Nonlinear parabolic systems in Musielack-Orlicz space

2 Musielak-Orlicz spaces Notations and properties 2 2.1 Musielak-Orlicz function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Musielak-Orlicz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Inhomogeneous Musielak-Orlicz-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Truncation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5


Introduction
From a physical point of view, the study of non-linear partial differential equations governed by general non-linear operators, with non-polynomial growths (growths described by N-functions or φ-functions) are considerable at the application level. For example, non-standard growth operators (φ(x, t) = |t| p(x) ), in this context include models from fluid mechanics (non-Newtonian fluid), image processing (see Rajagopal, Rusi cka [19]) for more details. For more general operators with growth described by modular functions φ(x, t), Polish and Czechoslovak mathematicians have developed a functional framework for this type of operators, the framework of modular spaces that is the extension of Orlicz spaces appeared in the literature in the 1930s, for more details we refer to (Musielak [18], Kovacic and Rakosnich [16]).
From a mathematical point of view, the weak formulation of the PDE's is very difficult in general due to the fact that the terms of the system are not well defined, so the uniqueness of the solution is not generally accessible (see Serrin's counter-example [22]) so a difficulty to find the physically observable solution, to overcome this problem we will use the notion of a renormalized solution first introduced by R.-J. Diperna and P.-L. Lions [9]. The approach of solving the system (S) and going through approximation theorems using the notion of general modular convergence see( [15]). Let Ω be an open subset of R N (N ≥ 2), and let T > 0, Q T = Ω × (0, T )). Consider the nonlinear parabolic system in Ω, A(u) = −div(a(x, t, u i , ∇u i )) is a Leray-lions operators defined on the Inhomogeneous Musielak-Orlicz-Sobolev spaces W 1,x 0 L ϕ (Q T ), where a : Ω × (0, T ) × R × R N → R N is a Carathéodory function and g i : Q T × R × R N → R is a Carathéodory functions, with sign conditions, the source terms f i is merely integrable.
The resolution of the system (S) within the framework of the Classical Sobolev spaces is well known, we cite as an example the works on the attractors of A. El hachimi and E. Elouardi [10,11]. For type (S) systems with degenerate operator we refer to [4,5]. And the resolution of the system (S) in the case of non-polynomial growths [20].
It our purpose to solve the system (S) in the case of operators with such general growths including non-standard and non-polynomial growths, we show the existence of at least one renormalized solution of the system (S). This is the case when we are dealing with non-linear parabolic system, as in the problem in Ω.
The plan of this paper: In the section 2 present the mathematical tools, which will be used in the following sections. In section 3, we give some useful Lemmas. In section 4, we give basics assumptions and the definition of a renormalized solution of (S). Finally, we establish Theorem 5.1 the existence of such a solution of the system (S) in section 5.
A function ϕ which satisfies the conditions Φ 1 and Φ 2 is called a Musielak-Orlicz function. For a Musielak-Orlicz function ϕ we put ϕ x (t) = ϕ(x, t) and we associate its non-negative reciprocal function ϕ −1 x , with respect to t, that is Let ϕ and γ be two Musielak-Orlicz functions, we say that ϕ dominate γ, and we write γ ≺ ϕ, near infinity (resp.globally) if there exist two positive constants c and t 0 such that for a.e. x ∈ Ω γ(x, t) ≤ ϕ(x, ct) for all t ≥ t 0 (resp. for all t ≥ 0 ).

Musielak-Orlicz space
For a Musielak-Orlicz function ϕ and a measurable function u : Ω → R, we define the functionnal The set K ϕ (Ω) = {u : Ω → R measurable : ̺ ϕ,Ω (u) < ∞} is called the Musielak-Orlicz class. The Musielak-Orlicz space L ϕ (Ω) is the vector space generated by K ϕ (Ω); that is, L ϕ (Ω) is the smallest linear space containing the set K ϕ (Ω). Equivalently For any Musielak-Orlicz function ϕ, we put ψ(x, s) = sup t≥0 (st − ϕ(x, s)). ψ is called the Musielak-Orlicz function complementary to ϕ (or conjugate of ϕ) in the sense of Young with respect to s. We say that a sequence of function u n ∈ L ϕ (Ω) is modular convergent to u ∈ L ϕ (Ω) if there exists a constant λ > 0 such that lim n→∞ ̺ ϕ,Ω ( un−u λ ) = 0. This implies convergence for σ(ΠL ϕ , ΠL ψ ) [see [6]]. In the space L ϕ (Ω), we define the following two norms which is called the Luxemburg norm, and the so-called Orlicz norm by where ψ is the Musielak-Orlicz function complementary to ϕ. These two norms are equivalent [6]. K ϕ (Ω) is a convex subset of L ϕ (Ω). The closure in L ϕ (Ω) of the set of bounded measurable functions with compact support in Ω is by denoted E ϕ (Ω). It is a separable space and (E ϕ (Ω)) * = L ϕ (Ω). We have E ϕ (Ω) = K ϕ (Ω), if and only if ϕ satisfies the ∆ 2 −condition for large values of t or for all values of t, according to whether Ω has finite measure or not. We define These functionals are convex modular and a norm on W 1 L ϕ (Ω), respectively. Then pair (W 1 L ϕ (Ω), u 1 ϕ,Ω ) is a Banach space if ϕ satisfies the following condition (see [18]).
There exists a constant c > 0 such that inf The space W 1 L ϕ (Ω) is identified to a subspace of the product α≤1 L ϕ (Ω) = • The Young inequality • The Hölder inequality We say that a sequence of functions u n converges to u for the modular convergence in The following spaces of distributions will also be used

Inhomogeneous Musielak-Orlicz-Sobolev spaces
Let Ω be a bounded Lipschitz domain in R N and let Q T = Ω×]0, T [ with some given T > 0. let ϕ be a Musielak-Orlicz function.For each α ∈ N N , denote by D α x the distributional derivative on Q T of order α with respect to the variable x ∈ R N . The inhomogeneous Musielak-Orlicz-Sovolev spaces of order 1 are defined as follows The last is a subspace of the first one, and both are Banach spaces under the norm We can easily show that they form a complementary system when Ω is a Lipschitz domain. These spaces are considered as subspaces of the product space ΠL ϕ (Q T ) which has (N + 1) copies. We shall also consider the weak topologies σ(ΠL ϕ , ΠE ψ ) and σ(ΠL ϕ , ΠL ψ ). If u ∈ W 1,x L ϕ (Q T ) then the function : t → u(t) = u(t, .) is defined on (0, T ) with values in W 1 L ϕ (Ω). If, further, u ∈ W 1,x E ϕ (Q T ) then this function is a W 1 E ϕ (Ω)-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 that when Ω is a Lipschitz domain then each element u of the closure of D(Ω) with respect Nonlinear parabolic systems in Musielack-Orlicz space 5 of the weak * topology σ(ΠL ϕ , ΠE ψ ) is limit, in W 1,x L ϕ (Q T ), of some subsequence (u i ) ∈ D(Ω) for the modular convergence, i.e. there exists λ > 0 such that for all |α| ≤ 1, this space will be denoted by . It is also, except for an isomorphism, the quotient of ΠL ψ by the polar set W 1,x 0 E ϕ (Q T ) ⊥ , and will be denoted by F = W 1,x L ψ (Q T ) and it is shown that this space will be equipped with the usual quotient norm where the inf is taken on all possible decomposition and is denoted by

Lemma 2.2. [17]. Let Ω be a bounded Lipschitz domain in R N and let ϕ and ψ be two complementary Musielak-Orlicz functions which satisfy the following conditions :
• There exists a constant c > 0 such that Under this assumptions D(Ω) is dense in L ϕ (Ω) with respect to the modular topology, D(Ω) is dense in W 1 0 L ϕ (Ω) for the modular convergence and D(Ω) is dense in W 1 0 L ϕ (Ω) for the modular convergence. Consequently, the action of a distribution S in in W −1 L ψ on an element u of W 1 0 L ϕ (Ω) is well defined. It will be denoted by < S, u >.

Truncation Operator
T k , k > 0, denotes the truncation function at level k defined on R by T k (r) = max(−k, min(k, r)). The following abstract lemmas will be applied to the truncation operators.
Lemma 2.4. Assume that Ω satisfies the segment property and let u ∈ W 1 0 L ϕ (Ω). Then, there exists a sequence u n ∈ D(Ω) such that Let Ω be an open subset of R N and let ϕ be a Musielak-Orlicz function satisfying (2.5) and the conditions of Lemma 2.2. We may assume without loss of generality that ϕ * its called the Sobolev conjugate function of ϕ (see [1] for the case of Orlicz function).

Theorem 2.5. ( [18])Let Ω be a bounded Lipschitz domain and let ϕ be a Musielak-Orlicz function satisfying (2.5), (2.6) and the conditions of Lemma 2.2. Then
where ϕ * is the Sobolev conjugate function of ϕ. Moreover, if φ is any Musielak-Orlicz function increasing essentially more slowly than ϕ * near infinity, then the imbedding
Proof. We adopte the same techniques as in [15].

Technical lemma
Lemma 3.1. [17] Under the assumptions of lemma 2.2, and by assuming that ϕ(x, t) decreases with respect to one of coordinate of x, there exists a constant c 1 > 0 which depends only on Ω such that

Essential assumptions
Let Ω be an bonded open subset of R N (N ≥ 2) satisfying the segment property, and let ϕ and γ be two Musielak-Orlicz functions such that ϕ and its complementary ψ satisfies conditions of Lemma 2.2 and γ ≺≺ ϕ. A : where a : Q T × R × R N → R N is Carathéodory function such that for a.e. x ∈ Ω and for all s ∈ R,ξ, ξ * ∈ The non linear terms g i : where c 2 (x, t) ∈ L 1 (Q T ) and b : R + → R is a continuous and nondecreasing. The and for almost every x ∈ Ω, for every s 1 , s 2 ∈ R, In this paper, for any measurable subset E of Q T , we denote by meas(E) the Lebesgue measure of E. For any measurable function v defined on Q T and for any real number s, χ {v<s} (respectively, for every function S in W 2,∞ (R) which is piecewise C 1 and such that S ′ has a compact support, we have
Proof. We divide the prof of Theorem 5.1 in 5 steps.
Step 1: Approximate problem. Let us introduce the following regularization of the data : for n > 0 and i = 1, 2 a n (x, t, s, ξ) = a(x, t, T n (s), ξ) a.e. in Q T , ∀s ∈ R, ∀ξ ∈ R N , (5.1) (Ω) as n tends to + ∞, Let us now consider the regularized problem ∂u i,n ∂t − div(a n (x, u i,n , ∇u i,n )) + g i,n (x, t, u i,n , ∇u i,n )) = f i,n (x, u 1,n , u 2,n ) in Q T , Since g i,n for i = 1, 2 is bonded for any fixed n, As a consequence, proving the existence of a weak solution u i,n ∈ W 1,x 0 L ϕ (Q T ) of (5.6)-(5.8) is an easy task (see e.g. [21]).
Step 2 : A priori estimates Proposition 5.2. assume that (4.1) are satisfied, and let u i,n be a solution of the approximate problem (1). Then for all k, n, we have Proof. Let τ ∈ (0, T ) and using T k (u i,n )χ (0,τ ) as a test function in problem (5.6), we get Qt ∂u i,n ∂t T k (u i,n )χ (0,τ ) dx dt + Qt a n (x, t, u i,n , ∇u i,n )∇T k (u i,n )dx dt implies that, (5.14) Due to the definition ofT k and (5.5), we have and (̺ ǫ θ ) ′ (s) ≥ 0, ∀s ∈ R, then, by using ̺ ǫ θ (u i,n ) as a test function in (5.5) and following [1], we can see that and, so by letting θ → 0 and using Fatou's a lemma, we deduce that g i,n (x, t, u i,n , ∇T k (u i,n )) is a bounded sequence in L 1 (Ω), then we obtain (5.11). By using (5.5) and (5.15) (5.11) permit to deduce from (5.13) that ΩT k (u i,n (τ ))dx + Qt a n (x, t, u i,n , ∇u i,n )∇T k (u i,n )dx dt Where here and below C 0 denote positive constants not depending on n and k. By using (5.26) and the fact thatT k (u i,n ) ≥ 0, permit to deduce that Qt a n (x, t, u i,n , ∇u i,n )∇T k (u i,n )dx dt ≤ kC 0 , (5.19) Which implies by virtu of (4.3) that We deduce from that above inequality (5.15) that ΩT k (u i,n (t))dx ≤ kC 0 , for almost any t ∈ (0, T ). And then, by (5.26), we conclude that T k (u i,n ) is bounded in W 1,x L ϕ (Q T ) independently of n and for any k ≥ 0, so there exists a subsequence still denoted by u n such that 20) weakly in W 1,x 0 L ϕ (Q T ) for σ(ΠL ϕ , ΠE ψ ) strongly in E ϕ (Q T ) and a.e in Q T . Since Lemma 3.1 and (5.26), we get also, Then Which implies that: Now we shall prove the following proposition.

Proposition 5.3. Let u i,n be a solution of the approximate problem, then :
for some X i,k ∈ (L ψ (Q T )) N lim m→+∞ lim n→+∞ m≤|ui,n|≤m+1 a i (x, t, u i,n , ∇u i,n )∇u i,n dxdt = 0 (5.23) Proof. The first we give the proof of (5.21) and (5.22). Consider now a function non decreasing ζ k ∈ C 2 (R) such that ζ k (s) = s for |s| ≤ k 2 and ζ k (s) = k for |s| ≥ k. Multiplying the approximate equation by ζ ′ k (u i,n ), we get Hence Corollary 3.3 implies that ζ k (u i,n ) is compact in L 1 (Q T ). Due to the choice of ζ k , we conclude that for each k, the sequence T k (u i,n ) converges almost everywhere in Q T , which implies that the sequence u i,n converge almost everywhere to some measurable function u i in Q T . Then by the same argument in [3], we have u i,n → u i a.e. Q T , (5.25) where u i is a measurable function defined on Q T .
Then, by (5.26), we conclude that T k (u i,n ) is bounded in W 1,x L ϕ (Q T ) independently of n and for any k ≥ 0, so there exists a subsequence still denoted by u n such that , strongly in E ϕ (Q T ) and a.e in Q T . Since Lemma 3.1 and (5.26), we get also, Now we shall to prove the boundness of a n (x, t, T k (u i,n ), ∇T k (u i,n )) in (L ψ (Q T )) N Let φ ∈ (E ϕ (Q T )) N with ||φ|| = 1. In view of the monotonicity of a, one easily has, using (4.1) and (5.26), we easily see that And so, we conclude that a n (x, t, T k (u i,n ), ∇T k (u i,n ) is bounded sequence in L ψ (Q T )) N , and we obtain (5.22).
The second we give the proof of (5.23). Considering the following function as test function in (5.5) we obtain, where U m in (u in )(r) = r 0 ∂u i,n ∂t T 1 (s − T m (s))ds and we use U m i,n (u i,0n (T )) ≥ 0 and (5.5) we obtain that, lim n→+∞ m≤ui,n≤m+1 a n (x, t, u i,n , ∇T k (u i,n ))∇T k (u i,n ) dx dt Step 3. Let υ i,j ∈ D(Q T ) be a sequence such that υ i,j → u i in W 1,x 0 L ϕ (Q T ) for the modular convergence. This specific time regularization of T k (υ i,j ) (for fixed k ≥ 0) is defined as follows. Let (α µ i,0 ) µ be a sequence of functions defined on Ω such that We just recall that, for the modular convergence as j → +∞. (5.36) for the modular convergence as µ → +∞. (5.37) Now, we introduce a sequence of increasing C ∞ (R)-functions S m such that, for any m ≥ 1 Through setting, for fixed K ≥ 0, we obtain upon integration, Next we pass to the limit as n tends to +∞ , j tends to +∞, µ tends to +∞ and then m tends to +∞, the real number K ≥ 0 being kept fixed. In order to perform this task we prove below the following results for fixed K ≥ 0 for any m ≥ 1.
Proof of (5.42): Proof. We can follow the same proof in [21].
Proof of (5.43): If we take n > m + 1, we get for any m ≥ 1 fixed QT S ′′ m (u i,n )a n (x, t, u i,n , ∇u i,n )∇u i,n W n i,j,µ dx dt a n (x, t, u i,n , ∇u i,n )∇u i,n dx dt, for any m ≥ 1, and any µ > 0. In view (5.38) and (5.39), we can obtain a n (x, t, u i,n , ∇u i,n )∇u i,n dx dt, for any m ≥ 1. Using (5.23) we pass to the limit as m → +∞ in (5.49) and we obtain (5.43). Proof of (5.44): Since g i,n (x, t, u i,n , ∇u i,n ) → g(x, t, u i,n , ∇u) a.e.in Q T , thanks to (4.4) and (5.17) and Vitali's theorem,it suffices to prove that g i,n (x, t, u i,n , ∇u i,n ) are uniformly equi-integrable in Q . Let E ⊂ Q T be a measurable subset of Q T ,then for any m > O, one has where C is the constant in (5.19), therefore, there exists m = m(ǫ) large enough such that E∩ui,n≤m where we have used (5.17) and (4.4), therefore, it is easy to see that there exists µ > 0 such that which shows that g i,n (x, t, u i,n , ∇u i,n are uniformly equi-integrable in Q are required. Moreover, we get For fixed n ≥ 1 and n > m + 1, we have In view (5.39),(5.40),(5.50) the theorem allow us to get, for lim n→+∞ QT we fixed m > 1, and using (5.37), we have Then we conclude the proof of (5.44). Proof of (5.45): For fixed n ≥ 1 and n > m + 1, we have we fixed m > 1, and using (5.37), we have Then we conclude the proof of (5.45).
Proof of (5.46): If we pass to the lim-sup when n ,j and µ tends to +∞ and then to the limit as m tends to +∞ in (5.41). We obtain using (5.42)-(5.43), for any K ≥ 0, Thanks to (5.39), we have in the right hand side of (5.51), for n > m + 1, Using (5.22), and fixing m ≥ 1, we get when n → +∞ .
Finally we should prove that u i satisfies (4.9).
Step 4: Passing to the limit. We first show that u satisfies (4.9) for n > m + 1. According to (5.58), one can pass to the limit as n → +∞ ; for fixed m ≥ 0 to obtain lim n→+∞ m≤|ui,n|≤m+1} a n (x, t, u i,n , ∇u i,n )∇u i,n dx dt a(x, t, u i , ∇u i )∇u i dx dt.
(5.62) Pass to limit as m tends to +∞ in (5.62) and using (5.23) show that u i satisfies (4.9). Now we shown that u i to satisfy (4.10)and (4.11). 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 supp S ′ ⊂ [−K, K]. the Pointwise multiplication of the approximate equation (1) by S ′ (u i,n ) leads to ∂S(u i,n ) ∂t − div S ′ (u i,n )a n (x, u i,n , ∇u i,n ) + S ′′ (u i,n )a n (x, u i,n , ∇u i,n )∇u i,n g i,n (x, t, u i,n , ∇u i,n )S ′ (u i,n ) − div S ′ (u i,n ) = f i,n (x, u 1,n , u 1,n )S ′ (u i,n ) (5.63) in D ′ (Q T ), for i = 1, 2. Now we pass to the limit in each term of (5.63).

∂S(ui,n) ∂t
: Since S(u i,n ) converges to S(u i ) a.e. in Q T and in L ∞ (Q T ) weak ⋆ and S is bounded and continuous. Then

∂S(ui,n) ∂t
converges to ∂S(ui) ∂t in D ′ (Q T ) as n tends to +∞.
Limit of div S ′ (u i,n )a n (x, t, u i,n , ∇u i,n ) . Since supp S ′ ⊂ [−K, K], for n > K, we have S ′ (u i,n )a n (x, t, u i,n , ∇u i,n ) = S ′ (u i,n )a n x, t, T K (u i,n ), ∇T K (u i,n ) a.e. in Q T .
Using the pointwise convergence of u i,n , (5.39),(5.22) and (5.57),imply that S ′ (u i,n )a n x, t, T K (u i,n ), ∇T K (u i,n ) ⇀ S ′ (u i )a x, t, T K (u i ), ∇T K (u i ) weakly in (L ψ (Q T )) N , for σ(ΠL ψ , ΠE ϕ ) as n → +∞, since S ′ (u i ) = 0 for |u i | ≥ K a.e. in Q T . And S ′ (u i )a x, t, T K (u i ), ∇T K (u i ) = S ′ (u i )a(x, t, u i , ∇u i ) a.e. in Q T .
Limit of S ′′ (u i,n )a n (x, t, u i,n , ∇u i,n )∇u i,n . Since supp S ′′ ⊂ [−K, K], for n > K, we have S ′′ (u i,n )a n (x, t, u i,n , ∇u i,n )∇u i,n = S ′′ (u i,n )a n x, t, T K (u i,n ), ∇T K (u i,n ) ∇T K (u i,n ) a.e. in Q T .
The pointwise convergence of S ′′ (u i,n ) to S ′′ (u i ) as n → +∞, (5.39) and (5.58) we have S ′′ (u i,n )a n (x, t, u i,n , ∇u i,n )∇u i,n ⇀ S ′′ (u i )a x, t, T K (u i ), ∇T K (u i ) ∇T K (u i ), weakly in L 1 (Q T ), as n → +∞. And Limit of S ′ (u i,n )g i,n (x, t, u i,n ): We have since supp S ′ ⊂ [−K, K]. Using (5.50), it's easy to see that Limit of f i,n (x, u 1,n , u 2,n )S ′ (u i,n ): Using that f i belongs to L 1 (Q T ), and (5.3) and (5.4), we have f i,n (x, u 1,n , u 2,n )S ′ (u i,n ) → f i (x, u 1 , u 2 )S ′ (u i ) strongly in L 1 (Q T ), as n → +∞. It remains to show that for i=1,2 S(u i ) satisfies the initial condition ((4.11)). To this end, firstly remark that, in view of the definition of S ′ ϕ , we have B ϕ (x, u i,n ) is bounded in L ∞ (Q T ). Secondly, by ((5.42)) we show that ∂S(u i,n ) ∂t is bounded in L 1 (Q T ) + W −1,x L ψ (Q T )). As a consequence, an Aubin's type Lemma (see e.g., [20], Corollary 4) implies that S(u i,n ) lies in a compact set of C 0 ([0, T ]; L 1 (Ω)) . It follows that, on one hand,Su i,n (t = 0) converges to S(u i )(t = 0) strongly in L 1 (Ω). On the order hand, the smoothness of S imply that S(ui, n)(t = 0) converges to S(u i )(t = 0) strongly in L 1 (Ω), we conclude that Su i,n )(t = 0) = S(u i,0n ) converges to S(u i )(t = 0) strongly in L 1 (Ω), we obtain S(u i )(t = 0) = S(u i,0 ) a.e. in Ω and for all M > 0, now letting M to +∞, we conclude that u i )(t = 0) = u i,0 ) a.e. in Ω.
As a conclusion, the proof of Theorem (5.1) is complete.