Strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces
Bol. Soc. Paran. Mat.
c SPM –ISSN-2175-1188 on line
SPM: www.spm.uem.br/bspm
(3s.) v. 33 1 (2015): 193–225.
ISSN-00378712 in press
doi:10.5269/bspm.v33i1.23083
Strongly Nonlinear Parabolic Problems in Musielak-Orlicz-Sobolev
Spaces
M. L. Ahmed Oubeid, A. Benkirane, and M. Sidi El Vally
abstract: We prove in this paper the existence of solutions of strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces. An approximation and
a compactness results in inhomogeneous Musielak-Orlicz-Sobolev spaces have also
been provided.
Key Words: Inhomogeneous Musielak-Orlicz-Sobolev spaces; parabolic problems; Compactness.
Contents
1 Introduction
193
2 Preliminaries
194
3 Approximation Theorem and Trace Result
199
4 Compactness Results
204
5 Existence Result
207
1. Introduction
Let Ω a bounded open subset of Rn and let Q be the cylinder Ω × (0, T ) with
some given T > 0.
We consider the strongly nonlinear parabolic problem
∂u
∂t + A(u) + g(x, t, u, ∇u) = f in Q
(1)
u(x, t) = 0 on ∂Ω × (0, T )
u(x, 0) = u0 (x) in Ω
where A = − div (a(x, t, u, ∇u)) is an operator of Leray-Lions type, g is a nonlinearity with the sign condition but any restriction on its growth.
This result generalizes analogous ones of Lions [21], Landes [18] when g ≡ 0 and
of Brezis-Browder [9], Landes.Mustonen [19] for g ≡ g(x, t, u). See also [7,8] for
related topics. In these results, the function a is supposed to satisfy a polynomial
growth condition with respect to u and ∇u.
In the case where a satisfies a more general growth condition with respect to u
and ∇u, it is shown in [12] that the adequate space in which (1) can be studied is
the inhomogeneous Orlicz-Sobolev space W 1,x LM (Q) where the N-function M is
2000 Mathematics Subject Classification: 46E35, 35K15, 35K20, 35K60
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style.
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M. L. Ahmed Oubeid, A. Benkirane, and M. Sidi El Vally
related to the actual growth of a . The solvability of (1) in this setting is proved
by Donaldson [12] for g ≡ 0 and by Robert [23] for g ≡ g(x, t, u) when A is
monotone, t2 ≪ M (t) and M satisfies a ∆2 condition and also by Elmahi [14]
for g = g(x, t, u, ∇u) when M satisfies a ∆′ condition and M (t) ≪ tN/(N −1) as
application of some LM compactness results in W 1,x LM (Q), see [13].
The solvability of (1) in this setting is proved by Elmahi-Meskine [16] for g ≡ 0
and for g ≡ g(x, t, u, ∇u) in [15], without assuming any restriction on the Nfunction M .
In a recent work, the authors [2] have established an existence result for problems of the form (1), when g ≡ 0, without assuming any restriction on the Musielak
function ϕ.
It is our purpose in this paper to prove the existence of solutions for problem
(1) in the setting of Musielak-Orlicz spaces for general Musielak function ϕ with
a nonlinearity g(x, t, u, ∇u) having natural growth with respect to the gradient.
In section 3 some new approximation result in inhomogeneous Musielak-OrliczSobolev spaces (see Theorem 3.2), and, on the other hand, to prove a trace result
(see Lemma 4.2). In Section 4, we establish L1 -compactness results in the inhomogeneous Musielak-Orlicz-Sobolev spaces W 1,x Lϕ (Q). Section 5 contains the main
result of this paper.
Our result generalizes that of the Elmahi-Meskine in [15] to the case of inhomogeneous 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)
= W m Lϕ (Ω). These spaces are called Variable exponent Lebesgue and Sobolev
spaces.
For some classical and recent results on elliptic and parabolic problems in Orliczsobolev spaces and a Musielak-Orlicz-Sobolev spaces, we refer to [1,2,3,6,12,14,15,
16,24].
2. Preliminaries
In this section we list briefly some definitions and facts about Musielak-OrliczSobolev spaces. Standard reference is [22]. We also include the definition of inhomogeneous 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
ϕ(x, t)
=
t
ϕ(x, t)
=
lim inf
t−→∞ x∈Ω
t
lim sup
t−→0 x∈Ω
0
0.
Musielak-Orlicz-Sobolev Spaces
195
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
−1
ϕ−1
x (ϕ(x, t)) = ϕ(x, φx ) = t.
For any two Musielak-Orlicz functions ϕ and γ we introduce the following ordering :
c) if there exists two positives constants c and T such that for almost everywhere
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 (sup
t→0 x∈Ω
ϕ(x, ct)
ϕ(x, ct)
) = 0 or lim (sup
)=0
t→∞ x∈ϕ γ(x, t)
γ(x, t)
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 measurability.
We define the functional
̺ϕ,Ω (u) =
Z
ϕ(x, |u(x)|)dx
Ω
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:
|u(x)|
Lϕ (Ω) = u : Ω → R mesurable /̺ϕ,Ω (
) < +∞, for some λ > 0
λ
Let
ψ(x, s) = sup{st − ϕ(x, t)},
t≥0
196
M. L. Ahmed Oubeid, A. Benkirane, and M. Sidi El Vally
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:
Z
|u(x)|
)dx, ≤ 1}.
||u||ϕ,Ω = inf{λ > 0/ ϕ(x,
λ
Ω
and the so-called Orlicz norm :
|||u|||ϕ,Ω = sup
||v||ψ ≤1
Z
|u(x)v(x)|dx.
Ω
where ψ is the Musielak-Orlicz function complementary to ϕ. These two norms are
equivalent [22].
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)
[22].
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 (...truncated)
