Entropy Solutions For Nonlinear Parabolic Inequalities Involving Measure Data In Musielak-Orlicz-Sobolev Spaces

A.Talha, A. Benkirane, M.S.B. Elemine Vall abstract: In this paper, we study an existence result of entropy solutions for some nonlinear parabolic problems in the Musielak-Orlicz-Sobolev spaces.


Introduction
Let Ω a bounded open subset of R N and let Q be the cylinder Ω × (0, T ) with some given T > 0. We consider the strongly nonlinear parabolic problem (P) where A : D(A) ⊂ W 1,x 0 L ϕ (Q) −→ W −1,x L ψ (Q) (see section 2) defined by A(u) = −div(a(x, t, u, ∇u)) is an operator of Leray-Lions type, where a is a Carathéodory function such that |a(x, t, s, ξ)| ≤ β h 1 (x, t) + ψ −1  x γ(x, ν|s|) + ψ −1 x ϕ(x, ν|ξ|) is satisfied, where b a positive function in L 1 (R + ) and h 2 ∈ L 1 (Q), and f ∈ L 1 (Q) and F ∈ (E ψ (Q)) N .Under these assumptions, the above problem does not admit, in general, a weak solution since the field a(x, t, u, ∇u) does not belong to (L 1 loc (Q)) N in general.To overcome this difficulty we use in this paper the framework of entropy solutions.This notion was introduced by Bénilan and al. [4] for the study of nonlinear elliptic problems.In the classical Sobolev spaces, the authors in [9,17] proved the existence of solutions for the problem (P) in the case where F ≡ 0, in [7] the authors had proved the existence of solutions for the problem (P) in the elliptic case.In the setting of Orlicz spaces, the solvability of (P) was proved by Donaldson [10] and Robert [18], and by Elmahi [12] and Elmahi-Meskine [13].In Musielak framework, recently M. L. Ahmed Oubeid, A. Benkirane and M. Sidi El Vally in [2] had studied the problem (P) in the Inhomogeneous case and the data belongs to L 1 (Q), in the elliptic case the authors in [1] proved the existence of weak solutions for the problem (P) where the data assume to be measure and g ≡ 0. It is our purpose in this paper to prove the existence of entropy solutions for problem (P) in the setting of Musielak Orlicz spaces for general Musielak function ϕ with a nonlinearity g(x, t, u, ∇u) having natural growth with respect to the gradient.Our result generalizes that of [13,1,2] to the case of inhomogeneous Musielak Orlicz Sobolev spaces.The plan of the paper is as follows.Section 2 presents the mathematical preliminaries.Section 3 we make precise all the assumptions on a, g, f and u 0 .Section 4 is devoted to some technical lemmas with be used in this paper.Section 5 we establish some compactness and approximation results.Final section is consecrate to define the entropy solution of (P) and to prove existence of such a solution.

Preliminary
In this section we list briefly some definitions and facts about Musielak-Orlicz-Sobolev spaces.Standard reference is [16].We also include the definition of inhomogeneous Musielak-Orlicz-Sobolev spaces and some preliminaries Lemmas to be used later.
Entropy solutions for nonlinear parabolic inequalities . . .Let Ω be an open set in R N and let ϕ be a real-valued function defined in Ω × R + , and satisfying the following conditions : a) ϕ(x, •) is an N-function convex, increasing, continuous, ϕ(x, 0) = 0, ϕ(x, t) > 0, ) is a measurable function.
A function ϕ, which satisfies the conditions a) and b) is called Musielak-Orlicz function.
For a Musielak-orlicz function ϕ we put ϕ x (t) = ϕ(x, t) and we associate its nonnegative reciprocal function ϕ −1 x , with respect to t that is The Musielak-orlicz function ϕ is said to satisfy the ∆ 2 -condition if for some k > 0 and a non negative function h integrable in Ω, we have When (2.1) holds only for t ≥ t 0 > 0; then ϕ said to satisfy ∆ 2 near infinity.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 almost all x ∈ Ω γ(x, t) ≤ ϕ(x, ct) for all t ≥ t 0 , ( resp.for all t ≥ 0 i.e. t 0 = 0).
We say that γ grows essentially less rapidly than ϕ at 0 (resp.near infinity), and we write γ ≺≺ ϕ, If for every positive constant c we have We define the functional where u : Ω −→ R a Lebesgue measurable function.In the following, the measurability of a function u : Ω −→ R means the Lebesgue measurability.The set is called the generalized Orlicz class.The Musielak-Orlicz space (or the generalized Orlicz spaces) 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 that is, ψ is the Musielak-Orlicz function complementary to ϕ in the sens of Young with respect to the variable s.
We define in the space L ϕ (Ω) 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 [16].
The closure in L ϕ (Ω) of the bounded measurable functions with compact support in Ω is denoted by E ϕ (Ω).A Musielak function ϕ is called locally integrable on Ω if ρ ϕ (tχ E ) < ∞ for all t > 0 and all measurable E ⊂ Ω with meas(E) < ∞.
Note that local integrability in the previous definition differs from the one used in L 1 loc (Ω), where we assume integrability over compact subsets.
We say that sequence of functions For any fixed nonnegative integer m we define and Entropy solutions for nonlinear parabolic inequalities . . .
, these functionals are a convex modular and a norm on W m L ϕ (Ω), respectively, and the pair W m L ϕ (Ω), m ϕ,Ω is a Banach space if ϕ satisfies the following condition [16] : (2. 3) The space W m L ϕ (Ω) will always be identified to a subspace of the product We denote by D(Ω) the space of infinitely smooth functions with compact support in Ω and by the space of functions u such that u and its distribution derivatives up to order m lie to E ϕ (Ω), and The following spaces of distributions will also be used : We say that a sequence of functions For ϕ and her complementary function ψ, the following inequality is called the Young inequality [16]: This inequality implies that |u| ϕ,Ω ≤ ρ ϕ,Ω (u) + 1. (2.5) In L ϕ (Ω) we have the relation between the norm and the modular (2.6) For two complementary Musielak Orlicz functions ϕ and ψ, let u ∈ L ϕ (Ω) and v ∈ L ψ (Ω), then we have the Hölder inequality [16] Ω u(x)v(x)dx ≤ u ϕ,Ω |v| ψ,Ω . (2.8)

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

Essential assumptions
Let Ω be a bounded open subset of R N satisfying the segment property and T > 0 we denote Q = Ω × [0, T ], and let ϕ and γ be two Musielak-Orlicz functions such that γ ≺≺ ϕ.
where a : a(x, t, s, ξ) where c(x, t) a positive function, c(x, t) ∈ E ψ (Q) and positive constants ν, α.Furthermore, let g(x, t, s, ξ) are satisfied, where b : 4. Some technical Lemmas Lemma 4.1.[5].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: i) There exist a constant c > 0 such that inf x∈Ω ϕ(x, 1) ≥ c.
ii) There exist a constant A > 0 such that for all x, y ∈ Ω with |x − y| ≤ 1 2 we have iii) 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 Consequently, the action of a distribution S in W −1 L ψ (Ω) on an element u of W 1 0 L ϕ (Ω) is well defined.It will be denoted by < S, u >.
To prove our result, it suffices to show that with d = max diam(Ω), 1 and diam(Ω) is the diameter of Ω.First, suppose that u ∈ D(Ω), then so by the Jensen integral inequality we obtain where f (σ) = ϕ x 1 , ..., x i0−1 , σ, x i0+1 , ..., x N , d ∂u ∂xi 0 (x 1 , ..., x i0−1 , σ, x i0+1 , ..., x N ) .By integrating with respect to x, we get , we can get it out of the integral to respect of x i0 and by the fact that σ is arbitrary, then by Fubini's Theorem we get For u ∈ W 1 0 L ϕ (Ω) according to Lemma 4.1, we have the existence of u n ∈ D(Ω) and λ > 0 such that e in Ω, ( for a subsequence still denote u n ).
Then, we have Lemma 4.6 (The Nemytskii Operator).Let Ω be an open subset of R N with finite measure and let ϕ and ψ be two Musielak Orlicz functions.Let f : Ω × R p −→ R q be a Carathodory function such that for a.e.x ∈ Ω and all s ∈ R p : where k 1 and k 2 are real positives constants and c(. ) is continuous from into (L ψ (Ω)) q for the modular convergence.Furthermore if c(•) ∈ E γ (Ω) and γ ≺≺ ψ then N f is strongly continuous from

Approximation and trace results
In this section, Ω be a bounded Lipschitz domain in R N with the segment property and I is a subinterval of R (both possibly unbounded) and Q = Ω × I.It is easy to see that Q also satisfies Lipschitz domain.We say that for the modular convergence for all |α| ≤ 1, and u 0 n −→ u 0 strongly in L 2 (Q).We shall prove the following approximation theorem, which plays a fundamental role when the existence of solutions for parabolic problems is proved.[2] Let ϕ be an Musielak-Orlicz function satisfies the assumption (4.1).
In order to deal with the time derivative, we introduce a time mollification of a function u ∈ W 1,x 0 L ϕ (Q).Thus we define, for all µ > 0 and all (x, t) where ũ(x, t) = u(x, t)χ [0,T ] (t).Throughout the paper the index ì always indicates this mollification.
Let λ > 0 large enough such that On the one hand, for a.e (x, t) ∈ Q, we have On the other hand, one has This implies that Entropy solutions for nonlinear parabolic inequalities . . .

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Let ε > 0 there exists k 0 > 0 such that ∀k > k 0 , we have and there exists µ 0 > 0 such that ∀µ > µ 0 and for all k > k 0 Then, we get Finely, by using (iii) of Lemma 4.1 and by letting µ −→ +∞, there exits 2. Since for all indice α such that |α| ≤ 1, we have D α x (u µ ) = (D α x u) µ , consequently, the first part above applied on each D α x u, gives the result.

Compactness Results
For each h > 0, define the usual translated τ h f of the function First of all, recall the following compactness results proved by the authors in [2].Lemma 6.1.Let ϕ be a Musielak function.Let Y be a Banach space such that the following continuous imbedding holds L 1 (Ω) ⊂ Y .Then for all ε > 0 and all λ > 0, there is Indeed, if the above assertion holds false, there is ε 0 > 0 and This gives, by setting w n = vn ∇vn Lϕ(Ω) , Since (w n ) n is bounded in W 1 0 L ϕ (Ω) then for a subsequence Thus, w n L 1 (Ω) is bounded and w n Y → 0 as n → +∞.We conclude w n → 0 in Y and that w = 0 implying that Integrating this over [0, T ] yields We also prove the following lemma which allows us to enlarge the space Y whenever necessary.
Moreover, there exists a finite sequence and hence F is relatively compact in L 1 (Q).Since γ ≪ ϕ then by using Vitali's theorem, it is easy to see that Proof Let γ and θ be Musielak functions such that γ ≪ ϕ and θ ≪ ϕ near infinity.For all 0 < t 1 < t 2 < T and all f ∈ F , we have where we have used the following continuous imbedding Since the imbedding with continuous imbedding.

Main results
For k > 0 we define the truncation at height k: T k : R −→ R by: We note also We define We consider the following boundary value problem We will prove the following existence theorem Let Ω be a bounded Lipschitz domain in R N , ϕ and ψ be two complementary Musielak-Orlicz functions satisfying the assumptions of Lemma 4.1 and ϕ(x, t) decreases with respect to one of coordinate of x, we assume also that (3.1)-(3.6)and (3.7) hold true.Then the problem (P) has at least one entropy solution of the following sense

Proof
Step 1 : Approximate problems Consider the following approximate problem where we have set g n (x, t, s, ξ) = T n (g(x, t, s, ξ)).Moreover, the sequence( . Thanks to theorem 5.1 of [2], there exists at least one solution u n of problem (P n ).
Step 2 : A priori estimates In this section we denote by c i , i = 1, 2, ... a constants not depends on k and n.For k > 0, consider the test function T k (u n ) in (P n ), we have On the one hand, let 0 < p < min(α, 1), (where α is the constant of (3.3)), then by using the Young's inequality, we have Combining (7.3) and (7.4), we obtain Using now (3.5) and (3.3) which implies that (7.6)In other hand, the first term of the left hand side of the last inequality, reads as Entropy solutions for nonlinear parabolic inequalities . . .

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Using the fact that S k (σ) > 0, |S k (u 0n | ≤ k|u 0n |, then (7.6) can be write as Hence by using (3.3), we have By using the Lemma 4.5, we have where c is the constant of Lemma 4.5. (7.9) Step 3 : Convergence in measure of (u n ) n Let k > 0 large enough, by using (7.8), we have Where c 4 is a constant not dependent on k ,hence For every λ > 0 we have Consequently, by (7.8) we can assume that (T k (u n )) n is a Cauchy sequence in measure in Q.
Let ε > 0, then by (7.10) there exists some k = k(ε) > 0 such that Which means that (u n ) n is a Cauchy sequence in measure in Q, thus converge almost every where to some measurable functions u.Then Step 4 : Boundedness of (a(•, Thanks to (7.7), we have On the other hand, for λ large enough (λ > β), we have by using (3.1). Q Now, since γ grows essentially less rapidly than ϕ near infinity ad by using the Remark 2.1, there exists r(k) > 0 such that γ(x, νk) ≤ r(k)ϕ(x, 1) and so we have Which implies that second term of the right hand side of (7.12) is bounded, consequently we obtain Hence by the theorem of Banach Steinhous the sequence (a(x, t, Which implies that, for all k > 0 there exists a function Step 5 : Modular convergence of truncations For the sake of simplicity, we will write only ε(n, j, µ, s) to mean all quantities (possibly different) such that For the remaining of this article, χ s and χ j,s will denoted respectively the characteristic functions of the sets Let 0 < p < min(1, α), by Young's inequality, we have Using now (3.3) on the last term of the last inequality, we get Which implies that, The first term of the left hand side of the last equality reads as The second term of the last equality can be easily to see that is positive and the third term can be written as thus by letting n, j −→ +∞, and since (α j k ) −→ T k (u) a.e. in Q and by using Lebesgue Theorem, Entropy solutions for nonlinear parabolic inequalities . . .

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Then, (7.14) can be write as On the other hand, Thus, by using the fact that Hence, by using (3.4), we get where b k = sup{b(s) : |s| ≤ k}.
Using now (7.8), there exists a constant c 3 > 0 depends on k such that Now, letting j to infinity, we obtain Hence, we get Then (7.20) becomes On the other hand, remark that for the second term of the last inequality, we have obviously that

+
We shall go to limit as n, j, µ and s to infinity in the last fifth integrals of the last side.
Step 7 : Passage to the limit Entropy solutions for nonlinear parabolic inequalities . . .

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Let v ∈ W 1,x 0 L ϕ (Q) such that ∂v ∂t ∈ W −1,x L ψ (Q) + L 1 (Q).There exists a prolongation v of v such that (see the proof of lemma ) By theorem , there exists a sequence (w j ) j in D(Ω × R) such that w j −→ v in W 1,x 0 L ϕ (Ω × R) and Thus , by using the modular convergence of j, we achieve this step.As a conclusion of Step 1 to Step 7, the proof of Theorem 7 is complete.