Strongly Nonlinear Parabolic Problems in Musielak-Orlicz-Sobolev Spaces

where A = − div (a(x, t, u,∇u)) is an operator of Leray-Lions type, g is a nonlinearity with the sign condition but any restriction on its growth. This result generalizes analogous ones of Lions [21], Landes [18] when g ≡ 0 and of Brezis-Browder [9], Landes.Mustonen [19] for g ≡ g(x, t, u). See also [7,8] for related topics. In these results, the function a is supposed to satisfy a polynomial growth condition with respect to u and ∇u. In the case where a satisfies a more general growth condition with respect to u and ∇u, it is shown in [12] that the adequate space in which (1) can be studied is the inhomogeneous Orlicz-Sobolev space W LM (Q) where the N-function M is


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 where A = − div (a(x, t, u, ∇u)) is an operator of Leray-Lions type, g is a nonlinearity with the sign condition but any restriction on its growth.
This result generalizes analogous ones of Lions [21], Landes [18] when g ≡ 0 and of Brezis-Browder [9], Landes.Mustonen [19] for g ≡ g(x, t, u).See also [7,8] for related topics.In these results, the function a is supposed to satisfy a polynomial growth condition with respect to u and ∇u.
In the case where a satisfies a more general growth condition with respect to u and ∇u, it is shown in [12] that the adequate space in which (1) can be studied is the inhomogeneous Orlicz-Sobolev space W 1,x L M (Q) where the N-function M is 2000 Mathematics Subject Classification: 46E35, 35K15, 35K20, 35K60 192 M. L. Ahmed Oubeid, A. Benkirane, and M. Sidi El Vally related to the actual growth of a .The solvability of (1) in this setting is proved by Donaldson [12] for g ≡ 0 and by Robert [23] for g ≡ g(x, t, u) when A is monotone, t 2 ≪ M (t) and M satisfies a ∆ 2 condition and also by Elmahi [14] for g = g(x, t, u, ∇u) when M satisfies a ∆ ′ condition and M (t) ≪ t N/(N −1) as application of some L M compactness results in W 1,x L M (Q), see [13].
The solvability of (1) in this setting is proved by Elmahi-Meskine [16] for g ≡ 0 and for g ≡ g(x, t, u, ∇u) in [15], without assuming any restriction on the Nfunction M .
In a recent work, the authors [2] have established an existence result for problems of the form (1), when g ≡ 0, without assuming any restriction on the Musielak function ϕ.
It is our purpose in this paper to prove the existence of solutions for problem (1) 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.In section 3 some new approximation result in inhomogeneous Musielak-Orlicz-Sobolev spaces (see Theorem 3.2), and, on the other hand, to prove a trace result (see Lemma 4.2).In Section 4, we establish L 1 -compactness results in the inhomogeneous Musielak-Orlicz-Sobolev spaces W 1,x L ϕ (Q).Section 5 contains the main result of this paper.
Our result generalizes that of the Elmahi-Meskine in [15] to the case of inhomogeneous Musielak-Orlicz-Sobolev spaces.
Let us point out that our result can be applied in the particular case when ϕ(x, t) = t p (x), in this case we use the notations L p(x) (Ω) = L ϕ (Ω), and W m,p(x) (Ω) = W m L ϕ (Ω).These spaces are called Variable exponent Lebesgue and Sobolev spaces.

Preliminaries
In this section we list briefly some definitions and facts about Musielak-Orlicz-Sobolev spaces.Standard reference is [22].We also include the definition of inhomogeneous Musielak-Orlicz-Sobolev spaces and some preliminaries Lemmas to be used later.
Musielak-Orlicz-Sobolev spaces : Let Ω be an open subset of R n .A Musielak-Orlicz function ϕ is a real-valued function defined in Ω × R + such that : a) ϕ(x, t) is an N-function i.e. convex, nondecreasing, continuous, ϕ(x, 0) = 0, ϕ(x, t) > 0 for all t > 0 and Now, let ϕ x (t) = ϕ(x, t) and let ϕ −1 x be the non-negative reciprocal function with respect to t, i.e the function that satisfies For any two Musielak-Orlicz functions ϕ and γ we introduce the following ordering : c) if there exists two positives constants c and T such that for almost everywhere x ∈ Ω : ϕ(x, t) ≤ γ(x, ct) for t ≥ T we write ϕ ≺ γ and we say that γ dominates ϕ globally if T = 0 and near infinity if T > 0.
d) if for every positive constant c and almost everywhere x ∈ Ω we have we write ϕ ≺≺ γ at 0 or near ∞ respectively, and we say that ϕ increases essentially more slowly than γ at 0 or near infinity respectively.
In the sequel the measurability of a function u : Ω → R means the Lebesgue measurability.
We define the functional is called the Musielak-Orlicz class (the generalized Orlicz class).The Musielak-Orlicz space (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).Equivelently: ψ is the Musielak-Orlicz function complementary to ( or conjugate of ) ϕ(x, t) in the sense of Young with respect to the variable s.
On the space L ϕ (Ω) we define the Luxemburg norm: and the so-called Orlicz norm : where ψ is the Musielak-Orlicz function complementary to ϕ.These two norms are equivalent [22].
The closure in L ϕ (Ω) of the set of bounded measurable functions with compact support in Ω is denoted by E ϕ (Ω).It is a separable space and E ψ (Ω) * = L ϕ (Ω) [22].
The following conditions are equivalent: g) ϕ has the ∆ 2 property.
We recall that ϕ has the ∆ 2 property if there exists k > 0 independent of x ∈ Ω and a nonnegative function h , integrable in Ω such that ϕ(x, 2t) ≤ kϕ(x, t) + h(x) for large values of t, or for all values of t, according to whether Ω has finite measure or not.
Let us define the modular convergence: we say that a sequence of functions For any fixed nonnegative integer m we define Now, the functional The pair W m L ϕ (Ω), ||u|| m ϕ,Ω is a Banach space if ϕ satisfies the following condition : as in [22].
The space W m L ϕ (Ω) will always be identified to a Let W m E ϕ (Ω) be the space of functions u such that u and its distribution derivatives up to order m lie in E ϕ (Ω), and let The following spaces of distributions will also be used: As we did for L ϕ (Ω), we say that a sequence of functions From [22], for two complementary Musielak-Orlicz functions ϕ and ψ the following inequalities hold: h) the young inequality : for all u ∈ L ϕ (Ω) and v ∈ L ψ (Ω).

Inhomogeneous Musielak-Orlicz-Sobolev spaces :
Let Ω an 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. and The last space 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 [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 ϕ , ΠL ψ ).
-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 t → u(t) ϕ,Ω is in L 1 (0, T ).The space W 1,x 0 E ϕ (Q) is defined as the (norm) closure in W 1,x E ϕ (Q) of D(Q).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, 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

Approximation Theorem and Trace Result
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.Definition 3.1.We say that u n → u in W −1,x L ψ (Q) + L 2 (Q) for the modular convergence if we can write for modular convergence for all |α| ≤ 1 and u α n → u α 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.Theorem 3.2.Let ϕ be an Musielak-Orlicz function satisfying the condition (1.7) of [5].
Q) and let ε > 0 be given.Writing ∂u ∂t = |α|≤1 D α x u α + u 0 , where u α ∈ L ψ (Q) for all |α| ≤ 1 and u 0 ∈ L 2 (Q), we will show that there exists λ > 0(depending only on u and N ) and there exists v ∈ D(Q) for which we can write The equation ( 3) flows from a slight adaptation of the arguments of [5], ( 4) and ( 5) flow also from classical approximation results.
Regrading the equation ( 6) it is enough to prove that D(Q) is dense in L ψ (Q) for this end.
We use the fact that the log-HÖlder continuity(commutes with the complementarity) i.e : if ϕ is log-HÖlder the its complementary ψ also it is, and proceed as in [5] (with ϕ and ψ interchanged ) and using of course R N +1 instead of R N and Q = Ω × (0, T ) instead of Ω.
These facts lead us to prove that is a measurable function with support in the ball B R = B(0, R) see [5]).And then we deduce that D(Q) is dense in L ψ (Q) for the modular convergence which gives the desired conclusion. ✷ is similar to the above arguments as in [5].
A first application of Theorem 3.2 is the following trace result generalizing a classical result which states that if u belong to L 2 (a, b; H 1 0 (Ω)) and ∂u ∂t belongs to After two consecutive reflection first with respect to t = b and then with respect to , setting u = η ũ, and using standard arguments (see [ [9], Lemme IV, Remarque 10, p. 158]), we have for the modular convergence.We have from which one deduces that v j is a Cauchy sequence in C(R, L 2 (Ω)), and since the limit of In order to deal with the time derivative, we introduce a time mollification of a function u ∈ L ϕ (Q).Thus we define, for all µ > 0 and all (x, t) where ũ(x, s) = u(x, s)χ (0,T ) (s) is the zero extension of u. ✷ Throughout the paper the index µ always indicates this mollification.
we deduce that u µ is measurable by Fubini's theorem.By Jensen's integral inequality [see [4]] we have, since For a.e.(x, t) ∈ Q we have On the other hand This implies that Let ε > 0. There exists k such that and there exists µ 0 such that (2) Since ∀α, |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.
, for the modular convergence).

Compactness Results
In this section, we shall prove some compactness theorems in inhomogeneous Musielak-Orlicz-Sobolev spaces which will be applied to get existence theorem for parabolic problems.
For each h > 0, define the usual translated First of all, recall the following compactness result proved by Simon [25].Lemma 4.1.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 This gives, by setting w n = vn ||∇vn|| Lϕ(Ω) : Thus )dx < ∞ for a.e t in (0, T ), and then Integrating this over (0, T ) yields and finally

✷
We also prove the following lemma which allows us to enlarge the space Y whenever necessary.
Since γ ≪ ϕ then by using Vitali's theorem, it is easy to see that F is relatively compact in E γ (Q).✷ Remark 4.3.(see [14]).
Proof: Let γ and θ be Musielak functions such that γ ≪ ϕ and θ ≪ ψ 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.By Remark 3 of [14], we deduce that Proof: .It is easily adapted from that given in [8] by using Theorem 4.4 and Remark 4.3 instead of Lemma 8 of [25].✷
where u 0 is a given function in L 2 (Ω).
We shall prove the following existence theorem.
Theorem 5.1.Assume that ( 8)-( 13) hold true.Then the problem ( 14) admits at least one weak solution and for v = u, which gives the energy equality As in the elliptic case (see, [6]), γ is introduced instead of ϕ in ( 8) is done only to guarantee the boundedness in L ψ (Q) of ψ −1 x γ(x, ϑ|u n |) and ψ −1 x γ(x, ϑ|∇u n |) whenever u n is bounded in W 1,x L ϕ (Q).In the elliptic case,one usually takes γ = ϕ in the term ψ −1 x γ(x, ϑ|u n |) since u n is bounded in a smaller space L θ (Ω) with ϕ ≪ θ; see [6].However, in the parabolic case, we cannot conclude that there is the boundedness.Nevertheless,we can take γ = ϕ if one of the following assertions holds true.
(2) A is monotone, that is A(u)−A(v), u−v ≥ 0 for all u, v ∈ D(A)∩W 1,x  0 L ϕ (Q).Indeed, suppose first that ϕ satisfies a △ 2 condition.Therefore (8) with now γ = ϕ, imply that, for all ε > 0, which allows us to deduce the boundedness in L ψ (Q) of a(x, t, u n , ∇u n ) and a(x, t, u n , ∇u n ).Assume now that A is monotone.We have, for all u n is bounded from above, thanks to the a priori estimates, , where C φ is a constant depending on φ but not n.Therefore, the Banach-Steinhauss theorem applies so that we can obtain the boundedness of Proof of Theorem 5.1.We divide the proof in four steps.
Step 1.A priori estimates.Consider the sequence of approximate problems: where g n (x, t, s, ξ) = T n (g(x, t, s, ξ)) and where for k > 0, T k means for the usual truncation operator at k defined on R by Note that g n (x, t, s, ξ)s ≥ 0, |g n (x, t, s, ξ)| ≤ |g(x, t, s, ξ)| and |g n (x, t, s, ξ)| ≤ n.Since g n is bounded for any fixed n > 0, there exists at last one solution u n of( 16),(the existence of u n can be obtained from Galerkin solutions corresponding to the Equation ( 16) as in [19], see Theorem 1 of [2] for more details).Note also that u ′ n , v is defined in the sense of distributions(where u (16) the test function u n , we get Where here and below C is a positive constant not depending on n.
By theorem 1 and theorem 5 of [3] we can say that: To prove that a(x, t, u n , In view of (9), we have Using ( 8) and ( 17), we easily see that and |u n | > 1 and using (11), we can write And then therefore Corollary 4.5 allows us to deduce that u n → u strongly in L 1 (Q).Thus, for some subsequence still denoted by u n and for some h ∈ (L ψ (Q)) N : Step 2. Almost everywhere convergence of gradients.Fix k > 0 and let φ(s) = s exp(δs 2 ), δ > 0.It is well known that when δ Let v j ∈ D(Q) be a sequence such that for the modular convergence (20) and let w i ∈ D(Ω) be a sequence which converges strongly to u 0 in L 2 (Ω).
where T k (v j ) µ is the mollification with respect to time of T k (v j ), see (6).Note that ω i µ,j is a smooth function having the following properties: Using in (16) In the sequel and throughout the paper, we will omit for simplicity the dependence on x and t in the function a(x, t, s, ξ) and denote ε(n, j, µ, i, s) all quantities (possibly different) such that and this will be the order in which the parameters we use will tend to infinity, that is, first n, then j, µ, i and finally s.Similarly,we will write only ε(n), or ε(n, j),... to mean that the limits are made only on the specified parameters.We will deal with each term of (21).First of all, observe that since for the modular convergence and so for the topology σ(ΠL ϕ , ΠLψ) as j → ∞, and finally and then, by Theorem 3.2, there exists a smooth function where Setting Φ(s) = s 0 φ(r)dr, it is easy to see that Φ(s) ≥ 0, Since, as σ → ∞, the last side goes to About I 2 (σ), we have, since (ω i µ,j ) ′ = µ(T k (v j ) − ω i µ,j ) and φ(s)s ≥ 0, Since, as σ → ∞, the last side goes to which is of form ε(n, j), we obtain For what concerns I 3 (σ), one has by integrating by parts we have and hence, by letting n → ∞ in the first integral of last side, where we have used the fact that (recall that Since the first integral of last side of ( 23) is of the form ε(j) while the second one is of the form ε(i), we deduce that Combining the estimates on each I i , we get For s > 0, set | ≤ s} and denote by χ s and χ s j the characteristic functions of Q s and Q s j , respectively.On the other hand, the second term of the left-hand side of ( 21) reads as We shall go to the limit as n, j, µ and s → ∞ in the last three integrals of the last side.
Starting with J 2 , we have by letting n → ∞ N by using (8) and Lebesgue theorem while ∇T k (u n ) ⇀ ∇T k (u) weakly in (L ϕ (Q)) N by (18).Letting j → ∞ in the first term of last side of of the above equality, one has, since a(T k (u), ∇T k (v j )χ s j ) → a(T k (u), ∇T k (u)χ s ) strongly in (E ψ (Q)) N by using ( 8), (20) and Lebesgue theorem while e in Q and is uniformly bounded by φ ′ (2k) we can let µ → ∞ in the first term of the last side to get and thus, by letting s → ∞, we conclude that J 2 = ε(n, j, µ, s).
About J 3 , we can write which gives by letting n → ∞, thanks to (18), so that, by letting j → ∞ in two first integrals last of the last side and using (20), in which we can let µ → ∞ to obtain Consequently, by letting s → ∞, For what concerns J 4 we have, as above, by letting first n then j and finally µ go to infinity : We conclude then that The third term of the left-hand side of( 21) can be estimated as Since c 2 (x, t) and d(x, t) belong to L 1 (Q) it is easy to see that On the other hand, the second term of the right-hand side of (26) reads as As above, by letting successively first n, then j, µ and finally s go to infinity, we can easily see that each one of last two integrals of the right-hand side of the last equality is of the form ε(n, j, µ) and then Combining ( 21),( 22),( 24),( 25) and ( 27), we get On the other hand, we have and,as it can be easily seen, each integral of the right-hand side is of the form ε(n, j, s), implying that For r ≤ s, we have hence, by passing to the limit sup over n, get and thus,as in the elliptic case(see [1]), there exists a subsequence also denote by u n such that ∇u n → ∇u a.e.inQ.
We deduce then that,for all k > 0 a Step 3. Modular convergence of the truncations and equi-integrability of the nonlinearities.
Thanks to (28) and (29), we can write and then in which we can pass to the limit as j, µ, i, s → ∞ to obtain Musielak-Orlicz-Sobolev Spaces

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On the other hand, Fatou's lemma implies and thus, as as n → ∞; implying by using ( 10) and Vitali's theorem that N for the modular convergence .
We shall now prove that g n (x, t, u n , ∇u n ) → g(x, t, u n , ∇u n ) strongly in L 1 (Q) by using Vitli's theorem.Since g n (x, u n , ∇u n ) → g(x, u n , ∇u n ) a.e. in Q,thanks to (17) Step 4. Passage to the limit and regularity of the solution.
There exists a prolongation v of v such that (see proof of Lemma1) and By Theorem1(see also Remark1), there exists a sequence (w j ⊂ D(Ω × R)) such that and for the modular convergence and ||w j || ∞,Ω×R ≤ (N + 2)||v|| ∞,Ω×R .Go back to approximate equations ( 16) and use w j χ (0,τ ) , for every τ ∈ [0, T ](which belongs to W We shall now go to the limit as j → ∞ in all terms of (37).In view of (33) and the fact that w j are uniformly bounded, there is problem to pass to the limit in last four terms of (37).For what concerns the first one,observe that,as in the proof of Lemma 3.4, we have w j → v in C([0, T ], L 2 (Ω)).Therefore, we can let j → ∞ in all terms of (37) to get (., t) is a Lebesgue measurable function
Qτ + Qτ a(x, t, u, ∇u)∇vdxdt+ Qτ g(x, t, u, ∇u)vdxdt = f, v Qτ ,which shows that u satisfies all properties of Theorem 5.1.It only remains to prove the energy equality.For that, we use, for a given k > 0, T k (u n ) as a test function in(16), to getu ′ n , T k (u n ) Qτ = − Qτ a(x, t, u n , ∇u n )∇T k (u n )dxdt − Qτ g n (x, t, u n , ∇u n )T k (u n )dxdt + f, T k (u n ) Qτ , which gives by setting S k (s) = s 0 T k (z)dz, Ω S k (u n (τ ))dx − Ω S k (u 0 )dx = − Qτ a(x, t, u n , ∇u n )∇T k (u n )dxdt − Qτ g n (x, t, u n , ∇u n )T k (u n )dxdt + f, T k (u n ) Qτ .(38) and (29), it suffices to prove that g n (x, t, u n , ∇u n ) are uniformly equi-integrable in Q.Let E ⊂ Q be a measurable subset of Q.We have for any m > 0 (x, t, T m (u n ), ∇T m (u n ))|dxdt (u n ), ∇T m (u n ))∇T m (u n )dxdt.By virtue of strong convergence (31) and the fact that c 2 (x, t), d(x, t) ∈ L 1 (Q), there exists ν such that|E| < ν ⇒ E∩{|un|≤m} |g n (x, t, u n , ∇u n )|dxdt ≤ ε 2 ∀n.(x, t, u n , ∇u n )|dxdt ≤ ε∀n,which shows that g n (x, t, u n , ∇u n ) are uniformly equi-integrable in Q as required.