Abstract
In this paper, we discuss the ndimensional diffraction problem for weakly coupled quasilinear parabolic system on
a bounded domain Ω, where the interfaces
MSC: 35R05, 35K57, 35K65.
Keywords:
diffraction problem; quasilinear parabolic system; interface; approximation method1 Introduction
Let Ω be a bounded domain in
In this paper, we consider the diffraction problem for quasilinear parabolic reactiondiffusion system in the form
where
repeated indices i or j indicate summation from 1 to n,
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 [15]). The linear diffraction problems have been treated by many researchers (see [110]). For the quasilinear parabolic and elliptic diffraction problems, when all of the
interfaces
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.
For any set S,
Let
where
We see that
For the vector functions with Ncomponents we denote the above function spaces by
Moreover, we recall the following.
Write u in the split form
The vector function
such that
The following hypotheses will be used in this paper:
(H) (i) ∂Ω and
Assume that for each
and
(ii) Assume that
where
(iii) There exist constant vectors
where
The vector functions
(iv) For each
For each
Definition 2.2 A function u is said to be a solution of (1.1) if u possesses the following properties: (i) For some
The main result in this paper is the following existence theorem.
Theorem 2.1Let Hypothesis (H) hold. Then problem (1.1) has a solutionuin
2.2 An example
We next give an example satisfying the conditions in Hypothesis (H).
Example 2.1 In problem (1.1), let
where
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
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
where
Then
For each
We find that these functions satisfy (2.2) and the hypothesis (iv) of (H). 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
3 The proof of the existence theorem
3.1 Preliminaries
Lemma 3.1The following statements hold true:
(i) For any given
(ii) There exists a positive number
Proof By (2.1), if
For any given
Hence, the conclusion in (ii) follows from the above relation by taking
For an arbitrary ε,
Lemma 3.2
Let functions
Then for any
Proof Since (3.2) is a special case of (3.4) with
Case 1. If
Case 2. If
Case 3. If
Case 4. If
3.2 The approximation problem of (1.1)
In this subsection, we construct a problem to approximate (1.1).
For each
It follows from hypothesis (iv) of (H), (3.2) and (3.5) that
We note that the functions
For each
with a sufficiently smooth nonnegative averaging kernel
where
These, together with (2.6), imply that
For any given ε,
where
We note that the interfaces in (3.11) are
Proposition 3.1Problem (3.11) has a unique piecewise classical solution
Proof Problem (3.11) is a special case of [[13], problem (1.1)] without time delays. Formulas (2.3) and (3.5) show that
3.3 The uniform estimates of
u
ε
In the following discussion, let
For each
Similarly, for any
Lemma 3.3There exist constants
Proof (3.16) follows from (3.14), (3.6), (3.7), (3.9) and [[1], Chapter V, Theorem 1.1 and Remark 1.2]. Setting
Lemma 3.4For any given
For any given
Proof Choose a subdomain B satisfying
Then the same proofs as those of [[13], formulas (3.30) and (3.31)] give (3.18) and (3.19). If
In the rest of this subsection, let
and the diffraction conditions on
Lemma 3.5Let
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 [[13], 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 ArzelaAscoli theorem it follows that we can
find a subsequence (we retain the same notation for it)
For any given
Since
For any given
By letting
For any given
Then
We next show that
By (3.27), (3.17), we get
and by (2.2), (3.22),
Since
and u satisfies the diffraction conditions on
In view of (3.30) u satisfies the diffraction conditions on
for some
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
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 (CSYXM1206).
References

Ladyzenskaya, OA, Solonnikov, VA, Ural’ceva, NN: Linear and Quasilinear Equations of Parabolic Type, Am. Math. Soc., Providence (1968)

Ladyzenskaya, OA, Ural’ceva, NN: Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968)

Ladyzenskaya, OA, Ryvkind, VJ, Ural’ceva, NN: Solvability of diffraction problems in the classical sense. Tr. Mat. Inst. Steklova. 92, 116–146 in Russian (1966)

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)

Kamynin, LI: On the linear Verigin problem. Dokl. Akad. Nauk SSSR. 150, 1210–1213 in Russian (1963)

Kamynin, LI: The method of heat potentials for a parabolic equation with discontinuous coefficients. Sib. Mat. Zh.. 4, 1071–1105 in Russian (1963)

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 secondorder elliptic and parabolic interface problems. J. Differ. Equ.. 184, 570–586 (2002). Publisher Full Text

Evans, LC: A free boundary problem: the flow of two immiscible fluids in a onedimmensional porous medium. I. Indiana Univ. Math. J.. 26, 914–932 (1977)

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 reactiondiffusion systems. Nonlinear Anal.. 55, 905–926 (2003). Publisher Full Text

Tan, QJ: Systems of quasilinear parabolic equations with discontinuous coefficients and continuous delays. Adv. Differ. Equ.. 2011, Article ID 925173. doi:10.1155/2011/925173 (2011)

Tan, QJ, Leng, ZJ: The method of upper and lower solutions for diffraction problems of quasilinear elliptic reactiondiffusion systems. J. Math. Anal. Appl.. 380, 363–376 (2011). Publisher Full Text

Pao, CV: Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York (1992)