Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 2, 1-19; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE RESULTS FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS IN ORLICZ SPACES

HICHAM REDWANE

Faculté des Sciences Juridiques, Économiques et Sociales. Université Hassan 1, B.P. 784. Settat. Morocco

Abstract. An existence result of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces is proved. No growth assumption is made on the nonlinearities.

1. Introduction

In this paper we consider the following problem:

\frac{∂b(x, u)}{∂t} − div(a(x, t, u, ∇u) + Φ(u)) = f in Ω × (0, T), (1.1)

b(x, u)(t = 0) = b(x, u_{0}) in Ω, (1.2)

u = 0 on ∂Ω × (0, T), (1.3)

where Ω is a bounded open subset of ℝ^{N} and T > 0, Q = Ω × (0, T). Let b be a Carathéodory function (see assumptions (3.1)-(3.2) of Section 3), the data f and b(x, u_{0}) in L^{1}(Q) and L^{1}(Ω) respectively, Au = −div(a(x, t, u, ∇u)) is a Leray-

Lions operator defined on W_{0}^{1,x}L_{M}(Ω), M is an appropriate N-function and which grows like M̄^{−1}M(β_{K}^{4}|∇u|) with respect to ∇u, but which is not restricted by any growth condition with respect to u (see assumptions (3.3)-(3.6)). The function Φ is just assumed to be continuous on ℝ.

Under these assumptions, the above problem does not admit, in general, a weak solution since the fields a(x, t, u, ∇u) and Φ(u) do not belong in (L_{loc}^{1}(Q)^{N} in general. To overcome this difficulty we use in this paper the framework of renormalized solutions. This notion was introduced by Lions and DiPerna [31] for the study of Boltzmann equation (see also [27], [11], [29], [28], [2]).

A large number of papers was devoted to the study the existence of renormalized solution of parabolic problems under various assumptions and in different contexts: for a review on classical results see [7], [30], [9], [8], [4], [5], [34], [12], [13], [14].

The existence and uniqueness of renormalized solution of (1.1)-(1.3) has been proved in H. Redwane [34, 35] in the case where Au = −div(a(x, t, u, ∇u)) is

a Leray-Lions operator defined on L^{p}(0, T; W_{0}^{1,p}(Ω)), the existence of renormalized solution in Orlicz spaces has been proved in E. Azroul, H. Redwane and M.

EJQTDE, 2010 No. 2, p. 1 Rhoudaf [32] in the case where b(x, u) = b(u) and where the growth of a(x, t, u, ∇u) is controlled with respect to u. Note that here we extend the results in [34, 32] in three different directions: we assume b(x, u) depend on x , and the growth of a(x, t, u, ∇u) is not controlled with respect to u and we prove the existence in Orlicz spaces.

1991 Mathematics Subject Classification. Primary 47A15; Secondary 46A32, 47D20.

Key words and phrases. Nonlinear parabolic equations. Orlicz spaces. Existence. Renormalized solutions.

The paper is organized as follows. In section 2 we give some preliminaries and gives the definition of N-function and the Orlicz-Sobolev space. Section 3 is devoted to specifying the assumptions on b, a, Φ, f and b(x, u_{0}). In Section 4 we give the definition of a renormalized solution of (1.1)-(1.3). In Section 5 we establish (Theorem 5.1) the existence of such a solution.

2. Preliminaries

Let M : ℝ^{+} → ℝ^{+} be an N-function, i.e., M is continuous, convex, with M(t) > 0 for t > 0, \frac{M(t)}{t} → 0 as t → 0 and \frac{M(t)}{t} → ∞ as t → ∞. Equivalently, M admits the representation : M(t) = ∫_{0}^{t} a(s) ds where a : ℝ^{+} → ℝ^{+} is nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞. The N-function \overline{M} conjugate to M is defined by \overline{M}(t) = ∫_{0}^{t} \overline{a}(s) ds, where \overline{a} : ℝ^{+} → ℝ^{+} is given by \overline{a}(t) = sup\{s : a(s) ≤ t\}.

The N-function M is said to satisfy the Δ_{2} condition if, for some k > 0,

(2.1)  M(2t) ≤ k M(t) for all t ≥ 0.

When this inequality holds only for t ≥ t_{0} > 0, M is said to satisfy the Δ_{2}-condition near infinity.

Let P and Q be two N-functions. P ≪ Q means that P grows essentially less rapidly than Q ; i.e., for each ε > 0,

(2.2)  \frac{P (t)}{Q(ε t)} → 0 as t → ∞.

This is the case if and only if,

(2.3)  \frac{Q^{−1}(t)}{P^{−1}(t)} → 0 as t → ∞.

We will extend these N-functions into even functions on all ℝ. Let Ω be an open subset of ℝ^{N}. The Orlicz class L_{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 :

(2.4) ∫_{Ω} M(u(x))dx < +∞ (resp. ∫_{Ω} M(\frac{u(x)}{λ})dx < +∞ for some λ > 0).

Note that L_{M}(Ω) is a Banach space under the norm

(2.5)  ∥u∥_{M,Ω} = inf \{λ > 0 : ∫_{Ω} M(\frac{u(x)}{λ})dx ≤ 1\}

and L_{M}(Ω) is a convex subset of L_{M}(Ω). The closure in L_{M}(Ω) of the set of bounded measurable functions with compact support in \overline{Ω} is denoted by E_{M}(Ω). The equality E_{M}(Ω) = L_{M}(Ω) holds if and only if M satisfies the Δ_{2}-condition, for all t or for t large according to whether Ω has infinite measure or not.

EJQTDE, 2010 No. 2, p. 2

The dual of E_{M}(Ω) can be identified with L\overline{M}(Ω) by means of the pairing ∫_{Ω} u(x)v(x)dx, and the dual norm on L\overline{M}(Ω) is equivalent to ∥.∥_{\overline{M},Ω}. The space L_{M}(Ω) is reflexive if and only if M and \overline{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. W^{1}L_{M}(Ω) (resp. W^{1}E_{M}(Ω)) 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

(2.6)  ∥u∥_{1,M,Ω} = ∑_{|α|≤1} ∥∇^{α}u∥_{M,Ω}.

Thus W^{1}L_{M}(Ω) and W^{1}E_{M}(Ω) can be identified with subspaces of the product of N + 1 copies of L_{M}(Ω). Denoting this product by ΠL_{M}, we will use the weak topologies σ(ΠL_{M}, ΠE\overline{M}) and σ(ΠL_{M}, ΠL\overline{M}). The space W_{0}^{1}E_{M}(Ω) is defined as the (norm) closure of the Schwartz space D(Ω) in W^{1}E_{M}(Ω) and the space W_{0}^{1}L_{M}(Ω) as the σ(ΠL_{M}, ΠE\overline{M}) closure of D(Ω) in W^{1}L_{M}(Ω). We say that u_{n} converges to u for the modular convergence in W^{1}L_{M}(Ω) if for some

λ > 0, ∫_{Ω} M(\frac{∇^{α}u_{n} − ∇^{α}u}{λ})dx → 0 for all |α| ≤ 1. This implies convergence

for σ(ΠL_{M}, ΠL\overline{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.

Let W^{−1}L\overline{M}(Ω) (resp. W^{−1}E\overline{M}(Ω)) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in L\overline{M}(Ω) (resp. E\overline{M}(Ω)). It is a Banach space under the usual quotient norm.

If the open set Ω has the segment property, then the space D(Ω) is dense in W_{0}^{1}L_{M}(Ω) for the modular convergence and for the topology σ(ΠL_{M}, ΠL\overline{M}) (cf. [21]). Consequently, the action of a distribution in W^{−1}L\overline{M}(Ω) on an element of W_{0}^{1}L_{M}(Ω) is well defined. For more details see [1], [23].

For K > 0, we define the truncation at height K, T_{K} : ℝ → ℝ by

(2.7)  T_{K}(s) = min(K, max(s, −K)).

The following abstract lemmas will be applied to the truncation operators.

Lemma 2.1. [21] Let F : ℝ → ℝ be uniformly lipschitzian, with F (0) = 0. Let M be an N-function and let u ∈ W^{1}L_{M}(Ω) (resp. W^{1}E_{M}(Ω)).

Then F (u) ∈ W^{1}L_{M}(Ω) (resp. W^{1}E_{M}(Ω)). Moreover, if the set of discontinuity points D of F′ is finite, then

\frac{∂}{∂x_{i}} F (u) = \{ \begin{matrix}F′(u)\frac{∂u}{∂x_{i}} & a.e. in \{x ∈ Ω : u(x) ∉ D\} \\ 0 & a.e. in \{x ∈ Ω : u(x) ∈ D\} \end{matrix}

Lemma 2.2. [21] Let F : ℝ → ℝ be uniformly lipschitzian, with F (0) = 0. We suppose that the set of discontinuity points of F′ is finite. Let M be an N-function, then the mapping F : W^{1}L_{M}(Ω) → W^{1}L_{M}(Ω) is sequentially continuous with respect to the weak* topology σ(ΠL_{M}, ΠE\overline{M}).

Let Ω be a bounded open subset of ℝ^{N}, T > 0 and set Q = Ω × (0, T). M be an N-function. For each α ∈ ℕ^{N}, denote by ∇_{x}^{α} the distributional derivative on Q of

EJQTDE, 2010 No. 2, p. 3 order α with respect to the variable x ∈ ℕ^{N}. The inhomogeneous Orlicz-Sobolev spaces are defined as follows,

(2.8)  \begin{matrix}W^{1,x}L_{M}(Q) = \{u ∈ L_{M}(Q) : ∇_{x}^{α}u ∈ L_{M}(Q) ∀ |α| ≤ 1\} \\ and W^{1,x}E_{M}(Q) = \{u ∈ E_{M}(Q) : ∇_{x}^{α}u ∈ E_{M}(Q) ∀ |α| ≤ 1\} \end{matrix}

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

(2.9)  ∥u∥ = ∑_{|α|≤1} ∥∇_{x}^{α}u∥_{M,Q}.

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) which have as many copies as there is α-order derivatives, |α| ≤ 1. We shall also consider the weak topologies σ(ΠL_{M}, ΠE\overline{M}) and σ(ΠL_{M}, ΠL\overline{M}). If u ∈ W^{1,x}L_{M}(Q) then the function : t ⟼ u(t) = u(t, .) is defined on (0, T) with values in W^{1}L_{M}(Ω). If, further, u ∈ W^{1,x}E_{M}(Q) then the concerned function is a W^{1}E_{M}(Ω)-valued and is strongly measurable. Furthermore the following imbedding holds: W^{1,x}E_{M}(Q) ⊂ L^{1}(0, T; W^{1}E_{M}(Ω)). The space W^{1,x}L_{M}(Q) is not in general separable, if u ∈ W^{1,x}L_{M}(Q), 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 W_{0}^{1,x}E_{M}(Q) is defined as the (norm) closure in W^{1,x}E_{M}(Q) of D(Q). We can easily show as in [22] that when Ω has the segment property, then each element u of the closure of D(Q) with respect of the weak * topology σ(ΠL_{M}, ΠE\overline{M}) is a limit, in W^{1,x}L_{M}(Q), of some subsequence (u_{i}) ⊂ D(Q) for the modular convergence; i.e., there exists λ > 0 such that for all |α| ≤ 1,

(2.10)  ∫_{Q} M(\frac{∇_{x}^{α}u_{i} − ∇_{x}^{α}u}{λ}) dx dt → 0 as i → ∞.

This implies that (u_{i}) converges to u in W^{1,x}L_{M}(Q) for the weak topology σ(ΠL_{M}, ΠL\overline{M}). Consequently,

(2.11)  \overline{D(Q)}^{σ(ΠL_{M} ,ΠE\overline{M})} = \overline{D(Q)}^{σ(ΠL_{M} ,ΠL\overline{M})}.

This space will be denoted by W_{0}^{1,x}L_{M}(Q). Furthermore, W_{0}^{1,x}E_{M}(Q) = W_{0}^{1,x}L_{M}(Q)∩ ΠE_{M}. Poincaré’s inequality also holds in W_{0}^{1,x}L_{M}(Q), i.e., there is a constant C > 0 such that for all u ∈ W_{0}^{1,x}L_{M}(Q) one has,

(2.12)  ∑_{|α|≤1} ∥∇_{x}^{α}u∥_{M,Q} ≤ C ∑_{|α|=1} ∥∇_{x}^{α}u∥_{M,Q}.

Thus both sides of the last inequality are equivalent norms on W_{0}^{1,x}L_{M}(Q). We have then the following complementary system

(2.13)  ( \begin{matrix}W_{0}^{1,x}L_{M}(Q) F \\ W_{0}^{1,x}E_{M}(Q) F_{0} \end{matrix} )

F being the dual space of W_{0}^{1,x}E_{M}(Q). It is also, except for an isomorphism, the quotient of ΠL\overline{M} by the polar set W_{0}^{1,x}E_{M}(Q)^{⊥}, and will be denoted by F =

EJQTDE, 2010 No. 2, p. 4 W^{−1,x}L\overline{M}(Q) and it is shown that,

(2.14)  W^{−1,x}L\overline{M}(Q) = \{f = ∑_{|α|≤1} ∇_{x}^{α}f_{α} : f_{α} ∈ L\overline{M}(Q)\}.

This space will be equipped with the usual quotient norm

(2.15)  ∥f∥ = inf ∑_{|α|≤1} ∥f_{α}∥_{\overline{M},Q}

where the infimum is taken on all possible decompositions

(2.16)  f = ∑_{|α|≤1} ∇_{x}^{α}f_{α}, f_{α} ∈ L\overline{M}(Q).

The space F_{0} is then given by,

(2.17)  F_{0} = \{f = ∑_{|α|≤1} ∇_{x}^{α}f_{α} : f_{α} ∈ E\overline{M}(Q)\}

and is denoted by F_{0} = W^{−1,x}E\overline{M}(Q).

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 of order 1 : W^{1,x}L_{M}(Q) and W_{0}^{1,x}L_{M}(Q).

3. Assumptions and statement of main results

Throughout this paper, we assume that the following assumptions hold true: Ω is a bounded open set on ℝ^{N} (N ≥ 2), T > 0 is given and we set Q = Ω × (0, T). Let M and P be two N-function such that P ≪ M.

(3.1) b : Ω × ℝ → ℝ is a Carathéodory function such that,

for every x ∈ Ω : b(x, s) is a strictly increasing C^{1}-function, with b(x, 0) = 0. For any K > 0, there exists λ_{K} > 0, a function A_{K} in L^{∞}(Ω) and a function B_{K} in L_{M}(Ω) such that

(3.2)  λ_{K} ≤ \frac{∂b(x, s)}{∂s} ≤ A_{K}(x) and |∇_{x}(\frac{∂b(x, s)}{∂s})| ≤ B_{K}(x),

for almost every x ∈ Ω, for every s such that |s| ≤ K.

Consider a second order partial differential operator A : D(A) ⊂ W^{1,x}L_{M}(Q) → W^{−1,x}L\overline{M}(Q) in divergence form,

A(u) = −div(a(x, t, u, ∇u))

where

(3.3) a : Ω × (0, T) × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function satisfying

for any K > 0, there exist β_{K}^{i} > 0 (for i = 1, 2, 3, 4) and a function C_{K} ∈ EM̄(Q) such that:

(3.4)  |a(x, t, s, ξ)| ≤ C_{K}(x, t) + β_{K}^{1} M̄^{−1}P (β_{K}^{2}|s|) + β_{K}^{3} M̄^{−1}M(β_{K}^{4}|ξ|)

EJQTDE, 2010 No. 2, p. 5 for almost every (x, t) ∈ Q and for every |s| ≤ K and for every ξ ∈ ℝ^{N}.

(3.5)  [a(x, t, s, ξ) − a(x, t, s, ξ^{∗})][ξ − ξ^{∗}] > 0
(3.6)  a(x, t, s, ξ)ξ ≥ αM(|ξ|)

for almost every (x, t) ∈ Q, for every s ∈ ℝ and for every ξ ≠ ξ^{∗} ∈ ℝ^{N}, where α > 0 is a given real number.

(3.7) Φ : ℝ → ℝ^{N} is a continuous function

(3.8) f is an element of L^{1}(Q).

(3.9)  u_{0} is an element of L^{1}(Ω) such that b(x, u_{0}) ∈ L^{1}(Ω).

Remark 3.1. As already mentioned in the introduction, problem (1.1)-(1.3) does not admit a weak solution under assumptions (3.1)-(3.9) (even when b(x, u) = u) since the growths of a(x, t, u, Du) and Φ(u) are not controlled with respect to u (so that these fields are not in general defined as distributions, even when u belongs to W_{0}^{1,x}L_{M}(Q).

4. Definition of a renormalized solution

The definition of a renormalized solution for problem (1.1)-(1.3) can be stated as follows.

Definition 4.1. A measurable function u defined on Q is a renormalized solution of Problem (1.1)-(1.3) if

(4.1)  T_{K}(u) ∈ W_{0}^{1,x}L_{M}(Q) ∀K ≥ 0 and b(x, u) ∈ L^{∞}(0, T; L^{1}(Ω)),

(4.2) ∫_{{(t,x)∈Q ; m≤|u(x,t)|≤m+1}} a(x, t, u, ∇u)∇u dx dt ⟶ 0 as m → +∞ ;

and if, for every function S in W^{2,∞}(ℝ), which is piecewise C^{1} and such that S′ has a compact support, we have

(4.3)  \frac{∂B_{S}(x, u)}{∂t} − div(S′(u)a(x, t, u, ∇u)) + S′′(u)a(x, t, u, ∇u)∇u
− div(S′(u)Φ(u)) + S′′(u)Φ(u)∇u = fS′(u) in D′(Q),

and

(4.4)  B_{S}(x, u)(t = 0) = B_{S}(x, u_{0}) in Ω,

where B_{S}(x, z) = ∫_{0}^{z} \frac{∂b(x, r)}{∂r}S′(r) dr.

The following remarks are concerned with a few comments on definition 4.1. EJQTDE, 2010 No. 2, p. 6 Remark 4.2. Equation (4.3) is formally obtained through pointwise multiplication of equation (1.1) by S′(u). Note that due to (4.1) each term in (4.3) has a meaning in L^{1}(Q) + W^{−1,x}L\overline{M}(Q).

Indeed, if K is such that suppS′ ⊂ [−K, K], the following identifications are made in (4.3).

⋆ B_{S}(x, u) ∈ L^{∞}(Q), because |B_{S}(x, u)| ≤ K∥A_{K}∥_{L^{∞}(Ω)}∥S′∥_{L^{∞}(ℝ)}.

⋆ S′(u)a(x, t, u, ∇u) identifies with S′(u)a(x, t, T_{K}(u), ∇T_{K}(u)) a.e. in Q. Since indeed |T_{K}(u)| ≤ K a.e. in Q. Since S′(u) ∈ L^{∞}(Q) and with (3.4), (4.1) we obtain that

S(u)a(x, t, T_{K}(u), ∇T_{K}(u)) ∈ (L\overline{M}(Q))^{N}.

⋆ S′(u)a(x, t, u, ∇u)∇u identifies with S′(u)a(x, t, T_{K}(u), ∇T_{K}(u))∇T_{K}(u) and in view of (3.2) and (4.1) one has

S′(u)a(x, t, T_{K}(u), ∇T_{K}(u))∇T_{K}(u) ∈ L^{1}(Q).

⋆ S′(u)Φ(u) and S′′(u)Φ(u)∇u respectively identify with S′(u)Φ(T_{K}(u)) and S′′(u)Φ(T_{K}(u))∇T_{K}(u). Due to the properties of S and (3.7), the functions S′, S′′ and Φ∘T_{K} are bounded on ℝ so that (4.1) implies that S′(u)Φ(T_{K}(u)) ∈ (L^{∞}(Q))^{N}, and S′′(u)Φ(T_{K}(u))∇T_{K}(u) ∈ (L_{M}(Q))^{N}.

The above considerations show that equation (4.3) takes place in D′(Q) and that

(4.5)  \frac{∂B_{S}(x, u)}{∂t} belongs to W^{−1,x}L\overline{M}(Q) + L^{1}(Q).

Due to the properties of S and (3.2), we have

(4.6)  |∇B_{S}(x, u)| ≤ ∥A_{K}∥_{L^{∞}(Ω)}|∇T_{K}(u)|∥S′∥_{L^{∞}(Ω)} + K∥S′∥_{L^{∞}(Ω)}B_{K}(x)

and

(4.7)  B_{S}(x, u) belongs to W_{0}^{1,x}L_{M}(Q).

Moreover (4.5) and (4.7) implies that B_{S}(x, u) belongs to C^{0}([0, T]; L^{1}(Ω)) (for a proof of this trace result see [30]), so that the initial condition (4.4) makes sense.

Remark 4.3. For every S ∈ W^{2,∞}(ℝ), nondecreasing function such that suppS′ ⊂ [−K, K] and (3.2), we have

(4.8) λ_{K}|S(r) − S(r′)| ≤ |B_{S}(x, r) − B_{S}(x, r′)| ≤ ∥A_{K}∥_{L^{∞}(Ω)}|S(r) − S(r′)|

for almost every x ∈ Ω and for every r, r′ ∈ ℝ.

5. Existence result

This section is devoted to establish the following existence theorem.

Theorem 5.1. Under assumption (3.1)-(3.9) there exists at at least a renormalized solution of Problem (1.1)-(1.3).

Proof. The proof is divided into 5 steps. □ EJQTDE, 2010 No. 2, p. 7 ⋆ Step 1. For n ∈ ℕ^{∗}, let us define the following approximations of the data:

(5.1)  b_{n}(x, r) = b(x, T_{n}(r)) + \frac{1}{n} r a.e. in Ω, ∀s ∈ ℝ,
(5.2)  a_{n}(x, t, r, ξ) = a(x, t, T_{n}(r), ξ) a.e. in Q, ∀s ∈ ℝ, ∀ξ ∈ ℝ^{N},
(5.3)  Φ_{n} is a Lipschitz continuous bounded function from ℝ into ℝ^{N},

such that Φ_{n} uniformly converges to Φ on any compact subset of ℝ as n tends to +∞.

(5.4) f_{n} ∈ C_{0}^{∞}(Q) : ∥f_{n}∥_{L^{1}} ≤ ∥f∥_{L^{1}} and f_{n} ⟶ f in L^{1}(Q) as n tends to +∞,

(5.5) u_{0n} ∈ C_{0}^{∞}(Ω) : ∥b_{n}(x, u_{0n})∥_{L^{1}} ≤ ∥b(x, u_{0})∥_{L^{1}} and b_{n}(x, u_{0n}) ⟶ b(x, u_{0}) in L^{1}(Ω)

as n tends to +∞.

Let us now consider the following regularized problem:

(5.6)  \frac{∂b_{n}(x, u_{n})}{∂t} − div(a_{n}(x, t, u_{n}, ∇u_{n}) + Φ_{n}(u_{n})) = f_{n} in Q,

u_{n} = 0 on (0, T) × ∂Ω, (5.7)

b_{n}(x, u_{n})(t = 0) = b_{n}(x, u_{0n}) in Ω. (5.8)

As a consequence, proving existence of a weak solution u_{n} ∈ W_{0}^{1,x}L_{M}(Q) of (5.6)- (5.8) is an easy task (see e.g. [25], [33]).

⋆ Step 2. The estimates derived in this step rely on usual techniques for problems of the type (5.6)-(5.8).

Proposition 5.2. Assume that (3.1)-(3.9) hold true and let u_{n} be a solution of the approximate problem (5.6) − (5.8). Then for all K, n > 0, we have

(5.9)  ∥T_{K}(u_{n})∥_{W_{0}^{1,x}L_{M} (Q)} ≤ K(∥f∥_{L^{1}(Q)} + ∥b(x, u_{0})∥_{L^{1}(Ω)}) ≡ CK,

where C is a constant independent of n.

(5.10)  ∫_{Ω} B_{K}^{n}(x, u_{n})(τ) dx ≤ K(∥f∥_{L^{1}(Q)} + ∥b(x, u_{0})∥_{L^{1}(Ω)}) ≡ CK,

for almost any τ in (0, T), and where B_{K}^{n}(x, r) = ∫_{0}^{r} T_{K}(s)\frac{∂b_{n}(x, s)}{∂s} ds.

(5.11) lim_{K→∞} meas{(x, t) ∈ Q : |u_{n}| > K} = 0 uniformly with respect to n.

Proof. We take T_{K}(u_{n})_{χ(0,τ)} as test function in (5.6), we get for every τ ∈ (0, T) (5.12)

\begin{matrix}⟨\frac{∂b_{n}(x, u_{n})}{∂t}, T_{K}(u_{n})_{χ(0,τ)}⟩ + ∫_{Q_{τ}} a_{n}(x, t, T_{K}(u_{n}), ∇T_{K}(u_{n}))∇T_{K}(u_{n}) dx dt \\ + ∫_{Q_{τ}} Φ_{n}(u_{n})∇T_{K}(u_{n}) dx dt = ∫_{Q_{τ}} f_{n}T_{K}(u_{n}) dx dt, \end{matrix}

which implies that, (5.13)

\begin{matrix}∫_{Ω} B_{K}^{n}(x, u_{n})(τ) dx + ∫_{Q_{τ}} a_{n}(x, t, T_{K}(u_{n}), ∇T_{K}(u_{n}))∇T_{K}(u_{n}) dx dt \\ + ∫_{Q_{τ}} Φ_{n}(u_{n})∇T_{K}(u_{n}) dx dt = ∫_{Q_{τ}} f_{n}T_{K}(u_{n}) dx dt + ∫_{Ω} B_{K}^{n}(x, u_{0n}) dx \end{matrix}

where, B_{K}^{n}(x, r) = ∫_{0}^{r} T_{K}(s)\frac{∂b_{n}(x, s)}{∂s} ds.

The Lipschitz character of Φ_{n}, Stokes formula together with the boundary condition (5.7), make it possible to obtain

(5.14)  ∫_{Q_{τ}} Φ_{n}(u_{n})∇T_{K}(u_{n}) dx dt = 0.

Due to the definition of B_{K}^{n} we have,

(5.15)  0 ≤ ∫_{Ω} B_{K}^{n}(x, u_{0n}) dx ≤ K ∫_{Ω} |b_{n}(x, u_{0n})| dx ≤ K∥b(x, u_{0})∥_{L^{1}(Ω)}.

By using (5.14), (5.15) and the fact that B_{K}^{n}(x, u_{n}) ≥ 0, permit to deduce from (5.13) that (5.16)

∫_{Q} a_{n}(x, t, T_{K}(u_{n}), ∇T_{K}(u_{n}))∇T_{K}(u_{n}) dx dt ≤ K(∥f_{n}∥_{L^{1}(Q)}+∥b_{n}(x, u_{0n})∥_{L^{1}(Ω)}) ≤ CK,

which implies by virtue of (3.6), (5.4) and (5.5) that,

(5.17)  ∫_{Q} M(∇T_{K}(u_{n})) dx dt ≤ K(∥f∥_{L^{1}(Q)} + ∥b(x, u_{0})∥_{L^{1}(Ω)}) ≡ CK.

We deduce from that above inequality (5.13) and (5.15) that

(5.18)  ∫_{Ω} B_{K}^{n}(x, u_{n})(τ) dx ≤ (∥f∥_{L^{1}(Q)} + ∥b(x, u_{0})∥_{L^{1}(Ω)}) ≡ CK.

for almost any τ in (0, T).

We prove (5.11). Indeed, thanks to lemma 5.7 of [21], there exist two positive constants δ, λ such that,

(5.19)  ∫_{Q} M(v) dx dt ≤ δ ∫_{Q} M(λ|∇v|) dx dt for all v ∈ W_{0}^{1,x}L_{M}(Q).

Taking v = \frac{T_{K}(u_{n})}{λ} in (5.19) and using (5.17), one has

(5.20)  ∫_{Q} M(\frac{T_{K}(u_{n})}{λ}) dx dt ≤ CK,

where C is a constant independent of K and n. Which implies that,

(5.21)  meas\{(x, t) ∈ Q : |u_{n}| > K\} ≤ \frac{C′K}{M(\frac{K}{λ})}.

where C′ is a constant independent of K and n. Finally,

lim_{K→∞} meas{(x, t) ∈ Q : |u_{n}| > K} = 0 uniformly with respect to n.

□

We prove de following proposition:

Proposition 5.3. Let u_{n} be a solution of the approximate problem (5.6)-(5.8), then

(5.22)  u_{n} → u a.e. in Q,
(5.23)  b_{n}(x, u_{n}) → b(x, u) a.e. in Q,
(5.24)  b(x, u) ∈ L^{∞}(0, T; L^{1}(Ω)),
(5.25) a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n})) ⇀ φ_{k} in (L\overline{M}(Q))^{N} for σ(ΠL\overline{M}, ΠE_{M})

for some φ_{k} ∈ (L\overline{M}(Q))^{N}.

(5.26)  lim_{m→+∞} lim_{n→+∞} ∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt = 0.

Proof. Proceeding as in [5, 9, 7], we have for any S ∈ W^{2,∞}(ℝ) such that S′ has a compact support (supp S′ ⊂ [−K, K])

(5.27)  B_{S}^{n}(x, u_{n}) is bounded in W_{0}^{1,x}L_{M}(Q),

and

(5.28)  \frac{∂B_{S}^{n}(x, u_{n})}{∂t} is bounded in L^{1}(Q) + W^{−1,x}L\overline{M}(Q),

independently of n.

As a consequence of (4.6) and (5.17) we then obtain (5.27). To show that (5.28) holds true, we multiply the equation for u_{n} in (5.6) by S′(u_{n}) to obtain

(5.29)  \frac{∂B_{S}^{n}(x, u_{n})}{∂t} = div(S′(u_{n})a_{n}(t, x, u_{n}, ∇u_{n}))

−S′′(u_{n})a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} + div(S′(u_{n})Φ_{n}(u_{n})) + f_{n}S′(u_{n}) in D′(Q).

Where B_{S}^{n}(x, r) = ∫_{0}^{r} S′(s)\frac{∂b_{n}(x, s)}{∂s} ds. Since supp S′ and supp S′′ are both

included in [−K, K], u^{ε} may be replaced by T_{K}(u_{n}) in each of these terms. As a consequence, each term in the right hand side of (5.29) is bounded either in W^{−1,x}L\overline{M}(Q) or in L^{1}(Q). As a consequence of (3.2), (5.29) we then obtain (5.28). Arguing again as in [5, 7, 6, 9] estimates (5.27), (5.28) and (4.8), we can show (5.22) and (5.23).

We now establish that b(x, u) belongs to L^{∞}(0, T; L^{1}(Ω)). To this end, recalling (5.23) makes it possible to pass to the limit-inf in (5.18) as n tends to +∞ and to obtain

\frac{1}{K} ∫_{Ω} B_{K}(x, u)(τ) dx ≤ (∥f∥_{L^{1}(Q)} + ∥b(x, u_{0})∥_{L^{1}(Ω)}) ≡ C,

EJQTDE, 2010 No. 2, p. 10 for almost any τ in (0, T). Due to the definition of B_{K}(x, s), and because of the pointwise convergence of \frac{1}{K} B_{K}(x, u) to b(x, u) as K tends to +∞, which shows that b(x, u) belongs to L^{∞}(0, T; L^{1}(Ω)).

We prove (5.25). Let φ ∈ (E_{M}(Q))^{N} with ∥φ∥_{M.Q} = 1. In view of the monotonicity of a one easily has, (5.30)

∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))φ dx dt ≤ ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n}) dx dt

+ ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), φ)[∇T_{k}(u_{n}) − φ] dx dt.

and (5.31)

− ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))φ dx dt ≤ ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n}) dx dt

− ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), −φ)[∇T_{k}(u_{n}) + φ] dx dt,

since T_{k}(u_{n}) is bounded in W_{0}^{1,x}L_{M}(Q), one easily deduce that a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))

is a bounded sequence in (L\overline{M}(Q))^{N}, and we obtain (5.25).

Now we prove (5.26). We take of T_{1}(u_{n} − T_{m}(u_{n})) as test function in (5.6), we obtain

(5.32) ⟨\frac{∂b_{n}(x, u_{n})}{∂t}, T_{1}(u_{n} −T_{m}(u_{n}))⟩+∫_{\{m≤|u_{n}|≤m+1\}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt

+ ∫_{Q} div [∫_{0}^{u_{n}} Φ(r)T′_{1}(r − T_{m}(r))] dx dt = ∫_{Q} f_{n}T_{1}(u_{n} − T_{m}(u_{n})) dx dt.

Using the fact that ∫_{0}^{u_{n}} Φ(r)T′_{1}(r−T_{m}(r)) dx dt ∈ W_{0}^{1,x}L_{M}(Q) and Stokes formula,

we get

(5.33)  ∫_{Ω} B_{n}^{m}(x, u_{n}(T)) dx + ∫_{{m≤|u_{n}|≤m+1}} a(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt

≤ ∫_{Q} |f_{n}T_{1}(u_{n} − T_{m}(u_{n}))| dx dt + ∫_{Ω} B_{n}^{m}(x, u_{0n}) dx,

where B_{n}^{m}(x, r) = ∫_{0}^{r} \frac{∂b_{n}(x, s)}{∂s}T_{1}(s − T_{m}(s)) ds.

In order to pass to the limit as n tends to +∞ in (5.33), we use B_{n}^{m}(x, u_{n}(T)) ≥ 0 and (5.4)-(5.5), we obtain that

(5.34)  lim_{n→+∞} ∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt

≤ ∫_{{|u|>m}} |f| dx dt + ∫_{{|u_{0}|>m}} |b(x, u_{0})| dx.

Finally by (3.8), (3.9) and (5.34) we obtain (5.26).

□

EJQTDE, 2010 No. 2, p. 11 ⋆ Step 3. This step is devoted to introduce for K ≥ 0 fixed, a time regularization w_{μ,j}^{i} of the function T_{K}(u) and to establish the following proposition:

Proposition 5.4. Let u_{n} be a solution of the approximate problem (5.6)-(5.8). Then, for any k ≥ 0:

(5.35)  ∇T_{k}(u_{n}) → ∇T_{k}(u) a.e. in Q,

(5.36)

a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n})) ⇀ a(x, t, T_{k}(u), ∇T_{k}(u)) weakly in (L\overline{M}(Q))^{N},

(5.37)  M(|∇T_{k}(u_{n})|) → M(|∇T_{k}(u)|) strongly in L^{1}(Q),

as n tends to +∞.

Let use give the following lemma which will be needed later:

Lemma 5.5. Under assumptions (3.1) − (3.9), and let (z_{n}) be a sequence in W_{0}^{1,x}L_{M}(Q) such that,

(5.38)  z_{n} ⇀ z in W_{0}^{1,x}L_{M}(Q) for σ(ΠL_{M}(Q), ΠE\overline{M}(Q)),
(5.39)  (a_{n}(x, t, z_{n}, ∇z_{n}))_{n} is bounded in (L\overline{M}(Q))^{N},
(5.40) ∫_{Q} [a_{n}(x, t, z_{n}, ∇z_{n}) − a_{n}(x, t, z_{n}, ∇zχ_{s})][∇z_{n} − ∇zχ_{s}] dx dt ⟶ 0,

as n and s tend to +∞, and where χ_{s} is the characteristic function of

Q_{s} = {(x, t) ∈ Q ; |∇z| ≤ s}.

Then,

(5.41)  ∇z_{n} → ∇z a.e. in Q,
(5.42)  lim_{n→∞} ∫_{Q} a_{n}(x, t, z_{n}, ∇z_{n})∇z_{n} dx dt = ∫_{Q} a(x, t, z, ∇z)∇z dx dt,

(5.43)  M(|∇z_{n}|) → M(|∇z|) in L^{1}(Q).

Proof. See [32]. □

Proof. (Proposition 5.4). The proof is almost identical of the one given in, e.g. [32]. where the result is established for b(x, u) = u and where the growth of a(x, t, u, Du) is controlled with respect to u. This proof is devoted to introduce for k ≥ 0 fixed, a time regularization of the function T_{k}(u), this notion, introduced by R. Landes (see Lemma 6 and Proposition 3, p. 230 and Proposition 4, p. 231 in [24]). More recently, it has been exploited in [10] and [15] to solve a few nonlinear evolution problems with L^{1} or measure data.

Let v_{j} ∈ D(Q) be a sequence such that v_{j} → u in W_{0}^{1,x}L_{M}(Q) for the modular convergence and let ψ_{i} ∈ D(Ω) be a sequence which converges strongly to u_{0} in L^{1}(Ω).

EJQTDE, 2010 No. 2, p. 12

Let w_{i,j}^{μ} = T_{k}(v_{j})_{μ}+e^{−μt}T_{k}(ψ_{i}) where T_{k}(v_{j})_{μ} is the mollification with respect to time of T_{k}(v_{j}), note that w_{i,j}^{μ} is a smooth function having the following properties:

(5.44)  \frac{∂w_{i,j}^{μ}}{∂t} = μ(T_{k}(v_{j}) − w_{i,j}^{μ}), w_{i,j}^{μ}(0) = T_{k}(ψ_{i}), |w_{i,j}^{μ}| ≤ k,
(5.45)  w_{i,j}^{μ} → T_{k}(u)_{μ} + e^{−μt}T_{k}(ψ_{i}) in W_{0}^{1,x}L_{M}(Q),

for the modular convergence as j → ∞.

(5.46)  T_{k}(u)_{μ} + e^{−μt}T_{k}(ψ_{i}) → T_{k}(u) in W_{0}^{1,x}L_{M}(Q),

for the modular convergence as μ → ∞.

Let now the function h_{m} defined on ℝ with m ≥ k by: h_{m}(r) = 1 if |r| ≤ m, h(r) = −|r| + m + 1 if m ≤ |r| ≤ m + 1 and h(r) = 0 if |r| ≥ m + 1.

Using the admissible test function φ_{n,j,m}^{μ,i} = (T_{k}(u_{n}) − w_{i,j}^{μ})h_{m}(u_{n}) as test function in (5.6) leads to

(5.47) ⟨\frac{∂b_{n}(x, u_{n})}{∂t}, φ_{n,j,m}^{μ,i}⟩ + ∫_{Q} a_{n}(x, t, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt

+ ∫_{Q} a_{n}(x, t, u_{n}, ∇u_{n})(T_{k}(u_{n}) − w_{i,j}^{μ})∇u_{n}h′_{m}(u_{n}) dx dt
+ ∫_{{m≤|u_{n}|≤m+1}} Φ_{n}(u_{n})∇u_{n}h′_{m}(u_{n})(T_{k}(u_{n}) − w_{i,j}^{μ}) dx dt
+ ∫_{Q} Φ_{n}(u_{n})h_{m}(u_{n})(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ}) dx dt = ∫_{Q} f_{n}φ_{n,j,m}^{μ,i} dx dt.

Denoting by ϵ(n, j, μ, i) any quantity such that,

lim_{i→∞} lim_{μ→∞} lim_{j→∞} lim_{n→∞} ϵ(n, j, μ, i) = 0.

The very definition of the sequence w_{i,j}^{μ} makes it possible to establish the following lemma.

Lemma 5.6. Let φ_{n,j,m}^{μ,i} = (T_{k}(u_{n}) − w_{i,j}^{μ})h_{m}(u_{n}), we have for any k ≥ 0:

(5.48)  ⟨\frac{∂b_{n}(x, u_{n})}{∂t}, φ_{n,j,m}^{μ,i}⟩ ≥ ϵ(n, j, μ, i),

where ⟨, ⟩ denotes the duality pairing between L^{1}(Q) + W^{−1,x}L\overline{M}(Q) and L^{∞}(Q) ∩ W_{0}^{1,x}L_{M}(Q).

Proof. See [34, 32]. □

Now, we turn to complete the proof of proposition 5.4. First, it is easy to see that (see also [32]):

(5.49)  ∫_{Q} f_{n}φ_{n,j,m}^{μ,i} dx dt = ϵ(n, j, μ),

(5.50)  ∫_{Q} Φ_{n}(u_{n})h_{m}(u_{n})(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ}) dx dt = ϵ(n, j, μ),

and

(5.51)  ∫_{{m≤|u_{n}|≤m+1}} Φ_{n}(u_{n})∇u_{n}(T_{k}(u_{n}) − w_{i,j}^{μ}) dx dt = ϵ(n, j, μ).

Concerning the third term of the right hand side of (5.47) we obtain that

(5.52)  ∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n}h′_{m}(u_{n})(T_{k}(u_{n}) − w_{i,j}^{μ}) dx dt

≤ 2k ∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt.

Then by (5.26). we deduce that, (5.53)

∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n}h′_{m}(u_{n})(T_{k}(u_{n}) − w_{i,j}^{μ}) dx dt ≤ ϵ(n, μ, m).

Finally, by means of (5.47)-(5.53), we obtain,

(5.54)  ∫_{Q} a_{n}(x, t, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt ≤ ϵ(n, j, μ, m).

Splitting the first integral on the left hand side of (5.54) where |u_{n}| ≤ k and |u_{n}| > k, we can write,

∫_{Q} a_{n}(x, t, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt
= ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt
− ∫_{{|u_{n}|>k}} a_{n}(x, t, u_{n}, ∇u_{n})∇w_{i,j}^{μ}h_{m}(u_{n}) dx dt.

Since h_{m}(u_{n}) = 0 if |u_{n}| ≥ m + 1, one has

(5.55)  ∫_{Q} a_{n}(x, t, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt

= ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt
− ∫_{{|u_{n}|>k}} a_{n}(x, t, T_{m+1}(u_{n}), ∇T_{m+1}(u_{n}))∇w_{i,j}^{μ}h_{m}(u_{n}) dx dt = I_{1} + I_{2}

In the following we pass to the limit in (5.55) as n tends to +∞, then j then μ and then m tends to +∞. We prove that

I_{2} = ∫_{Q} φ_{m}∇T_{k}(u)_{μ}h_{m}(u)_{χ_{{|u|>k}}} dx dt + ϵ(n, j, μ).

Using now the term I_{1} of (5.55), we conclude that, it is easy to show that,

(5.56)  ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u_{n}) − ∇w_{i,j}^{μ})h_{m}(u_{n}) dx dt

= ∫_{Q} [a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(v_{j})χ_{j}^{s})]

×[∇T_{k}(u_{n}) − ∇T_{k}(v_{j})χ_{j}^{s}]h_{m}(u_{n}) dx dt
+ ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(v_{j})χ_{j}^{s})[∇T_{k}(u_{n}) − ∇T_{k}(v_{j})χ_{j}^{s}]h_{m}(u_{n}) dx dt
+ ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(v_{j})χ_{j}^{s}h_{m}(u_{n}) dx dt
− ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇w_{i,j}^{μ}h_{m}(u_{n}) dx dt = J_{1} + J_{2} + J_{3} + J_{4},

where χ_{j}^{s} denotes the characteristic function of the subset

Ω_{s}^{j} = {(x, t) ∈ Q : |∇T_{k}(v_{j})| ≤ s}

In the following we pass to the limit in (5.56) as n tends to +∞, then j then μ then m tends and then s tends to +∞ in the last three integrals of the last side. We prove that

(5.57)  J_{2} = ϵ(n, j),
(5.58)  J_{3} = ∫_{Q} φ_{k}∇T_{k}(u)χ_{s} dx dt + ϵ(n, j),

and

(5.59)  J_{4} = − ∫_{Q} φ_{k}∇T_{k}(u) dx dt + ϵ(n, j, μ, s).

We conclude then that, (5.60)

∫_{Q} [a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))−a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})][∇T_{k}(u_{n})−∇T_{k}(u)χ_{s}] dx dt

= ∫_{Q} [a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})]
×[∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}]h_{m}(u_{n}) dx dt

+ ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)χ^{s}](1 − h_{m}(u_{n})) dx dt

− ∫_{Q} a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})[∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}](1 − h_{m}(u_{n})) dx dt.

Combining (5.48), (5.56), (5.57), (5.58), (5.59) and (5.60) we deduce, (5.61)

∫_{Q} [a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n}))−a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})][∇T_{k}(u_{n})−∇T_{k}(u)χ_{s}] dx dt

≤ ϵ(n, j, μ, m, s).

To pass to the limit in (5.61) as n, j, m, s tends to infinity, we obtain

(5.62) lim_{s→∞} lim_{n→∞} ∫_{Q} [a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a_{n}(x, t, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})]
×[∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx dt = 0.

EJQTDE, 2010 No. 2, p. 15 This implies by the lemma 5.5, the desired statement and hence the proof of Proposition 5.4 is achieved. □

⋆ Step 4. In this step we prove that u satisfies (4.2).

Lemma 5.7. The limit u of the approximate solution u_{n} of (5.6)-(5.8) satisfies

(5.63)  lim_{m→+∞} ∫_{{m≤|u|≤m+1}} a(x, t, u, ∇u)∇u dx dt = 0.

Proof. Remark that for any fixed m ≥ 0 one has

∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt

= ∫_{Q} a_{n}(x, t, u_{n}, ∇u_{n})[∇T_{m+1}(u_{n}) − ∇T_{m}(u_{n})] dx dt
= ∫_{Q} a_{n}(x, t, T_{m+1}(u_{n}), ∇T_{m+1}(u_{n}))∇T_{m+1}(u_{n}) dx dt
− ∫_{Q} a_{n}(x, t, T_{m}(u_{n}), ∇T_{m}(u_{n}))∇T_{m}(u_{n}) dx dt

According to (5.42) (with z_{n} = T_{m}(u_{n}) or z_{n} = T_{m+1}(u_{n})), one is at liberty to pass to the limit as n tends to +∞ for fixed m ≥ 0 and to obtain

(5.64)  lim_{n→+∞} ∫_{{m≤|u_{n}|≤m+1}} a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} dx dt

= ∫_{Q} a(x, t, T_{m+1}(u), ∇T_{m+1}(u))∇T_{m+1}(u) dx dt
− ∫_{Q} a(x, t, T_{m}(u), ∇T_{m}(u))∇T_{m}(u) dx dt
= ∫_{{m≤|u|≤m+1}} a(x, t, u, ∇u)∇u dx dt

Taking the limit as m tends to +∞ in (5.64) and using the estimate (5.26) it possible to conclude that (5.63) holds true and the proof of Lemma 5.7 is complete. □

⋆ Step 5. In this step, u is shown to satisfies (4.3) and (4.4). Let S be a function in W^{2,∞}(ℝ) such that S′ has a compact support. Let K be a positive real number such that supp(S′) ⊂ [−K, K]. Pointwise multiplication of the approximate equation (5.6) by S′(u_{n}) leads to

(5.65) \frac{∂B_{S}^{n}(x, u_{n})}{∂t} − div(S′(u_{n})a_{n}(x, t, u_{n}, ∇u_{n})) + S′′(u_{n})a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n}
− div(S′(u_{n})Φ(u_{n})) + S′′(u_{n})Φ(u_{n})∇u_{n} = fS′(u_{n}) in D′(Q),

where B_{S}^{n}(x, z) = ∫_{0}^{z} S′(r)\frac{∂b_{n}(x, r)}{∂r} dr.

It what follows we pass to the limit as n tends to +∞ in each term of (5.65). EJQTDE, 2010 No. 2, p. 16 ⋆ Since S′ is bounded, and B_{S}^{n}(x, u_{n}) converges to B_{S}(x, u) a.e. in Q and in L^{∞}(Q) weak ⋆. Then \frac{∂B_{S}^{n}(x,u_{n})}{∂t} converges to \frac{∂B_{S}(x,u)}{∂t} in D′(Q) as n tends to +∞.

⋆ Since suppS ⊂ [−K, K], we have

S′(u_{n})a_{n}(x, t, u_{n}, ∇u_{n}) = S′(u_{n})a_{n}(x, t, T_{K}(u_{n}), ∇T_{K}(u_{n})) a.e. in Q.

The pointwise convergence of u_{n} to u as n tends to +∞, the bounded character of S′, (5.22) and (5.36) of Lemma 5.4 imply that

S′(u_{n})a_{n}(x, t, T_{K}(u_{n}), ∇T_{K}(u_{n})) ⇀ S′(u)a(x, t, T_{K}(u), ∇T_{K}(u)) weakly in (L\overline{M}(Q))^{N},

for σ(ΠL\overline{M}, ΠE_{M}) as n tends to +∞, because S(u) = 0 for |u| ≥ K a.e. in Q. And the term S′(u)a(x, t, T_{K}(u), ∇T_{K}(u)) = S′(u)a(x, t, u, ∇u) a.e. in Q.

⋆ Since suppS′ ⊂ [−K, K], we have

S′′(u_{n})a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} = S′′(u_{n})a_{n}(x, t, T_{K}(u_{n}), ∇T_{K}(u_{n}))∇T_{K}(u_{n}) a.e. in Q.

The pointwise convergence of S′′(u_{n}) to S′′(u) as n tends to +∞, the bounded character of S′′ and (5.22)-(5.36) of Lemma 5.4 allow to conclude that

S′(u_{n})a_{n}(x, t, u_{n}, ∇u_{n})∇u_{n} ⇀ S′(u)a(x, t, T_{K}(u), ∇T_{K}(u))∇T_{K}(u) weakly in L^{1}(Q),

as n tends to +∞. And

S′′(u)a(x, t, T_{K}(u), ∇T_{K}(u))∇T_{K}(u) = S′′(u)a(x, t, u, ∇u)∇u a.e. in Q.

⋆ Since suppS′ ⊂ [−K, K], we have S′(u_{n})Φ_{n}(u_{n}) = S′(u_{n})Φ_{n}(T_{K}(u_{n})) a.e. in Q. As a consequence of (3.7), (5.3) and (5.22), it follows that:

S′(u_{n})Φ_{n}(u_{n}) → S′(u)Φ(T_{K}(u)) strongly in (E_{M}(Q))^{N},

as n tends to +∞. The term S′(u)Φ(T_{K}(u)) is denoted by S′(u)Φ(u).

⋆ Since S ∈ W^{1,∞}(ℝ) with suppS′ ⊂ [−K, K], we have S′′(u_{n})Φ_{n}(u_{n})∇u_{n} = Φ_{n}(T_{K}(u_{n}))∇S′′(u_{n}) a.e. in Q, we have, ∇S′′(u_{n}) converges to ∇S′′(u) weakly in L_{M}(Q)^{N} as n tends to +∞, while Φ_{n}(T_{K}(u_{n})) is uniformly bounded with respect to n and converges a.e. in Q to Φ(T_{K}(u)) as n tends to +∞. Therefore

S′′(u_{n})Φ_{n}(u_{n})∇u_{n} ⇀ Φ(T_{K}(u))∇S′′(u) weakly in L_{M}(Q).

⋆ Due to (5.4) and (5.22), we have f_{n}S(u_{n}) converges to fS(u) strongly in L^{1}(Q), as n tends to +∞.

As a consequence of the above convergence result, we are in a position to pass to the limit as n tends to +∞ in equation (5.65) and to conclude that u satisfies (4.3).

It remains to show that B_{S}(x, u) satisfies the initial condition (4.4). To this end, firstly remark that, S′ has a compact support, we have B_{S}^{n}(x, u_{n}) is bounded in L^{∞}(Q). Secondly, (5.65) and the above considerations on the behavior of the terms

EJQTDE, 2010 No. 2, p. 17 of this equation show that \frac{∂B_{S}^{n}(x, u_{n})}{∂t} is bounded in L^{1}(Q) + W^{−1,x}L\overline{M}(Q). As a consequence, an Aubin’s type Lemma (see e.g., [36], Corollary 4) (see also [16]) implies that B_{S}^{n}(x, u^{n}) lies in a compact set of C^{0}([0, T]; L^{1}(Ω)). It follows that, B_{S}^{n}(x, u_{n})(t = 0) converges to B_{S}(x, u)(t = 0) strongly in L^{1}(Ω). Due to (4.8) and (5.5), we conclude that B_{S}^{n}(x, u_{n})(t = 0) = B_{S}^{n}(x, u_{0n}) converges to B_{S}(x, u)(t = 0) strongly in L^{1}(Ω). Then we conclude that

B_{S}(x, u)(t = 0) = B_{S}(x, u_{0}) in Ω.

As a conclusion of step 1 to step 5, the proof of theorem 5.1 is complete.

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(Received June 11, 2009)

Faculté des Sciences Juridiques, Économiques et Sociales. Université Hassan 1, B.P.

764. Settat. Morocco

   E-mail address: redwane hicham@yahoo.fr