Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form
143 (2018)
MATHEMATICA BOHEMICA
No. 3, 225–249
ENTROPY SOLUTIONS TO PARABOLIC EQUATIONS IN
MUSIELAK FRAMEWORK INVOLVING NON COERCIVITY TERM
IN DIVERGENCE FORM
Mohamed Saad Bouh Elemine Vall, Ahmed Ahmed,
Abdelfattah Touzani, Abdelmoujib Benkirane, Fez
Received October 10, 2016. Published online October 17, 2017.
Communicated by Michela Eleuteri
Abstract. We prove the existence of solutions to nonlinear parabolic problems of the
following type:
∂b(u)
+ A(u) = f + div(Θ(x; t; u))
∂t
u(x; t) = 0
b(u)(t = 0) = b(u0 )
in Q,
on ∂Ω × [0; T ],
on Ω,
where b : R → R is a strictly increasing function of class C 1 , the term
A(u) = −div (a(x, t, u, ∇u))
is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions
of Musielak type, Θ : Ω × [0; T ] × R → R is a Carathéodory, noncoercive function which
satisfies the following condition: sup |Θ(·, ·, s)| ∈ Eψ (Q) for all k > 0, where ψ is the
|s|6k
Musielak complementary function of Θ, and the second term f belongs to L1 (Q).
Keywords: inhomogeneous Musielak-Orlicz-Sobolev space; parabolic problems; Galerkin
method
MSC 2010 : 58J35, 65L60
DOI: 10.21136/MB.2017.0087-16
225
1. Introduction
Our aim is to prove the existence of solutions u to the following nonlinear parabolic
problem:
(1.1)
∂b(u)
+ A(u) = f + div(Θ(x, t, u)) in Q,
∂t
u(x, t) = 0
b(u)(t = 0) = b(u0 )
on ∂Ω × [0, T ],
on Ω,
where Ω is an open subset RN which satisfies the segment property and Q = Ω×[0, T ],
T > 0, b : R → R is a strictly increasing function of class C 1 with b(0) = 0 and
lim b′ (t) = l < ∞, A(u) = −div(a(x, t, u, ∇u)) is a Leray-Lions operator det→±∞
fined on D(A) ⊂ W01,x Lϕ (Q) into its dual satisfying some conditions in Section 3,
ϕ is Musielak function and W01,x Lϕ (Q) is the Musielak space defined in Section 2,
f ∈ L1 (Q) and Θ : Ω × [0, T ] × R → R is a noncoercive function which satisfies the
following condition: sup |Θ(·, ·, s)| ∈ Eψ (Q) for all k > 0, where ψ is the comple|s|6k
mentary function of ϕ and Eψ (Q) is a Musielak space defined in Section 2.
Under our assumptions, the above problem does not admit, in general, a weak
solution since the field a(x, t, u, ∇u) does not belong to (L1loc (Q))N in general. To
overcome this difficulty we use in this paper the framework of entropy solutions.
This notion was introduced by Benilan et al. [9] for the study of nonlinear elliptic
problems.
In the classical Sobolev spaces, Aberqi et al. in [1] have proved the existence of
renormalized solutions (1.1) in the case where b(u) ≡ b(x, u) and Θ satisfies a growth
condition (for the definition of this notion of solution see [1], [20]), Redwane in [19]
has proved the existence of renormalized solutions of (1.1), where Θ(x, t, u) = Θ(u).
In the Sobolev variable exponent setting, Azroul, Benboubker, Redwane, and Yazough [6] have proved the existence result of renormalized solutions to a class of
nonlinear parabolic equations without sign condition involving nonstandard growth
in the particular case, where div(Θ(x, t, u)) = H(x, t, u, ∇u) and in the elliptic case
(see [8]).
In Orlicz framework, Redwane in [20] has proved the existence of renormalized
solutions of (1.1), where b(u) ≡ b(x, u) and Θ(x, t, u) = Θ(u), Hadj Nassar, Moussa
and Rhoudaf in [16] have studied the existence of renormalized solutions of (1.1) in
−1
W 1,x LM (Q), where b(u) ≡ b(x, u) and Θ satisfies |Θ(x, u)| 6 P P (|u|), where P
and P are two complementary Orlicz functions with P ≪ M . See also [7], [13],
and [14] for related topics. For some existing results for strongly nonlinear elliptic
and parablic equations in Musielak-Orlicz-Sobolev spaces see [2], [3], [4], [5], [21].
226
This research is divided into several parts. In Section 2 we recall some important
definitions and results of Musielak-Orlicz-Sobolev spaces. We introduce the assumptions that allow us to demonstrate our result in Section 3. Section 4 contains some
important and useful lemmas to prove our main result. In Section 5 we prove the
main result of this paper (Theorem 5.1) concerning the existence of solutions.
2. Preliminary
2.1. Musielak-Orlicz-Sobolev spaces. Let Ω be an open set in RN and let ϕ
be a real-valued function defined in Ω × R+ , and satisfiying the following conditions:
(a) ϕ(x, ·) is an N-function convex, increasing, continous, ϕ(x, 0) = 0, ϕ(x, t) > 0
for all t > 0, lim sup ϕ(x, t)t−1 = 0, lim inf ϕ(x, t)t−1 = ∞ .
t→∞ x∈Ω
t→0 x∈Ω
(b) ϕ(·, t) is a measurable function.
A function ϕ, which satisfies conditions (a) and (b) is called Musielak-Orlicz function.
For a Musielak-Orlicz function ϕ we put ϕx (t) = ϕ(x, t) and we associate its
nonnegative reciprocal function ϕx−1 with respect to t, that is
−1
ϕ−1
x (ϕ(x, t)) = ϕ(x, ϕx (t)) = t.
The Musielak-Orlicz function ϕ is said to satisfy the ∆2 -condition if for some k > 0
and a nonnegative function h integrable in Ω we have
(2.1)
ϕ(x, 2t) 6 kϕ(x, t) + h(x)
∀ x ∈ Ω and t > 0.
If (2.1) holds only for t > t0 > 0, then ϕ is said to satisfy ∆2 near infinity.
Let ϕ and γ be two Musielak-Orlicz functions. We say that ϕ dominates γ, and we
write γ ≺ ϕ, near infinity (or globally) if there exist two positive constants c and t0
such that for almost all x ∈ Ω
γ(x, t) 6 ϕ(x, ct)
∀ t > t0 ,
(or ∀ t > 0, i.e. t0 = 0).
We say that γ grows essentially less rapidly than ϕ at 0 (or near infinity), and we
write γ ≺≺ ϕ, if for every positive constant c we have
lim
t→0
γ(x, ct)
γ(x, ct)
= 0 (or lim sup
= 0).
t→∞ x∈Ω ϕ(x, t)
x∈Ω ϕ(x, t)
sup
R e m a r k 2.1 ([11]). If γ ≺≺ ϕ near infinity, then for all ε > 0 there exists
k(ε) > 0 such that for almost all x ∈ Ω we have
(2.2)
γ(x, t) 6 k(ε)ϕ(x, εt)
∀ t > 0.
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We define the functional
̺ϕ,Ω (u) =
Z
ϕ(x, |u(x)|) dx,
Ω
where u : Ω → R is a Lebesgue measurable function. In the following, the measurability of function u : Ω → R means the Lebesgue measurability. The set
Kϕ (Ω) = {u : Ω → R measurable: ̺ϕ,Ω (u) < ∞},
is called the generalized Orlicz class.
The Musielak-Orlicz space (or the generalized Orlicz space) Lϕ (Ω) is the vector
space generated by Kϕ (Ω), that is, Lϕ (Ω) is the smallest linear space containing the
set Kϕ (Ω). Equivalently,
n
Lϕ (Ω) = u : Ω → R
measurable: ̺ϕ,Ω
|u(x)|
λ
o
< ∞ for some λ > 0 .
We define the Musielak-Orlicz function complementary to ϕ in the sense of Young
with respect to the variable s as
ψ(x, s) = sup {st − ϕ(x, t)}.
t>0
We define in the space Lϕ (Ω) the two norms:
kukϕ,Ω = inf
|u(x)|
dx 6 1 ,
ϕ x,
λ
Ω
Z
λ > 0:
which is called the Luxemburg norm and the so called Orlicz norm defined as
|||u|||ϕ,Ω =
sup
kvkψ,Ω 61
Z
|u(x)v(x)| dx,
Ω
where ψ is the Musielak-Orlicz function complementary to ϕ and kvkψ,Ω is the Luxemburg norm of v associate to the Musielak function ψ. These two norms are equivalent (see [18]).
The closure in Lϕ (Ω) of the bounded measurable functions with compact support
in Ω is denoted by Eϕ (Ω). It is a separable space.
We say that a sequence of functions (...truncated)
