International Journal of Mathematics and Mathematical Sciences
Volume 2011, Issue 1 506857
Research Article
Open Access

On Degenerate Parabolic Equations

Mohammed Kbiri Alaoui, 

Corresponding Author

Mohammed Kbiri Alaoui

Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia kku.edu.sa

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First published: 18 October 2011
https://doi.org/10.1155/2011/506857
Academic Editor: Mihai Putinar

Abstract

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
( P )
where
(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, θ(0) = 0, and θ(u0) ∈ 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 respect to ξ in RN for almost every x in Ω) such that for all ξ, ξ* ∈ RN,   ξ ≠ ξ*,
(1.2)
(1.3)
(1.4)
There exist an N-function M such that
(1.5)
where c(x, t) belongs to and ki(i = 1,2) to R+, and α to .
Some examples of such operator are in particular the case where
(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
( E )
where β, ψ are monotone, ψ(s)  s is even and convex for s ≥ so > 0, |ψ(s)| → ∞,    | β(s)| → ∞ 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 θ(u), and, moreover, at the same time, P poses big problems, since in general we have not information on u but on θ(u). 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 a(x, 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; W1,p(Ω)) by an inhomogeneous Sobolev space W1,xLB 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.

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 M(t) > 0 for t > 0, M(t)/t → 0 as t → 0, and M(t)/t → ∞ as t → ∞. The N-function conjugate to M is defined by .

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,
(2.1)
Let Ω be an open subset of RN. The Orlicz class ℒM(Ω) (resp., the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that (resp., for some λ > 0).

Note that LM(Ω) is a Banach space under the norm ∥u∥M,Ω = inf  {λ > 0 : ∫ΩM(u(x)/λ)dx ≤ 1} and ℒM(Ω) is a convex subset of LM(Ω). The closure in LM(Ω) of the set of bounded measurable functions with compact support in is denoted by EM(Ω). In general, EM(Ω) ≠ LM(Ω) and the dual of EM(Ω) can be identified with by means of the pairing ∫Ωu(x)v(x)dx, and the dual norm on is equivalent to .

We say that un converges to u for the modular convergence in LM(Ω) if, for some λ > 0, ∫ΩM((un − u)/λ)dx → 0. This implies convergence for .

The inhomogeneous Orlicz-Sobolev spaces are defined as follows: . 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. We define the space . (For more details, see [9].)

For k > 0, we define the truncation at height k, Tk : R → R by
(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), θ(s) = s, the problem P admits at least one solution u in the following sense:

(3.1)
for all and for v = u.

Theorem 3.2. Under the hypothesis (1.2)–(1.5), the problem (P0) admits at least one solution u in the following sense:

(3.2)
for all .

Proof.

Step 1 (approximation and a priori estimate). Consider the approximate problem:

(3.3)
where −ΔM(u) = −div ((M(|∇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 (by Lemma 3.1).

Let .

Let v = θ(un)χ(0,τ) as test function, one has

(3.4)
then, (θ(un)) n bounded in .

There exist a measurable function v and a subsequence, also denoted (un), such that,

(3.5)

Let us consider the C2 function defined by

(3.6)

Multiplying the approximating equation by , we get

(3.7)
in the distributions sense. We deduce, then, ηk(un) is bounded in and ∂ηk(un)/∂t in . Then, ηk(un) is compact in L1(Q).

Following the same way as in [11], we obtain  θ(un)⇀θ(u), weakly in for , 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 , one has

(3.8)
By using the following decomposition:
(3.9)
and by the monotonicity of the operator defined by a and b, we obtain
(3.10)
by passage to the limit with a standard argument as in [10, 11], and using the above convergence of θ(un), we have
(3.11)
Taking now v = θ(u) − tψ, with 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 ∈ D(Q) with K ⊂ supp (φ) such that

(3.12)
Using φ as test function in (3.7), we get
(3.13)
The terms I2, I3, I4, I5 are bounded, so
(3.14)
Letting k tend to infinity, we have
(3.15)
We deal now with the following estimation which ends the proof.

For all compact K ⊂ Q,

(3.16)
Indeed, we differentiate the approximate problem with respect to xi, we multiply the obtained equation by , and one has the following equality in the distributions sense
(3.17)
which is equivalent to
(3.18)

We recall that , and are bounded on R,   (θ(un)) is bounded in , and (a(·, ∇θ(un))) is bounded in .

Using now the test function φ (defined below), we obtain, as for (3.14),

(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 θ(u(x, 0)) = θ(u0(x)) and θ(u0(x)) ∈ L2(Q).

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