Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 506857, 7 pages
doi:10.1155/2011/506857
Research Article
On Degenerate Parabolic Equations
Mohammed Kbiri Alaoui
Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia
Correspondence should be addressed to Mohammed Kbiri Alaoui, mka la@yahoo.fr
Received 31 March 2011; Accepted 28 July 2011
Academic Editor: Mihai Putinar
Copyright q 2011 Mohammed Kbiri Alaoui. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The paper deals with the existence of solutions of some generalized Stefan-type equation in the
framework of Orlicz spaces.
1. Introduction
In this paper, we deal with the following boundary value problems
∂u
Aθu f,
∂t
u 0,
in Q,
on ∂Q ∂Ω × 0, T ,
ux, 0 u0 x,
P in Ω,
where
Au − diva·, t, ∇u,
1.1
Q Ω × 0, T , T > 0, and Ω is a bounded domain of RN , with the segment property, f is
a smooth function, u0 ∈ L2 Ω, θ is a positive real function increasing but not necessarily
strictly increasing, θ0 0, and θu0 ∈ L2 Ω. a : Ω × RN → RN is a Carathéodory function
i.e., measurable with respect to x in Ω for every t, ξ in R × R × RN , and continuous with
ξ∗ ,
respect to ξ in RN for almost every x in Ω such that for all ξ, ξ∗ ∈ RN , ξ /
ax, t, ξξ ≥ αB|ξ|,
1.2
2
International Journal of Mathematics and Mathematical Sciences
ax, t, ξ − ax, t, ξ∗ ξ − ξ∗ > 0,
−1
|ax, t, ξ| ≤ cx, t k1 B Bk2 |ξ|.
1.3
1.4
There exist an N-function M such that
Bθt Mt,
1.5
where cx, t belongs to EB Q, c ≥ 0 and ki i 1, 2 to R , and α to R∗ .
Some examples of such operator are in particular the case where
a·, ∇θu B|∇θu|
|∇θu|2
∇θu,
1.6
where B is an N-function.
Many physical models in hydrology, infiltration through porous media, heat transport,
metallurgy, and so forth lead to the nonlinear equations systems of the form
∂t u ∇ψ ∇βu ,
E
where β, ψ are monotone, ψs s is even and convex for s ≥ so > 0, |ψs| → ∞, |βs| → ∞
for |s| → ∞ for the details see 1. Jäger and Kačur treated the porous medium systems
where β is strictly monotone in 2 and Stefan-type problems where β is only monotone. For
the last model, there exists a large number of references. Among them, let us mention the
earlier works 3–5 for a variational approach and 6 for semigroup.
In 7, a different approach was introduced to study the porous and Stefan
problems.The enthalpy formulation and the variational technique are used. Nonstandard
semidiscretization in time is used, and Newton-like iterations are applied to solve the
corresponding elliptic problems.
Due to the possible jumps of θ, problem P enters the class of Stefan problems. In the
present paper, we are interested in the parabolic problem with regular data. It is similar in
many respects to the so-called porous media equation. However, the equation we consider
has a more general structure than that in the references above.
Two main difficulties appear in the study of existence of solutions of problem P .
The first one comes from the diffusion terms in P since they do not depend on u but on
θu, and, moreover, at the same time, P poses big problems, since in general we have not
information on u but on θu. For the last reason, the authors in 8 define a new notion of
weak solution to overcome this problem.
In the above cited references, the authors have shown the existence of a weak solution
when the function ax, t, ξ was assumed to satisfy a polynomial growth condition with
respect to ∇u. When trying to relax this restriction on the function a·, ξ, we are led to replace
the space Lp 0, T ; W 1,p Ω by an inhomogeneous Sobolev space W 1,x LB built from an Orlicz
space LB instead of Lp , where the N-function B which defines LB is related to the actual
growth of the Carathéodory’s function.
International Journal of Mathematics and Mathematical Sciences
3
Our goal in this paper is, on the one hand, to give a generalization of E in the case
of one equation in the framework of Leray-Lions operator in Orlicz-Sobolev spaces. on the
second hand, we prove the existence of solutions in the BV Q space.
2. Preliminaries
Let M : R → R be an N-function, that is, M is continuous, convex, with Mt > 0 for t > 0,
Mt/t → 0 as t → 0, and Mt/t → ∞ as t → ∞. The N-function M conjugate to M is
defined by Mt sup{st − Ms : s > 0}.
Let P and Q be two N-functions. P Q means that P grows essentially less rapidly
than Q, that is, for each ε > 0,
P t
−→ 0,
Qεt
as t −→ ∞.
2.1
Let Ω be an open subset of RN . The Orlicz class LM Ω resp., the Orlicz space LM Ω is
defined
classes of real-valued measurable functions u on Ω such
as the set of equivalence
that Ω Muxdx < ∞ resp., Ω Mux/λdx < ∞ for some λ > 0.
Note that LM Ω is a Banach space under the norm u
M,Ω inf{λ > 0 :
Mux/λdx
≤ 1} and LM Ω is a convex subset of LM Ω. The closure in LM Ω of
Ω
the set of bounded measurable functions with compact support in Ω is denoted by EM Ω.
In general, EMΩ / LM Ω and the dual of EM Ω can be identified with LM Ω by means
of the pairing Ω uxvxdx, and the dual norm on LM Ω is equivalent to · M,Ω .
We say that un converges to u for the modular convergence in LM Ω if, for some λ > 0,
Mu
n − u/λdx → 0. This implies convergence for σLM , LM .
Ω
The inhomogeneous Orlicz-Sobolev spaces are defined as follows: W 1,x LM Q {u ∈
LM Q : Dxα u ∈ LM Q for all |α| ≤ 1}. These spaces are considered as subspaces of the
product space ΠLM Q which have as many copies as there are α-order derivatives, |α| ≤ 1.
σΠLM ,ΠLM We define the space W01,x LM Q DQ
. For more details, see 9.
For k > 0, we define the truncation at height k, Tk : R → R by
Tk s ⎧
⎪
⎨s,
if |s| ≤ k,
s
⎪
⎩k ,
|s|
if |s| > k.
2.2
3. Main Result
Before giving our main result, we give the following lemma which will be used.
Lemma 3.1 see 10. Under the hypothesis 1.2–1.4, θs s, the problem P admits at least
one solution u in the following sense:
u ∈ W01,x LB Q ∩ L2 Q,
∂u
,v a·, ∇u∇v fv dxdt,
∂t
Q
Q
for all v ∈ W01,x LB Q ∩ L2 Q and for v u.
3.1
4
International Journal of Mathematics and Mathematical Sciences
Theorem 3.2. Under the hypothesis 1.2–1.5, the problem (P0 ) admits at least one solution u in
the following sense:
u ∈ BVloc Q,
θu ∈ W01,x LB Q ∩ L2 Q,
∂u
,v a·, ∇θu∇v fv dxdt,
∂t
Q
Q
3.2
for all v ∈ W01,x LB Q.
Proof.
Step 1 approximation and a priori estimate. Consider the approximate problem:
∂un
1
− diva·, ∇θun − ΔM un f,
∂t
n
un x, 0 u0n x,
in Q,
3.3
in Ω,
where −ΔM u − divM|∇un |/|∇un |2 ∇un is the M-Laplacian operator and u0n is a
smooth sequence converging strongly to u0 in L2 Q.
The approximate problem has a regular solution un and in particular un ∈ W01,x LM Q
by Lemma 3.1. s
Let Θs 0 θtdt.
Let v θun χ0,τ as test function, one has
Ω
Θun τdx α
B|∇θun | ≤
Qτ
fθun Qτ
Ω
Θun 0dx,
3.4
then, θun n bounded in W01,x LB Q.
There exist a measurable function v and a subsequence, also denoted un , such that,
θun v,
a.e in Q and weakly in W01,x LB Q.
3.5
Let us consider the C2 function defined by
⎧
⎪
⎨s
k
,
2
ηk s ⎪
⎩k signs |s| ≥ k.
|s| ≤
3.6
International Journal of Mathematics and Mathematical Sciences
5
Multiplying the approximating equation by ηk un , we get
∂ηk un − div a·, ∇θun ηk un a·, ∇θun ηk un ,
∂t
1 M|∇un |
M|∇un |
1
∇un ηk un ∇un ηk un fηk un − div
2
n
n |∇un |2
|∇un |
3.7
in the distributions sense. We deduce, then, ηk un is bounded in W01,x LM Q and ∂ηk un /∂t
in W −1,x LM Q L2 Q. Then, ηk un is compact in L1 Q.
Following the same way as in 11, we obtain θun θu, weakly in W01,x LB Q for
σΠLB , ΠEB , strongly in L1 Q and a.e in Q.
Step 2 passage to the limit. Let set b·, ∇u M|∇u|/|∇u|2 ∇u.
Let v ∈ W01,x LB Q, one has
1
∂un
, θun − v b·, ∇un ∇θun − vdx
a·, ∇θun ∇θun − v ∂t
Q
Q n
fθun − vdxdt.
3.8
Q
By using the following decomposition:
a·, ∇θun ∇θun − v a·, ∇θun − a·, θ∇v∇θun − v
a·, θ∇v∇θun − v,
b·, ∇un ∇θun − v b·, ∇un − b·, ∇v∇θun − v b·, ∇v∇θun − v,
∇θun − v θ un ∇un − ∇v θ un ∇un − θ un ∇v θ un ∇v − ∇v,
3.9
and by the monotonicity of the operator defined by a and b, we obtain
1
∂un
, θun − v b·, ∇v∇θun − ∇v
a·, θ∇v∇θun − ∇v ∂t
n
Q
Q
1
fθun − v,
b·, ∇un − b·, ∇v θ un − 1 ∇v ≤
Q n
Q
3.10
by passage to the limit with a standard argument as in 10, 11, and using the above
convergence of θun , we have
∂u
, θu − v a·, ∇θv∇θu − ∇vdx ≤
fθu − v.
∂t
Q
Q
3.11
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International Journal of Mathematics and Mathematical Sciences
Taking now v θu − tψ, with ψ ∈ W01,x LB Q and t ∈ −1, 1, we deduce that u is solution
of the problem 1.2.
Step 3 u ∈ BVloc Q. Let K be a compact in Q, and let ϕ ∈ DQ with K ⊂ suppϕ such that
ϕ 1,
on K, ∇ϕ ≤ 1.
3.12
Using ϕ as test function in 3.7, we get
Q
∂ηk un ϕdxdt a·, ∇θun ∇ϕ · ηk un dxdt a·, ∇θun ϕ · ηk un dxdt
∂t
Q
Q
M|∇un |
M|∇un |
1
1
∇u
∇ϕ
·
η
∇un ϕ · ηk un dxdt
u
dxdt
n
n
k
n Q |∇un |2
n Q |∇un |2
3.13
I1 I 2 I 3 I 4 I 5
fηk un ϕdxdt.
Q
The terms I2 , I3 , I4 , I5 are bounded, so
∂ηk un ∂t dxdt ≤ C.
K
3.14
∂un
∂t x, tdxdt ≤ C.
3.15
Letting k tend to infinity, we have
K
We deal now with the following estimation which ends the proof.
For all compact K ⊂ Q,
|Dun x, t|dxdt ≤ C.
3.16
K
Indeed, we differentiate the approximate problem with respect to xi , we multiply the
obtained equation by ηk ∂xi un , and one has the following equality in the distributions sense
∂∂xi un 1
ηk ∂xi un − div∂xi a·, ∇θun ηk ∂xi un − ∂xi ΔM un ηk ∂xi un ∂t
n
∂xi fηk ∂xi un ,
3.17
International Journal of Mathematics and Mathematical Sciences
7
which is equivalent to
∂ηk ∂xi un 1
− div∂xi a·, ∇θun ηk ∂xi un − div∂xi b·, ∇un ηk ∂xi un ∂t
n
3.18
∂xi fηk ∂xi un .
We recall that ηk , ηk , and ηk are bounded on R, θun is bounded in W01,x LB Q, and
a·, ∇θun is bounded in LB Q.
Using now the test function ϕ defined below, we obtain, as for 3.14,
∂ηk ∂un dxdt ≤ C.
∂xi K ∂t
3.19
With the same way as above, we conclude the result, u ∈ BVloc Q.
Remark 3.3. As in Theorem 3.2, one can prove the same result in the case where we replace
the initial condition in the problem P by θux, 0 θu0 x and θu0 x ∈ L2 Q.
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