Nova S´erie

STRONGLY NONLINEAR PARABOLIC EQUATIONS
WITH NATURAL GROWTH TERMS AND L^{1} DATA

IN ORLICZ SPACES A. Elmahi and D. Meskine

Abstract: We prove compactness and approximation results in inhomogeneous
Orlicz–Sobolev spaces and look at, as an application, the Cauchy–Dirichlet problem
u^{0}+A(u) +g(x, t, u,∇u) =f ∈L^{1}. We also give a trace result allowing to deduce the
continuity of the solutions with respect to time.

1 – Introduction

Let Ω be a bounded open subset of R^{N} and let Q be the cylinder Ω×(0, T)
with some givenT >0 and let

A(u) =−div^{³}a(x, t, u,∇u)^{´}
be a Leray–Lions operator defined onL^{p}(0, T;W^{1,p}(Ω)).

Dall’aglio–Orsina [9] and Porretta [19] proved the existence of solutions for the following Cauchy–Dirichlet problem

∂u

∂t +A(u) +g(x, t, u,∇u) =f in Q,

u(x, t) = 0 on ∂Ω×(0, T),

u(x,0) =u_{0}(x) in Ω ,

(1)

Received: June 27, 2003.

AMS Subject Classification: 35K15, 35K20, 35K60.

Keywords and Phrases: inhomogeneous Orlicz–Sobolev spaces; parabolic problems; lack of compactness; approximation.

whereg is a nonlinearity with the following “natural” growth condition (of order p):

|g(x, t, s, ξ)| ≤ b(|s|)^{³}c(x, t) +|ξ|^{p}^{´}

and which satisfies the classical sign condition g(x, t, s, ξ)s≥0. The right hand
side f is assumed to belong to L^{1}(Q). This result generalizes analogous one of
Boccardo–Gallouet [4]. See also [5] and [6] for related topics. In all of these
results, the functionais supposed to satisfy a polynomial growth condition with
respect tou and ∇u.

When trying to relax this restriction ona(for example, ifahas exponential or
logarithmic growth with respect to ∇u) we are led to replace L^{p}(0, T;W^{1,p}(Ω))
with an inhomogeneous Sobolev spaceW^{1,x}L_{M}(Q) built from an Orlicz spaceL_{M}
instead ofL^{p} where the N-function M which definesL_{M} is related to the actual
growth ofa. The solvability of (1) in this setting is only proved in the variational
case i.e. where f belongs to the Orlicz space W^{−1,x}E_{M}(Q), see Donaldson [8]

for g ≡ 0 and Robert [20] for g ≡ g(x, t, u) when A is monotone, t^{2} ¿ M(t)
and M satisfies a ∆_{2} condition and also Elmahi [11] for g =g(x, t, u,∇u) when
M satisfies a ∆^{0} condition and M(t) ¿ t^{N/(N−1)} and finally the recent work
Elmahi–Meskine [13] for the general case.

It is our purpose in this paper to prove, in the case wheref belongs to L^{1}(Q),
the existence of solutions for parabolic problems of the form (1) in the setting of
Orlicz spaces by using a classical approximating method. Thus, and in order to
study the behaviour of the approximate solutions we call upon compactness tools,
so that, we first establish (in section 3)L^{1} compactness results nearly similar to
those of Simon [21] and Boccardo–Murat [6] and Elmahi [10].

Next, and when going to the limit in approximating problems, we have to reg- ularize an arbitrary test function by smooth ones with converging distributional time derivatives. This becomes possible thanks to the approximate theorem 3 which is slightly different from theorems 3 and 4 of [15] and will be also applied to get a trace result giving the continuity of such test functions with respect to time.

The plan of the paper is as follows: in Section 2 we recall some preliminaries concerning Orlicz–Sobolev spaces while in Section 3 we prove the compactness results in inhomogeneous Orlicz–Sobolev spaces.

Section 4 will be devoted to approximation results which allow us to overcome the difficulties which arise on time derivatives while in Section 5, we look at, as an application of all previous results, the solvability, in the framework of entropy solutions, of strongly nonlinear parabolic initial-boundary value problems of the

form (1), whose simplest model is the following

∂u

∂t −div µ

a(x, t, u)m(|∇u|)

|∇u| ∇u

¶

+g(x, t, u)m(|∇u|)|∇u| =f inQ,

u(x, t) = 0 on ∂Ω×(0, T),

u(x,0) =u0(x) in Ω ,

where 0< α≤a(x, t, s)≤β and wherem is any continuous function on [0,+∞) which strictly increases from 0 to +∞.

Note that, our existence result generalizes analogous ones of [9] and [19] (take
indeedm(t) =t^{p−1}, withp >1). Moreover, and contrary to [9] and [19], the proof
is achieved without extending the initial problem or assuming the positiveness of
either the dataf or the initial condition u_{0}.

2 – Preliminaries

2.1. LetM:R^{+}→R^{+} be an N-function, i.e. M is continuous, convex, with
M(t)>0 fort >0,M(t)/t→0 as t→0 andM(t)/t→ ∞ ast→ ∞.

Equivalently, M admits the representation: M(t) = ^{R}_{0}^{t}m(τ)dτ where m :
R^{+}→R^{+} is non-decreasing, right continuous, withm(0) = 0, m(t)>0 for t >0
andm(t)→ ∞ ast→ ∞.

The N-function M conjugate to M is defined by M(t) = ^{R}_{0}^{t}m(τ)dτ, where
m:R^{+} →R^{+} is given by m(t) = sup{s:m(s)≤t} (see [1], [16] and [17]).

We will extend these N-functions into even functions on all R.

The N-function M is said to satisfy a ∆_{2} condition if, for somek >0:

M(2t)≤k M(t) ∀t≥0 . (2)

when (2) holds only fort≥somet_{0} >0 thenM is said to satisfy the ∆_{2}condition
near infinity.

2.2. Let Ω be an open subset of R^{N}. The Orlicz classL_{M}(Ω) (resp. the Orlicz
spaceL_{M}(Ω)) is defined as the set of (equivalence classes of) real-valued measur-
able functionsuon Ω such that^{R}_{Ω}M(u(x))dx <+∞(resp.^{R}_{Ω}M(u(x)/λ)dx <+∞

for someλ >0).

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

kukM,Ω = inf

½ λ >0 :

Z

ΩM µu(x)

λ

¶ dx≤1

¾

andL_{M}(Ω) is a convex subset ofL_{M}(Ω).

The closure in L_{M}(Ω) of the set of bounded measurable functions with com-
pact support in Ω is denoted by EM(Ω). The equality EM(Ω) = LM(Ω) holds
if and only if M satisfies the ∆_{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_{M}(Ω) by means of the pairing
R

Ωu(x)v(x)dx, and the dual norm on L_{M}(Ω) is equivalent tok.k_{M ,Ω}.

The spaceL_{M}(Ω) is reflexive if and only ifM andM satisfy the ∆2 condition
(near infinity only if Ω has finite measure).

Two N-functions M and P are said to be equivalent (resp. near infinity),
if there exist reals numbersk_{1}, k_{2} >0 such that P(k_{2}t)≤M(t)≤P(k_{2}t) for all
t≥0 (resp. for all t≥ somet_{0}>0).

P¿M means thatP grows essentially less rapidly thanM, i.e. for eachε >0,
P(t)/(M(ε t))→0 ast→ ∞. This is the case if and only if M^{−1}(t)/P^{−1}(t)→0
ast→ ∞, therefore, we have the following continuous imbeddingL_{M}(Ω)⊂E_{P}(Ω)
when Ω has finite measure.

2.3. We now turn to the Orlicz–Sobolev spaces. W^{1}L_{M}(Ω) (resp.W^{1}E_{M}(Ω))
is the space of all functionsusuch thatu and its distributional derivatives up to
order 1 lie inL_{M}(Ω) (resp. E_{M}(Ω)). It is a Banach space under the norm:

kuk_{1,M,Ω} = ^{X}

|α|≤1

kD^{α}uk_{M,Ω} .

ThusW^{1}L_{M}(Ω) and W^{1}E_{M}(Ω) can be identified with subspaces of the product
of (N+ 1) copies ofL_{M}(Ω). Denoting this product by ΠL_{M}, we will use the weak
topologiesσ(ΠL_{M},ΠE_{M}) and σ(ΠL_{M},ΠL_{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_{M}) closure of
D(Ω) inW^{1}L_{M}(Ω).

We say that un converges to u for the modular convergence in W^{1}L_{M}(Ω) if
for some λ > 0, ^{R}_{Ω}M((D^{α}un−D^{α}u)/λ)dx → 0 for all |α| ≤ 1. This implies
convergence forσ(ΠLM,ΠL_{M}). Note that, ifun→u inLM(Ω) for the modular
convergence andv_{n}→v inL_{M}(Ω) for the modular convergence, we have

Z

Ωunvndx → Z

Ωuv dx as n→ ∞ . (3)

Indeed, letλ >0 and µ >0 such that Z

ΩM

µu_{n}−u
λ

¶

dx → 0 and Z

ΩM

µv_{n}−v
µ

¶

dx → 0
and, since u_{n}v_{n}−uv= (u_{n}−u) (v_{n}−v) +u_{n}v+uv_{n}−2uv, we obtain

1 λµ

¯

¯

¯

¯ Z

Ω(unvn−uv)dx

¯

¯

¯

¯ ≤

≤ Z

Ω

M

µu_{n}−u
λ

¶ dx +

Z

Ω

M

µv_{n}−v
µ

¶

dx + 1 λµ

¯

¯

¯

¯ Z

Ω

(u_{n}v+uv_{n}−2uv)dx

¯

¯

¯

¯ therefore, by lettingn→ ∞in the last side, we get the result.

IfM satisfies the ∆2condition (near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence.

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

If the open set Ω has the segment property, then the space D(Ω) is dense in
W_{0}^{1}L_{M}(Ω) for the modular convergence and thus for the topologyσ(ΠL_{M},ΠL_{M})
(cf. [14], [15]). Consequently, the action of a distributionT inW^{−1}L_{M}(Ω) on an
elementu ofW_{0}^{1}L_{M}(Ω) is well defined, it will be denoted byhT, ui.

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

Let M be an N-function. For each α ∈ N^{N}, denote by D^{α}_{x} the distributional
derivative on Qof order α with respect to the variable x∈R^{N}. The inhomoge-
neous Orlicz–Sobolev spaces of order 1 are defined as follows

W^{1,x}L_{M}(Q) = ^{n}u∈L_{M}(Q) : D_{x}^{α}u∈L_{M}(Q), ∀ |α| ≤1^{o}
and

W^{1,x}E_{M}(Q) = ^{n}u∈E_{M}(Q) : D^{α}_{x}u∈E_{M}(Q), ∀ |α| ≤1^{o}.

The latter space is a subspace of the former. Both are Banach spaces under the norm

kuk = ^{X}

|α|≤1

kD_{x}^{α}uk_{M,Q} .

We can easily show that they form a complementary system when Ω sat-
isfies the segment property. These spaces are considered as subspaces of the
product space ΠL_{M}(Q) which has (N+ 1) copies. We shall also consider the
weak topologies σ(ΠL_{M},ΠE_{M}) and σ(ΠL_{M},ΠL_{M}). If u ∈ W^{1,x}L_{M}(Q) then

the function: t 7−→ 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 u(., t) is a W^{1}E_{M}(Ω)-valued and is strongly
measurable. Furthermore the following continuous 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 from
(0, T) into W^{1}L_{M}(Ω). However, the scalar function t 7−→ kD_{x}^{α}u(t)k_{M,Ω} is in
L^{1}(0, T) for all |α| ≤1.

2.6. 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 [15] (see the proof of theorem 3 below) that
when Ω has the segment property then each elementuof the closure ofD(Q) with
respect to the weak ∗ topologyσ(ΠL_{M},ΠE_{M}) is limit, in W^{1,x}L_{M}(Q), of some
sequence (u_{n}) ⊂ D(Q) for the modular convergence i.e. there exists λ >0 such
that, for all|α| ≤1,^{R}_{Q}M((D^{α}_{x}u_{n}−D^{α}_{x}u)/λ)dx dt→0 whenn→ ∞, this implies
that (u_{n}) converges to u in W^{1,x}L_{M}(Q) for the weak topology σ(ΠL_{M},ΠL_{M}).

Consequently, D(Q)^{σ(ΠL}^{M}^{,ΠE}^{M}^{)} = D(Q)^{σ(ΠL}^{M}^{,ΠL}^{M}^{)}, this space will be denoted
byW_{0}^{1,x}L_{M}(Q). Furthermore, W_{0}^{1,x}E_{M}(Q) =W_{0}^{1,x}L_{M}(Q)∩ΠE_{M}.

Poincar´e’s inequality also holds in W_{0}^{1,x}L_{M}(Q) and then there is a constant
C >0 such that for all u∈W_{0}^{1,x}L_{M}(Q) one has

X

|α|≤1

kD_{x}^{α}uk_{M,Q} ≤ C ^{X}

|α|=1

kD_{x}^{α}uk_{M,Q} ,

thus both sides of the last inequality are equivalent norms on W_{0}^{1,x}LM(Q).

We have then the following complementary system

W_{0}^{1,x}L_{M}(Q) F
W_{0}^{1,x}E_{M}(Q) F_{0}

,

F being the dual space of W_{0}^{1,x}E_{M}(Q). It is also, up to an isomorphism,
the quotient of ΠL_{M} by the polar set W_{0}^{1,x}E_{M}(Q)^{⊥}, and will be denoted by
F = W^{−1,x}L_{M}(Q) and it is shown that W^{−1,x}L_{M}(Q) = {f = ^{P}_{|α|≤1}D_{x}^{α}f_{α} :
f_{α}∈L_{M}(Q)}. This space will be equipped with the usual quotient norm:

kfk= inf ^{X}

|α|≤1

kfαk_{M ,Q}

where the inf is taken over all possible decompositionsf=^{P}_{|α|≤1}D_{x}^{α}f_{α},f_{α}∈L_{M}(Q).

The space F_{0} is then given by F_{0} = {f = ^{P}_{|α|≤1}D_{x}^{α}f_{α} : f_{α} ∈ E_{M}(Q)} and is
denoted byF_{0}=W^{−1,x}E_{M}(Q).

3 – Compactness results

In this section, we shall prove some compactness theorems in inhomogeneous
Orlicz–Sobolev spaces which will be applied to study the behaviour of the ap-
proximating solutions for parabolic problems. These results, which are nearly
similar to those of Simon [21], Boccardo–Murat [6] and Elmahi [10], give onlyL^{1}
(and notL_{M}) compactness for sets in W^{1,x}L_{M}(Q). They are, however, sufficient
for applications to solve parabolic problems in Orlicz spaces of variational type
or withL^{1} data.

For eachh >0,define the usual translatedτ_{h}f of the function f by τ_{h}f(t) =
f(t+h).If f is defined on [0, T] thenτ_{h}f is defined on [−h, T −h].

First of all, recall the following compactness result proved by Simon [21].

Theorem 1. See [21]. Let B be a Banach space and let T > 0 be a fixed
real number. IfF ⊂L^{1}(0, T;B)is such that

½Z _{t}_{2}

t1

f(t)dt

¾

f

is relatively compact inB, for all 0< t_{1} < t_{2} < T .
(4)

kτ_{h}f −fk_{L}^{1}_{(0,T}_{;B)} →0 uniformly in f ∈F, when h→0.
(5)

ThenF is relatively compact in L^{1}(0, T;B).

Next, we prove the following lemma, which it can be seen as a “Orlicz” version
of the well known interpolation inequality related to the spaceL^{p}(0, T;W_{0}^{1,p}(Ω)).

Lemma 1. LetM be an N-function. LetY be a Banach space such that the
following continuous imbedding holds: L^{1}(Ω) ⊂Y. Then, for all ε > 0 and all
λ >0,there isC_{ε}>0 such that for allu∈W_{0}^{1,x}L_{M}(Q),with|∇u|/λ∈ L_{M}(Q),

kuk_{L}^{1}_{(Q)} ≤ ε λ
ÃZ

Q

M µ|∇u|

λ

¶

dx dt + T

!

+ C_{ε}kuk_{L}^{1}_{(0,T;Y}_{)} .

Proof: Since W_{0}^{1}L_{M}(Ω)⊂L^{1}(Ω) with compact imbedding, see [1], then, for
allε >0,there isC_{ε}>0 such that for all v∈W_{0}^{1}L_{M}(Ω):

kvk_{L}^{1}_{(Ω)} ≤ εk∇vk_{L}_{M}_{(Ω)}+C_{ε}kvk_{Y} .
(6)

Indeed, if the above assertion holds false, there isε_{0} >0 andv_{n}∈W_{0}^{1}L_{M}(Ω)
such that

kv_{n}k_{L}^{1}_{(Ω)} ≥ ε_{0}k∇v_{n}k_{L}_{M}_{(Ω)}+nkv_{n}k_{Y} .
This gives, by settingwn=vn/k∇vnk_{L}_{M}_{(Ω)}:

kw_{n}k_{L}^{1}_{(Ω)}≥ε_{0}+nkw_{n}k_{Y} , k∇w_{n}k_{L}_{M}_{(Ω)}= 1 .
Since (w_{n}) is bounded inW_{0}^{1}L_{M}(Ω) then for a subsequence,

wn* w inW_{0}^{1}LM(Ω) for σ(ΠLM,ΠE_{M}) and strongly in L^{1}(Ω).
Thuskw_{n}k_{L}^{1}_{(Ω)} is bounded and kw_{n}k_{Y} →0 asn→ ∞. We deduce that w_{n}→0
inY and thatw= 0 implying thatε_{0} ≤ kw_{n}k_{L}^{1}_{(Ω)} →0, a contradiction.

Using v=u(t) in (6) for allu∈W_{0}^{1,x}L_{M}(Q) with|∇u|/λ∈ L_{M}(Q) and a.e.

tin (0, T),we have

ku(t)k_{L}^{1}_{(Ω)} ≤ εk∇u(t)k_{L}_{M}_{(Ω)}+C_{ε}ku(t)k_{Y} .

Since^{R}_{Q}M(|∇u(x, t)|/λ)dx dt <∞ we have thanks to Fubini’s theorem,
Z

ΩM

µ|∇u(x, t)|

λ

¶

dx < ∞ for a.e. t in (0, T) and then

k∇u(t)k_{L}_{M}_{(Ω)} ≤ λ
ÃZ

Ω

M

µ|∇u(x, t)|

λ

¶

dx + 1

!

which implies that

ku(t)k_{L}^{1}_{(Ω)} ≤ ε λ
ÃZ

ΩM

µ|∇u(x, t)|

λ

¶

dx + 1

!

+ C_{ε}ku(t)k_{Y} .
Integrating this over (0, T) yields

kuk_{L}^{1}_{(Q)} ≤ ε λ
ÃZ

Q

M

µ|∇u(x, t)|

λ

¶

dx dt + T

!
+ C_{ε}

Z T 0

ku(t)k_{Y} dt
and finally

kuk_{L}^{1}_{(Q)} ≤ ε λ
ÃZ

Q

M µ|∇u|

λ

¶

dx dt + T

!

+ C_{ε}kuk_{L}^{1}_{(0,T}_{;Y}_{)} .

We also prove the following lemma which allows us to enlarge the space Y whenever necessary.

Lemma 2. Let Y be a Banach space such thatL^{1}(Ω)⊂Y with continuous
imbedding.

If F is bounded in W_{0}^{1,x}L_{M}(Q) and is relatively compact inL^{1}(0, T;Y) then
F is relatively compact in L^{1}(Q) (and also inE_{P}(Q) for all N-functionP ¿M).

Proof: Letε >0 be given. Let C >0 be such that ^{R}_{Q}M(|∇f|/C)dx dt≤1
for allf ∈F.

By the previous lemma, there existsC_{ε}>0 such that, for allu∈W_{0}^{1,x}L_{M}(Q)
with|∇u|/(2C)∈ L_{M}(Q),

ku(t)k_{L}1(Q) ≤ 2εC
4C(1 +T)

ÃZ

Q

M µ|∇u|

2C

¶

dx dt + T

!

+ C_{ε}kuk_{L}1(0,T;Y) .
Moreover, there exists a finite sequence (f_{i}) in F satisfying:

∀f ∈F, ∃f_{i} such that kf −f_{i}k_{L}1(0,T;Y)≤ ε
2Cε

so that

kf−f_{i}k_{L}^{1}_{(Q)} ≤ ε
2(1+T)

ÃZ

Q

M

µ|∇f − ∇f_{i}|
2C

¶

dx dt + T

!

+ C_{ε}kf −f_{i}k_{L}^{1}_{(0,T;Y}_{)}

≤ ε

and henceF is relatively compact inL^{1}(Q).

Since P ¿M then by using Vitali’s theorem, it is easy to see that F
is relatively compact inE_{P}(Q).

Lemma 3. (See [21]). Let B be a Banach space.

If f ∈ D^{0}(]0, T[;B) is such that ^{∂f}_{∂t} ∈ L^{1}(0, T;B) then f ∈ C(]0, T[, B) and
for allh >0

kτ_{h}f−fk_{L}^{1}_{(0,T}_{;B)} ≤ h

°

°

°

°

∂f

∂t

°

°

°

°L^{1}(0,T;B)

.

Remark 1. By lemma 4, if F ⊂ L^{1}(0, T;B) is such that ^{n}^{∂f}_{∂t} :f ∈F^{o} is
bounded inL^{1}(0, T;B) then

kτ_{h}f−fk_{L}^{1}_{(0,T;B)}→0 ash→0 uniformly with respect to f ∈F .

Lemma 4. (See [8]). The following continuous imbedding hold: W_{0}^{1,x}E_{M}(Q)

⊂L^{1}(0, T;W_{0}^{1}E_{M}(Ω)) and W^{−1,x}E_{M}(Q)⊂L^{1}(0, T;W^{−1}E_{M}(Ω)).

We shall now apply the previous results to prove some compactness theorems in inhomogeneous Orlicz–Sobolev spaces.

Theorem 2. LetM be an N-function. IfF is bounded in W_{0}^{1,x}L_{M}(Q) and
n∂f

∂t :f ∈F^{o}is bounded inW^{−1,x}L_{M}(Q) thenF is relatively compact inL^{1}(Q).

Proof: Let P and R be N-functions such that P ¿ M and R ¿ M near infinity.

For all 0< t1 < t2 < T and all f ∈F,we have

°

°

°

° Z t2

t1

f(t)dt

°

°

°

°W_{0}^{1}EP(Ω)

≤ Z T

0 kf(t)k_{W}^{1}

0EP(Ω)dt

≤ C_{1}kfk_{W}^{1,x}

0 EP(Q) ≤ C_{2}kfk_{W}^{1,x}

0 LM(Q) ≤ C where we have used the following continuous imbedding

W_{0}^{1,x}L_{M}(Q) ⊂ W_{0}^{1,x}E_{P}(Q) ⊂ L^{1}(0, T;W_{0}^{1}E_{P}(Ω)).

Since the imbedding W_{0}^{1}E_{P}(Ω)⊂L^{1}(Ω) is compact we deduce that

³Rt2

t1 f(t)dt^{´}

f∈F is relatively compact inL^{1}(Ω) and in W^{−1,1}(Ω) as well.

On the other hand ^{n}^{∂f}_{∂t} :f ∈F^{o} is bounded in W^{−1,x}L_{M}(Q) and in
L^{1}(0, T;W^{−1,1}(Ω)) as well, since

W^{−1,x}L_{M}(Q) ⊂ W^{−1,x}E_{R}(Q) ⊂ L^{1}(0, T;W^{−1}E_{R}(Ω)) ⊂ L^{1}(0, T;W^{−1,1}(Ω)) ,
with continuous imbedding.

By Remark 1, we deduce thatkτ_{h}f−fk_{L}^{1}_{(0,T;W}^{−1,1}_{(Ω))} →0 uniformly inf ∈F
whenh→0 and by using theorem 1,F is relatively compact inL^{1}(0, T;W^{−1,1}(Ω)).

Since L^{1}(Ω)⊂ W^{−1,1}(Ω) with continuous imbedding we can apply lemma 2
to conclude thatF is relatively compact in L^{1}(Q).

Corollary 1. Let M be an N-function.

Let (u_{n}) be a sequence of W^{1,x}L_{M}(Q) such that

u_{n}* u weakly in W^{1,x}L_{M}(Q) for σ(ΠL_{M},ΠE_{M})
and ∂u_{n}

∂t =h_{n}+k_{n} in D^{0}(Q)

with (hn) bounded in W^{−1,x}L_{M}(Q) and (kn) bounded in the space M(Q) of
measures onQ.

Then u_{n}→u strongly in L^{1}_{loc}(Q).

If further u_{n}∈W_{0}^{1,x}L_{M}(Q) thenu_{n}→u strongly inL^{1}(Q).

Proof: It is easily adapted from that given in [6] by using Theorem 2 and Remark 1 instead of lemma 8 of [21].

4 – Approximation and time mollification

In this section, Ω is an open subset of R^{N} with the segment property and
I is a subinterval of R(both possibly unbounded) and Q= Ω×I.

Definition 1. We say thatun→uinW^{−1,x}L_{M}(Q) +L^{1}(Q) for the modular
convergence if we can write

un = ^{X}

|α|≤1

D^{α}_{x}u^{α}_{n}+u^{0}_{n} and u = ^{X}

|α|≤1

D^{α}_{x}u^{α}+u^{0}

with u^{α}_{n} → u^{α} in L_{M}(Q) for the modular convergence ∀ |α| ≤ 1 and u^{0}_{n} → u^{0}
strongly inL^{1}(Q).

This implies, in particular, thatu_{n}→uinW^{−1,x}L_{M}(Q)+L^{1}(Q) for the weak
topologyσ(ΠL_{M} +L^{1},ΠL_{M} ∩L^{∞}) in the sense thathu_{n}, vi → hu, vi for all v ∈
W_{0}^{1,x}L_{M}(Q)∩L^{∞}(Q) where here and throughout the paperh, imeans for either
the pairing between W_{0}^{1,x}L_{M}(Q) and W^{−1,x}L_{M}(Q), or between W_{0}^{1,x}L_{M}(Q)∩
L^{∞}(Q) and W^{−1,x}L_{M}(Q) +L^{1}(Q); indeed,

hu_{n}, vi = ^{X}

|α|≤1

(−1)^{|α|}

Z

Q

u^{α}_{n}D^{α}_{x}v dx dt +
Z

Q

u^{0}_{n}v dx dt

and since for all |α| ≤1, u^{α}_{n} → u^{α} in L_{M}(Q) for the modular convergence, and
so forσ(L_{M}, L_{M}), we have

X

|α|≤1

(−1)^{|α|}

Z

Q

u^{α}_{n}D^{α}_{x}v dx dt +
Z

Q

u^{0}_{n}v dx dt →

→ ^{X}

|α|≤1

(−1)^{|α|}

Z

Qu^{α}D^{α}_{x}v dx dt +
Z

Qu^{0}v dx dt = hu, vi .
Moreover, if v_{n}→vinW_{0}^{1,x}L_{M}(Q) for the modular convergence and weakly*

inL^{∞}(Q), we have hu_{n}, v_{n}i → hu, vi asn→ ∞, see (3).

We shall prove the following approximation theorem which plays a fundamen- tal role when proving the existence of solutions for parabolic problems.

Theorem 3. If u ∈ W^{1,x}L_{M}(Q)∩L^{1}(Q) (resp. W_{0}^{1,x}L_{M}(Q)∩L^{1}(Q)) and

∂u/∂t∈W^{−1,x}L_{M}(Q) +L^{1}(Q) then there exists a sequence (v_{j}) inD(Q) (resp.

D(I,D(Ω))) such that

v_{j} →u in W^{1,x}L_{M}(Q) and ∂v_{j}

∂t → ∂u

∂t in W^{−1,x}L_{M}(Q) +L^{1}(Q)
for the modular convergence.

Proof: Letu∈W^{1,x}L_{M}(Q)∩L^{1}(Q) such that∂u/∂t∈W^{−1,x}L_{M}(Q)+L^{1}(Q)
and letε >0 be given. Writing ∂u/∂t=^{P}_{|α|≤1}D_{x}^{α}u^{α}+u^{0}, whereu^{α} ∈L_{M}(Q)
for all|α| ≤1 and u^{0} ∈L^{1}(Q), we will show that there exists λ >0 (depending
only on u and N) and there exists v ∈ D(Q) for which we can write ∂v/∂t =
P

|α|≤1D^{α}_{x}v^{α}+v^{0} withv^{α}, v^{0} ∈ D(Q) such that
Z

Q

M

µD^{α}_{x}v−D_{x}^{α}u
λ

¶

dx dt ≤ ε , Z

Q

M

µv^{α}−u^{α}
λ

¶

dx dt ≤ ε (7)

∀ |α| ≤1 and kv^{0}−u^{0}k_{L}^{1}_{(Q)} ≤ε .
We will process as in [15] (see the proofs of Theorem 3 and Theorem 4). Since
the approximation ofu and D^{α}_{x}u can be obtained in the same way, we will only
show that the approximation also holds for the time derivative. Thus, we consider
ϕ∈ D(R^{N}^{+1}) with 0 ≤ϕ≤1, ϕ = 1 for |(x, t)| ≤1 andϕ= 0 for |(x, t)| ≥2.

Letϕ_{r}(x, t) =ϕ((x, t)/r) and letu_{r} =ϕ_{r}u.

On the one hand, we have

∂u_{r}

∂t = ϕ_{r}

X

|α|≤1

D_{x}^{α}u^{α}+u^{0}

+1 r

∂ϕ

∂t

µ(x, t) r

¶ u

= ^{X}

|α|≤1

D_{x}^{α}(ϕ_{r}u^{α}) +

−1 r

X

|α|=1

D_{x}^{α}ϕ
µ(x, t)

r

¶
u^{α}

+

·1 r

∂ϕ

∂t

µ(x, t) r

¶

u+ϕ_{r}u^{0}

¸

:= u^{1}_{r}+u^{2}_{r}+u^{3}_{r} .

Whenr→ ∞, we have, by Lemma 5 of [15],u^{1}_{r}→^{P}_{|α|≤1}D^{α}_{x}u^{α} inW^{−1,x}L_{M}(Q)
for the modular convergence and, by direct examination, u^{2}_{r} → 0 strongly in
L_{M}(Q) and u^{3}_{r} → u^{0} strongly in L^{1}(Q). Hence, we can choose λ > 0 (namely
such that D^{α}_{x}u/λ ∈ L_{M}(Q) and u^{α}/λ ∈ L_{M}(Q) for all |α| ≤ 1) and r > 0 such
that

Z

Q

M^{³}(D_{x}^{α}u_{r}−D_{x}^{α}u)/λ^{´}dx dt ≤ ε ∀ |α| ≤1,
Z

Q

M(u^{2}_{r}/λ)dx dt ≤ ε
ku^{3}_{r}−u^{0}k_{L}^{1}_{(Q)} ≤ε and

Z

Q

M^{³}(ϕ_{r}u^{α}−u^{α})/λ^{´}dx dt ≤ ε ∀ |α| ≤1.
(8)

On the other hand, let ψ_{i} be a C^{∞} partition of unity onQ subordinate to a
covering{Ui} of Q satisfying the properties of lemma 7 of [15] and consider the
translated function (ψivr)ti defined by (ψivr)ti(x, t) = (ψivr)((x, t) +tiyi) where
y_{i} is the vector associated to U_{i} by the segment property. Let ρ_{σ} be a mollifier
sequence in R^{N}^{+1}, that is, ρ_{σ} ∈ D(R^{N+1}), ρ_{σ}(x, t) = 0 for |(x, t)| ≥ σ, ρ_{σ} ≥ 0
and ^{R}_{R}N+1ρ_{σ} = 1. Extending u_{r} outside Q by zero, we see that ψ_{i}u_{r} vanishes
identically for alli≥ somei_{r}. As in [15], we define

v =

ir

X

i=1

(ψ_{i}u_{r})_{t}_{i}∗ρ_{σ}_{i} ∈ D(Q) .
Clearly, we have

∂v

∂t =

ir

X

i=1

(ψiu^{1}_{r})ti∗ρσi+

ir

X

i=1

(ψiu^{2}_{r})ti∗ρσi+

ir

X

i=1

(ψiu^{3}_{r})ti∗ρσi+

ir

X

i=1

µ∂ψ_{i}

∂t ur

¶

ti

∗ρσi

and since

ir

X

i=1

(ψiu^{1}_{r})ti∗ρσi =

ir

X

i=1

ψi

X

|α|≤1

D^{α}_{x}(ϕru^{α})

ti

∗ρσi =

=

ir

X

i=1

X

|α|≤1

D_{x}^{α}(ψ_{i}ϕ_{r}u^{α})

ti

∗ρ_{σ}_{i} −

ir

X

i=1

X

|α|=1

(D^{α}_{x}ψ_{i})ϕ_{r}u^{α}

ti

∗ρ_{σ}_{i}

= ^{X}

|α|≤1

Ã _{i}_{r}
X

i=1

³D_{x}^{α}(ψ_{i}ϕ_{r}u^{α})^{´}

ti

∗ρ_{σ}_{i}

!

−

ir

X

i=1

X

|α|=1

(D^{α}_{x}ψ_{i})ϕ_{r}u^{α}

ti

∗ρ_{σ}_{i}
we deduce that

∂v

∂t = ^{X}

|α|≤1

D_{x}^{α}v^{α}+v^{2}+v^{3}
where, as it can be easily seen

v^{α} =

ir

X

i=1

(ψiϕru^{α})ti∗ρσi ∀ |α| ≤1 ,

v^{2} =

ir

X

i=1

(ψ_{i}u^{2}_{r})_{t}_{i}∗ρ_{σ}_{i} −

ir

X

i=1

X

|α|=1

D_{x}^{α}(ψ_{i})ϕ_{r}u^{α}

ti

∗ρ_{σ}_{i}

v^{3} =

ir

X

i=1

(ψ_{i}u^{3}_{r})_{t}_{i}∗ρ_{σ}_{i} +

ir

X

i=1

µ∂ψ_{i}

∂t u_{r}

¶

ti

∗ρ_{σ}_{i} .

Now, for each i= 1, ..., i_{r}, we can choose (see lemma 5 of [15]) 0 < t_{i} < 1 and
ρσi =ρi such that

Z

Q

M
ÃÃ_{i}_{r}

X

i=1

(ψ_{i}D_{x}^{α}u_{r})_{t}_{i}∗ρ_{i}−D_{x}^{α}u_{r}

! /λ

!

dx dt ≤ ε ∀ |α| ≤1, Z

Q

M^{³}(v^{2}−u^{2}_{r})/λ^{´}dx dt ≤ ε,
(9)

kv^{3}−u^{3}_{r}k_{L}^{1}_{(Q)} ≤ ε ,
Z

QM
ÃÃ_{i}_{r}

X

i=1

(ψiϕru^{α})ti ∗ρi−ϕru^{α}

! /λ

!

dx dt ≤ ε ∀ |α| ≤1. Combining (8) and (9), we get the result.

The case where u ∈ W_{0}^{1,x}L_{M}(Q)∩L^{1}(Q) can be handled similarly without
essential difficulty as it is mentioned in the proof of theorem 4 of [15].

Remark 2. The assumptionu∈L^{1}(Q) in theorem 3 is needed only whenQ
has infinite measure, since else, we have L_{M}(Q) ⊂L^{1}(Q) and so W^{1,x}L_{M}(Q)∩
L^{1}(Q) =W^{1,x}L_{M}(Q).

Remark 3. If, in the statement of theorem 3 above, one takes I =R,
we have that D(Ω×R) is dense in {u ∈ W_{0}^{1,x}L_{M}(Ω×R)∩L^{1}(Ω×R) : ∂u/∂t ∈
W^{−1,x}L_{M}(Ω×R) +L^{1}(Ω×R)} for the modular convergence. This trivially fol-
lows from the fact thatD(R,D(Ω))≡ D(Ω×R).

A first application of theorem 3 is the following trace result (see [19], Theorem 1.1, for the case of ordinary Sobolev spaces).

Lemma 5. Let a < b∈Rand Ω be a bounded open subset ofR^{N} with the
segment property. Then

½

u∈W_{0}^{1,x}L_{M}(Ω×(a, b)) : ∂u/∂t∈W^{−1,x}L_{M}(Ω×(a, b)) +L^{1}(Ω×(a, b))

¾

⊂

⊂ C([a, b], L^{1}(Ω)).

Proof: Letu∈W_{0}^{1,x}LM(Ω×(a, b)) such that∂u/∂t∈W^{−1,x}L_{M}(Ω×(a, b)) +
L^{1}(Ω×(a, b)). After two consecutive reflections first with respect to t = b and
then with respect tot=a:

ˆ

u(x, t) = u(x, t)χ_{(a,b)}+u(x,2b−t)χ_{(b,2b−a)} on Ω×(a,2b−a)
and

˜

u(x, t) = ˆu(x, t)χ_{(a,2b−a)}+ ˆu(x,2a−t)χ_{(3a−2b,a)} on Ω×(3a−2b,2b−a),
we get a function ˜u∈W_{0}^{1,x}L_{M}(Ω×(3a−2b,2b−a)) with∂˜u/∂t∈W^{−1,x}L_{M}(Ω×
(3a−2b,2b−a)) +L^{1}(Ω×(3a−2b,2b−a)). Now, by letting a functionη∈ D(R)
with η = 1 on [a, b] and suppη ⊂ (3a−2b,2b−a), we set u = ηu; therefore,˜
by standard arguments (see [7], Lemme IV and Remarque 10 p. 158), we have:

u=u on Ω×(a, b),u∈W_{0}^{1,x}L_{M}(Ω×R)∩L^{1}(Ω×R) and∂u/∂t∈W^{−1,x}L_{M}(Ω×R)+

L^{1}(Ω×R).

Let now v_{j} the sequence given by theorem 3 corresponding tou, that is,
v_{j} →u in W_{0}^{1,x}L_{M}(Ω×R)

and

∂vj

∂t → ∂u

∂t in W^{−1,x}L_{M}(Ω×R) +L^{1}(Ω×R)
for the modular convergence.

Throughout this paper, we denoteT_{k}the usual truncation at heightkdefined
on R by T_{k}(s) = min(k,max(s,−k)) and S_{k}(s) = ^{R}_{0}^{s}T_{k}(t)dt its primitive. We
have,

Z

ΩS1(vi−vj)(τ)dx = Z

Ω

Z _{τ}

−∞T1(vi−vj)
µ∂v_{i}

∂t−∂v_{j}

∂t

¶

dx dt → 0 as i, j→ ∞,
from which, by following [19], one deduces that v_{j} is a Cauchy sequence in
C(R, L^{1}(Ω)) and hence u∈C(R, L^{1}(Ω)). Consequently,u∈C([a, b], L^{1}(Ω)).

In order to deal with the time derivative, we introduce a time mollification of
a functionu∈L_{M}(Q). Thus we define, for allµ >0 and all (x, t)∈Q

u_{µ}(x, t) = µ
Z _{t}

−∞

˜

u(x, s) exp^{³}µ(s−t)^{´}ds
(10)

where ˜u(x, s) =u(x, s)χ_{(0,T}_{)}(s) is the zero extension of u.

Throughout the paper the index µalways indicates this mollification.

Proposition 1. If u ∈ L_{M}(Q) then u_{µ} is measurable in Q and ∂u_{µ}/∂t =
µ(u−u_{µ}) and ifu∈ L_{M}(Q)then

Z

QM(uµ)dx dt ≤ Z

QM(u)dx dt .

Proof: Since (x, t, s) 7−→ u(x, s) exp(µ(s−t)) is measurable in Ω×[0, T]×

[0, T],we deduce thatu_{µ}is measurable by Fubini’s theorem. By Jensen’s integral
inequality we have, since^{R}_{−∞}^{0} µexp(µs)ds= 1,

M
µZ _{t}

−∞µu(x, s) exp˜ ^{³}µ(s−t)^{´}ds

¶

= M

µZ _{0}

−∞µexp(µs) ˜u(x, s+t)ds

¶

≤ Z 0

−∞

µexp(µs)M(˜u(x, s+t))ds which implies

Z

QM(uµ(x, t))dx dt ≤ Z

Ω×R

µZ _{0}

−∞µexp(µs)M(˜u(x, s+t))ds

¶ dx dt

≤ Z 0

−∞µexp(µs) µZ

Ω×RM(˜u(x, s+t))dx dt

¶ ds

≤ Z 0

−∞

µexp(µs) µZ

Q

M(u(x, t))dx dt

¶ ds

= Z

Q

M(u)dx dt .

Furthermore

∂u_{µ}

∂t = lim

θ→0

1

θ(e^{−µθ}−1)u_{µ}(x, t) + lim

θ→0

1 θ

Z t+θ t

u(x, s)e^{µ(s−(t+θ))}ds

= −µu_{µ}+µu .
Proposition 2.

1) If u ∈ L_{M}(Q) then u_{µ} → u as µ → +∞ in L_{M}(Q) for the modular
convergence.

2) If u ∈ W^{1,x}L_{M}(Q) then u_{µ} → u as µ → +∞ in W^{1,x}L_{M}(Q) for the
modular convergence.

Proof: 1) Let (ϕ_{k}) ⊂ D(Q) such that ϕ_{k} → u in L_{M}(Q) for the modular
convergence. Letλ >0 large enough such that

u

λ ∈ L_{M}(Q) and
Z

QM

µϕ_{k}−u
λ

¶

dx dt → 0 as k→ ∞ . For a.e. (x, t)∈Qwe have

|(ϕ_{k})_{µ}(x, t)−ϕ_{k}(x, t)| = 1
µ

¯

¯

¯

¯

∂ϕ_{k}

∂t (x, t)

¯

¯

¯

¯

≤ 1 µ

°

°

°

°

∂ϕ_{k}

∂t

°

°

°

°_{∞}
.
On the other hand

Z

Q

M

µu_{µ}−u
3λ

¶

dx dt ≤ 1 3

Z

Q

M

µu_{µ}−(ϕ_{k})_{µ}
λ

¶

dx dt + 1 3

Z

Q

M

µ(ϕ_{k})_{µ}−ϕ_{k}
λ

¶ dx dt + 1

3 Z

Q

M

µϕ_{k}−u
λ

¶ dx dt

≤ 1 3

Z

QM

µ(ϕ_{k}−u)µ

λ

¶

dx dt + 1 3

Z

QM

µ(ϕ_{k})µ−ϕ_{k}
λ

¶ dx dt + 1

3 Z

Q

M

µϕ_{k}−u
λ

¶ dx dt . This implies that

Z

QM

µuµ−u 3λ

¶

dx dt ≤ 2 3 Z

QM

µϕ_{k}−u
λ

¶

dx dt+ 1 3M

µ 1 µλ

°

°

°

°

∂ϕ_{k}

∂t

°

°

°

°∞

¶

meas(Q). Let ε >0.There exists ksuch that

Z

Q

M

µϕ_{k}−u
λ

¶

dx dt ≤ ε

and there existsµ_{0} such that
M

µ 1 µλ

°

°

°

°

∂ϕ_{k}

∂t

°

°

°

°_{∞}

¶

meas(Q)≤ε for all µ≥µ_{0} .
Hence

Z

QM

µu_{µ}−u
3λ

¶

dx dt ≤ ε for all µ≥µ_{0} .

2) Since∀α, |α| ≤1,we haveD^{α}_{x}(u_{µ}) = (D^{α}_{x}u)_{µ}, consequently, the first part
above applied on eachD_{x}^{α}u,gives the result.

Remark 4. If u ∈ E_{M}(Q), we can choose λ arbitrary small since D(Q) is
(norm) dense inE_{M}(Q).Thus, for allλ >0

Z

Q

M

µu_{µ}−u
λ

¶

dx dt → 0 as µ→+∞

andu_{µ}→u strongly inE_{M}(Q). Idem forW^{1,x}E_{M}(Q).

Proposition 3. If u_{n} → u in W^{1,x}L_{M}(Q) strongly (resp. for the modular
convergence) then (u_{n})_{µ} → u_{µ} in W^{1,x}L_{M}(Q) strongly (resp. for the modular
convergence).

Proof: For allλ >0 (resp. for some λ >0), Z

Q

M

ÃD_{x}^{α}((u_{n})_{µ})−D^{α}_{x}(u_{µ})
λ

!

dx dt ≤ Z

Q

M

ÃD^{α}_{x}(u_{n})−D_{x}^{α}(u)
λ

!

dx dt → 0
as n→ ∞ ,
then (un)µ→uµinW^{1,x}LM(Q) strongly (resp. for the modular convergence).

5 – Existence theorem

Let Ω be a bounded open subset of R^{N} (N ≥2) with the segment property,
T >0 and set Q= Ω×(0, T). LetM be an N-function.

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

A(u) = −diva(x, t, u,∇u)

where a: Ω×[0, T]×R×R^{N}→ R^{N} is a Carath´eodory function satisfying for a.e.

(x, t)∈Ω×[0, T] and all s∈R, ξ6=ξ^{∗}∈R^{N}:

|a(x, t, s, ξ)| ≤ β(|s|)^{³}c_{1}(x, t) +M^{−1}M(γ|ξ|)^{´}
(11)

ha(x, t, s, ξ)−a(x, t, s, ξ^{∗})^{i}[ξ−ξ^{∗}] > 0
(12)

a(x, t, s, ξ)ξ ≥ α M(|ξ|) (13)

where c_{1}(x, t) ∈ E_{M}(Q), c_{1}≥0; β : [0,+∞) → [0,+∞) a continuous and non-
decreasing function; α, γ >0.

Note that, (13) written forξ=εζ, ε >0,and the fact thatais a Carath´eodory function, imply that

a(x, t, s,0) = 0 for almost every (x, t)∈Q and every s∈R.

Let g : Ω×[0, T]×R×R^{N}→ R be a Carath´eodory function satisfying for a.e.

(x, t)∈Ω×(0, T) and for alls∈R, ξ∈R^{N} :

|g(x, t, s, ξ)| ≤ b(|s|)^{³}c_{2}(x, t) +M(|ξ|)^{´}
(14)

g(x, t, s, ξ)s ≥ 0 (15)

where c_{2}(x, t)∈L^{1}(Q) and b:R^{+}→R^{+} is a continuous and nondecreasing
function. Furthermore let

f ∈L^{1}(Q) .
(16)

Throughout this paperh, imeans for either the pairing betweenW_{0}^{1,x}L_{M}(Q)∩

L^{∞}(Q) andW^{−1,x}L_{M}(Q)+L^{1}(Q) or betweenW_{0}^{1,x}L_{M}(Q) andW^{−1,x}L_{M}(Q) and
Q_{τ} = Ω×(0, τ) for τ ∈[0, T].

Consider, then, the following parabolic initial-boundary value problem:

∂u

∂t +A(u) +g(x, t, u,∇u) =f in Q

u(x, t) = 0 on ∂Ω×(0, T)

u(x,0) =u_{0}(x) in Ω

(17)

whereu_{0} is a given function in L^{1}(Ω).

Let us now precise in which sense the problem (17) will be solved. Thus, we state, as in [19], the following