Renormalized solutions for nonlinear parabolic problems with L 1 − data in orlicz-sobolev spaces

u = 0 on ∂Ω× (0, T ), u(x, 0) = u0(x) in Ω. (1.1) where Ω is a bounded open subset of R , N ≥ 1, T > 0 and QT is the cylinder Ω × (0, T ). The operator A(u) = −div(a(x, t, u,∇u)) is a Leray-Lions operator defined in W 1,x 0 LM (QT ). In the case where A is a Leary-Lions operator defined on L(0, T ;W (Ω)), Dall’aglio-Orsina [18] and Porretta [28] proved the existence of solutions for the problem (1.1), where g is a nonlinearity with the following ”natural” growth condition (of order p) |g(x, t, s, ξ)| ≤ d(|s|) ( c1(x, t) + |ξ| p ) ,

(1.1) where Ω is a bounded open subset of R N , N ≥ 1, T > 0 and Q T is the cylinder Ω × (0, T ).The operator A(u) = −div(a(x, t, u, ∇u)) is a Leray-Lions operator defined in W 1,x  0 L M (Q T ).In the case where A is a Leary-Lions operator defined on L p (0, T ; W 1,p (Ω)), Dall'aglio-Orsina [18] and Porretta [28] proved the existence of solutions for the problem (1.1), where g is a nonlinearity with the following "natural" growth condition (of order p) |g(x, t, s, ξ)| ≤ d(|s|) c 1 (x, t) + |ξ| p , and which satisfies the classical sign condition g(x, t, s, ξ)s ≥ 0. The right hand side f is assumed to belong to L 1 (Q).This result generalizes analogous one of Boccardo-Gallouët [13], see also [12] and [14] for related topics.In all of these results, the and finally the recent work Elmahi-Meskine [23] for the general case.A large number of papers was devoted to the study the existence of renormalized solution of parabolic problems under various assumptions and in different contexts: for a review on classical results see [6,7,9,10,15,16,17,28].
In the case where Φ = 0, the existence of entropy solutions for parabolic problems of the form (1.1) in the setting of Orlicz spaces has been proved in A. Elmahi and D. Meskine [23] in the case where f belongs to L 1 (Q) and g be a carathéodory function satisfying |g(x, t, s, ξ)| ≤ b(|s|) c(x, t) + M (|ξ|) g(x, t, s, ξ)s ≥ 0.
It is our purpose, in this article, to prove the existence of renormalized solution for the problem (1.1) in the setting of the Orlicz Sobolev space W 1,x L M (Q), the nonlinearity g satisfying the sign condition and the function Φ is just assumed to be continuous on R.
Let us briefly summarize the contents of this article.In section 2 we give some preliminaries and gives the definition of N -function and the Orlicz-Sobolev space.Section 3 is devoted to specifying the assumptions on a, Φ, g, f and the definition of a renormalized solution of (1.1).In Section 4 we establish (Theorem 4.1) the existence of such a solution.

Preliminaries
Let Ω be a bounded open subset of R N with the segment property.Let M : R + → R + be an N -function, i.e., M is continuous, convex, with M (t) > 0 for t > 0, M(t) t → 0 as t → 0 and M(t) t → ∞ as t → ∞.Equivalently, M admits the representation : M (t) = t 0 m(s)ds where m : R + → R + is non-decreasing, right continuous, with m(0) = 0, m(t) > 0 ∀t > 0 and m(t) → ∞ as t → ∞.The N -function M conjugate to M is defined by M (t) = t 0 m(s)ds, where m : R + → R + is given by m(t) = sup{s : m(s) ≤ t} ( see [1,5,26]).We will extend these N -functions into even functions on all R. The N -function M is said to satisfy the ∆ 2 condition if, for some k > 0: M (2t) ≤ kM (t) ∀t ≥ 0. (2.1) Renormalized solutions for nonlinear parabolic problems

59
When this inequality holds only for t ≥ t 0 > 0, M is said to satisfy the ∆ 2 condition near infinity.Let P, Q be two N -functions, P ≪ Q means that P grows essentially less rapidly than Q; i.e. for each ε > 0 (2.2) This is the case if and only if Let Ω be an open subset of R N .The Orlicz class L M (Ω) (resp.the Orlicz space L M (Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that : Not that L M (Ω) is a Banach space under the norm: and L M (Ω) is a convex subset of L M (Ω).The closure in L M (Ω) of the set of bounded measurable functions with compact support in Ω is denoted by E M (Ω).
The equality E M (Ω) = L M (Ω) holds if and only if M satisfies the ∆ 2 condition, for all t or for t large according to whether Ω has infinite measure or not.The dual of E M (Ω) can be identified with L M (Ω) by means of the pairing Ω u(x)v(x)dx, and the dual norm on L M (Ω) is equivalent to .M,Ω .The space L M (Ω) is reflexive if and only if M and M satisfy the ∆ 2 −condition, for all t or for t large, according to whether Ω has infinite measure or not.We now turn to the Orlicz-Sobolev space.
is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L M (Ω) (resp.E M (Ω)).This is a Banach space under the norm Thus W 1 L M (Ω) and W 1 E M (Ω) can be identified with subspaces of the product of N + 1 copies of L M (Ω).Denoting this product by ΠL M (Ω), we will use the weak topologies σ(ΠL M , ΠE M ) and σ(ΠL M , ΠL M ).The space W 1 0 E M (Ω) is defined as the (norm) closure of the Schwartz space D(Ω) in W 1 E M (Ω) and the space We say that u n converges to u for the modular convergence in This implies convergence for σ(ΠL M , ΠL M ).If M satisfies the ∆ 2 −condition on R + (near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence.Let W −1 L M (Ω) (resp.W −1 E M (Ω)) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in L M (Ω) (resp.E M (Ω)).It is a Banach space under the usual quotient norm.
If the open set Ω has the segment property, then the space D(Ω) is dense in W 1 0 L M (Ω) for the modular convergence and for the topology σ(ΠL M , ΠL M ) (cf. [24,25]).Consequently, the action of a distribution T in W −1 L M (Ω) on an element u of W 1 0 L M (Ω) is well defined.It will be denoted by T, u .For k > 0, we define the truncation at height k, T k : R → R by (2.7) The following lemmas can be found in [4].
Lemma 2.1.Let F : R → R be uniformly Lipschitzian, with Lemma 2.2.Let F : R → R be uniformly Lipschitzian, with F (0) = 0.I suppose that the set of discontinuity points of F ′ is finite.Let M be an N -function, then the mapping F : is sequentially continuous with respect to the weak * topology σ(ΠL M , ΠE M ).
Let Ω be a bounded open subset of R N , T > 0 and set Q = Ω × (0, T ).Let M be an N -function.For each α ∈ N N , denote by ∇ α x the distributional derivative on Q of order α with respect to the variable x ∈ R N .The inhomogeneous Orlicz-Sobolev spaces of order 1 are defined as follows The latter space is a subspace of the former.Both are Banach spaces under the norm We can easily show that they form a complementary system when Ω satisfies the segment property.These spaces are considered as subspaces of the product space ΠL M (Q) which has (N + 1) copies.We shall also consider the weak topologies σ(ΠL M , ΠE M ) and σ(ΠL -valued and is strongly measurable.Furthermore the following continuous imbedding holds: The space We can easily show as in [25] (see the proof of theorem 3 below) that when Ω has the segment property then each element u of the closure of D(Q) with respect to the weak * topology σ(ΠL M , ΠE M ) is limit, in W 1,x L M (Q); of some sequence (u n ) ⊂ D(Q) for the modular convergence i.e. there exists λ > 0 such that, for all |α| 1, .11)This implies that (u n ) converges tou in W 1,x L M (Q) for the weak topology σ(ΠL M , ΠL M ).Consequently, this space will be denoted by Poincaré's inequality also holds in W 1,x 0 L M (Q) and then there is a constant C > 0 such that for all u ∈ W 1,x 0 L M (Q) one has thus both sides of the last inequality are equivalent norms on W 1,x 0 L M (Q).We have then the following complementary system   F being the dual space of W 1,x 0 E M (Q).It is also, up to an isomorphism, the quotient of ΠL M by the polar set W 1,x 0 E M (Q) ⊥ , and will be denoted by This space will be equipped with the usual quotient norm: where the inf is taken over all possible decompositions (2.17) The space F 0 is then given by and is denoted by Remark 2.3.We can easily check, using lemma 2.1, that each uniformly lipschitzian mapping F , with F (0) = 0, acts in inhomogeneous Orlics-Sobolev spaces of order 1: Proof: (See [22]) ✷

Basic assumptions, definition of a renormalized solution
Throughout the paper, we assume that the following assumptions hold true: Let Ω is a bounded open set of R N (N ≥ 2 ), T > 0 is given and we set Q T = Ω×(0, T ).Let M and P be two N −functions such that P ≪ Q.
where c 2 (x, t) ∈ L 1 (Q) and d : R + → R + is a continuous and nondecreasing function.Furthermore let As already mentioned in the introduction, problem (1.1) does not admit a weak solution under assumptions 3.1-3.7 since the growths of a(x, t, u, ∇u) and Φ(u) are not controlled with respect to u (so that these fields are not in general defined as distributions, even when u belongs to Throughout this paper , means for either the pairing between The definition of a renormalized solution for problem (1.1) can be stated as follows.

Definition 3.2. A measurable function u defined on
and if, for every function S ∈ W 2,∞ (R), which is piecewise C 1 and such that S ′ has a compact support, we have ) is formally obtained through pointwise multiplication of (1.1) by S ′ (u).However, while a(x, t, u, ∇u), Φ(u) and g(x, t, u, ∇u) does not in general make sense in 1.1, all the terms in (3.11) have a meaning in D ′ (Ω) and the following identifications are made in (3.11): and with (3.1), (3.9) we obtain that and by the same arguments as above we get • S ′ (u)Φ(u) and S ′′ (u)Φ(u)∇u respectively identify with S ′ (u)Φ(T K (u)) and S ′′ (u)Φ(T K (u))∇T K (u).Due to the properties of S and Φ is a continuous function, the functions S ′ , S ′′ and Φ • T K are bounded on R so that (3.9)

Statements of results
This section is devoted to establish the following existence theorem: Proof: The proof of Theorem 4.1 is done in 6 steps.
For n ∈ N * , let us define the following approximation of a, g, Φ and f : such that Φ n uniformly converges to Φ on any compact subest of R as n −→ +∞.
Let us now consider the approximate problem: in Ω. (4.6)Note that g n (x, t, u n , ∇u n ) satisfies the following conditions Since g n is bounded for any fixed n, as a consequence, proving of a weak solution ) is an easy task (see e.g.[21,27] ).
The estimates derived in this step rely on usual techniques for problems of the type (4.6).Proposition 4.2.Assume that (3.1)-(3.7)are satisfied, and let u n be a solution of the approximate problem (4.6).Then for all ℓ, n > 0, we have where C g is a positive constant not depending on n.
Proof: We take T ℓ (u n )χ (0,τ ) as test function in (4.6), we get for every τ ∈ (0, T ) which implies that where The Lipshitz character of Φ n , Stokes formula together with the boundary condition u n = 0 on (0, T ) × Ω, make it possible to obtain Due to the definition of T ℓ and (4.4) we have and so by letting θ −→ 0 and using Fatou's lemma, we deduce that g n (x, t, u n , ∇u n ) is a bounded sequence in L 1 (Q T ), then we obtain iii).By using (4.5), (4.10), (4.11) and iii), permit to deduce from (4.9) that ) where here and below C i denote positive constants not depending on n and ℓ.By using (4.14) and the fact that T ℓ (u n ) ≥ 0, permit to deduce that which implies by virtue of (3.3) that We deduce from that above inequality (4.14) that On the other hand, thanks to Lemma 5.7 of [25], there exists two positive constants δ, λ such that ) and using (4.16), one has which implies that for some in the sense of distributions.This implies, thanks to (4. 19) and the fact that Due to the choice of ζ ℓ , we conclude that for each ℓ, the sequence T ℓ (u n ) converges almost everywhere in Q T , which implies that u n converges almost everywhere to some measurable function u in Q T .Therfore, following [7,8,10,11,28], we can see that there exists a measurable function u ∈ L ∞ (0, T ; L 1 (Ω)) such that for every ℓ > 0 and a subsequence, not relabeled, and strongly in L 1 (Q T ) and a. e. in Q T .
We prove that In view of the monotonicity of a one easily has, which gives (4.28) On the other hand, using (3.1), we see that And so, by using the dual norm of (L M (Q T )) N we conclude that Using the fact that as n tends to +∞.
Let use give the following lemma which will be needed later: Lemma 4.6.Assume that (3.1)-(3.7)are satisfied, and let z n be a sequence in as n and s tend to +∞, and where χ s is the characteristic function of

.44)
Proof: See [23].✷ Proof: (Proposition 4.5).The proof is almost identical of the one given in, e.g.[23].where the result is established for the growth of a(x, t, u, Du) is controlled with respect to u.This proof is devoted to introduce for ℓ ≥ 0 fixed, a time regularization of the function T ℓ (u), this notion, introduced by R. Landes (see Lemma 6 and Proposition 3, p. 230 and Proposition 4, p. 231 in [27]).More recently, it has been exploited in [13] and [18] to solve a few nonlinear evolution problems with L 1 or measure data.
for the modular convergence and let ψ ı be a sequence which converges strongly to u 0 in L 1 (Ω).
Let ω β ı, = T ℓ (υ  ) β +exp −βt T ℓ (ψ ı ) where T ℓ (υ  ) β is the mollification with respect to time of T ℓ (υ  ), note that ω β ı, is a smooth function having the following properties: ) for the modular convergence as  → ∞, for the modular convergence as ı → ∞.Let now the function ρ m defined on R with m ≥ ℓ by: Using the admissible test function ϕ β,m ı,,n as test function in (4.6) leads to The very definition of the sequence ω β ı, makes it possible to establish the following lemma.
where , denotes the duality pairing between Proof: See ( [22]).✷ Now, we turn to complete the proof of Proposition 4.5.First, it is easy to see that Indeed, by the almost everywhere convergence of u n , we have that (4.52) Then we deduce that Similarly, Lebesgue's convergence theorem shows that as n → +∞, and as n → +∞.On the other hand, by using the modular convergence of ω β ı, as  → +∞ and letting β tend to infinity, we get Concerning the third term of the right hand side of (4.48) we obtain that (4.56) Then by (4.31) we deduce that,

.57)
We now turn to the fourth term of the left hand side of (4.49).We can write On the other hand, the second term of the right hand side of (4.58) reads as where χ s  denotes the characteristic function of the subset And, as above, by letting first n then , β and finally s go to infinity, we can easily see that each one of last two integrals is of the form ǫ(n, β, ).This implies that .61) where we have used the fact that, since (4.62) By letting n → +∞ which implies that, by letting  → +∞ so that, by letting β → +∞ Using now the term I 1 of (4.62), we conclude that, it is easy to show that, ) As before, in the following we pass to the limit in (4.64): first we let n tends to +∞, then  then β then m tends tends to +∞.Starting with J 2 , observe first that as n tends to +∞.We get denoting by χ s the characteristic function of the subset By letting n → +∞ and since a(x, t, T ℓ (u n ), ∇T ℓ (u n )) ⇀ ϕ ℓ weakly in (L M (Q T )) N we have which gives by letting  → +∞ and since v  → T ℓ (u) in W 1,x 0 L M (Q T ) for the modular convergence, we have (4.66)implying that, by letting β → +∞, J 3 = QT ϕ ℓ ∇T ℓ (u)χ s dx dt+ ǫ(n, , β), and thus (4.67) Concerning J 4 we can write which implies that, by letting  → +∞, By letting β → +∞ we obtain and so, thanks to (4.35 Passing to the limit in n and  in the last three terms on the right-hand side of the last equality, we get On the other hand, we have It is easy to see that the last terms of the last equality tend to zero as n → +∞, which implies that ≤ ǫ(n, , β, m, s). (4.72) To pass to the limit in (4.72) as n, , m, s tend to infinity, we obtain then as a consequence of (4.36), it follows that ∇u n converges to ∇u in measure and therefore, always reasoning for a subsequence, Step 4: Equi-integrability of the nonlinearitie g n (x, t, u n , ∇u n ).
We shall now prove that g n (x, t, u n , ∇u n ) → g(x, t, u, ∇u) strongly in L 1 (Q T ) by using Vitali's theorem.Since g n (x, t, u n , ∇u n ) → g(x, t, u, ∇u) a.e. in Q T , thanks to (4.22) and (4.74), it suffices to prove that g n (x, t, u n , ∇u n ) are uniformly equi-integrable in Q T .Let E ⊂ Q T be a measurable subset of Q T .We have for any m > 0 : where we have used (3.4) and (4.13).Therefore, it is easy to see that there exists δ > 0 such that which shows that g n (x, t, u n , ∇u n ) are uniformly equi-integrable in Q T as required.
Step 5: In this step we prove that u satisfies (3.9).
Lemma 4.9.The limit u of the approximate solution u n of (4.(4.77) Taking the limit as m tends to +∞ in (4.77) and using the estimate (4.31) it possible to conclude that (4.76) holds true and the proof of Lemma 4.9 is complete.✷ Step 6: In this step, u is shown to satisfies (3.11) .Let S be a function in W 2,∞ (R) such that S ′ has a compact support.Let K be a positive real number such that suppS As a consequence of (3.6), (4.3) and (4.22), it follows that: as n tends to +∞.The term S ′ (u)Φ(T K (u)) is denoted by S ′ (u)Φ(u).
◮ Since S ∈ W 1,∞ (R) with suppS ′ ⊂ [−K, K], we have we have, ∇S ′′ (u n ) converges to ∇S ′′ (u) weakly in (L M (Q T )) N as n tends to +∞, while Φ n (T K (u n )) is uniformly bounded with respect to n and converges a. e. in Q T to Φ(T K (u)) as n tends to +∞.Therefore S ′′ (u n )Φ n (u n )∇u n ⇀ Φ(T K (u))∇S ′′ (u) weakly in L M (Q T ).
◮ Due to (4.5) and (4.22), we have f n S ′ (u n ) converges to f S ′ (u) strongly in L 1 (Q T ), as n tends to +∞.
◮ Due to (4.22) and the fact that g n (x, t, u n , ∇u n ) → g(x, t, u, ∇u) strongly in L 1 (Q T ), we have g n S ′ (u n ) converges to gS ′ (u) strongly in L 1 (Q T ), as n tends to +∞.
As a consequence of the above convergence result, we are in a position to pass to the limit as n tends to +∞ in equation (4.78) and to conclude that u satisfies (3.11).Remark that, S ′ has a compact support, we have S(u n ) is bounded in L ∞ (Q T ). by (4.78) and the above considerations on the behavior of the terms of this equation show that ∂S(un)   ∂t is bounded in L 1 (Q T )+W −1,x L M (Q T ). a consequence, an Aubin's type Lemma (see e.g., [30], Corollary 4) (see also [23]) implies that S(u n )(t = 0) lies in a compact set of C 0 ([0, T ]; L 1 (Ω)).It follows that, S(u n )(t = 0) converges to S(u)(t = 0) strongly in L 1 (Ω).Due to (4.4), we conclude that S(u n )(t = 0) = S(u n (x, 0)) converges to S(u)(t = 0) strongly in L 1 (Ω).Then we conclude that S(u)(t = 0) = S(u 0 ) in Ω.
As a conclusion of step 1 to step 6, the proof of theorem 4.

( 4 .
60) Splitting the first integral on the left hand side of (4.57) where |u n | ≤ ℓ and |u n | > ℓ, we can write,
Let u u be a solution of the approximate problem(4.6).Then Considering the following function ϕ = T 1 (u n − T m (u n )) as test function in (4.6) we obtain, , β, s).
QTa n (x, t, T ℓ .73) This implies by the lemma 4.6, the desired statement and hence the proof of Proposition 4.5 is achieved.