Doubly Nonlinear Parabolic Systems In Inhomogeneous Musielak- Orlicz-Sobolev Spcaes

Volume 2, Issue 5, Page No 180-192, 2017

Author’s Name: Ahmed Aberqi^1,a), Mhamed Elmassoudi², Jaouad Bennouna², Mohamed Hammoumi²

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¹University of Fez, National school of applied sciences.Department of Electric and computer engineering, Fez, Morocco
²University of Fez, Faculty of Sciences Dhar El Mahraz. Laboratory LAMA, Department of Mathematics, B.P 1796 Atlas Fez, Morocco

^a)Author to whom correspondence should be addressed. E-mail: aberqi_ahmed@yahoo.fr

Adv. Sci. Technol. Eng. Syst. J. 2(5), 180-192 (2017); DOI: 10.25046/aj020526

Keywords: Parabolic system, Museilack-Orlicz spaces, Renormalized solutions

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Abstract

In this paper, we discuss the solvability of the nonlinear parabolic systems associated to the nonlinear parabolic equation: $\frac{\partial b_{i}(x,u_{i})}{\partial t} -div(a(x,t,u_{i},\nabla u_{i}))- \phi_{i}(x,t,u_{i})) +f_{i}(x,u_{1},u_{2})=0$ where the function $b_{i}(x,u_{i})$ verifies some regularity conditions, the term $\Big(a(x,t,u_{i},\nabla u_{i})\Big)$ is a generalized Leray-Lions operator and $\phi_{i}$ is a Caratheodory function assumed to be Continuous on $u_i$ and satisfy only a growth condition. The source term $f_{i}(t,u_{1},u_{2})$ belongs to $L^{1}(\Omega\times(0,T))$ .

Received: 01 June 2017, Accepted: 12 July 2017, Published Online: 29 December 2017

Full Text

Introduction

Given a bounded-connected open set Ωof R^N(N = 2), with Lipschitz boundary ∂Ω, Q_T= Ω × (0,T ) is the generic cylinder of an arbitrary finite hight, T < +∞. We prove the existence of a renormalized solutions for the nonlinear parabolic systems:

∂bi∂t(x,ui) − div(a(x,t,ui, ∇ui)) − div(Φi(x,t,ui))

+f_i(x,u₁,u₂) = 0 in Q_T, (1.1) u_i= 0 on (0,T ) × ∂Ω,

b_i(x,u_i)(t = 0) = b_i(x,u_i,₀) in Ω,

where i=1,2. Here the vector field a : Ω × _R× R^N→ _R^Nis a Caratheodory function such that´

− div(a(x,t,u_i, ∇u_i)) is a Leray-lions operator defined from the Inhomogeneous Musielak-Orlicz-Sobolev

Spcaes W₀^1,xL_ϕ(Q_T)into its dual W ^−1,xL_ψ(Q_T). Let b_i: Ω×_R→ _Ra Caratheodory function such that for every´ x ∈ Ω, b_i(x,.) is a strictly increasing C¹-function, the divegential term Φ_i(x,t,u_i) is a Caratheodory function´ satisfy only a polynomial growth with respect to the anisotropic N-function ϕ (see (4.6)), the data u_0,iis in L¹(Ω) such that b_i(.,u_0,i) in L¹(Ω) and the source f_iis a

Caratheodory function satisfy the assumptions ((4.7)-´ (4.10)). When problem ((1.1)) is investigated, there is a difficulty due to the fact that the data b₁(x,u₀¹(x)) and b )) only belong to L¹and the functions a(x,t,u_i, ∇u_i),Φ_i(x,t,u_i) and f_i(x,u₁,u₂) do not belong to (L^Nin general, so that proving existence of weak solution seems to be an arduous task, and we cannot use the Stocks formula in the a priori estimates of the nonlinearity ,Φ_i(x,t,u_i). In order to overcome this difficulty, we work with the framework of renormalized solutions (see Definition 3.1). One of the models of applications of these operators is the system of Boussinesq:

) in Q_T

in Q_T

u(t = 0) = u₀, b(θ)(t = 0) = b(θ₀) on Ω

u = 0 θ = 0 on ∂Ω × (0,T )

Equation first equation is the motion conservation equation, the unknowns are the fields of displacement u : Q_T→ _R^Nand temperature θ : Q_T→ _R, The field ε(u) = ¹₂(∇u +(∇u)^t) is the strain rate tensor.

It is our purpose, in this paper to generalize the result of ([1], [2], [3]) and we prove the existence of a renormalized solution of system (1.1).

The plan of the paper is as follows: In Section 2 we give the framework space, in Section 3 and 4 we give some useful Lemmas and basics assumptions. In Section 5 we give the definition of a renormalized solution of (1.1), and we establish (Theorem 5.1) the existence of such a solution.

2 Preliminaries

2.1 Musielak-Orlicz function

Let Ω be an open subset of R^N(N ≥ 2), and let ϕ be a real-valued function defined in Ω × _R+and satisfying conditions:

Φ₁:ϕ(x,.) is an N-function for all x ∈ Ω (i.e. convex, non-decreasing, continuous, ϕ(x,0) = 0

,ϕ(x,0) > 0 for t > = 0 and

limt→∞ infx∈Ω ϕ(x,tt ) = ∞). Φ₂:ϕ(.,t) is a measurable function for all t ≥ 0. A function ϕ which satisfies the conditions Φ₁and Φ₂is 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_x⁻¹, with respect to t, that is ϕ_x⁻¹(ϕ(x,t)) = ϕ(x,ϕ_x⁻¹(t)) = t

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 t₀such that for a.e. x ∈ Ω γ(x,t) ≤ ϕ(x,ct) for all t ≥ t₀(resp. for all t ≥ 0 ). We say that γ grows essentially less rapidly than ϕ at 0(resp. near infinity) and we write γ ≺≺ ϕ, for every positive constant c, we have

(resp.lim_t

Remark 2.1 [4]. If γ ≺≺ ϕ near infinity,then ∀ > 0 there exist k such that for almost all x ∈ Ω, we have

(x,t) ^∀t ^≥0

2.2 Musielak-Orlicz space

For a Musielak-Orlicz function ϕ and a measurable function u : Ω → _R, we define the functionnal

%_ϕ,dx.

The set K_ϕ(Ω) = {u : Ω → _Rmesurable :

%_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

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 u ⁻u

This implies convergence for σ(ΠL_ϕ,ΠL_ψ)[see [5]].

In the space L_ϕ(Ω), we define the following two norms

ku k^o

which is called the Luxemburg norm, and the socalled Orlicz norm by

k|u |kϕ,Ω = supkvkψ≤1RΩ |u(x)v(x)|dx

where ψ is the Musielak-Orlicz function complementary to ϕ. These two norms are equivalent [8]. 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

W¹L_ϕ(Ω) = {u ∈ L_ϕ(Ω) : D^αu ∈ L_ϕ(Ω), ∀α ≤ 1}

W¹E_ϕ(Ω) = {u ∈ E_ϕ(Ω) : D^αu ∈ E_ϕ(Ω), ∀α ≤ 1}

where α = (α₁,…,α_N),|α | = |α₁| + … + |α_N| and : D^αu denote the distributional derivatives. The space W¹L_ϕ(Ω) is called the Musielak-Orlicz-Sobolev space. Let

%ϕ,Ω(u) = P|α|≤1%ϕ,Ω(Dαu) and ku k1ϕ,Ω = inf{λ > 0 :

These functionals are convex modular and a norm on W¹L_ϕ(Ω), respectively. Then pair

(W¹L_ϕ(Ω), ku k¹_ϕ,_Ω) is a Banach space if ϕ satisfies the following condition [6].

There exists a constant c > 0 such that inf ϕ(x,1) > c

x∈Ω The space W¹L_ϕ(Ω) is identified to a subspace of the product ^Q_α≤1L_ϕ(Ω) = ^QL_ϕWe denote by D (Ω) the Schwartz space of infinitely smooth functions

with compact support in Ω and by D (Ω) the restriction of D (R) on Ω. The space W₀¹L_ϕ(Ω) is defined as the σ(ΠL_ϕ,ΠE_ψ) closure of D (Ω) in W¹L_ϕ(Ω) and the space W₀¹E_ϕ(Ω) as the(norm) closure of the Schwartz space D (Ω) in W¹L_ϕ(Ω). For two complementary Musielak-Orlicz functions ϕ and ψ, we have (See [7]).

The Young inequality:

L_ϕ(Ω) = {u : Ω → R mesurable : %_ϕ,_Ω( ) < ∞, for some λ > 0} λ

• The Holder inequality¨ For any Musielak-Orlicz function ϕ, we put

ψ(x,s) = sup_t≥₀(st ⁻ϕ(x,s)). _R

ψ is called the Musielak-Orlicz function comple- _Ωu u k_ϕ,_Ωk|v |k_ψ,_Ωfor all mentary to ϕ (or conjugate of ϕ) in the sense of u ∈ L_ϕ(Ω),v ∈ L_ψ(Ω)

st ≤ ϕ(x,s)+ψ(x,t) for all s,t ≥ 0 , x ∈ Ω.

We say that a sequence of functions u_nconverges to u for the modular convergence in W¹L_ϕ(Ω) (respectively in W₀¹L_ϕ(Ω)) if, for some λ > 0.

uⁿ

lim %ϕ,Ω λ

n→∞

The following spaces of distributions will also be used

W −1Lψ(Ω) = nf ∈ D 0 (Ω) : f = X(−1)αDαfα

α≤1

where f_α∈ L_ψ(Ω)

and

W −1Eψ(Ω) = nf ∈ D 0 (Ω) : f = X(−1)αDαfα

α≤1

where f_α∈ E_ψ(Ω)

2.3 Inhomogeneous Musielak-OrliczSobolev spcaes:

Let Ω be a bounded Lipschitz domain in R^Nand let

Q = Ω×]0,T [ with some given T > 0. let ϕ be a

Musielak-Orlicz function.For each α ∈ N^N, denote by D_x^αthe distributional derivative on Q_Tof order α with respect to the variable x ∈ R^N. The inhomogeneous Musielak-Orlicz-Sovolev spaces of order 1 are defined as follows

W1,xLϕ(Q) = nu ∈ Lϕ(Q) : ∀|α | ≤ 1, Dxαu ∈ Lϕ(Q)o

W1,xEϕ(Q) = nu ∈ Eϕ(Q) : ∀|α | ≤ 1, Dxαu ∈ Eϕ(Q)o

The last is a subspace of the first one, and both are Banach spaces under the norm

ku k = kD_xαu k_ϕ,Q

|α|≤m

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) 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,xL_ϕ(Q) then the function : t 7→ u(t) = u(t,.) is defined on (0,T ) with values in W¹L_ϕ(Ω). If, further, u ∈ W^1,xE_ϕ(Q) then this function is a W¹E_ϕ(Ω)-valued and is strongly measurable. Furthermore the following imbedding holds: W^1,xE_ϕ(Q) ⊂ L¹(0,T ;W¹E_ϕ(Ω)). The space W^1,xL_ϕ(Q) is not in general separable, if u ∈ W^1,xL_ϕ(Q), we can not conclude that the function u(t) is measurable on (0,T ). However, the scalar function t 7→ u(t) = ku(t)k_ϕ,_Ωis in L¹(0,T ). The space W₀^1,xE_ϕ(Q) is defined as the (norm) closure in W^1,xE_ϕ(Q) of D (Ω). We can easily show that when Ω is a Lipschitz domain then each element u of the closure of D (Ω) with respect of the weak^∗topology σ(ΠL_ϕ,ΠE_ψ) is limit, in W^1,xL_ϕ(Q), of some subsequence (u_i) ∈ D (Ω) for the modular convergence, i.e. there exists λ > 0 such that for all |α | ≤ 1,

Z ϕ(x,(Dxαui − Dxαu)dxdt → 0 as i → ∞

_Qλ

, this implies that (u_i) converge to u in W^1,xL_ϕ(Q) for the weak topology σ(ΠL_ϕ,ΠL_ψ). Consequently

D (Q)σ(ΠL_ϕ,ΠE_ψ) = D (Q)σ(ΠL_ϕ,ΠL_ψ)

, this space will be denoted by W). Furthermore WE_ϕ. We have the following complementary system F being the dual space of W₀^1,xE_ϕ(Q). It is also, except for an isomorphism, the quotient of ΠL_ψby the polar set W, and will be denoted by F = W^1,xL_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 F₀is then given by F₀= W ^−1,xE_ψ(Q).

Lemma 2.1 [4]. Let Ω be a bounded Lipschitz domain in R^Nand let ϕ and ψ be two complementary MusielakOrlicz functions which satisfy the following conditions:

There exists a constant c > 0 such that

inf ϕ(x,1) > c (2.1) x∈Ω

∃A > 0 such that for all x,y ∈ Ω with |x , we have

for all t ≥ 1. (2.2)

ϕ(y,1)dx < ∞ (2.3)

Ω

∃C > 0 such that ψ(y,t) ≤ C a.e. in Ω

(2.4)

Under this assumptions D (Ω) is dense in L_ϕ(Ω) with respect to the modular topology, D (Ω) is dense in

W₀¹L_ϕ(Ω) for the modular convergence and D (Ω) is dense in W₀¹L_ϕ(Ω) for the modular convergence.

Consequently, the action of a distribution S in W ⁻¹Lψ(Ω) on an element u of W₀¹L_ϕ(Ω) is well defined. It will be denoted by < S,u >.

2.4 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 2.2 [4]. Let F : R → _Rbe uniformly Lipschitzian,with F(0) = 0.Let ϕ be an Musielak-Orlicz function and let u(resp.u ∈ W¹E_ϕ(Ω)). Then F(u) ∈ W¹L_ϕ(Ω)(resp.u ).Moreover,if the set of discontinuity points D of F⁰is finite,then

∂ F(u) = ⁽F⁰(x)_∂x^∂u_ia.e. in ^{x ^∈Ω; u(x) < D ^}

∂x_i0 a.e. in {x ∈ Ω; u(x) ∈ D }

Lemma 2.3 Suppose that Ω satisfies the segment property and let u. Then, there exists a sequence u_n^{∈ D}(Ω) such that

u_n→ u for modular convergence in W.

Furthermore, if u then ku_nk_∞≤

(N +1)ku k_∞.

Let Ω be an open subset of R^Nand let ϕ be a

Musielak-Orlicz function satisfying :

dt = ∞ a.e. x ∈ Ω (2.5)

and the conditions of Lemma 2.1. We may assume

without loss of generality that

Z 1 ϕ^x−1(t)dt < ^∞a.e. x ^∈Ω (2.6)

_N₊₁

0 t N

Define a function ϕ^∗: Ω × [0, ∞) by ϕ^∗(x,s) =

R s

dt x ∈ Ω and s ∈ [0, ∞). ₀t _N

ϕ^∗its called the Sobolev conjugate function of ϕ (see

[1] for the case of Orlicz function).

Theorem 2.1 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 2.1. Then

W01Lϕ(Ω) ,→ Lϕ∗ (Ω)

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

W₀¹L_ϕ(Ω) ,→ L_φ(Ω) is compact

Corollary 2.1 Under the same assumptions of theorem

5.1, we have

W01Lϕ(Ω) ,→,→ Lϕ(Ω)

Lemma 2.4 If a sequence un u_n∈ L_ϕ(Ω) converges a.e. to u and if u_nremains bounded in L_ϕ(Ω), then u ∈ L_ϕ(Ω) and u_n* u for σ(L_ϕ(Ω),E_ψ(Ω)).

Lemma 2.5 Let u_n,u ∈ L_ϕ(Ω). If u_n→ u with respect to the modular convergence, then u_n* u for σ(L_ϕ(Ω),L_ψ(Ω)).

See ([8]).

3 Technical lemma

Lemma 3.1 Under the assumptions of lemma 2.1, and by assuming that ϕ(x,t) decreases with respect to one of coordinate of x, there exists a constant c₁> 0 which depends only on Ω such that

Z Z

ϕ(x, |u |)dx (3.1)

Ω

Theorem 3.1 Let Ω be a bounded Lipschitz domain and let ϕ be a Musielak-Orlicz function satisfying the same conditions of Theorem 5.1. Then there exists a constant λ > 0 such that

ku k

4 Essential assumptions

Let Ω be an open subset of R^N(N ≥ 2) satisfying the segment property,and let ϕ and γ be two MusielakOrlicz functions such that ϕ and its complementary ψ satisfies conditions of Lemma 2.1 and γ ≺≺ ϕ.

b_i: Ω × R → R

is a Caratheodory function such that for every´ x ∈ Ω,

(4.1)

b_i(x,.) is a strictly increasing C ¹(R)-function and b_i∈ L^∞(Ω × _R) with b_i(x,0) = 0. Next for any k > 0, there exists a constant λⁱ_k> 0 and functions Aⁱ_k∈ L^∞(Ω) and Bⁱ_k∈ L_ϕ(Ω) such that:

) and x ∂bi∂s(x,s) ≤ Bik(x)

a.e. x ∈ Ω and ∀ |s | ≤ k. (4.2)

A : → W ⁻¹L_ψ(Q_T) defined by

A(u) = ⁻diva(x,t,u, ^∇u),where a : Q ^×_R^×_R^N^→_R^Nis Caratheodory function such that for a.e.´ x ∈ Ω and for

all s ∈ R,ξ,ξ∗ ∈ RN ,ξ , ξ∗

(A₁) : |a(x,t,s,ξ)

|ξ |))),

β > 0, c(x) ∈ E_ψ(Ω),	(4.3)
(A₂) : (a(x,t,s,ξ) − a(x,s,ξ^∗)(ξ − ξ^∗) > 0,	(4.4)
(A₃) : a(x,t,s,ξ).ξ ≥ αϕ(x, \|ξ \|).	(4.5)

Φ(x,s,ξ) : Ω × IR × IR^N→ IR^Nis a Caratheodory func-´ tion such that

|Φ_i(x,t,s), (4.6)

f_i: Ω × _R× _R→ _Ris a Caratheodory function with´ f₁(x,0,s) = f₂(x,s,0) = 0 a.e. x ∈ Ω, ∀s ∈ _R. (4.7) and for almost every x ∈ Ω, for every s₁,s₂∈ _R, sign(s_i)f_i(x,s₁,s₂) ≥ 0. (4.8)

The growth assumptions on f_iare as follows: For each K > 0, there exists σ_K> 0 and a function F_Kin L¹(Ω) such that

^|f₁(x,s₁,s₂)^{| ≤}F_K(x)+σ_K^|b₂(x,s₂)^|(4.9) a.e. in Ω, for all s₁such that |s₁| ≤ K, for all s₂∈ _R. For each K > 0, there exists λ_K> 0 and a function G_Kin L¹(Ω) such that

^|f₂(x,s₁,s₂)^{| ≤}G_K(x)+λ_K^|b₁(x,s₁)^|, (4.10)

for almost every x ∈ Ω, for every s₂such that |s₂| ≤ K, and for every s₁∈ _R.

Finally, we assume the following condition on the initial data u_i,₀: for i=1,2.

u_i,₀is a measurable function such that b_i(.,u_i,₀) ∈ L¹(Ω), (4.11)

In this paper, for K > 0, we denote by T_K: r 7→ min(K,max(r, −K)) the truncation function at height K. 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 and for any real number s,χ_{v<s} (respectively, χ_{v=s},χ_{v>s}) denote the characteristic function of the set {(x,t) ∈ Q_T; v(x,t) < s } (respectively, {(x,t) ∈ Q_T;v(x,t) = s }, {(x,t) ∈ Q_T;v(x,t) > s }).

Definition 4.1 A couple of functions (u₁,u₂) defined on Q is called a renormalized solution of (4.1)–(4.11)if for i = 1,2 the function u_isatisfies

T_K and b_i(x,u_i) ^∈L^∞(0,T ;L¹(Ω)),

(4.12)

Z a(x,t,u_i, ^∇u_i)^∇u_idxdt ^→0 as m ^→+^∞,

{ m≤|u_i|≤m+1}

(4.13) For every function S in W^2,∞(R) which is piecewise C¹and such that S⁰has a compact support,we have

(x,t,u_i, ∇u_i))

+S⁰⁰(u_i)a(x,t,u_i, ^∇u_i)^∇u_i

+div(S⁰(u_i)φ_i(x,t,u_i)) − S⁰⁰(u_i)φ_i(x,t,u_i)∇u_i

+f_i(x,u₁,u₂)S⁰(u_i) = 0, (4.14)

B_i,S(x,u_i)(t = 0) = B_i,S(x,u_i,₀) in Ω, (4.15)

where B_i,S(r) = ^R₀^rb_i⁰(x,s)S⁰(s)ds.

Due to (4.12), each term in (4.14) has a meaning in

W −1,xLψ(QT )+L1(QT ).

Indeed, if K such that suppS ⊂ [−K,K], the following identifications are made in (4.14)

B_i,S(x,u_i) ∈ L^∞(Q_T), since |B_i,S(x,u_i)| ≤

K kAiK kL∞(Ω) kS0 kL∞(R)

S⁰(u_i)a(x,t,u_i, ∇u_i) can be identified with S⁰(u_i)a(x,t,T_K(u_i), ∇T_K(u_i)) a.e. in Q_T. Since indeed |T_K(u_i)| ≤ K e. in Q_T, . As a consequence of (4.3) , (4.12) and S⁰(u_i) ∈ L^∞(Q_T) , it follows that

S⁰(u_i)a(x,T_K(u_i), ^∇T_K(u_i)) ^∈(L_ψ(Q_T))^N.

S⁰(u_i)a(x,t,u_i, ∇u_i)∇u_ican be identified with

S⁰(u_i)a(x,t,T_K(u_i), ∇T_K(u_i))∇T_K(u_i) a.e. in Q_T.with (4.2) and (4.12) it has

S⁰(u_i)a(x,t,T_K(u_i), ^∇T_K(u_i))^∇T_K(u_i) ^∈L¹(Q_T)

S⁰(u_i)Φi(u_i) and S⁰⁰(u_i)Φi(u_i)∇u_irespectively identify with S⁰(u_i)Φ_i(T_K(u_i)) and S⁰⁰(u_i)Φ(T_K(u_i))∇T_K(u_i). In view of the properties of S and (4.6), the functions S⁰,S⁰⁰and Φ ◦ T_Kare bounded on R so that (4.12) implies that S⁰(u_i)Φi(T_K(u_i)) ∈ (L^∞(Q_T))^Nand S⁰⁰(u_i)Φ_i(T_K(u_i))∇T_K(u_i) ∈ (L_ψ(Q_T))^N.
S⁰(u_i)f_i(x,u₁,u₂) identifies with S⁰(u_i)f₁(x,T_K(u₁),u₂)

a.e. in Q_T

(or S⁰(u_i)f₂(x,u₁,T_K(u₂)) a.e. in Q_T). Indeed, since |T_K(u_i)| ≤ K a.e. in Q_T, assumptions (4.9) and (4.10) and using (4.12) and of S⁰(u_i) ∈ L^∞(Q), one has

S⁰(u₁)f₁(x,T_K(u₁),u₂) ^∈L¹(Q_T) and S⁰(u₂)f₂(x,u₁,T_K(u₂)) ^∈L¹(Q_T).

As consequence, (4.14) takes place in D⁰(Q_T) and that

Due to the properties of S and (4.2)

B_i,S. (4.17)

Moreover (4.16) and (4.17) implies that B_i,S(x,u_i) ∈ C⁰([0,T ],L¹(Ω)) so that the initial condition (4.15) makes sense.

5 Existence result

We shall prove the following existence theorem

Theorem 5.1 Assume that (4.1)–(4.11) hold true. There at least a renormalized solution (u₁,u₂) of Problem (1.1).

We divide the prof in 5 steps.

Step 1: Approximate problem.

Let us introduce the following regularization of the data: for n > 0 and i = 1,2

1 b_i,n(x,s) = b_i(x,T_n(s))+ s ∀s ∈ R, (5.1) n

a_n(x,t,s,ξ) = a(x,t,T_n(s),ξ) a.e. in Ω, ^∀s ^∈_R, ^∀ξ ^∈_R^N, (5.2)

Φi,n(x,t,s) = Φi,n(x,t,T_n(s)) a.e. (x,t) ∈ Q_T, ∀s ∈ IR.

(5.3)

f₁,n(x,s₁,s₂) = f₁(x,T_n(s₁),s₂) a.e. in Ω, ^∀s₁,s₂^∈R,

(5.4)

f₂,n(x,s₁,s₂) = f₂(x,s₁,T_n(s₂)) a.e. in Ω, ^∀s₁,s₂^∈R,

(5.5)

ui,,bi,n(x,ui,0n) → bi(x,ui,0) inL1(Ω)

as n tends to +∞ (5.6)

Let us now consider the regularized problem

∂bi,n∂t(x,ui,n) − div(an(x,ui,n, ∇ui,n)) − div(Φi,n(x,t,ui,n))

+f_i,n(x,u₁,n,u₂,n) = 0 in Q_T,	(5.7)
u_i,n= 0 on (0,T ) × ∂Ω,	(5.8)
b_i,n(x,u_i,n)(t = 0) = b_i,n(x,u_i,0n) in Ω.	(5.9)

In view of (5.1), for i = 1,2, we have

∂bi,n(x,s) ≥ 1, |bi,n(x,s)| ≤ max |bi(x,s)| +1 ∀s ∈ R,

∂s n |s|≤n

In view of (4.9)-(4.10), f_1,nand f_2,nsatisfy: There exists F_n∈ L¹(Ω),G_n∈ L¹(Ω) and σ_n> 0,λ_n> 0, such

that

^|f₁,n(x,s₁,s₂)^{| ≤}F_n(x)+σ_nmax_|s|≤_n^|b_i(x,s)^|

a.e. in x ^∈Ω, ^∀s₁,s₂^∈R,

^|f₂,n(x,s₁,s₂)^{| ≤}G_n(x)+λ_nmax_|s|≤_n^|b_i(x,s)^|

a.e. in x ∈ Ω, ∀s₁,s₂∈ _R. As a consequence, proving the existence of a weak solution u_i,n) of

(5.7)-(5.9) is an easy task (see e.g. [9]).

Step2:A priori estimates.

Let t ∈ (0,T ) and using T_k(u_i,n)χ_(0,t)as a test function in problem (5.7), we get:

RΩBni,k(x,ui,n(t))dx+RQt an(x,t,ui,n, ∇ui,n)∇Tk(ui,n)dxdt

+ φ_i,n(x,t,u_i,n)∇T_k(u_i,n)dxdt (5.10)

Q_t

Z Z

+ fi,nTk(ui,n)dxdt ≤ Bni,k(x,ui,0n)dx,

Q_tΩ ⁿ(x,r) = Z r ∂bi,n(x,s)T_k(s)ds. where B_i,k∂s

₀

Due to definition of Bⁿ_i,kwe have:

Z Z

Bni,k(x,ui,n(t))dx ≥ λ2n Ω |Tk(ui,n)|2dx, ∀k > 0,

Ω

(5.11)

and

Z Z

Bni,k(x,ui,0n)dx ≤ k |bi,n(x,ui,0n)|dx (5.12)

Ω

≤ k ||bi(x,ui,0)||L1(Ω), ∀k > 0.

In view of (4.8), we have ^RQ_tf_i,nT_k(u_i,n)dxdt ≥ 0 Also, we obtain with Young inequality:

φi,n(x,t,ui,n)∇Tk(ui,n)dxdt Q_t

= φi,n(x,t,ui,n)∇Tk(ui,n)dxdt

{|u_i,n|≤k} Z

≤ ψ(x, ^α0i φ_i,n(x,t,u_i,n))dxdt

{|ui,n|≤k}

Z dxdt

≤ ψ(x, (x, |k |))dxdt

{|u_i,n|≤k} α₀i

dxdt

≤ ψ(x, (x, |k |))dxdt

{|u_i,n|≤k} α₀i

Z dxdt

then

φi,n(x,t,Tk(ui,n))∇Tk(ui,n)dxdt

Q_t

≤ C_i,k+^α₀ⁱϕ(x, ∇T_k(u_i,n))dxdt (5.13)

Q_t

We conclude that ^λZ |Tk(u )|²dx +αⁱZ ϕ(x, ^∇T_k(u_i,n)dxdt

i,n

2 Ω Qt

ϕ(x, ∇Tk(ui,n))dtdx +Ci,k +k ||bi(x,ui,0n)||L1(Ω)

Q_t

Then

λ ZZ

|Tk ϕ(x, ∇Tk(ui,n))dtdx ≤ Ci.k

2 ΩQt

Choosing α₀ⁱsuch that

0 < α₀ⁱ< min(1,αⁱ) we get

Z ϕ(x, ∇T_k(u_i,n))dxdt ≤ C_i.k (5.14)

Q_t

Then, by (5.14), we conclude that T_k(u_i,n) is bounded in W^1,xL_ϕ(Q_T) independently of n and for any k ≥ 0, so there exists a subsequence still denoted by u_nsuch that

Tk(ui,n) → ψi,k (5.15)

weakly in W₀^1,xL_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 (5.14),we get also,

ϕ(x,k) meas {|ui,n | > k } ∩ BR × [0,T ]

Z T Z ≤ ϕ(x,T_k(u_i,n))dxdt

0 {|ui,n|>k}∩BR

≤ ϕ(x,T_k(u_i,n))dxdt

Q_T

≤ diamQ_Tϕ(x, ∇T_k(u_i,n))dxdt

Q_T

Then

{|ui,n | > k } ∩ BR × [0,T ] ≤ diamQ_T.C_i.k

meas ϕ(x,k)

Which implies that:

lim meas {|u_i,n| > k } ∩ B_R× [0,T ] = 0. uniformly

k→+∞

with respect to n.

Now we turn to prove the almost every con-

vergence of u_i,n, b_i,n(x,u_i,n) and convergence of ai,n(x,t,Tk(ui,n), ∇Tk(ui,n)).

Proposition 5.1 Let u_i,nbe a solution of the approximate problem, then:

ui,n → ui a.e in QT .

bi,n(x,ui,n) → bi(x,ui) a.e in QT

(5.16)

b_i(x,u_i) ^∈L^∞(0,T ,L¹(Ω)).

an(x,t,Tk(ui,n), ∇Tk(ui,n)) * Xi,k

(5.17)

in (L_ψ(Q_T))^Nfor σ(ΠL_ψ,ΠE_ϕ)

(5.18)

for some X_i,k^∈(L_ψ(Q_T))^N

lim→+∞n→lim+∞ m≤|ui,n|≤m+1ai(x,t,ui,n, ∇ui,n)∇ui,ndxdt = 0

(5.19)

Proof of (5.16) and (5.17):

Consider now a function non decreasing g_k∈ C²(IR) such that g_k(s) = s for |s and g_k(s) = k for |s | ≥ k. Multiplying the approximate equation by g

get

∂Bi,nk,g(x,ui,n) 0

∂t − div an(x,t,ui,n, ∇ui,n)gk(ui,n)

+a_n(x,t,u_i,n,u_i,n(5.20)

0 00

+ div φi,n(x,t,ui,n)gk(ui,n) − gk (ui,n)φi,n(x,t,ui,n)∇ui,n

0 0

+f_i,ng_k(u_n) = 0 in D (Q_T) ^i,nZ z ∂bi,n(x,s) 0

where B_k,g(x,z) = 0 ∂s g_k(s)ds.

Using (5.20),we can deduce that g_k(u_i,n) is bounded in W₀^1,xL_ϕ(Q_T) and _∂tis bounded in L¹(Q_T)+

W ^−1,xL_ψ(Q_T) independently of n. thanks to (4.6) and properties of g_k, it follows that

| φi,n(x,t,un)gk0 (ui,n)dxdt |

Q_T

≤ kgk0 k∞ ci(x,t)ψ−1ϕ(x,Tk(ui,n))dxdt

Q_T

≤ kgk0 k ci(x,t)dxdt ≤ Ci,k1

Q_TBy (5.13), we get

| gk00(ui,n)φi,n(x,t,ui,n)∇ui,ndxdt |

Q_T

≤ kgk0 k∞Ci,k +c0i ψ(x, ∇Tk(ui,n))dxdt ≤ Ci,k2

Q_T

where C_i,k¹and C_i,k²constants independently of n. we conclude that ∂g^k_∂t^(ui,n)is bounded in L¹(Q_T) + W ^−1,xL_ψ(Q_T) for k < n. which implies that g_k(u_i,n) is compact in L¹(Q_T).Due to the choice of g_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,nconverge almost everywhere to some measurable function u_iin Q_T.

Then by the same argument in [9], we have

u_i,n→ u_ia.e. Q_T, (5.21)

where u_iis a measurable function defined on Q_T. and b_i,n(x,u_i,n) → b_i(x,u_i) a.e. in Q_Tby (5.15) and (5.21) we have

T_k(u_i,n) → T_k(u_i) (5.22)

weakly in W₀^1,xL_ϕ(Q_T) for σ(ΠL_ϕ,ΠE_ψ) strongly in E_ϕ(Q_T) and a.e in Q_T.

We now show that b_i(x,u_i) ∈ L^∞(0,T ;L¹(Ω)). Indeed using ¹_εT_ε(u_i,n) as a test function in (5.7),

1 Z ε (x,ui,n)(t)dx + 1 Z an(x,ui,n, ∇ui,n)∇Tε(ui,n)dxdt bi,n

ε Ω ε QT

1 Z 1 Z

− Φi,n(x,t,ui,n)∇Tε(ui,n)dxdt + fi,n(x,u1,n,u2,n)Tε(ui,n) ε QT ε QT

1 Z

(x,u_i,₀n)dx,

(5.23)

for almost any t in (0,T ). Where, b_i,n^ε(r) =

R r 0

₀bi,n(s)Tε(s)ds. Since a_nsatisfies (4.5) and f_i,nsatisfies (4.8), we get

R bε (x,ui,n)(t)dx ≤ RQT Φi,n(x,t,ui,n)∇Tε(ui,n)dxdt

Ω i,n

+ b_i,n^ε(x,u_i,₀n)dx, (5.24)

Ω

By Young inequality and (4.6), we get

Z Z

Φi,n(x,t,u_i,n)^∇T_ε(u_i,n)dxdt ^≤ψ(x,Φi,n(x,t,u_i,n))dxdt

QT |ui,n|≤ε

+ ϕ(x, ∇T_ε(u_i,n))dxdt

|u_i,n|≤ε

.meas(Q_T)+ (ϕ(x, ∇T_ε(u_i,n))dxdt

|u_i,n|≤ε

(5.25)

Using the Lebesgue’s Theorem and ϕ(x, ∇T_ε(u_i,n)) ∈

W₀^1,xL(Q_T) in second term of the left hand side of the

(5.25) and Letting ε → 0in (5.24)we obtain

|b_i,n(x,u_i,n)(t)| dx ≤ kb_i,n(x,u_i,0n)k_L1(_Ω) (5.26)

Ω for almost t ∈ (0,T ). thanks to (5.6) , (5.16), and passing to the limit-inf in (5.26), we obtain b_i(x,u_i) ∈ L^∞(0,T ;L¹(Ω)). Proof of (5.18) :

Following the same way in([10]),we deduce that a_n(x,t,T_k(u_i,n), ∇T_k(u_i,n)) is a bounded sequence in (L_ψ(Q_T))^N,and we obtain (5.18).

Proof of (5.19) :

Multiplying the approximating equation (5.7) by the test function θ_m(u_i,n) = T_m+1(u_i,n) − T_m(u_i,n)

B_i,m(x,u_i,n(T ))dx +

Ω

an(x,t,ui,n, ∇ui,n)∇θm(ui,n)dxdt

Q_T

+ φ_i,n(x,t,u_i,n)^∇θ_m(u_i,n)dxdt (5.27)

Q_T

Z Z

+ fi,nθm(ui,n)dxdt ≤ Bi,m(x,ui,0n)dx,

Q_TΩ

where B_i,mds.

By (4.6),we have

φi,n(x,t,ui,n)∇θm(ui,n)dxdt

Q_T

≤ ψ(x,(x, |u_i,n|))dxdt m≤|u_i,n|≤m+1

+ ϕ(x, ∇θ_m(u_i,n))dxdt m≤|u_i,n|≤m+1

Also ^RQ_Tf_i,nθ_m(u_i,n)dxdt ≥ 0in view of (4.8).Then, The same argument in step 2 ,we obtain,

ϕ(x, ∇u_i,n)dxdt

Q_T

^≤C_iZ ψ(x, β −1ϕ(x, |ui,n |))dxdt ψ

m≤|u_i,n|≤m+1

+ B_i,m(x,u_i,0n)dx

Ω

Where C_i= _αi¹₋where 0 .

passing to limit as n → +∞ ,since the pointwise convergence of u_i,nand strongly convergence in L¹(Q_T) of B_i,m(x,u_i,₀n) we get

→lim+∞ Q_Tϕ(x, ∇u_i,n)dxdt n

≤ C_iψ(x,(x, |u_i|))dxdt m≤|u_i|≤m+1

+ B_i,m(x,u_i,0)dx

Ω

By using Lebesgue’s theorem and passing to limit as m → +∞, in the all term of the right-hand side, we get Z

lim→+∞n→lim+∞ m_≤|u_i_|≤m+1ϕ(x, ∇u_i,n)dxdt = 0 (5.28) m and the other hand, we have

lim→+∞n→lim+∞ Q_Tφ_i,n(x,t,u_i,n)^∇θ_m(u_i,n)dxdt m

≤ lim→+∞n→lim+∞ m_≤|u_i_|≤m+1ϕ(x, ∇θ_m(u_i,n))dxdt m

+ lim→+∞n→lim+∞ m≤|ui,n|≤m+1ψ(x,φ_i,n(x,t,u_i,n))dxdt m

Using the pointwise convergence of u_i,nand by Lebesgue’s theorem,in the second term of the right side ,we get

^→lim+^∞m≤|u_i,n|≤m+1ψ(x,φ_i,n(x,t,u_i,n))dxdt

= ψ(x,φ_i(x,t,u_i))dxdt, m≤|u_i|≤m+1

and also ,by Lebesgue’s theorem

lim^→+^∞m≤|u_i|≤m+1ψ(x,φ_i(x,t,u_i))dxdt = 0 (5.29)

m we obtain with (5.28) and (5.29),

lim→+∞n→lim+∞ Q_Tφ_i,n(x,t,u_i,n)∇θ_m(u_i,n)dxdt = 0 m

then passing to the limit in (5.27), we get the (5.19). Step 3: Let υ_i,j∈ D (Q_T) be a sequence such that υ_i,j→ u_iin W₀^1,xL_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. µ

Let (α_i,₀)_µbe a sequence of functions defined on Ω such that

) for allµ > 0 (5.30)

kαi,0 kL∞(Ω) ≤ k for allµ > 0.

and α_i,₀converges to T_k(u_i,₀) a.e. in Ω and_µ¹kα_i,^µ0 k_ϕ,_Ωconverges to 0 µ → +∞. For k ≥ 0 and µ > 0, let us consider the unique solution (T_k) of the monotone problem:

∂(Tk(υi,j))µ 0

+µ((T_k(υ_i,j))_µ− T_k(υ_i,j)) = 0 in D (Ω),

∂t

(5.31)

(T_k(υ_i,j))_µ(t = 0) = α_i,₀in Ω. (5.32) Remark that due to

∂(Tk(υi,j))µ ∈ W1,x

∂t ₀L_ϕ(Q_T) (5.33)

We just recall that,

(T_k(υ_i,j))_µ→ T_k(u_i) a.e. in

(Tk(υi,j))µ → (Tk(ui))µ

Q_T, weakly∗

in W01,xLϕ(QT )

for fixed K ≥ 0:

lim Z ∂bi,Sm(ui,n) ,Sm0 (ui,n)Wi,j,µn E ≥ 0, D liminf lim

µ→+∞ j→+∞n→+∞ Q_T∂t

in L^∞(Q_T), (5.41)

lim lim lim S

(5.34) →+∞ →+∞ZQT 0 (ui,n)Φi,n(x,t,ui,n)∇Wi,j,µn = 0,

µ→+∞j n n

(5.42)

(5.35) _Z

µ→lim+∞j→lim+∞n→lim+∞ QT Sm00 (ui,n)Wi,µn Φi,n(x,t,ui,n)∇ui,n = 0,

(5.43)

lim lim lim lim S⁰⁰(u )W ⁿa (x,t,u , ∇u )∇u = 0

for the modular convergence as j → +∞.

(T_k(u_i))_µ→ T_k(u_i) in W₀^1,xL_ϕ(Q_T) (5.36)

for the modular convergence as µ → +∞.

||(Tk(υi,j))µ ||L∞(Q_T) ≤ max(||(Tk(ui))||L∞(Q_T), ||^α₀||L∞(Ω)) ≤ k

(5.37)

∀ µ > 0 , ∀ k > 0. Now, we introduce a sequence of increasing C^∞(R)-functions S_msuch that, for any

m ≥ 1

(5.44)

lim lim lim f_i,n(x,u₁,n,u₂,n)S^{0 n}

µ→+∞j→+∞n→+∞ QT m(ui,n)Wi,j,µ = 0.

(5.45)

limsup a(x,t,u_i,n, ^∇T_K(u_i,n))^∇T_K(u_i,n)dxdt n→+∞ Q_T

(5.46)

≤ X_i,K∇T_K(u_i)dxdt. (5.47)

Q_T

QT [a(x,t,Tk(ui,n), ∇Tk(ui,n)) − a(x,t,Tk(ui,n), ∇Tk(ui))]

[∇T (u ) − ∇T (u )]dxdt → 0. (5.48)

_m→_Qm i,n _i,j,µn i,n i,n i,n

_{k i,n k i}S_m(r) = r for |r | ≤ m,,

Through setting, for fixed K ≥ 0,

Lemma 5.1

Z ∂bi,n(x,ui,n),Sm0 (ui,n)Wi,j,µn Edxdt (n,j,µ,m), D

_QT ∂t

(5.38) Proof of (5.41):

^kS.

Wi,j,µn = TK(ui,n)−TK(υi,j)µ and Wi,µn = TK(ui,n)−TK(ui)µ (5.49)

(5.39)

we obtain upon integration,

Z D∂bi,Sm(ui,n) n Edxdt

∂t ,Wi,j,µ

+ Sm0 (ui,n)an(x,uin, ∇ui,n)∇Wi,j,µn dxdt

Q_T

+ Sm00 (ui,n)Wi,j,µn an(x,ui,n, ∇ui,n)∇ui,n dxdt

Z^QT(5.40)

+ Φi,n(x,t,ui,n)Wi,j,µn dxdt

Q_T

+ Sm00 (ui,n)Wi,j,µn Φi,n(x,t,ui,n)∇ui,n dxdt

Q_T

+ fi,n(x,u1,n,u2,n)Sm0 (ui,n)Wi,j,µn dxdt = 0

Q_T

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 See [23]. Proof of (5.42):

If we take n > m+1, we get

φi,n(x,t,ui,n)Sm0 (ui,n) = φi(x,t,Tm+1(ui,n))Sm0 (ui,n) Using (4.6), we have:

ψ(φ_i,n(x,t,T_m(x,t,T_m+1(u_i,n)))

≤ (m+1)ψ(kc(x,t)k_L^∞(Q_T)ψ⁻¹M(m+1))

Then φ_i,n(x,t,u_n)S_m(u_i,n) is bounded in L_ψ(Q_T), thus, by using the pointwise convergence of u_i,nand Lebesgue’s theorem we obtain φ_i,n(x,t,u_i,n)S_m(u_i,n) → φ_i(x,t,u_i)S_m(u_i) with the modular convergence as n → +∞, then φi,n(x,t,ui,n)Sm(ui,n) → φ(x,t,ui)Sm(ui) for σ(^QL_ψ,^QL_ϕ).

In the other hand ∇Wi,j,µn = ∇Tk(ui,n) − ∇(Tk(υi,j))µ for converge to ∇T_k(u_i) − ∇(T_k(υ_i,j))_µweakly in (L_ϕ(Q_T))^N,then

φi,n(x,t,ui,n)Sm(ui,n)∇Wi,j,µn dxdt

Q_T

→ φi(x,t,ui)Sm(ui)∇Wi,j,µ dxdt

Q_T

as n ^→+^∞.

By using the modular convergence of W_i,j,µas j → +∞ and letting µ tends to infinity, we get (5.42).

Proof of (5.43):

For n > m +1 > k , we have ∇u_i,nS

a.e. in Q_T. By the almost every where convergence of ui,n we have Wi,j,µn → Wi,j,µ in L∞(QT ) weak-

* and since the sequence (φ_i,n(x,t,T_m₊₁(u_i,n)))_nconverge strongly in E_ψ(Q_T) then

φi,n(x,t,Tm+1(ui,n)) Wi,j,µn → φi(x,t,Tm+1(ui)) Wi,j,µ

converge strongly in E_ψ(Q_T)as n → +∞.By virtue of

∇T_m+1(u_n) → ∇T_m+1(u_i) weakly in (L_ϕ(Q_T))^Nas n →

+∞ we have

φi,n(x,t,Tm+1(ui,n))∇ui,nSm00 (ui,n)Wi,j,µn dxdt

m≤|u_i,n|≤m+1

→ φ(x,t,ui))∇uiWi,j,µ dxdt m≤|u_i|≤m+1 as n ^→+^∞

with the modular convergence of W_i,j,µas j → +∞ and letting µ → +∞ we get (5.43).

Proof of (5.44):

For any m ≥ 1 fixed, we have

Z 00 n dxdt

Sm(ui,n)an(x,t,ui,n, ∇ui,n)∇ui,nWi,j,µ

Q_T

≤ kSm00 kL∞(R) kWi,j,µn kL∞(QT ) an(x,t,ui,n, ∇ui,n)

{m≤|u_i,n|≤m+1}

× ∇ui,n dxdt,

for any m ≥ 1, and any µ > 0. In view (5.37) and (5.38), we can obtain

limsup Sm00 (ui,n)an(x,t,ui,n, ∇ui,n)∇ui,nWi,j,µn dxdtn→+∞ Q_T

≤ 2Klimsup an(x,t,ui,n, ∇ui,n)∇ui,n dxdt,

n→+∞ {m≤|u_i,n|≤m+1}

(5.50)

for any m ≥ 1. Using (5.19) we pass to the limit as m → +∞ in (5.50) and we obtain (5.44).

Proof of (5.45):

For fixed n ≥ 1 and n > m+1, we have

f1,n(x,u1,n,u2,n)^S_m⁰(u1,n) = f₁(x,T_m+1(u₁,n),T_n(u₂,n))^S_m⁰(u₁,n), f2,n(x,u1,n,u2,n)^S_m⁰(u2,n)

= f₂(x,T_n(u₁,n),T_m+1(u₂,n))^S_m⁰(u₂,n),

In view (4.9),(4.10),(5.22) and Lebesgue’s the theorem allow us to get, for

_n→lim₊∞ _Q_Tf_i,n(x,u₁,n,u₂,n)Sm⁰(ui,n)^Wi,j,µⁿdxdt

= fi(x,u1,u2)Sm0 (ui)Wi,j,µ dxdt

Q_T

Using (5.35), we follow a similar way we get as j →

+∞,

→lim+∞ Q_Tfi(x,u1,u2)Sm0 (ui)Wi,j,µ dxdt

= f_i(x,u₁,u₂)S_m⁰(u_i)(T_K(u_i) ⁻T_K(u_i)_µ)dxdt

Q_Twe fixed m > 1, and using (5.36), we have

µ→lim+∞ Q_Tf_i(x,u₁,u₂)S_m⁰(u_i)(T_K(u_i) − T_K(u_i)_µ)dxdt = 0

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 (5.40). We obtain using (5.41)-(5.45), for any K ≥ 0,

mlim^→+^∞limsupµ→+∞ limsupj→+∞ limsupn→+∞ Q_TS_m⁰(u_i,n)a_n(x,t,u_i,n, ∇u_i,n)

∇TK(ui,n) − ∇TK(υi,j)µ dxdt ≤ 0.

Since

Sm0 (ui,n)an(x,t,ui,n, ∇ui,n)∇TK(ui,n)

= an(x,t,ui,n, ∇ui,n)∇TK(ui,n) for n > K and K ≤ m. Then, for K ≤ m,

limsup an(x,t,ui,n, ∇ui,n)∇TK(ui,n)dxdt n→+∞ Q_T

≤ m µ→+∞ j→+∞ n +∞ Q_T^S_m0 (u_i,n) (5.51)

an(x,ui,n, ∇ui,n)∇TK(υi,j)µ dxdt.

Thanks to (5.38), we have in The right hand side of

(5.51), for n > m+1,

Sm0 (ui,n)an(x,t,ui,n, ∇ui,n)

= ^S_m(u_i,n)a x,t,T_m+1(u_i,n), ^∇T_m+1(u_i,n) a.e. in Q_T.

Using (5.18), and fixing m ≥ 1, we get

Sm0 (ui,n)an(ui,n, ∇ui,n) * Sm0 (ui)Xi,m+1 weakly in (Lψ(QT ))N .

when n → +∞ .

We pass to limit as j → +∞ and µ → +∞, and using

(5.35)-(5.36)

limsuplimsuplimsup S_m⁰(u_i,n))a_n(x,t,u_i,nµ→+∞ j→+∞ n→+∞ Q_T

,∇ui,n))∇TK(υi,j)µ dxdt

= Sm0 (ui)Xi,m+1 ∇TK(ui)dxdt

Q_T

= Xi,m+1 ∇TK(ui)dxdt

Q_T

(5.52)

where K ≤ m, since S_m⁰(r) = 1 for |r | ≤ m. Since (1.1), (4.4) and (5.22), imply that the function

On the other hand, for K ≤ m, we have a_K(x,s,ξ) is continuous and bounded with respect to

s. Then we conclude that (5.57).

a x,t,T_mProof of (5.58):

Using (4.5) and (5.48), for any K ≥ 0 and any T ⁰< T ,

= a x,t,T_K, we have

a.e. in Q_T. Passing to the limit as n → +∞, we obtain ^ha(x,t,T_K(u_i,n, ∇T_K(u_i,n)) − a(x,t,T_K(u_i,n), ∇T_K(u))ⁱ

Xi,m+1χ{|ui|<K} = Xi,Kχ{|ui|<K} a.e. in QT −{|ui | = K } for K ≤ n.

(5.53) ^{× h}∇T_K(u_i,n) − ∇T_K(u_i)ⁱ→ 0 (5.60)

Then

∇TK(ui) = XK ∇TK(ui) a.e. in QT . (5.54) strongly inOn the other hand with (5.22), (5.18), (5.56) andL¹(Q_T⁰) as n → +∞ .

Xm+1

(5.57), we get Then we obtain (5.46).

Proof of (5.48):

Let K ≥ 0 be fixed. Using (4.5) we have a x,t,T_K(u_i,n), ^∇T_K(u_i,n) ^∇T_K(u_i)

h i

a(x,t,TK(ui,n), ∇TK(ui,n)) − a(x,t,TK(ui,n), ∇TK(ui))

Q_T* a x,t,T_K(u_i), ^∇T_K(u_i) ^∇T_K(u_i)

h i

∇TK(ui,n) − ∇TK(ui) dxdt ≥ 0,

(5.55) weakly in L¹(Q_T),

In view (1.1) and (5.22), we get

a x,t,T (u ), ∇T (u ) ∇T (u )

a(x,t,T_K(u_i,n), ^∇T_K(u_i)) ^→a(x,t,T_K(u_i), ^∇T_K(u_i)) a.e. in

as n → +∞, and by (4.2) and Lebesgue’s theorem, we obtain

a x,t,T_K(u_i,n), ^∇T_K(u_i) ^→a x,t,T_K(u_i), ^∇T_K(u_i)

(5.56)

strongly in (L_ψ(Q_T))^N. Using (5.46), (5.22), (5.18) and (5.56), we can pass to the lim-sup as n → +∞ in

(5.55) to obtain (5.48).

To finish this step, we prove this Lemma: Lemma 5.2 For i = 1,2 and fixed K ≥ 0, we have

X_i,K= a xt,,T_K(u_i), ∇T_K(u_i) a.e. in Q. (5.57)

Also, as n ^→+^∞,

x,t,T_K* a x,t,T_K(u_i),DT_K

(5.58)

weakly in L¹(Q_T).

Proof of (5.57): It’s easy to see that a_n(x,t,T_K(u_i,n),ξ) = a(x,t,T_K(u_i,n),ξ) = a_K(x,t,T_K(u_i,n),ξ)

a.e. in Q_Tfor any K ≥ 0, any n > K and any ξ ∈ _R^N.

In view of (5.18), (5.48) and (5.56) we obtain

→lim+∞ Q_TaK x,t,TK(ui,n), ∇TK(ui,n) ∇TK(ui,n)dxdt n

= Xi,K ∇TK(ui)dxdt.

Q_T

(5.59)

Q_T, K i,n K i K i,n

* a x,t,T_K(u_i), ^∇T_K(u_i) ^∇T_K(u_i) weakly in L¹(Q_T),

a x,t,TK(ui,n), ∇TK(ui) ∇TK(ui)

^→a x,t,T_K(u_i), ^∇T_K(u_i) ^∇T_K(u_i),

strongly in L¹(Q), as n → +∞. It’s results from (5.60), for any K ≥ 0 and any T ⁰< T ,

a x,t,TK(ui,n), ∇TK(ui,n) ∇TK(ui,n)

* a x,t,T_K(u_i), ∇T_K(u_i) ∇T_K(u_i) (5.61)

weakly in L¹(Q_T0 ) as n → +∞.then for T ⁰= T , we have (5.58). Finally we should prove that u_isatisfies (4.13).

Step 4:Pass to the limit. we first show that u satisfies (4.13)

a(x,t,ui,n, ∇ui,n)∇ui,n dxdt

m≤|u_i,n|≤m+1}

= an(x,t,ui,n, ∇ui,n) ∇Tm+1(ui,n) − ∇Tm(ui,n) dxdt Q_T

= an x,t,Tm+1(ui,n), ∇Tm+1(ui,n) ∇Tm+1(ui,n)dxdt

Q_T

− an x,t,Tm(ui,n), ∇Tm(ui,n) ∇Tm(ui,n)dxdt Q_T

Aberqi / Advances in Science, Technology and Engineering Systems Journal Vol. 1, No. 1, XX-YY (2016)

for n > m +1. According to (5.58), one can pass to the limit as n → +∞ ; for fixed m ≥ 0 to obtain

^→lim+^∞m≤|u_i,n|≤m+1} an(x,t,ui,n, ∇ui,n)∇ui,n dxdt

= a x,t,T_m+1(u_i), ^∇T_m+1(u_i) ^∇T_m+1(u_i)dxdt

⁻a x,t,T_m(u_i), ^∇T_m(u_i) ^∇T_m(u_i)dxdt Q

= a(x,t,u_i, ∇u_i)∇u_idxdt m≤|u_i|≤m+1}

(5.62) Pass to limit as m tends to +∞ in (5.62) and using

(5.19) show that u_isatisfies (4.13).

Now we shown that u_ito satisfy (4.14)and (4.15). Let S be a function in W^2,∞(R) such that S⁰has a compact support. Let K be a positive real number such that suppS⁰⊂ [−K,K]. the Pointwise multiplication of the approximate equation (1.1) by S⁰(u_i,n) leads to

− div S (ui,n)an(x,ui,n, ∇ui,n) ∂t

+S00(ui,n)an(x,ui,n, ∇ui,n)∇ui,n

0 (5.63)

− div S (ui,n)Φi,n(x,t,ui,n)

+S00(ui,n)Φi,n(x,t,ui,n)∇ui,n

= fi,n(x,u1,n,u1,n)S0(ui,n)

in D⁰(Q_T), for i = 1,2.

Now we pass to the limit in each term of (5.63).

Limit of ∂Bni,S_∂t(ui,n): Since∞ Bn (ui,n) converges to Bi,S(ui) i,S

a.e. in Q_Tand in L (Q_T) weak ? and S is bounded

D⁰(Q_T) as n tends to +∞.

Limit of div S (u_i,n)a_n(x,t,u_i,n, ^∇u_i,n) :

suppS⁰⊂ [−K,K], for n > K, we have

Since

∂Bni,S(ui,n)

and continuous. Then _∂tconverges toin

S0(ui,n)an(x,t,ui,n, ∇ui,n)

= S (ui,n)an x,t,TK(ui,n), ∇TK(ui,n)

a.e. in Q_T. Using the pointwise convergence of u_i,n

, (5.38),(5.18) and (5.57), imply that

S (ui,n)an x,t,TK(ui,n), ∇TK(ui,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 S⁰(u_i) = 0 for |u_i| ≥ K a.e. in Q_T. And

0 0

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 S00(ui,n)an(x,t,ui,n, ∇ui,n)∇ui,n. Since suppS⁰⁰⊂ [−K,K], for n > K, we have

S00(ui,n)an(x,t,ui,n, ∇ui,n)∇ui,n

= S00(ui,n)an x,t,TK(ui,n), ∇TK(ui,n) ∇TK(ui,n) a.e. in QT .

The pointwise convergence of S⁰⁰(u_i,n) to S⁰⁰(u_i) as n → +∞, (5.38) and (5.58) we have

S00(ui,n)an(x,t,ui,n, ∇ui,n)∇ui,n

₀₀

* S (u_i)a x,t,T_K(u_i), ∇T_K(u_i) ∇T_K(u_i) weakly in L¹(Q_T), as n → +∞, and

₀₀

S (u_i)a x,t,T_K(u_i), ^∇T_K(u_i) ^∇T_K(u_i)

= S⁰⁰(u_i)a(x,t,u_i, ∇u_i)∇u_ia.e.in Q_T.

Limit of S⁰(u_i,n)Φ_i,n(x,t,u_i,n): We have

S0(ui,n)Φi,n(x,t,ui,n)

= S0(ui,n)Φi,n(x,t,TK(ui,n))

a.e.in Q_T, Since suppS⁰⊂ [−K,K].Using (4.5), (5.24) and (5.16), it’s easy to see that

S⁰(u_i,n)Φ_i,n(x,t,u_i,n) * S⁰(u_i)Φ_i(x,t,T_K(u_i)) weakly for σ(ΠL_ψ,ΠL_ϕ) as n → +∞. And S⁰(u_i)Φi(x,t,T_K(u_i)) =

S⁰(u_i)Φ_i(x,t,u_i) a.e. in Q_T.

Limit of S00(ui,n)Φi,n(x,t,ui,n)∇ui,n: Since S0 ∈

W^1,∞(R) with suppS⁰⊂ [−K,K], we have

S00(ui,n)Φi,n(x,t,ui,n)∇ui,n = Φi,n(x,t,TK(ui,n))∇S0(TK(ui,n)) a.e. in Q_T. The weakly convergence of truncation allows us to prove that

S00(ui,n)Φi,n(x,t,ui,n)∇ui,n * Φi(x,t,ui)∇S0(ui), strongly in L¹(Q_T).

Limit of f_i,n(x,u_1,n,u_2,n)S⁰(u_i,n): Using (4.9), (4.10),

(5.4) and (5.5), we have

f_i,n(x,u₁,n,u₂,n)S⁰(u_i,n) → f_i(x,u₁,u₂)S⁰(u_i) strongly in

L¹(Q_T), as n → +∞.

It remains to show that for i=1,2 B_S(x,u_i) satisfies the initial condition (4.15).

To this end, firstly remark that, in view of the definition of S_ϕ⁰, we have B_ϕ(x,u_i,n) is bounded in L^∞(Q_T).

∂Bϕ(x,ui,n)

Secondly, by (5.41) we show that is

∂t

bounded in L¹(Q_T) + W ^−1,xL_ψ(Q_T)). As a consequence, an Aubin’s type Lemma (see e.g., [11], Corollary 4) implies that B_ϕ(x,u_i,n) lies in a compact set of

C⁰([0,T ];L¹(Ω)) .

It follows that, on one hand,B_ϕ(x,u_i,n)(t = 0) converges to B_ϕ(x,u_i)(t = 0) strongly in L¹(Ω). On the order hand, the smoothness of B_ϕimply that B_ϕ(x,u_i,n)(t = 0) converges to B_ϕ(x,u_i)(t = 0) strongly in L¹(Ω), we conclude that B_ϕ(x,u_i,n)(t = 0) = B_ϕ(x,u_i,_0n) converges to B_ϕ(x,u_i)(t = 0) strongly in

L¹(Ω), we obtain B_ϕ(x,u_i)(t = 0) = B_ϕ(x,u_i,₀) a.e. in Ω and for all M > 0, now letting M to +∞, we conclude that b(x,u_i)(t = 0) = b(x,u_i,₀) a.e. in Ω.

As a conclusion, the proof of Theorem (5.1) is complete.

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Doubly Nonlinear Parabolic Systems In Inhomogeneous Musielak- Orlicz-Sobolev Spcaes