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

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^N is 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 → _R a 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,i is in L¹(Ω) such that b_i(.,u_0,i) in L¹(Ω) and the source f_i is 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 L¹ and 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^N in 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^N and 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 Φ₁ and Φ₂ is called a Musielak-Orlicz function.

For a Musielak-Orlicz function ϕ we put ϕ_x(t) = ϕ(x,t) and we associate its non-negative reciprocal function ϕ_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 t₀ such 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

Z

%_ϕ,dx.

The set K_ϕ(Ω) = {u : Ω → _R mesurable :

%_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

u

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_ϕ We 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:

Lϕ(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) = supt≥0(st − ϕ(x,s)).                                                                                    R ψ is called the Musielak-Orlicz function comple-  Ωu  u kϕ,Ω k|v |kψ,Ω for all mentary to ϕ (or conjugate of ϕ) in the sense of u ∈ Lϕ(Ω),v ∈ Lψ(Ω)

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

We say that a sequence of functions u_n converges 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

o

                                    where      f_α ∈ L_ψ(Ω)

and

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

α≤1

o

                                    where      f_α ∈ E_ψ(Ω)

2.3    Inhomogeneous Musielak-OrliczSobolev spcaes:

Let Ω be a bounded Lipschitz domain in R^N and 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^α the distributional derivative on Q_T of order α with respect to the variable x ∈ R^N . The inhomogeneous Musielak-Orlicz-Sovolev spaces of order 1 are defined as follows

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

X

                                 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,Ω is 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 →  ∞

             _Q                                 λ

, 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_ψ by 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 F₀ is then given by F₀ = W ^−1,xE_ψ(Q).

Lemma 2.1 [4]. Let Ω be a bounded Lipschitz domain in R^N and 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

                                               _A         

                           for all       t ≥ 1.          (2.2)

Z

                                                  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 → _R be 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 F⁰ is finite,then

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

       ∂x_i                              0                    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_n k_∞ ≤

(N +1)ku k_∞.

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

Musielak-Orlicz function satisfying :

Z

                                            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, ∞). ₀ t _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_n remains 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^N is 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) ∈ Eψ(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^N is a Caratheodory func-´ tion such that

                              |Φ_i(x,t,s),                      (4.6)

f_i : Ω × _R × _R → _R is 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_i are as follows: For each K > 0, there exists σ_K > 0 and a function F_K in L¹(Ω) such that

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

T_K       and               b_i(x,u_i) ^∈ L^∞(0,T ;L¹(Ω)),

(4.12)

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

{ m≤|u_i|≤m+1}

(4.13) For every function S in W^2,∞(R) which is piecewise C¹ and such that S⁰ has 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^r b_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_i can 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_i respectively 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,S⁰⁰ and Φ ◦ T_K are bounded on R so that (4.12) implies that S⁰(u_i)Φi(T_K(u_i)) ∈ (L^∞(Q_T ))^N and 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))

+fi,n(x,u1,n,u2,n) = 0         in QT ,	(5.7)
ui,n = 0       on (0,T ) × ∂Ω,	(5.8)
bi,n(x,ui,n)(t = 0) = bi,n(x,ui,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,n and f_2,n satisfy: 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_n max_|s|≤_n ^|b_i(x,s)^|

a.e. in x ^∈ Ω, ^∀s₁,s₂ ^∈ R,

       ^|f₂,n(x,s₁,s₂)^{|     ≤} G_n(x)+λ_n max_|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

Z

                +             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_t                Ω 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,k we 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_t f_i,nT_k(u_i,n)dxdt ≥ 0 Also, we obtain with Young inequality:

Z

φi,n(x,t,ui,n)∇Tk(ui,n)dxdt Q_t

Z

                 =                      φi,n(x,t,ui,n)∇Tk(ui,n)dxdt

{|u_i,n|≤k} Z

1

                  ≤                  ψ(x, ^α0i φ_i,n(x,t,u_i,n))dxdt

{|ui,n|≤k}

Z dxdt

Z

1

                   ≤                   ψ(x,          (x, |k |))dxdt

                          {|u_i,n|≤k}               α₀i

Z

dxdt

Z

1

                   ≤                   ψ(x,          (x, |k |))dxdt

                          {|u_i,n|≤k}               α₀i

Z dxdt

then

Z

φi,n(x,t,Tk(ui,n))∇Tk(ui,n)dxdt

Q_t

Z

                 ≤ C_i,k +^α₀ⁱ                    ϕ(x, ∇T_k(u_i,n))dxdt                 (5.13)

Q_t

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

i,n

            2 Ω                                                          Qt

Z

             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 α₀ⁱ such 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_n such 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

Z

≤             ϕ(x,T_k(u_i,n))dxdt

Q_T

Z

≤ diamQ_T                      ϕ(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,n be 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)
bi(x,ui) ∈ L∞(0,T ,L1(Ω)). an(x,t,Tk(ui,n), ∇Tk(ui,n)) * Xi,k	(5.17)
in       (Lψ(QT ))N for         σ(ΠLψ,ΠEϕ)	(5.18)

for some X_i,k ^∈ (L_ψ(Q_T ))^N

Z

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

m

(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,n Z 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 _∂t is 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

Z

                        |         φi,n(x,t,un)gk0 (ui,n)dxdt |

Q_T

Z

               ≤ kgk0 k∞                  ci(x,t)ψ−1ϕ(x,Tk(ui,n))dxdt

Q_T

Z

            ≤ kgk0 k         ci(x,t)dxdt ≤ Ci,k1

Q_T By (5.13), we get

Z

                  |          gk00(ui,n)φi,n(x,t,ui,n)∇ui,ndxdt |

Q_T

Z

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

Q_T

where C_i,k¹ and C_i,k² constants 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,n converge almost everywhere to some measurable function u_i in Q_T .

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

                                      u_i,n → u_i a.e. Q_T ,                            (5.21)

where u_i is a measurable function defined on Q_T . and b_i,n(x,u_i,n) → b_i(x,u_i) a.e. in Q_T by (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_ε T_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

₀ bi,n(s)Tε(s)ds. Since a_n satisfies (4.5) and f_i,n satisfies (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

Z

                                  +         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

Z

                         +                  ϕ(x, ∇T_ε(u_i,n))dxdt

|u_i,n|≤ε

Z

                                                .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

Z

                |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)

Z

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

Ω

Z

an(x,t,ui,n, ∇ui,n)∇θm(ui,n)dxdt

Q_T

Z

               +              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_T                                                               Ω

where B_i,mds.

By (4.6),we have

Z

φi,n(x,t,ui,n)∇θm(ui,n)dxdt

Q_T

Z

≤ ψ(x,(x, |u_i,n |))dxdt m≤|u_i,n|≤m+1

Z

+      ϕ(x, ∇θ_m(u_i,n))dxdt m≤|u_i,n|≤m+1

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

Z

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

                                    Z

                                 +          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,n and strongly convergence in L¹(Q_T ) of B_i,m(x,u_i,0n) we get

Z

→lim+∞        Q_T ϕ(x, ∇u_i,n)dxdt n

Z

≤ C_i           ψ(x,(x, |u_i |))dxdt m≤|u_i|≤m+1

                                     Z

                                  +         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

Z

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

Z

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

Z

+ 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,n and by Lebesgue’s theorem,in the second term of the right side ,we get

Z

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

n

Z

=    ψ(x,φ_i(x,t,u_i))dxdt, m≤|u_i|≤m+1

and also ,by Lebesgue’s theorem

Z

        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),

Z

lim→+∞n→lim+∞                Q_T φ_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_i in 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)_µ be a sequence of functions defined on Ω such that

           )     for allµ > 0            (5.30)

µ

kαi,0 kL∞(Ω) ≤ k for allµ > 0.

µ

and α_i,0 converges to T_k(u_i,0) a.e. in Ω and_µ¹ kα_i,^µ0 k_ϕ,Ω converges 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)

µ

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

∂(Tk(υi,j))µ ∈ W1,x

                         ∂t       ₀        L_ϕ(Q_T )                  (5.33)

We just recall that, (Tk(υi,j))µ → Tk(ui)             a.e. in (Tk(υi,j))µ → (Tk(ui))µ      QT ,      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→+∞          QT                   ∂t in L∞(QT ),                                                                                                (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) Z lim           lim           lim           lim          S00 (u      )W n             a (x,t,u    , ∇u         )∇u          = 0 for the modular convergence as         j → +∞. T               (Tk(ui))µ → Tk(ui)       in       W01,xLϕ(QT )              (5.36) for the modular convergence as        µ → +∞. µ ||(Tk(υi,j))µ ||L∞(QT ) ≤ max(||(Tk(ui))||L∞(QT ), ||α0 ||L∞(Ω)) ≤ k (5.37) ∀ µ > 0 , ∀ k > 0. Now, we introduce a sequence of increasing C∞(R)-functions Sm such that, for any m ≥ 1 (5.44) Z lim        lim     lim              fi,n(x,u1,n,u2,n)S0                             n µ→+∞j→+∞n→+∞              QT                                                         m(ui,n)Wi,j,µ = 0. (5.45) Z limsup a(x,t,ui,n, ∇TK(ui,n))∇TK(ui,n)dxdt n→+∞ QT (5.46) Z ≤              Xi,K ∇TK(ui)dxdt.                      (5.47) QT R 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→      Q        m       i,n            i,j,µ n                     i,n            i,n            i,n k           i,n                    k                      i Sm(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

QT

Z

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

Q_T

Z

+            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

Z

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

Q_T

Z

+               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,n and 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))_µ weakly in (L_ϕ(Q_T ))^N ,then

Z

                            φi,n(x,t,ui,n)Sm(ui,n)∇Wi,j,µn       dxdt

Q_T

Z

                  →              φi(x,t,ui)Sm(ui)∇Wi,j,µ dxdt

Q_T

as n ^→ +^∞.

By using the modular convergence of W_i,j,µ as 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)))_n converge 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 ))^N as n →

+∞ we have

Z

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

m≤|u_i,n|≤m+1

Z

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

with the modular convergence of W_i,j,µ as 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

Z

≤ 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

Z

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

Z

≤ 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

Z

        _n→lim₊∞      _QT f_i,n(x,u₁,n,u₂,n)Sm⁰ (ui,n)^Wi,j,µⁿ        dxdt

Z

                    =            fi(x,u1,u2)Sm0 (ui)Wi,j,µ dxdt

Q_T

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

+∞,

Z

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

j

Z

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

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

Z

µ→lim+∞           Q_T f_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,

Z

mlim^→+^∞limsupµ→+∞ limsupj→+∞ limsupn→+∞ Q_T S_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,

Z

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

Z

      ≤ 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)

0

= ^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)

Z

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

,∇ui,n))∇TK(υi,j)µ dxdt

Z

          =             Sm0 (ui)Xi,m+1 ∇TK(ui)dxdt

Q_T

Z

          =             Xi,m+1 ∇TK(ui)dxdt

Q_T

(5.52)

where K ≤ m, since Sm0 (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                                                     aK(x,s,ξ) is continuous and bounded with respect to s. Then we conclude that (5.57). a x,t,TmProof of (5.58): Using (4.5) and (5.48), for any K ≥ 0 and any T 0 < T , = a x,t,TK,                                                 we have a.e. in QT . Passing to the limit as n → +∞, we obtain                            ha(x,t,TK(ui,n, ∇TK(ui,n)) − a(x,t,TK(ui,n), ∇TK(u))i Xi,m+1χ{|ui|<K} = Xi,Kχ{|ui|<K}                       a.e. in QT −{|ui | = K } for K ≤ n. (5.53)                                 × h∇TK(ui,n) −    ∇TK(ui)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(QT 0 ) 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,TK(ui,n), ∇TK(ui,n) ∇TK(ui) Z h                                                                                               i a(x,t,TK(ui,n), ∇TK(ui,n)) − a(x,t,TK(ui,n), ∇TK(ui)) QT                                                                                                                                                                                                         * a x,t,TK(ui), ∇TK(ui) ∇TK(ui) h                                     i ∇TK(ui,n) −         ∇TK(ui) dxdt ≥ 0, (5.55)       weakly in L1(QT ), 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_T for any K ≥ 0, any n > K and any ξ ∈ _R^N .

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

               ^Z           

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

Z

     =             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_T 0 ) as n → +∞.then for T ⁰ = T , we have (5.58). Finally we should prove that u_i satisfies (4.13).

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

Z

a(x,t,ui,n, ∇ui,n)∇ui,n dxdt

m≤|u_i,n|≤m+1}

      ^Z                                      

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

      ^Z          

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

Q_T

          ^Z          

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

1. 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

Z

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

n

            ^Z      

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

Q

                ^Z      

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

Z

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

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

(5.19) show that u_i satisfies (4.13).

Now we shown that u_i to satisfy (4.14)and (4.15). Let S be a function in W^2,∞(R) such that S⁰ has 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

0

    − 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_T and in L (Q_T ) weak ? and S is bounded

D0(QT ) as n tends to +∞.
0 Limit        of           div S (ui,n)an(x,t,ui,n, ∇ui,n) : suppS0 ⊂ [−K,K], for n > K, we have	Since

∂Bni,S(ui,n)

and continuous. Then           _∂t                                      converges toin

S0(ui,n)an(x,t,ui,n, ∇ui,n)

0

= 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

0

S (ui,n)an x,t,TK(ui,n), ∇TK(ui,n)

0

* 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_i              a.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_ϕ imply 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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