2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 203–220. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Strongly nonlinear parabolic initial-boundary value problems in Orlicz spaces ^{∗}

Abdelhak Elmahi

Abstract

We prove existence and convergence theorems for nonlinear parabolic problems. We also prove some compactness results in inhomogeneous Orlicz-Sobolev spaces.

1 Introduction

Let Ω be a bounded domain in ℝ^{N}, T > 0 and let

A(u) = ∑_{|α|≤1}(−1)^{|α|}D^{α}A_{α}(x, t, u, ∇u)

be a Leray-Lions operator defined on L^{p}(0, T; W^{1,p}(Ω)), 1 < p < ∞. Boccardo and Murat [5] proved the existence of solutions for parabolic initial-boundary value problems of the form

\frac{∂u}{∂t} + A(u) + g(x, t, u, ∇u) = f in Ω × (0, T),  (1.1)

where g is a nonlinearity with the following growth condition

g(x, t, s, ξ) ≤ b(|s|)(c(x, t) + |ξ|^{q}), q < p,  (1.2)

and which satisfies the classical sign condition g(x, t, s, ξ)s ≥ 0. The right hand side f is assumed (in [5]) to belong to L^{p′}(0, T; W^{−1,p′}(Ω)). This result generalizes the analogous one of Landes-Mustonen [14] where the nonlinearity g depends only on x, t and u. In [5] and [14], the functions A_{α} are assumed to satisfy a polynomial growth condition with respect to u and ∇u. When trying to relax this restriction on the coefficients A_{α}, we are led to replace L^{p}(0, T; W^{1,p}(Ω)) by an inhomogeneous Sobolev space W^{1,x}L_{M} built from an Orlicz space L_{M} instead of L^{p}, where the N-function M which defines L_{M} is related to the actual growth of the A_{α}’s. The solvability of (1.1) in this setting is proved by Donaldson [7] and Robert [16] in the case where g ≡ 0. It is our purpose in this paper, to prove existence theorems in the setting of the inhomogeneous Sobolev space W^{1,x}L_{M} by applying some new compactness results in Orlicz spaces obtained under the assumption that the N- function M(t) satisfies Δ′-condition and which grows less rapidly than |t|^{N/(N−1)}. These compactness results, which we are at first established in [8], generalize those of Simon [17], Landes-Mustonen [14] and Boccardo-Murat [6]. It is not clear whether the present approach can be further adapted to obtain the same results for general N-functions.

∗Mathematics Subject Classifications: 35K15, 35K20, 35K60.

Key words: Orlicz-Sobolev spaces, compactness, parabolic equations.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

For related topics in the elliptic case, the reader is referred to [2] and [3].

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(τ)dτ 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}(τ)dτ, where \overline{a} : ℝ^{+} → ℝ^{+} is given by \overline{a}(t) = sup\{s : a(s) ≤ t\} [1, 11, 12].

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

M(2t) ≤ k M(t) for all t ≥ 0,  (2.1)

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,

\frac{P (t)}{Q(ε t)} → 0 as t → ∞.

This is the case if and only if

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

An N-function is said to satisfy the △′-condition if, for some k_{0} > 0 and some t_{0} ≥ 0:

M(k_{0}tt′) ≤ M(t)M(t′), for all t, t′ ≥ t_{0}.  (2.2)

It is easy to see that the △′-condition is stronger than the △_{2}-condition. The following N-functions satisfy the △′-condition: M(t) = t^{p}(Log^{q} t)^{s}, where 1 < p < +∞, 0 ≤ s < +∞ and q ≥ 0 is an integer (Log^{q} being the iterated of order q of the function log).

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:

∫_{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

∥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.

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

∥u∥_{1,M,Ω} = ∑_{|α|≤1} ∥D^{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{D^{α}u_{n}−D^{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. [9, 10]). 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 k > 0, we define the truncation at height k, T_{k} : ℝ → ℝ by

T_{k}(s) = \begin{cases}s & if |s| ≤ k \\ ks/|s| & if |s| > k. \end{cases}  (2.3)

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

Lemma 2.1 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 of F′ is finite, then

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

Lemma 2.2 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}).

Proof By the previous lemma, F (u) ∈ W^{1}L_{M}(Ω) for all u ∈ W^{1}L_{M}(Ω) and

∥F (u)∥_{1,M,Ω} ≤ C ∥u∥_{1,M,Ω},

which gives easily the result. □

Let Ω be a bounded open subset of ℝ^{N}, T > 0 and set Q = Ω×]0, T [. Let m ≥ 1 be an integer and let M be an N-function. For each α ∈ N^{N}, denote by D_{x}^{α} the distributional derivative on Q of order α with respect to the variable x ∈ ℝ^{N}. The inhomogeneous Orlicz-Sobolev spaces are defined as follows

W^{m,x}L_{M}(Q) = {u ∈ L_{M}(Q) : D_{x}^{α}u ∈ L_{M}(Q) ∀|α| ≤ m}
W^{m,x}E_{M}(Q) = {u ∈ E_{M}(Q) : D_{x}^{α}u ∈ E_{M}(Q) ∀|α| ≤ m}

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

∥u∥ = ∑_{|α|≤m} ∥D_{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| ≤ m. We shall also consider the weak topologies σ(ΠL_{M}, ΠE\overline{M}) and σ(ΠL_{M}, ΠL\overline{M}). If u ∈ W^{m,x}L_{M}(Q) then the function : t ⟼ u(t) = u(t, .) is defined on [0, T] with values in W^{m}L_{M}(Ω). If, further, u ∈ W^{m,x}E_{M}(Q) then the concerned function is a W^{m}E_{M}(Ω)-valued and is strongly measurable. Furthermore the following imbedding holds: W^{m,x}E_{M}(Q) ⊂ L^{1}(0, T; W^{m}E_{M}(Ω)). The space W^{m,x}L_{M}(Q) is not in general separable, if u ∈ W^{m,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}^{m,x}E_{M}(Q) is defined as the (norm) closure in W^{m,x}E_{M}(Q) of D(Q). We can easily show as in [10] 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 limit, in W^{m,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| ≤ m,

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

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

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

∑_{|α|≤m} ∥D_{x}^{α}u∥_{M,Q} ≤ C ∑_{|α|=m} ∥D_{x}^{α}u∥_{M,Q}.

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

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

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

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

This space will be equipped with the usual quotient norm

∥f∥ = inf ∑_{|α|≤m} ∥f_{α}∥_{\overline{M},Q}

where the infimum is taken on all possible decompositions

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

The space F_{0} is then given by

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

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

Remark 2.3 We can easily check, using [10, lemma 4.4], 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 Galerkin solutions

In this section we shall define and state existence theorems of Galerkin solutions for some parabolic initial-boundary problem.

Let Ω be a bounded subset of ℝ^{N}, T > 0 and set Q = Ω×]0, T [. Let

A(u) = ∑_{|α|≤m}(−1)^{|α|}D_{x}^{α}(A_{α}(u))

be an operator such that

A_{α}(x, t, ξ) : Ω × [0, T] × ℝ^{N_{0}} → ℝ is continuous in (t, ξ), for a.e. x ∈ Ω

and measurable in x, for all (t, ξ) ∈ [0, T] × ℝ^{N_{0}},

where, N_{0} is the number of all α-order’s derivative, |α| ≤ m.

(3.1)

|A_{α}(x, s, ξ)| ≤ χ(x)Φ(|ξ|) with χ(x) ∈ L^{1}(Ω) and Φ : ℝ^{+} → ℝ^{+} increasing.

(3.2)

∑_{|α|≤m} A_{α}(x, t, ξ)ξ_{α} ≥ −d(x, t) with d(x, t) ∈ L^{1}(Q), d ≥ 0.  (3.3)

Consider a function ψ ∈ L^{2}(Q) and a function \overline{u} ∈ L^{2}(Ω) ∩ W_{0}^{m,1}(Ω). We choose an orthonormal sequence (ω_{i}) ⊂ D(Ω) with respect to the Hilbert space L^{2}(Ω) such that the closure of (ω_{i}) in C^{m}(\overline{Ω}) contains D(Ω). C^{m}(\overline{Ω}) being the space of functions which are m times continuously differentiable on \overline{Ω}. For V_{n} = span⟨ω_{1}, …, ω_{n}⟩ and

∥u∥_{C^{1,m}(Q)} = sup \{|D_{x}^{α}u(x, t)|, |\frac{∂u}{∂t}(x, t)| : |α| ≤ m, (x, t) ∈ Q\}

we have

D(Q) ⊂ \overline{\{∪_{n=1}^{∞}C^{1}([0, T], V_{n})\}}^{C^{1,m}(Q)}

this implies that for ψ and \overline{u}, there exist two sequences (ψ_{n}) and (\overline{u}_{n}) such that

ψ_{n} ∈ C^{1}([0, T], V_{n}), ψ_{n} → ψ in L^{2}(Q).  (3.4)
\overline{u}_{n} ∈ V_{n}, \overline{u}_{n} → \overline{u} in L^{2}(Ω) ∩ W_{0}^{m,1}(Ω).  (3.5)

Consider the parabolic initial-boundary value problem

\frac{∂u}{∂t} + A(u) = ψ in Q,

D_{x}^{α}u = 0 on ∂Ω×]0, T [, for all |α| ≤ m − 1,

u(0) = \overline{u} in Ω.

(3.6)

In the sequel we denote A_{α}(x, t, u, ∇u, …, ∇^{m}u) by A_{α}(x, t, u) or simply by A_{α}(u).

Definition 3.1 A function u_{n} ∈ C^{1}([0, T], V_{n}) is called Galerkin solution of (3.6) if

∫_{Ω} \frac{∂u_{n}}{∂t} φdx + ∫_{Ω} ∑_{|α|≤m} A_{α}(u_{n}).D_{x}^{α}φdx = ∫_{Ω} ψ_{n}(t)φdx

for all φ ∈ V_{n} and all t ∈ [0, T]; u_{n}(0) = \overline{u}_{n}.

We have the following existence theorem.

Theorem 3.2 ([13]) Under conditions (3.1)-(3.3), there exists at least one Galerkin solution of (3.6).

Consider now the case of a more general operator

A(u) = ∑_{|α|≤m}(−1)^{|α|}D_{x}^{α}(A_{α}(u))

where instead of (3.1) and (3.2) we only assume that

A_{α}(x, t, ξ) : Ω × [0, T] × ℝ^{N_{0}} → ℝ is continuous in ξ, for a.e. (x, t) ∈ Q
and measurable in (x, t) for all ξ ∈ ℝ^{N_{0}}.  (3.7)
|A_{α}(x, s, ξ)| ≤ C(x, t)Φ(|ξ|) with C(x, t) ∈ L^{1}(Q).  (3.8)

We have also the following existence theorem

Theorem 3.3 ([14]) There exists a function u_{n} in C([0, T], V_{n}) such that \frac{∂u_{n}}{∂t} is in L^{1}(0, T; V_{n}) and

∫_{Q_{τ}} \frac{∂u_{n}}{∂t} φ dx dt + ∫_{Q_{τ}} ∑_{|α|≤m} A_{α}(x, t, u_{n}).D_{x}^{α}φ dx dt = ∫_{Q_{τ}} ψ_{n}φ dx dt

for all τ ∈ [0, T] and all φ ∈ C([0, T], V_{n}), where Q_{τ} = Ω × [0, τ]; u_{n}(0) = \overline{u}_{n}.

4 Strong convergence of truncations

In this section we shall prove a convergence theorem for parabolic problems which allows us to deal with approximate equations of some parabolic initialboundary problem in Orlicz spaces (see section 6). Let Ω, be a bounded subset of ℝ^{N} with the segment property and let T > 0, Q = Ω×]0, T [. Let M be an N-function satisfying a Δ′ condition and the growth condition

M(t) ≪ |t|^{\frac{N}{N−1}}

and let P be an N-function such that P ≪ M. Let A : W^{1,x}L_{M}(Q) → W^{−1,x}L\overline{M}(Q) be a mapping given by

A(u) = − div a(x, t, u, ∇u)

where a(x, t, s, ξ) : Ω × [0, T] × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function satisfying for a.e. (x, t) ∈ Ω×]0, T [ and for all s ∈ ℝ and all ξ, ξ^{∗} ∈ ℝ^{N}:

|a(x, t, s, ξ)| ≤ c(x, t) + k_{1}\overline{P}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}M(k_{4}|ξ|)  (4.1)
[a(x, t, s, ξ) − a(x, t, s, ξ^{∗})][ξ − ξ^{∗}] > 0 if ξ ≠ ξ^{∗}  (4.2)
αM(\frac{|ξ|}{λ}) − d(x, t) ≤ a(x, t, s, ξ)ξ  (4.3)

where c(x, t) ∈ E\overline{M}(Q), c ≥ 0, d(x, t) ∈ L^{1}(Q), k_{1}, k_{2}, k_{3}, k_{4} ∈ ℝ^{+} and α, λ ∈ ℝ_{∗}^{+}. Consider the nonlinear parabolic equations

\frac{∂u_{n}}{∂t} − div a(x, t, u_{n}, ∇u_{n}) = f_{n} + g_{n} in D′(Q)  (4.4)

and assume that:

u_{n} ⇀ u weakly in W^{1,x}L_{M}(Q)for σ(ΠL_{M}, ΠE\overline{M}),  (4.5)
f_{n} → f strongly in W^{−1,x}E\overline{M}(Q),  (4.6)
g_{n} ⇀ g weakly in L^{1}(Q).  (4.7)

We shall prove the following convergence theorem.

Theorem 4.1 Assume that (4.1)-(4.7) hold. Then, for any k > 0, the truncation of u_{n} at height k (see (2.3) for the definition of the truncation) satisfies

∇T_{k}(u_{n}) → ∇T_{k}(u) strongly in (L_{M}^{loc}(Q))^{N}.  (4.8)

Remark 4.2 An elliptic analogous theorem is proved in Benkirane-Elmahi [2].

Remark 4.3 Convergence (4.8) allows, in particular, to extract a subsequence n′ such that:

∇u_{n′} → ∇u a.e. in Q.

Then by lemma 4.4 of [9], we deduce that

a(x, t, u_{n′}, ∇u_{n′}) ⇀ a(x, t, u, ∇u) weakly in L\overline{M}(Q))^{N} for σ(ΠL\overline{M}, ΠE_{M}).

Proof of Theorem 4.1 Step 1: For each k > 0, define S_{k}(s) = ∫_{0}^{s} T_{k}(τ)dτ. Since T_{k} is continuous, for all w ∈ W^{1,x}L_{M}(Q) we have S_{k}(w) ∈ W^{1,x}L_{M}(Q) and ∇S_{k}(w) = T_{k}(w)∇w. So that, by mollifying as in [6], it is easy to see that for all φ ∈ D(Q) and all v ∈ W^{1,x}L_{M}(Q) with \frac{∂v}{∂t} ∈ W^{−1,x}L\overline{M}(Q) + L^{1}(Q), we have

⟨⟨\frac{∂v}{∂t} , φT_{k}(v)⟩⟩ = − ∫_{Q} \frac{∂φ}{∂t}S_{k}(v) dx dt.  (4.9)

where ⟨⟨, ⟩⟩ means for the duality pairing between W_{0}^{1,x}L_{M}(Q) + L^{1}(Q) and W^{−1,x}L\overline{M}(Q) ∩ L^{∞}(Q). Fix now a compact set K with K ⊂ Q and a function φ_{K} ∈ D(Q) such that 0 ≤ φ_{K} ≤ 1 in Q and φ_{K} = 1 on K. Using in (4.4) v_{n} = φ_{K}(T_{k}(u_{n}) − T_{k}(u)) ∈ W^{1,x}L_{M}(Q) ∩ L^{∞}(Q) as test function yields

\begin{matrix}⟨⟨\frac{∂u_{n}}{∂t} , φ_{K}T_{k}(u_{n})⟩⟩ − ⟨⟨\frac{∂u_{n}}{∂t} , φ_{K}T_{k}(u)⟩⟩ \\ + ∫_{Q} φ_{K}a(x, t, u_{n}, ∇u_{n})[∇T_{k}(u_{n}) − ∇T_{k}(u)]dx dt \\ + ∫_{Q}(T_{k}(u_{n}) − T_{k}(u))a(x, t, u_{n}, ∇u_{n})∇φ_{K} dx dt \\ = ⟨⟨f_{n}, v_{n}⟩⟩ + ⟨⟨g_{n}, v_{n}⟩⟩. \end{matrix}  (4.10)

Since u_{n} ∈ W^{1,x}L_{M}(Q) and \frac{∂u_{n}}{∂t} ∈ W^{−1,x}L\overline{M}(Q) + L^{1}(Q) then by (4.9),

⟨⟨\frac{∂u_{n}}{∂t} , φ_{K}T_{k}(u_{n})⟩⟩ = − ∫_{Q} \frac{∂φ_{K}}{∂t} S_{k}(u_{n}) dx dt.

On the other hand since (u_{n}) is bounded in W^{1,x}L_{M}(Q) and \frac{∂u_{n}}{∂t} = h_{n}+g_{n} while g_{n} is bounded in L^{1}(Q) and so in M(Q) and h_{n} = div a(x, t, u_{n}, ∇u_{n}) + f_{n} is bounded in W^{−1,x}L\overline{M}(Q), then by [8, Corollary 1], u_{n} → u strongly in L_{M}^{loc}(Q). Consequently, T_{k}(u_{n}) → T_{k}(u) and S_{k}(u_{n}) → S_{k}(u) in L_{M}^{loc}(Q). So that

∫_{Q} \frac{∂φ_{K}}{∂t} S_{k}(u_{n}) dx dt → ∫_{Q} \frac{∂φ_{K}}{∂t} S_{k}(u) dx dt

and also ∫_{Q}(T_{k}(u_{n}) − T_{k}(u))a(x, t, u_{n}, ∇u_{n})∇φ_{K} dx dt → 0 as n → ∞. Furthermore ⟨⟨f_{n}, v_{n}⟩⟩ → 0, by (4.6). Since g_{n} ∈ L^{1}(Q) and T_{k}(u_{n}) − T_{k}(u) ∈ L^{∞}(Q),

⟨⟨g_{n}, φ_{K}(T_{k}(u_{n}) − T_{k}(u))⟩⟩ = ∫_{Q} g_{n}φ_{K}(T_{k}(u_{n}) − T_{k}(u)) dx dt

which tends to 0 by Egorov’s theorem.

Since φ_{K}T_{k}(u) belongs to W_{0}^{1,x}L_{M}(Q) ∩ L^{∞}(Q) while \frac{∂u_{n}}{∂t} is the sum of a bounded term in W^{−1,x}L\overline{M}(Q) and of g_{n} which weakly converges in L^{1}(Q) one has

⟨⟨\frac{∂u_{n}}{∂t} , φ_{K}T_{k}(u)⟩⟩ → ⟨⟨\frac{∂u}{∂t}, φ_{K}T_{k}(u)⟩⟩ = − ∫_{Q} \frac{∂φ}{∂t}S_{k}(u) dx dt.

We have thus proved that

∫_{Q} φ_{K}a(x, t, u_{n}, ∇u_{n})[∇T_{k}(u_{n}) − ∇T_{k}(u)] dx dt → 0 as n → ∞. (4.11)

Step 2: Fix a real number r > 0 and set Q_{(r)} = {x ∈ Q : |∇T_{k}(u)| ≤ r} and denote by χ_{r} the characteristic function of Q_{(r)}. Taking s ≥ r one has:

\begin{aligned}0 ≤ & ∫_{Q_{(r)}} φ_{K}[a(x, t, u_{n}, ∇T_{k}(u_{n})) − a(x, t, u_{n}, ∇T_{k}(u))] \\ & × [∇T_{k}(u_{n}) − ∇T_{k}(u)] dx dt \\ & ≤ ∫_{Q_{(s)}} φ_{K}[a(x, t, u_{n}, ∇T_{k}(u_{n})) − a(x, t, u_{n}, ∇T_{k}(u))] \\ & × [∇T_{k}(u_{n}) − ∇T_{k}(u)] dx dt \\ = & ∫_{Q_{(s)}} φ_{K}[a(x, t, u_{n}, ∇T_{k}(u_{n})) − a(x, t, u_{n}, ∇T_{k}(u)χ_{s})] \\ & × [∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx dt \\ ≤ & ∫_{Q} φ_{K}[a(x, t, u_{n}, ∇T_{k}(u_{n})) − a(x, t, u_{n}, ∇T_{k}(u)χ_{s})] \\ & × [∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx dt \\ = & ∫_{Q} φ_{K}a(x, t, u_{n}, ∇u_{n})[∇T_{k}(u_{n}) − ∇T_{k}(u)] dx dt \\ & − ∫_{Q} φ_{K}[a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇T_{k}(u_{n}))] \\ & × [∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx dt \\ & + ∫_{Q} φ_{K}a(x, t, u_{n}, ∇u_{n})[∇T_{k}(u) − ∇T_{k}(u)χ_{s}] dx dt \\ & − ∫_{Q} φ_{K}a(x, t, u_{n}, ∇T_{k}(u)χ_{s})[∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx dt. \end{aligned}  (4.12)

Now pass to the limit in all terms of the right-hand side of the above equation.

By (4.11), the first one tends to 0. Denoting by χ_{G_{n}} the characteristic function of G_{n} = {(x, t) ∈ Q : |u_{n}(x, t)| > k}, the second term reads

∫_{Q} φ_{K}[a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, 0)]χ_{G_{n}}∇T_{k}(u)χ_{s} dx dt  (4.13)

which tends to 0 since [a(x, t, u_{n}, ∇u_{n})−a(x, t, u_{n}, 0)] is bounded in (L\overline{M}(Q))^{N}, by (4.1) and (4.5) while χ_{G_{n}}∇T_{k}(u)χ_{s} converges strongly in (E_{M}(Q))^{N} to 0 by Lebesgue’s theorem. The fourth term of (4.12) tends to

\begin{aligned}− ∫_{Q} & φ_{K}a(x, t, u, ∇T_{k}(u)χ_{s})[∇T_{k}(u) − ∇T_{k}(u)χ_{s}] dx dt \\ & = ∫_{Q\Q_{(s)}} φ_{K}a(x, t, u, 0)∇T_{k}(u) dx dt \end{aligned}  (4.14)

since a(x, t, u_{n}, ∇T_{k}(u)χ_{s}) tends strongly to a(x, t, u, ∇T_{k}(u)χ_{s}) in (E\overline{M}(Q))^{N} while ∇T_{k}(u_{n})−∇T_{k}(u)χ_{s} converges weakly to ∇T_{k}(u)−∇T_{k}(u)χ_{s} in (L_{M}(Q))^{N} for σ(ΠL_{M}, ΠE\overline{M}).

Since a(x, t, u_{n}, ∇u_{n}) is bounded in (L\overline{M}(Q))^{N} one has (for a subsequence still denoted by u_{n})

a(x, t, u_{n}, ∇u_{n}) ⇀ h weakly in (L\overline{M}(Q))^{N} for σ(ΠL\overline{M}, ΠE_{M}).  (4.15)

Finally, the third term of the right-hand side of (4.12) tends to

∫_{Q\Q_{(s)}} φ_{K}h∇T_{k}(u) dx dt.  (4.16)

We have, then, proved that

\begin{aligned}0 ≤ & lim sup_{n→∞} ∫_{Q_{(r)}} φ_{K}[a(x, t, u_{n}, ∇T_{k}(u_{n})) − a(x, t, u_{n}, ∇T_{k}(u))] \\ & × [∇T_{k}(u_{n}) − ∇T_{k}(u)] dx dt \\ ≤ & ∫_{Q\Q_{(s)}} φ_{K}[h − a(x, t, u, 0)]∇T_{k}(u) dx dt. \end{aligned} (4.17)

Using the fact that [h − a(x, t, u, 0)]∇T_{k}(u) ∈ L^{1}(Ω) and letting s → +∞ we get, since |Q \ Q_{(s)}| → 0,

∫_{Q_{(r)}} φ_{K}[a(x, t, u_{n}, ∇T_{k}(u_{n})) − a(x, t, u_{n}, ∇T_{k}(u))][∇T_{k}(u_{n}) − ∇T_{k}(u)] dx dt

(4.18)

which approaches 0 as n → ∞. Consequently

∫_{Q_{(r)}∩K}[a(x, t, u_{n}, ∇T_{k}(u_{n}))−a(x, t, u_{n}, ∇T_{k}(u))][∇T_{k}(u_{n})−∇T_{k}(u)] dx dt → 0

as n → ∞. As in [2], we deduce that for some subsequence ∇T_{k}(u_{n}) → ∇T_{k}(u) a.e. in Q_{(r)} ∩K. Since r, k and K are arbitrary, we can construct a subsequence (diagonal in r, in k and in j, where (K_{j}) is an increasing sequence of compacts sets covering Q), such that

∇u_{n} → ∇u  a.e. in Q.  (4.19)

Step 3: As in [2] we deduce that

∫_{Q} φ_{K}a(x, t, u_{n}, ∇u_{n})∇T_{k}(u_{n}) dx dt → ∫_{Q} φ_{K}a(x, t, u, ∇u)∇T_{k}(u) dx dt

as n → ∞, and that

a(x, t, u_{n}, ∇T_{k}(u_{n}))∇T_{k}(u_{n}) → a(x, t, u, ∇T_{k}(u))∇T_{k}(u) strongly in L^{1}(K).

(4.20)

This implies that (see [2] if necessary): ∇T_{k}(u_{n}) → ∇T_{k}(u) in (L_{M}(K))^{N} for the modular convergence and so strongly and convergence (4.8) follows.

Note that in convergence (4.8) the whole sequence (and not only for a subsequence) converges since the limit ∇T_{k}(u) does not depend on the subsequence.

5 Nonlinear parabolic problems

Now, we are able to establish an existence theorem for a nonlinear parabolic initial-boundary value problems. This result which specially applies in Orlicz spaces generalizes analogous results in of Landes-Mustonen [14]. We start by giving the statement of the result.

Let Ω be a bounded subset of ℝ^{N} with the segment property, T > 0, and Q = Ω×]0, T [. Let M be an N-function satisfying the growth condition

M(t) ≪ |t|^{\frac{N}{N−1}} ,

and the △′-condition. Let P be an N-function such that P ≪ M. Consider an operator A : W_{0}^{1,x}L_{M}(Q) → W^{−1,x}L\overline{M}(Q) of the form

A(u) = − div a(x, t, u, ∇u) + a_{0}(x, t, u, ∇u)  (5.1)

where a : Ω × [0, T] × ℝ × ℝ^{N} → ℝ^{N} and a_{0} : Ω × [0, T] × ℝ × ℝ^{N} → ℝ are Carathéodory functions satisfying the following conditions, for a.e. (x, t) ∈ Ω × [0, T] for all s ∈ ℝ and ξ ≠ ξ^{∗} ∈ ℝ^{N}:

\begin{matrix}|a(x, t, s, ξ)| ≤ c(x, t) + k_{1}\overline{P}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}M(k_{4}|ξ|), \\ |a_{0}(x, t, s, ξ)| ≤ c(x, t) + k_{1}\overline{M}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}P (k_{4}|ξ|), \end{matrix}  (5.2)
[a(x, t, s, ξ) − a(x, t, s, ξ^{∗})][ξ − ξ^{∗}] > 0,  (5.3)
a(x, t, s, ξ)ξ + a_{0}(x, t, s, ξ)s ≥ αM(\frac{|ξ|}{λ}) − d(x, t)  (5.4)

where c(x, t) ∈ E\overline{M}(Q), c ≥ 0, d(x, t) ∈ L^{1}(Q), k_{1}, k_{2}, k_{3}, k_{4} ∈ ℝ^{+} and α, λ ∈ R_{∗}^{+}. Furthermore let

f ∈ W^{−1,x}E\overline{M}(Q)  (5.5)

We shall use notations of section 3. Consider, then, the parabolic initialboundary value problem

\frac{∂u}{∂t} + A(u) = f in Q

u(x, t) = 0 on ∂Ω×]0, T [ u(x, 0) = ψ(x) in Ω.

(5.6)

where ψ is a given function in L^{2}(Ω). We shall prove the following existence theorem.

Theorem 5.1 Assume that (5.2)-(5.5) hold. Then there exists at least one weak solution u ∈ W_{0}^{1,x}L_{M}(Q) ∩ L^{2}(Q) ∩ C([0, T], L^{2}(Ω))of (5.6), in the following sense:

\begin{matrix}− ∫_{Q} u\frac{∂φ}{∂t} dx dt + [∫_{Ω} u(t)φ(t)dx]_{0}^{T} + ∫_{Q} a(x, t, u, ∇u).∇φ dx dt \\ + ∫_{Q} a_{0}(x, t, u, ∇u).φ dx dt = ⟨f, φ⟩ \end{matrix}  (5.7)

for all φ ∈ C^{1}([0, T], L^{2}(Ω)).

Remark 5.2 In (5.6), we have u ∈ W_{0}^{1,x}L_{M}(Q) ⊂ L^{1}(0, T; W^{−1,1}(Ω)) and \frac{∂u}{∂t} ∈ W^{−1,x}L\overline{M}(Q) ⊂ L^{1}(0, T; W^{−1,1}(Ω)). Then u ∈ W^{1,1}(0, T; W^{−1,1}(Ω)) ⊂ C([0, T], W^{−1,1}(Ω)) with continuity of the imbedding. Consequently u is, possibly after modification on a set of zero measure, continuous from [0, T] into W^{−1,1}(Ω) in such a way that the third component of (5.6), which is the initial condition, has a sense.

Proof of Theorem 4.1 It is easily adapted from the proof given in [14]. For convenience we suppose that ψ = 0. For each n, there exists at least one solution u_{n} of the following problem (see Theorem 3.3 for the existence of u_{n}):

u_{n} ∈ C([0, T], V_{n}), \frac{∂u_{n}}{∂t} ∈ L^{1}(0, T; V_{n}), u_{n}(0) = ψ_{n} ≡ 0 and,

for all τ ∈ [0, T], ∫_{Q_{τ}} \frac{∂u_{n}}{∂t} φ dx dt + ∫_{Q_{ε}} a(x, t, u_{n}, ∇u_{n}).∇φ dx dt

+ ∫_{Q_{ε}} a_{0}(x, t, u_{n}, ∇u_{n}).φ dx dt = ∫_{Q_{ε}} f_{n}φ dx dt, ∀φ ∈ C([0, T], V_{n}).

(5.8)

where f_{k} ⊂ ∪_{n=1}^{∞}C([0, T], V_{n}) with f_{k} → f in W^{−1,x}E\overline{M}(Q). Putting φ = u_{n} in (5.8), and using (5.2) and (5.4) yields

\begin{matrix}∥u_{n}∥_{W_{0}^{1,x}L_{M} (Q)} ≤ C, ∥u_{n}∥_{L^{∞}(0,T;L^{2}(Ω))} ≤ C \\ ∥a_{0}(x, t, u_{n}, ∇u_{n})∥_{L\overline{M} (Q)} ≤ C and ∥a(x, t, u_{n}, ∇u_{n})∥_{L\overline{M} (Q)} ≤ C. \end{matrix} (5.9)

Hence, for a subsequence

u_{n} ⇀ u weakly in W_{0}^{1,x}L_{M}(Q) for σ(ΠL_{M}, ΠE\overline{M})and weakly in L^{2}(Q), a_{0}(x, t, u_{n}, ∇u_{n}) ⇀ h_{0}, a(x, t, u_{n}, ∇u_{n}) ⇀ h in L\overline{M}(Q) for σ(ΠL\overline{M}, ΠE_{M})

(5.10)

where h_{0} ∈ L\overline{M}(Q) and h ∈ (L\overline{M}(Q))^{N}. As in [14], we get that for some subsequence u_{n}(x, t) → u(x, t) a.e. in Q (it suffices to apply Theorem 3.9 instead of Proposition 1 of [14]). Also we obtain

− ∫_{Q} u\frac{∂φ}{∂t} dx dt + [∫_{Ω} u(t)φ(t)dx]_{0}^{T} + ∫_{Q} h∇φ dx dt + ∫_{Q} h_{0}φ dx dt = ⟨f, φ⟩,

for all φ ∈ C^{1}([0, T]; D(Ω)). The proof will be completed, if we can show that

∫_{Q}(h∇φ + h_{0}φ) dx dt = ∫_{Q}(a(x, t, u, ∇u)∇φ + a_{0}(x, t, u, ∇u)φ) dx dt (5.11)

for all φ ∈ C^{1}([0, T]; D(Ω)) and that u ∈ C([0, T], L^{2}(Ω)). For that, it suffices to show that

lim_{n→∞} ∫_{Q}(a(x, t, u_{n}, ∇u_{n})[∇u_{n} −∇u]+a_{0}(x, t, u_{n}∇u_{n})[u_{n} −u]) dx dt ≤ 0. (5.12)

Indeed, suppose that (5.12) holds and let s > r > 0 and set Q^{r} = {(x, t) ∈ Q : |∇u(x, t)| ≤ r}. Denoting by χ_{s} the characteristic function of Q^{s}, one has

0 ≤ ∫_{Q^{r}} [a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u)][∇u_{n} − ∇u] dx dt
≤ ∫_{Q^{s}} [a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u)][∇u_{n} − ∇u] dx dt
= ∫_{Q^{s}} [a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u.χ_{s})][∇u_{n} − ∇u.χ_{s}] dx dt
≤ ∫_{Q} [a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u.χ_{s})][∇u_{n} − ∇u.χ_{s}] dx dt
= ∫_{Q} a_{0}(x, t, u_{n}, ∇u_{n})(u_{n} − u) − ∫_{Q} a(x, t, u_{n}, ∇u_{n}.χ_{s})[∇u_{n} − ∇u.χ_{s}] dx dt
+ ∫_{Q} [a(x, t, u_{n}, ∇u_{n})(∇u_{n} − ∇u) + a_{0}(x, t, u_{n}, ∇u_{n})(u_{n} − u)] dx dt
+ ∫_{Q\Q^{s}} a(x, t, u_{n}, ∇u_{n})∇u dx dt.

(5.13)

The first term of the right-hand side tends to 0 since (a_{0}(x, t, u_{n}, ∇u_{n})) is bounded in L\overline{M}(Q) by (5.2) and u_{n} → u strongly in L_{M}(Q). The second term tends to ∫_{Q\Q^{s}} a(x, t, u_{n}, 0)∇u dx dt since a(x, t, u_{n}, ∇u_{n}.χ_{s}) tends strongly in (E\overline{M}(Q))^{N} to a(x, t, u, ∇u.χ_{s}) and ∇u_{n} ⇀ ∇u weakly in (L_{M}(Q))^{N} for σ(ΠL_{M}, ΠE\overline{M}). The third term satisfies (5.12) while the fourth term tends to ∫_{Q\Q^{s}} h∇u dx dt since a(x, t, u_{n}, ∇u_{n}) ⇀ h weakly in (L\overline{M}(Q))^{N} for σ(ΠL\overline{M}, ΠE_{M}) and M satisfies the △_{2}-condition. We deduce then that

0 ≤ lim sup_{n→∞} ∫_{Q^{s}}[a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u)][∇u_{n} − ∇u] dx dt
≤ ∫_{Q\Q^{s}}[h − a(x, t, u, 0)]∇u dx dt → 0 as s → ∞.

and so, by (5.3), we can construct as in [2] a subsequence such that ∇u_{n} → ∇u a.e. in Q. This implies that a(x, t, u_{n}, ∇u_{n}) → a(x, t, u, ∇u) and that a_{0}(x, t, u_{n}, ∇u_{n}) → a_{0}(x, t, u, ∇u) a.e. in Q. Lemma 4.4 of [9] shows that h = a(x, t, u, ∇u) and h_{0} = a_{0}(x, t, u, ∇u) and (5.11) follows. The remaining of the proof is exactly the same as in [14]. □

Corollary 5.3 The function u can be used as a testing function in (5.6) i.e.

\frac{1}{2}[ ∫_{Ω}(u(t))^{2}dx]_{0}^{τ} +∫_{Q_{τ}} [a(x, t, u, ∇u).∇u+a_{0}(x, t, u, ∇u)u] dx dt = ∫_{Q_{τ}} fu dx dt

for all τ ∈ [0, T].

The proof of this corollary is exactly the same as in [14].

6 Strongly nonlinear parabolic problems

In this last section we shall state and prove an existence theorem for strongly nonlinear parabolic initial-boundary problems with a nonlinearity g(x, t, s, ξ) having growth less than M(|ξ|). This result generalizes Theorem 2.1 in Boccardo-Murat [5]. The analogous elliptic one is proved in Benkirane-Elmahi [2].

The notation is the same as in section 5. Consider also assumptions (5.2)- (5.5) to which we will annex a Carathéodory function g : Ω × [0, T] × ℝ × ℝ^{N} → ℝ^{N} satisfying, for a.e. (x, t) ∈ Ω × [0, T] and for all s ∈ ℝ and all ξ ∈ ℝ^{N}:

g(x, t, s, ξ)s ≥ 0  (6.1)
|g(x, t, s, ξ)| ≤ b(|s|)(c′(x, t) + R(|ξ|))  (6.2)

where c′ ∈ L^{1}(Q) and b : ℝ^{+} → ℝ^{+} and where R is a given N-function such that R ≪ M. Consider the following nonlinear parabolic problem

\frac{∂u}{∂t} + A(u) + g(x, t, u, ∇u) = f in Q,

u(x, t) = 0 on ∂Ω × (0, T),
u(x, 0) = ψ(x) in Ω.

(6.3)

We shall prove the following existence theorem.

Theorem 6.1 Assume that (5.1)-(5.5), (6.1) and (6.2) hold. Then, there exists at least one distributional solution of (6.3).

Proof It is easily adapted from the proof of theorem 3.2 in [2] Consider first

g_{n}(x, t, s, ξ) = \frac{g(x, t, s, ξ)}{1 + \frac{1}{n}g(x, t, s, ξ)}

and put A_{n}(u) = A(u)+g_{n}(x, t, u, ∇u), we see that A_{n} satisfies conditions (5.2)- (5.4) so that, by Theorem 5.1, there exists at least one solution u_{n} ∈ W_{0}^{1,x}L_{M}(Q) of the approximate problem

\frac{∂u_{n}}{∂t} + A(u_{n}) + g_{n}(x, t, u_{n}, ∇u_{n}) = f in Q

u_{n}(x, t) = 0 on ∂Ω×]0, T [
u_{n}(x, 0) = ψ(x) in Ω

(6.4)

and, by Corollary 5.3, we can use u_{n} as testing function in (6.4). This gives

∫_{Q}[a(x, t, u_{n}, ∇u_{n}).∇u_{n} + a_{0}(x, t, u_{n}, ∇u_{n}).u_{n}] dx dt ≤ ⟨f, u_{n}⟩

and thus (u_{n}) is a bounded sequence in W_{0}^{1,x}L_{M}(Q). Passing to a subsequence if necessary, we assume that

u_{n} ⇀ u weakly in W_{0}^{1,x}L_{M}(Q) for σ(ΠL_{M}, ΠE\overline{M})  (6.5)

for some u ∈ W_{0}^{1,x}L_{M}(Q). Going back to (6.4), we have

∫_{Q} g_{n}(x, t, u_{n}, ∇u_{n})u_{n} dx dt ≤ C.

We shall prove that g_{n}(x, t, u_{n}, ∇u_{n}) are uniformly equi-integrable on Q. Fix m > 0. For each measurable subset E ⊂ Q, we have

∫_{E} |g_{n}(x, t, u_{n}, ∇u_{n})|

≤ ∫_{E∩{|u_{n}|≤m}} |g_{n}(x, t, u_{n}, ∇u_{n})| + ∫_{E∩{|u_{n}|>m}} |g_{n}(x, t, u_{n}, ∇u_{n})|

≤ b(m) ∫_{E}[c′(x, t) + R(|∇u_{n}|)] dx dt + \frac{1}{m} ∫_{E∩\{|u_{n}|>m\}} |g_{n}(x, t, u_{n}, ∇u_{n})| dx dt

≤ b(m) ∫_{E}[c′(x, t) + R(|∇u_{n}|)] dx dt + \frac{1}{m} ∫_{Q} u_{n}g_{n}(x, t, u_{n}, ∇u_{n}) dx dt

≤ b(m) ∫_{E} c′(x, t) dx dt + b(m) ∫_{E} R(\frac{|∇u_{n}|}{λ′}) dx dt + \frac{C}{m}

Let ε > 0, there is m > 0 such that \frac{C}{m} < \frac{ε}{3}. Furthermore, since c′′ ∈ L^{1}(Q) there exists δ_{1} > 0 such that b(m) ∫_{E} c′′(x, t) dx dt < \frac{ε}{3}. On the other hand, let μ > 0 such that ∥∇u_{n}∥_{M,Q} ≤ μ, ∀n. Since R ≪ M, there exists a constant K_{ε} > 0 depending on ε such that

b(m)R(s) ≤ M(\frac{ε}{6} \frac{s}{μ}) + K_{ε}

for all s ≥ 0. Without loss of generality, we can assume that ε < 1. By convexity we deduce that

b(m)R(s) ≤ \frac{ε}{6}M(\frac{s}{μ}) + K_{ε}

for all s ≥ 0. Hence

b(m) ∫_{E} R(\frac{|∇u_{n}|}{λ′}) dx dt ≤ \frac{ε}{6} ∫_{E} M(\frac{|∇u_{n}|}{μ}) dx dt + K_{ε}|E|

≤ \frac{ε}{6} ∫_{Q} M(\frac{|∇u_{n}|}{μ}) dx dt + K_{ε}|E|

≤ \frac{ε}{6} + K_{ε}|E|.

When |E| ≤ ε/(6K_{ε}), we have

b(m) ∫_{E} R(\frac{|∇u_{n}|}{λ′}) dx dt ≤ \frac{ε}{3}, ∀n.

Consequently, if |E| < δ = inf(δ_{1}, \frac{ε}{6K_{ε}}) one has

∫_{E} |g_{n}(x, t, u_{n}, ∇u_{n})| dx dt ≤ ε, ∀n,

this shows that the g_{n}(x, t, u_{n}, ∇u_{n}) are uniformly equi-integrable on Q. By Dunford-Pettis’s theorem, there exists h ∈ L^{1}(Q) such that

g_{n}(x, t, u_{n}, ∇u_{n}) ⇀ h weakly in L^{1}(Q).  (6.6)

Applying then Theorem 4.1, we have for a subsequence, still denoted by u_{n},

u_{n} → u, ∇u_{n} → ∇u a.e. in Q and u_{n} → u strongly in W_{0}^{1,x}L_{M}^{loc}(Q). (6.7)

We deduce that a(x, t, u_{n}, ∇u_{n}) ⇀ a(x, t, u, ∇u) weakly in (L\overline{M}(Q))^{N} for σ(ΠL_{\overline{M},}ΠL_{M}) and since \frac{∂u_{n}}{∂t} → \frac{∂u}{∂t} in D′(Q) then passing to the limit in (6.4) as n → +∞, we obtain

\frac{∂u}{∂t} + A(u) + g(x, t, u, ∇u) = f in D′(Q).

This completes the proof of Theorem 6.1.

References

[1] R. Adams, Sobolev spaces, Ac. Press, New York, 1975.

[2] A. Benkirane and A. Elmahi, Almost everywhere convergence of the gradients to elliptic equations in Orlicz spaces and application, Nonlinear Anal., T. M. A., 28 n^{∘} 11 (1997), pp. 1769-1784.

[3] A. Benkirane and A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Analysis T.M.A., (to appear).

[4] A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. Henri Poincaré, 5 n^{∘} 4 (1988), pp. 347-364.

[5] L. Boccardo and F. Murat, Strongly nonlinear Cauchy problems with gradient dependent lower order nonlinearity, Pitman Research Notes in Mathematics, 208 (1989), pp. 247-254.

[6] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients, Nonlinear analysis, T.M.A., 19 (1992), n^{∘} 6, pp. 581-597.

[7] T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial-boundary value problems, J. Diff. Equations, 16 (1974), 201-256.

[8] A. Elmahi, Compactness results in inhomogeneous Orlicz-Sobolev spaces, Lecture notes in Pure and Applied Mathematics, Marcel Dekker, Inc, 229 (2002), 207-221.

[9] J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), pp.163-205.

[10] J.-P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), pp.17-24.

[11] M. Krasnosel’skii and Ya. Rutickii, convex functions and Orlicz spaces, Noordhoff Groningen, 1969.

[12] A. Kufner, O. John and S. Fucík, Function spaces, Academia, Prague, 1977.

[13] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh sect. A. 89 (1981), 217-137.

[14] R. Landes and V. Mustonen, A strongly nonlinear parabolic initialboundary value problem, Ark. f. Mat. 25 (1987), 29-40.

[15] Morrey C. B. Jr., Multiple integrals in the calculus of variation, Springer-Verlag, Berlin-Heidelberg-NewYork, 1969.

[16] Robert J., Inéquations variationnelles paraboliques fortement non linéaires, J. Math. Pures Appl., 53 (1974), 299-321.

[17] Simon J., Compact sets in the space L^{p}(0, T; B), Ann. Mat. pura. Appl. 146 (1987), 65-96.

Abdelhak Elmahi Department de Mathematiques, C.P.R. B.P. 49, Fès - Maroc e-mail: elmahi abdelhak@yahoo.fr