Rothe time-discretization method for a nonlinear parabolic problem in Orlicz-Sobolev spaces

In this paper, we prove the existence and uniqueness of entropy solutions for the following equations in Orlicz spaces: ∂u∂t-div(ax,∇u(x,t))+β(u)=finQT=]0.T[×Ωu=0onΣT=]0.T[×∂Ωu(0,.)=0inΩ,(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{c} \frac{\partial u}{\partial t}-div\Big (a \left( x,\nabla u(x,t)\right) \Big )+ \beta (u)=f\text { in }Q_{T}= ] 0.T[ \times \Omega \\ u=0\text { on }\Sigma _{T}=] 0.T[ \times \partial \Omega \\ u(0,.)=0 \text { in }\Omega , \end{array} \right. \qquad (1) \end{aligned}$$\end{document}where f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f$$\end{document} is an element of L1(QT)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}( Q_{T} )$$\end{document}, the term -div(a(x,∇u(x,t)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\text{ div }\Big (a(x,\nabla u(x,t))\Big )$$\end{document} is a Leray-Lions operator on W01,xLM(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_0^{1,x}L_M(\Omega )$$\end{document}, with M(.) does not satisfy the Δ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _2$$\end{document} condition and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} is a continuous non decreasing real function defined on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} with β(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (0)=0$$\end{document}. The investigation is made by approximation of the Rothe method which is based on a semi-discretization of the given problem with respect to the time variable.


3 Introduction
The study of variational problems where the operator associated to a PDEs satisfies the non-polynomial growth conditions instead of having the usual p-structure arouses much interest with the development of applications to electro-rheological fluids as an important class of non-Newtonian fluids (sometimes referred to as smart fluids). The electro-rheological fluids are characterized by their ability to drastically change the mechanical properties under the influence of an external electromagnetic field. A mathematical model of electro-rheological fluids was proposed by Rajagopal and Ruzicka (we refer to [31,33] for more details). Another important application is related to image processing [29] where this kind of the diffusion operator is used to underline the borders of the distorted image and to eliminate the noise.
Let Ω be a bounded open subset of ℝ d , d ≥ 2 , with segment property and A(u) = −div a(x, ∇u(x, t)) is a Leray-Lions operator defined on W 1,x 0 L M (Q T ) , where M is an N-function without assuming a Δ 2 -condition on M.
We consider the following nonlinear parabolic problem: where the data f ∈ L 1 (Q T ) and is taken as continuous non decreasing real function everywhere defined on ℝ with (0) = 0. Under these assumptions, the above problem does not admit, in general, a weak solution since the fields a(x, ∇u(x, t)) do not belong to (L 1 loc (Q T )) d . To overcome this difficulty we use the framework of entropy solutions was introduced by Benilan and al. [4] for the nonlinear elliptic problems. The concept of entropy solution in the case parabolic problem was obtained in [28], At the same the equivalent notion of renormalized solution has been introduced by Lions and Di Perna [6] for the study of Boltzmann equation, this notion was then adapted to elliptic vesion by Boccardo, J.-L. Diaz, D. Giachetti, F. Murat [5].
Note that in the case of classical Sobolev spaces W 1,p J. Droniou and A. Prignet in [7] demonstrated the equivalence between entropy and renormalized solutions for parabolic equations (see also [21]), however in our opinion in the case of Orlicz spaces that this result is not known.
Recently, Gwiazda and al. studied the problem 1.1 in the Musielak Orlicz space in [18] but they assumed more restrained conditions on the N-function M and M * namely M(x, ) ≥ | | 1+ and M * ∈ Δ 2 . In [11] and [17] the authors have studied a problem of the type 1.1 by using the Galerkin classical method to demonstrate the existence of the solution, contrary to our strategy which consists of using the Rothe method to approche the problem 1.1 with a sequences of elliptic problems.
Precisely in this work we use the Rothe's method as a mean ingredient to prove our principal result. Rothe's method introduced by E. Rothe in 1930 and it has been used and developed by many authors, e.g P.P. Mosolov, K. Rektorys in linear and quasilinear

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Rothe time-discretization method for a nonlinear parabolic… parabolic problems. Nonlinear and abstract parabolic problems have been studied by Kaũr et al., using similar technique to that of Rothe's method. For the more complete references we refer the reader to [2,8,9,[16][17][18]20] and [32].
Our aim is to prove the existence and uniqueness of an entropy solution for the problem 1.1 in Orlicz spaces using Rothe time-discretization method, which is based on a semidiscretization of the given problem with respect to the time variable . We will approximate the parabolic problem 1.1 by a sequence of the following elliptic problems: We will get our results by a semi discretization process, then we prove uniqueness and stability results for the semi-discretized problem. Within the proof we use truncation methods, monotonicity arguments techniques, the integration by parts formula, Young inequality which have been adapted to non-reflexive Orlicz spaces.
This paper is divided into four sections. In the next one, we recall some well-known preliminaries, properties, and results of Orlicz-Sobolev spaces. Section (3) contains the basic assumptions and the main result. Finally, in Sect. (4) we prove the existence and uniqueness of entropy solution to the nonlinear parabolic equation 1.

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: The N-function M is said to satisfy the Δ 2 condition if, for some k > 0, When this inequality holds only for t ≥ t 0 > 0 , M is said to satisfy the Δ 2 condition near infinity.

Young's inequality
Let Ω be an open subset of ℝ d . The Orlicz class  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: The space L M (Ω) is reflexive if and only if M and M * satisfy the Δ 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 d + 1 copies of L M (Ω) . Denoting this product by ΠL M , we will use the weak topologies (ΠL M , ΠE M * ) and (ΠL M , ΠL M * ) . The space W 1 0 E M (Ω) is defined as the (norm) closure of the Schwartz space (Ω) in W 1 E M (Ω) and the space W 1 0 L M (Ω) as the (ΠL M , ΠE M * ) closure of (Ω) in W 1 L M (Ω) . We say that u n converges to u for the This implies convergence for (ΠL M , ΠL M * ) . If M satisfies the Δ 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 (Ω) is dense in W 1 0 L M (Ω) for the modular convergence and for the topology (ΠL M , ΠL M * ) (cf. [13]). 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 [1,22].
For k > 0 , we define the truncation at height k, T k ∶ ℝ → ℝ by and its primitive by ‖D u‖ M,Ω .
Rothe time-discretization method for a nonlinear parabolic… it is pretty obvious that J k (s) ≥ 0 and J k (s) ≤ k|s|.
We have as in the paper [3], 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 T ) which have as many copies as there is -order derivatives, | | ≤ 1 . We shall also consider the weak topologies (ΠL M , ΠE M * ) and (ΠL M , ΠL M * ) . If then the concerned function is a W 1 E M (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 t ↦ ‖u(t)‖ M,Ω is in L 1 (0, T) . The space . We can easily

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show as in [14] that when Ω has the segment property, then each element u of the closure of (Q T ) with respect of the weak * topology (ΠL M , ΠE M * ) is a limit, in W 1,x L M (Q T ) , of some subsequence (u i ) ⊂ (Q T ) for the modular convergence; i.e., there exists > 0 such that for all | | ≤ 1, Thus both sides of the last inequality are equivalent norms on W 1,x 0 L M (Q T ) . We have then the following complementary system F being the dual space of W 1,x 0 E M (Q T ) . It is also, except for an isomorphism, the quotient of ΠL M * by the polar set W 1,x 0 E M (Q T ) ⊥ , and will be denoted by F = W −1,x L M * (Q T ) and it is shown that, 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, and is denoted by .
Rothe time-discretization method for a nonlinear parabolic… 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 In the sequel we have to use the following results which concern mollification with respect to time and space variable and some trace results.

Lemma 2.6 [11]
Let Ω be a bounded open subset of ℝ d with the segment property. Then, 1. The statement of lemma 2.5 generalizes that of Simon in Orlicz-Sobolev Spaces. 2. While lemma 2.6 generalizes the trace theorem in this general setting. 3. Let us mention that the following trace result, hold true: for the modular convergence,(see [11]), such trace result generalizes the following classical result, Let Ω be a bounded open subset of ℝ d and let M be an N-function. Consider a second-order partial differential operator where a ∶ Ω × ℝ d → ℝ d is a Caratheodory function satisfying for almost every x ∈ Ω, and for every v, v * ∈ D(A) , such that ∇v ≠ ∇v * , we have .
We have the following lemma: Lemma 2.8 [26] Under assumptions (10)(11)(12), and let (z n ) be a sequence in as n and s tend to +∞, and where s is the characteristic function of Then, Consider the following non-linear elliptic problem: where Proposition 2. 9 Assume that (10)(11)(12) and (20)(21) hold true, then the following problem (17) ∇z n →∇z a.e. in Ω.
g is a continuous and non-decreasing function on ℝ such that g(0) = 0.

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Rothe time-discretization method for a nonlinear parabolic… has at least one solution u.
Since g n is bounded for any fixed n > 0 , there exists at least one solution u n of the problem (2.21) (see Proposition 13 in [15]).
Using the same technics and the same steps as in the paper [10], we deduce that the problem (2.17) has at least one solution. □

Assumptions
Let Ω be a bounded open set of ℝ d (d ≥ 2) with the segment property, T > 0 is given real number and Q T =]0.T[×Ω.
Consider a second order partial differential operator where M is an N-function without assuming a Δ 2 -condition on M.
We consider the following nonlinear parabolic problem: where a ∶ Ω × ℝ d → ℝ d Caratheodory function satisfying for almost every (x, t) ∈ Q T , and for every v, v * ∈ D(A) , such that ∇v( Remark 3. 1 We can take u(0, .) = u 0 such that u 0 ∈ L 1 (Ω) and we use regularization as in [25] Throughout this paper ⟨. , .⟩ means for either the pairing between And also for simplicity, we write a(x, ∇u) instead of a(x, ∇u(x, t)). (24)(25)(26), and let (z n ) be a sequence in W 1,x 0 L M (Q T ) such that, as n and s tend to +∞, and where s is the characteristic function of Then, Proof Using the same argument in [3] we get the result. □

Lemma 3.2 Under assumptions
is non decreasing continuous functions on ℝ such that (0) = 0.

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Rothe time-discretization method for a nonlinear parabolic…

The semi-discrete problem
By using the semi-discretization in time by the implicit Euler method for the parabolic problem (1) we get, We will sometimes use the shorter notation ‖ .

Definition 3.3
An entropy solution to the discretized problems (35) is a sequence of measurable functions (u n ) 0≤n≤N such that u 0 = 0 and u n is defined by induction as an entropy solution of the problem

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By assumption (28)  Since u 1 T k (u 1 ) k ≥ 0, and by using the Fatou's lemma with k → 0 , we deduce that with C not depending on k.

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In the same way and using hypothesis (28) Rothe time-discretization method for a nonlinear parabolic… For simplicity, we write u = u 1 and v = v 1 .
Let h > k , for the solution u we take = T h (v) as test function, and for the solution v we take = T h (u) as test function.
By summing up the two inequalities, and letting h go to infinity, we find by applying Lebesque's theorem that where and By applying hypothesis (28)

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Rothe time-discretization method for a nonlinear parabolic… we deduce that where According to ( 48) , we have and since f n ∈ L 1 (Ω) , u n ∈ L 1 (Ω) for all n ∈ ℕ , u 0 = v 0 = 0 and the fact that lim Rothe time-discretization method for a nonlinear parabolic…

Stability results
Proposition 3.7 Let (u n ) 0≤n≤N , N ∈ ℕ be an entropy solution of the approximate problem (35), then there exists a positive constants C 1 (f ), C 2 (f ) and C(f, k) independents on N, such that for all n = 1, . .

.N, we have
Proof : Proof of (i) and (ii): We take = 0 as test function in (37), we obtain Since then thus implies that

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Let k → 0, we get by the properties of T k and the Fatou lemma, that Summing (86) from i = 1 to i = n , we obtain Proof of (iii):

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Rothe time-discretization method for a nonlinear parabolic… where Then, by using (26), we deduce from the inequalities (88-90) that From (87), (88) and taking k = 1 , we get Then by applying the Lebesgue's dominated convergence theorem as h → 0 , we obtain Summing (94) from i = 1 to i = n , we have Proof of (iv): We take = 0 as test function in (37), we obtain by using the assumption 3.3, we get (92)

Existence result
In this section we give the notion of entropy solution for the nonlinear parabolic problem (1.1) and we state the main result of this paper. The main result is the following: Theorem 3. 9 Under assumption (24)-(28), the nonlinear parabolic problem (1) has a unique entropy solution.

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Rothe time-discretization method for a nonlinear parabolic… and a piecewise constant function where t n ∶= n and u n is unique solution of problem (35). We have for any t ∈]t n−1 , t n ], n = 1, . .., N that Remark 4. 1 Using the Rothe's approximations, we can write the problem approximated of (1) as follows: By using the stability results, we will prove the following a priori estimates. (24)-(28), there exists a constant C i (T, f ), (i = 1, ..., 6) not depending on N such that for all N ∈ ℕ, we have:

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Rothe time-discretization method for a nonlinear parabolic… Proof of (iii): From (98) and (79), we have then Proof of (iv): For t ∈]t n−1 , t n ] , using (99) and (81) we get Proof of (v): We have then Proof of (vi): Using the same technical as in the above, it is easy to show that U N (t) ∶= u n for all t ∈]t n−1 , t n ], Step 2: Convergence of the sequence (∇T k (U N )) N∈ℕ in L M (Q T ). According to (102), (104) and using Aubin-Lions compactness lemma (see [19]), we deduce that the sequence (U N ) N∈ℕ is relatively compact in L 1 (Q T ) , then there exists u in

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Rothe time-discretization method for a nonlinear parabolic… • In this step we prove the almost everywhere convergence of the gradients and the modular convergence of the truncations.
Proof By using the same technics as in the proof of proposition (5.5) in paper [25], we can easily prove the desired result.

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Rothe time-discretization method for a nonlinear parabolic… Summing up the two inequalities (122, 123) and letting h → ∞ , we obtain where i.e Since we have

then the inequality (126) becomes
On the one hand, since T k (U On the other hand, we have lim Using the same technique used in proof of the uniqueness theorem (3.6), we prove that lim Since U N → u a.e , then U N → u in C([0, T], L 1 (Ω)). • Step 5: Passage to the limit. It remains to show that the function u is the entropy solution of problem ( 1).
Rothe time-discretization method for a nonlinear parabolic… then u is an entropy solution of the nonlinear parabolic problem (1), and this completes the proof of existence of the entropy solution.
• Let v be another entropy solution of nonlinear parabolic problem (1) . We take = T h (U N ) as test function. Passing to the limit with h → ∞ , we obtain then where On the other hand, taking = T h (v) as test function in (121), and by letting h to ∞ , we obtain and (145) where Summing up the two inequalities ( 145, 147), we obtain where II N (k, h) = II N 1 (k, h) + II N 2 (k, h) . By using the results of convergence and the hypothesis (28), we get and Combining (148, 150), we deduce that By applying the same technique used in the proof of the uniqueness theorem (3.6), we prove that then since lim k→0 J k (s) k = |s| , then lim k→0 J k (v(t)−u(t)) k = |v(t) − u(t)|. Therefore, according to Fatou's lemma, we obtain Hence the uniqueness of the entropy solution of the problem (1), and the proof of theorem (3.9) is complete.