = 0, from which the assertion (30) follows thanks to assumption (3). Now notice that it results where C 0 = p −2 (|Ω|+ u 0 2 L 2 (Ω) ) −1 and S is the Sobolev constant (hence a constant depending only on N ). As a matter of fact we have where C 0 is the constant defined above. By the previous inequality and the Sobolev inequality in W 1,1 0 (Ω) we deduce Now (33) follows by the previous inequality passing to the limit as ε that goes to zero. By (30) and (33) we deduce that where and hence assuming p < m we have Thus the following interpolation inequality holds true where λ satisfies Recalling that (31) holds true for every p > 1, hence also for p = m, we obtain Using the previous inequality and (9) in (37) we deduce where λ is as before. By (40) we have (if u 0 L m (Ω) = 0, otherwise the assertion is evidently true) that By (41) and (35) we obtain where we have set Integrating (42) we deduce and hence, being λ ∈ (0, 1), we have Notice that by (9) and the previous inequality we obtain Hence the assertion follows by passing to the limit as n → +∞ (using Fatou's Lemma) and setting , since by definition of λ it results Proof of Theorem 3.2. Proceeding exactly as in the proof of Theorem 3.1 we deduce that (35) holds true, which, under the assumption N = 2, reads where, as before, Hence the assertion follows by integrating the previous inequality and passing again to the limit on n.

4.
Uniqueness. We prove here the following uniqueness result. (47) Then if u n and z n solve the following problems in Ω; in Ω; (49) where u 0,n (x) and z 0,n (x) belong to L ∞ (Ω), then the following estimate holds true Remark 6. Notice also that if we choose u 0,n (x) = T n (u 0 ), then the solution of the approximating problem (8) (constructed in Section 2 to prove the existence of a solution of our problem) satisfies u n L ∞ (ΩT ) ≤ u 0,n L ∞ (Ω) ≤ n, ⇒ T n (u n ) = u n , and hence u n solution of (8) coincides with the solution of (48).

Remark 7.
Indeed the proof of Theorem 4.1 (and hence also that of Corollary 1 below) works for more general problems where the structure assumption (3) is replaced by the following weaker assumption under the assumption that there exist the solutions u n and z n of (48) and (49), respectively, in L ∞ (Ω T ) ∩ L 2 (0, T ; H 1 0 (Ω)). An immediate consequence of Theorem 4.1 is that the solution of problem (2) obtained by approximation is unique. We notice that in the "classical case γ = 0" the uniqueness of solutions obtained by approximations can be found in [9]. Corollary 1. Let the structure conditions (3) and (47) hold true. Assume that u and v are obtained, respectively, as limit a.e. in Ω T of the sequences {u n } n∈N and {v n } n∈N solutions of (48) and (49). Finally assume that the initial data u 0,n (x) and z 0,n (x) converge both to u 0 in L 1 (Ω). Then we have that u = v a.e. in Ω T .
Proof of Theorem 4.1. As in [2] and [5] take T h (u n − z n ) as test function in (48) and in (49) where h > 0 is arbitrarily fixed. Subtracting the obtained equations we deduce that Ω Ψ h (u n − z n )(t)dx + [a(x, t, u n ) − a(x, t, z n )] ∇z n ∇T h (u n − z n ).
Notice that we have, since by (3) a(x, t, s) ≥ 0, t 0 Ω a(x, t, u n )∇(u n − z n )∇T h (u n − z n ) ≥ 0, while for the second integral in the right-hand side of (53) we deduce, thanks to assumption (47), Moreover, noticing that it results we can estimate the first and the last integral in (52), as follows Ω Ψ h (u 0,n − z 0,n )dx ≤ h Ω |u 0,n − z 0,n |.
Hence, putting together the previous inequalities we deduce Ω (u n − z n )T h (u n − z n )(t) − h 2 2 |Ω| ≤ Lh Hence letting h go to zero we deduce (50).