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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 506857, 7 pages
http://dx.doi.org/10.1155/2011/506857
Research Article

On Degenerate Parabolic Equations

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

Received 31 March 2011; Accepted 28 July 2011

Academic Editor: Mihai Putinar

Copyright © 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.

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 where , and is a bounded domain of , with the segment property, is a smooth function, is a positive real function increasing but not necessarily strictly increasing, , and is a Carathéodory function (i.e., measurable with respect to in for every in , and continuous with respect to in for almost every in ) such that for all , There exist an N-function such that where belongs to and to , and to .

Some examples of such operator are in particular the case where where 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 where are monotone, is even and convex for , for (for the details see [1]). Jäger and Kačur treated the porous medium systems where is strictly monotone in [2] and Stefan-type problems where is only monotone. For the last model, there exists a large number of references. Among them, let us mention the earlier works [35] 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 , problem 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 . The first one comes from the diffusion terms in since they do not depend on but on , and, moreover, at the same time, poses big problems, since in general we have not information on but on . 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 was assumed to satisfy a polynomial growth condition with respect to . When trying to relax this restriction on the function , we are led to replace the space by an inhomogeneous Sobolev space built from an Orlicz space instead of , where the N-function which defines 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 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 space.

2. Preliminaries

Let be an N-function, that is, is continuous, convex, with for , as , and as . The N-function conjugate to is defined by .

Let and be two N-functions. means that grows essentially less rapidly than , that is, for each , Let be an open subset of . The Orlicz class (resp., the Orlicz space ) is defined as the set of (equivalence classes of) real-valued measurable functions on such that (resp., for some .

Note that is a Banach space under the norm and is a convex subset of . The closure in of the set of bounded measurable functions with compact support in is denoted by . In general, and the dual of can be identified with by means of the pairing , and the dual norm on is equivalent to .

We say that converges to for the modular convergence in if, for some , . This implies convergence for .

The inhomogeneous Orlicz-Sobolev spaces are defined as follows: . These spaces are considered as subspaces of the product space which have as many copies as there are -order derivatives, . We define the space . (For more details, see [9].)

For , we define the truncation at height by

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), , the problem admits at least one solution in the following sense: for all and for .

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

Proof. Step 1 (approximation and a priori estimate). Consider the approximate problem: where is the -Laplacian operator and is a smooth sequence converging strongly to in .
The approximate problem has a regular solution and in particular (by Lemma 3.1).
Let .
Let as test function, one has then, bounded in .
There exist a measurable function and a subsequence, also denoted , such that,
Let us consider the function defined by
Multiplying the approximating equation by , we get in the distributions sense. We deduce, then, is bounded in and in . Then, is compact in .
Following the same way as in [11], we obtain, weakly in for , strongly in and a.e in .
Step 2 (passage to the limit). Let set .
Let , one has By using the following decomposition: and by the monotonicity of the operator defined by and , we obtain by passage to the limit with a standard argument as in [10, 11], and using the above convergence of , we have Taking now , with and , we deduce that is solution of the problem (1.2).
Step 3 (). Let be a compact in , and let with such that Using as test function in (3.7), we get The terms are bounded, so Letting tend to infinity, we have We deal now with the following estimation which ends the proof.
For all compact , Indeed, we differentiate the approximate problem with respect to , we multiply the obtained equation by , and one has the following equality in the distributions sense which is equivalent to
We recall that , and are bounded on is bounded in , and is bounded in .
Using now the test function (defined below), we obtain, as for (3.14), With the same way as above, we conclude the result, .

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 by and .

References

  1. J. R. Esteban and J. L. Vázquez, “Homogeneous diffusion in R with power-like nonlinear diffusivity,” Archive for Rational Mechanics and Analysis, vol. 103, no. 1, pp. 39–80, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  2. W. Jäger and J. Kačur, “Solution of porous medium type systems by linear approximation schemes,” Numerische Mathematik, vol. 60, no. 3, pp. 407–427, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. Blanchard and G. A. Francfort, “A few results on a class of degenerate parabolic equations,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 18, no. 2, pp. 213–249, 1991. View at Google Scholar · View at Zentralblatt MATH
  4. H. Brézis and M. G. Crandall, “Uniqueness of solutions of the initial-value problem for utΔφ(u)=0,” Journal de Mathématiques Pures et Appliquées, vol. 58, no. 2, pp. 153–163, 1979. View at Google Scholar
  5. A. Damlamian, “Some results on the multi-phase Stefan problem,” Communications in Partial Differential Equations, vol. 2, no. 10, pp. 1017–1044, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. P. Bénilan, Equations d'evolution dans un espace de Banach quelconque et applications, these d'Etat, Orsay, France, 1972. View at Zentralblatt MATH
  7. J. Kačur, A. Handlovičová, and M. Kačurová, “Solution of nonlinear diffusion problems by linear approximation schemes,” SIAM Journal on Numerical Analysis, vol. 30, no. 6, pp. 1703–1722, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Carrillo, J. I. Díaz, and G. Gilardi, “The propagation of the free boundary of the solution of the dam problem and related problems,” Applicable Analysis, vol. 49, no. 3-4, pp. 255–276, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. Donaldson, “Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems,” Journal of Differential Equations, vol. 16, pp. 201–256, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Elmahi and D. Meskine, “Parabolic equations in Orlicz spaces,” Journal of the London Mathematical Society, vol. 72, no. 2, pp. 410–428, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Porretta, “Existence results for nonlinear parabolic equations via strong convergence of truncations,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 177, pp. 143–172, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet