International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 2 (2014), 174-191
http://www.etamaths.com

PARABOLIC EQUATIONS IN MUSIELAK-ORLICZ-SOBOLEV

SPACES

M.L. AHMED OUBEID1, A. BENKIRANE1 AND M. SIDI EL VALLY2,∗

Abstract. We prove in this paper the existence of solutions of nonlinear
parabolic problems in Musielak-Orlicz-Sobolev spaces. An approximation and

a trace results in inhomogeneous Musielak-Orlicz-Sobolev spaces have also

been provided.

1. INTRODUCTION

Let Ω a bounded open subset of Rn and let Q be the cylinder Ω × (0,T) with
some given T > 0.

This paper is concerned with the existence of solutions for boundary value prob-
lems for quasi-linear parabolic equations of the form




∂u
∂t

+ A(u) = f in Q
u(x,t) = 0 on ∂Ω × (0,T)
u(x, 0) = u0(x) in Ω

(1)

where A is a Leray-Lions operator of the form:

A(u) = − div (a(x,t,u,∇u)) + a0(x,t,u,∇u),
with the coefficients a and a0 satisfying the classical Leray-Lions conditions.
Consider first the case where a and a0 have polynomial growth with respect to u and
∇u. Therefore A is a bounded operator from Lp(0,T,W 1,p0 (Ω)), 1 < p < ∞,into its
dual. In this setting, it is well known that problems of the form (1) were solved by
Lions [16], and Brzis and Browder [9] in the case where p ≥ 2, and by Landes [14]
and Landes and Mustonen [15] when 1 < p < 2. See also [6, 7] for related topics.

In the case where a and a0 satisfy a more general growth with respect to u and
∇u (for example of exponential or logarithmic type), it is shown in [10] that the
adequate space in which (1) can be studied is the inhomogeneous Orlicz-Sobolev
space W 1,xLM (Q), where the N-function M is related to the actual growth of a
and a0. The solvability of (1) in this setting was proved by Donaldson [10] and
Robert [18] when A is monotone, and by Elmahi [11] and Elmahi-Meskine [12].

2010 Mathematics Subject Classification. 46E35, 35K15, 35K20, 35K60.
Key words and phrases. Inhomogeneous Musielak-Orlicz-Sobolev spaces; parabolic problems;

Musielak-Orlicz function.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

174



PARABOLIC EQUATIONS 175

Our purpose in this paper is to prove existence theorems for the problem (1)
in the setting of inhomogeneous Musielak-Orlicz-Sobolev spaces W 1,xLϕ(Q) by ap-
plying some new approximation result in inhomogeneous Musielak-Orlicz-Sobolev
spaces (see Theorem 1), as it is done in the setting of Orlicz-Sobolev spaces (see
[12]), which allows us, on the one hand, to regularize a test function by smooth ones
with converging time derivatives (and thus enlarge the set of test functions in order
to cover the solution u and then get the energy equality), and, on the other hand,

to prove a trace result (see Lemma 3) which states that if u ∈ W 1,x0 Lϕ(Q)∩L
2(Q)

such that ∂u
∂t
∈ W−1,xLψ(Q) + L2(Q), then u ∈ C([0,T],L2(Ω)), showing that the

assumption u0 ∈ L2(Ω) cannot be weakened.
Our result generalizes that of the Elmahi-Meskine in [12] to the case of inhomo-

geneous Musielak-Orlicz-Sobolev spaces.

Let us point out that our result can be applied in the particular case when
ϕ(x,t) = tp(x), in this case we use the notations Lp(x)(Ω) = Lϕ(Ω), and W

m,p(x)(Ω) =
WmLϕ(Ω). These spaces are called Variable exponent Lebesgue and Sobolev
spaces.

For some classical and recent results on elliptic and parabolic problems in Orlicz-
sobolev spaces and a Musielak-Orlicz-Sobolev spaces, we refer to [1, 2, 5, 10, 11, 12].

2. PRELIMINARIES

In this section we list briefly some definitions and facts about Musielak-Orlicz-
Sobolev spaces. Standard reference is [17]. We also include the definition of inho-
mogeneous Musielak-Orlicz-Sobolev spaces and some preliminaries Lemmas to be
used later.

Musielak-Orlicz-Sobolev spaces : Let Ω be an open subset of Rn.
A Musielak-Orlicz function ϕ is a real-valued function defined in Ω×R+ such that
:

a): ϕ(x,t) is an N-function i.e. convex, nondecreasing, continuous, ϕ(x, 0) =
0, ϕ(x,t) > 0 for all t > 0 and

lim
t−→0

sup
x∈Ω

ϕ(x,t)

t
= 0

lim
t−→∞

inf
x∈Ω

ϕ(x,t)

t
= 0.

b): ϕ(., t) is a Lebesgue measurable function

Now, let ϕx(t) = ϕ(x,t) and let ϕ
−1
x be the non-negative reciprocal function with

respect to t, i.e the function that satisfies

ϕ−1x (ϕ(x,t)) = ϕ(x,φ
−1
x ) = t.

For any two Musielak-Orlicz functions ϕ and γ we introduce the following or-
dering :



176 OUBEID, BENKIRANE AND VALLY

c): if there exists two positives constants c and T such that for almost every-
where x ∈ Ω :

ϕ(x,t) ≤ γ(x,ct) for t ≥ T
we write ϕ ≺ γ and we say that γ dominates ϕ globally if T = 0 and near
infinity if T > 0.

d): if for every positive constant c and almost everywhere x ∈ Ω we have

lim
t→0

(sup
x∈Ω

ϕ(x,ct)

γ(x,t)
) = 0 or lim

t→∞
(sup
x∈ϕ

ϕ(x,ct)

γ(x,t)
) = 0

we write ϕ ≺≺ γ at 0 or near ∞ respectively, and we say that ϕ increases
essentially more slowly than γ at 0 or near infinity respectively.

In the sequel the measurability of a function u : Ω 7→ R means the Lebesgue mea-
surability.

We define the functional

%ϕ,Ω(u) =

∫
Ω

ϕ(x, |u(x)|)dx

where u : Ω 7→ R is a measurable function.
The set

Kϕ(Ω) = {u : Ω → R mesurable /%ϕ,Ω(u) < +∞} .
is called the Musielak-Orlicz class (the generalized Orlicz class).

The Musielak-Orlicz space (the generalized Orlicz spaces) Lϕ(Ω) is the vector
space generated by Kϕ(Ω), that is, Lϕ(Ω) is the smallest linear space containing
the set Kϕ(Ω).
Equivelently:

Lϕ(Ω) =

{
u : Ω → R mesurable /%ϕ,Ω(

|u(x)|
λ

) < +∞, for some λ > 0
}

Let

ψ(x,s) = sup
t≥0
{st−ϕ(x,t)},

ψ is the Musielak-Orlicz function complementary to ( or conjugate of ) ϕ(x,t) in
the sense of Young with respect to the variable s.

On the space Lϕ(Ω) we define the Luxemburg norm:

||u||ϕ,Ω = inf{λ > 0/
∫

Ω

ϕ(x,
|u(x)|
λ

)dx,≤ 1}.

and the so-called Orlicz norm :

|||u|||ϕ,Ω = sup
||v||ψ≤1

∫
Ω

|u(x)v(x)|dx.

where ψ is the Musielak-Orlicz function complementary to ϕ. These two norms are
equivalent [17].

The closure in Lϕ(Ω) of the set of bounded measurable functions with compact

support in Ω is denoted by Eϕ(Ω). It is a separable space and Eψ(Ω)
∗ = Lϕ(Ω)

[17].



PARABOLIC EQUATIONS 177

The following conditions are equivalent:

e): Eϕ(Ω) = Kϕ(Ω)

f ): Kϕ(Ω) = Lϕ(Ω)
g): ϕ has the ∆2 property.

We recall that ϕ has the ∆2 property if there exists k > 0 independent of x ∈ Ω
and a nonnegative function h , integrable in Ω such that ϕ(x, 2t) ≤ kϕ(x,t) + h(x)
for large values of t, or for all values of t, according to whether Ω has finite measure
or not.

Let us define the modular convergence: we say that a sequence of functions
un ∈ Lϕ(Ω) is modular convergent to u ∈ Lϕ(Ω) if there exists a constant k > 0
such that

lim
n→∞

%ϕ,Ω(
un −u
k

) = 0.

For any fixed nonnegative integer m we define

WmLϕ(Ω) = {u ∈ Lϕ(Ω) : ∀|α| ≤ m Dαu ∈ Lϕ(Ω)}

where α = (α1,α2, ...,αn) with nonnegative integers αi; |α| = |α1|+|α2|+...+|αn|
and Dαu denote the distributional derivatives.
The space WmLϕ(Ω) is called the Musielak-Orlicz-Sobolev space.

Now, the functional

%ϕ,Ω(u) =
∑
|α|≤m

%ϕ,Ω(D
αu),

for u ∈ WmLϕ(Ω) is a convex modular. and

||u||mϕ,Ω = inf{λ > 0 : %ϕ,Ω(
u

λ
) ≤ 1}

is a norm on WmLϕ(Ω).
The pair 〈WmLϕ(Ω), ||u||mϕ,Ω〉 is a Banach space if ϕ satisfies the following condition
:

there exist a constant c > 0 such that inf
x∈Ω

ϕ(x, 1) ≥ c,

as in [17].

The space WmLϕ(Ω) will always be identified to a σ(ΠLϕ, ΠEψ) closed subspace
of the product

∏
|α|≤m Lϕ(Ω) =

∏
Lϕ.

Let Wm0 Lϕ(Ω) be the σ(ΠLϕ, ΠEψ) closure of D(Ω) in W
mLϕ(Ω).

Let WmEϕ(Ω) be the space of functions u such that u and its distribution deriva-
tives up to order m lie in Eϕ(Ω), and let W

m
0 Eϕ(Ω) be the (norm) closure of D(Ω)

in WmLϕ(Ω).



178 OUBEID, BENKIRANE AND VALLY

The following spaces of distributions will also be used:

W−mLψ(Ω) = {f ∈ D′(Ω); f =
∑
|α|≤m

(−1)|α|Dαfα with fα ∈ Lψ(Ω)}

W−mEψ(Ω) = {f ∈ D′(Ω); f =
∑
|α|≤m

(−1)|α|Dαfα with fα ∈ Eψ(Ω)}

As we did for Lϕ(Ω), we say that a sequence of functions un ∈ WmLϕ(Ω) is
modular convergent to u ∈ WmLϕ(Ω) if there exists a constant k > 0 such that

lim
n→∞

%ϕ,Ω(
un −u
k

) = 0.

From [17], for two complementary Musielak-Orlicz functions ϕ and ψ the follow-
ing inequalities hold :

h) : the young inequality :

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

i) : the Hölder inequality :∣∣∣∣
∫

Ω

u(x)v(x) dx

∣∣∣∣ ≤ ||u||ϕ,Ω|||v|||ψ,Ω.
for all u ∈ Lϕ(Ω) and v ∈ Lψ(Ω).

Inhomogeneous Musielak-Orlicz-Sobolev spaces :

Let Ω an bounded open subset of Rn and let Q = Ω×]0,T[ with some given T ¿
0. Let ϕ be a Musielak function. For each α ∈ Nn , denote by Dαx the distributional
derivative on Q of order α with respect to the variable x ∈ Rn. The inhomogeneous
Musielak-Orlicz-Sobolev spaces of order 1 are defined as follows.

W 1,xLϕ(Q) = {u ∈ Lϕ(Q) : ∀|α| ≤ 1 Dαxu ∈ Lϕ(Q)}
and

W 1,xEϕ(Q) = {u ∈ Eϕ(Q) : ∀|α| ≤ 1 Dαxu ∈ Eϕ(Q)}
The last space is a subspace of the first one, and both are Banach spaces under the
norm

‖u‖ =
∑
|α|≤m

‖Dαxu‖ϕ,Q.

We can easily show that they form a complementary system when Ω is a Lipschitz
domain [4]. These spaces are considered as subspaces of the product space ΠLϕ(Q)
which has (N + 1) copies. We shall also consider the weak topologies σ(ΠLϕ, ΠEψ)
and σ(ΠLϕ, ΠLψ). If u ∈ W 1,xLϕ(Q) then the function : t 7−→ u(t) = u(t, .) is
defined on (0,T) with values in W 1Lϕ(Ω). If, further, u ∈ W 1,xEϕ(Q) then this
function is a W 1Eϕ(Ω)-valued and is strongly measurable. Furthermore the follow-
ing imbedding holds : W 1,xEϕ(Q) ⊂ L1(0,T; W 1Eϕ(Ω)). The space W 1,xLϕ(Q)
is not in general separable, if u ∈ W 1,xLϕ(Q), we can not conclude that the func-
tion u(t) is measurable on (0,T). However, the scalar function t 7→ ‖u(t)‖ϕ,Ω is



PARABOLIC EQUATIONS 179

in L1(0,T). The space W
1,x
0 Eϕ(Q) is defined as the (norm) closure in W

1,xEϕ(Q)
of D(Q). We can easily show as in [4] that when Ω a Lipschitz domain then each
element u of the closure of D(Q) with respect of the weak * topology σ(ΠLϕ, ΠEψ)
is limit, in W 1,xLϕ(Q), of some subsequence (ui) ⊂ D(Q) for the modular conver-
gence; i.e., there exists λ > 0 such that for all |α| ≤ 1,∫

Q

ϕ(x, (
Dαxui −Dαxu

λ
)) dxdt → 0 as i →∞,

this implies that (ui) converges to u in W
1,xLϕ(Q) for the weak topology σ(ΠLM, ΠLψ).

Consequently

D(Q)
σ(ΠLϕ,ΠEψ)

= D(Q)
σ(ΠLϕ,ΠLψ)

,

this space will be denoted by W
1,x
0 Lψ(Q). Furthermore, W

1,x
0 Eϕ(Q) = W

1,x
0 Lϕ(Q)∩

ΠEϕ.

Poincaré’s inequality also holds in W
1,x
0 Lϕ(Q) i.e. there is a constant C > 0

such that for all u ∈ W 1,x0 Lϕ(Q) one has∑
|α|≤1

‖Dαxu‖ϕ,Q ≤ C
∑
|α|=1

‖Dαxu‖ϕ,Q.

Thus both sides of the last inequality are equivalent norms on W
1,x
0 Lϕ(Q). We

have then the following complementary system(
W

1,x
0 Lϕ(Q) F

W
1,x
0 Eϕ(Q) F0

)
,

F being the dual space of W
1,x
0 Eϕ(Q). It is also, except for an isomorphism,

the quotient of ΠLψ by the polar set W
1,x
0 Eϕ(Q)

⊥, and will be denoted by F =
W−1,xLψ(Q) and it is shown that

W−1,xLψ(Q) =
{
f =

∑
|α|≤1

Dαxfα : fα ∈ Lψ(Q)
}
.

This space will be equipped with the usual quotient norm

‖f‖ = inf
∑
|α|≤1

‖fα‖ψ,Q

where the inf is taken on all possible decompositions

f =
∑
|α|≤1

Dαxfα, fα ∈ Lψ(Q).

The space F0 is then given by

F0 =
{
f =

∑
|α|≤1

Dαxfα : fα ∈ Eψ(Q)
}

and is denoted by F0 = W
−1,xEψ(Q).

The following technical lemmas are important for the proof of our main result.
Lemma 1. If u ∈ W 1,10 (Ω), then ||uσ −u||1,Ω ≤ σ||∇u||1,Ω, where uσ = u∗ρσ and
where (ρσ) is a mollifier sequence in RN .



180 OUBEID, BENKIRANE AND VALLY

Lemma 2. Let ϕ be an Musielak-Orlicz function. Let (un) be a bounded

sequence in W
1,x
0 Lϕ(Q) ∩ L

∞(0,T; L1(Ω)). If un(t) ⇀ u(t) weakly in L
1(Ω) for

almost every t ∈ [0,T], then un → u strongly in L1(Q).

Proof.For each v ∈ W 1,x0 Lϕ(Q), denote vσ(x,t) =
∫
RN v(y,t)ρσ(x−y)dy, where

v(y,t) = 0 if y /∈ Ω and where (ρσ) is a mollifier sequence in RN .
Since un(t) ⇀ u(t) weakly in L

1(Ω), we have unσ(x,t) → uσ(x,t) almost every-
where in Q
and unσ(t) → uσ(t) strongly in L1(Ω) for almost every t ∈ [0,T],we have∫

Ω

|un(t) −uk(t)|dx ≤
∫

Ω

|un(t) −unσ(t)|dx +
∫

Ω

|unσ(t) −ukσ(t)|dx +
∫

Ω

|ukσ(t) −uk(t)|dx

≤ σ(
∫

Ω

|∇un(t)|dx +
∫

Ω

|∇uk(t)|dx) + ||unσ(t) −ukσ(t)||1,Ω

Integrating this over [0,T] yields

∫
Q

|un(t) −uk(t)|dxdt ≤ σ(
∫
Q

|∇un(t)|dxdt +
∫
Q

|∇uk(t)|dxdt) +
∫ T

0

||unσ(t) −ukσ(t)||1,Ωdt

which gives, since Lϕ(Q) ⊂ L1(Q) with continuous imbedding,∫
Q

|un(t) −uk(t)|dxdt ≤ σC1(||∇un||ϕ,Q + ||∇uk||ϕ,Q) +
∫ T

0

||unσ(t) −ukσ(t)||1,Ωdt

where C1 and C2 are constants which do not depend on n and k such that
||v||1,Q ≤ C1||v||ϕ,Q for all v ∈ Lϕ(Q) and ||∇un||ϕ,Q ≤ C2 for all n. Consequently,
we obtain : ∫

Q

|un(t) −uk(t)|dx ≤ 2C1C2σ +
∫ T

0

||unσ(t) −ukσ(t)||1,Ωdt.

Since ||unσ(t) −ukσ(t)||1,Ω → 0 almost everywhere in [0,T] when n,k →∞
and ||unσ(t)||L1(Ω) ≤ ||un(t)||L1(Ω) ≤ C uniformly with respect to n and t ∈
[0,T],we deduce by using Lebesgue’s theorem that∫ T

0

||unσ(t) −ukσ(t)||1,Ωdt → 0

as n,k →∞ implying, since σ is arbitrary, that
∫
Q
|un(t)−uk(t)|dxdt → 0 when n

and k →∞.
Hence (un) is a Cauchy sequence in L

1(Q) and thus un → u strongly in L1(Q).

3. APPROXIMATION AND TRACE RESULTS

In this section, Ω is a bounded Lipschitz domain in RN and I is a subinterval
of R ( possibly unbounded) and Q = Ω × I. It is easy to see that Q also satisfies
Lipschitz domain.

Definition. We say that un → u in W−1,xLψ(Q) + L2(Q) for the modular
convergence if we can write

un =
∑
|α|≤1

Dαxu
α
n + u

0
n and u =

∑
|α|≤1

Dαxu
α + u0



PARABOLIC EQUATIONS 181

with uαn → uα in Lψ(Q) for modular convergence for all |α| ≤ 1
and uαn → uα strongly in L2(Q).

We shall prove the following approximation theorem, which plays a fundamental
role in the prove of our main results.
Theorem 1. If u ∈ W 1,xLϕ(Q) ∩L2(Q) (respectively W

1,x
0 Lϕ(Q) ∩L

2(Q))

and ∂u
∂t
∈ W−1,xLψ(Q) + L2(Q), then there exists a sequence (vj) in D(Q) (respec-

tively D(I,D(Ω)) ) such that vj → u in W 1,xLϕ(Q) ∩L2(Q) and
∂vj
∂t
→ ∂u

∂t
in W−1,xLψ(Q) + L

2(Q) for the modular convergence.

Proof. Let u ∈ W 1,xLϕ(Q) ∩L2(Q) such that ∂u∂t ∈ W
−1,xLψ(Q) + L

2(Q)

and let ε > 0 be given. Writing ∂u
∂t

=
∑
|α|≤1 D

α
xu

α + u0, where uα ∈ Lψ(Q)
for all |α| ≤ 1 and u0 ∈ L2(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
=
∑
|α|≤1 D

α
xv

α + v0 with

vα,v0 ∈D(Q) such that∫
Q

ϕ(x,
Dαxv −Dαxu

λ
)dxdt ≤ ε,∀|α| ≤ 1,(2)

||v −u||L2(Q) ≤ ε,(3)

||v0 −u0||L2(Q) ≤ ε,(4) ∫
Q

ψ(x,
vα −uα

λ
)dxdt ≤ ε,∀|α| ≤ 1,(5)

The equation (2) flows from a slight adaptation of the arguments of [4],
(3) and (4) flow also from classical approximation results.
Regrading the equation (5) it is enough to prove that D(Q) is dense in Lψ(Q),
for this end we use the fact that the log-Hölder continuity(commutes with the

complementarity) i.e :if ϕ is log-HÖlder the its complementary ψ also it is, and
proceed as in [4] (with ϕ and ψ interchanged ) and using of course RN+1 instead
of RN and Q = Ω × (0,T) instead of Ω.
These facts lead us to prove that

||Kεf||ψ,Q ≤ C||f||ψ,Q,∀f ∈ Lψ(Q)

(with Kεf(x,t) = k
−1
ε

∫
Q
Kε(x − y)f(kεy,t)dy ,Kε(x) = 1εN K(

x
ε
) and K(x) is a

measurable function with support in the ball BR = B(0,R) see [4]).
And then we deduce that D(Q) is dense in Lψ(Q) for the modular convergence
which gives the desired conclusion.
The case of W

1,x
0 Lϕ(Q) ∩L

2(Q) is similar to the above arguments as in [4].

Remark 1. If, in the statement of Theorem 1, one consider Ω×R instead of Q,
we have D(Ω×R) is dense in u ∈ W 1,x0 Lϕ(Ω×R)∩L

2(Ω×R) : ∂u
∂t
∈ W 1,x0 Lψ(Ω×

R) + L2(Ω × R) for the modular convergence. This follows trivially from the fact
that D(R,D(Ω)) ≡D(Ω ×R).
A first application of Theorem 1 is the following trace result generalizing a clas-
sical result which states that if u belong to L2(a,b; H10 (Ω)) and

∂u
∂t

belongs to

L2(a,b; H−1(Ω)), then u is in C([a,b],L2(Ω)).



182 OUBEID, BENKIRANE AND VALLY

Lemma 3. Let a < b ∈ R and let Ω be a bounded Lipschitz domain in RN .
Then
{u ∈ W 1,x0 Lϕ(Ω×(a,b))∩L

2(Ω×(a,b)) : ∂u
∂t
∈ W−1,xLψ(Ω×(a,b))+L2(Ω×(a,b))}

is a subset of C([a,b],L2(Ω)).

Proof. Let u ∈ W 1,x0 Lϕ(Ω × (a,b)) ∩ L
2(Ω × (a,b)) such that W−1,xLψ(Ω ×

(a,b)) + L2(Ω × (a,b)). After two consecutive reflection first with respect to t = b
and then with respect to t = a,
û(x,t) = u(x,t)χ(a,b) + u(x, 2b− t)χ(b,2b−a) on Ω × (a, 2b−a)
ũ(x,t) = û(x,t)χ(a,2b−a) + û(x, 2a− t)χ(3a−2b,a) on Ω × (3a− 2b, 2b−a),
we get a function ũ ∈ W 1,x0 Lϕ(Ω × (3a− 2b, 2b−a)) ∩L

2(Ω × (3a− 2b, 2b−a))
such that ∂ũ

∂t
∈ W−1,xLψ(Ω × (3a− 2b, 2b−a)) + L2(Ω × (3a− 2b, 2b−a)).

Now, by letting a function η ∈D(R) with η = 1 on [a,b] and supp η ⊂ (3a−2b, 2b−
a), setting u = ηũ, and using standard arguments (see [[8],Lemme IV,Remarque

10,p.158]), we have u = u on Ω × (a,b) ũ ∈ W 1,x0 Lϕ(Ω × R) ∩ L
2(Ω × R) ∂ũ

∂t
∈

W−1,xLψ(Ω ×R) + L2(Ω ×R).
Now let vj ∈D(Ω ×R) be the sequence given by Theorem 1 corresponding to u,
that is,

vj → u ∈ W
1,x
0 Lϕ(Ω×R)∩L

2(Ω×R) and
∂vj
∂t
→

∂u

∂t
∈ W−1,xLψ(Ω×R)+L2(Ω×R)

for the modular convergence.
We have∫

Ω

(vi(τ) −vj(τ))2dx = 2
∫

Ω

∫ τ
−∞

(vi −vj)(
∂vi
∂t
−
∂vj
∂t

)dxdt → 0, as i,j →∞

from which one deduces that vj is a Cauchy sequence in C(R,L2(Ω)), and since the
limit
of vj in L

2(Ω × R) is u, we have vj → u inC(R,L2(Ω)). Consequently, u ∈
C([a,b],L2(Ω)).

4. EXISTENCE RESULT

Let Ω be a bounded Lipschitz domain in RN (N ≥ 2) , T > 0 and set Q =
Ω × (0,T).
Throughout this section, we denote Qτ = Ω × (0,τ) for every τ ∈ [0,T].
Let ϕ and γ two Musielak-Orlicz functions such that γ � ϕ.
Consider a second-order operator A : D(A) ⊂ W 1,xLϕ(Q) → W−1,xLψ(Q) of the
form

A(u) = −div(a(x,t,u,∇u)) + a0(x,t,u,∇u),

where a : Ω× [0,T]×R×RN → RNa0 : Ω× [0,T]×R×RN → R are Carathodory
functions, for almost every(x,t) ∈ Ω × [0,T] and all s ∈ R,ξ 6= ξ∗ ∈ RN ,



PARABOLIC EQUATIONS 183

|a(x,t,s,ξ)| ≤ β(c(x,t) + ψ−1x γ(x,ϑ|s|) + ψ
−1
x ϕ(x,ϑ|ξ|))(6)

|a0(x,t,s,ξ)| ≤ β(c(x,t) + ψ−1x γ(x,ϑ|s|) + ψ
−1
x ϕ(x,ϑ|ξ|))(7)

(a(x,t,s,ξ) −a(x,t,s,ξ∗))(ξ − ξ∗) > 0(8)

a(x,t,s,ξ)ξ + a0(x,t,s,ξ)s ≥ αϕ(x,
|ξ|
λ

) −d(x,t)(9)

with c(x,t) ∈ Eψ(Q),c ≥ 0,d(x,t) ∈ L1(Q),α,β,ϑ > 0. Furthermore, let

f ∈ W−1,xEψ(Q)(10)
Consider then the following parabolic initial-boundary value problem.


∂u
∂t

+ A(u) = f in Q
u(x,t) = 0 on ∂Ω × (0,T)
u(x, 0) = u0(x) in Ω

(11)

where u0 is a given function in L
2(Ω).

We shall prove the following existence theorem.

Theorem 2. Assume that (6)-(10) hold true. Then there exists at least one

distributional solution u ∈ D(A)∩W 1,x0 Lϕ(Q)∩C(([0,T],L
2(Ω)) of (11) satisfying

u(x, 0) = u0(x) for almost every x ∈ Ω. Furthermore, for all τ ∈ [0,T], we have

〈
∂φ

∂t
,u〉Qτ + [

∫
Ω

u(t)φ(t)dx]τ0 +

∫
Qτ

[a(x,t,u,∇u)∇φ + a0(x,t,u,∇u)φ]dxdt = 〈f,φ〉Qτ(12)

for every φ ∈ W 1,x0 Lϕ(Q) ∩L
2(Q) with ∂φ

∂t
∈ W−1,xLψ(Q) + L2(Q)

and for φ = u, which gives the energy equality

1

2

∫
Ω

u2(τ)dx−
1

2

∫
Ω

u20dx +

∫
Qτ

[a(x,t,u,∇u)∇u + a0(x,t,u,∇u)u]dxdt = 〈f,u〉Qτ

Remark 2. Note that all the terms in (12) make sense. Indeed, it easy to see that
the first, third, and fourth terms are well defined.
For the second one, we have by the trace result in Lemma 3 that φ ∈ C([0,T],L2(Ω)),
from which we can easily show that the second term of (12) makes sense.
Note also that taking φ ∈D(Q) in (12) shows that u is a distributional solution of
(11).

Remark 3. If a0 ≡ 0 and a(x,t,s,ξ) ≡ a(x,t,ξ) does not depend on s, then
the solution u is unique. Ineed, let v ∈ W 1,x0 Lφ(Q) ∩L

2(Q) be another solution of
(11). Using u − v as a test function in both equations corresponding to u and v
with τ = T, we get

1

2
[

∫
Ω

(u(t) −v(t))2dx]τ0 +
∫
Qτ

[a(x,t,u,∇u) −a(x,t,u,∇v)][∇u−∇v]dxdt = 0,(13)

which implies that, by (8) and the fact that u(0) = v(0),∇u = ∇v. This gives,
again by(13),
u(t) = v(t) for almost every t ∈ [0,T] and hence u = v.

Remark 4. Note that the trace result in Lemma3 shows that the assumption
u0 ∈ L2(Ω) cannot be weakened in order to get a distributional solution for the
Cauchy-Dirichlet problem (11).



184 OUBEID, BENKIRANE AND VALLY

Remark 5. As in the elliptic case (see, [5]), γ is introduced instead of ϕ in
(6) and (7) only to guarantee the boundedness in Lψ(Q) of ψ

−1
x γ(x,ϑ|un|) and

ψ−1x γ(x,ϑ|∇un|) whenever un is bounded in W 1,xLϕ(Q).

In the elliptic case, one usually takes γ = ϕ in the term ψ−1x γ(x,ϑ|un|) since un
is bounded in a smaller space Lθ(Ω) with ϕ � θ; see [5].
However, in the parabolic case, we cannot conclude that there is the boundedness.
Nevertheless, we can take γ = ϕ if one of the following assertions holds true.
(1) ϕ satisfies a 42 condition near infinity.
(2) A is monotone, that is 〈A(u)−A(v),u−v〉≥ 0 for all u,v ∈ D(A)∩W 1,x0 Lϕ(Q).
Indeed, suppose first that ϕ satisfies a 42 condition.Therefore (6) and (7),now with
γ = ϕ, imply that, for all ε > 0,

|a(x,t,s,ξ)| ≤ βε(cε(x,t) + ψ−1x ϕ(x,ε|s|) + ψ
−1
x ϕ(x,ε|ξ|)),

|a0(x,t,s,ξ)| ≤ βε(cε(x,t) + ψ−1x ϕ(x,ε|s|) + ψ
−1
x ϕ(x,ε|ξ|))

which allows us to deduce the boundedness in Lψ(Q) of a(x,t,un,∇un) and a(x,t,un,∇un).
Assume now that A is monotone. We have, for all φ ∈ W 1,x0 Eϕ(Q),〈A(un) −
A(φ),un − φ〉 ≥ 0. This gives 〈A(un),φ〉 ≤ 〈A(un),un〉− 〈A(φ),un − φ〉, which
implies that, since un is bounded in W

1,x
0 Lϕ(Q) and 〈A(un),un〉 is bounded from

above, thanks to the a priori estimates,

〈A(un),φ〉≤ Cφ for all φ ∈ W
1,x
0 Eϕ(Q),

where Cφ is a constant depending on φ but not n. Therefore, the Banach-Steinhauss
theorem applies so that we can obtain the boundedness of A(un) in W

−1,xLψ(Q).
Proof of Theorem 2. We will use a Galerkin method due to Landes and Musten
[15]. For the Galerkin method, we choose a sequence {w1,w2, .....} in D(Ω) such
that

⋃∞
n=1 Vn with

Vn = span{w1,w2, ....,wn}
is dense in Hm0 (Ω) with m sufficiently large such that H

m
0 (Ω) is continuously em-

bedded in C1(Ω). For any v ∈ Hm0 (Ω), there exists a sequence (vk) ⊂
⋃∞
n=1 Vn

such that vk → v in Hm0 (Ω) and C1(Ω) too.
We denote further Vn = C([0,T],Vn). It is easy to see that the closure of

⋃∞
n=1 Vn

with respect to the norm

||v||C1,0(Q) = sup|α|≤1{|Dαxv(x,t)| : (x,t) ∈ Q}

contains D(Ω). This implies that, for any f ∈ W−1,xEψ(Q), there exists a sequence
(fk) ⊂

⋃∞
n=1 Vn such that fk → f strongly in W

−1,xEψ(Q). Indeed,let ε > 0 be
given . Writing f =

∑
|α|≤1 D

α
xf

α for all |α| ≤ 1, there exists gα ∈ D(Q) such
that ||fα −gα||ψ,Q ≤ ε(2N+2) . Moreover, by setting g =

∑
|α|≤1 D

α
xg

α, we see that

g ∈D(Q), and so there exists φ ∈
⋃∞
n=1 Vn such that ||g−φ||∞,Q ≤

ε
(2meas(Q))

. We

deduce then that

||f −φ||W−1,xLψ(Q) ≤
∑
|α|≤1

||fα −gα||ψ,Q + ||g −φ||ψ,Q

For any u0 ∈ L2(Ω), there is a sequence u0k ⊂
⋃∞
n=1 Vn such that u0k → u0 in

L2(Ω).



PARABOLIC EQUATIONS 185

We divide the proof into three steps.

Step 1( a priori estimates): As in [15], by using [[14],Lemma1], we find that
there exists a Gelerkin solution un of (13) in the following sense.

un ∈Vn,
∂un
∂t
∈ L1(0,T; Vn),un(0) = u0n,(14)

and for all φ ∈Vn and all τ ∈ [0,T]∫
Qτ

∂un
∂t

φdxdt +

∫
Qτ

a(x,t,un,∇un)∇φdxdt +
∫
Qτ

a0(x,t,un,∇un)φdxdt =
∫
Qτ

fφdxdt.

Letting φ = un in (13) with τ = T and using (9) yields

||un||W1,x0 Lϕ(Q) ≤ C, ||un||L∞(0,T;L2(Ω)) ≤ C,∫
Q

a(x,t,un,∇un)∇undxdt +
∫
Q

a0(x,t,un,∇un)undxdt ≤ C,

where here and below C denotes a constant not depending on n. Using (7) and the
fact that γ � ϕ, it is easy to see that a0(x,t,un,∇un) is bounded in Lψ(Q). This
implies that ∫

Q

a(x,t,un,∇un)∇undxdt ≤ C.

To prove that a(x,t,un,∇un) is bounded in (Lψ(Q))N , let φ ∈ (Eϕ(Q))N , with
||φ||ϕ,Q = 1. In view of (8), we have∫

Q

[a(x,t,un,∇un) −a(x,t,un,φ)][∇un −φ]dxdt ≥ 0,

which gives∫
Q

[a(x,t,un,∇un)φdxdt ≤
∫
Q

a(x,t,un,∇un)∇undxdt−
∫
Q

a(x,t,un,φ)[∇un −φ]dxdt,

and since, thinks to (6), a(x,t,un,φ) is uniformly bounded in (Lψ(Q))
N , we deduce

that ∫
Q

a(x,t,un,∇un)φdxdt ≤ C for all φ ∈ (Eϕ(Q))N, ||φ||ϕ,Q = 1,

which implies that, by the use of the dual norm of (Lψ(Q))
N,a(x,t,un,∇un)

is bounded in Lψ(Q))
N . Hence, for a subsequence and some h0 ∈ Lψ(Q),h ∈

(Lψ(Q))
N ,

un ⇀ u in W
1,x
0 Lψ(Q) for σ(ΠLϕ, ΠEψ) and weakly in L

2(Q),
a0(x,t,un,∇un) ⇀ h0,a(x,t,un,∇un) ⇀ h in Lψ(Q) for σ(ΠLψ, ΠEϕ).
As in [15], we get un(t) ⇀ u(t) in L

1(Ω) for almost every t ∈ [0,T], and then,
by using Lemma2, we deduce that un → u strongly in L1(Q) and that, for some
subsequence still denoted by un, un → u almost everywhere in Q.



186 OUBEID, BENKIRANE AND VALLY

Step 2( almost everywhere convergence of the gradients):For every τ ∈ (0,T]
and for all φ ∈ C1([0,T],D(Ω)), we get from (10)∫

Qτ

u
∂φ

∂t
dxdt + [

∫
Ω

u(t)φ(t)dx]τ0 +

∫
Qτ

h∇φ +
∫
Qτ

h0φdxdt = 〈f,φ〉Qτ ,(15)

and then, by choosing τ = T and taking φ to be arbitrary in D(Q), we have
∂u
∂t
∈ W−1,xLψ(Q).

Consider now the prolongation of u to Ω × R as int the proof of Lemma3. We see
that there exists a sequence vk in D(Ω ×R) such that
vk → u in W

1,x
0 Lϕ(Q) ∩ L

2(Q) and ∂vk
∂t
→ ∂u

∂t
in W−1,xLψ(Q) + L

2(Q) for the

modular convergence and so ( see the proof of Lemma 3), vk → u in C([0,T],L2(Ω)),
which implies that, in particular,
u ∈ C([0,T],L2(Ω)). Consequently,

lim
k→∞

∫
Q

∂vk
∂t

(vk −u)dxdt = 0,

which gives, by the use of the fact that ∂vk
∂t
∈ Eψ(Q),

lim
k→∞

lim
n→∞

∫
Q

∂vk
∂t

(vk −un)dxdt = 0.

This implies that

lim sup
k→∞

lim sup
n→∞

∫
Q

∂un
∂t

(vk −un)dxdt ≤ 0.

Since∫
Q
∂un
∂t

(vk −un)dxdt = −12 [
∫

Ω
(un(t) −vk(t))2dx]T0 +

∫
Q
∂vk
∂t

(vk −un)dxdt
≤ 1

2
||u0n −vk(0)||2L2(Ω) +

∫
Q
∂vk
∂t

(vk −un)dxdt

and u0n → u0 in L2(Ω) and vk(0) → u(0) in L2(Ω)( note that u(0) = u0 since
un(0) ⇀ u(0) in L

1(Ω)).
From (14) and (15), we have

lim sup
n→∞

(

∫
Ω

[a(x,t,un∇un)∇un −h∇vk + a0(x,t,un,∇un)un −h0vk]dxdt)

≤ lim sup
n→∞

〈fn,un〉−〈f,vk〉 +

lim sup
n→∞

(−
∫
Q

∂un
∂t

undxdt) −
∫
Q

∂vk
∂t

udxdt + [

∫
Ω

u(t)vk(t)dx]
T
0

= 〈f,u−vk〉 + lim sup
n→∞

∫
Q

∂un
∂t

(vk −un)dxdt

Where we have used the fact that

−
∫
Q

∂vk
∂t

(u)dxdt + [

∫
Ω

u(t)vk(t)dx]
T
0

= lim
n→∞

(−
∫
Q

∂vk
∂t

(un)dxdt + [

∫
Ω

un(t)vk(t)dx]
T
0 )

= lim
n→∞

∫
Q

∂un
∂t

(vk)dxdt.



PARABOLIC EQUATIONS 187

We deduce that

lim sup
k→∞

lim sup
n→∞

(

∫
Ω

[a(x,t,un,∇un)∇un −h∇vk + a0(x,t,un∇un)(un −vk)]dxdt) ≤ 0(16)

Since, as can be easily seen,
limn→∞

∫
Ω

(a(x,t,un,∇un)∇vk−∇vk+a0(x,t,un∇un)vk)dxdt) =
∫
Q

(h∇vk+h0vk)dxdt.

In the sequel, and for any r > 0 and any k ∈ N, we denote by χrk χ
r the

characteristic functions of {(x,t) ∈ Q : |∇vk| ≤ r} and {(x,t) ∈ Q : |∇u| ≤ r},
respectivly. We also denote by ε(n,k,s) all quantities (possibly different) depending
on l such that

lim
s→∞

lim
k→∞

lim
n→∞

ε(n,k,s) = 0,

and this will be the order in which the parametres we use will tend to infinity, that
is, first n, then k, and finally s. Similarly, we will write onlyε(n), or ε(n,k),...to
mean that the limits are only on the specified parametrers. We have, for any l > 0,∫

{|un|≤l}
[a(x,t,un,∇un) −a(x,t,un,∇u.χs)][∇un −∇u.χs]dxdt

−
∫
{|un|≤l}

[a(x,t,un,∇un) −a(x,t,un,∇vk.χsk)][∇un −∇vk.χ
s
k]dxdt

=

∫
{|un|≤l}

a(x,t,un,∇vk.χsk)[∇un −∇vk.χ
s
k]dxdt

+

∫
{|un|≤l}

a(x,t,un,∇un)[∇vk.χsk −∇u.χ
s]dxdt∫

{|un|≤l}
a(x,t,un,∇u.χs)[∇u.χs −∇un]dxdt

:= I1 + I2 + I3.

We shall go to the limit in all integrals Ii (for i = 1, 2, 3) as first n, then k and
finally s tend to infinity. Starting with I1 and letting n →∞ yields

I1 =

∫
{|u|≤l}

a(x,t,u,∇vk.χsk)[∇u−∇vk.χ
s
k]dxdt + ε(n)

Since χ{|un|≤l}a(x,t,un,∇vk.χ
s
k) → χ{|u|≤l}a(x,t,u,∇vk.χ

s
k) strongly in (Eψ(Q))

N

by (6) and the Lebesgue theorem while ∇un →∇u in (Lϕ(Q))N . This implies, by
letting k →∞ in the integral of last side, that

I1 =

∫
{|u|≤l}∩{|∇u|>s}

a(x,t,u, 0)∇udxdt + ε(n,k),

from which we get I1 = ε(n,k,s), since the first term of the last side goes to 0 as
s →∞.
For I2, we have, by letting n →∞,

I2 =

∫
{|u|≤l}∩{|∇u|>s}

h[∇vk.χsk −∇u.χ
s]dxdt + ε(n),

and so, by letting k → ∞ in the integral of last side and using the fact that
∇vkχsk →∇uχ

s strongly in (Eϕ(Q))
N , we deduce that I2 = ε(n,k) .



188 OUBEID, BENKIRANE AND VALLY

For the third term I3, we have, by letting n →∞,

I3 = −
∫
{|u|≤l}∩{|∇u|>s}

a(x,t,u, 0)∇udxdt + ε(n,k),

and since the first term of the last side tends to zero as s → ∞ , we obtain
I3 = ε(n,k,s).
We have then proved that∫

{|un|≤l}
[a(x,t,un,∇un) −a(x,t,un,∇u.χs)][∇un −∇u.χs]dxdt

=

∫
{|un|≤l}

[a(x,t,un,∇un) −a(x,t,un,∇vk.χsk)][∇un −∇vk.χ
s
k]dxdt + ε(n,k,s).

For all s ≥ r > 0 and all l ≥ δ > 0 , we have

0 ≤
∫
{|un|≤δ}∩{|∇u|≤r}

[a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt(17)

≤
∫
{|un|≤l}∩{|∇u|≤s}

[a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt

≤
∫
{|un|≤l}

[a(x,t,un,∇un) −a(x,t,un,∇u.χs)][∇un −∇u.χs]dxdt

=

∫
{|un|≤l}

[a(x,t,un,∇un) −a(x,t,un,∇vk.χsk)][∇un −∇vk.χ
s
k]dxdt + ε(n,k,s)

= −
∫
{|un|≤l}

a(x,t,un,∇vk.χsk)[∇un −∇vk.χ
s
k]dxdt

+

∫
Q

[a(x,t,un,∇un)(∇un −∇vk) + a0(x,t,un,∇un)(un −vk)]dxdt

−(
∫
{|un|≤l}

a(x,t,un,∇un)[∇un −∇vk]dxdt +
∫
Q

a0(x,t,un,∇un)(un −vk)]dxdt)

+

∫
{|un|≤l}∩{|vk|>s}

a(x,t,un,∇un)∇vkdxdt + ε(n,k,s)

:= J1 + J2 + J3 + J4 + ε(n,k,s).

We shall go to the limit sup first over n and next over k and finally over s in all
integrals of the last side.
First of all, note that J1 = −I1 = ε(n,k,s) and that, thanks to (16),

lim sup
k→∞

lim sup
k→∞

J2 ≤ 0.

The third integral reads
J3 = −

∫
{|un|>l}

[a(x,t,un,∇un)(∇un −∇vk) + a0(x,t,un,∇un)(un −vk)]dxdt
−
∫
{|un|≤l}

a0(x,t,un,∇un)(un −vk)dxdt,
and, by using (9),

J3 ≤
∫
{|un|>l}

[a(x,t,un,∇un)∇vk + a0(x,t,un,∇un)vk]dxdt

−
∫
{|un|>l}

d(x,t)dxdt−
∫
{|un|≤l}

a0(x,t,un,∇un)(un −vk)dxdt,



PARABOLIC EQUATIONS 189

which gives
lim supn→∞J3 ≤

∫
{|u|≥l}(h∇vk + h0vk)dxdt−

∫
{|u|≥l}d(x,t)dxdt−

∫
{|u|≤l}h0(u−

vk)dxdt, where we have used the strong convergence of χ{|un|>l}|∇vk| and χ{|un|>l}|vk|
and χ{|un|≤l}un in Eϕ(Q)
as n →∞. This implies that

lim sup
k→∞

lim sup
n→∞

J3 ≤
∫
{|u|≥l}

(h∇u + h0u)dxdt−
∫
{|u|≥l}

d(x,t)dxdt

since vk → u in W
1,x
0 Lφ(Q) for the modular convergence. For J4, we have

lim
n→∞

J4 =

∫
{|u|≤l}∩{|∇vk|>s}

h∇vkdxdt

since χ{|un|≤l}∩{|∇vk|>s}∇vk → χ{|u|≤l}∩{|∇vk|>s}∇vk strongly in (Eϕ(Q))
N as

n →∞. This implies that

lim
k→∞

lim
n→∞

J4 =

∫
{|u|≤l}∩{|u|≥s}

h∇udxdt ≤
∫
{|u|≥s}

|h∇u|dxdt

and thus

lim sup
s→∞

lim
k→∞

lim n →∞J4 ≤ 0.

Combining these estimates with (17) and passing to the limit sup first over n, then
over k, and finally over s, we deduce that

0 ≤ lim sup
n→∞

∫
{|un|≤δ,|∇u|≤r}

[a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt

≤
∫
{|un|≥l}

(h∇u + h0u−d(x,t))dxdt,

in which we can let l →∞ to get
limn→∞

∫
{|un|≤δ,|∇u|≤r}

[a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt = 0, and
thus, as the elliptic case (see [1]), we deduce that, for a subsequence still denoted
by un,

∇un →∇u a.e in Q,
and so h = a(x,t,u,∇u) and h0 = a0(x,t,u,∇u). Therefore, we get for every
τ ∈ (0,T] and for all φ ∈ C1([O,τ],D(Ω)),

−
∫
Qτ

u
∂φ

∂t
dxdt + [

∫
Ω

u(t)φ(t)dx]τ0 +

∫
Qτ

[a(x,t,u,∇u)∇φ + a0(x,t,u,∇u)φ]dxdt = 〈f,φ〉Qτ .(18)

Step 3 ( passage to the limit): Let v ∈ W 1,x0 Lϕ(Q) ∩ L
2(Q) such that ∂v

∂t
in

W−1,xLψ(Q) + L
2(Q). There exists a prolongation v of v such that (see proof of

Lemma 3)

v = v on Q,v ∈ W 1,x0 Lϕ(Ω ×R) ∩L
2(Ω ×R),

∂v

∂t
∈ W−1,xLψ(Ω ×R) + L2(Ω ×R).(19)

By Theorem 1 (see also Remark 1),there exists a sequence wj ⊂ D(Ω × R) such
that

wj → v ∈ W
1,x
0 Lϕ(Ω ×R) ∩L

2(Ω ×R) and
∂wj
∂t
→

∂v

∂t
∈ W−1,xLψ(Ω ×R) + L2(Ω ×R)(20)



190 OUBEID, BENKIRANE AND VALLY

for the modular convergence.
Letting φ = wjχ(0,τ) ( which belongs to C

1([0,τ],D(Ω))) as a test function in (18),
we get, for every τ ∈ (0,T],

−
∫
Qτ

u
∂wj
∂t

dxdt + [

∫
Ω

u(t)wj(t)dx]
τ
0 +

∫
Qτ

[a(x,t,u,∇u)∇wj + a0(x,t,u,∇u)wj]dxdt = 〈f,wj〉Qτ .(21)

We shall now go to the limit as j →∞ in all terms of (21). In view of (20), there is
no problem with passing to the limit in the first and last three terms. For the second
one, observe that, as in the proof of Lemma 3, we have wj → v in C([0,T],L2(Ω)),
and since, for all t ∈ [0,T],u(t) is in L2(Ω), we have, for every t ∈ [0,T],∫

Ω

u(t)wj(t)dx →
∫

Ω

u(t)v(t)dx.

Letting j →∞ in both sides of (21)

−〈
∂v

∂t
,u〉Qτ + [

∫
Ω

u(t)v(t)dx]τ0 +

∫
Qτ

[a(x,t,u,∇u)∇v + a0(x,t,u,∇u)v]dxdt = 〈f,v〉Qτ .

To prove the energy equality, it suffices to take v = u in the above equality (note

that this is possible since u ∈ W 1,x0 Lϕ(Q) ∩L
2(Q) and ∂u

∂t
∈ W−1,xLψ(Q)).

This gives

−〈
∂u

∂t
,u〉Qτ + [

∫
Ω

u2(t)dx]τ0 +

∫
Qτ

[a(x,t,u,∇u)∇u + a0(x,t,u,∇u)u]dxdt = 〈f,u〉Qτ ,

and since, as can by easily seen,

〈
∂u

∂t
,u〉Qτ =

1

2
(

∫
Ω

u2(τ)dx−
∫

Ω

u20dx),

we get the desired equality. This completes the proof of Theorem 2.

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1Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz,
B. P. 1796 Atlas Fès, Maroc

2Department of Mathematics, Faculty of Science, King Khalid University,Abha 61413,

Kingdom of Saudi Arabia

∗Corresponding author