EXISTENCE RESULTS FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS IN ORLICZ SPACES Faculté des Sciences Juridiques,

An existence result of a renormalized solution for a class of non- linear parabolic equations in Orlicz spaces is proved. No growth assumption is made on the nonlinearities.

Under these assumptions, the above problem does not admit, in general, a weak solution since the fields a(x, t, u, ∇u) and Φ(u) do not belong in (L 1 loc (Q) N in general.To overcome this difficulty we use in this paper the framework of renormalized solutions.This notion was introduced by Lions and DiPerna [31] for the study of Boltzmann equation (see also [27], [11], [29], [28], [2]).
The existence and uniqueness of renormalized solution of (1.1)-(1.3)has been proved in H. Redwane [34,35] in the case where Au = −div a(x, t, u, ∇u) is a Leray-Lions operator defined on L p (0, T ; W 1,p 0 (Ω)), the existence of renormalized solution in Orlicz spaces has been proved in E. Azroul, H. Redwane and M. EJQTDE, 2010 No. 2, p. 1 Rhoudaf [32] in the case where b(x, u) = b(u) and where the growth of a(x, t, u, ∇u) is controlled with respect to u.Note that here we extend the results in [34,32] in three different directions: we assume b(x, u) depend on x , and the growth of a(x, t, u, ∇u) is not controlled with respect to u and we prove the existence in Orlicz spaces.
The paper is organized as follows.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 b, a, Φ, f and b(x, u 0 ).In Section 4 we give the definition of a renormalized solution of (1.1)- (1.3).In Section 5 we establish (Theorem 5.1) the existence of such a solution.

Preliminaries
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 a(s) ds where a : R + → R + is nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞.The N -function M conjugate to M is defined by M (t) = t 0 a(s) ds, where a : R + → R + is given by a(t) = sup{s : a(s) ≤ t}.
The N-function M is said to satisfy the ∆ 2 condition if, for some k > 0, (2.1) When this inequality holds only for t ≥ t 0 > 0, M is said to satisfy the ∆ 2 -condition near infinity.Let P and 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, We will extend these N-functions into even functions on all R. 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 : (2.4) 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 , we will use the weak topologies σ(ΠL M , ΠE M ) and σ(ΠL M , ΠL M ).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.
) 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. [21]).Consequently, the action of a distribution in W −1 L M (Ω) on an element of W 1 0 L M (Ω) is well defined.For more details see [1], [23].
The following abstract lemmas will be applied to the truncation operators.
Lemma 2.1.[21] Let F : R → R be uniformly lipschitzian, with F (0) = 0. Let M be an N -function and let u ∈ [21] Let F : R → R be uniformly lipschitzian, with F (0) = 0. We 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 ).M be an N -function.For each α ∈ N N , denote by ∇ α x the distributional derivative on Q of EJQTDE, 2010 No. 2, p. 3 order α with respect to the variable x ∈ N N .The inhomogeneous Orlicz-Sobolev spaces are defined as follows, (2.8) The last space is a subspace of the first one, and both are Banach spaces under the norm, (2.9) 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 have as many copies as there is α-order derivatives, |α| ≤ 1.
We shall also consider the weak topologies σ(ΠL M , ΠE M ) and σ(ΠL -valued and is strongly measurable.Furthermore the following imbedding holds: we can not conclude that the function u(t) is measurable on (0, T ).However, the scalar function We can easily show as in [22] that when Ω has the segment property, then each element u of the closure of D(Q) with respect of the weak * topology σ(ΠL M , ΠE M ) is a limit, in W 1,x L M (Q), of some subsequence (u i ) ⊂ D(Q) for the modular convergence; i.e., there exists λ > 0 such that for all |α| ≤ 1, (2.10) This space will be denoted by 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 (2.13) It is also, except for an isomorphism, the quotient of ΠL M by the polar set W 1,x 0 E M (Q) ⊥ , and will be denoted by F = EJQTDE, 2010 No. 2, p. 4 This space will be equipped with the usual quotient norm where the infimum is taken on all possible decompositions The space F 0 is then given by, (2.17) 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 Orlicz-Sobolev spaces of order 1 :

Assumptions and statement of main results
Throughout this paper, we assume that the following assumptions hold true: Ω is a bounded open set on R N (N ≥ 2), T > 0 is given and we set Q = Ω × (0, T ).Let M and P be two N -function such that for almost every x ∈ Ω, for every s such that |s| ≤ K. Consider a second order partial differential operator A : for any K > 0, there exist for almost every (x, t) ∈ Q and for every |s| ≤ K and for every ξ ∈ R N .
for almost every (x, t) ∈ Q, for every s ∈ R and for every ξ = ξ * ∈ R N , where α > 0 is a given real number.
Remark 3.1.As already mentioned in the introduction, problem (1.1)-(1.3)does not admit a weak solution under assumptions (3.1)-(3.9)(even when b(x, u) = u) since the growths of a(x, t, u, Du) 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

Definition of a renormalized solution
The definition of a renormalized solution for problem (1.1)-(1.3)can be stated as follows.
and if, for every function S in W 2,∞ (R), which is piecewise C 1 and such that S ′ has a compact support, we have The following remarks are concerned with a few comments on definition 4.
the following identifications are made in (4.3).
⋆ S ′ (u)a(x, t, u, ∇u)∇u identifies with S ′ (u)a x, t, T K (u), ∇T K (u) ∇T K (u) and in view of (3.2) and (4.1) one has ⋆ 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 (3.7), the functions S ′ , S ′′ and Φ•T K are bounded on R so that (4.1) implies that The above considerations show that equation (4.3) takes place in D ′ (Q) and that Due to the properties of S and (3.2), we have and (4.7) B S (x, u) belongs to W 1,x 0 L M (Q).Moreover (4.5) and (4.7) implies that B S (x, u) belongs to C 0 ([0, T ]; L 1 (Ω)) (for a proof of this trace result see [30]), so that the initial condition (4.4) makes sense.Remark 4.3.For every S ∈ W 2,∞ (R), nondecreasing function such that suppS ′ ⊂ [−K, K] and (3.2), we have for almost every x ∈ Ω and for every r, r ′ ∈ R.

Existence result
This section is devoted to establish the following existence theorem.
such that Φ n uniformly converges to Φ on any compact subset of R as n tends to +∞. (5.4) Let us now consider the following regularized problem: (5.6) As a consequence, proving existence of a weak solution u n ∈ W 1,x 0 L M (Q) of (5.6)-(5.8) is an easy task (see e.g.[25], [33]).

⋆
Step 2. The estimates derived in this step rely on usual techniques for problems of the type (5.6)-(5.8).
EJQTDE, 2010 No. 2, p. 9 where C ′ is a constant independent of K and n.Finally, lim K→∞ meas (x, t) ∈ Q : |u n | > K = 0 uniformly with respect to n.
We prove de following proposition: Proposition 5.3.Let u n be a solution of the approximate problem (5.6)- (5.8), then (5.26) lim Proof.Proceeding as in [5,9,7], we have for any (5.28) As a consequence of (4.6) and (5.17) we then obtain (5.27).To show that (5.28) holds true, we multiply the equation for u n in (5.6) by S ′ (u n ) to obtain (5.29) Where B n S (x, r) = r 0 S ′ (s) ∂b n (x, s) ∂s ds.Since supp S ′ and supp S ′′ are both included in [−K, K], u ε may be replaced by T K (u n ) in each of these terms.As a consequence, each term in the right hand side of (5.29) is bounded either in As a consequence of (3.2), (5.29) we then obtain (5.28).Arguing again as in [5,7,6,9] estimates (5.27), (5.28) and (4.8), we can show (5.22) and (5.23).
We now establish that b(x, u) belongs to L ∞ (0, T ; L 1 (Ω)).To this end, recalling (5.23) makes it possible to pass to the limit-inf in (5.18) as n tends to +∞ and to obtain 1 for almost any τ in (0, T ).Due to the definition of B K (x, s), and because of the pointwise convergence of 1 K B K (x, u) to b(x, u) as K tends to +∞, which shows that b(x, u) belongs to L ∞ (0, T ; L 1 (Ω)).
EJQTDE, 2010 No. 2, p. 11 Step 3.This step is devoted to introduce for K ≥ 0 fixed, a time regularization w i µ,j of the function T K (u) and to establish the following proposition: Proposition 5.4.Let u n be a solution of the approximate problem (5.6)- (5.8).Then, for any k ≥ 0: as n tends to +∞.
Let use give the following lemma which will be needed later: Lemma 5.5.Under assumptions (3.1) − (3.9), and let (z n ) be a sequence in as n and s tend to +∞, and where χ s is the characteristic function of Then, (5.41) ∇z n → ∇z a.e. in Q, Proof.See [32].
Proof.(Proposition 5.4).The proof is almost identical of the one given in, e.g.[32].where the result is established for b(x, u) = u and where the growth of a(x, t, u, Du) is controlled with respect to u.This proof is devoted to introduce for k ≥ 0 fixed, a time regularization of the function T k (u), this notion, introduced by R. Landes (see Lemma 6 and Proposition 3, p. 230 and Proposition 4, p. 231 in [24]).More recently, it has been exploited in [10] and [15] to solve a few nonlinear evolution problems with L 1 or measure data.Let v j ∈ D(Q) be a sequence such that v j → u in W 1,x 0 L M (Q) for the modular convergence and let ψ i ∈ D(Ω) be a sequence which converges strongly to u 0 in L 1 (Ω).
EJQTDE, 2010 No. 2, p. 12 Let w µ i,j = T k (v j ) µ +e −µt T k (ψ i ) where T k (v j ) µ is the mollification with respect to time of T k (v j ), note that w µ i,j is a smooth function having the following properties: (5.44) Let now the function h m defined on R with m ≥ k by: Using the admissible test function ϕ µ,i n,j,m = (T k (u n ) − w µ i,j )h m (u n ) as test function in (5.6) leads to (5.47) Denoting by ǫ(n, j, µ, i) any quantity such that, The very definition of the sequence w µ i,j makes it possible to establish the following lemma.
Lemma 5.6.Let ϕ µ,i n,j,m = (T k (u n ) − w µ i,j )h m (u n ), we have for any k ≥ 0: (5.48) where , denotes the duality pairing between [34,32].Now, we turn to complete the proof of proposition 5.4.First, it is easy to see that (see also [32]): (5.49) n,j,m dx dt = ǫ(n, j, µ), (5.50) Concerning the third term of the right hand side of (5.47) we obtain that (5.52) Then by (5.26).we deduce that, (5.53) Finally, by means of (5.47)-( 5.53), we obtain, (5.54) Splitting the first integral on the left hand side of (5.54) where In the following we pass to the limit in (5.55) as n tends to +∞, then j then µ and then m tends to +∞.We prove that Using now the term I 1 of (5.55), we conclude that, it is easy to show that, (5.56) where χ s j denotes the characteristic function of the subset In the following we pass to the limit in (5.56) as n tends to +∞, then j then µ then m tends and then s tends to +∞ in the last three integrals of the last side.We prove that (5.57)

and
(5.59) We conclude then that, (5.60) Combining (5.48), (5.56), (5.57), (5.58), (5.59) and (5.60) we deduce, (5.61) s).To pass to the limit in (5.61) as n, j, m, s tends to infinity, we obtain , one is at liberty to pass to the limit as n tends to +∞ for fixed m ≥ 0 and to obtain (5.64) lim  It what follows we pass to the limit as n tends to +∞ in each term of (5.65).EJQTDE, 2010 No. 2, p. 16 ⋆ Since S ′ is bounded, and B n S (x, u n ) converges to B S (x, u) a.e. in Q and in L ∞ (Q) weak ⋆.Then ⋆ Since suppS ⊂ [−K, K], we have S ′ (u n )a n (x, t, u n , ∇u n ) = S ′ (u n )a n x, t, T K (u n ), ∇T K (u n ) a.e. in Q.
The pointwise convergence of u n to u as n tends to +∞, the bounded character of S ′ , (5.22) and (5.36) of Lemma 5.4 imply that S ′ (u n )a n x, t, T K (u n ), ∇T K (u n ) ⇀ S ′ (u)a x, t, T K (u), ∇T K (u) weakly in (L M (Q)) N , for σ(ΠL M , ΠE M ) as n tends to +∞, because S(u) = 0 for |u| ≥ K a.e. in Q.And the term S ′ (u)a x, t, T K (u), ∇T K (u) = S ′ (u)a(x, t, u, ∇u) a.e. in Q.
⋆ Since suppS ′ ⊂ [−K, K], we have S ′′ (u n )a n (x, t, u n , ∇u n )∇u n = S ′′ (u n )a n x, t, T K (u n ), ∇T K (u n ) ∇T K (u n ) a.e. in Q.
⋆ Due to (5.4) and (5.22), we have f n S(u n ) converges to f S(u) strongly in L 1 (Q), 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 (5.65) and to conclude that u satisfies (4.3).
It remains to show that B S (x, u) satisfies the initial condition (4.4).To this end, firstly remark that, S ′ has a compact support, we have B n S (x, u n ) is bounded in L ∞ (Q).Secondly, (5.65) and the above considerations on the behavior of the terms EJQTDE, 2010 No. 2, p. 17
∂B n S (x,un) ∂t converges to ∂BS (x,u) ∂t in D ′ (Q) as n tends to +∞.
2010 No.2, p. 15This implies by the lemma 5.5, the desired statement and hence the proof of Proposition 5.4 is achieved.⋆Step 4. In this step we prove that u satisfies (4.2).
{m≤|un|≤m+1} a n (x, t, u n , ∇u n )∇u n dx dt In this step, u is shown to satisfies (4.3) and (4.4).Let S be a function inW 2,∞ (R) such that S ′ has a compact support.Let K be a positive real number such that supp(S ′ ) ⊂ [−K, K].Pointwise multiplication of the approximate equation (5.6) by S ′ (u n ) leads to ′ (u n )a n (x, t, u n , ∇u n ) + S ′′ (u n )a n (x, t, u n , ∇u n )∇u n t, T m (u), ∇T m (u) ∇T m (u) dx dt = {m≤|u|≤m+1} a(x, t, u, ∇u)∇u dx dt Taking the limit as m tends to +∞ in (5.64) and using the estimate (5.26) it possible to conclude that (5.63) holds true and the proof of Lemma 5.7 is complete.⋆Step 5.