Diffraction problems for quasilinear parabolic systems with boundary intersecting interfaces
© Tan and Pan; licensee Springer. 2013
Received: 19 January 2013
Accepted: 9 April 2013
Published: 22 April 2013
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.
Keywordsdiffraction problem quasilinear parabolic system interface approximation method
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.
For any set S, denotes its closure. The symbol means that and .
We see that .
For the vector functions with N-components we denote the above function spaces by , , , and , respectively.
Moreover, we recall the following.
such that is nondecreasing in , and is nonincreasing in for all .
- (H)(i) ∂ Ω and , , are of for some exponent and there exist and such that for every open ball centered at and radius ,
- (ii)Assume that(2.2)
- (iii)There exist constant vectors and , , such that(2.3)
- (iv)For each , , , , (), . There exist a positive nonincreasing function and a positive nondecreasing function for such that(2.4)
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.
Theorem 2.1 Let Hypothesis (H) hold. Then problem (1.1) has a solution u in .
2.2 An example
We next give an example satisfying the conditions in Hypothesis (H).
with , and , , , and are all positive constants for , .
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
- (i)For any given , if for some , then
- (ii)There exists a positive number such that for any given , if , then
Proof By (2.1), if and , then . Thus for each , . Again by (2.1) we get . This proves the result in (i).
Hence, the conclusion in (ii) follows from the above relation by taking . □
Proof Since (3.2) is a special case of (3.4) with for all , we only prove (3.4).
Case 1. If , then the conclusion of (i) in Lemma 3.1 implies that and for all . (3.1) yields that and for . These, together with (3.3), imply that .
Case 2. If , then the conclusion of (ii) in Lemma 3.1 shows that and for all . Hence, and for all . Again by (3.3) we get .
Case 3. If and for some , then Lemma 3.1 yields that , for all , and that , for all . Hence, and for . Therefore, .
Case 4. If for some , then it follows from Lemma 3.1 that and for all , and that . Again by the conclusion of (i) in Lemma 3.1 we have and for all . Hence, , and for . Thus, . □
3.2 The approximation problem of (1.1)
In this subsection, we construct a problem to approximate (1.1).
We note that the functions , and are continuous on , and are allowed to be discontinuous on .
We note that the interfaces in (3.11) are () which do not intersect with ∂ Ω. In view of (3.10), the compatibility condition on holds.
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). □
3.3 The uniform estimates of
In the following discussion, let be an arbitrary open ball of radius ρ with center at , and let be an arbitrary cylinder of the form .
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)]. □
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).
Since and are arbitrary, then u satisfies the equations in (3.11) for .
and u satisfies the diffraction conditions on in (3.23).
for some . Therefore, u is a solution of (1.1). This completes the proof of Theorem 2.1.
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).
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