In this paper, we discuss the n-dimensional diffraction problem for weakly coupled quasilinear parabolic system on a bounded domain Ω, where the interfaces () are allowed to intersect with the outer boundary ∂Ω and the coefficients of the equations are allowed to be discontinuous on the interfaces. The aim is to show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces, which do not intersect with ∂Ω.
MSC: 35R05, 35K57, 35K65.
Keywords:diffraction problem; quasilinear parabolic system; interface; approximation method
Let Ω be a bounded domain in with boundary ∂Ω (), and let Ω be partitioned into a finite number of subdomains () separated by , where , , are interfaces, which do not intersect with each other. For any , set
In this paper, we consider the diffraction problem for quasilinear parabolic reaction-diffusion system in the form
repeated indices i or j indicate summation from 1 to n, is the unit normal vector to Γ (the positive direction of is fixed in advance), the symbol denotes the jump of a quantity across , and the coefficients , and are allowed to be discontinuous on . In the following, we refer to the conditions on in (1.1) as diffraction conditions.
The diffraction problems often appear in different fields of physics, ecology, and technics. In some of them, the interfaces are allowed to intersect with the outer boundary ∂Ω (see [1-5]). The linear diffraction problems have been treated by many researchers (see [1-10]). For the quasilinear parabolic and elliptic diffraction problems, when all of the interfaces do not intersect with ∂Ω, the existence and uniqueness of the solutions have been investigated in [11-14] by Leray-Schauder principle and the method of upper and lower solutions. In this paper, we investigate the existence of solutions of (1.1) when the interfaces are allowed to intersect with ∂Ω. In this case, because of the existence of the intersection of Γ and ∂Ω, the methods in [11-14] can not be extended. We shall show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces which do not intersect with ∂Ω.
The plan of the paper is as follows. In Sect. 2, we give the notations, hypotheses and an example, and state the existence theorem of the solutions. Section 3 is devoted to the proof of the existence theorem.
2 The hypotheses, main result and example
2.1 The notations, hypotheses and main result
First, let us introduce more notations and function spaces.
where , , intersect with the outer boundary ∂Ω, and , do not intersect with ∂Ω. Assume that the domain Ω is partitioned into subdomains , , separated by interfaces , and partitioned into , , separated by . The interface of and is . Then . Set
Moreover, we recall the following.
Write u in the split form
The following hypotheses will be used in this paper:
(ii) Assume that
Definition 2.2 A function u is said to be a solution of (1.1) if u possesses the following properties: (i) For some , , . For any given and , there exists , , such that and , , ; (ii) u satisfies the equations in (1.1) for , , the diffraction conditions for and the parabolic boundary conditions for .
The main result in this paper is the following existence theorem.
2.2 An example
We next give an example satisfying the conditions in Hypothesis (H).
Example 2.1 In problem (1.1), let
The outer boundary of domain is a circle of radius 10 with the center at the origin, whereas the interface curves are two parabolas and a smaller circle of radius 1 (see Figure 1). We see that and intersect with ∂Ω, and does not.
Figure 1. The example of the domain and the interfaces for.
For the coefficients of the equations and the boundary values in (1.1) we set
It follows from these inequalities that there exist positive constant vector M, such that m and M satisfy (2.3). Furthermore, the vector functions , , are mixed quasimonotone in with the same index vector . The above arguments show that the conditions in Hypothesis (H) can be satisfied.
3 The proof of the existence theorem
Lemma 3.1The following statements hold true:
3.2 The approximation problem of (1.1)
In this subsection, we construct a problem to approximate (1.1).
with a sufficiently smooth nonnegative averaging kernel that is equal to zero for and is such that . Then from the hypothesis (iv) of (H) and [, Chapter II] we know that for each , is in , is in (), in and in . Thus,
These, together with (2.6), imply that
Proof Problem (3.11) is a special case of [, problem (1.1)] without time delays. Formulas (2.3) and (3.5) show that , are a pair of bounded and coupled weak upper and lower solutions of (3.11) in the sense of [, Definition 2.2]. We find that the conditions of [, Theorem 4.1] are all fulfilled. Then from [, Theorem 4.1], we obtain that problem (3.11) has a unique piecewise classical solution in possessing the properties in (3.12). □
Proof (3.16) follows from (3.14), (3.6), (3.7), (3.9) and [, Chapter V, Theorem 1.1 and Remark 1.2]. Setting in (3.14) and using Cauchy’s inequality, we can obtain (3.17). □
Then the same proofs as those of [, formulas (3.30) and (3.31)] give (3.18) and (3.19). If , then for some , . Hence, the conclusion in (3.20) follows from (3.18), (3.19), (3.21) and the same argument as that for [, formula (3.37)]. □
In the rest of this subsection, let be an arbitrary fixed number in , and let be an arbitrary fixed subdomain satisfying , and . We next investigate the uniform estimates in the neighborhood of . Let be any point of . [, Chapter 3, Section 16] and  show that there exists a ball with center at such that we can straighten out by introducing a local coordinate system . Our assumptions concerning Γ imply that we can divide into a finite number of pieces and to introduce for each of them coordinates y. Since the investigations in the rest of this subsection are local properties, we can assume without loss of generality that the interface lies in the plane . Then by (3.4), when the coefficients of problem (3.11) can be represented in the form
Proof It follows from (3.22) and Hypothesis (H) that
and from the equations in (3.11) that
Then using (3.13), (3.15), (3.22), (3.27) and (3.28), we can prove (3.24)-(3.26) by a slight modification of the proofs of [, formulas (3.30) and (3.31)]. The detailed proofs are omitted. □
3.4 The proof of Theorem 2.1
From estimates (3.16), (3.17) and the Arzela-Ascoli theorem it follows that we can find a subsequence (we retain the same notation for it) such that converges in to u and converges weakly in to for each . Then and . Furthermore, the parabolic boundary conditions for in (3.11) imply that u satisfies the parabolic boundary conditions in (1.1).
By (3.27), (3.17), we get
and by (2.2), (3.22),
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the reviewers and the editors for their valuable suggestions and comments. The work was supported by the research fund of Department of Education of Sichuan Province (10ZC127) and the research fund of Chengdu Normal University (CSYXM12-06).
Yi, FH: Global classical solution of Muskat free boundary problem. J. Math. Anal. Appl.. 288, 442–461 (2003). Publisher Full Text
Druet, PE: Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface. Weierstrass Institute for Applied Analysis and Stochastics, WIAS Preprint No. 1571 (2010)
Chen, Z, Zou, Z: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math.. 79, 175–202 (1998). Publisher Full Text
Huang, J, Zou, J: Some new a priori estimates for second-order elliptic and parabolic interface problems. J. Differ. Equ.. 184, 570–586 (2002). Publisher Full Text
Rivkind, V, Ural’tseva, N: Classical solvability and linear schemas for the approximated solutions of the diffraction problem for quasilinear elliptic and parabolic equations. Mat. Zametki. 3, 69–111 in Russian (1972)
Boyadjiev, G, Kutev, N: Diffraction problems for quasilinear reaction-diffusion systems. Nonlinear Anal.. 55, 905–926 (2003). Publisher Full Text
Tan, QJ, Leng, ZJ: The method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction-diffusion systems. J. Math. Anal. Appl.. 380, 363–376 (2011). Publisher Full Text